Design of Human Powered Vehicles (Bicycle Drive Train)

Design of Human-Powered Vehicles by Mark Archibald

© 2016, The American Society of Mechanical Engineers (ASME), 2 Park Avenue, New York, NY 10016, USA (www.asme.org) All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher.

INFORMATION CONTAINED IN THIS WORK HAS BEENBELIEVED OBTAINED BY AMERICAN SOCIETY OF MECHANICAL ENGINEERS FROM SOURCES TO BETHE RELIABLE. HOWEVER, NEITHER ASME NOR ITS AUTHORS OR EDITORS GUARANTEE THE ACCURACY OR COMPLETENESS OF ANY INFORMATION PUBLISHED IN THIS WORK. NEITHER ASME NOR ITS AUTHORS AND EDITORS SHALL BE RESPONSIBLE FOR ANY ERRORS, OMISSIONS, OR DAMAGES ARISING OUT OF THE USE OF THIS INFORMATION. THE WORK IS PUBLISHED WITH THE UNDERSTANDING THAT ASME AND ITS AUTHORS AND EDITORS ARE SUPPLYING INFORMATION BUT ARE NOT ATTEMPTING TO RENDER ENGINEERING OR OTHER PROFESSIONAL SERVICES. IF SUCH ENGINEERING OR PROFESSIONAL SERVICES ARE REQUIRED, THE ASSISTANCE OF AN APPROPRIATE PROFESSIONAL SHOULD BE SOUGHT. ASME shall not be responsible for statements or opinions advanced in papers or printed in its publications (B7.1.3). Statement from the Bylaws. For authorization to photocopy material for internal or personal use under those circumstances not falling within the fair use provisions of the Copyright Act, contact the Copyright Clearance Center (CCC), 222 Rosewood Drive, Danvers, MA 01923, Tel: 978-750-8400, www.copyright.com. Requests for special permission or bulk reproduction should be addressed to the ASME Publishing Department or submitted online at https://www.asme.org/shop/books/book-proposals/ permissions.

ASME Press books are available at special quantity discounts to use as premiums or for use in corporate training programs. For more information, contact Special Sales at CustomerCare@ asme.org. Library of Congress Cataloging-in-Publication Data

Names: Archibald, C. Mark, author. Title: Design of human-powered vehicles / by C. Mark Archibald. Description: New York : ASME Press, [2016] | Includes bibliographical references and index. Identifiers: LCCN 2016005632 | ISBN 9780791861103 Subjects: LCSH: Velocipedes--Design and construction. | Bicycles--Design and construction. | Human powered vehicles--Design and construction. Classification: LCC TL400 .A73 2016 | DDC 629.227/2--dc23 LC record available at http://lccn.loc. gov/2016005632

TABLE OF CONTENTS Acknowledgments ................................................................................................................ ix Preface ................................................................................................. ................................ xi Chapter 1: Rationale for Human-Powered Vehicle Design and Use ........................................... 1 Recreation .........................................................................................................2 Competition .......................................................................................................2 Economics .........................................................................................................4 Fitness and Health ............................................................................................5 Mobility ..............................................................................................................8 Environment......................................................................................................8

Chapter 2: Overview of Human-Powered Vehicles ................................................................. 13

A Brief Historical Perspective ........................................................................13 Land Vehicle Applications and Functions .....................................................16 Land Vehicle Configurations...........................................................................17

Chapter 3: General Structured Design of HPV’s .................................................................... 25 General Structured Design of HPVs...............................................................25 Target Speeds for Tandem Bicycle ................................................................30 Example Vehicle Design Specification ...........................................................31 Functional Requirements ...............................................................................32 Outline of Design Process...............................................................................38

Chapter 4: Physiology of Human Power Generation............................................................... 41 Muscle Structure and Function ......................................................................42 Nutrition ..........................................................................................................49 Body Systems during Exercise .......................................................................52 Maximal Oxygen Consumption ......................................................................57 Anaerobic Threshold.......................................................................................58 Appendix: Calculating the CO2 Production Rate as a Function of External Work .............................................................................................58

iii

Design of Human-Powered Machines

Chapter 5: The Human-Machine Interface ............................................................................. 61 Comfort ............................................................................................................61 Mechanisms for Power Transfer to Human-Powered Vehicles.....................63 Optimal Body Position for Leg Cranks ...........................................................64 Crank Length and Physiological Efficiency ...................................................67 Hand Cranks ....................................................................................................69 Unusual Mechanisms for Human Power Transfer .........................................71 Safety ...............................................................................................................72

Chapter 6: Manufacturing Processes and Materials ............................................................. 77 Wrought Metals—Overview ............................................................................78 Cast Metals—Note ..........................................................................................81 Non-Metal Materials—Overview ....................................................................81 Frame Materials and Manufacturing Processes.............................................85 Steel .................................................................................................................86 Stainless Steel .................................................................................................89 Aluminum ........................................................................................................89 Titanium ..........................................................................................................91 Fiber Reinforced-Polymer Composites ..........................................................92 Other Frame Materials ....................................................................................96 Frame Manufacturing Processes ....................................................................97 Brazing .............................................................................................................99 Bonding ............................................................................................................99 Other Frame Processes (Monocoque, etc.) .................................................100 Fairing or Shell Materials ..............................................................................101 Fairing Hardware ..........................................................................................104 Summary ........................................................................................................104

Chapter 7: Road Loads ................................................................................................ ........105 Review of Equilibrium Equations .................................................................105 SAE Vehicle Coordinate System for Vehicle Dynamics ..............................106 Static Loads on Level Ground ......................................................................107 Static Loads on a Grade ................................................................................110 Steady Motion Road Loads ...........................................................................112 Basic Loads in a Steady Turn .......................................................................115 Acceleration and Braking .............................................................................117 Power-Limited Acceleration .........................................................................118 Traction-Limited Acceleration .....................................................................119 Inertia Coefficient .........................................................................................121 Load Transfer during Acceleration and Braking .........................................124

iv

Table of Contents

Braking ..........................................................................................................125 Braking Performance ....................................................................................128

Chapter 8: Speed and Power Models ................................................................................... 133 Drive Train Efficiency ...................................................................................134 Aerodynamic Drag ........................................................................................135 Rolling Resistance .........................................................................................138 Frictional Losses in Wheel Bearings ............................................................139 Changes in Potential Energy ........................................................................140 Changes in Kinetic Energy ........................................................................... 140 Power Models ................................................................................................141 Cubic Power Model .......................................................................................144 Applications of Power Models ......................................................................146

Chapter 9: Aerodynamic Drag ............................................................................................. 151 Causes of Aerodynamic Drag .......................................................................151 Lift and Induced Drag ...................................................................................157 Computing Drag Force .................................................................................158 Factors Affecting the Drag Coefficient ........................................................159 Estimation of the Drag Coefficient ...............................................................162 Drag Coefficients for Various Vehicles .........................................................163

Chapter 10: Bicycle Handling Performance .........................................................................167 Bicycle Stability.............................................................................................167 Bicycle Handling ...........................................................................................173 Patterson’s Method .......................................................................................175

Chapter 11: Multi- Track Vehicle Handling Performance ....................................................... 195 Multi-Track Vehicle Handling .......................................................................195 Definitions and Nomenclature ......................................................................196 Low-Speed Cornering ...................................................................................197 High-Speed Cornering ..................................................................................207 Lateral Load Transfer and Rollover Threshold............................................214 Summary of Multi-Track Handling Characteristics .....................................225 Appendix 11-1 Kinematic Solution of the Track Rod Steering Mechanism ......................................................................................226 Appendix 11-2 Derivation of Rollover Threshold for Tadpole Trike ..........228

Chapter 12: Drive Train Design ........................................................................................... 233 Gearing ..........................................................................................................233 Recumbent Drivetrains .................................................................................236 Drive Train Configurations ...........................................................................238 Drive Train Design ........................................................................................242

v


Drive Train Technologies ..............................................................................243 Efficiency of Chain Drives ............................................................................246 Chain Drive Design .......................................................................................251

Chapter 13: Land Vehicle Frames and Structures ............................................................... 253 Functional Requirements .............................................................................253 Yielding ..........................................................................................................254 Fatigue ...........................................................................................................254 Inadequate Stiffness .....................................................................................256 Finite Element Modeling of Frames .............................................................256 FEA Modeling—Idealizations, Loads, Constraints, and Validation ............ 258 Beam Elements .............................................................................................259 Shell Elements ..............................................................................................262 Summary of Frame Analysis Using Shell Idealizations ...............................264 Solid Elements ..............................................................................................264 Boundary Conditions: Loads and Constraints .............................................265 Load Cases.....................................................................................................265 Vertical Drop .................................................................................................266 Horizontal Impact (CPSC or ISO Frame Test) ............................................267 Maximum Acceleration .................................................................................269 Hill Climb .......................................................................................................271 Maximum Front Braking ...............................................................................271 Maximum Rear Braking ................................................................................273 Steady-State Pedaling ...................................................................................274 Other Load Cases ..........................................................................................274 FEA Load Verification...................................................................................274 Initial Design Using Beam Idealizations .......................................................274 Detailed Frame Design .................................................................................275 Frame Compatibility with Components .......................................................276 Bottom Bracket Shells ..................................................................................276 Head Tubes ....................................................................................................277 Steerer Tubes ................................................................................................278 Dropout Locknut-to-Locknut Dimensions ...................................................279 Integral Derailleur Hangar Dimensions........................................................280

Chapter 14: Bicycle Components ......................................................................................... 281 Wheels and Tires ...........................................................................................282 Drivetrain Components.................................................................................290 Headsets ........................................................................................................296 Stems .............................................................................................................298 Handlebars.....................................................................................................299

vi

Table of Contents

Brakes ............................................................................................................299 Compatibility of Levers and Brakes .............................................................304 Summary ........................................................................................................305

Index .................................................................................................... ............................. 307 About the Author ...................................................................................................... ........... 311

vii

ACKNOWLEDGMENTS I am deeply indebted to the many people who—directly or indirectly— assisted with this book. First, many, many thanks go to Aaron Williams, who thoughtfully and thoroughly reviewed many of the chapters. The book is much improved as a result of his help. Many of my students have assisted in various projects over the years that contributed data for this book. Among them, I would particularly like to thank Tyler Baker, Cameron Daugherty, Gretchen Robinson, and Liz Casteel, each of whom spent many hours conducting experiments and analyzing data that was used in this text. I must also thank Dr. Mark Reuber, who set me on the journey of writing this book and, of course, my family that put up with my long days and requests for proofreading.

ix

PREFACE Matlab Computer Programs

Several chapters in this book refer to short computer code written in Matlab. Matlab is a technical computing environment that includes a high-level programming language. The script files and functions mentioned in the text are available for download to all purchasers of this book. Navigate to the ftp site ftp://dohpv.gcc.edu. To download the files, log in as anonymous, and enter your email address as your password. At this time, all files require a valid Matlab license and installation. Many universities with engineering or science programs have site licenses for Matlab. If you are affiliated with a university, you may well have a license available. Otherwise, see http://www.mathworks.com/pricing-licensing/ for purchasing Matlab. (Student licenses are quite reasonably priced, and home licenses are only a little more expensive.) All files include a help section that will provide instructions for use. In some cases, multiple versions of a program may be provided. The functionality may be the same, but usage may differ. For example, a program may be offered as both a script and a function. Please make use of the programs and use them to design your ideal vehicle. I hope you find both the text and the programs useful and helpful in your design efforts.

xi

CHAPTER

RATIONALE FOR HUMAN-POWERED

1

VEHICLE DESIGN AND USE

T

his book is about the design of vehicles with wheels that are powered by human muscles alone. These can provide affordable, sustainable, and healthy transportation to people around the globe. The term Human-Powered Vehicle, or HPV, is sometimes used to denote a sub-class of vehicles including only high-performance bicycles or tricycles equipped with aerodynamic fairings. More generally, the term refer to any semi-recumbent bicycle. But the term should properly refer to any means of carriage, conveyance, or transport that is powered solely by human muscles. Manufacturers of bicycles, canoes, kayaks, and scooters do not market their products as HPVs, but surely all of these qualify for the name. Hybrid human-powered vehicles such as mopeds and electric bikes use human power in addition to other sources. While these vehicles are outside our definition of HPVs, they are certainly similar, in both technology and philosophy. Human-powered vehicles were srcinally designed for transportation, and that is still their most important use. HPVs today provide clean, quiet, and efficient transportation. In most developed countries, and in particular the United States, the primary transportation systems are powerful and inefficient, generating large amounts of air and noise pollution. HPVs may be chosen simply because it is pleasurable to travel quietly through the countryside, experiencing nature rather than blocking it out behind steel and glass. They may be chosen because in some cases HPVs provide mobility that no other vehicle can. Couriers in congested cities use bicycles because they are faster. Campers and fishermen in areas such as Minnesota’s Boundary Waters or Ontario’s Algonquin Park may choose a canoe because no other vehicle can traverse the lakes, rivers, and portages quite so

1


well. Many choose human power because it is significantly less expensive than other alternatives, or perhaps because it is good for their health. Athletes and those with a competitive bent can find many venues for racing. Perhaps the most compelling reason to use HPVs is sustainability: the environmental footprint of HPVs is typically much, much smaller than that of other modes of transportation. Despite these commonalities, HPVs are used by a variety of different people for a wide range of diverse reasons, including recreation, competition, cost, health, transportation, and concern for the environment. This book is limited to design of land human-powered vehicles. There are many reasons why the design and use of such vehicles is beneficial. In developed countries, using an HPV in lieu of an automobile (or in lieu of a second automobile for a family) can save $5,000 to $10,000 each year, while improving health and reducing emissions of greenhouse gasses and pollutants. Greenhouse gas emission will be reduced by more than 4,000 kg per year due to the corresponding reduction in energy consumption of more than 17,000 kWh. In addition, infrastructure for cycling is far less costly than highways designed for automotive traffic. It is appropriate to look more deeply into the benefits of HPV use.

Recreation HPVs, including both land and water vehicles, are frequently used for recreation. Often a bike ride or a canoe trip is a social event with friends and family. Quiet streets and rural roads can offer excellent cycling. The number of bicycle paths is increasing in many parts of the United States as abandoned railroads are converted into rail-trails and as local, state, and national parks provide more bike trails and paths. These facilities provide scenic routes for day rides, and can provide a sense of security for young riders, their parents, and others who are concerned about riding in traffic. Increasingly, the trails are long enough to use for multi-day trips. Streams, rivers, lakes, and coastal waters provide a rich range of environments for canoeists, kayakers, and rowers. A quiet pond or small stream may be an ideal place to get away for a while with a small paddle craft, while white

water offers kayakers thrills and challenges. Many regions of the country have waterways that are restricted to human-power, either through law or in practice due to the nature of the lake or river.

Competition

Racing HPVs has likely existed as long as human-powered vehicles themselves. It is easy to imagine a group of tough and intrepid cyclists racing their

2

Rationale for Human-Powered Vehicle Design and Use

high-wheeled ordinary cycles over roughly paved roads in the nineteenth century. As bicycles became more advanced, the competition undoubtedly became faster, but perhaps no keener. Today, bicycle racing is an extremely popular sport in many parts of the world, especially Europe. Competitors can find venues for racing a variety of human-powered land and water craft, and several competitions have involved aircraft. HPV organizations provide many venues for racing recumbent vehicles, including very fast streamliners. Often these events showcase technological developments and design innovation. Many are local or regional events, sponsored by clubs. Of the more traditional races, the most well known is the Tour-de-France, an event restricted to diamond-frame bicycles. Two notable races that permit recumbent bicycles and streamliners are the Race Across America and the World Human-Powered Speed Challenge. The Race Across America is one of the toughest races in the world. Competing individuals or teams start in California and race to New Jersey, with minimal sleep. The team record is slightly over 5 days for a faired recumbent bicycle, while the individual record is a little over 8 days. The World Human-Powered Speed Challenge held in Battle Mountain, Nevada, has hosted most of the land HPV speed records in recent years. On September 17, 2015, Todd Reichert set the men’s world record for the 200 meter flying start time trial with a speed of 137.9 kph 1 (85.71 mph). This is quite remarkable, considering that top speeds for conventional racing bikes are usually under 50 kph (31 mph) and for recreational cyclists around 30 kph (19 mph). For vehicle engineers, racing is a means of validating and proving new designs and design modifications. Competitive cyclists tend to be strong and to ride frequently. They demand the best performance from each vehicle system and often ride vehicles to the limits of performance. Components and systems that continue to operate and function well throughout training and racing generally function well for many years of less rigorous use. In recent years, cycling component manufacturers have competed to develop better, lighter, race-worthy parts and systems. The most successful designs become top-tier components seen on the best competition vehicles. Lower-tier components benefit as the best technologies trickle down through product lines. The bicycle or HPV consumer is the ultimate beneficiary of this process, as the quality of lower-end components has increased significantly over the last few years wit hout a concomitant increase in cost.

1

IHPVA announcements, http://ihpva.org/home/, Accessed October 10, 2015.

3


Economics

While the physical challenges and excitement of racing appeal to some, economics attracts more people to HPVs than perhaps any other reason. Human-powered vehicles, especially bicycles, are substantially less expensive to purchase, own, and operate than other vehicles. In the United States, a significant number of people with limited or no income use bicycles for transportation simply because they are affordable. This group includes students, of course, but it also includes many of our nation’s poor. It is not hard to find a rideable bicycle at a garage sale or thrift store for less than $25.00. What many people with incomes well above the poverty level do not realize is just how large transportation costs can be, particularly with a transportation infrastructure that favors personal automobiles. The cost differential can be calculated relatively easily. Consider a commuter that lives seven miles from her workplace. Additional driving brings her yearly average up to 15,000 miles. She bought the car after graduation from college for $18,000, paying $4,000 down and financing the rest at six% interest. On average, the car gets 22 miles per gallon, and her average price for fuel is $2.60. Maintenance costs her on average five cents per mile. She is more fortunate than many city workers, as she has free parking both at work and home. Including insurance at $350 per year, her total operating costs are very close to average, about 45 cents per mile. See Table 1-1 for more details and assumptions. She decides to investigate how much money she would save if she sold her car and bought a bicycle. Her bicycle would cost $1500, plus an additional $250 for clothing and accessories. She would spend about $725 each year on bicycle maintenance, sports foods and drinks, and accessories. Because she makes some long-distance trips that would not be practical for the bike, she spends about $400 per year on automobile rental. Since she rides regularly, she also cancelled her $216 gym membership. Annual cost for both car and bike are plotted in Figure 1-1. The first year, she would save over $8,000, primarily because of the large down payment on the car. During loan repayment, she would save over $5,900 per year, but after the loan is paid off, her annual savings is still almost $2000. Each year she places the savings in a certificate of deposit earning four percent interest. At the end of 10 years she would have over $50,000 in the bank thanks to her bicycle commute. The example is quite realistic, and the savings are realizable. In this case, the yearly savings is more than 12% of the United States median family income. For many people, this is a very significant amount. Different scenarios may result in different savings, but in virtually all cases, the savings are large. In some cases, the savings after 10 years can approach $100,000.

4


Table 1-1 Data for cost comparison of bicycle and automobile Input Variables

Value

InitialCostofAuto

18000

Financedamount Loanperiod Interest GasMileage Annual Miles driven Gasolineprice Insurance Maintenance Sold at Salvage value

Units

$

14000 $ 4 years 6 %APR 22 MPG 15000 miles 2.60 $/gal 350 $/year 0.05 $/mile

Input Variables

Initialcostofbike Cost accessories/ clothing Maintenance Clothingmaintenance Autorental(trips) Sports drinks/snacks GymSavings Soldat Salvagevalue

Value

1500

Units

$

250 $ 150 $/year 75 $/year 400 $/year 500 $/year 216 $/year 10 years 0 $

10 years 1500 $

Savings interest rate

4

%APR

Assumptions: Auto and bike replaced with identical vehicle

Notes:

Difference placed in savings Auto loan is paid off on schedule Bicycle is bought with cash Bicycle is not insured Auto rental covers transportations costs for trips that would be driven in auto only Annual bike mileage is usually less than comparable auto mileage Savings based on the entire difference placed in savings account

Fitness and Health

Human-powered vehicles used on a regular basis can significantly improve health and fitness. Health problems related to sedentary lifestyles affect a significant portion of the world’s population. In 2006, Dr. Barry Popkin, in a presentation to the International Association of Agricultural Economists, announced that the number of overweight people around the globe exceeds the number of hungry people.2 This is particularly a problem in developed countries such as the United 2

Story from BBC NEWS: http://news.bbc.co.uk/go/pr/fr/-/2/hi/health/4793455.stm Published: 2006/08/15 09:06:27 GMT, accessed 2007/08/20.

5


Figure 1-1 Cost comparison of bicycle and automobile

States. Sedentary lifestyles can lead to obesity, with many associated health risks including diabetes and cardiovascular disease. In contrast, regular physical activity can reduce the likelihood of obesity and increase overall health. Many people in the United States recognize the benefits of regular exercise. People who exercise regularly tend to live longer, more active lives. Regular aerobic exercise keeps the cardiovascular system in good shape, prevents or reduces high blood pressure, and reduces levels of potentially harmful LDL cholesterol. Long-term regular exercise may also raise HDL (good) cholesterol levels. Moderate levels of exercise improve the immune system (although very intense exercise may actually impair immune system function.) Exercise also elevates mood and feelings. Brain levels of endorphins, serotonin, and dopamine are raised with either brief, intense exercise or longer, moderate exercise. These benefits have been shown to reduce the effects of depression and lead to improved feelings of well-being. Long-term exercise, coupled with a good diet, is effective for weight loss. Even brief, but regular, periods of exercise can be beneficial, and cycling is particularly effective. Generally, exercise does not make overweight people hungrier. Regular cycling or HPV use provides all of these health benefits. Gym memberships and fitness clubs are quite popular, and work well for many. Others, however, find it difficult to make time for a workout, or drop out after a few sessions. Recreational bicycling or HPV riding is an enjoyable way to exercise, but—as with

6


gym memberships—it can be difficult to make time for it. An excellent alternative is to use HPVs for commuting or errands around town. The incremental time required to complete a trip by HPV is much smaller than that required for an exercise program, and can be done on a daily basis. Some consider HPV transportation a means of getting time for free. If a commute takes 20 minutes by automobile and 50 minutes by HPV, the trip adds 30 minutes to the commute, but provides 50 minutes of exercise. Compared to spending an hour in the gym, this provides an extra 20 minutes per day. Using HPVs as transportation leads to a much healthier lifestyle than that of driving every trip. Counterintuitively, blood levels of toxins from automobile exhaust products are higher in motorists than in cyclists in the same environment—another health benefit of cycling. Bicycle and HPVs are also much safer than many people realize. Based on statistics for fatal accidents, for every hour of operation a motorist is approximately twice as likely to die in an accident as a cyclist. By comparing on an hourly basis, compensation is made for the different number of miles traveled and the different speeds of automobiles and bicycles. The likely cause is the substantial difference in kinetic energies. A light car at 45 mph has about one hundred times the kinetic energy of a cyclist, while an SUV at interstate speeds has about a thousand times the energy. Most bicycle accidents are falls, in which the rider is not seriously injured. HPVs can be made even safer than bicycles. A long wheelbase recumbent bicycle with under-seat steering is perhaps the safest type of non-faired bicycle. The risk of a forward tip-over is negligible, and the energy of a frontal collision is absorbed by the bicycle and the rider’s legs (rather than his head). A fairing, or shell around the rider, can provide additional protection. Shells are used to reduce aerodynamic drag and to enclose vehicle systems which sometimes include rollover protection. A well designed shell will also provide protection against abrasions in the event of a fall. The safety advantage of enclosed HPVs was illustrated in 2003 when Sam Whittingham experienced a front tire blowout at 82 mph during a speed record attempt. He slid, spun, and went airborne before coming to rest 250 yards away. He was shaken, but walked away from the accident. It is quite possible to design everyday HPVs and velomobiles to be exceptionally safe. There is a well-established body of literature documenting the health benefits of cycling. Johan de Hartog3 and colleagues conducted a study in the Netherlands investigating the health benefits and risks of cycling as compared to car driving. 3

Johan de Hartog, J., Boogaard, H., Nijland, H., and Hoek, G., 2010, “Do the Health Benefits of Cycling Outweigh the Risks?”,Environmental Health Perspectives, 118(8), 1109–1116.

7


They found that the beneficial effects of exercise exceeded the health risks due to both accidents and exposure to air pollution. Cyclists in general lived longer than car drivers. A number of other studies support the claimed health benefits of cycling.4,5 These studies indicate that the general perception in the United States that cycling is inherently dangerous is probably inaccurate.

Mobility

Human-powered vehicles provide excellent mobility for local trips. Mobility is the ability to get to a destination in an efficient and relatively quick manner. Road HPVs are particularly adept at local trips in both rural and urban areas, with mobility sometimes approaching or exceeding that of motor vehicles. For example, in heavily congested areas, bicycles are often faster than automobiles. The bicycle messenger business exists primarily because of this fact. Mountain bicycles can provide outstanding mobility in areas without roads. Canoes and kayaks excel in waterways that other boats cannot navigate, such as shallow, rocky waters or waters choked with vegetation. They are also easily carried across portages, making them ideal for wilderness trips in regions with many lakes and rivers. The majority of automobile trips in the United States are short, and many trips are very short—less than one or two miles. Most of these trips could be completed by human-power with little or no lost time. In areas that have limited parking for automobiles and in congested areas, bicycles are much more convenient, often allowing the operator to ride right up to his or her destination.

Environment

Environmental concerns provide a compelling reason to use human-powered vehicles. In the developed world, transportation systems account for a very significant fraction of air pollution. Emissions from motor vehicles include toxins and greenhouse gasses (GHGs.) In the United States, the prevalence of personal automobiles and the relatively low cost of motor fuel have led to a transportation infrastructure that is predominantly based on highway vehicles. These vehicles are generally inefficient, resulting in more pollution per passenger-mile than other transportation modes, such as rail. A second consequence is the rapid growth and 4

Pucher, J., Dill, J., Handy, S., 2010, “Infrastructure, Programs, and Policies to Increase Bicycling: An International Review”, Preventive Medicine, 50, S106–S125. 5 Yang, L., Sahlqvist, S., McMinn, A., Griffin, S. J., and Ogilvie, D., 2010, “Interventions to Promote Cycling: Systematic Review”, c5293, British Medical Journal, 341.

8


development of suburban and rural areas. This has put increased pressure on wetlands and wildlife habitat, as well as encouraged longer commutes from home to workplace. The result is a combination of increased air pollution and decreased natural resources. Automobiles are also noisy, which can also be considered a type of pollution of the environment. Transportation accounts for a significant fraction of air pollution and greenhouse gas emissions in the United States. On-road mobile sources account for 51% of all carbon monoxide emissions, 29% of all hydrocarbon emissions, 34% of all nitrogen oxides, and 10% of all particulate emissions. Cars and motorcycles contribute more than half of the on-road mobile sources for carbon monoxide and hydrocarbons, while gasoline trucks and diesel vehicles account for most of the on-road nitrogen oxides and particulates. Emissions include combustion products and fuel evaporation. One third of the greenhouse gas emissions are produced by mobile sources. For every liter of gasoline burned, 2.3 kg of carbon dioxide equivalents are released into the atmosphere (19.4 lb CO2E per US gallon burned). Greenhouse gas concentrations have risen sharply since the start of the industrial

Many scientists are concerned about the significant increase in the concentration of CO2 and other GHGs in the atmosphere. Since the preindustrial era, atmospheric concentrations of CO2 have increased by nearly 30 percent and CH4 concentrations have more than doubled. There is a growing international scientific consensus that this increase has been caused, at least in part, by human activity, primarily the burning of fossil fuels (coal, oil, and natural gas) for such activities as generating electricity and driving cars. Moreover, in international scientific circles a consensus is growing that the buildup of CO2 and other GHGs in the atmosphere will lead to major environmental changes such as (1) rising sea levels that may flood coastal and river delta communities; (2) shrinking mountain glaciers and reduced snow cover that may diminish fresh water resources; (3) the spread of infectious diseases and increased heat-related mortality; (4) possible loss in biological diversity and other impacts on ecosystems; and (5) agricultural shifts such as impacts on crop yields and productivity. Although reliably detecting the trends in climate due to natural variability is difficult, the most accepted current projections suggest that the rate of climate change attributable to GHGs will far exceed any natural climate changes that have occurred during the last 1,000 years.

9


Figure 1-2 CO2 Production from typical cyclist on a city bike

era, leading to increasing average global temperatures. The US EPA summarized the likely or possible environmental changes, some of which are already visible:6 Significantly increasing the number and use of human-powered vehicles used for transportation can alleviate the environmental damage from the transportation sector. This is a practical and effective solution that could be easily and quickly implemented. HPVs produce no pollutants during use, and a small fraction of the greenhouse gas emitted by automobiles. Figure 1-2 shows the greenhouse gas emissions of automobiles on a per-mile basis as a function of miles-per-gallon. The average automobile fuel mileage is about 22 mpg. The average automobile emits 418 grams of greenhouse gas every mile. An HPV driver exhales carbon dioxide, but in much smaller amounts. Carbon dioxide emissions for a typical city bicycle with a 77 kg rider are plotted as a function of speed in Figure 1-3. For a given trip, greenhouse gas emissions are reduced by about two orders of magnitude. Any alternative transportation system must not only reduce emissions, but must also be affordable and use existing technology. Bicycles and HPVs use existing infrastructure—roadways and bicycle paths—and would require no additional capital outlay. In fact, HPVs inflict less damage to roads than automobiles, so it is conceivable that infrastructure costs could actually decline with increased HPV 6

SOLID WASTE MANAGEMENT AND GREENHOUSE GASES A Life-Cycle Assessment of Emissions and Sinks, 3rd EDITION, US EPA, September 2006.

10


Figure 1-3 Greenhouse gas emissions from gasoline-powered motor vehicles

use. Cost to consumers could also be significantly reduced, as discussed above. Human-powered vehicles fill a unique role in sustainable transportation alternatives. No other option can provide quantifiable reduction in air pollutants and greenhouse gas emissions with available and affordable technology that uses existing infrastructure. HPVs are, for the present, critical to achieving a sustainable transportation system.

11

CHAPTER

OVERVIEW OF HUMAN-POWERED VEHICLES

2

A Brief Historical Perspective Humans are mobile. People like to move, to travel, to roam. In the dawn of human history, there were no vehicles, and all transportation was by our own— human—power. Probably the very first vehicles were floating logs or vegetative mats carried by currents or waves. When an early human first kicked, paddled, or poled one of these, human-powered vehicles came into existence. Simple paddled or rowed watercraft was quite likely the earliest human-powered vehicles. By the time of the ancient Greeks, human-powered boat technology had advanced to large rowed vessels—biremes and triremes—that were up to 40 m (130 ft) long and powered by a crew of up to 170 men. The ancient Greek trireme, with three rows of rowers, probably was capable of seven or eight knots for short bursts.1 The technology, as always, was driven by need—in this case the need for a fast, powerful military vessel. For millennia, human muscles, along with wind and currents, were a dominant source of power for boats. For land transportation, animal power as well as human power was extensively used prior to the industrial revolution. Wheeled vehicles probably date back at least 5,000 years. While these may have been primarily pulled by animals, surely some were pulled by humans—perhaps creating a primitive version of the rickshaw. With the advent of steam and internal combustion engines, and later electric motors and batteries, most high-power transportation modes switched from muscle to alternative power sources. This transition, which occurred during second half of the eighteenth century, marked

1

Abbot, Allan V. and Wilson, David Gordon, 1995, Human-Powered Vehicles, Human Kinetics, Champaign, IL, Chap 2.

13


a blossoming of new technologies, including the bicycle, the automobile, and the soon-to-fly airplane. Muscle power (both animal and human), steam power, and power from the new internal-combustion engine were all viewed as viable options for transportation and industry. Electric power was also rapidly developing, although the power grid was still extremely limited and batteries were heavy and low-powered. Human mobility was increasing significantly, and vehicles of all types were rapidly developing. These heady times saw the bicycle develop from a rudimentary hobby-horse type device into a practical, clean, and fast means of personal transportation. The bicycle srcinated with the Draisenne (Figure 2-1), invented by German Baron Karl von Drais early in the nineteenth century. This device had two wheels, the front one steerable. It was propelled by kicking the ground, similar to a young child’s bicycle. By 1880, the bicycle had evolved into the ordinary, or highwheel, bicycle of familiar lore. This vehicle, sometimes called a penny-farthing, was driven by cranks connected directly to the front wheel. In order to obtain high speeds, the wheel diameter was increased to quite large sizes—up to 1.5 m (60 in). The large wheel allowed higher speeds, but was limited by rider leg length. These bicycles, with a dangerous tendency to pitchover (throwing the rider over the handlebars during hard braking), were soon replaced with the safety bicycle, driven by a chain drive which permitted higher gearing and hence smaller wheels. This lowered the center of gravity and moved the front wheel forward, significantly reducing pitchover accidents. The chain drive, in conjunction with the invention of the pneumatic tire, created a strong demand for bicycles in the last decade of the nineteenth century. The bicycle industry experienced rapid and extensive growth, and spawned many technological improvements that carried over to the automobile and aviation industries in the early twentieth century. 2 The late nineteenth and early twentieth centuries were exciting times for the bicycle industry. Cycling was immensely popular. Cycling was the most advanced form of transportation for local trips, and bicycle racing was quite popular. Cycling technology advanced dramatically, with many new developments in structural designs, components, and systems. Recumbent seating, tricycles, and tandems were all tried in a variety of configurations. However, by the 1930s international racing rules prohibited all but conventional upright frame designs. The bicycle industry by and large followed the racing standards, and the upright bicycle design became the standard.

Herlihy, David V., 2004, Bicycle, The History, Yale University Press, New Haven and London. 2

14

Overview of Human-Powered Vehicles

Figure 2-1 Draisenne or Hobby Horse

Although the basic configuration of the modern multi-speed bicycle did not change dramatically during the twentieth century, cycling technologies continued to develop. Advancements in materials, processes, and components were dramatic. Perhaps the bigger story of that century is the shift in transportation modes and public perception of bicycles. At the turn of the twentieth century, bicycles were seen primarily as affordable, fast transportation. As the automobile became more affordable—most dramatically in the United States, and to a lesser extent in Europe—and the network of roads (srcinally paved for cyclists!) improved, public perception of the bicycle shifted from a desirable primary mode of transportation to a sporting device or child’s toy. Recreational cycling increased in the United States at the end of the twentieth century, perhaps due to the introduction of the mountain bicycle in the 1980s. Cycling as a primary transportation mode in the United States remains low, but is experiencing slow growth. In Europe, particularly northern European countries, cycling for transportation is much more common than in the US, and bicycle sales exceed automobile sales. In both Europe and the US, regional planners are much more likely—and often required by law—to consider infrastructure investments for cyclists than they were several decades ago. This is in contrast with China, where the bicycle has been a ubiquitous and affordable transportation mode for quite some time. As China’s economic growth continues, there is a mode shift away from bicycles to private automobiles.

15


Land Vehicle Applications and Functions Human-powered vehicles meet a diverse variety of transportation needs around the globe. Unlike the United States, where cycling, along with canoeing and kayaking, are seen primarily as recreational activities, many people choose human power as the most viable transportation alternative. The choice of transportation mode, is—and always has been—based on the balance of needs and resources. For many, human-powered vehicles are simply the best option. Factors that affect mode choice include the available transportation infrastructure, performance and functional needs, the local environment, public transportation policy, and economics. In some areas of Africa, public transportation is very limited, the road network is poor, and private automobiles are very expensive. Bicycles are highly valued for transportation. In Copenhagen, Denmark, high costs of living and taxes make automobiles quite expensive; an excellent infrastructure for cycling and public transportation exists; and cycling is perceived as a viable and affordable—and socially acceptable—option. As a result, 38% of trips into downtown Copenhagen are by bicycle, and 50% of trips within the central city are by cyclists. In Cairo, flatbread is delivered by cycle couriers that reportedly balance up to 50 kg of bread while they work their way through congested traffic.3 The entire bike messenger business, common in many US cities, exists because cycling is often faster than automobile travel in congested urban traffic. Land human-powered vehicles are used primarily for personal transportation: one person getting from one place to another. Some vehicles can carry passengers. Passengers may be young children in a vehicle designed and marketed for family transportation or they could be paying customers in a pedicab. Tandems allow both riders to pedal; some vehicles allow three or more riders to contribute to propulsion. In some cases, the passenger is a medical patient, and the vehicle is a human-powered ambulance. Several of these are in operation in Namibia and other African nations. Rather than carry people, some vehicles are cargo haulers, designed to transport relatively heavy loads. Several commercial models are available that are designed to carry over 90 kg of cargo. Many industrial factories use work cycles to move people or light goods around large plants. Trailers are also widely used with cycles to transport both children and cargo. Bicycle courier, or bike messenger, services are offered in many urban areas, where cyclists can deliver light parcels or letters much quicker than alternative transportation modes. These cyclists are able to negotiate around heavy automotive traffic and congestion in order to make prompt deliveries. Somewhat

3

Marthaler, Claude, 2006, “Riding with the Breadmen,” Velovision, 22, pp. 10–12.

16


larger parcels are routinely carried in panniers, trunks, baskets or backpacks by cyclists during the daily routines of life, such as shopping, commuting to school or work, or a picnic in the park. Many bicycles are used for recreation or competition. These cycles are often designed to be fast and sporty, with little consideration for cargo, weather protection, or reduced maintenance cycles. Professional bicycle racing is a very popular sport in some areas of the world, and famous races, such as the Tour de France, draw very large viewing audiences worldwide. Recumbent bicycles and multi-track vehicles are not generally allowed in these races, so alternative venues have developed specifically for these vehicles. The Race Across America—a race in which contestants ride across the United States from the Pacific to the Atlantic coasts—permit both recumbents and tandems, but they compete in different classes from the upright single bicycles. The annual World Human-Powered Speed Challenge, held at Battle Mountain, Nevada sees thefastest human-powered vehicles in the world—sleek recumbent streamliners that have demonstrated speeds exceeding 38 m/s. The current record, set on September 17, 2015 by T odd Reichert is 38.3 m/s (85.71 mph). Recreational riders cycle for pleasure: the sounds, sights and smells of spring ride along a country road can be immensely rewarding. Some recreational riders also ride for fitness, reaping health and fitness benefits in an enjoyable sport. More adventurous riders use human-powered vehicles for extended tours lasting from a few days to many months. Touring cycles are usually rugged, with reliable tires and equipment suitable for varied and perhaps hostile terrain. They generally have good cargo capacity and are designed to be comfortable for long hours of riding.

Land Vehicle Configurations The arrangement of elements, such as wheels, seat, drive train, and steering system, define the configuration of a vehicle. For land human-powered vehicles, the number and position of the wheels and the body position of the rider are the most significant configuration options. Bicycles have two wheels, tadpole tricycles have three wheels with one in the rear and two in front, delta tricycles have one wheel in front and two aft, and quadricycles have four wheels. Body positions are most often identified as upright or recumbent, although there is considerable variation within each classification. The word recumbent means “lying down.” In cycling, the more accurate term would be semi-recumbent, referring to leaning back in a reclined position. The term is shortened in the vernacular to “recumbent,” and sometimes the even shorter “bent,” and is often used as a noun to denote any vehicle in which the rider sits in a semi-recumbent attitude.

17


Vehicle configuration is determined during the conceptual stage of design. In the vast majority of cases, the configuration is immutable once the design is in production. There are exceptions, however, and some vehicles are designed to be convertible by the user. A few production bicycles can be converted from a long wheelbase recumbent with the front wheel in front of the cranks to a short wheel base, for example. Aftermarket kits are available to convert some bicycle models into tricycles. (The author once had the opportunity to ride a bespoke vehicle that could convert between a recumbent and upright riding position. The conversion could be made on-the-fly, while riding the vehicle. It was quite fun to ride, if of questionable utility.) Configuration design is an important task that must be completed early in the design sequence. The following descriptions provide examples of some fairly typical configurations, along with some general comments. The familiar upright bicycle has two wheels, and the rider sits on a saddle in an upright position or a forward-leaning crouch. Steering is generally direct, with several different styles of handlebars. Typically, the rear wheel is driven, and the front wheel is steered. The differences between upright bicycles are generally more in the details than in configuration. Mountain bicycles, city bikes, and racing bicycles, while different, all share a similar configuration. Road bicycles usually have drop-style handlebars and the rider is positioned in a crouch that minimizes air resistance and provides good power transfer. These bicycles usually use narrow tires and caliper brakes. Mountain bicycles position the rider in a more upright position, with straight or riser handlebars. They usually have wide-range gearing and can accommodate wide tires. Linear pull, disc, or cantilever brakes are used as these types better accommodate the wide tires. Touring bicycles usually have wide-range gearing, a longer wheelbase, and ample fittings for racks and bottle cages. They may have any style of handlebar, but are designed for heavy use and relative comfort for long hours in the saddle. Figure 2-2 shows an upright bicycle designed to be folded and transported in a suitcase. This bicycle requires some disassembly prior to transport, but can be easily checked on an airline or train as ordinary luggage. Two companies known for their high-quality travel bikes are Bike Friday and Airnimal. Other folding bicycles are designed to fold quickly and be stored in a very compact space. These are frequently used by urban commuters that either have little storage space or use multi-mode transport, carrying bikes onto the subway or bus. Some of these models fold into a compact package within seconds. Brompton and Dahon are examples of companies that specialize in folding bicycles. Recumbent bicycles also have two wheels, but the rider sits in a seat rather than on a saddle. Recumbent bicycles tend to be more specialized than upright bicycles. The fastest bicycle in the world is a recumbent. The most comfortable

18


Figure 2-2 Upright folding travel bicycle

bike in the world is a recumbent. The safest bicycle in the world is most likely a recumbent. However, these may be three very different bicycles. This illustrates the great diversity of configurations for recumbent cycles. Most recumbent bicycles have rear-wheel drive, although some use front-wheel drive to avoid lengthy chains and idlers. Steering may be direct, meaning the handlebar connects directly to the fork, or indirect, meaning the fork and handlebar are connected via a linkage of some type. There are many options for handlebar type and position. Some have the handlebar mounted under the seat, while others may curve over the rider’s legs or extend nearly to the rider’s chest. The last is sometimes colloquially called the “begging hamster” position. The wheelbase has great significance for recumbent bikes as it affects rider position, handling, and vehicle structure. Long-wheelbase recumbents often have the front wheel located in front of the crank. They tend to be larger and heavier than comparable short-wheelbase models, but may be easier to ride and somewhat more stable. Short-wheelbase bikes may have sportier handling and be lighter and more compact. The crank is usually in front of, or above, the front wheel. Seat construction and position can vary greatly between vehicles. Seats can be rigid molded composites, made of mesh slings, or a combination of the two. Position varies from a fairly upright sitting position to very reclined. In general, the more upright positions are more comfortable, and are used on recreational vehicles, while the more reclined seats reduce aerodynamic drag and are frequently used on performance bikes. Hard or rigid seats are also often used on high-performance vehicles, and provide a firm back to resist hard pedal forces. Sling seats provide both breathability and some suspension, and are generally considered more comfortable. Many seats have mesh back rests and a padded, rigid seat bottom. The position of the cranks relative to the seat also affects rider position. The most aerodynamic vehicles place the cranks

19


Figure 2-3 Long-wheelbase recumbent bicycle

above the seat bottom and have very reclined seat backs. Lower crank positions and more upright seating are used when comfort is more important. The following figures illustrate several examples of recumbent bicycles. Figure 2-3 depicts a long-wheelbase bicycle with over seat direct steering, a padded hard seat bottom and mesh seat back. This bicycle would likely be well suited for rides through the countryside, where maneuvering is less important than a comfortable high-speed cruise. A short-wheelbase bike is shown in Figure 2-4. This bicycle also has direct steering, but uses a pivoting handlebar mast that allows easy mounting, dismounting, and comfortable hand positioning. Note the hardshell seat. The vehicle in Figure 2-5 is a low-racer streamliner, designed to be fast and aerodynamic. The entire bicycle fits within a compact, streamlined fairing. It has front-wheel drive and a steeply reclined hard shell seat. This bicycle would be very impractical in urban traffic, but can be quite fast on the race track. Note that in the figure, only one-half of the fairing is depicted in order to show the bicycle inside the fairing. Gunnar Fehlau presents a nice overview and comparison of many configuration options for recumbent bicycles.4 His book is easy to read, and includes a brief description of each option, along with bullet lists of advantages and disadvantages. Tricycles have three wheels, with either upright or recumbent rider positioning. Tadpole tricycles have two wheels in front and one wheel in back. Usually, the front wheels are steered and the rear wheel provides drive power. Tadpole trikes tend to provide the best handling response, but the steering mechanism is

4

Fehlau, Gunnar, 2006, The Recumbent Bicycle, 2nd Ed., Out Your Backdoor Press.

20


Figure 2-4 Short-wheelbase recumbent bicycle

Figure 2-5 Short-wheelbase low-racer streamliner. This bicycle has front-wheel

drive

more complicated. Underseat steering via handlebars and an Ackerman linkage (described in Chapter 16) can provide very nice steering response, although direct steering via handle bars connected to the kingpins is also a common option. The example shown in Figure 2-6 is a recumbent tadpole tricycle with direct steering. Like most tricycles, tadpoles are stable when stopped or moving very slowly. This is a distinct advantage for riders with balance problems, or riders that spend a great deal of time at very low speeds or hauling very heavy loads up hills. Performance tadpole trikes usually have long drive trains, often routed below the seat with idlers. Low seating provides more stability during high-speed cornering,

21


Figure 2-6 Tadpole recumbent tricycle with direct steering and hard shell seat

although not all tadpoles have low seating. A few upright tadpole tricycles have been produced, although most models are recumbents. In contrast to the tadpole configuration, delta tricycles have two wheels in back and one in front. Steering is much simpler than the tadpole, but drive systems can be more complex. Delta trikes with rear wheel drive can be driven by only one wheel or both wheels. If both rear wheels are driven, a mechanism to allow the outside wheel to speed up during turns prevents wheel slip and the resultant handling problems. This can be done by clutches or a much more expensive differential. Some delta tricycles are driven with only one rear wheel, avoiding the problem altogether. These vehicles produce a turning moment under heavy pedaling which can be a problem in low-speed, high-torque situations. In some conditions—such as a hard, steep climb on a wet road—this can cause the front wheel to lose traction and control. Front-wheel drive is another option, but this usually results in pedals that swing during maneuvering or twisting chains. Delta tricycles have a tendency toward oversteer as speed increases. This results in poorer handling (and in extreme cases less stability) than tadpole trikes. Quadricycles, or quads for short, have four wheels. Quads can have better stability and handling than either tadpole or delta tricycles. In addition, there are essentially no restrictions on wheelbase, since the wheels do not interfere w ith either the seat or the cranks. However, quads have both the drive train complexities of the delta trike and the steering complexities of the tadpole.This usually results in a more complex and potentially heavier vehicle. Accurate wheel alignment both front and rear is required for good performance. There are a great many configuration

22


Figure 2-7 Delta recumbent tricycle with under-seat steering

variations possible with quads. Figure 2-8 shows a recumbent quadricycle designed to use either hand cranks or leg cranks. This is a multi-configuration design: the leg cranks can be replaced with stirrups for handcycling, or the hand cranks could be replaced with a conventional handlebar for leg cranking only. A minor drivetrain modification allowed both drive options to be used simultaneously. The examples in this chapter illustrate some of the many configurations that are possible with land human-powered vehicles. The number and position of the wheels, position of the rider, number of riders, drivetrain type and wheels driven, handlebar position and steering system design, and other options define the configuration. An understanding of vehicle systems and configuration options is very beneficial to the design engineer at the outset of a new vehicle project. Additional information for the interested reader:

For an excellent history of the bicycle, see: Herlihy, David V., 2004, Bicycle, The History, Yale University Press, New Haven and London. The following book contains illustrations of early bicycles and (mostly) components. Originally published in Japan, there is essentially no text, only figures.

100 Years of Bicycle Component and Accessory Design Authentic Reprint Edition of The Data Book, Van der Plas Publications, San Francisco, 1998.

23


Figure 2-8 Recumbent quadricycle with options for either arm or leg cranks

24

CHAPTER

GENERAL STRUCTURED DESIGN

3

OF HPV’s

D

esigning a new human-powered vehicle from scratch is both stimulating and rewarding. It may be quite challenging, and the road from idea to realization can be more convoluted than linear. A pre-established structure, or plan, for the design process can be helpful to the designer, in much the way an outline aids a writer. This chapter defines and describes a general structure for HPV design. It is a broad template, applicable to many types of human-powered vehicles. The objective is to provide an outline or plan for the design process. Each section includes a description of tasks that should be completed, followed by a brief outline, which can be used as a checklist.

General Structured Design of HPVs The design process for human-powered vehicles is similar to that for any other product. The design progresses through stages, from formulation of the problem statement to prototype testing. These stages are depicted in Figure 3-1 and described throughout this chapter. The central column in Figure 3-1 represents the eight stages of design. These are essentially the same steps used in the design of any product. The left column represents requirements that must be completed at each stage. These are specific to human-powered vehicle design. A large portion of this book is devoted to these topics. The right column illustrates the iterative nature of design. The bubble labeled “verify mission” is a reminder of the importance of keeping a sharp focus on the mission of the vehicle. Although this chapter divides the design process into eight distinct stages, this is for organizational simplification only. The design process is continuous and

25


Customer Info

Define Design Parameters

1. Define Vehicle Mission

2. Product Design Specification

Verify Mission

Determine Target Values Configuration Gross Geometry 3. Develop Vehicle Concepts

Drive Train Frame Geometry Special Features

4. Concept Selection

Iterate As Required

5. Parametric Design

Iterate As Required

6. Detailed Design

Iterate As Required

Geometry Analysis Testing

Drawings Manufacturing Components 7. Prototype Fabrication

8. Performance Evaluation

Figure 3-1 Diagram of the design process

contains many iterations, changes, and modifications. Frequently, stages overlap chronologically. For example, some parts may be in detailed design stage or even prototype fabrication stage while other parts are still in the parametric design stage. Nonetheless, the general workflow follows the stages described here and depicted in Figure 3-1.

26

General Structured Design of HPV’s

STEP 1. Define the mission of the vehicle A clear mission statement for the proposed vehicle is invaluable. It is far too easy for a design to gradually shift direction mid-stream, resulting in a vehicle that may not be what the customer wants. A well-formulated mission statement forces the designer to focus on the most important aspects of the project, and maintains that focus throug hout the design process. This is the best way to ensure that the customer gets what he or she orders. A mission statement should clearly and concisely state the primary functional requirements for the vehicle. For example, a vehicle designed for year-round commuting in northern temperate might have the following mission statement: This vehicle shall provide reliable year-round single-person transportation in northern urban and suburban regions. This statement is brief and contains quite a bit of information. The word reliable indicates that the vehicle will not break down very frequently. The vehicle must be practical for the transportation needs of a single individual. This probably implies that cargo space be included. Other items can be inferred, such as lights and reflectors for night transport. Northern urban and suburban regions will undoubtedly include weather and environmental hazards such as rain, snow, poor pavement, and probably salt. While much of this could be inferred from the statement, additional sentences can clarify and remove ambiguity from the statement. This vehicle shall provide reliable year-round single-person transportation in northern urban and suburban regions. Expected usage includes personal transport, commuting, shopping, and recreation. The operator must be provided reasonable protection against the elements, and vehicle maintenance should be minimized. The vehicle should be comfortable, easy to operate, and easy to propel. Expected environmental conditions include cold, heat, rain, snow, and salt. The vehicle should be safe and legal for day or night travel. a. Obtain customer information If the user of the vehicle is not the designer, it is quite important to consult actual or typical customers and potential users of the vehicle. This should be done early in the design, and continue throughout the design process. The mission statement should be developed in conjunction with customer input. STEP 2. Develop vehicle design specifications Design specifications define the attributes of a successful vehicle. In engineering design, these are more generally called product design specifications, or, collectively, the PDS. The design specifications should clearly identify the qualitative and quantitative attributes

27


required or desired in the final design. They are used as a reference throughout the entire design process. For example, a specification for the maximum load capacity of a vehicle is used in vehicle performance calculations, and structural analyses, and vehicle testing. The vehicle design specification should include the purpose, function, and intended market for the vehicle; functional performance requirements; a-priori design constraints; and any additional requirements. Specifications should be specific with regard to outcomes without limiting design options. That is, the requirements should be clearly identified without placing any constraints on how they will be physically realized. This gives the designer guidance without limiting creativity. Developing the vehicle design specifications requires gathering information, identifying parameters, and setting target values. 1. Information Sources The product design specification should not be written without adequate information. Sources of information include customer data (Step 1), codes, standards, and warranty or historical data. Information should also be gathered from vehicles with similar missions or technologies. Particular emphasis should be placed on customer desires, whether the customer be a single individual or a population. This information is obtained with user surveys and interviews. QFD is particularly useful for this. Historical and current solutions should be thoroughly investigated as well. Many designers have made the mistake of re-inventing old technologies, sometimes with the srcinal faults or limitations. Vehicles that serve similar missions should be investigated and ridden if possible. The new design should show a distinct advantage over other vehicles in performance, cost, comfort, or other attributes. 2. Design Parameters Design parameters should be identified while developing the vehicle specifications. A well-defined set of critical design parameters provides a template for the new vehicle design. Three classes of parameters are defined: control variables, envelope variables, and performance metrics. Identifying these parameters from scratch for each design is time consuming and runs the risk of omissions. The alternative— to start with a slate of pre-determined parameters—reduces design time and potentially improves the quality of the final vehicle. a. Design Control Variablesinclude the parameters over which the designer has complete control. Examples include wheelbase, castor angle, wheel size, etc. In general, these variables should not be specified

28


in the PDS unless there is a strong a-priori reason for doing so. The values of these parameters will be determined during the design process, and will depend on the conceptual design solutions. b. Design Envelope Variables describe attributes over which the designer has no direct control, but whose values are required in the design process. These typically depend on the operator and environmental conditions. Examples include limits on operator size and weight, design vehicle speed envelope, etc. Many of these will be specified in the PDS, either with a target value or a range of values. c. Performance metrics are used to evaluate how well a proposed design meets the mission and design specifications. Objective, quantifiable performance metrics are said to be hard, while subjective metrics are soft. Examples of hard metrics include vehicle weight, coefficients of drag and rolling resistance, power required to operate the vehicle, etc. Aesthetics, ride quality, etc. are soft metrics. Performance metrics should be included in the vehicle design specifications. For many vehicles, the performance specifications are among the most important. 3. Determine target valuesValues must be determined or f the parameters identified in Step 3. The starting point is always the design specifications— the design envelope variables and performance metrics. These are incorporated into the product design specification, which is the driving force for vehicle design. The importance of this step cannot be overemphasized! A poor job at this stage will result in ambiguity in later stages of the design. The result will likely not meet the customer’s desires, or will fail to perform as required. The target values used in the specifications should be stated in a way that directly relates to analytical and experimental results. During the design process, the performance of proposed concepts is evaluated and compared to the specification. A specification that is expressed in a form that cannot be evaluated with reasonable accuracy is not useful. Consider vehicle top speed. For most vehicles, setting a numeric specification for vehicle top speed simply does not make sense. There are many factors that affect top speed, including the strength of the driver, road and wind conditions, and grade. Most human-powered vehicles will never operate at their theoretical top speed. (Exceptions are vehicles for which the achievement of that speed is critical to the success of the vehicle.) Much more important than top speed is the overall efficiency of the vehicle, which can be expressed in a variety of ways. For a hard specification, the

29


speed envelope under specified conditions may be appropriate, as the following example for an unfaired tandem bicycle illustrates.

Target Speeds for Tandem Bicycle Conditions: 600 W Power Total from Both Riders Smooth Asphalt Pavement • • • •

24 m/s with a 5% down slope and negligible wind 6 km/h, 5% up slope with negligible wind 14 m/s on a level surface with negligible wind 11 m/s on a level surface with 5 m/s head wind

First, determine Design Envelope variable values. Use customer data, field experience, and knowledge of the operational environment to determine design envelopes. An obvious example is to determine values for height, weight, fitness level and potential power output of the design operator or operators. Chapter 4 includes data on the power a human can produce with leg cranks, and is relevant to this task. Second, determine values (or ranges) for all the primary performance metrics. The product design specification (PDS) can then be prepared. Note that soft metrics can be quantified with appropriate scales. For example “Overall Appearance of Vehicle” may have a minimum value of 4 based on a scale of 1 to 5. In general, values for the design control variables will be determined in a later step. These values will be selected in order to obtain the desired performance characteristics defined by the design envelope variables and the performance target metrics. Design control variable values should only be set at this point if there is a valid a-priori reason for doing so. CHECK: Before proceeding, go back to the mission statement for the vehicle and verify that all target values are consistent with it. If not, return to Step 2 and revise the values. Following is an example of a vehicle design specification. This specification is 1

based loosely on the format presented by George Dieter. The interested reader is referred to that reference for more information on general mechanical engineering design.

1

Dieter, Geroge E., Engineering Design A Materials and Processing Approach 3rd Ed., McGraw-Hill, 2000.

30


Example Vehicle Design Specification Mission Statement This vehicle shall provide reliable year-round single-person transportation in temperate to cool urban and suburban regions. Expected usage includes personal transport, commuting, shopping, and recreation. The operator must be provided reasonable protection against the elements, and vehicle maintenance should be minimized. The vehicle should be comfortable, easy to operate, and easy to propel. Expected environmental conditions include cold, heat, rain, snow, and salt. The vehicle should be safe and legal for day or night travel. In-Use Purposes and Market Product Title: Practical Velomobile Purpose/Function To provide clean, quiet, and efficient personal transportation for typical daily tasks regardless of season or time of day. Unintended Uses Operation on very rough terrain Carrying passengers Special Features Weather protection against rain, snow, cold and heat Cargo capacity for routine shopping or work-related gear Sufficient lighting and reflectors for safe and legal night transportation Reduced maintenance, especially in salty environments Competitors Production velomobiles are direct competitors. Areas that can be improved over production velomobiles include cost reduction, weight reduction, improved ingress and egress, and visibility. Production recumbent bicycles and tricycles are also competitors. Areas for improvement include weatherization, winter performance, drag reduction, and maintenance. Upright bicycles may also be competitors. These vehicles are lower cost and offer excellent maneuverability, visibility, and operational flexibility in urban traffic. A challenge is to retain as many of these advantages as possible with a weatherized vehicle.

31


Functional Requirements Performance Requirements Turn Radius: Turn radius shall not exceed that required to negotiate a turn on 1.52 meter wide sidewalks with no rounded corners. That is, the minimum turn radius shall be

R

1   =W   ) − 1 cos(45   °

R

1 1  +T  − − 2 1 cos(45 )

°

= 3.4142 W − 2.9142 T

where R is the centerline turn radius, W is the sidewalk width (1.52 m) and T is the vehicle wheel track in meters. Braking: The vehicle shall be ca pable of making a complete stop from a speed of 25 kph within no more than 6 meters on level, dry, smooth asphalt. Speed and Power: With a 75 kg driver supplying no more than 180 watts on smooth asphalt surfaces, the vehicle should be capable of the following: 13 m/s 30 m/s 3.3 m/s 10 m/s

Level Road 5% Down slope 5% Up slope Level Road

No wind No wind No wind 5 m/s head wind

No more than No more than No more than No more than

180 Watts 180 Watts 180 Watts 180 Watts

Mechanical Drive Train Efficiency: The average mechanical efficiency of the drivetrain should be 90–95% with clean, new components. Mechanical efficiency is defined as: h = Prear wheel/Pcrank. Ingress/Egress: The vehicle should permit easy, unassisted ingress and egress of driver, as well as easy loading of cargo. Parking: The vehicle should be equipped with a device to prevent rolling while parked. Theft Protection: The vehicle shall have a convenient means to prevent theft. Driver’s View:The driver shall have views of the road and surrounding areas sufficient to permit safe operation in traffic.

32


Physical Requirements Cargo Volume: At least 60 L in no more than two compartments. Cargo Mass: The vehicle must be capable of carrying at least 25 kg cargo. Maximum Vehicle Weight:25 kg. Ground Clearance: For a multi-track vehicle only, ground clearance must be sufficient to pass over a 7.5 cm cube anywhere between the outermost wheels without touching the vehicle (except for additional wheels inside the track). Frontal Area: Total frontal area of the vehicle (including maximum size design rider) shall be such that the fully faired vehicle can achieve a drag area (CDA) of not more than .150 m2. Design Operators: The vehicle shell shall be capable of fitting riders with normal proportions ranging from 145 to 155 cm height and weighing no more than 72 kg. Seating and drive train components may be replaced as necessary to accommodate different sized riders. No special clothing shall be required to enter, operate, or exit the vehicle. Components with potential to stain clothing, such as chains, shall be covered or located outside of the driver compartment. Regulations: Meet or exceed all safety requirements contained in 16CFR1512A and ISO 4210. Service Environment Region: The vehicle shall provide comfortable and safe transport in temperate to cold climates in urban, suburban, and rural areas. Road Surface: The vehicle shall be operable without significant service or life penalty on road surfaces ranging from smooth asphalt or concrete to gravel or loose dirt, including rough or broken asphalt, grass, chip seal, etc. Weather: The vehicle shall be operable without undue safety penalty in rain, snow, slush, mud or on ice. Tires and other components may be changed to meet weather extremes. Consideration should be given to reducing corrosion due to riding in wet, muddy, or salty (northern winter roads) environments. Temperature: All vehicle systems should be operable between –25°C and 40°C ambient temperature.

33


Safety Hazards: There should be no hazards such as sharp edges, open tubes, or pinch-points that could harm the operator or bystanders during normal use of the vehicle, including operation, ingress and egress, and moving the vehicle. Crashworthiness: The vehicle shall be able to sustain a head-on collision from 1.3 m/s with no permanent deformation to the vehicle frame or steering mechanism and no damage that would render the vehicle inoperable. Vehicle fairings should withstand normal handling of the vehicle, including a person leaning on the fairing. STEP 3. Develop vehicle concepts 1. Determine vehicle configuration The vehicle configuration must be determined very early in the design stage. Configuration implies both form and principle features. The number and location of wheels, driven wheels, steered wheels, and type of steering are examples of configuration options for a land vehicle. A configuration for a solo land vehicle might be a short wheel base semi-recumbent bicycle with under seat steering and rear wheel drive. A boat configuration might be an open monohull single operator boat with stern propeller drive. Choosing a configuration places limits on control variable values, constraining the design. Values become easier to establish and to interpret. See Chapter 2 for examples of vehicle configurations. CHECK: Before proceeding, check that the configuration is compatible with the mission statement. If not, repeat this step and develop a new configuration. 2. Determine vehicle gross geometry Tentatively assign control variable values to establish the gross vehicle geometry. These parameters should be sufficient to define the vehicle size and basic performance characteristics. Strive for values that will obtain performance goals. Known design principles, background knowledge, experience, comparisons with similar vehicles, and any other source of information may be used to help determine the values. In most cases, values are determined based on performance criteria. For example, the wheelbase, center-of-gravity range, headtube angle, and fork offset for a bicycle concept should be selected to provide acceptable handling and stability characteristics. Gross geometry is usually based on handling performance, as described in Chapters 10 and 11.

34


3. Design the drive train and establish chain lineDetermine the type of drive train and the physical locations of drive components. If a chain is used, the best approach is to establish the chain line early, then zealously guard against frame or component encroachment. In most cases, the chain line can be established as soon as gross vehicle geometry is known. The frame and components can then be designed around the chain. (In some cases, preliminary frame geometry must be developed simultaneously with the chain line. This is clearly the case when the chain will be routed internally through the frame.) Drive train design is described in Chapter 12. 4. Establish initial frame geometry Frame geometry is driven by the gross vehicle geometry, drive train, and expected load cases for the vehicle. At this stage, viable conceptual gross geometry is required. The type of frame—monocoque or space frame—and preliminary location of frame members is required. Final geometry, tube sizing, and optimization will take place in a later stage. Chapter 13 discusses frames and frame analyses. 5. Special features Special features should be included as viable concepts. These features may include cargo areas, lighting systems, fairings or shells, landing gear, etc. It is best to address these features during the concept stage, rather than wait until later stages. This approach provides better system integration and improved vehicle quality. STEP 4. Select Optimal Concept The goal of the concept selection process is to choose the best of all conceptual designs for further development. Concept selection should be based on rational decision making processes, which are described in many engineering design texts, including Dieter.2 Ideally, several viable concepts should be considered. Each should be evaluated based on its merits, using the most quantitative measures available. The concepts should be developed to comparable levels of detail. Once the optimal concept is selected, the design can proceed with confidence that the proposed design solution is the best available. STEP 5. Complete Parametric Design Part geometry is established and verified in the parametric design stage. This is the stage where frames, shells, and special-purpose parts are designed, validated, and finalized for prototyping. For example, the gross geometry of the frame, along with tubing material, diameters, 2

Ibid.

35


and wall thickness is established in this stage. The geometric shape, size and material for all custom-designed components should also be designed. The three major aspects of this stage are (1) determination of part geometry and material, (2) analytical validation of the geometry and material, and (3) experimental testing required to validate geometry and material. Initial geometry should be based on functional requirements, such as vehicle performance, strength, and life specifications. Aesthetics of the design are must also be addressed. Appearance and style often make the difference between a successful design and a mediocre one. For the functional requirements, use relevant analyses to assess the success of the proposed geometry and material selection. This includes handling (Chapters 15 and 16), structural requirements of the frame (Chapter 18), and possibly power/speed characteristics (Chapters 12–14). Often, data required for an analysis is not available. In this case, experimentation may be required to obtain the necessary data. For example, the designer of a bamboo frame should know the flexural strength of the bamboo used. Due to the many variables involved—such as species, treatment, moisture content, etc., few published tables are able to give reliable values. An experiment can provide the necessary information. The design is iterative. Analyses and optimization studies are performed on trial geometry, the geometry and/or material is updated, and the analyses are repeated. The cycle continues until satisfactory results are obtained. Optimization methods are available in many cases to speed convergence. It is important during this process to keep the mission of thevehicle in mind. It is far too easy to shift focus during this stage, developing a possibly fine vehicle that does not meet the objectives of the project. This is known as “mission drift” and most designers are susceptible to it. Performance analysis is a key component of the parametric design stage. Performance, in this context, should be interpreted broadly so as to include all aspects of the design. The product design specifications set performance goals: analyses conducted at this stage predict if those goals will be reached or exceeded. Ideally, the design should continue iteration until all specifications are met. In the real world, compromises must sometimes be made, particularly with very ambitious design goals.

STEP 6. Complete Detailed Design Detailed dimensions, tolerances, specifications, and other detail items are completed in the Detailed Design stage. This is a critical stage, requiring diligence and attention to detail. Incorrect specifications can lead to non-functional parts, cost overruns, and lost time. The three primary tasks include development of working drawings, manufacturing plans, and specification of components. Detailed working drawings are needed for production of the frame and special-purpose parts. Drawings should include all dimensions required to fully

36


specify the part, but no redundant dimensions. Tolerances should be as large as function will allow. It is quite possible to increase the cost of a part by an order of magnitude or more by specifying a smaller tolerance than is required. As a rough rule of thumb, the cost doubles for every additional decimal place used in a tolerance. A tolerance of 0.01 mm will generally require grinding, and will be approximately twice as expensive as a tolerance of 0.10 mm. Likewise, surface finish and other special requirements should be used only where necessary for fit or function of the parts. Note that dimensions on welded structures such as vehicle frames can be very difficult to hold to close tolerances. Manufacturing plans should be prepared for each part and assembly. They include the process used to make the component, raw materials, equipment required and any special tooling required. Special tooling is very significant, as it needs to be made prior to manufacturing the part. Effective time and money allocations depend on good manufacturing plans. Component selection should be finalized during this stage. Wheels and tires, drivetrain components, brakes, etc. should be specified. A very wide range of commercial components is available, with an equally wide range of price and performance. The selected components should match the mission of the vehicle while also meeting budget constraints. Often, components must be carefully selected to work together for optimal performance. See Chapters 17 and 19 for more details on component functionality and selection. STEP 7. Fabricate Prototype On completion of the design, a prototype vehicle should be constructed and evaluated. In some cases, a single bespoke vehicle is the goal of the design process, and the prototype is the final product. Occasionally, one or more prototypes must be constructed during the design stage in order to test specific characteristics of the vehicle. This is usually only required for very innovative designs, when a design deviates significantly from existing vehicles. Prototypes take time and money to complete. To be most effective, testing should be planned in advance, and the prototype(s) built to accommodate the required testing. For example, a prototype meant to test a new transmission might be constructed by modifying an existing vehicle to use the novel drivetrain. This would permit testing the transmission sooner with less expense and effort. STEP 8. Complete Functional and Performance Testing The design must be validated with experimental testing for functionality and performance. A significant investment of time, money, and effort goes into the design and construction of a new vehicle. Inadequate or poorly planned experiments

37


reduce the value of that investment—valuabl e knowledge of the effectiveness of design elements and features is lost. To retain the maximum value, a comprehensive plan for testing should be developed and implemented. Testing should include functionality and performance of all systems and the overall vehicle, with the goal of answering the following questions: Do the systems function as designed? How well do they function? How could functionality be enhanced, or how can non-functional systems be made functional? Does performance meet or exceed specifications? How could performance be improved? Results of testing should be compared to the design specifications and the analytical predictions. This step closes the loop of the design process, and makes future design and development efforts much more productive and fruitful. Knowledge of the successful and non-successful design elements is very valuable. The most effective experimental program follows these guidelines: • • • • • •

Use the Product Design Specification as a guide for planning experiments. Develop a plan for each experiment. Each experiment should include an explicit objective. Each experiment should assess one or more design specifications. Wherever possible, conduct a statistical analysis of the data. Compare results with design specifications and analytical predictions. Attempt to understand the cause of discrepancies.

•

Outline of Design Process STEP 1. Define the mission of the vehicle 1. Obtain customer information STEP 2. Develop vehicle design specifications 1. Information Sources 2. Design Parameters a. Design Control Variables b. Design Envelope Variables c. Performance metrics 3. Determine target values 4. CHECK: Before proceeding, go back to the mission statement for the vehicle and verify that all target values are consistent with it. If not, return to Step 2 and revise the values.

38


STEP 3. Develop vehicle concepts 1. Determine vehicle configuration a. CHECK: Before proceeding, check that the configuration is compatible with the mission statement. 2. Determine vehicle gross geometry 3. Design the drive train and establish chain line 4. Establish initial frame geometry 5. Special Features STEP 4. Select optimal concept 1. Use rational decision-making techniques 2. Ensure each concept is compatible with mission and meets minimum design specification requirements STEP 5. Complete parametric design 1. Validate vehicle geometry Preliminary performance analyses should be used to validate the vehicle gross geometry. (frame dimensions, hull shape) a. Use handling and performance analyses b. Verify top-level performance objectives are met, iterate as required c. If performance objectives cannot be met with configuration, go back to Step 3 d. CHECK: Performance and handling must be consistent with mission 2. Complete Structural and other analyses and optimization a. Frame or shell type and structure (e.g., diamond frame vs. monotube) b. Member sizes/shapes c. Materials and conditions d. Processes 3. Determine testing needed to complete an optimal design STEP 6. Complete detailed design 1. Complete detailed drawings of frame and all special-purpose parts 2. Complete manufacturing plans 3. Select components STEP 7. Fabricate prototype STEP 8. Complete functional and performance testing 1. Compare results with design specifications and analytical predictions a. Attempt to understand the cause of discrepancies

39


2. Test the following categories: a. Validate drive train i. Chain management and interference ii. Gearing requirements iii. Impedance match b. Determine load cases and loads

c.

d. e.

f.

g.

i. CPSC Fork and frame test ii. Maximum Acceleration iii. Road/water obstructions iv. Hill climb v. Maximum braking—front vi. Maximum braking—rear Validate frame/hull structure i. Structural analysis (FEM, etc.) ii. Optimization iii. Modify and iterate as required Review component compatibility i. Modify and iterate as required Complete details i. CHECK: Interference, chain management ii. Features—brake bosses, braze-ons, seats/saddles, etc. iii. Cable/control routing and hardware Complete drawings i. Detailed SP part drawings ii. Assembly drawings iii. BOM Prototype fabrication and testing i. Fabrication and Assembly ii. Functionality testing iii. Performance testing iv. Market testing

h. Modify and iterate or advance to production planning

40

CHAPTER

PHYSIOLOGY OF HUMAN POWER

4

GENERATION

T

he human is the power plant for all human-powered vehicles. Although the human engine is often compared to an internal combustion engine, its operation and performance characteristics are quite different. Both engines convert the chemical energy of fuel into mechanical energy, and both produce about four times as much waste heat as useful power. The similarity ends here, however. The internal combustion engine is a heat engine, operating on a thermodynamic cycle and subject to the second law of thermodynamics. This cycle is limited by the difference in temperatures between the hottest and coolest parts of the cycle. In contrast, the human engine, while producing heat as a by-product, is not a heat engine. The energy conversion is completely chemical, and the human is not subject to the second law limitations. Performance characteristics are quite different as well. The internal combustion engine can produce no zero-speed torque. As the engine speed increases, the torque increases to a maximum, then decreases at high RPM. In contrast, a human achieves maximum torque or force when the speed is zero. Torque decreases with speed to a value of essentially zero at high speed. In this respect, the human is much more like a permanent magnet DC motor than an internal combustion heat engine. Carbohydrates, fats, and proteins are fuels for the human power plant. Carbohydrates are converted to energy easier and more quickly than fats, and are the principle fuels for the athlete. At low power levels where rapid metabolism of fuel to energy is not required, fats can become an important fuel source as well. Proteins are used more for body repair than for fuel, but can be metabolized and provide some energy. Proper nutrition includes more than fuel sources, however. Vitamins and minerals are essential for good nutrition, but do not contribute any energy.

41


The human power plant produces waste energy in the form of heat. The chemical reactions that produce muscular energy are exothermic, producing extra heat. For every Watt of useful power produced, 3 to 4 W of waste heat must be dissipated. This heat is carried away from the muscles by the blood, and is dissipated by sweat and increased blood flow to the skin. A simple definition of efficiency is the ratio of work done to total energy generated. The efficiency of the human power plant varies somewhat between individuals, but is approximately 25%. η=

Work Output Total Generated Energy

≈

0.25

(4-1)

It is perhaps worth noting that a heat engine operating at 25% efficiency with an ambient temperature of 15°C would require a hot temperature of at least 111°C. The human body could not survive this temperature. This calculation, based on Carnot efficiency, clearly illustrates that the human body is not a heat engine in a thermodynamic sense.

Muscle Structure and Function Muscle Structure Humans and other animals use skeletal muscles, tendons, and bones to generate forces and to absorb energy imparted to the body. Muscles and tendons are the actuators that move skeletal joints. Neurons trigger the contraction of muscle fibers, which produces forces within the muscle. For skeletal muscles this force acts across a joint, which can result in motion. The structure and function of the muscle-tendon unit are quite complex, and the following description is simplified. A more detailed overview of muscle-tendon structure and function can be found in Hawkins.1 Muscles are complex bundles of individual cells or fibers. Most muscle cells are quite long—several centimeters in some cases—requiring multiple nuclei. The individual fibers are grouped into small units called fasciculi, which in turn are grouped into larger units that make up the muscle. The entire muscle is encased in a sheath called the epimysium. Tendons connect the muscle to bone. Blood vessels and capillary networks in the muscle provide oxygen, which is required for oxidative metabolism. Neurons provide control inputs to the muscle fibers.

1

Hawkins, David, Biomechanical Systems: Techniques & Applications, Vol. III, Musculoskeletal Models & Techniques, Cornelius Leondes, Ed., CRC Press, 2001.

42

Physiology of Human Power Generation

Skeletal muscles—the only type of interest here—are under control of the somatic nervous system, that is, they are thus under voluntary control. Each individual neuron controls many muscle fibers, usually distributed throughout the muscle. The neuron and all its associated muscle cells are called a motor unit. An electrical signal is sent to the neuron either from the brain or from reflex pathways. The signal is transmitted to the muscle fiber’s motor end point by chemical means. If the signal is strong enough, the muscle fibers contract. Contractions of mammalian skeletal muscles are not modulated—they are either fully contracted or relaxed. After firing, the motor unit must wait for a period of time known as the refractory period before it can fire again. To exert a greater force, either more motor units are activated or the firing frequency is increased. In the former case, a greater fraction of fibers in the muscle are contracting at any given time. For the latter case, each motor unit is repeatedly activated. Once the firing frequency exceeds the inverse of the refractory period, the motor unit will be continuously activated. Several different types of muscle fibers exist, often within a given muscle. Slow oxidative (SO) muscle fibers are energized by aerobic metabolism and enervated by small, low threshold, slow conducting neurons. Fast glycolytic (FG) fibers are energized by anaerobic metabolism and enervated by large, high threshold, fast conducting neurons. The slow muscle fibers are generally recruited first due to the low activation threshold. As the name implies, SO fibers respond more slowly to neural stimulation, but can contract repeatedly without fatigue. They are smaller and weaker than glycolytic fibers. FG fibers respond more quickly and are stronger, but fatigue much more rapidly than SO fibers. Exercise of moderate intensity of extended duration predominantly relies on SO fibers, while short bursts of high intensity effort use FG fibers. In humans, the two types of fibers exist together within a given muscle in an approximately 50/50 ratio. Individuals with a slightly higher fraction of SO fibers tend to excel at endurance sporting events, while those with a higher fraction of FG fibers excel at sprint events. Training can improve performance in either type of event, but cannot change the ratio of muscle fibers. SO fibers are also called slow twitch or red fibers, and are darker in color that FG fibers. The characteristic red color is due to the rich supply of oxygen-storing myoglobin in the slow fibers. Fast gycolytic fibers are light in color and are also known as fast-twitch fibers. Some animals, such as the domestic chicken, have muscles comprised predominantly of only one type. The breast muscles of a chicken are white FG fibers. Chickens only fly for short bursts which require high effort. The leg muscles are dark SO fibers, as the chicken spends most of its time walking on the ground.

43


Muscular Energy Supply Adenosine Triphosphate, or ATP, is the energy source for all muscle functions. (In addition to force generation, ATP provides energy for cellular repair, synthesis of proteins, and ion transport.) It is produced by mitochondria within the cell cytoplasm. Also contained in the cytoplasm are glycogen, lipids, and enzymes. Glycogen is a polymer form of glucose. It can be rapidly converted into ATP to provide energy for the muscle. Partial conversion occurs without oxygen, or anaerobically. Complete conversion is aerobic, requiring oxygen supplied by the bloodstream. This process is slower, but much more efficient. Lipids (fat) can also be converted to ATP, but only in the presence of oxygen and at a rather slow rate. ATP is provided to the muscle through several mechanisms. A limited amount of ATP is stored within the muscle itself. This can provide energy for only about eight muscle twitches, or two to five seconds of maximal exercise. This is the energy source used for very high power levels of short duration, such as cyclist in a sudden attack or a weightlifter hoisting a heavy load. There are no harmful byproducts produced by this mechanism, but the ATP must be replaced. At the onset of intense exercise, the most common mechanism for replenishing ATP in the muscle is through phosphocreatine (PCr). PCr combines with a byproduct of ATP to create additional ATP and creatine. Muscles store enough PCr for several hundred muscle twitches, or about 10 seconds of maximal effort. While this is longer than ATP stores, it is still far too brief for sustained effort. The process is anaerobic as it does not require oxygen, and it also produces no detrimental byproducts. The muscular ATP and PCr mechanisms are rapid and anaerobic. They are both used for very brief, very intense bouts of exercise, and predominantly recruit FG muscle fibers. Additional mechanisms for producing ATP must be used for exercise lasting more than a few seconds. Which mechanism is used by the body depends on the availability of sufficient oxygen in the muscle cell, which is supplied by the bloodstream. Anaerobic glycolysis can provide ATP to the muscle through partial breakdown of glucose, or its polymerized form glycogen. Glycogen is stored within the muscle and does not circulate around the body. Glucose is transported by the bloodstream to muscles, organs, and the brain. Anaerobic glycolysis only partially converts glycogen and produces two ATP molecules for each glucose molecule. This represents only about 7% of the energy available in the glucose molecule. Byproducts of this reaction include pyruvic acid and Hydrogen. If sufficient oxygen is present, aerobic phosphorylation occurs, completing the glucose breakdown and producing 34 ATP molecules for each glucose molecule. In this case, only benign byproducts—CO2 and H2O—are produced.

44


However, if there is insufficient oxygen, the pyruvic acid combines with the hydrogen to form lactic acid. Lactic acid can be transported out of the muscle by the bloodstream, where it is converted back into glycogen and stored. However, this process is slow. If exercise intensity is great enough, the production of lactic acid exceeds the body’s ability to remove it, and it begins to accumulate. When lactic acid accumulates in the muscle, fatigue and pain set in, restricting or stopping exercise. Anaerobic glycolysis provides energy for only a few minutes of high intensity exercise, after which lactic acid buildup induces fatigue. In contrast, if exercise intensity is slightly lower, metabolism is primarily aerobic, and the rate of lactic acid production is less than the rate at which the bloodstream can remove it from the muscle. In this case, exercise can continue until the muscle glycogen is depleted, usually one to two hours of hard aerobic effort. If carbohydrates are consumed during exercise, the glycogen stores can be replenished, and exercise can last much longer.

Process

Description

Aerobic

FUEL + O2 ® ATP + Phosphorylation H2O + CO2

Anaerobic Glycolysis (Glycogenolysis)

CHO ® ATP + LACTIC ACID

NetATP 34

Comments CO2 carried away by blood and exhaled H2O absorbed by muscle NO Fatiguing by-products

2

Lactic acid builds up in muscle Lactic acid can prevent muscle contraction With time and O2 lactic acid converts to glucose

Fats, or lipids, can also be metabolized as fuel for muscles. Lipids are stored throughout the body, and can provide enough fuel for days of lower intensity exercise. However, the metabolism of fats is very slow, and requires more oxygen than glycogen. Higher intensity exercise suppresses the metabolism of fats. However, training at moderate intensity over time will improve the body’s ability to metabolize fat, and helps to make it a viable fuel for low to moderate intensity exercise. Fats are a significant source of energy for exercise lasting more than about 3 hours.

45


Summary of Energy Supply Mechanisms

Duration MuscleATP

Anaerobic

3seconds

Fatigue By-products None

PCr

Anaerobic

10 seconds

Anaerobic Glycolytic

Anaerobic

3 minutes

Lactic Acid

Aerobic Phosphorylation

Aerobic

2 hours

None

Lipids

Aerobic

Days(lowintensity)

None

None

Muscle fibers require energy to contract, whether or not any external work is done. Isometric exercise does no work in the thermodynamic sense, but does require energy and creates heat as a byproduct. When a muscle fiber elongates while activated, work is done on the muscle. This also requires energy. For example, consider slowly lowering a heavy weight. Although the weight does work on the body during the process, the muscles are still expending energy. Endurance athletes should strive to minimize isometric exercise (squeezing bicycle handlebars, for example) to maximize performance. This will help to minimize fatigue and maximize the external work done by the body. Power-Duration Relationship During exercise, muscles fatigue over time. Fatigue is caused by either a buildup of lactic acid or by depletion of fuel faster than it can be supplied. The onset of fatigue depends on both the exercise intensity and the duration. Very intense, anaerobic exercise can result in very rapid fatigue, whereas low to moderate intensity exercise can continue for many hours. For intense exercise, the onset of fatigue occurs when lactic acid builds up within the muscle. During intense anaerobic exercise, the production rate of lactic acid exceeds the rate at which the bloodstream can remove it from the muscle. The lactic acid interferes with muscle chemistry, making it more difficult for muscle fibers to contract. It also creates a painful burning sensation within muscle. Fatigue sets in quickly. When the exercise intensity is somewhat lower, lactic acid is removed from the muscle fast enough to prevent buildup. Fatigue then occurs when energy demands exceed the supply. When muscle glycogen is depleted, the musc ular “fuel tank” is empty, and fatigue sets in. Consumption of carbohydrates during exercise can maintain blood glucose levels and postpone fatigue. However, once glycogen stores are depleted and blood glucose levels drop, fatigue significantly slows or stops exercise. This is described by athletes as “bonking” or “hitting the wall.”

46


POWER-DURATION DATA FOR LEG CRANKS

1200

MAXIMUM MIURA 2009 NASA HEALTHY MEN

1000

NASA ATHLETES

) 800 s tt a (W r 600 e w o P 400 200 0

-1

10

0

10

1

10

2

10

3

10

TIME (minutes)

Figure 4-1 Power output with leg cranks vs exercise duration1

Figure 4-1 illustrates the relationship between exercise time and intensity. For exercise up to 30 minutes or so, a hyperbolic relationship exists between cycling intensity and time to fatigue. The solid curves on the plot use this model, which can be expressed as

= P W

1 ′ ×  CP + t 

(4-2)

Where CP is the critical power and W¢ is the curvature constant. This model is not accurate for very intense but brief bouts of exercise, or for exercise lasting more than an hour. However, it is quite appropriate for rides of moderate duration. The figure includes a curve loosely based on a conjectured maximum power-duration relationship for extraordinary athletes with optimal equipment.2 Miura et al.3 presented results from testing healthy subjects (male and female) with and without prior work. While prior work did provide a slight increase in the power for a given duration, the data shown is for no prior work. Also shown are data from a NASA study on healthy men and champion athletes. The healthy men

1

Data for curves Max Power curves taken from Wilson (Reference 3). Abbot, Allen, and David Gordon Wilson, Human-Powered Vehicles, Human Kinetics, 1995. 3 Miura, Akira et al., “The Effect Of Prior Heavy Exercise On The Parameters Of The PowerDuration Curve For Cycle Ergometry” Appl. Physiol. Nutr. Metab. 34: 1001–1007, 2009. 2

47


were young, physically active, fit, and accustomed to doing work. Poole et al. 4 presented data for healthy young males that fell in the range between the NASA healthy men and Miura’s data, lending credence to the values shown in the figure. Note that at exercise duration over about 1 hour, the maximum power falls below the critical power or asymptotic value. Perhaps an exponential model would be more appropriate for extended duration cycling (beyond one hour). An average young, healthy man can be expected to produce between 350 and 400 Watts for one minute, or about 180 to 2800 W for 10 minutes. Women, inactive men, and older people will likely produce less power at any give duration. Gender, Height, and Age FactorsThe peak power achievable by an athlete is strongly dependent on active muscle mass. Larger cyclists have greater muscle mass, and can generate greater power. Maximum power also decreases with age, perhaps due to reduces muscle mass. The designer of a human-powered vehicle will generally have some idea of the height and possibly the gender of the rider, so an understanding of how these factors affect power is of interest. Wohlfart and Farazdaghi5 studied male and female subjects in order to revise reference values for clinical ramp tests on cycle ergometers. Subjects cycled on an ergometer, initially at low power. The power was increased every 20 seconds by 5 W until exhaustion, when the maximum power attained was recorded. Equation 4-3 presents their formulas for predicting the max power, in Watts, as a function of age and height for both men and women. In these formulas, height is given in meters and age in years.

WMAX ,MEN

=

WMAX ,WOMEN

=

244.6 × height −92.1 1 + e 0.038 ( age 77.3) 137.7 × height −23.1 ×

−

(4-3)

1 + e 0.064 ( age 75.9) ×

−

Data from Equation 4-3 is plotted in Figure 4-2, which shows the reduced max power of women as compared to men. This data can be used to estimate design power, which can be used for performance predictions and, to a limited extent, component sizing, and strength.

4

Poole, David C. et al., “Metabolic and Respiratory Profile of the Upper Limit for Prolonged Exercise in Man,” Ergonomics, 31(9), 1265–1279, 1988. 5 Wohlfart, Bjorn, and Gholam R. Farazdaghi, “Reference Values for the Physical Work Capacity on a Bicycle Ergometer for Men—A Comparison With A Prevous Study On Women,” Clinical Physiology and Functional Imaging 23(3), 166–170, 2003.

48


MAXIMAL CYCLING LOAD FOR MEN (W) 2

MAXIMAL CYCLING LOAD FOR WOMEN (W) 2

34

0

1.9 32

3

0

0

2

2

0 30 ) (m1.7 0 t h 28 g i e 0 H1.6 26

6

4

0 2

0

8 1

1.9 2

0

0 2

0

1

2

20

0 8 1

2

1

20 1

8

6

0

0

40 50 AGE (years)

) m ( 1.7 t h g i e H1.6

4 1

0

1

0

8

0

20

1.5

70

0

0 18

1.4

60

0 6

0

6 1

0 24

0

1.8

0 0 2

30

0 2

0 22

0 4 2

1.4 20

2

0 8

1.8

1.5

0 24

0

20

0 4 1

0 2 1

0 16 30

40

50

60

70

AGE (years)

Figure 4-2 Maximal cycling loads for men and women as functions of age and height

Nutrition Proper nutrition is essential for the well-performing human power plant. Nutrition includes fuel for the muscles, but also includes vitamins and minerals that help the body recover from strenuous exercise and repair damage. The athlete that disregards nutrition will at best perform below her potential, and may possibly injure herself or make herself sick. It is particularly important when physical activity must be repeated day after day, as is the case for many cycling endurance events like ultra-marathons and stage races. In this case, a full recovery must be made in time to ride the next morning. Good nutrition involves appropriate consumption of fluids, carbohydrates, fats, proteins, vitamins and minerals. Water is arguably the most important nutrient for the human body. Although a person can live quite a long time without food, without water death occurs in just a few days. When a person loses water amounting to just two percent of his body weight, his body will not be able to control core temperatures adequately. Perfor6

mance is degraded by up to 7%. As water loss continues, endurance is reduced, strength is reduced, and—in severe cases—cramps, heat exhaustion, heatstroke, and death may possibly result. Water loss from the body occurs through breathing, sweating, and urination. An average, non-athletic person may lose up to 12 cups of fluid each day. An athlete can lose much more, particularly in hot weather. The importance of drinking 6

Burke, Edmund R., Serious Cycling, 2nd Ed. Human Kinetics, 2002.

49


water has been emphasized in the popular press in recent years, and most athletes know that they should drink regularly. What is not widely recognized is the relationship of sodium levels and fluid intake. Proper fluid intake is that which will result in no weight change over the course of extended exercise. This amount depends on the individual and on ambient conditions, so general guidelines may not be adequate.7 Sodium intake should be proportional to fluid intake. Sweat and urine (during exercise) contain approximately equal fractions of sodium, about 50 milliequivalents per liter. This is less than that supplied by most sports drinks, so it is possible to remain properly hydrated and still be hyponatremic (sodium levels too low). Extreme forms of hyponatremia can lead to serious consequences, including death. This was illustrated dramatically by the death of a participant in the 2002 Boston Marathon.8 Salt tablets or salty snacks such as pretzels may be needed during extended exercise in hot weather. Diet must provide adequate fuel for extended exercise of moderate or higher intensity. The three fuels used by the body are carbohydrates, fats, and protein. Carbohydrates are metabolized rapidly, and are the preferential fuel for high exertion levels. Fats can provide a tremendous amount of fuel, but only very slowly. They are an important fuel source for extended low to moderate exercise. Proteins are used in prolonged moderate to high intensity exercise. This is generally not good, as proteins are needed for other functions within the muscle. Of the three fuels, fats provide the most calories per gram. Fat contains about 9 kcal per gram, whereas carbohydrates and protein each provide only 4 kcal per gram. (The Calorie, spelled with a capital C on food labels, is actually a kilocalorie of energy. One Joule is equivalent to 4.187 kcal or Calories.)

Fuel

7

Energy Content (Calories per Gram)

Fat

9

Carbohydrate

4

Protein

4

Weschler, Lulu, “Water and Salt Intake During Exercise,” online, accessed August 19, 2005, http://www.ultracycling.com/nutrition/hyponatremia2.html. 8 Portsmouth Herald Portsmouth Herald Mass News: “Medical report says runner’s death linked to excess fluid intake,” Portsmouth, NH Tuesday, August 13, 2002.

50


A typical athlete working at 70% of his aerobic capacity will require about 60 to 80 grams of carbohydrate per hour of exercise. Initially his body will use mainly muscle glycogen and fat in roughly equal fractions, with blood glucose supplying about one seventh of the total. After 2 to 3 hours, the glycogen stores are depleted. Blood glucose and fat become the principle fuel sources, with muscle protein making up a significant minority fraction—again about one seventh 9

of the total. At this point, consumption of carbohydrates is essential to prevent depletion of blood glucose. Glucose is the only energy source for the brain, so low glucose levels can lead to lightheadedness and headaches. Protein should also be consumed to replenish protein lost as fuel. During extended exercise, it is essential to consume sufficient fluids, electrolytes, carbohydrates, and protein. Some fat may also be consumed, particularly for lower intensity activities. However, fat can interfere with and slow down the process of replenishing muscle glycogen. During and after high-intensity exercise, fat should be avoided. The active athlete must be attentive to her entire diet, not only during exercise, but throughout the day. The total daily caloric requirements vary from person to person, and depend on exercise levels and duration. A typical thirty year old sedentary male requires 2400 Calories per day, while a 30-year-old sedentary female requires 2000 Calories per day. An active 30-year old of either sex would require an additional 400 to 600 Calories per day. 10 In contrast, endurance athletes in extended events such as the Tour de France may require caloric intake over 6000 Calories per day, and under extreme conditions 9000 Calories per day. It is very difficult to maintain an energy balance under these conditions. Fortunately, athletes can recover on rest days. An overall diet for the athlete should be comprised of approximately 60% carbohydrate, 25% fat and 15% protein. Carbohydrates include both sugars (simple carbohydrates) and starches, or complex carbohydrates. The glycemic index rates carbohydrate foods based on how quickly and effectively the body can process them. Foods with high glycemic indices will cause a rapid rise in blood glucose. This helps speed the process of replenishing muscle glycogen stores, which is very helpful during recovery after strenuous exercise.

9

Burke, Edmund R. and Ed Pavelka, The Complete Book of Long-Distance Cycling, Rodale, 2000. 10 USDA, Dietary Requirements for Americans, “Chapter 2: Adequate Nutrients within Calorie Needs,” accessed online August 22, 2005, http://www.health.gov/dietaryguidelines/ dga2005/document/html/chapter2.htm. Wilmore, Jack H., and David L. Costill, 2004, Physiology of Sport and Exercise, 3rd Ed., Human Kinetics.

51


Fat is also an important nutrient. Fats not only provide energy, they also aid the absorption of vitamins A, D, E, and K and help transport nutrients in and out of cells. The fat burned as fuel is not sub-cutaneous fat, but rather it is located within the muscles. Endurance athletes “train” their bodies to use fat more efficiently by extended low to moderate intensity exercise. This helps the body to use a higher fraction of fat as opposed to muscle glycogen, thus extending endurance. Athletes requiring 5000 to 6000 Calories per day find it difficult to maintain an energy balance with less than about 25% fat in their diet. Intake should be balanced between saturated, monounsaturated and polyunsaturated fats. Because most diets contain significant calories from fat, athletes do not need to deliberately consume fatty foods. They should be consumed in moderation in order to provide about 25% of the daily Caloric intake. Vitamins and minerals provide no energy for exercise, but are a vital component of an athlete’s diet. Vitamins can act as enzymes to speed metabolic processes within the body. Perhaps more importantly for the athlete, some vitamins, such as C, E, and beta carotene, act as antioxidants. Free radicals—toxic compounds that can damage cell structure and function—occur naturally within the body. Intense exercise stimulates the formation free radicals, potentially exacerbating the damage they cause. Antioxidants help prevent the damage. The minerals zinc and selenium also act as antioxidants. Other minerals are equally important in forming the structural components of bones and teeth. Protein is used to create enzymes and other compounds that are vital for body functions. They can also provide some energy for the athlete. About 15% of the athlete’s total daily caloric intake should be protein. (The FDA daily recommendation is 0.8 grams per kilogram of body weight. The active athlete needs more than this, about 1.5 grams per kilogram, which is about 15% of the daily caloric requirements described previously.) Proteins are made up of amino acids. Amino acids are either essential, meaning that they must be provided by diet, or nonessential. The body can produce nonessential amino acids. Complete proteins contain all of the amino acids, while incomplete proteins contain only some of them. Grains, vegetables, and nuts contain only incomplete proteins, while meat, fish, eggs, and dairy products contain complete proteins. An athlete should be sure to consume a well-rounded diet that includes foods from the latter group.

Body Systems during Exercise Exercise places increased demands on the body, and body systems respond, sometimes dramatic ally, to meet those needs. Short term respo nses to a particular bout of exercise are known as acute responses. Acute response includes

52


changes in heart rate, cardiac output, blood flow, oxygen consumption, and other changes. The body also responds to long-term exercise, providing the basis for training regimens. Acute response to a particular bout of exercise varies from on individual to the next, and also depends on environmental factors, making specific predictions difficult. Some factors that affect acute response to exercise include temperature, humidity, ambient light level, noise levels, time and size of last meal, sleep quantity and quality, and even time of day. Even so, the acute response of the human body to exercise is general well understood. During exercise, all body systems work in concert to ensure that physiological needs are effectively met. The active muscles require more fuel and oxygen, and produce more waste products which must be removed. Additional heat is produced in the body that must be dissipated. The cardiovascular system must work harder to provide more blood and oxygen to the active muscles. The lungs must take in additional oxygen. A whole series of complex and interrelated changes occur through out the body to ensure it can perform at its maximum level. During exercise, the active muscles require increased blood flow both to supply needed substances such as oxygen and glucose and to remove harmful products such as lactic acid. At rest, an average adult heart will pump about 5 L of blood per minute. During intense exercise, output increases to 20 to 40 L per minute. The extra blood flow comes in part from increased cardiac output, and partially by re-distributing the distribution of blood away from organs to the active muscles. When a person begins intense exercise, the heart starts beating faster. (In some cases, the heart rate increases even before exercise begins—an anticipatory response.) At rest, heart rate for healthy adults is in the range of 60 to 80 beats per minute. Heart rate increases linearly with exercise intensity up to some maximum level. The maximum heart rate is the highest heart rate achievable by an individual during all-out exercise. It varies from person to person, but is correlated with age. Equation 4-4 provides an approximate maximum heart ra te as a fun ction of age. This formula is an average value, and individuals can deviate from the predicted value.

HRMAX = 208 – (0.7 ´ AGE)

(4-4)

This formula is somewhat more accurate than subtracting age from 220, which is a common alternative formula for maximum heart rated. During a workout at a constant sub-maximal load, heart rate will increase and then level off at a plateau known as the steady-state heart rate. Usually, the steady-state heart rate is achieved within one to two minutes of exercise or change in exercise intensity.

53


For prolonged exercise, particularly under heat stress, the heart rate may drift upward over time. The increased heart rate accounts for some of the increased cardiac output, or blood flow rate out of the heart. The remainder is due to increased stroke volume. Stroke volume is the volume of blood ejected from the left ventricle with each heartbeat. Up to about 50% of maximal effort, stroke volume increases linearly. Thereafter, it appears to reach a plateau. The increased cardiac output is due to both increased heart rate and stroke volume from the onset of exercise up to about 50% of maximal effort, and to increased heart rate alone above 50% effort. Blood pressure increases with exercise intensity. The contraction phase of the heart chambers, in which blood is expelled from the heart, is known as systole. Diastole is the resting phase, in which the heart muscles relax. Blood pressure measurements consist of two numbers, the systolic and diastolic pressures. Systolic blood pressure is the maximum pressure in the arteries, and corresponds to systole of the heart. It is the higher number reported for blood pressure. Diastolic blood pressure is the lower number, corresponding to diastole of the heart. During exercise, the systolic pressure increases linearly with exercise intensity, but the diastolic blood pressure remains essentially constant. This helps drive the blood through the vascular system to the muscles that need it. At maximum exertion, systolic pressure nearly doubles over resting levels, from 120 mmHg to over 200 mmHg. Higher values up to 250 mmHg have been recorded for some athletes. Blood flow patterns also change significantly with exercise. At rest, 15 to 20% of the blood flows to the muscles, with the remainder going to organs such as the liver, kidneys, stomach, and brain. During intense exercise, up to 85% of the blood goes to the active muscles. This can amount to a 25-fold increase. Additional flow is also sent to the skin for cooling. If a large meal has been eaten prior to exercise, the digestive system and the active muscles are in competition for blood. Some blood will be diverted from the muscles to the gastrointestinal system, which can adversely affect performance. Most of the increased muscular flow comes from increased cardiac output, but at high levels of exercise blood flow is actually reduced to the organs. Two mechanisms are responsible for changes in blood flow, auto regulation and extrinsic neural control. Auto regulation occurs at a local level, without intervention of the nervous system. Increased demands for oxygen stimulate the arterioles, or small arteries, to dilate, permitting increased blood flow and hence more oxygen to be delivered to the active muscles. Extrinsic neural control redistributes blood throughout the whole body, and is controlled by the sympathetic nervous system. Vessels leading to organs are constricted, reducing the blood flow. These mechanisms effectively increase blood flow to the muscles with high demand, and reduce flow to organs, maximizing performance.

54


Delivery of oxygen to active muscles is one of the primary roles of the cardiovascular system. At rest, arterial blood flowing from the heart contains about 20 ml of oxygen for every 100 ml of blood, while the veins contain about 14 ml per 100 ml blood. The difference between these values—about 6 ml 2 O /100 ml blood—is the arterial-venous difference, or the amount of oxygen actually consumed by the body. During exercise, the arterial oxygen content changes little, but the venous oxygen decreases. As the active muscles demand more oxygen from the blood, the venous oxygen content approaches zero in veins serving the active muscles. Since blood returning from organs and inactive muscles will have higher oxygen content, blood in returning to the right atrium of the heart rarely drops as low as 4 ml oxygen per 100 ml blood. Other changes in blood also occur, including decreased plasma volume fraction, increased viscosity, and decreased pH. Ventilation, or the volume of air inhaled per unit time, increases significantly with exercise. Ventilation increases from around 5 L per minute at rest 100 to L per minute or more during intense exercise. Larger individuals may approach 200 L per minute. Initially, the increased ventilation comes from greater tidal volume. Breathing is deeper, rather than faster, resulting in a greater volume of air inhaled with each breath. As exercise intensity increases, the breathing rate begins to increase. After termination of exercise, the pulmonary ventilation does not immediately return to the resting rate. It takes several minutes to recover resting levels. The reason for the increased ventilation is, of course, to increase the oxygen delivered to the active muscles. In low-intensity steady-state exercise, ventilation and oxygen consumption are proportional to exercise intensity. Oxygen consumption is designated V O2, while ventilation and carbon dioxide production are designated VE and V CO2, respectively. As exercise intensity increases above about 50% of maximum effort, oxygen consumption continues to increase linearly, while ventilation increases at a faster rate. This is due to the accumulation of carbon dioxide in the body, brought on by the onset of anaerobic metabolism. The increased ventilation is the body’s method of removing excessive CO 2. The point at which the anaerobic metabolism begins is known as the anaerobic threshold. Oxygen consumption continues to increase above the anaerobic threshold to a maximum value, VO2 max, at the limit of performance. Since oxygen consumption increases in proportion to exercise intensity, it can provide good estimates of energy expenditure during exercise. (The most precise measurements of metabolic energy expenditure require the subject to be placed in what is essentially a large calorimeter. The energy balance is precisely measured to determine metabolic energy. This is of course an expensive and often impractical method. Hence alternatives such as oxygen consumption are very attractive.) The slope of the energy/V O2 curve depends on the type of food being

55


METABOLIC ENERGY AS FUNCTION OF OXYGEN CONSUMPTION 30 RER = 0.71, A = 4.69, 100% FATS RER = 0.85, A = 4.87, 50/50 FATS/CARBS RER = 1.00, A = 5.05, 100% CARBS

25

) 20 n i m /l a c k (

15

Y G R E N E 10

E = A*VO2 5

0

1

1.5

2

2.5 3 3.5 4 OXYGEN CONSUMPTION (liters/min)

4.5

5

Figure 4-3 Energy as function of oxygen consumption

oxidized. More oxygen is required to metabolize fats than carbohydrates. If only fats are used as fuel, the slope of the energy/ V O2 curve is slightly flatter that if only carbohydrates are metabolized. Instruments are available to measure ventilation, oxygen consumption, and carbon dioxide production either in the laboratory or in the field. The respiratory exchange ratio is the ratio of carbon dioxide produced to oxygen consumed, or

RER

VCO2 =

(4-5)

O V 2

The value of the respiratory exchange ratio depends on the mix of fuels used for energy, ranging from about 0.71 for fuel provided solely by fats to 1.0 for carbohydrates alone. A 50/50 mix of carbohydrates and fats produces a RER of about 0.85. Thus, the RER can be used to obtain the slope of the energy/ V O2. The relationship is

A = 1.2414 +× 3.8086 RER

56

(4-6)


The slope A in Equation 4-6 has units of kcal/liter O2. The metabolic energy production rate during a bout of exercise can then be determined by E

Where

E

=

A × VO2

(4-7)

is given in kilocalories per minute and V O2 in liters of oxygen per

minute. Due to the relatively low efficiency of the human body, most of the energy produced must be dissipated as heat. Only about 25% or so produces useful work output.

Maximal Oxygen Consumption At rest, an average person consumes oxygen at a rate of about 0.3 L per minute. When walking at a brisk 1.5 m/s, oxygen consumption increases to about 1.8 L per minute for the average person. A competitive cyclist riding a racing bicycle at 12 m/s may consume 4.8 L per minute and approach his maximum limit for oxygen consumption. The maximum rate of oxygen consumption for an individual is known as his or her V O2 max, generally considered to be the best measure of aerobic fitness and endurance. Because body size affects oxygen consumption, maximum oxygen uptake is often reported per unit body mass. Typical units for V O2 max are milliliters oxygen per kilogram of body mass per minute. Elite cyclists have V O2 max values in the range of 70 to 80 ml/kg/min, while lesser, but trained, cyclists have values in the 50 to 60 ml/kg/min range. Maximal oxygen uptake does depend on the muscles used—larger active muscle mass leads to greater oxygen consumption. The highest recorded V O2 max of 94 ml/kg/min was obtained by a Nordic skier—a sport that utilizes many muscles and large muscle mass. 11 Women generally have lower V O2 max values than men, probably due to body composition differences, blood hemoglobin content, and social factors. Maximal oxygen uptake cannot be improved to a significant extent by training. People that are young and sedentary may realize some increase due to training, but trained athletes can only see modest if any increase. V O2 max changes with age. For active people, it may remain constant from adulthood up to about the age of 40, and then decreases to about half that value by age 80. Sedentary lifestyles result in a decline starting much earlier in life, perhaps by age 20.

11

Ibid, p. 142.

57


Anaerobic Threshold Maximal oxygen uptake is not highly correlated with cycling performance. In road races, strategy, teamwork, and riding skills are much more important than a high V O2 max. In time trials, the correlation improves, but is still weak, because it is difficult or impossible for a cyclist to maintain a power output corresponding to V O2 max long enough for most races. In contrast, the fraction of V O2 max that can be sustained during extended exercise is an excellent indicator of steadystate endurance performance. The best cyclists can pedal for extended periods at high fractions of their V O2 max. The anaerobic threshold is a better indicator of extended endurance performance than maximal oxygen uptake. The exercise intensity at which lactate is generated faster than it can be removed is known as the Onset of Blood Lactate Accumulation, or OBLA. The OBLA is a good predictor of time trial performance for cyclists. As noted, V O2 max is relatively insensitive to training. The same is not true for OBLA. Endurance performance can certainly be improved with training, and the OBLA can be improved. The OBLA is very similar to the anaerobic threshold (AT), although their strict definitions differ. At the anaerobic threshold, any increased effort will be anaerobic, and oxygen demands exceed what the body can provide. The OBLA can be determined by taking blood samples during exercise, which is useful in the laboratory, but not particularly desirable as part of a regular training regimen. Fortunately, there is a very good correlation with heart rate, which is easy to measure during exercise. Today, most training programs are based on heart rate measurements.

Appendix Calculating the CO 2 Production Rate as a Function of External Work Abbot and Wilson12 use the following formula to predict oxygen consumption during cycling with leg cranks: V O2

= 12.2

P

+3.5

where:  O is in ml/min V 2 P is mechanical output power in Watts W is body mass in kilograms 12

Ibid, p. 20.

58

W

(4-8)


The respiratory exchange ratio is given above as

RER

VCO2 =

(4-9)

V O2

Then the rate of CO2 generation should be

 CO V RE2RV =

 O2 *RER

*(12.2 P

=

W3+.5

)

(4-10)

VCO2 is in ml/min. To convert to kg/min, use the ideal gas law pV = nRT, with n

m =

MW

and R = 8314.3 J/kmol-K, or more conveniently, R = 82.06 atm-ml/mol-K.

Then,

m

MW *p V *

=

(4-11)

R *T

Where p is pressure in atm, V is volume in ml, and T is temperature in Kelvins. The molecular weight of CO2 is 44 g/mol or .044 kg/mol. At STP ( p = 1 atm, T = 298.18 K) Equation 4-11 becomes

m

.044 *1   = Ve =V1.798  82.06 * 298.18 

−6

(4-12)

where m is given in kilograms. Substituting into Equation 4-10 and multiplying by 60 m/hour gives the kilograms of CO 2 produced in one hour of exercise at a power level P:

 CO2 m

=

e RQ 1.079

4

−

P W *(12.2

3.5

+

)

(4-13)

 CO2 has units of kg CO 2/hour. Where m

59

CHAPTER

THE HUMANMACHINE

5

INTERFACE

S

uccessful human-powered vehicles are comfortable, easy to control, and have efficient drive systems. A well-designed vehicle interfaces with the rider in a natural, intuitive, and comfortable way. The rider is considered during development of all vehicle systems, so that riding is a pleasure rather than an uncomfortable chore. Comfort, safety, and ease of operation are important attributes for any type of vehicle, regardless of power source. They become crucial with human-powered vehicles because performance depends on an effective interface between the human body and the vehicle. The interface includes all points of contact, controls, and all points of interaction between rider and vehicle. The seat or saddle, pedals, and handlebars are prime examples of contact points. Controls include shift and brake levers, light switches, bell, latches, and similar devices. The rider also interfaces visually with the vehicle when monitoring the cycle computer, GPS, or gear indicators.

Comfort Contact points must be designed to fit the rider’s body in a comfortable manner. Saddles, handlebars, and pedals are the most crucial contact points for upright bicycles. Anyone who has ridden an upright bicycle for an extended period of time understands the importance—and subjectivity—of a comfortable saddle. Often beginning riders look for soft, compliant saddles. This is usually a mistake, as the most comfortable saddles are generally firm. However, saddle selection is very personal, and there are many options on the market for even the most sensitive rider. Handlebars are also important for rider comfort, particularly when a significant portion of the rider’s weight is carried by the arms. The fit should allow

61


a slight bend in the elbow during riding. Multiple hand positions helps reduce fatigue of the hands and wrists. Drop-style road bars accommodate several hand positions—on the drops, on the brake hoods, or on the top of the bar. Flat or riser bars offer fewer options. Bar ends can usually be added to these bars to provide additional hand positions. Alternative bar styles, such as any of the trekking-style or mustache bars can provide additional options for hand positions. Handlebar grips or tape should be comfortably padded to absorb vibration. Egonomic grips are now available by several manufacturers, and can significantly improve comfort. Pedals should be positioned properly and should grip the shoe without restricting blood circulation to the foot. There are many styles of clipless pedals on the market that work very well in terms of both comfort and performance. The older style toe clips and straps provide less performance than clipless pedals, and can contribute to numbness for some people. Platform pedals are good for casual riding or around-town errands, although some models provide very little grip for the shoes. Recumbent vehicles are often considered more comfortable than upright bikes, but contact points are no less important. The seat is a prime example. Hard-shell seats offer good performance in terms of rigidity and power transfer to the pedals, but must fit the rider’s body. A firm foam pad is often used for additional comfort. Rigid seats do not breathe well, and may lead to some discomfort on long or hot rides. Rigid seats are also usually designed for a reclined position. Using them in a more vertical position is likely to be uncomfortable. Mesh seats breathe better and provide improved shock absorption, and may be quite comfortable. However they provide poorer power transmission due to increased compliance. Many seats have a mesh back and rigid padded bottom, providing a good compromise in comfort and performance. If the rider’s position is much reclined, a head rest is required for comfort. Holding one’s head up on a laid-back recumbent is very tiring after only a short ride. Hand numbness is not usually a problem on recumbents, but handlebar design is still important. Handlebars should be positioned for easy arm reach. On many vehicles, attention is required to ensure the handlebars do not interfere with the legs during pedaling. This can be a challenge, especially for vehicles with cranks positioned higher than the seat. Arms should be relaxed and elbows bent during operation. Pedals are more crucial on recumbents than for upright bicycles, particularly with high crank positions. If the shoe is not gripped well, it tends to slide down and off the pedal. This causes the rider to work harder to just hold his or her legs up, increasing fatigue and discomfort. Foot numbness can also be a greater problem on recumbents. Clipless pedals are arguably more important for comfort and performance on recumbents than for upright vehicles. An alternative is BMX-style pedals with pins that provide good grip even on street

62

The Human-Machine Interface

shoes. These pedals provide better grip than platform pedals (although not so good as clipless pedals), but can be quite painful when the pedal strikes a shin inadvertently. Well-designed, ergonomic controls can make a significant difference in terms of safety and ease-of-use. Brake and shift levers should be sized for the rider’s hand and positioned for easy reach. Short-throw levers are available for children or those with small hands. The reach is adjustable on many brake levers, particularly those for flat or riser handlebars. Shifting and braking should be easily actuated with the hands in a normal riding position, without moving the hands from the handlebar. Other controls—such as a bell, light switches, ventilation controls, or retractable landing gear, should be within easy reach and should operate easily. Consider mechanical advantage in the design of control levers. The force required should be appropriate for the task. Latches for canopy or doors should be easy to reach and simple to operate. Data systems, such as cycle computers and GPS units, should be placed where they are easy to read, but do not obstruct road views for the rider.

Mechanisms for Power Transfer to Human-Powered Vehicles Effective human-powered machines require that muscle force be combined with efficient motion and sufficient active muscle mass. The combination of force and motion produce power, but the mechanism must be designed for the physiological and kinematic constraints of the human body to be effective. The total power a particular person can generate for any given time depends on the active muscle mass. Arms can provide more power than hands alone, and legs can provide more power than arms. Nordic skiing and rowing are examples of sports that employ large muscle mass by using many muscles in both arms and legs for propulsive power. The moderate to high power levels required for vehicles demand sufficient muscle mass, so vehicular mechanisms usually use the larger leg muscles. In addition, the operation of most human-powered machines also requires additional tasks that must not be impeded by the power-producing mechanism. For example, a cyclist must steer, scan for traffic and road hazards, and signal in addition to pedaling the vehicle. For any mechanism, force, stroke, and cadence are the relevant mechanical parameters, while joint angles, joint forces, and cadence are physiological parameters. Unfortunately, even for a given mechanism there is not unique set of physiological parameters that correspond with a given set of mechanical parameters. This is because of variations in the way the human body can move to match the motion of a given machine. For example, there are several different pedaling

63


styles that can be used to power the same bicycle. Each style may employ different ankle and knee angles, even though the crank length and cadence remain the same. Frequently, cyclists will switch pedaling styles while riding, either consciously or unconsciously. This may reduce fatigue by using slightly different muscle groups or using muscles in different ways. This considerably complicates the question of optimal mechanisms. It may be that a particular mechanism is optimal for only one style. Circular cranks powered by leg muscles are by far the most common mechanism currently in use to transfer power from human to land or air vehicles. The large leg muscle mass, continuous motion, and efficient mechanical mechanisms involved make this a particularly effective means of transferring power from the human body to a machine. Circular leg cranks are used on most wheeled land vehicles.

Optimal Body Position for Leg Cranks The position and orientation of the rider’s body and the angles of the hip, knee, and ankle affect cycling performance. Assessing the effect is difficult, as there are complex interactions between the biomechanics of cycling, the rider’s physical condition, acclimatization to a particular cycling position, and environmental factors. The optimum orientation likely varies from one person to the next, and may change over time due to training and other variables. A significant amount of research has been conducted in an attempt to understand the factors involved and determine optimal cycling parameters. Joint angles and range of motion of joints during cycling are important variables. Figure 5-1 defines a few of the significant position and joint angles. The body configuration angle (BCA) is the angle between a line drawn from crank axis to the hip and from the hip to the mid-plane of the torso. The BCA plays a significant role in cycling performance. The rider shown in the figure could be rotated about the crank axis while maintaining a constant BCA. The orientation of the body relative to the crank is defined by the hip orientation angle (HOA), defined as the angle between the horizontal and a line running from the crank axis to the hip joint. If the crank is below the hip, as shown in the figure, the hip orientation angle is positive. Thus recumbent vehicles with cranks above the hips have negative hip orientation angles. Upright cyclists have large positive values for the HOA. The knee angle (KA) is the angle between a line drawn from the hip joint through the knee joint and a line from the knee to the ankle joint. This is also an important cycling parameter. The backrest angle (BA) is the angle between the back of the seat and the horizontal. It is similar to, but slightly different from,

64


Figure 5-1 Illustration of body joint angle definitions

the trunk angle, measured between the horizontal and a line running from hip to shoulder. The body configuration angle is significant for optimal cycling performance. Reiser et al.1 investigated the effects of body configuration angle on peak power and average power during a Wingate test (a short-duration anaerobic cycling test conducted on an ergometer). They found that in the recumbent position, an optimal BCA occurs between 130° and 140°. This is the same range used by upright cyclists on mountain bicycles, but somewhat greater than the BCA for upright road cyclists. Several other studies have shown comparable results. Interestingly, Too’s2 work indicated an optimal BCA of approximately 115°, which is closer to that used by road cyclists. Reiser speculates that the difference in results may be due to the different types of cycling the subjects were used to. Reiser’s subjects that rode upright bicycles use mountain bikes. Presumably, Too’s subjects used road bicycles with drop handlebars. (Too also tested subjects with the backrest angle fixed at 90°, a very unusual position. It is possible the unusual position could affect the results.) It may very well be that the optimum BCA depends on the training history of the rider. However, a BCA in the range of 130° to 140° appears to be near optimal for most riders. The BCA of the cyclist in Figure 5-1 is 135°. The hip orientation angle significantly affects aerodynamic drag on the vehicle. If the HOA is negative, the crank is above the hips of the rider. At a hip orientation 1

Reiser II, Raoul F., Michael L. Peterson, and Jeffrey P. Broker, 2001, “Anaerobic Cycling Power Output With Variations in Recumbent Body Configuration,”Journal of Biomechanics, 17, 204–216. 2 Too, Danny, 1991, “The Effect of Hip Position/Configuration on Anaerobic Power and Capacity in Cycling,” International Journal of Sport Biomechanics, 7, 359–370.

65


angle of negative 15° or so, the feet of the rider do not extend below the body during pedaling. This reduces the frontal area, and hence the aerodynamic drag. However, there is some evidence that negative hip orientation angles reduce the peak power output of the rider. An optimal compromise may exist between the aerodynamic advantages of a high crank and the power advantages of a lower crank. Power required to overcome drag increases with the cube of speed. Figure 5-2 shows the power required to ride a recumbent HPV on level ground with two different hip orientation angles, 5°, (the position shown in Figure 5-1) and –15°. As speed increases, the advantages of a more streamlined position become quite clear. For this example, the power required at 8 meters per second is reduced by 40 Watts (19%) relative to the 5° HOA position. This is a substantial reduction in drag, and is most likely more important than the increase in power production in the more upright position. (The simulation assumed that only the frontal area changed, that is, the drag coefficient remained constant between the two positions. This may be inaccurate, and the difference may be somewhat less than predicted. However, the general trends are accurate.) More research may be required to optimize the HOA relative to both power production and power demand.

POWER REQUIRED TO RIDE HPV AS FUNCTION OF SPEED 400 HOA = -15 deg

350

HOA = + 5 deg

300 250 s tt a W R 200 E W O P 150 100 50 0

0

1

2

3

4

5

6

7

8

9

10

SPEED m/s

Figure 5-2 Power requirements for HPV with different HOA values

66


Crank Length and Physiological Efficiency In upright cycling, crank arm length is typically varied over only a very small range. Mountain bikes usually use crank arms between 170 and 175 mm, while road bikes are slightly shorter, about 170 mm. If the crank length is increased much more than 175 mm, bottom bracket height must be increased to avoid striking a pedal on the ground during turns. Shorter cranks require increasing saddle height if the minimum knee angle is to be maintained. To prevent uncomfortably high riding positions, the bottom bracket height should be lowered. However, there is little evidence that short crank arms provide an advantage in upright cycling, except possibly for people with short legs or children. If the crank arm length is selected to be proportional to leg length, the range of knee and hip angles can remain comparable to “average” cyclists. Children’s bicycles often use very long crank arms relative to the rider’s leg length, frequently using 170 mm arms despite the noticeable difference in leg lengths between children and adults. This apparently does not affect peak power production in children 3; however, it does lead one to wonder if the resulting large joint angle ranges are hard on joint health. It is likely that bicycle manufacturers use stock components rather than develop child-specific components for cost reasons, and may have little concern for physiological effects. The question of crank length becomes much more interesting for recumbent cycles. The constraints on upright cycle crank arm length do not apply to recumbents— it is feasible to use very short or extremely long cranks. With positions that have cranks above the seat, gravity makes pedaling at high cadences more difficult unless shorter cranks are used. Conversely, extra-long crank arms permit very high torques to be applied at relatively low cadences. Some riders of recumbent vehicles prefer very short crank arms—as short as 110 to 140 mm—when riding vehicles with cranks located above the seat height. Unlike upright cycling, short cranks have several definitive advantages for recumbent vehicles. A significant benefit of recumbent vehicles is the reduced aerodynamic drag. Shortening the crank arms reduces the area required for the feet and pedals, enabling smaller and more aerodynamic velomobile or streamliner fairings. Peak chain loads and frame stress/deflection due to hard pedaling can be reduced with short cranks. Shorter cranks also reduce the hip, knee, and ankle joint ranges of motion during pedaling (the maximum joint angles depend on the seat-pedal distance and are not dependent on crank length). When the seat position is adjusted such that the 3

Martin, J.C., R.M. Malina, and W.W. Spirduso, 2002, “Effects of Crank Length on Maximal Cycling Power and Optimal Pedaling Rate of Boys Aged 8–11 Years,”European Journal of Applied Physiology, 86(3), 215–221.

67


maximum leg extension is constant with respect to crank length, knees bend to a less acute angle with shorter cranks. The leg muscles exert a force on the joints through a smaller range of joint angles. 4 Some individuals with knee problems have reported that short crank arms make pedaling more enjoyable because of decreased stress on the knees. Longer crank arms increase forces on tendons in the knee which can cause knee pain and injury.5 A number of studies have investigated the effects of crank length on both upright and recumbent cycling. Crank arm length has a smaller effect on peak power, fatigue, and metabolic efficiency than other factors such as cadence. However, crank length and cadence are interrelated. As crank length is reduced, the optimal cadence increases. However, cadence appears to be more important than crank length. That is, the penalty (in power, metabolic efficiency, or fatigue) for pedaling at a non-optimal cadence is much greater than the penalty for pedaling with a non-optimal crank length. This is fortuitous for the vehicle designer, since crank arm length is a design parameter and cadence is an operational parameter. The vehicle engineer can specify crank arm length based on factors other than bio-mechanics, such as fit within a fairing, avoiding component interference or customer preference. For upright cycling, the standard crank length of 170 mm is likely near optimal for peak anaerobic power output, although the optimum is dependent on leg length. As leg length increases, the optimum crank arm length also increases. Inbar found that a leg length to crank arm length ratio of 6.28 optimizes peak power in a 30 second Wingate power test.6 Other studies have produced similar results. The effect of crank length on recumbent cycling is somewhat more uncertain, but it appears that the optimal length with respect to peak anaerobic power is somewhat less than that for upright cycling, perhaps in the 145 mm range. Baker and Archibald found that metabolic efficiency is largely independent of crank arm length in the recumbent position, at least for crank arms in the range of 115 to 170 mm.7 This means that the metabolic cost of cycling does not depend on crank

4

Too, D. and G.E., Landwer, 2004. “The Biomechanics of Force and Power Production in Human Powered Vehicles,”Human Power, 55, 3–6. 5 Asplund, C. and P., Pierre, 2004, “Knee Pain and Bicycling,” The Physician and Sportsmedicine, 32(4), 2–3. 6 Inbar, O., R. Dotan, T. Trousil, and Z. Dvir, 1983, “The Effect of Bicycle Crank-Length Variation upon Power Performance,” Ergonomics, 26(12), 1139–1146. 7 Baker, Tyler and Mark Archibald, 2011, “The Effect of Crank Length on Delta Efficiency in Recumbent Cycling,” Proceedings of the 2011 ASEE North Central & Illinois-Indiana Section Conference.

68


arm length. For steady-state, submaximal pedaling, the crank arm length is not important from a power or endurance standpoint. Fitting into a fairing or knee comfort may be much more important reasons for selecting crank arm length.

Hand Cranks Hand operated vehicles are often used by people with lower limb disabilities. Rim-powered wheelchairs have been used for many years, and are still quite common. Hand cranks have several significant advantages over handrim-wheelchairs, and have become increasingly popular for recreation, competition, and therapy in recent years. Hand cranks allow users to achieve about twice the power output of handrim wheelchairs.8 Special cranks are available for individuals with upperbody impairment, making hand cycling more comfortable and efficient. These devices make cycling possible for people with more severe limitations. Many of these individuals can only achieve 15 to 25 W with handrim devices—insufficient for practical transportation. Hand cranks can make muscle-powered transportation possible for many tetraplegics. This not only provides independence, but also provides aerobic exercise that might otherwise not be possible. Data for optimal hand-crank parameters is even more sparse than for leg cranks. Identifying the most efficient cranking mode, crank arm length, and cadence is helpful to competitive hand cyclists, therapists, and anyone with lower limb disability. Most hand cycles currently in use have synchronous cranks, in which both arms extend simultaneously. On hand-crank ergometers, some studies have shown that asynchronous cranks are more efficient. However, van der Woude found synchronous cranking to be more efficient when subjects used a hand cycle on a treadmill.9 The difference is likely due to the requirement for steering a hand cycle, as opposed to an ergometer. Asynchronous cranking leads to oscillations of the front wheel, with additional effort required to maintain a straight track with the vehicle. An ergometer with a fixed crank is not steerable, and hence not affected by steering inputs. Most hand-cycle athletes prefer asynchronous hand cranking, probably for the same reason.

8

Janssen, Thomas W.J., Annet J. Dallmeijer, and Lucas H.V. van der Woude, 2001, “Physical Capacity and Race Performance of Handcycle Users,” Journal of Rehabilitation Research and Development, 38(1), 33–40. 9 Van der Woude, Luc H.V., I. Bosmans, B. Bervoets, and H.E.J. Veeger, 2000, “Handcycling: Different Modes and Gear Ratios,”Journal of Medical Engineering and Technology, 24(6), 242–249.

69


Optimal cadence and crank arm length appear to follow trends similar to leg cranks, although there is some disagreement in the literature. Physiological efficiency during sub-maximal hand cranking is somewhat more sensitive to crank length than leg cranking. As with leg cranks, optimal crank lengths are probably proportional to limb length. It appears that crank arms in the range of 180 to 190 mm are optimal for most people. Crank width seems to be less important than crank arm length. Kramer found that crank length equal 10 to 26% of his subjects’ forward reach were optimal for peak power generation. This length averaged 190 mm for his study. He also found a clear inverted u-shaped relationship between cadence and peak power. Cadence at peak power decreased from 125 RPM to 107 RPM as crank length increased from 19% to 26% of forward reach. These values are rather high, but the power output levels were also higher than in many other studies. Optimal cadence does depend on power level: higher cadences are required to produce very high power levels. In sub-maximal cycling, where efficiency is more important than peak power, optimal cadences are likely lower. Van der Woude predicted an optimal cadence of around 50 RPM, based on his own data and other published data. His study used 180 mm cranks, and only tested subjects at relatively low cadences, with a maximum of 45 RPM. Therefore, he could have under-estimated the optimal cadence. Goosey-Tolfrey studied the effects of crank length and cadence, and determined that 180 mm arm cranks were more efficient than 220 mm cranks during sub-maximal cycling. However, she was not able to detect a statistically significant difference in efficiency due to cadence.11 She also noted that based on the literature, crank length differences of 40mm are required to elicit a significant difference in efficiency. As with leg cranks, crank length—a design parameter—is of greater interest than cadence—an operational parameter. Based on the literature, crank lengths around 180 mm are likely optimal for most people. Riders with longer arms may prefer longer cranks. Riders preferring high cadences may be more comfortable with somewhat shorter crank lengths. The length of the crank arms may have a more significant effect on peak power than efficiency, so optimal cranks are crucial for events in which peak power is required. These events include sprints,

10

Kramer, Christian, Lutz Hilker, and Harald Bohm, 2009, “Influence of Crank Length and Crank Width on Maximal Hand Cycling Power and Cadence,”European Journal of Applied Physiology, 106, 749–757. 11 Goosey-Tolfrey, Victoria L., Helen Alfano, and Neil Fowler, 2008, “The Influence of Crank Length and Cadence on Mechanical Efficiency in Hand Cycling,”European Journal of Applied Physiology, 102, 189–194.

70


short time trials, and sometimes hill climbing. If arms are used for both steering and cranking, synchronous cranking should be used.

Unusual Mechanisms for Human Power Transfer Cranks are by far the most common type of mechanism for transferring human power to a vehicle. They have been used on bicycles since the earliest days, as well as on other types of human-powered equipment. Many other types of mechanisms have been tried over the years. No doubt many more will be attempted in the future. Relatively few alternatives have shown significant promise, however. The currently produced Rowbike (see http://www.rowbike.com/) uses a rowing motion to power a recumbent bicycle. Both arm and leg muscles are used for propulsion, giving the rider a whole-body workout while riding. The active muscle mass is greater than for leg-cranks, so total power can be expected to increase somewhat. Oxygen requirements may limit the benefit of greater muscle mass for many people. Linear drives and elliptical drives have been introduced by several innovators with little long-term success. With a linear drive, the feet move back and forth in a straight line. Elliptical drives are a compromise between a linear drive and normal circular cranks, in which the feet move in an elongated ellipse. Both types of drives have been used with recumbent streamliners to reduce the space required for pedaling, and hence the size of the fairing. Linear drives require linear bearings, which are inherently less efficient than rotational bearings. Further, they introduce additional complexity into the mechanism. The added complexity and reduced mechanical efficiency are potential problems for linear drives. From a biomechanics standpoint, linear drives require accelerating and decelerating the legs every cycle. Kinetic energy in the legs and feet cannot be conserved throughout the cycle as can be done (at least partially) with circular cranks. This means that linear drives are less efficient from both a mechanical and bio-mechanical viewpoint, and the benefits are questionable at best. They should only be considered if aerodynamic drag can be demonstrably reduced by a significant fraction. Elliptical drives have an aerodynamic advantage similar to linear drives, without quite so many problems. It is possible to construct a drive in which the feet move in a high-aspect ratio ellipse using only rotational bearings. However, the mechanism will be complicated, requiring chains or gears and additional joints to achieve the desired motion. The biomechanics of elliptical drives are probably only slightly more efficient than linear drives. Neither drive type is recommended. It is worth noting that an elliptical drive is not the same as a circular drive with elliptical chainrings, a different drive alternative. With an elliptical drive, the pedals

71


move in an elliptical path. In contrast, an ordinary crankset can be outfitted with elliptical chainrings to alter the effective gear ratio within the pedal cycle. In this case, the pedals and feet still move in a circular path. The goal is apparently to reduce the dead spots at either extreme in the pedal cycle. Other chainring shapes have been introduced over the years, with no clear advantage. For a number of years, Shimano manufactured non-circular chainrings called Biopace rings. Currently, Rotor produces non-circular chainrings marketed as Q-rings. Non-circular chainrings probably do not incur the penalties of linear or elliptical drives, and some riders may prefer them. However, they have not been definitively shown to provide a performance benefit. Levers, either for legs or arms, have also been used. Human-powered boats have used levers in the form of paddles and oars for millennia. Some very early bicycles used levers rather than cranks, coupled with cable drives. Hydraulic drives using levers to actuate hydraulic pumps have also been proposed. Hydraulic drives of any type are much less efficient than chain drives, and incur a significant performance penalty. Levers suffer many of the same drawbacks as linear and elliptical drives—greater complexity, reduced mechanical efficiency, and reduced bio-mechanical effectiveness. Other drive mechanisms include scooters, in which propulsion is attained by kicking the ground. The compact foldable scooters popular in the last decade or so can provide practical transportation, although not nearly as efficient or fast as bicycles. Bicycles for young children are produced without cranks, and with a seat height low enough to push them with the feet. This presumably helps to teach balance at an early age. Efficiency is not a significant concern. Circular cranks remain the dominant means of power transfer to land humanpowered vehicles. The bio-mechanics of cycling with circular cranks are efficient, and the mechanical design is simple and straightforward. It is probable that the circular crank will remain a standard well into the future.

Safety The design of any human-powered vehicle must address safety. For commercially sold vehicles, safety is mandated by codes and standards such as 16CFR1512A in the United States, ISO 4210 for Europe, and JIS D9414 in Japan. For any vehicle, including prototypes and home-builts, safety should be considered throughout the design. There are two types of safety features to consider: preventive and protective. Preventive features are incorporated in vehicle design for the specific purpose of reducing the likelihood of accident or injury. Adequate visibility of the road, structural integrity, and steering system reliability

72


are obvious examples of preventive safety features. Protective features reduce the likelihood of injury or the severity of an injury in the event of an accident. Operationally, helmet use is a prime example of a protective feature. Helmets do not prevent accidents, but have been clearly shown to reduce the likelihood of serious head injury when an accident does occur. For the design engineer, roll-over protection systems are examples of protective features. It is important to note that injury can occur without an accident. Poorly placed components or sharp edges can lead to cuts, scrapes, and bruises by simply operating the vehicle in a normal manner. This is clearly not acceptable for any vehicle. Other injuries such as muscle strain and joint damage are possible if the vehicle does not fit the rider well. The designer is responsible for ensuring that all reasonable precautions for protecting the rider have been addressed in the design stage. This includes, but is not limited to, compliance with safety standards. Safety design starts with hazard analysis. What can harm the rider of the vehicle? Accidents comprise an important portion of the answer. The Consumer Product Safety Commission identified the following accident patterns for bicycles: 12 • • • • •

Collisions with other vehicles, including motor vehicles Loss of Control Mechanical and Structural Problems Entanglement of hands, feet, or clothing Foot slippage from pedal.

In addition to accidents, injury can be incurred due to the following • • • •

Sharp edges and puncture hazards Poorly placed components or frame elements Unguarded chains, spokes, or other moving components Loose or improperly secured components or parts

Table 5-1 lists accident classes and common hazards along with recommended remediation during vehicle design. Although a bit general, the table can be used as a checklist for safety items. Streamliners and velomobiles have additional requirements due to the enclosed body. In an accident, it is less likely that the rider will be thrown clear of the vehicle. Rollover protection systems and side protection may be required.

12

Consumer Product Safety Commission Document 346, “Bicycle Fact Sheet.”

73


Table 5-1 Remediation Checklist for Safety during the Design Stage H a za r d Collisions with other vehicles, including motor vehicles

DesignRemediation Ensure rider has good visibility of the road Ensure vehicle is easily visible to others Ensure vehicle has acceptable handling qualities for escape maneuvers Ensure vehicle has adequate braking capacity

Loss of control

Ensure handlebar and steering system is robust and damage-tolerant Ensure vehicle is stable and has acceptable handling qualities Verify wheels are properly tensioned and trued Verify all controls are secure and operational

Mechanical and structural problems

Verify frame and fork strength and rigidity Ensure components are mounted correctly and fasteners properly torqued Test frame, fork and all components thoroughly Verify handlebar and seat are properly secured Provide keepers to prevent loss of a chain

Entanglement of hands, feet, or clothing

Provide guards for any location where entanglement is likely or possible Route chains and moving components away from the rider’s reach

Foot slippage from pedal

Specify pedals with adequate grip or clipless pedals

Sharp edges and puncture hazards

Avoid sharp edges on frame members, mounting tabs and brackets, shells, and components Cap all open tube ends Ensure screws are correct length and do not protrude more than 3 threads Ensure all cable ends are capped or soldered (Continued)

74


Table 5-1 (Continued) Hazard Poorly placed components or frame elements

Unguarded chains, spokes, or other moving components Loose or improperly secured components or parts

Design Remediation Verify components do not interfere with normal operation of the vehicle Adequately test ingress, egress and operation to ensure no hazards exist Provide guards for any location where hands, feet, or clothing might become caught

Specify torque requirements for all fasteners Rigorously test all special-purpose components and mounts Specify thread locking compound where necessary

Rollover protection should clear the head of the rider, and provide protection in the event the vehicle rolls inverted or falls and slides top-first into an obstacle. These systems are sometimes used on competitive vehicles to reduce injuries during accidents. Design parameters include deflection, stress at peak load, and energy absorption. A rider restraint system must be used for roll over protection systems to be effective. Side protection may be provided by the fairing or shell of the vehicle. If a vehicle falls and skids across the road, these systems protect the rider from abrasion. Aramid composite fairings provide excellent abrasion resistance, superior to many other common fairing materials.

75

CHAPTER

MANUFACTURING PROCESSES AND

6

MATERIALS

F

ortunate is the rider whose steed is well-designed, finely tuned, and carefully crafted from quality materials. These machines exhibit a beauty and elegance that transcend the functionality of chains and cogs, wheels and pedals. The muddy wrack of the street is left behind as each pedal stroke is felt as a buoyant surge of acceleration, each twist of the handlebar in a crisp, nimble turn. Imagine cresting a mountain ridge as the early morning sun cuts across the valley, tendrils of mist still clinging to the slopes, plunging down and around switchbacks, tuned to nature, in tune with your swift and sure ride. Such experiences are richer and felt more keenly when the vehicle is carefully designed and crafted. Many of the qualities that make a vehicle stand out are related to materials and, less visibly but no less important, the process used to craft those materials. The choice of material and manufacturing process for a vehicle frame, shell, or component has significant consequences in terms of cost, quality, and performance. Processes and materials cannot be selected independently, as all materials have a limited range of processing options, of which only a few may be economically feasible. The optimal combination of material and processing method provides the best balance of cost, performance, and fabrication effort with respect to the goals and mission of the project. Always, the decision is a compromise. The best performing materials may be difficult or expensive to process, for example. This chapter outlines some of the most common materials used for human-powered vehicle frames, shells, and (to a lesser extent) components. The scope is limited to an overview, with emphasis on the relative merits and drawbacks of each material or process. Tables and descriptions should provide guidance for selecting materials and processes suitable for a builder’s goals, budget, and fabrication capabilities.

77


Wrought Metals—Overview Wrought metals and their alloys are extensively used in human-powered vehicles due to favorable mechanical, physical, and processing properties. Wrought metals are formed or shaped in the solid or plastic state by processes such as rolling, drawing, forging and extruding. In contrast, cast metals are formed by pouring or injecting liquid metal into a mold. Generally, wrought metals have a more refined grain structure and improved mechanical properties relative to cast metals of similar chemical composition. Improvements in strength, ductility, and toughness may be achieved through the forming process. This makes wrought alloys well suited for human-powered vehicles, which require high strength and stiffness coupled with low weight. Many parts are made of these materials, including both frames and components. Some material properties, such as density and modulus of elasticity, are primarily dependent on the type of the material and vary only slightly across alloy variations, heat treatments, and amount of cold work. Other properties, such as strength, are highly dependent on the condition of the material, and can vary greatly for the same chemistry depending on the history of the particular product. For example, cold-drawn alloy steel tubing is stronger than hot-rolled bar stock of the same alloy. Many metals can be further strengthened by appropriate heat treatment. The properties of several wrought alloys used in human-powered vehicles are shown in Table 6-1. Most of this data is also plotted in Figure 6-1, providing a visual comparison of several materials. In order to make comparisons easier, all properties have been normalized relative to low-carbon steel. Composite measures for mechanical and process properties are used to provide data that is as complete as possible while remaining easy to interpret. Mechanical properties scores are based on the product of fatigue strength (at 10E7 cycles), tensile strength, and modulus of elasticity. The processibility score is based on the product of formability, machinability, solder/brazability, and weldability ratings. The relative CO2 footprint is based on primary production of the metal, and is a good indicator of ecological merit. CO 2 emissions are generally proportional to energy requirements. The remaining columns are self-described. It is worth noting that in the first four columns, higher scores indicate more desirable (improved properties), whereas in the last three columns, lower scores are more desirable. Values in the table vary widely, and there is no clear optimal material when looking at all seven property classes. Bear in mind that this comparison is of necessity high-level, and specific products and processes may alter the results. Property values are averages for each

78

Manufacturing Processes and Materials

y ti s n e D t n ir p t o

y r a m F o ri 2 P O ( C

1 6 le b a T

sl a t e M t h g u o r W d e t c e l e S f o s e it r e p o r P l a ir e t a M

f o s u l u d o M e u g it a F

l a c i n a h c e M

o t y ti c it s a l E o t tg h n e rt S y ti li b is s e c o r P s e it r e p o r P

g 3 K m

1 9 6 , 2

5 1 2 , 8

3 4 7 , 8

7 4 ,8 7

7 4 ,8 7

7 4 ,8 7

7 4 ,8 7

3 4 ,8 7

4 9 ,5 4

0 1 ,7 1

g

9 7 . 2 1

6 5 . 3

7 1 . 4

0 8 . 1

3 0 . 2

1 8 . 1

1 8 . 1

7 9 . 4

4 3 . 6 4

1 3 . 5 3

D S U

1 0 . 1

4 6 . 2

9 7 . 3

5 2 . 0

7 2 . 0

5 2 . 0

5 2 . 0

0 8 . 2

8 .5 0 1

8 .4 1

a 3 P /m M N

8 2 3 2 . 0

9 2 0 .1 0

3 3 8 .0 0

5 4 2 .2 0

3 8 2 .2 0

5 4 2 .2 0

0 5 2 .2 0

8 5 1 .2 0

4 2 1 .2 0

8 0 2 .2 0

a 3m P / K N

3 5 4 .3 0

2 0 0 .2 0

5 6 .1 0

7 6 4 .4 0

1 5 .4 0

0 4 6 .2 0

3 1 0 .4 0

2 3 9 .3 0

7 4 1 .1 1

1 7 0 .7 0

x e d n I

5 4 . 1 3 1

1 2 . 7 4 4

1 4 . 6 4 3

0 3 . 7 8 3

0 0 . 0 0 3

0 3 . 7 8 3

0 3 . 7 8 3

0 0 . 0 5 1

8 9 . 0 3

0 0 . 0 2 1

x e d n I

2 7 4 .2 .2 .4 8 4 7 2 1

5 .9 7 4 2

9 .7 3 6 2

2 .0 9 6

8 .8 4 6 1

9 .7 8 2 2

9 .5 7 2 2

7 5 . 5

.) d rP o

e c ir P o it a R y ti s n e D io t a R y ti s n e D

l a ir e t a M

g

K K

g n i n e d r a h e g A

l e e n s st o y rb l y lo l lo e a c e la z -a l ss n h e t A a o g s w r r i o B B H L

l e e t s n o b r a c w o L

n o rb a c m u i d e M

l e e ts s l se e l n te s ia t S

s y llo a m u i n a ti T

m u i s e n g a M

s y o ll a

79


WROUGHT METALS COMPARISON RELATIVE TO LOW CARBON STEEL

1

10

X 0 E 10 D N I E C N A M R O F -1 R 10 E P

Age-hardening wrought Al-alloys Low alloy steel Low carbon steel Stainless steel Titanium alloys Wrought magnesium alloys

-2

10

MECHPROP

PROCESS

FSTODRATIO

E TO DRATIO

1/PRICE

1/CO

2

1/DENSITY

Figure 6-1 Properties of wrought metals relative to low carbon steel

material, so a particular material product may exhibit either better or poorer values. However, the data does provide a very good glimpse into the overall properties of these materials. For example, wrought aluminum alloys are generally not as strong as steel, have poorer fatigue strength, and are one-third as stiff. However, some alloys of aluminum are stronger than some alloys of steel. In terms of manufacturability, aluminum is generally easier to form (in the soft tempers) and to machine (in hard tempers) than steel, but is more difficult to weld and braze. The latter brings down the overall relative processibility score for aluminum alloys. The data is particularly well suited for comparing vehicle frame materials, where joining members is important. Tabulated data can be difficult to assimilate—often a figure is more informative. Figure 6-1 is based on selected materials from Table 6-1, but the values are normalized to that of low carbon steel. Within each property set, all values were divided by that of low carbon steel to obtain the relative values. This makes comparisons between materials easier to discern on the figure. Also, the reciprocals of the last three properties—price, CO 2 footprint, and density—are plotted so that higher values on the plot are always better. The resulting curves are easily compared. Based on performance alone, alloy steel, stainless steel, and titanium are good choices. However, both titanium and stainless steel are difficult to process, expensive, and produce more CO 2 during material production. Titanium is the extreme case, with excellent performance, but extremely poor manufacturability, cost, and CO2 emissions. Taken as a whole, this explains why most of the world’s bicycles have steel frames, but high-performance (and higher price-point) frames are often made of carbon fiber, aluminum, or titanium.

80


Cast Metals—Note Cast metals have fewer human-powered vehicle applications than wrought metals do. In general, cast metals have poorer mechanical properties than do wrought metals of similar composition, making wrought metals more attractive from a performance standpoint. However, cast metals provide the designer with much more design freedom—very complex geometries can be obtained with castings. This also makes it easier to combine several parts with different functions into a single part. (In a production environment, this is very good—leading to fewer parts to inventory and assemble lead to reduced cost and improved quality. It is hard to realize these benefits for bespoke vehicles, however.) Castings may also be less expensive than machined components if production quantities are high enough. The cost of molds and patterns required for casting can be considerable, particularly for the more accurate casting processes such as die casting or investment casting. Cast components for human-powered vehicles include parts used in some components, where production quantities are high enough to justify the tooling expense. Frame parts for brazed steel bicycle frames often use investment cast lugs, dropouts, and similar items. (Although lugs can also be formed by stamping, a less expensive process if quantities are high.)

Non-Metal Materials—Overview Non-metals include polymers (plastics), elastomers, natural materials, ceramics, glasses and most composites. The vast number of non-metallic materials make a comprehensive overview impossible. Literally thousands of materials are available for human-powered components, fairings, and even frames. For this overview, a small selection of polymers, polymer composites, and natural materi als are briefly described including properties typical of their type. Relatively few glasses and ceramics are used in human-powered vehicles, and so those materials, although many have remarkable properties, are not included except for glass-reinforced polymer composites. The range of material properties for non-metals is extremely wide, with some materials surpassing the performance of the best metals, and others well below metals. Few characteristics are common to all of them, although all but carbon exhibit low thermal and electrical conductivity. In general, polymers and most natural materials such as wood are significantly less strong and less rigid than metals, and often have lower densities. They are often easier to process than metals, either by molding or machining. On the other hand, reinforced polymer composites can be competitive with metals in terms of strength and stiffness, but with a lower density. Table 6-2 shows the properties of a few selected non-metal

81


2 6 le b a T

sl a i rt e a M l a r u t a N d n a ,s e its o p m o C ,s r e m y l o P d e t c e l e S f o s ie tr e p o r P l a i rt e a M

t in r O p C to o F 2

y ti s n e D

g 3 K m

5 3 0 9 1 , 6 1

9 4 5 , 1

0 9

6 5 8 , 1

) . d o r P P (

g g K K

0 3 8 . 3

4 1 3 . 0

6 0 .6 4 3

6 0 6 . 3

8 8 9 . 9

6 1 . 1

4 7 . 0

9 8 . 7 1

1 3 . 1

2 .1 3 1

7 3 2 6 .2 4 . 0 3

9 0 . 9

1 0 . 0

3 .5 1

0 0 2 1 . 0

6 .3 0

6 .1 1

9 .0 0

3 3 . 0

9 .4 5 1

6 6 . 5

5 7 . 7

3 .9 6

6 9 . 0 1

2 3 . 3

2 .9 2 6

0 .0 8 3 ,0 0 5

0 0 . 0

8 5 . 8 2 8

y r a m ir

f o s u l u d o M

y ti c it s a l E

y ti s n e D o t

e u g it a F

h t g n e rt S

y ti s n e D o t y ti li b is s e c o r P

l a c i n a h c e M

s ie tr e p o r P

e c ir P

D S U

o it a R

a 3m P / M N

o it a R

a P m / K N

x e d n I

x e d n I

l a ir e t a M

82

3

e e n i d a t u b e ilr it n lo y r c A

) S B A ( e n e r ty s

o o b m a B

ix rt a m y x o p e , P R F C

m a o F r e ) m y ic l p o P rto e l o si ib ( x le F

ix rt a m y x o p ) e D ,P M ( R F G

) ic p rto o si (


9 8 7 4 2 , 7 1

0 3 ,1 1

9 0 ,3 1

9 8 ,1 1

9 0 ,4 1

5 4 ,0 1

4 1 5

4 1 5

0 2 .6 3

6 2 8 . 0

0 7 9 . 7

1 2 .1 3 2

2 9 7 . 6

0 5 0 . 4

5 9 7 . 3

7 7 3 . 0

7 7 3 . 0

0 8 . 0

6 2 . 0

4 9 . 1

7 9 . 4 4

7 2 . 1

0 4 . 1

2 4 . 1

3 .4 0

3 .4 0

0 4 . 0

5 7 . 1

5 3 . 0

1 4 . 0

4 3 . 0

5 3 . 0

3 .2 0

0 .2 0

1 .5 2

2 2 .1 .1 0 0

7 .3 0

2 .2 0

6 .1 0

7 5 .1 .1 0 0

2 .0 0

5 .3 0

9 .4 5 1

2 .3 7 1

9 .4 5 1

9 .4 5 1

9 .4 5 1

9 .4 5 1

9 .4 5 1

5 2 . 2 1

5 2 . 2 1

5 5 . 9

1 4 . 6

8 .4 2 5

4 .9 3 3

1 .3 2 1

7 6 . 1 2

1 .4 4

1 0 . 0

5 9 . 5 4

s c il o n e h P

d o o w y l P

,s n o l y N ( s e d i m ) a A y l P o P

n e o t e rk e h t re e h t e y l o P

) K E E P (

r y c a th e m l y h t e m y l o P

) A M M P , ic l ry c A ( e t la

e n e l y th e m y x o y l o P

) M O P ,l a t e c A (

) S P ( e n e r y t s y l o P

s so r c a , e n i p : d o o w tf o S

g n lo a , e n i p : d o n i o ra w g tf o S

in ra g

83


materials. Several polymers are included, along with polymer composites and a few natural materials such as pine wood and bamboo. All columns except for the processibility index are defined as in Table 6-1. The non-metal processibility index is the product of moldability and machinability, processes more applicable to non-metals than forming, welding and brazing. Figure 6-2 plots the properties for selected non-metals in order to provide a visual comparison. All properties were normalized to ABS plastic, chosen as typical of engineered polymers. As with Table 6-1, the inverse of price, CO2 footprint, and density is plotted. Note the vast differences in the material properties between materials—CFRP, or carbon fiber reinforced polymer, has a mechanical properties index more than 1000 times that of ABS! The figure clearly shows the performance advantage of carbon composite (CFRP) and to a lesser extent glass composite (GFRP, also known as fiberglass) over other non-metal materials. Fiber reinforced polymers—of which there are many types—are used structurally for frames and shells, most notably for high-performance vehicles. Many racing bikes have carbon composite frames, and almost all record-class streamliners use composite shells. As the figure shows, these materials have high mechanical properties, high stiffness-to-weight and fatigue-strength-to-weight ratios, although they tend to cost more and have a higher CO 2 footprint than many of the other materials shown. Plastics, such as ABS and Nylon, are used in components and small parts where performance is not critical. They are easily molded into many shapes, but this requires expensive tooling and high production quantities. Both materials can be printed on a 3D printer for low production and custom parts. Printing in ABS is particularly common, although printed parts will generally not attain the

5

NON-METALS COMPARISON RELATIVE TO ACRYLONITRILE BUTADIENE STYRENE (ABS)

10

Acrylonitrile butadiene styrene (ABS)

4

10

Bamboo CFRP, epoxy matrix (isotropic)

3

GFRP, epoxy matrix (isotropic)

X 10 E D N I 2 E 10 C N A M 1 R 10 O F R E 0 P 10

Plywood Polyamides (Nylons, P A) Softwood: pine, along grain

-1

10

-2

10

MECHPROP

PROCESS

FSTODRATIO

E TODRATIO

1/PRICE

1/CO2

1/DENSITY

Figure 6-2 Material properties of selected non-metals relative to ABS plastic

84


properties of molded parts. Natural materials such as wood and bamboo have been used for human-powered vehicle frames and shells. Search the internet for “wooden bicycle” or “wooden velomobile” to see examples of beautiful, fully functional vehicles. Bamboo in particular can have high performance with relatively low cost. In most bamboo bicycle frames, composite joints are used to join the bamboo tubes. Even soft woods such as pine can offer high good performance, particularly if properly protected from moisture.

Frame Materials and Manufacturing Processes Material selection always involves compromise. The ideal frame would be light, strong, corrosion resistant, fatigue resistant and inexpensive. In addition, the frame would be laterally and longitudinally stiff (where deformation would impede performance) but compliant in the vertical direction for rider comfort. It is certainly possible to design for any of these attributes, and even for many combinations. All cannot be achieved in one vehicle frame, however. The quality of bicycle frames currently on the market is actually quite remarkable. Some expensive frames meet almost all of the other criteria, while inexpensive frames can still offer very good performance. A well-designed titanium frame can offer light weight, high strength and fatigue life, good stiffness, and corrosion resistance, but will be quite expensive. On the other hand, an inexpensive aluminum frame can still provide a lightweight, responsive ride. The choice of material depends on service loads and requirements, material properties, cost, and other factors. Service loads and requirements include forces and moments acting on the structure, resulting stress and deflection, design life, requirements for wear, abrasion, and corrosion resistance, and so forth. These should be included, either directly or indirectly, in the product design specification. Material properties include strength (ultimate strength, yield strength, shear strength, fatigue strength, etc.) elasticity, toughness, ductility, density, corrosion resistance, abrasion resistance, and others. Cost is always an important factor, but raw material cost may not be as important as finished product cost. Sometimes a more expensive, but easily processed, material can produce a less expensive part than that made from a lower cost material that is difficult to process. Other factors include available forms, sizes and shapes of t he raw material, process limitations, potential failure modes, and even customer appeal. Titanium and carbon fiber are quite attractive to cycling consumers. These materials may entice a customer due to their exotic, high-performance appeal—even if the actual difference in performance relative to less expensive materials is minor.

85


Steel Steel is by far the most common material ever used for bicycle and humanpowered vehicle frames. Steel is strong, stiff, and inexpensive compared to other frame materials, making it an ideal material for many vehicle designs. It also has excellent fatigue properties. Many manufacturing processes are suitable for steel, including assembly processes such as welding and brazing. The two greatest drawbacks to steel are its tendency to corrode (rust) and its relatively high density. Designing with steel is straightforward, and generally easier than other materials. Steel is also considered to be more environmentally benign than aluminum or titanium. Carbon steel is the most inexpensive material commonly used for frames. Due to its relatively low strength, it is generally only used for inexpensive frames where cost reduction is more important than performance. Cold-worked carbon steel is available in round, square, and rectangular tubes, as well as strip, plate, and other forms. Cold-drawn carbon steel tubing is often used for manufacturing low-cost bicycle and human-powered vehicle frames. A wide range of tubular carbon steel products are available on the market, with different material compositions, manufacturing processes, and mechanical properties. Products with an annular cross-section are broadly classified as either pipe or tube. The term pipe designates a tubular product made to specific dimensional sizes and generally used for pipelines and connections. It is not suitable for vehicle frames. Tube is used in heat exchangers and for structural applications. It is available in both hot and cold processed products, and may be seamless or welded. Cold-drawn tubing is generally available is thinner gauges, and has improved mechanical and dimensional properties. Tubing suitable for vehicle frames is classified as structural or mechanical tube, and several ASTM standards apply to these products. ASTM A500 is cold-formed welded or seamless carbon steel structural tubing in round, rectangular, square or other shapes. It has a tensile strength ranging from 310 Mpa (45 ksi) to 427 Mpa (62 ksi) and yield strengths ranging from 228 to 317 Mpa (33 to 46 ksi). This is a fairly low-strength tube. Mechanical tubing is often preferred for vehicle frames, due to improved properties. ASTM A519 is cold-drawn, seamless mechanical carbon steel tubing, available in various grades with tensile strengths ranging from 331 Mpa (48 ksi) for 1020 steel in the annealed condition to 552 MPa (80 ksi) for 1045 stress-relieved tubes. There are several other specifications for carbon steel tube, and strength properties can vary significantly between different tube types and conditions. The design engineer should be aware of the specific product and standard used to ensure safety and performance. Most carbon steel tubing is readily weldable and easily brazed. It is usually bendable,

86


although tubes with high diameter to wall-thickness ratios can be difficult to bend with simple equipment. Higher quality steel frames use chromium-molybdenum steels due to their higher strength, good weldability, and availability in thin gauges. Seamless aircraft alloy tubing made to military specification MIL-T-6736B is an example. This is a tubular product made from 4130 alloy steel. It is available in many outside diameters from 4.76 mm (3/16 in) up to 90 mm (3.5 in), although smaller and larger diameters may be available as well. Wall thickness ranges from .7 mm (.028 in) up to 12.7 mm (.5 in), again with lighter and heavier gauges sometimes available. For many human-powered vehicle frames, wall thickness of .9 mm (.035 in) is suitable. In the normalized condition, tensile strength is usually 655 MPa (95 ksi) and yield strength 517 MPa (75 ksi). Thin-wall chrome moly tubing is very difficult to bend, although it is possible in the smaller diameters. Bends in tubes with large diameter-to-wall thickness ratios may only be possible with specialized bending equipment that uses mandrels. MIL-T-6736B tubes can be easily welded and brazed, although skill is required to weld very thin-walled sections. When brazing or welding, care must be taken to avoid rapid cooling, as the material may become locally brittle. It is possible to obtain 4130 tubing in a hardened condition. Heat-treated tubes have increased strength and reduced ductility. They require more skill during fabrication and are generally more expensive than tubing obtained in the normalized condition. This tubing is available through a variety of distributors, and is usually three to five times as expensive as carbon steel tubes. Several companies offer high-performance bicycle tubes in steel, including Reynolds Technology in the United Kingdom, Columbus in Italy, and True Temper in the United States. These products are engineered for upright diamond-frame bikes, and offer high strength alloys and heat treated tubes, butted tubes and bicycle-specific size tubes. Butted tubes have a wall thickness that changes along the length of the tube. These are used to optimize weight by providing a heavier wall near welds and high-stress areas, while using a thinner wall in the less-stressed portions of the tube. Bicycle tube sets are used to fabricate lightweight, high-performance steel diamond frame bicycles, and some products are not applicable to non-traditional frames such as recumbent bicycles and tricycles. Nonetheless, Frame designers of non-traditional frames can make good use of some of these products. Head tubes and seat tubes are made in sizes compatible with standard bicycle headsets and seatposts, and are convenient to use if applicable. If a conventional bicycle saddle is to be used, the inside diameter of the seat tubes is critical for correct seat post fit. Use of an engineered seat tube assures a good fit without reaming or shims. Threadless and threaded steerer tubes are available that are butted to provide good strength at the fork crown. Fork blades

87


Figure 6-3 Double-butted bicycle tube

are available in round and oval sections, with either straight or bent ends. Tapered and bent fork blades, known as unicrown blades, are available for building forks without a separate crown. Figure 6-4 illustrates a fork made with unicrown blades. The larger size unicrown blades can sometimes be used on a recumbent frame to support a cantilevered rear wheel. Chain stays and seat stays are available when a conventional rear triangle is included in the frame design. Information regarding products from Reynolds, Columbus, and True Temper can be found at these companies’ websites. Be aware that the very high strength, heat-treated tube products require extra skill during welding or brazing, and should generally be avoided by the beginner. Reynolds: http://reynoldstechnology.biz/ Columbus: http://www.columbustubi.com/eng/1.htm True Temper: http://www.alphaqbike.com/performance_tubing/bike_ tubing.asp Bicycle-specific frame components other than tubes can also be purchased for many applications. For upright diamond-frame bikes, lugs are available for brazing lightweight steel frames. These are not particularly useful for recumbents and other non-traditional geometries, however. Examples of frame components that

Figure 6-4 Fork made with unicrown fork blades

88


are frequently useful for all vehicle designers include front and rear dropouts, disc brake mounts, cantilever or linear pull brake bosses, water-bottle braze-ons, cable stops, and rack bosses. Seat bolt bosses are used for securing the seat post in the seat tube, but can be used effectively for vehicles with cranks mounted on booms that fit inside a main tube. Henry James in the United States and Ceeway in the United Kingdom each sell a variety of these components. Henry James is also a distributor for True Temper bicycle tubes. While it is often unnecessary to use these components, they may save time and effort in many cases. The brazeons add a professional, finished look as well as providing important functionality. Ceeway: Henry James:

http://www.ceeway.com/ http://www.henryjames.com/

Stainless Steel Stainless steel is used in many frame components and occasionally for frame tubes themselves. Stainless steel frame components include lugs, dropouts, and S&S frame tube couplers. Other bicycle components made from stainless steel include spokes, pedal spindles, and fasteners. Stainless steel’s greatest virtue is corrosion resistance. Depending on the alloy and condition, it can be quite strong and hard. It has high rigidity, high strength, and good fatigue life. Ultimate strength for materials used in frames can range from 620 to 1280 MPa (90 to 185 ksi), with stiffness and density approximately that of steel, 200 GPa and 8 g/cm3 (29 Mpsi and 2.9 lb/in3). Stainless steel is significantly more expensive than carbon or alloy steel, costing approximately five to ten times as much as carbon steel.

Aluminum Aluminum alloys are widely used in modern bicycles, and can result in lightweight, stiff frames. The density of aluminum is around 2.7 g/cm3 (0.098 lb/ in3), roughly one-third that of steel, but the range of strength overlaps that of steel. Aluminum alloys thus have very favorable strength-to-weight ratios. It is a common misconception that aluminum frames are inherently lighter than steel frames. Stiffness and fatigue properties of aluminum alloys result in frames that are much closer in weight to comparable steel frame than the difference in density would suggest. While well-designed aluminum frames are quite light, lowerquality frames are often heavier than good-quality steel frames. Good design is the key to a successful aluminum frame. Compared to steel, aluminum requires more design effort to achieve strength, performance, and life objectives.

89


Aluminum alloys exhibit combinations of properties that make them wellsuited for performance vehicle frames, but also more challenging to design. The relatively low density of these alloys is well known. Many alloys exhibit good strength; the strongest aluminum alloys exceed the strength of some carbon steels. (It should be noted that the strongest steels are three time as strong as the strongest aluminum alloys, however. Steels span an exceptionally wide range of strengths.) Unlike steel, however, aluminum alloys do not exhibit a fatigue endurance limit. Design for fatigue loading is thus an important consideration. The modulus of elasticity of aluminum is 71 GPa (10.3 Mpsi), or about one-third as stiff as steel. Although aluminum is a highly reactive metal due to its location in the electromotive series, all alloys readily form a tightly bonded aluminum oxide surface film. This film provides excellent protection against corrosion in many environments. If the oxide layer is damaged, it quickly re-forms and continues to protect the material. Aluminum is readily processable by most modern manufacturing methods. In the soft tempers, it is quite formable, although some alloys will work-harden rapidly. Tubes of almost any cross-section can also be easily extruded, and custom sections in relatively small quantities may be economically feasible. In some cases, as little as 80 kg can justify a custom extrusion. Tubes that taper or change shape along their length may be hydroformed, as can be seen in many commercial aluminum frames. Aluminum alloys generally have excellent machinability ratings, and high cutting speeds and feeds may be used. Most—but not all— alloys are weldable, an important consideration in frame design. Gas tungsten-arc or gas metal-arc welding are typically used for fusion welding. Aluminum alloys 6061 and 7005 have excellent weldability. Alloy 2024, which has excellent mechanical properties, has very limited weldability. In general, the 1000, 3000, and 5000 series non-heat-treatable alloys are readily weldable, along with 6000 series heat-treatable alloys. The 2000 and 4000 series alloys are generally more difficult to weld, and only a few 7000 series alloys are weldable. Alloy 7005 was specifically designed to be weldable. Heat-treated alloys, such as 6061-T6, may be welded, but the as-welded strength is slightly less than the unwelded T4 condition. This is due to partial annealing in the heat-affected zone. The as-welded joint is thus weaker and less ductile than the parent material. Solution heat treatment and artificial aging after welding can restore most of the strength, but some ductility loss will remain. Vehicle frames are sometimes annealed after welding in order to straighten the frame, and subsequently solution heat treated and artificially aged to obtain the T6 condition.

90


The two alloys of aluminum most often used for human-powered vehicle frames are 6061-T6 and 7005-T6. Aluminum alloy 6061 is a general-purpose alloy widely available in many forms, including extruded tube. It has good strength in the T6 condition—over 300 MPa ultimate strength and around 275 MPa yield strength. Fatigue strength is 97 MPa at 500 million cycles of fully reversed bending (R. R. Moore test). In softer tempers, it has good formability. It is typically welded with GTAW using ER4043 filler rod. Alloy 7005 is a high-strength alloy designed specifically to be welded. It is available in extruded tubes, as well as sheet and plate. Typical strength in the T6 temper is: 372 MPa ultimate strength, 317 MPa yield strength, and 150 MPa fatigue strength at 500 million cycles. It is typically welded using ER5356 filler rod.

Titanium Titanium alloys are used for expensive, performance-oriented vehicle frames. Titanium alloys exhibit high strength-to-weight ratios and are very resistant to corrosion, making these alloys very attractive for human-powered vehicles. However, titanium is quite expensive, costing up to 35 times as much as carbon steel. Alloys can be very strong, with typical strengths for bicycle tubes in the range of 650 to 1030 MPa (95–150 ksi). The modulus of elasticity is only about half that of steel, around 100 GPa (15 Mpsi), requiring care during design to ensure adequate frame stiffness is obtained. The low density of 4.5 g/cm 3 (.162 lb/in3), coupled with good strength properties, make titanium a prized material for frames. Titanium alloys also exhibit good fatigue properties; like steel they have an endurance limit. Unfortunately, titanium alloys are notoriously difficult to work, with poor machinability and poor formability. The two titanium alloys most often used for vehicles are Ti6Al4V and Ti3Al2.5V. Ti6Al4V provides a good balance of mechanical properties and processability. It is produced in several mill forms, and is widely used in many different applications. In the annealed condition, tensile strength is 895 MPa (130 ksi), although forgings and solution treated aged bar can achieve 1035 MPa (150 ksi). Ti6Al4V is the most commonly used titanium alloy, and should be considered for many structural applications. Tube forming demands good formability, not a typical characteristic of any titanium alloy. Ti3Al2.5V is frequently used for the production of tubing, and is available for frame tubes. It is most often used in the annealed condition or in the cold-worked and stress relieved condition. Annealed, it has a tensile strength of 655 MPa (95 ksi), while cold worked and stress relieved products have tensile strength of 895 MPa (130 ksi).

91


Processing of titanium and its alloys is generally much more difficult than steel, aluminum or copper alloys. As with many other materials, the properties of the final product depend strongly on processing. Titanium can be forged, and forged components have improved tensile strength, fatigue strength, creep resistance, and toughness. Extrusion is possible, although difficult. Most titanium alloys are strain hardened by cold work, and also exhibit significant springback, making forming difficult. Successful formed products may exhibit improved strength and reduced ductility. Titanium products can be formed with powder metallurgy, with P/M product achieving nearly the strength of other processes. Titanium components can be joined by welding, bonding, brazing, or mechanical fasteners. Brazing is even more difficult than for aluminum alloys, and requires special materials and procedures. Welding titanium alloys requires an inert atmosphere on both sides of the joint. This is due to its very high reactivity at elevated temperatures. For vehicle frames, this can be accomplished with an inert gas purge on the inside of the tubes. Gas tungsten arc welding (GTAW) is the most common type of welding, particularly for the light gauges used in human-powered vehicles. The high cost and processing difficulty of titanium demand a high level of engineering in the design stage. Additional time and effort to complete a detailed design is justified. Finite element analyses should be completed in more detail; manufacturing methods should be carefully evaluated, and design details completed prior to prototype fabrication. Initial prototypes used to verify gross geometry, fit, and ergonomics may be fabricated of less expensive material, such as steel, prior to investing in the final titanium frame. It is inappropriate to simply switch materials from steel or aluminum to titanium without a rigorous analysis.

Fiber Reinforced-P olymer Composites Fiber reinforced-polymer composite materials have tremendous potential for high-performance vehicle structures. These materials can be strong, rigid, and lightweight. They offer the knowledgeable designer tremendous design freedom, both in terms of geometry and material properties. Macroscopic mechanical properties can be “fine-tuned” to meet service requirements, including anisotropic stiffness and strength. However, it is challenging to realize these benefits. Highlevel engineering and excellent process control are required to fully realize the potential of these materials. Failures can be catastrophic, and defects may be far from obvious. Fiber reinforced-polymer composite materials are used for frames, forks, fairings, and components. Monocoque frames, in which the fairing skin is a structural

92


member, are often made with composites. These materials all use a fiber reinforcement embedded in a polymer matrix. The resulting structure has mechanical properties that exceed those of the constituent materials. A variety of processing methods are used, with differing tooling and labor costs, applicable part sizes, and properties. The materials are often described based on the reinforcement material, such as carbon, aramid, or glass. Though common, this is not strictly accurate as the resins used for the polymer matrix do affect part quality, properties, and cost. Most reinforced polymer composite materials are fabricated in layers, which may be made of different materials or different orientations of the same material. Reinforcement is available in chopped strands, strands, mats, uni-directional cloth, and woven cloth, as well as several other options. Strength is usually greatest in the direction of the fibers. To achieve good strength properties in different directions, multiple layers of unidirectional fabric can be used with different orientations. More frequently, woven cloth is used for bi-directional strength. The woven fabric is readily available and provides a more tolerant product for those who don’t have the scale and resources to thoroughly test and refine composite parts. Sometimes different forms—or even different materials—are used in different layers to obtain desired part-specific properties. Woven fabric of both aramid and carbon can be used to obtain both the high strength and stiffness of carbon and the abrasion resistance of aramid. This combination is advantageous in fairings. In conjunction with geometric design freedom, this allows parts to be optimized for their specific function. For example, chainstays can be designed to provide high lateral and longitudinal stiffness, while providing some compliance in the vertical direction. This ensures good power transfer and handling, yet provides a smoother, more comfortable ride for the operator. The details of such an optimization are well beyond the scope of this text, however. Large plate-like structures often must be reinforced to provide adequate stiffness and strength, particularly under bending (flexural) loads. A good way to improve the flexural stiffness and flexural strength, without adding much extra weight, is to use sandwich construction. A lightweight core is “sandwiched” between two layers of FRP. This is quite useful for fairings, which consist of large thin sections that are prone to bending. The improved rigidity and strength provided by a core can be quite noticeable, yet the weight penalty can be slight. Small closed sections, such as frame tubes, have adequate stiffness by virtue of their shape, and generally do not require cores. A core material, made of balsa wood, foam, or honeycomb, is sandwiched between two layers of resin/reinforcement material. Figure 6-5 shows the effect of core thickness on flexural strength, stiffness, and weight of a finished part. In

93


Figure 6-5 Effect of core thickness on strength, stiffness, and weight 1

each case the thickness of the reinforcement/resin layers is the same; only the core thickness changes. The numeric values assume a high-performance core material that is very lightweight. Similar results are obtained with foam or balsa wood. This technique is very useful for large structural shells such as streamliner or velomobile monocoque fairings. The specification for the number of layers, the orientation of the layers, and the presence and location of core materials is known as the layup schedule. Highly engineered composite parts require much attention to optimal specification of the layup schedule. Done well, this results in a highly optimized structure meeting the desired strength and rigidity requirements with minimum weight. The best design may still prove deficient if manufacturing cannot maintain consistent high quality production. The quality and reliability of composite manufacturing slowed the widespread use of these materials in bicycles for several decades. Fortunately, there is now enough expertise and experience with composites that these materials are used successfully by many manufacturers. Nonetheless, quality manufacturing is crucial for reliable composite structures. Carbon fiber reinforcement is used for parts demanding high strength and/or high stiffness. Some manufacturers offer finished carbon fiber tubes which can be assembled into frames by bonding. In addition to frames and forks, aerodynamic fairings, handlebars, seatposts, crank arms and other parts are made from carbon fiber. The actual carbon fibers used for reinforcement can be very strong, stiff, and light, with tensile strengths as high as 5,580 MPa (810 ksi), elastic modulus approaching (in some cases exceeding) 300 GPa(43 Mpsi) and a density of about 1.8 g/cm3 (.065 lb/in3). However, these values are generally not achieved when the fibers are used in a polymer matrix. Thefiber-resin ratio, resin type, and layup willaffect the properties of the final part. Typical values for strength, stiffness and density

Marshall, Andrew C., 1998, Composite Basics—5, 5th Ed., Marshall Consulting, Walnut Creek, CA, USA. 1

94


3 in CFRP parts are 550–1050 MPa, 69–150 GPa, and 1.5–1.6 g/cm respectively. As a comparison, this is about the same strength as steel, but at one fifth the density. Property values quoted for composite materials should be treated only as general information, and should not be used for design without testing. Actual values depend on type of fiber and resin, layup schedule and fiber orientation, and manufacturing quality control, and can vary substantially from the values quoted here. (This

applies to all types of reinforced-polymer composite materials.) Density is affected by the resin-fiber ratio—resin in excess of that required adds no strength, but does increase density and hence the final part weight. These impressive properties account for the popularity of carbon fiber composites. Other properties are not quite so good. CFRP has poor impact performance and poor abrasion resistance. Failure modes can be abrupt, with no yielding—resulting in potentially serious accidents. Aramid fiber, more commonly known by the DuPont trade name Kevlar, is used less often for vehicle frames, but is frequently used in fairings and human-powered boat hulls. In an epoxy matrix, the tensile strength can exceed 1,000 MPa (145 ksi). Unlike carbon, aramid has excellent abrasion resistance, and is often used in aerodynamic shells to provide abrasion resistance in the event of an accident and resulting slide over pavement. It is available in a variety of weaves, including woven carbon/aramid mixtures. The carbon improves rigidity and strength, while the aramid provides abrasion resistance. Glass has good availability at modest cost, good mechanical properties, and good manufacturing properties. It is not generally used in frame tubes, as is carbon fiber. Glass tubular construction is possible, but the resulting frame would be less rigid and heavier than a carbon fiber frame. It is frequently used in monocoque vehicles and fairings, where the lower cost makes it an attractive alternative to aramid or carbon. The two types of glass fibers used in human-powered vehicles are E-glass and S-glass. E-glass has been available for many decades, and offers very good performance at moderate price. S-glass (S for structural) is often used where structural performance justifies the higher cost. E-glass epoxy composites can achieve tensile strengths over 1000 MPa (150 ksi) with a modulus of 45 GPa (6.5 Mpsi). S-glass is a bit better, with tensile strengths of 1800 MPa (260 ksi) and modulus of 55 GPa (8.1 Mpsi). The density of glass composite is a bit greater than carbon or aramid, about 2.6 g/cm 3 (.095 lb/in3), comparable to aluminum alloys. The resins used for reinforced-polymer composites are usually thermoset polymers. Epoxies, polyester and vinylester are the most common resins. Epoxy resins work well with all types of reinforcement material, and offer the best strength and chemical resistance. Epoxy resins can be up to three times stronger than other resins, and they shrink less than polyester or vinylester during curing. Epoxies are also more expensive. Due to its strength and ability to bond with glass, aramid, and

95


Table 6-3 Table 1 Reinforced-Polymer Composite Material Properties Material

Density g/cm3

UTS MPa

Carbon-Epoxy, UD 0°

1.5–1.6

Carbon-Epoxy, BX or QI

1.5–1.6

519–725

Epoxy/S-glass fiber, UD 0°

1.84–1.97

260–283

FS MPa

1610–2005 887–1305

Cycles Elongation Modulus million % GPa 10

1.3–1.5

57–61

246–421

10

.8–1.1

68–75

18–27

10

1.0–1.6

16–17

Epoxy/S-glass 1.84–1.97 1700–1760 339–527 fiber, BX or QI

10

3.5–3.7

48

Kevlar-Epoxy, UD 0°

1.38

1100–1380 605–897

10

1.7

60–80

Kevlar-Epoxy, QI

1.38

355–392

10

.374–.42

23.5–30.9

70.7–98.5

Notes: UD -- Unidirectional BX -- Biaxial

QI -- Quasi-Isotropic Laminate Source: Compiled from CES EduPack 2016 by Granta Design Limited

carbon, it is used for most HPV parts applications. Most high-performance fairings are made with epoxy resin. However, on a commercial scale polyester resins are the most widely used. It is easy to work with and the least expensive of the three options. The cure rate for polyester resins can be controlled with the amount of catalyst added. However, it does not bond well with aramid or carbon fibers, and should not be used with these fibers for structural parts. Vinylester resin is easy to work with and is suitable for glass, carbon and aramid. In terms of cost and strength, it is between epoxy and polyester.

Other Frame Materials Human-powered vehicle frames have been successfully made of many other materials, including both metals and non-metals. Magnesium has been used for bicycle frames to a very limited extent. The low density and relatively good strength make it attractive, and the few frame manufacturers tout its ability to damp vibrations. However, it has poor fatigue properties, design and manufacturing is more difficult than aluminum, and there few companies producing tubes. Frames may also be made of wood. Wooden frames have been used since the earliest days of

96


cycling, but the properties of wood do not favor high-performance vehicles. However, wooden frames can be beautiful works of art. Laminated wood can provide a functional and attractive structure. Currently, most wooden frames are built by hobbyists that prefer impressive style to high performance. Bamboo is a fascinating material, with the potential to provide a sustainable product with good to excellent ride qualities. High-end road, mountain, and tandem bicycles made from hemp-lugged bamboo that are beautiful and functional are currently on the market. For best results the bamboo must be treated to prevent splitting and ensure adequate life. However, it has been used to construct inexpensive bicycle frames in regions of the world—such as many parts of Africa— where bamboo grows naturally and there is a need for inexpensive transportation.

Frame Manufacturing Processes Welding Most steel, aluminum, and titanium frames today are assembled using gastungsten arc welding (GTAW, formerly TIG), although gas-metal arc (GMAW or MIG) is used in higher-production environments. Both carbon steel and chromiummolybdenum steel are readily welded with either process. GTAW is preferred for low production levels, as it provides high-quality welds and good arc control. This is particularly important for welding thin-walled tubes. Disadvantages of GTAW are low filler material deposition rate, higher equipment cost, and greater skill required. For welding thin-wall (.9 mm (.035 inch)) 4130 tubing, Lincoln Electric recommends GTAW welding with 2% thoriated tungsten electrode, DC straight polarity (DC electrode negative) with 20 to 40 amps. The recommended filler rod is .9 mm (.035 inch) diameter ER80S-D2.2 ER70S-2 will also work, although the weld will be less strong and more ductile. Some welders prefer the more expensive 310, 312 or 309 stainless steel filler rod when welding 4130 steel. Prior to welding it is very important to mechanically clean the surfaces to be welded to remove all oxides and burrs, and then to wipe down with acetone to remove oils. Parts should fit closely together prior to welding. For best results, the gap between parts should not exceed .25 mm (.010 inch.). Electrodes used with GTAW welding are made of tungsten, either pure or alloyed with other elements. The addition of 1% or 2% thorium improves the electron emissivity, increasing the current carrying capacity of the electrode. Thoriated electrodes also produce a more stable arc and the arc is easier to start than pure tungsten. The

2

Lincoln Electric, TIG Weld 4130, AV-492 6/01.

97


tip retains its shape well, and is more resistant to contamination. Unfortunately, thorium is slightly radioactive, a concern when grinding the electrode tips. In the last few decades electrodes alloyed with cerium have begun to replace thoriated electrodes. Ceriated electrodes provide the benefits of thorium, but are not radioactive. They perform as well as, or better than, thoriated electrodes at lower current levels. Both ceriated and thoriated electrodes are generally used with DC welding, although ceriated electrodes will also provide good AC results. Lanthanum is also used as an alternative to thorium, and has good performance at low amperage. Pure tungsten electrodes are less expensive, but carry less current and have a low resistance to contamination. They are primarily used with AC welding, as pure tungsten forms and holds a ball at the end. Sometimes zirconium is used as an alloy agent. Zirconium electrodes have properties between thorium and pure tungsten, and are often used with AC welding. The electrode shape and size is also important. The size is determined by the required current; larger diameters can carry higher amperages. The shape depends on the amperage, polarity, and whether the current is AC or DC. Welding with the electrode negative (also known as DC straight polarity) is preferred, as it places the most heat at the workpiece rather than the electrode. In this case, the tip should be ground into a truncated cone, as shown in Figure 6-6. Welding manuals can provide specific angles and tip diameters for specific current ranges. In either AC or DC electrode positive welding, the tip should be hemispherical. Most materials, including steel, titanium and stainless steel, are welded with DC electrode negative. Aluminum and magnesium develop oxides that are quite stable, and have melting points well above the base metal. AC is used for these materials. The straight polarity portion of the wave provides good penetration and heat at the workpiece, while the electrode positive portion of the cycle provides good cleaning action that removes the oxide layer.

Figure 6-6 Electrode tip shapes for GTAW welding

98


Brazing Steel is also easily brazed. There are two types of brazed bicycle assembly methods—fillet brazing and brazed lugged construction. Many excellent, lightweight steel bicycle frames have been built using lugs, such as those produced by Henry James and Ceeway. Frame tubes are inserted into lugs at each joint, and the tubes are silver brazed to the lugs. This method permits very thin-wall, highstrength tubes to be joined without butting, and—if brazed by a competent frame builder—results in the strongest joints compared with welding or fillet brazing. In the past, many steel production bicycles used brazed lugged construction with stamped lugs. Today, this method is primarily used by low-volume manufacturers and custom frame builders, using investment-cast lugs. Some of the best handmade bicycles today use brazed lugged construction. Unfortunately, lugs are manufactured for particular tube diameters and frame angles, so it isprimarily used with traditional diamond frame bicycles. Fillet brazing joins tubes without lugs by building up a brazed fillet around the joint. Brass brazing rod is used. Fillet brazing can be used with carbon or chromiummolybdenum steel, does not require expensive welding equipment or lugs, and is applicable to all frame geometries. A skilled braze technician can achieve very high joint strength by building up a fillet of appropriate thickness. Although the brazing material is not as strong as 4130 steel, extra thickness can provide superior strength.3 Tests have shown that lugged, brazed joints can achieve the highest strength; fillet brazed joints next, and welded joints lowest strength. However, all three processes produce satisfactory frames. Silver brazing is frequently used to attached fittings and small parts to steel frames. Water bottle bosses, rack bosses and eyelets, pump hooks, and derailleur mounts are common examples. Silver brazing requires simple equipment.

Bonding Adhesive bonding can be used with virtually any frame material. For some materials, such as carbon composites or bamboo, it is the principle method of joining frame tubes. The Windcheetah tadpole trike is an example of a bonded aluminum frame. 3

While skilled technicians can achieve brazed joint strengths exceeding that of GTAW welded joints, novices may not be able to achieve these results. The author’s mechanical engineering students just learning to braze and weld typically achieve higher strength joints with GTAW welding than with fillet brazing. Joint cleanliness, heat control, and proper fillet size and shape are important factors affecting joint strength. Whether welding or brazing, maximum joint strength depends on the skill of the fabricator.

99


With aluminum, the bonding can speed the assembly process and avoid thermal distortion and subsequent straightening operations. Also, alloys that are not weldable can be used effectively in bonded structures. Other advantages of bonding include rapid assembly time and the ability to bond dissimilar materials. Bonding is used less frequently with steel vehicle due to the ease with which it can be welded and brazed. With any bonded joint, and particularly with aluminum members, diligent research and care should be used to select the correct adhesive and joint types. Improperly bonded joints can—and have—led to frame failures and accidents. The strength of bonded joints can equal or exceed the strength of the frame members, but this requires good joint design, selection of an appropriate adhesive, and high assembly quality control. Bonded joints are strongest when loaded in shear, compression, or tension, and weakest when loaded in peel or cleavage. Peel occurs when the loading tends to pull the adhesive apart from one end, much like peeling tape from a surface. Cleavage occurs when a force tends to separate the members starting from one end. Peel and cleavage should be avoided in bonded joint design. Socket joints are particularly useful for bonded frame construction. Lugs are made by casting or other processes, into which the frame tubes are inserted. Once the lugs are manufactured, the frame assembly is relatively quick and easy. In lug design, the diameter of the tube is more important than the length of the socket, since the shear stress is concentrated near the end of the joint. Diametral clearance should be small, and close tolerances may need to be maintained to achieve quality joints and to reduce un-necessary adhesive costs. The success of bonded joints depends on good joint design, proper surface preparation, appropriate adhesive selection, and good assembly and curing practice. The details of bonded joint design and adhesive selection are beyond the scope of this text, and in-depth research is recommended. There is a plethora of adhesive options available on the market, and selecting the optimal one can be a daunting task. Adhesives are available in a variety of forms, including both one and two part liquids, pastes, tapes and films. Check for compatibility with the materials to be bonded, cure time, strength, resistance to moisture and other environmental exposure, and service temperatures. Discussing specific requirements with a technical representative from an adhesive manufacturer is recommended. Be sure to follow recommended surface preparation treatments, which as a minimum involve careful cleaning. Aluminum and many polymers required additional preparation.

Other Frame Processes (Monocoque, etc.) Vehicle frames can be constructed with a variety of other methods as well. Frames have been cast whole, but this usually results excess weight and reduced

100


strength. More recently, composite frames have been molded as one entire part. This is sometimes referred to as monocoque construction, although the meaning is somewhat different from the more common usage of the term. Monocoque traditionally refers to a body in which the skin is a structural member. For humanpowered vehicles, this can mean a streamliner or velomobile in which the shell is an integral part of the structure. Monocoque vehicles can be lightweight and structurally efficient, if correctly designed and fabricated. They can be made of many materials, although reinforced-polymer composites are most often used. Aluminum is also occasionally used in monocoque construction, similar to aluminum aircraft.

Fairing or Shell Materials Velomobiles and streamliners fully enclose the rider within a shell, either for aerodynamics or weather protection, or both. Many other vehicles include devices that reduce aerodynamic drag, protect the rider, or provide other functionality, such as cargo space. Examples include tailboxes, partial front fairings, and combinations. These fairings, or shells, can be made of many different materials, with very different price/labor/performance characteristics. Reinforced-polymer composites, fabric, plastic, aluminum wood and foam have been successfully used. Reinforced—polymer composite materials are widely used for fairings, and for good reasons. These materials are previously described, so a brief list of advantages and disadvantages should be sufficient: Advantages of reinforced-polymer composite fairings include: • • • • • • •

Significant design freedom with respect to fairing shape Excellent aerodynamics are possible Very smooth surface finish is possible (but requires skill to achieve) Can be very lightweight, depending on material and design Good to excellent rigidity Good to excellent strength Aramid has excellent abrasion resistance

Disadvantages include: • • •

•

High-quality fairings require either extensive labor or expensive tooling Advanced composite materials can be expensive Poor quality control during fabrication can lead to structural weakness and failure Environmental emissions during manufacture may be significant

101


Sheet aluminum is used for monocoque velomobiles and sometimes for partial fairings and tailboxes. One notable advantage is that no mold is required, as it is with composites. Aluminum monocoque vehicles can provide strong and relatively lightweight fairings, although not as strong or light as the best reinforced polymer composite fairings. Stiffness can be quite good if properly designed. Construction usually involves forming the sheet intoshapes and riveting together, although bonding and welding are possible. Gussets and ribs may be used internally to improve stiffness and strength. Aerodynamic curved shapes can bemade by stretch forming or stamping, but both processes require expensive tooling.Shapes that require only straight bends and cylindrical or conical shapes are much easier to manufacture, and can usually be realized. These vehicles are structurally complete, and do not require a separate internal frame. They are also impervious to solar UV radiation, which can damage reinforced polymer composites. However, they do require sheet metal working tools, and without expensive tooling some shapes are not possible. Unreinforced plastics are used extensively for transparent windshields, canopies and partial front fairings, and less often for tailboxes and full shells. In general, plastics are less expensive, and have poorer mechanical properties than aluminum or composites. Rigidity and strength are significantly lower than for composites. Abrasion resistance is also low—much less than aramid or fiberglass. A plastic fairing is usually heavier and less rigid than other materials, but may cost less. Due to the low stiffness and strength, plastics are generally not used for monocoque construction—most plastic fairings are mounted to a traditional framed vehicle, usually a tricycle. Clear plastic is used where visibility is important. Many models of clear front fairings are available commercially, such as the models made by Zzipper for both recumbent and upright bicycles. Bubble canopies and velomobile windshields also use clear plastic, often thermoformed to obtainthe desired shape. Thermoforming, which can be used for either clear or colored plastic, involves heating a plastic sheet, draping it over a form with the desired shape, and using a vacuum to hold the plastic in place until it cools. For clear parts, the tool used must be smooth and highly polished to prevent visible tool marks in the finished product that could obstruct vision. An alternative method is to form the plastic without a tool by using heat and gravity. These products provide good clarity, but the shape is less accurate. It is also possible to produce full fairings by thermoforming. The fairing must be formed in several pieces that are subsequently joined together, as thermoforming cannot produce an enclosed shape. Thermoforming involves relatively modest tooling cost, but does require specialized equipment for forming. Many thermoforming vendors will not have equipment large enough to produce a full fairing. Another plastic process used for fairing production is rotational molding,

102


in which plastic pellets are melted in a closed, slowly rotating mold. A full fairing can be produced in one step with this process, although holes for windows, door, and wheels must be cut out. Rotational molding is used to make the shell of the Rotovelo velomobile. Neither thermoforming nor rotational molding is feasible for home builders or one-off prototypes. Corrugated plastic, such as that manufactured under the trade name Coroplast, has been used by home builders to make fairings that offer a good compromise in performance, cost, and fabrication labor. This material is often used for temporary signs, as it is inexpensive and relatively durable and stiff. The corrugations provide good stiffness at a relatively light weight, advantages for fairings. These products cannot achieve the design freedom available with composites, but can be heated and bent into useful shapes. Panels may be joined by tape, glue, or an internal support. Figure 6-7 shows a velomobile with a shell made of corrugated plastic. John Tetz popularized the use of foam for aerodynamic fairings. His articles posted on www.recumbents.com have inspired many home builders to fashion fairings from cross-linked polyethylene foam. Foam fairings are very lightweight, relatively inexpensive, fairly easy to fabricate, and are quiet and damage-tolerant. These fairings are not structural, so they are built around a traditional vehicle chassis, usually a tricycle. Builders have claimed weights for full foam fairings slightly over 3.2 kg (7 lb). Compared to quality composite fairings, foam is less rigid and has a poorer surface finish. Both factors affect the aerodynamic performance to some extent. While it is unlikely that a new world record will be set using a foam fairing, it can be a good material for everyday use. Fabric, wood, and other materials can also be used for fairings. Lightweight lycra fairings are made to attach to clear plasticfront fairings. These body socks often attach with hook-and-loop fasteners to the fairing, seat, and either a tailbox or specialpurpose frame on the rear of the vehicle. They can be removed and stowed in small

Figure 6-7 Example of a home built fairing built from corrugated plastic sheets

103


trunk bag if desired. The areodyamic advantage is less than for more rigid fairings, but still may be noticeable. They also provide a degree of weather protection. Other fabric fairings are stretched over a rigid frame, similar to fabric aircraft construction. When durability is not a concern, such as a single race event, model airplane covering such as Monokote has been used. This material is very lightweight, but cannot withstand the rigors of daily riding.Aircraft fabric such as Ceconite is much more durable, but requires a strong frame for mounting. Thin wooden strips or veneer can be formed into a fairing shape, often covered with fiberglass or carbon for added strength and rigidity. When skillfully crafted, these canbe beautiful works of art, as well as functional for weather protection and aerodynamics.

Fairing Hardware Regardless of the fairing material, some type of hardware is required to secure it to the vehicle frame and to allow hatches and doors to operate. Simple, lightweight hinges can be made of cloth or tape bonded to the door and frame. This results in an inexpensive hinge that can function well, but may not have the durability of more rugged hardware. In the event of damage, they can usually be replaced fairly easily. Similarly, inexpensive latches can be made from elastic cords or loops that attach to hooks. At the other extreme is aerospace hardware, including hinges and latches. These function well and can have good durability, but are quite expensive and not readily available. Fairing panels and components can be attached to the frame by screws, quarter-turn fasteners, hook-and-loop fasteners and other devices. Dzus makes very rugged quarter-turn fasteners that can be excellent for attaching panels that must be removable for service or access. They have excellent resistance to vibration, but the fit must be properly engineered for the thickness of the panel and frame.

Summary Selection of materials and processes for human-powered vehicle frames, shells, and components is an important task, affecting vehicle production cost, performance, quality, and customer appeal. No single material is best—each as merits and challenges. Compatibility of process and material is essential, and the choice should be consistent with the mission of the vehicle. The information in this chapter is just a starting point for design decision related to materials and processes. Of necessity, much has been omitted. However, it should provide the basis for preliminary design decisions and a springboard to more detailed knowledge of manufacturing process and materials.

104

CHAPTER

ROAD LOADS

7

R

oad loads are the external forces acting on a vehicle during operation. They arise from gravity, the interaction between tires and the road surface, and from air flowing past the moving vehicle. Understanding them is essential for the vehicle designer, as they affect almost all aspects of the design, from vehicle dynamics and handling to power-speed models and structural analyses. Figure 7-1 illustrates road loads acting on a bicycle riding in a straight line on level ground. Aerodynamic drag, rolling resistance of the tires, traction forces and braking forces all act parallel to the longitudinal axis of the bicycle. For a bicycle traveling at a constant speed on a level road, the sum of the drag and rolling resistance will exactly equal the tractive force produced by the driving wheel. If the road is not level, a component of the weight will also act in the longitudinal direction, pushing the bike downhill or restraining it during a climb.

Review of Equilibrium Equations Frequently throughout this book equilibrium equations will be solved to determine unknown reaction forces and moments. These equations stem from Newton’s second law, which states that the sum of forces acting on a body is equal to the mass of the body times its acceleration, or F = ma

∑ In a Cartesian coordinate system, this equation can be applied in each of the coordinate directions x, y, and z. For static equilibrium, the acceleration, and hence the right-hand side, is equal to zero. Quasi-static analyses treat the negative of the right-hand side (– ma) as if it were an externally applied force. This quantity is often called the inertial force. Quasi-static analyses are quite useful for many problems related to human-powered vehicles.

105


Figure 7-1 Forces acting on a bicycle during level riding at constant speed (arrows

are not proportional to magnitude)

Newton’s second law is valid for moments as well as forces provided the righthand side is replaced with the product of the mass moment of inertia and the angular acceleration. Hence:

∑M = Iα As with forces, this equation is valid for moments about each coordinate axis, giving three more equations. There are a total of six equilibrium equations for three-dimensional systems, and hence up to six unknown forces or moments can be found. When all forces lie within a common plane, three of the six equations are eliminated, leaving two force equations and one moment equation. The six equilibrium equations are:

∑F ∑F

x

max =

y

ma =

∑F

maz =

z

M I ∑ M I ∑ x

y

xxx

=

y

=

yy α y

=

α

M I ∑ z

zzz

α

SAE Vehicle Coordinate System for Vehicle Dynamics Figure 7-2 depicts the vehicle coordinate system defined per SAE J670e, which is used throughout this book for all topics related to vehicle dynamics (with the exception of Patterson’s method for predicting bicycle handling). The vehicle

106

Road Loads

Figure 7-2 Vehicle coordinate system (SAE Convention)

coordinate system is located at the center of gravity with the x-axis directed forward, y-axis to the driver’s right, and the z-axis downward. These correspond to the longitudinal, lateral, and vertical directions respectively. The corresponding rotations about these axes are termed roll, pitch, and yaw. Rotation about the x-axis, such as a lean to either right or left, is known as roll. Yaw is a rotation about the vertical z-axis, for example from a heading of east to one of north. Pitch is rotation about the lateral y-axis. Vehicles with suspension systems may pitch forward during heavy braking or when riding over a bump.

Static Loads on Level Ground For the simple case of a stationary vehicle on level ground, the total weight of the vehicle, including the rider, and the vertical forces at each wheel are the parameters of interest. The static weight fraction is ratio of weight on one wheel to that of the entire loaded vehicle. Weight fraction affects handling and stability, and frequently crops up in many analyses, so it is good to understand the concept and how to calculate it. Gravity acts on the entire vehicle, but for analytical purposes may be modeled as a point force acting through the center of gravity of the vehicle. The center of gravity is located at a location b forward of the rear axle and height h above the ground. Referring to Figure 7-3, it is seen that the total vehicle weight W is reacted by vertical forces at each wheel. The loads on each wheel can be found using Newton’s second law. First, the load on the front wheel is found by summing moments about the contact patch of the rear wheel. Since the vehicle is stationary, there are no accelerations and the right-hand side is zero.

107


Figure 7-3 Static loads on a level, stationary vehicle

Wb = 0 – Wf L

(7-1)

Given the total vehicle weight, the wheelbase, and the center of gravity location, the unknown force on the front wheel can be found

Wf

=

Wb L

=

mgb L

(7-2)

With the vertical load on the front wheel known, the load on the rear wheel can be found either by summing moments about the front tire patch or by summing forces in the vertical (z) direction.

Wr

W (L b) −

=

L

( L) b mg

=

−

=−

L

(W

Wf )

(7-3)

As noted, the weight fractions are obtained by dividing each wheel load by the total vehicle weight. Usually, the weight fraction is specified as the front axle weight fraction or rear axle weight fraction, and may be expressed as a percentage. The front and rear weight fractions are given by:

108

fR

=

fF

=

WR W WF W

×

100%

×

100%

(7-4)

Road Loads

On level ground, the wheel loads WF and WR are always directed upward, opposing gravity. In this case, they are perpendicular to the road, and are also known as the normal forces on the tires. By definition, WF and WR are taken to be normal to the road surface, or acting in the z-direction relative to the vehicle coordinate system. In this case, the sum of the wheel loads is less than the total weight of the vehicle, as will be shown.

Example 7-1 A bicycle with a 104 cm wheelbase is weighed with a rider by placing scales under each wheel. The rear wheel scale reads 491 N, while the scale under the front wheel reads 415 N. Find the total weight of the bicycle and rider and determine the horizontal location of the center of gravity. Solution: The total weight is found by W = WR + WF = 491 + 415 = 906 N. The horizontal location of the center of gravity can be found solving Equation 7-2 for b:

b

=

WF L

415 × 1.04 =

W

=

.476 m

906

The center of gravity is 0.476 meters forward of the rear wheel axle. The front and rear static weight fractions are useful indicators of the static weight distribution between the front and rear axles. The weight fractions for this example are given by: WR

fR

=

fF

=

W

491 =

WF W

906

=

415 =

906

=

.542

.458

This shows that 54% of the weight of the vehicle is on the rear axle, and 46% is on the front axle.

109


Static Loads on a Grade The weight fraction shifts when a vehicle is sitting on a slope. The analysis is similar except that longitudinal as well as vertical loads occur at the wheels. The vertical wheel forces are reduced and the horizontal force is produced by brakes or by applying pressure to the pedals so as to prevent rolling backwards. Slopes on roadways are usually specified as percent grade. Thus a road that rises 50 m over 1 km has a moderately steep grade of 0.05 or 5%. The inclination angle of the road is the arctangent of the grade.

q = tan–1(G)

(7-5)

Typically, road grades are fairly small. Grades exceeding 12% are rare, even in mountainous regions. For small grades, the inclination angle and its tangent are approximately equal, and with little loss of accuracy one may assume q = G (where q is measured in radians). This simplification incurs less than 0.5% error for a 12% grade and only 1.3% error for an extremely steep 20% grade. The forces are shown in Figure 7-4. The longitudinal force shown on the rear wheel could occur at either wheel, depending on which brakes are applied. This analysis is not affected by which wheel is used, although it may be important in other contexts. The wheel loads Wf and Wr are found by summing moments about the rear tire patch. (W cos(q))b – (W sin(q))h = Wf L

Figure 7-4 Static loads on a grade

110

Road Loads

This equation can be solved for the front wheel load Wf as before. Likewise, the reaction at the rear wheel patch can be found by summing forces about the front tire patch. The resulting equations are:

Wf =

W (bc os( θ) − hsin( θ )) L

((b − Wr = W L

(7.6)

h )cos( θ + ) θ sin( )) L

Equations 7-6 are the general equation for calculating vertical loads on the front and rear axles. If the grade is zero, they reduce to Equations 7-2 and 7-3. However, they can be simplified by using small angle approximations sin( q) @ q @ G, (with the angle expressed in radians), and cos( q) @ 1.

Wf Wr

=

=

W (b

−hG

)

L W (( L b − ) hG + )

(7.7)

L

Once the front and rear wheel loads are known, the weight fraction can be calculated using Equation 7-4. Going up a hill increases the load on the rear wheel and reduces the load on the front wheel. Conversely, going down a hill increases the load on the front wheel. Vehicles with high centers of gravity will experience more of a weight shift. An upright road bike climbing a 10% grade may experience a 20% to 25% decrease in the front wheel load. (This assumes the rider stays in the saddle. Standing on the pedals can shift the CG forward significantly during hill climbing.) A low-racer on the same grade will probably experience less than 10% decrease in front wheel load compared with that on level ground. Figure 7-5 shows the fractional reduction in front axle load as a function of grade and the dimensionless ratio of the height to position from rear tire patch of the vehicle center of gravity (h/b). Typical values of h/b for upright bicycles, recumbent, and very low, laid-back recumbents are indicated. When riding up a steep hill with poor traction conditions, there is a possibility of losing steering control due to the loss of load on the front wheel. This is particularly true with vehicles that have lightly loaded front axles, such as many long-wheelbase recumbents. Long-wheelbase delta tricycles that drive a single rear wheel are likely candidates for this problem, due to the low front weight

111


FRACTIONAL CHANGE IN FRONT TIRE LOAD WITH GRADE

2.5

-0 0 .1

/b h io t 2 a r

-0

0 .0 5

n io t a c o l 1.5 to t h g i e h G O 1

-0

-0

-0

.1

-0

.2 5 -0

.2

-0

5

-0

.3

-0

5

.3

.2

-0 .3 5

RECUMBENT -0 .1 5

.0 5

-0 . 2 -0 .1

4

.3

.2

-0 .0 5 2

UPRIGHT

5

-0 .1

0.5 0

5

.4

-0

-0

.1

.4

6

8

10

LOW RACER 5

-0 .1 12

14

16

18

20

GRADE, %

Figure 7-5 Fractional change in front tire load as function of grade and h/b ratio

fraction coupled with the turning moment produced by the single drive wheel under heavy torque conditions. More severe problems arise when riding downhill. Braking also shifts the vertical load to the front wheel. Braking on a downhill slope can reduce the load on the rear wheel to zero, potentially resulting in a dangerous accident with the rider pitching over the handlebars. A high center of gravity exacerbates this problem, which is more common with upright bicycles than recumbents.

Steady Motion Road Loads A vehicle moving in a straight line at constant velocity is said to have steady motion. That is, all vehicle acceleration terms are zero. There are four groups of forces acting on a vehicle in steady motion—propulsive forces, retarding forces, weight and normal forces. These forces are shown in Figure 7-1. Propulsive forces provide the forward thrust on the vehicle. For vehicles driven by wheels, this is a tractive force, which acts at the tire patch of each driving wheel. Retarding forces include all forces resisting forward motion of the vehicle, such as aerodynamic drag, rolling resistance, and braking forces. Weight can be considered to act downward through the center of gravity of the vehicle. The normal reaction forces act through the tire contact patches in a direction normal to the road surface. As any cyclist has experienced, if the vehicle is moving up or down a grade, a component of the weight becomes a significant retarding or propulsive force.

112

Road Loads

For a vehicle traveling at constant speed, the propulsive forces must equal the retarding forces. The tractive force is a function of the driving wheel torque T and drive wheel rolling diameter d, and is given by:

Fx

=

2T

(7-8)

d

At typical cruising speeds for most vehicles, aerodynamic drag is the most important retarding force. For example, a 70-kg man riding an unfaired long-wheel base recumbent bicycle over asphalt would expend a total of 158 Watts in order to maintain 8.4 m/s on level ground without wind. He would expend 117 Watts to overcome aerodynamic drag, but only 35 Watts to overcome rolling resistance. If our rider switched to an upright touring bike, the power required to overcome drag would increase to 165 Watts due to the greater drag force. The drag force depends strongly on the speed of the vehicle, increasing with the square of the speed. It also depends on the shape and size of the vehicle. ρ

FAERO = CAV D 2

2

(7-9)

Where ρ is air density, A is the frontal or surface area of the vehicle, and CD is known as the drag coefficient. The drag coefficient takes into account the shape and surface texture of the vehicle. V is the air velocity, which is equal to the vehicle speed if there is no wind. The drag force acts through the aerodynamic center of pressure which is not easily computed. Aerodynamic forces can also produce vertical forces, which can be significant at high speeds. If the vertical force is directed up, it is termed lift; otherwise it is called a downforce. Drag and drag coefficients are discussed in detail in Chapter 9. There is some resistance to motion as tires roll over pavement. On very smooth surfaces, tires deform to support the load, creating a flat patch of contact between the tire and the flat road. Tires also deform as they roll over irregularities in the road surface. In addition, if the surface is soft, such as sand or soft dirt, the surface itself is deformed. The energy dissipated in these deformations gives rise to a retarding force known as rolling resistance. Rolling resistance is directly proportional to the vertical load on the wheel, but also depends on wheel diameter, tire construction and tread, inflation pressure, and the roughness and properties of the road surface. Larger diameter wheels can roll over surface irregularities easier, and thus have lower rolling resistance. Thin-walled tires with slick tread have significantly less rolling resistance than do knobby tires designed for off-road

113


traction. The tread, particularly knobby tread patterns, deforms significantly, increasing the hysteretic energy losses and rolling resistance. Rolling resistance is given by:

FRR = WCRR

(7-10)

Where W is the total weight of the vehicle and CRR is the coefficient of rolling resistance. It can range from less than .002 for high-pressure racing tires on smooth track surfaces to almost .02 for wide, low-pressure tires on rough roads. Typical road bike tires on typical asphalt may have a CRR value around .005 or so. Equation 7-10 is applicable to either a single wheel or the entire vehicle. If all wheels are the same size, use the same tires, and are inflated to the same pressure, CRR will be the same for all wheels. In that case, if W is the total vehicle weight, the calculated force applies to the whole vehicle; if it is the vertical load on one wheel, the force applies to that wheel only. For vehicles with different size wheels or different tires, the rolling resistance coefficient CRR will vary between wheels. In that case, the rolling resistance force is the sum of the product of the vertical force on a wheel with the coefficient of rolling resistance for that wheel.

FRR =

∑F

C

zi RRi ,

(7-11)

,

i

In practice Equation 7-10 can be used for a vehicle with differing tire sizes/ types as long as CRR is determined for the entire vehicle. Generally, Equation 7-11 is only used if it is known that the coefficients of rolling resistance vary significantly between wheels. Braking forces FB are retarding forces that act at the tire contact patch in a direction parallel with the road surface. As expected, they can be considerably greater in magnitude than any of the other retarding forces. Braking forces are treated in more detail later in this chapter. Problems involving steady-motion road loads frequently involve determination of the tractive force required to maintain steady speed on level roads or climbing hills. For steep descents, the braking force required to maintain constant speed can also be found. Usually, a cyclist is not braking and pedaling simultaneously, so either Fx or FB is taken to be zero. For vehicles with steady motion on level ground, the equilibrium equation for forces in the longitudinal direction is

Fx – FAERO – F=RR0– FB

114

(7-12)

Road Loads

On a grade, an additional term isrequired in Equation 7-12 to account for thegravitational force pushing the vehicle downhill. The complete term ismg sin(tan–1 G), but the small angle approximationmgG again incurs less than a 2% error at a grade of 20%. Thus, for steady motion up or down a hill Equation 7-12 becomes:

Fx – FAERO – FRR – FB –=mgG 0

(7-13)

Equation 7-13 describes the longitudinal forces acting on a vehicle traveling at constant speed. It is an important equation for modeling vehicle loads acting on a vehicle, and is the basis for the power-speed models developed in Chapter 8. In conjunction with Equations 7-6 both longitudinal and lateral loads can be analyzed. Lateral forces are developed during cornering, and are the subject of the next section.

Basic Loads in a Steady Turn Lateral forces act on a vehicle to produce and maintain a turn. In accordance with Newton’s first law, a body moving in a curvilinear path must be acted on by an external force. For a turning vehicle, the tires sustain cornering forces that cause the vehicle to turn. These forces depend on tire properties and angles, and are discussed in detail in Chapter 11. A steady turn is one in which neither the vehicle speed nor the turn radius changes. During a steady turn the tire forces produce the lateral acceleration required for curvilinear motion. This force is directed toward the inside of the turn, and is opposed by centrifugal force, which is just the inertial force –may. The lateral acceleration ay is related to the vehicle speed and turn radius by: ay

V2 =

(7-14)

R

Where V is the speed of the vehicle and R is the turn radius, measured from the turn center to the center of gravity of the vehicle. The total side load on the wheels is given by Newton’s second law as Fy = may. This load is shared by the front and rear axles in proportion to the normal load, or weight, on each axle, or

 ay V2  A  W = f W   gR    g V2   ay = r  W = r W FyrW A  gR    g

= Fyf W

f

 =  =

f

y

(7-15) r

y

115


Figure 7-6 Lateral forces on a turning vehicle

Equations 7-15 describe the lateral loads acting on a vehicle. For convenience, they are expressed in three different forms. These equations complete the set of fundamental equilibrium equations applied to a vehicle in steady motion. Equations 7-6, 7-13, and 7-15 describe the forces on such a vehicle in the Z, X, and Y coordinate dimensions respectively. In a steady turn, a rolling moment is produced that tends to overturn the vehicle. The moment is produced because Fyf and Fyr act in the ground plane, but the countering centrifugal force may acts through the center of gravity of the vehicle, located at a height h above the ground. The rolling moment is the product of the total lateral force and the CG height, Fyh. A tricycle or quadricycle resists this moment by an increased vertical load on the outside wheels, known as lateral load transfer. A bicycle must lean into the turn to reduce or eliminate this moment. The lean angle that eliminates this moment is found by summing moments about the longitudinal axis of the vehicle.

V2  θ = tan −1    gR 

116

(7-16)

Road Loads

A rider that maintains this lean angle will experience no rolling moment. If she does not maintain this angle, she will have to shift her weight to prevent rolling and a potential accident. When the ideal lean angle is realized, the force on the wheels is purely radial. This is very good for bicycle wheels, which are much stronger in the radial direction than in the lateral direction. Note, however, that this does not reduce or eliminate the side load on the bicycle as a whole. There is always a side load on a turning vehicle. The dynamics of a turning vehicle are much more complex than the preceding discussion might indicate. Cornering performance depends on the reaction between the tire and the road surface, tire slip angles, steering linkage construction, and other factors. Cornering is discussed in more detail in Chapter 11.

Acceleration and Braking Acceleration and braking both result in changes in the forward velocity of the vehicle. Equation 7-13 is no longer valid, since the right-hand term is no longer zero, but max. Acceleration performance for human-powered vehicles is limited either by available power or by low traction conditions. Usually, available power is the limiting factor. Only on low traction surfaces, such as ice or loose gravel, is traction-limited acceleration a problem. Brakes on human-powered vehicles are almost invariably much more powerful than the human rider. A vehicle usually can stop from a given speed in a much shorter distance than it can accelerate up to that speed. Like acceleration, braking performance has two possible limiting conditions: slipping on a low-traction surface or forward pitch-over. The acceleration/braking analysis is quasi-static, meaning that we can use Equation 7-13 but must now include an inertial term max on the right-hand side.

Fx – FAERO – FRR – FB – mgG = max

(7-17)

The inertial term is the product of total vehicle mass m and longitudinal acceleration ax. Equation 7-17 is valid for both acceleration and braking—the acceleration term is positive if the vehicle speed is increasing and negative if the speed is decreasing. During acceleration, the braking force FB is usually zero and the traction force Fx is limited either by the power of the rider or the available traction. During braking, the traction force is generally zero. Under most conditions, the braking force can be much larger than the maximum tractive effort created by rider power. A strong rider starting from a full stop and pedaling hard might achieve

117


5 m/s2 (about 0.5 G’s) peak acceleration, but this value could not be sustained more than a second or so. Braking on a hard, dry road surface can easily achieve decelerations of 0.8 G’s or higher, and high braking levels can be maintained as long as necessary to stop.

Power-Limited Acceleration

This chapter primarily treats forces. Power is the topic of Chapter 8. However, in considering acceleration, power must be included as it is a limiting factor. The drive power is the power that pushes the vehicle forward—the power at the drive wheel contact patch. It is denoted by P¢ and is related to the trac tion force Fx and the vehicle velocity by:

Fx

=

P′

(7-18)

V

(The drive power is less than the total power supplied by the operator(s) because of mechanical losses in the drivetrain, as will be described in Chapter 8.) Substituting Equation 7-18 into 7-17 and solving for the acceleration illustrates the relationship between acceleration, drive power, and the longitudinal road loads already discussed.

ax

=

1 P ′ − F (AERO + F + m  V

mgG

RR

 

)

(7-19)

Acceleration is frequently expressed terms of G’s, where one G is equal to the acceleration of gravity. This provides a more intuitive measure of acceleration in terms of our normal experience in Earth’s gravitational field. For example, a bicycle and rider weigh 750 Newtons and accelerate at 1.96 m/s2. It may be rather difficult to have an intuitive feel for this number. To obtain acceleration inG’s, 2 divide a by the acceleration due to gravity (9.81 m/s or 32.2 ft/s2). In this example, =

ax

1.96 =

=

´

0.2

Ax g 9.81 G. The inertial forcemax is then easily obtained by 750 0.2 = 150 Newtons. In this text, lower case a is used for acceleration in units of length per time squared, and upper case A is used for acceleration in dimensionless G’s. Equation 7-19 can be converted by dividing both sides by g.

Ax =

118

1 P′ − (+AERO F+ mg  V

F

RR

 

mgG )

(7-20)

Road Loads

Either equation shows that acceleration depends on the available drive power in excess of that required to overcome drag, rolling resistance and hill climbing. If

P′ V

is greater than the sum of the longitudinal forces, power is available to ac-

celerate the vehicle.

Traction-Limited Acceleration In low-traction conditions such as icy roads, acceleration performance may be limited by traction rather than power. In this case, the drive wheel slips, producing little or no drive power. This is not common, but does occur with smooth tires on snow, mud, and even wet grass, particularly if the drive wheel is lightly loaded. It usually occurs during heavy pedal forces, and the resulting sudden spin of the pedals can be painful. However, under normal conditions human-powered vehicles do not have enough power to reach the traction limit. If μ is the coefficient of friction between the tire and the road, the maximum traction force is given by

Fx,max = µWDW

(7-21)

Where WDW is the vertical load on the driving wheel. (For rear-wheel drive vehicles, WDW = Wr, and is given by Equation 7-13. For front-wheel drive vehicles WDW = Wf. Some multitrack vehicles with a driving wheel on only one side will have a different value.) The maximum acceleration under traction-limiting conditions is then found by replacing the power term in Equation 7-20 with μ WDW. The result, expressed in G’s, is given by Equation 7-22.

AxMAX ,

1

=µ

mg

−DW  W

+AERO (F +

RR

F

mgG )

(7-22)

On level ground, the grade G is zero, the last term is not applicable. Large accelerations for human-powered vehicles typically occur at low speeds, where aerodynamic drag is of little consequence. If rolling resistance is also neglected, it is seen that the acceleration limit in G’s is numerically equal to the coefficient of friction.

Ax,max = m

(7-23)

Computationally, Equations 7-20 and 7-22 may be more convenient if multiplied through by the vehicle weight. The rolling resistance and grade terms are then simplified. Doing so results in the following equations for maximum acceleration.

119


Power Limited:

Traction Limited:

AMAX x, A MAX x,

 P ′ =   −  mgV   µW  =  DW−  mg 

F  + AERO + RR C mg  F  + AERO + RR C mg 

G (7-24) G

It is worth noting that loss of traction can also occur while climbing on wet or loose surfaces. Upright bicycles ascending steep grades on wet roads or loose surfaces are particularly prone to slipping when the rider stands on the pedals. Standing shifts weight to the front wheel, reducing rear wheel traction. Pulling a trailer up a grade with poor traction conditions also exacerbates the problem. The traction force must be sufficient to pull the trailer, but the trailer weight does not bear on the drive wheel. Recumbent vehicles are generally somewhat less likely to lose traction under power, but can still do so when conditions are poor.

Example 7-2 A 70 kg cyclist attempts to ride a 12 kg upright bicycle towing a 25 kg trailer up an 8% grade at 2 m/s. The combined bike/trailer rolling resistance coefficient is 0.01, and the trailer load is balanced such that the tongue weight is negligible. A light rain has fallen on an oily road surface, reducing the friction coefficient to .18. The bicycle wheelbase is 1 m. The cyclist stands on the pedals, shifting the center of gravity to 65 cm forward of the rear axle and 110 cm above the ground. What is the maximum possible acceleration? Seated, the center of gravity is 41 cm forward of the rear axle and 99 cm above the ground. What is the maximum possible acceleration when seated? Solution: At 2 m/s, the drag force can be neglected. To determine the weight on the rear wheel (the drive wheel) use Equation 7-7. However, the mass used in this calculation should not include the trailer mass. Since the tongue weight is negligible, all the trailer weight is supported by the trailer wheels. (However, the trailer weight does contribute to the rolling resistance and gravity terms.) Substituting Equation 7-7 into the first term in Equation 7-24 gives

µ µ=

120

Wr mg

L( b −+hG ) L

= 

  

−(1.0 + ×.65) 1.10 .08 .18  = 1.0 

  .0788 

Road Loads

The maximum acceleration is then given, in G’s, by Equation 7-24 as–.011 G:

Ax = .0788 – .01 – .08 = –.011 The negative sign indicates that the bicycle is slowing down. Even if rolling resistance is omitted, the bicycle would continue slowing down, eventually coming to a stop. The rider will be unable to climb to the top of the hill. If the rider sits on the saddle, however, the center of gravity shifts aft and down. The maximum acceleration is then

Ax = .1205 – .01 – .08 = .03 In this case, the acceleration is positive, albeit small. The rider will be able to continue up the hill. If the cyclist used panniers rather than a trailer, the cargo weight would have increased the load on the rear wheel and made climbing the slick hill more feasible.

Inertia Coefficient The expressions for maximum acceleration are useful for estimating acceleration limits. However, for accurate calculation of acceleration under a given set of conditions the inertial effects of wheels and moving drive train components must be included. This will become important in the power-speed models described in Chapter 8. During acceleration, a certain amount of energy is required to accelerate the wheels, chain, and crankset. This energy is stored as kinetic energy in these components, but this effect was not included in Equation 7-24. The actual acceleration obtained with a given supply power will be slightly less than predicted due to this effect. For the greatest accuracy, the inertia of components should be included the analysis. The wheels have the largest mass moment of inertia of all the drivetrain components, and hence contribute most to the acceleration capabilities of the bicycle. To include the contribution of the wheels, sum torques on the driving wheel:

Tc = Fxr + Ia

(7-25)

121


The parameters I, α, and r are the mass moment of inertia, the angular acceleration, and the rolling radius of the drive wheel. This equation can be converted to a power equation by multiplying through by the angular velocity of the driving wheel

Tcw = Fxrw + Iαw

Noting that rw = V, α

=

ax

and

ω =

r

PDW =

FxrV + r

I

V r

this expression becomes

axV = + 2 r

IaxV

P′

r

2

(7-26)

where PDW is the power supplied to the drive wheel and r is the driving wheel radius. The non-driven wheels must also be accounted for. If they are the same size as the drive wheel, the moments of inertia can simply be added and Equation 7-26 can be used as-is. However, if they are different sizes, the relationship between ax and a is different, and must be accounted for separately. The total supplied power then becomes PS

= P ′ + 

I R xa V 2

rR

+F xI

a 2

rF

V

 

(7-27)

The power to accelerate the crankset is Icacwc where the subscripts indicate the crankset inertia, angular acceleration, and angular velocity, respectively. The angular velocity and acceleration at the crank is related to those at the drivewheel through the velocity ratio, which is the ratio of teeth on the wheel sprocket to those on the chainring. Hence ωC = ωDW

αC = α DW

NW NC NW NC

Where the subscript DW refers to the drive wheel, and NW and NC are the number of teeth on the freewheel and crank sprocket, respectively. The power to accelerate the crankset is thus IC

122

 axV NW  r 2  N  DW  C 

2

Road Loads

Adding this term to Equation 7-27 give the total power supplied by the operator to accelerate from initial speed V with an acceleration ax.

PS

 = +P ′ 

 I R I F + 2 + 2  rR F r

2   1  NW   axV   r 2  N  DW   C  

IC

(7-28)

The terms in the curly braces depend only on the vehicle parameters, and not on operational or environmental conditions. A single value can be computed that remains constant for a given vehicle. It is convenient do so, and the resulting parameter is known as the inertia coefficient. The inertia coefficient CI is defined for convenience to make acceleration problems easier to write and solve. Specifically, the inertia coefficient is defined as

CI

I I I N  =  R2 +F 2 C+  W2  r r r DW    R F

  NC

2

  

(7-29)

Substituting CI into Equation 7-28 gives a simplified form.

PS = [P¢ + CIaxV]

(7-30)

The inertia coefficient CI provides an insight into the effects of rotational inertia on vehicle acceleration. It has units of mass, which makes direct comparisons to the vehicle mass possible. For example, a bicycle and rider have a mass of 80 kg and CI of 4 kg. This means that the effect of the inertia of the wheels and drivetrain on acceleration performance is equal to that of adding 4 kg, or 5%, to the mass of the bicycle. For maximum acceleration performance, reducing weight is important, but reducing rotating weight will have a greater benefit. The inertia coefficient definition may be changed to include more or less detail without affecting Equation 7-30. For example, the crank inertia can be neglected by omitting the third term in Equation 7-29. Additional accuracy, particularly for vehicles with long chains, can be obtained by including the inertia of the chain. However, this is generally a fairly complicated calculation. The inertia coefficient is used in the power models developed in Chapter 8. It should be noted that actual acceleration performance is affected strongly by the ability to effectively supply power to the drive wheel. The relationship between pedal force, cadence, acceleration, and speed is quite important. Humans pedaling leg cranks have a fairly narrow cadence range over which maximum power can be

123


attained. The curve of power with respect to cadence is shaped like an inverted U, with a peak in the range of 90 or more RPM. Human-Powered vehicles are equipped with a range of gears to enable riders to maintain optimal cadence over a wide speed range. High accelerations necessarily imply rapid speed changes, so shifts must be made during acceleration to maintain an acceptable gear. However, shifting usually requires an interruption of power to the drive wheels, and thus adversely affects acceleration performance. (Modern derailleurs and cassettes are capable of shifting under full load, so in some cases the interrupted power flow is minimized.) Acceleration performance is thus strongly influenced by the quality of the shift system and the spacing of gears. This is addressed more fully in Chapter 12.

Load Transfer during Acceleration and Braking The vertical load on each axle changes when the vehicle accelerates or decelerates. The change in axle load is known as longitudinal load transfer. Since the total weight of the vehicle remains constant, the sum of the axle loads is also constant, but the weight fraction on each axle depends on the acceleration and the height of the center of gravity. During hard braking, load is transferred from the rear axle to the front axle, whereas during acceleration, load is transferred to the rear axle. The load shift is due primarily to the moment produced by the inertial force –max acting through the center of gravity. The negative sign indicates that this “force” is directed aft (negative x-direction) for a positive acceleration. For braking, the acceleration is negative. The loads on each wheel can be found by summing moments about the front and rear tire patches, as was done for the acceleration limits. In that case, the longitudinal forces and the limiting longitudinal accelerations were of interest. Here, the vertical wheel loads are of interest. The applicable free-body diagram is shown in Figure 7-7. Summing moments about the rear wheel contact patch and then about the front wheel contact patch and solving each for the unknown wheel force gives the equations for the front and rear wheel loads during acceleration. Note that dimensionless acceleration is used in these equations.

The term 



W L

W

Wf

=

Wr

=

L W L

b ( Gh − Ah −

x

(L(−b+G +)h

Ah

)

x

)

(7-31)

  Ax h is the load that is shifted due to acceleration. For the same 

acceleration, long, low vehicles will experience less load shift than high short

124

Road Loads

Figure 7-7 Free body diagram used to find the longitudinal load shift during braking

vehicles. Low racers incur relatively little load shift, while upright bicycles may experience a significant amount. Equations 7-31 are valid for both positive and negative accelerations (increasing and decreasing speed, respectively). As noted, human-powered vehicles generally are not capable of achieving high positive accelerations under most conditions. Braking does produce large negative accelerations, and the load shift from rear to front axle may be very significant for vehicles with high or forward centers of gravity. In extreme cases, this can lead to vehicle pitchover—the vehicle flips forward, usually sending the rider over the handlebars in a dangerous accident. Upright bicycles and tadpole tricycles tend to be more prone to braking pitchover. Tadpole trikes often have a center of gravity closer to the front axle than the rear (large b) while upright bikes have a high cg (large h). To be strictly complete, the drive torque reaction acting on the vehicle frame also contributes somewhat to the load shift. Under heavy load, there can be a significant torque at the rear wheel. This is particularly true at very low speed when the operator is exerting a large force on the pedals. The frame reacts against the wheel torque, which also unloads the front wheel. This is a small effect, and may generally be neglected.

Braking Peak braking forces on human-powered vehicles can be much larger than peak acceleration forces from hard pedaling. The ability to stop in a short distance

125


Example 7-3 A 77 kg cyclist rides a 14 kg long wheelbase recumbent bicycle. The wheelbase is 1.6 m, while the center of gravity position and height are 56 cm and 75 cm respectively. Compare the front wheel load fraction for riding at constant speed on level ground with that for 0.5 G braking on a 6% down grade. Solution: For riding on level ground, the front wheel load is

Wf

=

mgb L

=

(77 + 14) ×9.91 ×.56 1.60

=

312 N

The corresponding weight fraction is

ff

=

Wf W

312 =

893

=

.35

This is a typical value for long wheelbase vehicles, which tend to have lightly loaded front wheels. For braking on a 6% downslope, the front wheel load and the corresponding front wheel weight fraction is: W Wf = h Gh b A L ff

=

Wf W

(− =

−

547 893

=

=

x

)

893 ×(.56 + × .05 = .75 N .6 .75) 547 1.60

+

0.61

Note that both the grade term and the deceleration term are positive, unlike the first Equation 7-31. This is because the grade is negative (downhill) and the acceleration in negative (slowing). The increased load on the front axle is rather significant in this example, although the rear wheel is not fully unloaded. It is instructive to look at the braking and down hill components separately, to see which has the greater effect. A total of 234 Newtons was transferred to the front axle, of which 25 N were due to the down grade and 209 N were due to braking. This illustrates how important braking can be on vehicle loads.

126

Road Loads

while maintaining control is an important safety attribute of any vehicle. Good braking performance is particularly important when riding in traffic in order to avoid collisions. Even with good brakes, most upright bicycles require a longer stopping distance than a typical automobile with the same initial speed. Stopping power for recumbent bicycles is often superior to a comparable upright bike, due to the lower CG height and the greater distance between the CG and the front axle. For either type, the dangers of inadequate stopping distance must still be addressed during the design stage. Like acceleration, the maximum braking force is limited by either loss of traction or by shifting so much weight to the front wheel that the rear wheel is lifted off the ground. The first case results in reduced braking performance and possible loss of control, either of which can lead to an accident. In good traction conditions, the forward weight shift may cause the rider to be thrown over the handlebars, potentially sustaining serious head or neck injuries. The longitudinal load shift from the rear wheel to the front wheel always occurs during braking. The pitchover threshold, at which the vertical force on the rear wheel just goes to zero, limits braking when traction is good. This can be found from Equation 7-31 by setting the rear wheel load to zero and solving for the acceleration. Assuming level ground, the term Gh may be omitted, giving the following braking limit:

Ax ,Unload rear wheel

(L = −

−

h

b)

(7-32)

The negative sign simply indicates that the acceleration is negative, that is, the vehicle is braking. Some skillful riders can brake hard, lifting the rear wheel off the ground while maintaining control of the vehicle. More often this results in a hazardous accident in which the rider is thrown over the handlebars. While it is possible to exceed this limit on recumbent vehicles, it is more common with upright bicycles and tadpole trikes. In some cases, traction is not sufficient for a vehicle to pitch over. In this case, the vehicle skids prior to unloading the rear wheel. Usually, the rear wheel begins to skid first, as the vertical load is reduced. Once skidding starts, braking is adversely affected and the deceleration is decreased. This usually precludes both skidding and pitchover from occurring. However, if the front brakes have good traction and the CG is forward, the rear wheel can skid and, with increased front braking, the vehicle can still pitch over. Tadpole trikes are more prone to this than many other vehicles.

127


On level ground, traction limited deceleration is found by noting that FB,max = mmg. Applying Newton’s second law gives mmg = –max. Dividing through by mg, and solving for the deceleration gives the skid limit for braking.

Ax,skid = –m

(7-33)

This equation differs from that for traction-limited acceleration, Equation 7-23, only in sign. The maximum deceleration (in G’s) is just equal to the braking coefficient. Both limiting braking equations are reproduced here for clarity.

Pitchover Limited: Traction Limited:

Ax ,MAX

Ax ,MAX

= − 

L

−b   

h

(7-34)

= −µ

Due to the load transfer during deceleration, it is unusual for most vehicles to lose traction on both axles at the same time. With bicycles, the rear wheel generally loses traction first, due to the reduced vertical load. Maximum braking performance will occur at a deceleration that just keeps the rear wheel in contact with the ground or just keeps the wheels from skidding. Exceeding these limits can increase stopping distance and risk loss of control of the vehicle.

Braking Performance Braking performance may be measured either by the time or the distance to come to a full stop from a given initial speed. Equation 7-17 can be used to analyze stopping distance by setting the drive force to zero and assuming a constant deceleration. Assume that the braking force is constant during the deceleration. Also assume a constant grade and that the rolling resistance is independent of velocity. The relevant equation is then {FB + FRR + mgG} + FAERO = max

(7-35)

The terms in the curly braces can be taken as constant retarding forces, designated FD for convenience. The drag force cannot be combined with the remaining forces because it is dependent on the squared velocity of the vehicle, and hence does not remain constant during the braking process. However, the magnitude of the drag force is small compared to the braking force, particularly at stops made

128

Road Loads

from low to moderate speeds. Under these conditions, drag can be neglected with little loss of accuracy. For simplicity, initially assume a stop made from a low initial speed such that aerodynamic drag can be neglected. WithFAERO = 0 and this simplification, and assuming that FD is constant, the deceleration will also be constant. The distance and time required to stop can be calculated based by noting that FD

= −

m dV dt

dV ax

= −

dt

, so

. Integrating and solving for t gives the time to decelerate:

ts

1 =

ax

(V0

−

VF )

(7-36)

Where V0 is the initial speed and VF is the final speed of the vehicle. Letting VF = 0 gives the time required to come to a full stop.

 1





1

t SD =   V0 =  V0  ax   gAx 

(7-37)

Likewise, the distance required to come to a full stop is given by: (7-38) Note that the stopping time is proportional to the initial speed, whereas the stopping distance is proportional to the speed squared. If the initial speed is doubled, the stopping time is doubled, but the distance required to stop is quadrupled. For stops made from high initial speeds, aerodynamic drag becomes significant. With the drag term included in Equation 7-35, the deceleration is not constant, even for constant FD. The stopping distance with drag included is given by x SD

 m    ρC A   = V   ln  1 +   ρC A    2F   D

D

2 o

(7-39)

D

Neglecting the drag term, as in Equation 7-38, always results in overestimating the stopping distance. If used in design, this results in an extra margin of safety. The difference between Equation 7-38 and Equation 7-39 increases with initial speed—the faster the initial speed the more wind resistance will help to stop the vehicle. Likewise, a more streamlined the vehicle (low drag area CD A) is affected less by drag and requires more distance to stop. At low initial speeds the difference between the two equations is slight.

129


Example 7-4 A pedicab loaded with two passengers must make an emergency brake to stop from 8 m/s on a downhill grade of 3%. The total vehicle mass is 300 kg. The coefficient of rolling resistance Crr is .008 and the drag area CD A is 1.5 m2. The braking friction coefficient is 0.80. The pedicab has a wheelbase of 2.0 m, with a center of gravity located 0.40 meters forward of the rear axle and 1.3 m above the ground. Determine the time and distance to stop under maximum braking. Solution: First, the limiting deceleration must be determined. The pitchover limit is given by L

Ax ,pitchover

=

−

2.0 0.40

b

h

−

=

=

1.3

1.23

The skid limit is similarly given by:

Ax,skid = m = 0.80 Since Ax,skid < Ax,pitchover, the limiting factor is a skid. Assume that the pedicab driver is sufficiently skilled so as to maintain braking just prior to initiating a skid. The deceleration is then 0.80 Gs. Next, find the retarding forces due to the downhill gradient and rolling resistance.

Frr

=

mgCrr=

×

Fgrade =mgG=

300 9.81 0.008 × =

23.5 N

300 9.81 × × − = −( .03)

88.3 N

Notice that the force due to the grade is negative because the pedicab is rolling downhill. Neglect aerodynamic drag for the time being. The total retarding force FD is then equal to max, as in Equation 7-35. The braking force is thus given by

FB =F D− F−rr F FB

130

mA−g −xF

= grade

Frr

grade

300 9.81 ( 88.3) × −0.80 − − 23.5 =

=×

2419 N

Road Loads

This is a very large braking force. Pedicabs must be equipped with powerful brakes in order to stop the large loads they must carry. The time and distance to stop, neglecting air resistance, is given by Equations 7-37 and 7-38:

 1    1 Vo =  0.8 × 9.81  8 = 1.5 s A g  x  1  1   2 xs =  Vo2 =   8 = 9.2m   2 × 0.89× .81   2Ax g  ts = 

Compare the stopping distance above with that obtained by Equation 7-39, which takes into account aerodynamic drag. Assuming a standard atmosphere with ρ = 1.226, the stopping distance is:

xs

 m    ρC A   =  ln  1 +  V   ρC A    2F    ×  300    1.2 1.5 = ln 1 + 8 8.9   =m    1.2 ×1.5     2 ×2354  D

D

xs

2 o

D

2

A loaded pedicab is a large, heavy, and has a large drag area. A vehicle such as this one would reach terminal velocity on a 3% grade at about 8.5 m/s, so the operator in this case would not need to work to obtain the 8 m/s speed. However, even though drag was significant and the initial speed relatively slow (for most HPVs), the difference between including and neglecting drag was only 30 cm over 9 m of stopping distance, or 2.8%.

131

CHAPTER

SPEED AND POWER MODELS

8

C

ycling performance can be modeled and predicted fairly easily once the road forces are understood. Road loads such as aerodynamic drag, rolling resistance, and hill climbing resist the rider’s efforts. If the rider is generating more power at the cranks than the total resistive power, the vehicle will speed up. Conversely, if she is generating less power, the vehicle will slow down. Performance models relate the vehicle speed and ride conditions to the power required of the rider. They can predict performance for competitive events and the potential for success in record-breaking efforts. They are also useful for determining if a vehicle is “fast” or “easy to pedal.” In this case, “fast” is taken to mean that under typical operating conditions and with an average rider generating average power, the vehicle is faster than an average or typical vehicle. “Easy to pedal” may mean the same thing, but often the implication is that at average speeds the power required is less than that for an average vehicle. The external factors that resist motion include aerodynamic drag, rolling resistance, bearing and drive train friction, and changes in potential and kinetic energy. Martin et al.1 conducted a study to determine if mathematical models using these terms can accurately predict a cyclist’s power requirements during cycling. Their findings demonstrated that such models can do so with high accuracy. Of course, the accuracy of any model depends on the accuracy of the parameter values used. An accurate estimate of drag power, for example, requires an accurate value of the drag coefficient. Parameter estimation in the design stage can be difficult, and is discussed in other chapters. The power required to ride a vehicle under a given set of conditions is the sum of the road forces acting on the vehicle, each multiplied by the road velocity of 1

Martin, James C., Douglas L. Milliken, John E. Cobb, Kevin L. McFadden, and Andrew R. Coggan, “Validation of a Mathematical Model for Road Cycling Power,” Journal of Applied Mechanics, 14, 276–291, 1988.

133


the vehicle. Drive train friction is not a road load, and is accounted for by dividing the required power by the drive train efficiency. The road power formula in a generic form is given in Equation 8-1.

PTOTAL

1 = P { AERO P+ P+RR P+ η

P + WB

∆ΚΕ

PE∆

}

(8-1)

where

η = Mechanical efficiency of the drive train PAERO

= Aerodynamic drag power

PRR

= Rolling resistance power

PWB

= Wheel bearing friction power

P∆KE

= Change in kinetic energy

P∆PE

= Change in potential energy

Many of these terms have been discussed in the previous sections. They will be reviewed here for clarity and consistency.

Drive Train Efficiency All mechanical drive systems incur some power losses. This is due to friction in the chain, pedal and bottom bracket bearings, idlers, and gears, if so equipped. (Frame flexure and torsion due to the cyclical pedal loads can also contribute to power losses.) These loads are generally not evaluated individually, but are lumped together as system losses. When actually measuring power output while riding, the instruments used may or may not include the drive train power losses. Some commercially available systems use an instrumented rear hub, while others use instrumented cranks. (Other methods, ranging from chain vibration to heart rate, have also been used to measure power, usually with less accuracy.) If power is measured at the crank, drive train power losses are included in the power measurement. Hub based systems measure power downstream from the drive train and will not capture these losses. That is, the cyclist is actually generating somewhat more power than indicated by the power meter. Drive train efficiency is defined as the ratio of the power available at the drive wheel divided by the power supplied by the crank.

134

Speed and Power Models

η=

PDrive

Wheel

(8-2)

PCrank

Drive train efficiencies for typical bicycle chain drives in good condition and properly lubricated are quite good, often exceeding 95% and under ideal conditions exceeding 98%. The best hub gears approach these efficiencies, particularly at high power levels. Intermediate drives, chain guides and other components sometimes used on recumbent and front-wheel drive vehicles cangenerally be expected to reduce efficiency. Drive train efficiencies are discussed in more depth in Chapter 12.

Aerodynamic Drag Aerodynamic drag is the most significant factor in the power equation for moderate and high speed riding. Only for low speed vehicles or on steep grades does its significance diminish. The formula for computing aerodynamic drag presented in Chapter 7 is given as if the vehicle were placed in a wind tunnel, with the wind always coming from directly in front of the vehicle. On the road, this is seldom the case. The wind component produced by the motion of the bicycle (known as the ground speed, be added to theonnatural wind VW toofobtain the relative VG) must wind velocity VR. The drag force depends the component the relative wind velocity that is tangent to the direction of the vehicle’s motion. This is the headwind component, designated VH. The power, however, is the product of the drag force (which depends on the head wind speed VH) and the ground speed VG. Thus both velocity terms appear in the drag power formula:

PAERO

1

=  Vρ 2

2

 

C V HA (

D

)

G

(8-3)

Given the direction of the wind DW and the direction of the vehicle DV, the headwind component can be found by

VH = VG + VW cos(DW – DB)

(8-4)

Where VG is the vehicle speed over the road, VW is the wind speed, and DW and DB are respectively the directions of the wind and the bicycle. The bicycle direction is the compass heading that the vehicle is going, while the wind direction is the compass direction from which the wind is blowing. The term VW cos(DW – DB)

135


is the headwind component of the natural wind. The crosswind component, also known as the normal wind component, is given by:

VCW = VW sin(DW – DB)

(8-5)

The relative wind strikes the vehicle at some angle to the direction of motion of the vehicle. This angle is known as the aerodynamic yaw angle, and is given by YA

(VV D +cos(  V  D = tan −1  CW =tan −1 G W  W B   VH  VW DWDB sin(

)−

− )



(8-6)

The aerodynamic yaw angle may be significant because drag area—the product of drag coefficient and area—will vary with wind yaw angle. Depending on the vehicle, the drag area may increase or decrease. The drag force and consequently the power to overcome drag usually increase, particularly at high yaw angles, even if the headwind component remains constant. This is why a cyclist riding with a direct crosswind will not ride as fast as on a calm day for the same power output. A noticeable tailwind component must be present to help the rider. Generally, the wind direction must be more than 100º from the direction of the vehicle to be beneficial. Streamliners can be exceptions to this rule. It is possible for a streamliner to develop a thrust force in the presence of a crosswind. The thrust force can more than compensate for the increased drag, making even small yaw angles advantageous. However, even some unfaired road bikes may experience a slight reduction in drag area with small yaw angles. Equation 8-3 is valid for crosswind conditions, provided that the drag area CDA is determined as a function of the aerodynamic yaw angle YA. This is best accomplished with wind tunnel testing which is often costly and difficult. Modeling crosswind effects is generally less important for designers of unfaired vehicles. However, designers of streamliners and HPVs with large fairings that might behave adversely in crosswind conditions should assess the effects of aerodynamic yaw on vehicle handling and performance.

Example 8-1 A cyclist on an unfaired bicycle travels 30º northeast at 10 m/s. The wind is from the south (180º) at 6 m/s. Find the relative headwind and crosswind components, the magnitude of the relative wind, and the aerodynamic yaw angle. Also find the power required to overcome aerodynamic drag if the cyclist’s drag area AFCD = 0.30 m2.

136


Solution:

DW is the direction from which the wind is blowing and DB is the direction the cyclist is riding. Thus, DW = 180° and DB = 30°. Then VH

= 10+ 6cos(180 −° °= 30 ) 4.8 m/s

VCW

VR

Y

= 6sin(180 −° °=30 ) 3.0 m/s

= 4.8 2 + 3.0 2 =8.1 m/s

 3.0  = tan −1   =32 °  4.8 

To find the drag power, the air density must be known. Assume a standard day at sea level. The standard air density is then 1.226 kg/m3. The power required to overcome drag is given by Equation 8-3.

PAERO

ρ =  V  2

2



CA V(

H

 

D

)

G

= 

1.226  4.8 2(.3)10 42 =W 2

The tailwind significantly reduces the drag power. In still wind conditions, the cyclist would need to produce 184 W to maintain 10 m/s. The brisk tailwind reduced aerodynamic power requirements by 76%.

The spokes of rotating wheels also produce aerodynamic drag. Large numbers of round spokes on small diameter wheels are poorest, while disk or aero spoked wheels are much better. For example, the test bicycle studied by Martin et al. used an aerodynamic disk rear wheel and 24 aerodynamic spokes on the front wheel. The drag due to wheel rotation was 1.7% of the total aerodynamic drag, and accounted for less than 3 watts of power at moderate speeds. (Note that an aerodynamic rim shape is designed for the flow of air over the wheel due to the vehicle’s motion, not due to the wheel rotation.) The easiest way to include the effects of wheel rotation on overall drag is to define a drag area of the wheels, ADW. This term is analogous to the vehicle drag area, AD, but applies strictly to the drag force due to wheel rotation. The units are those

137


of area, and the term in general must be experimentally determined for a given set of wheels. The significance of this term is small, and for many analyses may be omitted with little consequence. When high accuracy is required—predicting the likelihood of a world record time for example—the term should be included in the power equation. Designating the drag area of the wheels as ADW, the drag power due to wheel rotation is PDW

1  =  VρA V  2  2

H

(8-7)

DW G

The total aerodynamic drag power is then the sum of Equations 8-3 and 8-7.

PAERO

1 =  Vρ 2

2

 

C AA ( DV H

+

DW

)

G

(8-8)

Rolling Resistance Rolling resistance is often the dominant retarding force for low speeds, knobby, low-pressure tires on soft surfaces. It is always present, giving an effect similar to riding up a slight grade. The power lost to rolling resistance depends on the load on the wheel, and on tire/surface properties including tire construction, pressure, size, road surface, and possibly vehicle speed. Wilson 2 reviews several theories of rolling resistance. The effects of the tire/surface properties are usually lumped together in a coefficient of rolling resistance, as discussed in Chapter 7. Strictly, the component of load normal to the road is used for rolling resistance calculations. That is, if the vehicle is riding up or down a slope, the normal forces, rather than the axle weights, should be used to estimate rolling resistance. As noted in Chapter 7 however, the difference is not significant for typical road grades. The power required to overcome rolling resistance is given by:

PRR = (mg cos(tan–1G))(CRR)VG

2

(8-9)

Wilson, David Gordon, Bicycling Science, 3rd Ed., MIT Press, Cambridge, MA, 2004.

138


where

G

=

CRR

mg

Gradient of the road =

=

Coefficient of rolling resistance Vehicle weight

If the effects due to the grade are neglected, Equation 8-9 simplifies to

PRR = mgCRRVG

(8-10)

The difference between the true term cos(tan –1 G) and the small angle approximation of unity is 0.5% for a 10% grade and less than 2% for a very steep 20% grade. Frequently the uncertainty of CRR is significantly greater than this error, implying the simpler formula is sufficient for all but the most precise calculations.

Frictional Losses in Wheel Bearings Power lost to friction in the wheel bearings is generally small, and for many analyses may be neglected. For the best accuracy in performance prediction, the term should be included, however. The bearing losses depend strongly on the wheel diameter. Larger wheels rotate more slowly for a given road speed than do smaller wheels. Dahn et al.3 reports bicycle wheel bearing frictional torque as

T = 0.015 + 0.00005n

(8-11)

where T n

=

Torque (N-m)

=

Speed (RPM)

Recalling that power is torque times speed, and making the necessary conversions to relate wheel speed to vehicle speed, the power (in Watts) lost to bearing friction per wheel is PBF

= 

.030

d

 .006  + G G V  V 2  πd  

(8-12)

3

Dahn, K., L. Mai, J. Poland, and C. Jenkens, “Frictional Resistance in Bicycle Wheel Bearings,” Cycling Science, 3(3), 28–32.

139


Where d is the wheel diameter in meters, and VG is given in meters per second. Note the unit dependence of Equation 8-12 !

Changes in Potential Energy The power to climb a steep grade easily becomes the dominant retarding force on a cyclist, as most riders have experienced. It is strictly dependent on the vehicle weight and rate of ascent, given by

PDPE = mg sin(tan–1 G)VG

(8-13)

For grades less than 10%, sin(tan–1 G) » G, so DPE = mgGVG is a good approximation.

Changes in Kinetic Energy Kinetic energy changes whenever the vehicle’s velocity changes. The average power required to change speed is the change in kinetic energy divided by the time required for the acceleration or deceleration, KE

=

1 mV 2

2 G

∆KE ∆t

. Recalling that

, formulas for the average power due to changes in kinetic energy are

easily derived.

(8-14)

Note that Equation 8-14 includes the inertia of the drivetrain and wheels. It gives the total change in kinetic energy corresponding to a change in speed. This equation is useful if we know the initial and final speeds and the time interval for acceleration, but sometimes the instantaneous power required to change speed is more important. The instantaneous power required to accelerate from a speed VG with an acceleration ax is given by

PDKE = (m + CI)axVG

140

(8-15)


Power Models Equation 8-1 is used to model power-speed relationships for a vehicle, or for guiding investigations into how to best improve performance. All terms are functions of speed. Given that all applicable parameters are known, the power required of the operator can be predicted for any given speed. Likewise, if power is known, the resulting speed can be predicted. These models can be quite accurate, if sufficient care is used to evaluate the model parameters. Within a vehicle development or improvement cycle, power models are used to identify the best areas for improvement. The best areas to work on are those that will provide significant performance enhancements without undue development costs. The power model can quantitatively compare the performance benefits of a given improvement—a 5% reduction in vehicle weight, for example. A vehicle with a mass of 80 kg climbing a 5% grade at 4 m/s would require 194 Watts. A 5% reduction in mass, to 76 kg, would require only 185 Watts, a 4.6% reduction in power. While a 4 kg reduction may be feasible for a heavy vehicle, it could be a daunting task for lightweight vehicles. Power models have been used quite successfully to predict performance in time trial competitive events. (During road races, drafting and race strategy are more important performance elements, and so the power models are much less likely to predict finish times.) A few years ago, these models were used to successfully predict the ability of an elite-class cyclist to break the world hour record. Predicted speed for the hour record attempt was 52.880 km/hr, and actual speed of the successful record attempt was 53.040 km/hr—a difference of only 160 meters (Padilla et al., 2000).4 In this case, the applicable model parameters, along with the subject’s physiological parameters, were very carefully and accurately measured prior to the record attempt. Equation 8-16 summarizes the terms that appear in the power equation. All are functions of vehicle speed. Note that small angles have been assumed for road grades, and the bearing friction term is split into front and rear tires, with df and dr the front and rear wheel diameters, respectively. (For tricycles or quads, multiply the appropriate term by two.)

4

Padilla, Sabino, Inigo Mujika, Francisco Angulo, and Juan Jose Goiriena, “Scientific Approach to the 1-h Cycling World Record: a Case Study,”Journal of Applied Physiology, 89, 1522–1527, 2000.

141


PAERO

1  ( DV + =  Vρ CH2AA 2 

DW

)

G

PRR =mgCRRVG PBF

=

 .030  .006  + V +G + f  d  πd f 2

.030   V V  dr

.006



GG

πdr 2

(8-16)



P∆PE =mgGV G P∆KE

=m ( C+ aI VxG)

Example 8-2 A cyclist on a recumbent bicycle travels south (180º) 8 m/s. The wind is from the south-southeast (150º) at 5 m/s. The drag area is AFCD = 0.4 m2. The road is rough asphalt with a coefficient of rolling resistance Crr = .010. The combined mass of the rider and bicycle is 90 kg; the inertia coefficient for the two 25-622 wheels and the crank is CI = 2.1 kg. The drivetrain efficiency is approximately 95%. Find the power the cyclist must expend to maintain his speed. How much additional power is required if the rider maintains speed on a slight 1% up grade? How much additional power would be required to accelerate to 10 m/s within 15 seconds on level ground? Solution:

DW is the direction from which the wind is blowing and DB is the direction the cyclist is riding. Thus, DW = 150° and DB = 180°. Then VH = 8 + 5 cos(150 ° – 180 °) = 12.3 m/s The power required to overcome drag is: PAERO

 ρ =  V  2

2

CAV (

H

  

D

)

G

 = 

1.226 12.3 2(.4)8 298 = W 2

The power required to overcome rolling resistance is:

PRR = mgCRRVG = 90 ´ 9.81 ´ 0.010 ´ 8 = 70.6 W

142


Bearing friction results in power losses of:

The power lost to wheel bearing friction is small, and may often be neglected. For high accuracy, such as predicting success in racing events, it should be included however. Riding at a constant speed on level ground, no energy is expended to change potential or kinetic energy, so the total power required is:

1

PTOTAL = P { AERO P +P + η

=

RR

WB

+ }

1 {298 = 70.6 3.59} 372W .95

+

A healthy adult male could maintain this power level for more than a minute, while an elite cyclist could maintain this power for more than an hour. If the rider maintained speed while climbing a slight 1% grade, the increase in power due to changes in potential energy is

PDPE = mgGVG = 90*9.81*.01*8 = 70.6 W Bringing the total required power up to 443 Watts. Back on level ground, the rider accelerates to 10 m/s in 15 seconds. The additional power to accelerate is given by:

This brings the total power to 483 Watts, a level most non-competitive riders could maintain only for a short time.

143


Cubic Power Model Each of the terms in Equation 8-16 is a polynomial function of ground speed. The aerodynamic drag terms are cubic with respect to speed, while other terms are linear or quadratic. A polynomial model is convenient for assessing speed/ power requirements. Substituting Equation 8-4 for VH in Equation 8-16 and inserting these terms into Equation 8-1 gives a cubic polynomial expression for the power required as a function of speed. 3

2

PTOTAL =C V1G C+ V

CV

G 2

(8-17)

+G 3

Where the coefficients are

(8-18)

These coefficients are sufficient if acceleration is constant and known. If it is desired to find the acceleration resulting from a given power and speed, the acceleration term in the coefficient C3 must be isolated. This is accomplished by splitting C3 into a sum of two new coefficients C4 and C5, such

C3 = C4 + C5ax

(8-19)

where 1

C4

=

C5

=

ρ

η  2CA( A +D 1

η

2

DW

D)(cos( −DVW

+mgC+) ) BW

RR

++

1 0.030mgG  d f

(m + CI )

1 dr

 

 (8-20)

The power polynomial then becomes 3

PTOTAL = C V1G+ CV 2

144

+ G

2

+ C (a V 4 5C

x G

)

(8-21)


Solving for the acceleration gives: ax

 P  C C4 C  2   =  VGtotal − −1−− V G VG    C 5 C5 C5  C5 

1

2

(8-22)

Equations 8-21 and 8-17 provide mathematical models for evaluating and predicting performance of wheeled land vehicles. They can be used during vehicle design to predict performance.

Example 8-3 Plot the power required as a function of speed for the rider in the previous example, if the road is level and he rides at a constant speed. Solution: Determine the values of the coefficients using Equation 8-18:

The power function for this example is a cubic polynomial with these coefficients:

PTOT = .2581V 3 + 2.2442V 2 + 14.2271V

145


Figure 8-1 plots the power function for this example. Total power, and the power to overcome aerodynamic drag, rolling resistance, and bearing friction are included. It is clear from the figure that power requirements increase dramatically as speed increases. This is primarily due to the drag term. At speeds above two meters per second, drag is the largest of the retarding forces. Above 5 m/s, it is significantly larger than the other terms.

POWER REQUIRED TO RIDE HPV AS FUNCTION OF SPEED 700 TOTAL POWER

600

AERO POWE R ROLLING RESISTANCE POWER BEARING FRICTION POWER

500

s tt a 400 W R E W300 O P 200

100

0 0

1

2

3

4

5

6

7

8

9

10

SPEED m/s

Figure 8-1 Power function for the example HPV

Applications of Power Models Vehicle designers can use the power model of Equation 8-1 to assess the effect of design changes on vehicle performance. For example, the effects of different tires on performance can be evaluated if the rolling resistance coefficient and tire mass are known (or able to be approximated). Design decisions can be made based on quantitative performance predictions rather than speculation. Many design decisions involve trade-offs between performance and cost. Power models can provide guidance as to where money is best invested in order to achieve the correct balance of cost and performance. Design concepts for entire vehicles or for components can be evaluated and compared early in the design cycle.

146


Power models are also commonly used to predict cycling performance. Coaches and athletes use the power models developed in this chapter to predict performance for competitive events such as time trials and track races. In these events, the athlete competes against the clock, which is directly related to the ability to overcome the road loads described in Chapter 7. In these events, the power models can provide accurate predictions of performance. The following examples illustrate the application of power models to aid design decisions and predict athletic performance.

Example 8-4 A cyclist will attempt an hour time trial in a streamlined recumbent bicycle on an indoor velodrome track. Previous tests have shown that the cyclist can maintain 350 watts for one hour. The drag coefficient for the vehicle is .25 with a frontal area of .35 m 2. The drag area of the spokes is .01 m 2. The coefficient of rolling resistance over the smooth track surface is 0.002. The total mass of vehicle and rider is 95 kg, and the inertia coefficient is 3.8 kg. The vehicle uses a 23-622 rear tire and a 28-451 front tire. The drive train efficiency is 95%. How far is the cyclist expected to ride in one hour? Solution: For this problem, power is given and speed must be determined. Equation 8-17 should be used to solve for speed. The power equation coefficients must be computed with Equation 8-18. Assume that the cyclist will be allowed a rolling start, so that the power to accelerate is not significant. Also assume a standard atmospheric density of 1.226 kg/m 3. The first coefficient depends on the aerodynamic parameters.

C1 =

ρ

C(A + A D

2η

=

)

DW

1.226 × + =(.25 .35 .01) .0629 × 20.95

Since there is no wind in the indoor velodrome, VW = 0. The diameter of a wheel is given approximately as the sum of the bead seat diameter and twice the section width, or d = BSD + 2 ´ SW, or dr = 622 + 2 ´ 23 = 668 and df = 451 + 2 ´ 28 = 507 mm.

147


C2

1 = ρ C +A( AD η

)cos( − DVW+

DWD

1   0.006 = + 0  + .95  π ×  0.668

  .006  πd 2 πd 2   f   r 

.006 

+)

B W

 0.006 =  0.507 π×

2

2

 0.01233 

The third coefficient is likewise simplified since not only is there zero wind, but there is no grade or acceleration either.

The power equation then becomes PTOTAL =C V1G 3C+ V 350

CV2

G2 3 G

= 0.0629 × +

V

+G 3 2

0.01233 × + × VG 2.0716

VG

This equation can be solved for VG using most scientific calculators or a computer root-finding algorithm. In this case, the solution is V = 17.1 m/s. In one hour, the cyclist will go 17.0 ´ 3600 = 61,334 m or 61.3 km.

Example 8-5 The cyclist in the previous example rides his vehicle at 14 m/s on the velodrome track. How much power is required? If the cyclist is able to increase power to 400 watts from this speed, what is the resulting acceleration? Solution: To determine the required power, the coefficients calculated previously can be used. The power is given directly by Equation 8-17. 3

PTOTAL =C V1G C+ V P14

148

=0.0629 × +14

2

CV

G 2 3

+G 3

0.01233 × + 14 ×=

2

2.0716 14

204 Watts


A power increase to 400 watts would essentially double the power output. To find the resulting acceleration, the coefficients C4 and C5 must be calculated using Equation 8-20.

The resulting acceleration is obtained using Equation 8-22

ax

ax

P  C  C C  =  VGtotal − −1− −V4G VG 2 C  5 C5 C5  C5  400   −1 2.0716 =   − 14 −  −   104    104 

     

2

1

0.01233 

= 104

14

0.0629 14 104

2

0.1346m/s

2

The acceleration resulting from a near doubling of power to 400 watts is thus .135 m/s2 or 0.013 Gs.

In some modeling problems, power is known or assumed and an initial speed is known. The acceleration will decrease with increasing speed, as shown by Equation 8-22. The speed will increase, but at a decreasing rate up to that speed corresponding to zero acceleration. An iterative solution is required for these problems. The general procedure is to first compute the initial acceleration as in the previous example. Then compute the speed after a short time interval Δ t assuming constant acceleration for the duration of the time interval, v = v0 + a ´

149


ACCELERATION PERFORMANCE AT CONSTANT POWER 18 17.5 17 16.5 ) s / m ( 16 D E E 15.5 P S

15 14.5 14 13.5 0

20

40

60

80

100 120 TIME (s)

140

160

180

Figure 8-2 Speed as a function of time for constant power acceleration from 14 m/s

D

t. Compute the instantaneous acceleration again at the new speed, and iterate as required. This is a task best suited for computer calculation. The figure above shows how the speed of the vehicle in the example increases while the cyclist pedals at a constant 400 W. The initial speed is 14 m/s. After 2 minutes at this power level, the speed has almost stabilized, and is within .5% of the speed that can just be maintained at 400 watts, 17.9 m/s. The curve was computed by iterating with a time step of Dt = 0.01 s. However, it is worth noting that a cyclist is rarely able to actually ride this type of profile, which assumes a step increase in power from an initial level to some new, constant level. The power during a hard acceleration actually increases at a decreasing rate due to, among other things, the changing torque/speed relationship at the cranks. A parabolic rise might better model the actual power profile during heavy acceleration.

150

CHAPTER

AERODYNAMIC DRAG

9

A

t typical speeds on level roads, the aerodynamic drag force is the principle retarding force acting on the bicycle. This is true even though the actual value of the force is small. For example, a cyclist riding a typical road bike at 8 m/s over a level asphalt road will expend about 80% of her effort to overcome drag (Figure 9-1). At 11 m/s, the fraction increases to 90% (although her total power output more than doubles). If our cyclist climbs into a world-record class streamlined HPV, she will find that at comparable speeds, the drag is almost negligible. She could ride at 11 m/s using only one-sixth the power required for her road bike at that speed. The drag would represent about 28% of the (substantially lower) total power she is exerting. At the high speeds for which these vehicles are designed, however, the drag power fraction is about 80%. The conclusion is clear: aerodynamic drag is a very important factor for all but the slowest land human-powered vehicles Figure 9-2. For the competitor, minimizing drag is critical, and often makes the difference between winning or losing a time trial. The tourist and recreational rider will find faster or easier rides more enjoyable, and arguably safer. Likewise the commuter will reap benefits of a more aerodynamic vehicle. Thus, a good design goal is toalways strive for reduced drag.

Causes of Aerodynamic Drag 1

The total aerodynamic drag acting on a vehicle is attributed to several causes. Form drag—the dominant type of drag on unfaired vehicles—is caused by the air pressure difference between the front and rear of the vehicle. Essentially, high-pressure air just pushes against the front of the vehicle and rider. If a vehicle is streamlined, form drag is reduced and skin friction drag becomes most

1

Tamai, Goro, “Aerodynamic Drag Components,”Human Power, Vol 12 n 2, Spring 1998.

151


POWER FOR TYPICAL BIKE 100% 80% 60% 40% 20% 0% 8

11

m/s

m/s

Other Drag Power

Figure 9-1 Aerodynamic power fraction for a typical road bicycle

100%

POWER FOR STREAMLINED HPV

80% 60% 40% 20% 0% 8m/s

11 m/s

33 m/s

Other Drag Power

Figure 9-2 Aerodynamic power fraction for a streamlined HPV

important. It is caused by the friction of the airstream as it flows past the surfaces of the vehicle. Interference drag is produced by flow over protrusions, bumps, holes, etc. One example is mirrors on the outside of a streamliner, which can reduce the effectiveness of an aerodynamic fairing. Interference drag can become significant, particularly for vehicles designed for extremely low drag. Induced drag is generated by any body that produces lift—a net force perpendicular to the free airstream. Induced drag is important for airplane wings and similar surfaces, but is less important for most human-powered land vehicles. Most bicycles and human-powered vehicles are unfaired or have only partial fairings. Form drag—also known as pressure drag—is the dominant retarding force, caused by the difference in pressure between the front and the rear of the bicycle. The pressure in an airstream is given by Bernoulli’s Equation, which

152

Aerodynamic Drag

states that the total air pressure is equal to the sum of the ambient, or static, pressure and the dynamic pressure due to the air velocity.2

Ptotal = Pstatic + Pdynamic The static pressure is simply the ambient barometric pressure. A pressure tap placed anywhere along the surface of an HPV would measure only the static pressure. The dynamic pressure depends on the velocity of the airstream and is given by

Pdynamic

=

ρ

2

V2

where ρ = mass density of the air V =

velocity of the airstream

On a calm day, the airstream velocity in front of the vehicle is equal to the speed of the vehicle over the road. (It is sometimes more convenient to consider the vehicle as stationary and the air flowing past it, as if in a wind tunnel. This is essentially equivalent, from a fluid dynamics standpoint, to a moving HPV on a windless road.) Streamlines represent the flow of a line of air over the vehicle. (Imagine a small massless speck of dust in the air. The path it follows will trace out a streamline.) Well in front of the vehicle, all streamlines are parallel and have the same velocity, and hence the same pressure. Near the front of the vehicle, the streamlines must split to go around it. At some point there will be a streamline that strikes the vehicle straight ahead and comes to a complete stop, or stagnates. According to Bernoulli’s Equation, the total pressure will not change, so the static pressure at the stagnation point must equal the total pressure of the free airstream. This produces a high pressure area on the front of the vehicle. Pressure affects the flow velocity and direction. Visualize an isolated stream of air flowing in a straight line, say from left to right. A pressure gradient exists in the direction of flow, such that pressure increases to the right. There is no change in pressure, or pressure gradient, in the transverse direction. According to Bernoulli’s Principle, the flow must slow down as the pressure increases. The velocity decreases with increasing pressure, but the air continues to flow in a straight line. A pressure gradient parallel to flow thus causes changes in airstream speed. Now consider a pressure gradient that varies across the flow direction, say increasing from bottom to top. This causes the airstream to turn toward the low pressure side, or turn downward in this example. Transverse pressure gradients 2

Gillespie, Thomas D., Fundamentals of Vehicle Dynamics, SAE, 1992.

153


Figure 9-3 Invicid flow over a cylinder

cause a change in flow direction. In summary, pressure gradients acting on an airstream will always result in a change in fluid speed or direction—or both. To understand what happens as the flow progresses around the vehicle, two rather bold assumptions are initially made. First, that a bicycle rider in an upright position can be modeled as a vertica l cylinder. This may be reasonable for a crude approximation of drag (omitting the bicycle itself) but is used here to simplify the discussion and improve visualization. The second assumption is more drastic—that the air is frictionless, with a viscosity of zero. This assumption is clearly unrealistic, and in fact leads to zero drag. It is enlightening, however, to understand how air would flow in such a condition. After a brief discussion of frictionless flow, this assumption will be discarded and the differences between the ideal and real case explored. Now visualize flow around our idealized rider—or vertical cylinder. Figure 9-3 shows an end view of the cylinder, (or a top view of the rider). Streamline A leads to the stagnation point S, which has zero veloc ity and a very high pressure. This line is a pressure “ridge”—the pressure increases toward the stagnation pointand decreases to either side. The pressure gradient is positive in the direction parallel to the flow, but is zero in the transverse direction. Pressure increases and velocity decreases toward the stagnation point, where velocity is zero and pressure is maximum. Streamline B is very close to streamline A, and the pressure gradients are similar, but not the same. Pressure increases as the flow nears the cylinder, but the transverse pressure gradient is now negative to the left of the streamline. That is, the pressure in front of and to the right of the streamline is greater than that to the left of it and the air slows and starts to turn left to pass around the cylinder. As the air flows upward toward point C, the pressure decreases and the velocity increases, pushing the air to the right. Point C is an inflection point, where the direction of the airstream becomes concave down in order to

154

Aerodynamic Drag

follow the profile of the cylinder. For this to occur, the pressure gradient must reverse. Starting with point C, the pressure on the right of the airstream is lower than that on the left. The speed of the airstream continues to increase and the pressure to decrease until point D is reached. At this point the pressure is very low—below ambient—and the velocity is maximum—above the free stream velocity. The frictionless air continues to flow down the back side of the cylinder. As it does so, the pressure increases and velocity decreases. A second inflection point occurs at E, where again the pressure gradient reverses. The airstream turns left, away from the cylinder, and continues to slow down. Well downstream of the cylinder both the pressure and the velocity are equal to that of the free airstream. The flow is symmetric. Since the high pressure on the front of the cylinder is balanced by the high pressure on the back side, there is no net pressure difference and hence no pressure drag. Also, there is no resistance to the air as it slides over the cylinder, so there is no skin friction drag. Unlike this thought experiment, real air does have viscosity, and therefore friction. The viscosity affects the flow patterns significantly, and produces both pressure drag and skin friction drag. Discard the rather presumptuous assumption of frictionless flow, and see how real air flows over the same cylinder. Figure 9-4 shows streamlines of a viscous fluid flowing over a cylinder, as modeled using computational fluid dynamics (CFD). The colors represent flow velocity. Note how the upstream side (to the left of the cylinder in the figure) looks similar to that of Figure 9-3, but the downstream side looks very different. Figure 9-5 depicts simplified viscous flow around the cylinder. The stagnation point S is similar to the inviscid example. Airstream B initially looks similar to Figure 9-3, but now a thin layer of air next to the cylinder surface, called the boundary layer, is formed because of friction between the air and the surface of the cylinder. The boundary layer starts at the stagnation point and continues around the cylinder. At the surface, friction causes the flow velocity to be zero.

Figure 9-4 CFD model of viscous flow over a cylinder

155


Figure 9-5 Viscous flow over a cylinder

Between the surface and the outside of the boundary layer, the velocity increases until it achieves the full airstream velocity at the edge of the boundary layer. Initially the boundary layer can be thought of as being comprised of smooth, laminar sheets of air that do not interfere with each other. The velocity of the “sheets” increases linearly from the surface to the boundary edge. This is known as a laminar boundary layer. As the flow moves over the surface, the boundary layer thickness increases. At some point, (C in the figure) the laminar sheets start to break up, and the boundary layer flow becomes turbulent. The thickness of the boundary layer continues to grow. The nature of the boundary layer can have a significant affect on the drag of a vehicle. It is a very important topic, and will be addressed in more detail below. From the stagnation point to point D, the pressure is decreasing and the velocity is increasing. This tends to force the boundary layer along in the direction of flow, and prevents it from growing too large. After point D, however, the pressure starts to increase, and the boundary layer is impeded. The boundary layer grows in thickness, and the velocity is decreased. At some point, known as theseparation point, the boundary layer flow will stop entirely. Beyond this point, flow may even reverse, meaning that air at the surface is flowing backward! The region behind the separation point is characterized by turbulence, vortices, and low pressure. The low pressure behind the vehicle caused by separated flow is the root cause of form drag. The high pressure region in front of the rider, coupled with the low pressure region due to separation, mean that a net force will act on the rider, retarding her progress. Form drag is proportional to the frontal area of the vehicle and the dynamic pressure. From an energy standpoint, the vortices produced after separation contain a large amount of kinetic energy. Clearly reducing the size of the separation region is important for reducing drag. Streamlining does just this, as will be seen.

156

Aerodynamic Drag

The friction between the boundary layer and the surface produces a shear force at the surface that retards the motion of the vehicle—the skin friction drag. For blunt objects, like our cylinder—or a real cyclist on an unfaired bike—this is very small compared to the form drag. However, for streamlined bodies it is usually the dominant cause of drag. Skin friction drag is proportional to the wetted area—the total area exposed to flow—and the dynamic pressure. Boundary layers with laminar flow have significantly less skin friction drag than turbulent boundary layers. The smooth sheets of air flow over the surface with minimal resistance. If the laminar boundary layer remains attached—no separation— drag will be minimized. However, laminar flow is more unstable than turbulent boundary layers, and often leads to early separation. The resulting increase in pressure drag can result in greater total drag than an attached turbulent boundary layer. Designers of aircraft and high-speed HPVs have striven to achieve the benefits of laminar flow, but the task is difficult. The laminar flow can be interrupted very easily, from road vibrations or a dirty surface, for example. Hence, many designers try to force turbulent flow, which delays separation and reduces pressure drag. Airflow over protrusions from a streamlined body disrupts the air flow by creating local separation, turbulence, and vortices. This is the source of interference drag. Even gaps between body panels or the junction of the windshield can disrupt the flow. Larger protrusions, such as mirrors, can have a significant drag penalty. For an unstreamlined bicycle, much of the total drag is due to interference drag over the handlebars, frame, cables, etc.

Lift and Induced Drag The airflow around the cylinder was symmetric on either side. The pressure distribution was also symmetric, and thus there was no net force acting in the transverse direction. Consider what would happen if the cylinder was squashed in on one side, creating an asymmetrical shape such as that shown.

Figure 9-6 Asymmetrical shape produces net lateral force in a flow field

157


In this case the pressure distribution on either side of the body would not be symmetrical. The net force due to the air pressure would be at some oblique angle to the free stream airflow. The component of this force parallel to the flow is induced drag, and the component acting perpendicular to the flow is called lift. Induced drag exists whenever a body is creating lift. For example, it always exists for an airplane in normal flight. Road vehicles can produce lift and induced drag, which affect the load distribution on the wheels. For many commuter HPVs these are relatively small effects, but they certainly can become significant for high speeds, and should be considered for competitive speed HPVs.

Computing Drag Force Recalling that form drag is caused by the high-pressure region in front of t he vehicle coupled with the low-pressure region behind it, the drag force can be estimated. On a simplistic level, this force should be equal to the dynamic pressure of the free airstream times the frontal area of the vehicle. Measured values of drag force differ from this value, however, due to a variety of reasons. To account for these various factors, an extra parameter, known as the drag coefficient , is included in the equation. Thus, the drag force is generally computed as

ρ

FD = C DF AVF I

2

2

(9-1)

where FD

=

CDF AF

Aerodynamic Drag Force

=

=

Drag Coefficient

Frontal Projection Area

For streamlined vehicles, skin-friction drag is generally more significant than form drag. In this case the area that is used is the total wetted surface area, and drag is computed as ρ

FD = C DS AVS I 2

where As = Wetted Surface Area

158

2

(9-2)

Aerodynamic Drag

Equations 14-1 and 14-2 both have the same form. For a given vehicle, drag may be computed by either method—they are equivalent. The product CD A is the same for both cases, although CDF ¹ CDS and AF ¹ AS. Care must be exercised to always use the correct area that corresponds with the way in which the drag coefficient CD was determined. For the remainder of this book, the second subscript will be dropped from the drag coefficient, as it is implied by the subscript on the area. Frequently, the product CD A is of more interest than either individual parameter. This product is known as the drag area, and is frequently cited in aerodynamic studies in the literature. A particular vehicle may have a lower drag coefficient than a similar vehicle, but its larger area may cause it to incur greater drag. Use of the drag area avoids this confusion, as it is always directly proportional to the drag force.

Factors Affecting the Drag Coefficient The drag coefficient for a vehicle is affected by a number of variables, including:

• • • • •

The shape of the vehicle The surface texture and smoothness Any protrusions in the airflow The Reynolds number Devices that affect boundary layer flow

Shape: From the previous discussion, it is clear that streamlined shapes that reduce the separation area have lower drag coefficients. A well-designed streamlined HPV can have a drag coefficient two orders of magnitude less than an unfaired bicycle. The rear of the vehicle should ideally taper to a point or edge. A compromise solution—particularly for very tapered shapes—is to truncate the taper by rounding the aft end of the fairing. This produces a small separation area (with an increase in form drag), but eliminates a significant surface area of the vehicle. The aerodynamic penalty may be slight, and the size and weight of the vehicle can be reduced. The front of the vehicle should be gently rounded, with a fairly low stagnation point. The stagnation point height is important, as it determines which streamlines pass above or below the vehicle. Changes in the height of the stagnation point can affect CD by up to 15% in automobiles. (Although the effect may be reduced for most HPVs, due to their relative narrowness.)

159


Surface Texture: A smooth surface will delay the onset of turbulent flow in the boundary layer and reduce surface friction. Streamliners that strive for laminar flow must have very smooth and clean surfaces. However, due to the instability of laminar flow—its propensity to separate early—turbulent flow is sometimes induced early. This can actually reduce drag (in some cases) by delaying separation. This is one reason for the dimples on a golf ball. There may thus be cases where a slightly rougher surface may be beneficial. Still, smooth surfaces are generally preferred. Protrusions: Protrusions in the airflow create significant interference drag. On streamliners, they should be eliminated as much as possible. For unfaired or partially faired vehicles, elimination of protrusions is not possible, but they can still be minimized. Cables can be routed inside the frame, for example. If a protrusion must exist—handlebars or frame members for example—it should be streamlined. Junctions between parts should be faired smooth. Placing a wheel in the wake of a frame member can help, at can be seen on many time trial bicycles. Reynolds Number: The Reynolds number (Re) is the ratio of viscous to inertial forces in an airstream. Drag coefficients generally have a very strong dependence on this dimensionless number. This is clearly illustrated in Figure 9-7, which plots CD for various simple shapes as a function of Reynolds number. Reynolds number is given by

Re

=

VD

(9-3)

ν

where V

=

Free Airstream Velocity

D

=

Characteristic Length

ν =

Kinematic Viscosity

The characteristic length is a representative number for the general size of the vehicle. Generally, for unstreamlined shapes D is taken as a transverse dimension, such as the width. For example, a cylinder normal to the flow would use the diameter for D. Streamlined shapes typically use the length of the vehicle. Generally, use the width for bodies that exhibit predominantly form drag and length for bodies that exhibit predominantly skin-friction drag. The exact dimension used is not critical, but should be specified. For example, it might be reported that “Reynolds number, based on length, is 1.4e6.”

160

Aerodynamic Drag

8

s e re g e

S T S E T L E N N U T D IN W M O R F A T A D G A R D

V P H d e iln m a e rt S

ip h rs i A

d 0 ,r e d n li y C r a l u b tu i m e S

s e re g e d 0 8 1 ,r e d n li y C r la u b tu i m e S

r e d lin y C r la u rc i C

1 : 2 r, e d in l y C ic t lip l E

1 : 8 r, e d in l y C ic t lip l E

e b u T e r a u q S

e b u T r a l u g n ira T

re e h p S c i n o s b u S

7

0 1

0 1

0 1

6 – 3

6

0 1

R E B M U N 'S 5 0 D 1 O L N Y E R

4

0 1

3

0 1

2

1

-1

0

0 1I T NE

C I FF E OC GA RD

. 1 5 4 o N tr o p e

0 1

-2

0 0 1 1

g n it s e t l e n n u t d n i w m fro d e n i a t b o s e p a h s s u io r a v r o f st n e i c fi f e o c g a r D 7 9 e r u g i F

r e w o P . J

: A tr a P 8 1 l2 o V .s r g n E . h c e M .t s n I . c o r P ,”s e l c i h e V d re e w o P n a m u H f o s c i m a n y d o r e A “ ,r e v a e

R A C A N ” s, e g a d n e p p A d n a s e c n . a r 9 e 1 b 6 . tu o o r N P tr y p b o d e e t R c A e ff C A A N s a s,” s e ie p d a o h B S e le n li p i m m a S e f tr o S rs o e w d T in f ly o g C ra o f .W D g D e ra . h D M T “ “ d ,. ,.F n H . .a a R rI W , . ,t y C e o s ,e b d l b n i y A 4L 5K 3

. 5 8 5 1 R C tr o p e R A S A N ,” g a r D e r e h p S c i n o s b u S n o ts c fe

. 4 0 0 ,2 4 5 1 – 1 4 1 ,

y g r e n E d n a

fE e c n le u b r u T d n a m u u in t n o -C n o N f o t n e m re u s a e M “ ,. A . N , in r a Z 6

161


The transition from laminar flow to turbulent flow depends on the Reynolds number. Above about Re = 3.0e3, a laminar boundary layer will spontaneously become turbulent. The curve for a cylinder in Figure 9-7 levels off at about this value. This is due to the transition from laminar to turbulent flow. If laminar flow could be maintained at higher Reynolds numbers, the knee would be shifted down and right, giving lower drag coefficients. Typical values of Reynolds number (based on width) for bicycles and unfaired HPVs range from about 1.2e5 to 1e6. For streamliners, the number increases to about 5e6, due to the higher speeds and use of the vehicle length as a basis. Inspection of Figure 9-7 shows that this is precisely the range where CD is most sensitive to changes in Reynolds number. Devices that Affect Boundary-Layer Flow:Because of the instability of laminar flow, some designers try to force the boundary layer to become turbulent. The reduction in form drag offsets the increase in skin friction drag. Trip wires running around the surface can trigger turbulent flow, and are sometimes used. Boundary-layer suction is sometimes used for the opposite effect—to prolong the laminar flow. As the laminar boundary encounters an adverse pressure gradient, it becomes thicker, with the layers closest to the skin having little momentum. Holes or slits in the surface can be used to suck away these layers, resulting in a boundary layer that is thinner and remains attached longer. The power required to maintain the suction is, in theory, much less than the power lost to non-laminar flow.

Estimation of the Drag Coefficient The HPV designer is very concerned with estimating CD—or at least minimizing it. This is particularly true for high-performance vehicles. The drag coefficient is very difficult to estimate accurately in most cases. The tools available to the designer are experimentation, computational fluid dynamics, and approximation based on published data for bodies with similar shapes. Carefully conducte d experiments are by far the most accurate—many would argue the only accurate—way to obtain drag data. Tests are typically conducted in wind tunnels, (although some experimenters have used road tests). They require an accurate model or prototype of the vehicle, and are generally very expensive and time consuming. If scale models are used for testing, the airstream velocity must be scaled on the basis of the Reynolds number (described below). As a design tool, the time penalty is large: model fabrication, test scheduling and execution, data analysis, redesign and repeat. If budget and time constraints do not preclude wind tunnel tests, they can provide invaluable data.

162

Aerodynamic Drag

Computer modeling with computational fluid dynamics (CFD) software can provide estimates of drag coefficients. However, drag coefficients can be quite sensitive to subtle changes in conditions which may be missed in the model. This is particularly true in the range of Reynolds numbers where most HPVs operate. CFD software is not trivial to use and interpret, and is usually fairly expensive. (It is generally far less expensive than wind tunnel tests, particularly when the cost is amortized over several projects.) In the hands of a skilled analyst, CFD can be a valuable tool, with the potential of reducing the build-test-modify cycle. For critical applications, tests should still be conducted. Many designers will not have the budget for wind tunnel testing or the expertise for CFD. Approximations of the drag can be obtained by using published data. For example, a rider on an upright bicycle was crudely approximated as a vertical cylinder. If we assume that the cylinder diameter is approximately 0.35 m, and the speed of the bike is 8 m/s, the Reynolds number (at standard conditions) will be about 1.9e5. Many basic fluid dynamics texts provide graphs for the drag coefficient of a cylinder.7 In this case, the coefficient is slightly less than one. This corresponds very well with published data for upright cyclists. Wilson reports drag coefficients 4 8 for an upright commuting bike and a touring bike to be 1.15 and 1.0 respectively. The example may lead to false optimism regarding drag estimation. First, it is a very crude method, and will only provide ballpark figures. Secondly, the drag curve for a cylinder exhibits a marked drop at a Reynolds number of about 2e5. It is unlikely that an upright cyclist would realize the nearly five fold reduction in drag coefficient predicted for a cylinder simply by increasing her speed a few meters per second. Of course, streamlined vehicles should be compared with streamlined shapes or published data on streamliners. Caution should be used here as well—if comparison is made to a well-designed and well-crafted vehicle CD may be underestimated. Poor craftsmanship can cause significant increases in drag, as discussed above. Nonetheless, approximating drag coefficients by comparison with published data can be useful. It is inexpensive with respect to both time and money, and will often get results “in the ballpark.”

Drag Coefficients for Various Vehicles Drag measurements for various bicycles and HPVs have been published by several investigators. Figure 9-8 shows a comparison of the drag area CD A for 7

Robertson, John A. and Crowe, Clayton T., Engineering Fluid Mechanics, 3rd ed, Houghton Mifflin, 1985. 8 Wilson, David G., Bicycling Science, 3rd Ed., MIT Press, 2004.

163


S V P H D N A S E L C Y C I B F O A E R A G A R D

.7 0

.6 0

.5 0

.4 0

.3 0

.2 0

AE RA GA RD

164

.1 0

0

ytil it tn U y G r et n r eeP a str cir T o eki pS B gn ni a iri t n aF uo ll M uF ec aR ,r e tn ec aci aR rT B da WS o 6 R ec 1V s gnaR ui d iri aR aF tn or TT F, r t i usr u et nac ir ai P. d T pi c nI nir xe P da tn C e 20 B 02 B , ec WS aR t r e cej o dt S gnso r po iri M re aF A ra eR VP ,B re H WS ca rw xu l o VPH Lderi F 6 VP 89 aF 5 H 1 M 20 02

s V P H s u o ri a v r fo a re a g a r D 8 9 e r u g i F

Aerodynamic Drag

a wide range of vehicles. Some caution should be exercised in interpreting the results, however. Some of the studies used wind tunnels, while others used coastdown tests. The speed (and hence Reynolds number) varied for the different tests. Measures of variability are in general not published, and hence the standard deviation of the tests is not known. The wind tunnel tests are assumed to have higher accuracy. The values shown for the Road Race, Time Trial and Individual Pursuit upright bicycles are averages of several different models that were measured in wind tunnel tests. These are most likely accurate average values for these types of bicycles at the professional level. Partially faired and unfaired recumbent bicycles are poorly represented in the figure. Recumbents are made in a wide range of styles, and it is reasonable to expect a wide range of drag area values. It is expected that well-designed, unfaired performance recumbents should have drag areas smaller than the most aerodynamic upright bicycles. The figure does not include enough data to demonstrate this, however. The two recumbents with the lowest drag areas each include partial fairings. Further, few high-performance recumbents are included. Only one tricycle is included. Several conclusions can be drawn from the figure, despite the reservations. •

•

•

•

•

Streamliners unquestionably have the lowest drag areas. World-class streamlined racers in particular have extremely little drag. Upright bicycles with riders sitting in an upright position have the largest drag areas. This category includes mountain bikes, European-style commuter bikes, and road bikes while riding with hands on the uprights. Performance upright bicycles can reduce drag over a standard road bicycle significantly. Recumbent bicycles, with some exceptions, generally rank with the best upright bicycles. Partial fairings can reduce drag noticeably.

165

CHAPTER

BICYCLE HANDLING

10

PERFORMANCE

Bicycle Stability Bicycles, despite being unstable at rest, are remarkably stable when in motion. Some have claimed that the invention of the bicycle was a revolution in thought, demanding extraordinary foresight to predict that a single-track vehicle could be stable enough to ride. It is much more likely that when Baron Karl Von Drais built his Draisienne—the first two-wheeled single track vehicle known—he had no notion of it being stable. Probably, he expected it would require balancing with the feet (which also provided propulsion by pushing against the ground). Perhaps the Baron experienced a moment of surprise and astonishment when the vehicle coasted unsupported down a grade without falling over! The ability of a two-wheeled vehicle to remain upright at speed, even though it cannot do so when stopped, is fascinating, and to many, a mystery. A common misconception, or perhaps oversimplification, is that the gyroscopic effect of the wheels makes a bike stable. Gyroscopic effects do play a role in bicycle stability, but it is relatively minor, particularly at low speeds. Many stable, rideable bicycles with little or no gyroscopic effects have been built. Consider the foldable scooters that have been popular with children and students in recent years. These two-wheeled vehicles have very small and light wheels. The angular momentum of the wheels must be extremely small compared to a bicycle, and almost negligible, yet they are easily ridden, even a low speeds. In 1970, David Jones1 built a bicycle with a counter rotating wheel that exactly cancelled the rotating inertia of the front wheel. His goal was to build an “unrideable” bicycle, in order to better understand bicycle stability. The bicycle turned out to be easily

1

Jones, David E. H., 1970,“The Stability of the Bicycle,” Physics Today, April 1970, pp. 34–40.

167


rideable, although the “feel” was a bit strange. Numerous “ice bikes” have been built by replacing the front wheel of a bicycle with a skate blade. These bikes can be stable and rideable, although they do look a bit odd. It is a very interesting sight to see a Bachetta highracer zip past with the rider high in the air and only a small blade in place of the front wheel. Figure 10-1 shows an upright bicycle with a skate blade in place of the front wheel. These examples illustrate that gyroscopic effects alone do not make a bicycle stable, yet they do contribute to stability. Any rider that has replaced light racing tires with heavy touring or winter tires has noticed the effect of the increased angular momentum. The bicycle does “feel” more stable, less likely to fall over. This perception is supported by analytical models, as will be seen. It is important to note that a bicycle does not have to be stable to be rideable. Skilled riders can balance on a stationary bike, ride backwards, balance on one wheel or ride a unicycle, and so forth. Also, virtually all bicycles have a low-speed range of instability that riders routinely operate in, if only briefly. However, a stable bicycle will remain upright on its own, without active control by the rider, whereas an unstable bicycle requires rider control to remain balanced.

Figure 10-1 Ice bike with skate blade in lieu of front wheel

168

Bicycle Handling Performance

To understand how a bicycle can remain upright, consider something that is not a bicycle at all: a single wheel. If a bicycle wheel is placed upright on the ground, it will quickly fall over, just as a bicycle falls over. In both cases, the wheel or bike is behaving more or less like an inverted pendulum, which might balance for a while but then falls over in a very unstable manner. Spin the wheel to give it some angular momentum and set it on a flat surface, and it behaves very differently. The wheel will roll away, initially staying upright and moving in essentially a straight line. As the rotational speed decreases, the wheel starts to lean and begins moving in a slowly tightening spiral until at some point the wheel collapses. This is due to gyroscopic effects acting in conjunction with ground forces and centrifugal acceleration.2 The action of a bicycle is different. Spin the wheels on a bicycle and release it on a flat surface, and the bicycle will initially take off in a straight line, similar to the wheel. Unlike the wheel, the bicycle most likely does not simply start a spiral leading to collapse. Instead, it first leans one way, then the other, weaving back and forth. If the bicycle is rolling within the right range of speeds, it will actually right itself after being disturbed, say by bumping the handlebars. Below this speed range, the weaving behavior becomes greater with each oscillation until the bike falls over. This is clearly different from the rolling wheel. What, then, is the difference between a rolling bicycle and a rolling wheel? The interaction between the tire and the ground is crucial in both cases. The wheel—whether attached to a bicycle or rolling free—can move about the surface on which it rolls. It is not constrained to move in a straight line. Nonetheless, there must be some lateral constraint on the wheel to keep it from slipping sideways. Dynamicists call anything that restricts the motion of a body a constraint. The number of ways a body can move is known as the degrees of freedom of that body. Any rigid body in three-dimensional space can move in six ways unless it is constrained: three translations (moving in the X, Y, and Z directions) and three rotations (rotating about X, Y, and Z). The wheel rolling on a flat surface cannot move vertically due to gravity pushing it down against the surface. This is a constraint, and eliminates one of the six degrees of freedom of the rolling wheel—vertical translation. It is spinning about its axis of rotation, and there is nothing to prohibit it from leaning side to side or turning right or left, so none of the rotational motions are constrained. Clearly, the wheel is free to move in the 2

At some point after launch, the wheel will start to lean either to the right or left. Gyroscopic effects cause the wheel to precess about the vertical axis, turning in the direction of the lean. Simultaneously, a lateral force due to the (now) off-center CG is developed at the contact patch between the wheel and the ground due to the lateral friction of the tire. This force causes the tire to turn, and also produces a moment tending to reduce the lean.

169


forward direction as it rolls. But is the wheel free to move right and left on the surface? This turns out to be a very interesting question. As the wheel rolls, it can move right or left, but the lateral friction between the ground and the wheel does not permit direct lateral motion of the wheel; that is, it cannot slide to either side. In order to move right or left, the wheel must turn, and then roll to one side. There is a constraint, but the constraint does not prevent the wheel from moving anywhere on the surface. The wheel has five degrees of freedom (translation on the plane of the surface and rotations about all three axes) but two non-redundant constraints! However, the lateral friction constraint does not really prohibit lateral motion, it just restricts how that lateral motion occurs (the rolling wheel must turn in order to move left or right). Thus it does not reduce the number of degrees of freedom. This is known as a non-holonomic constraint, a term that tends to crop up frequently in bicycle dynamics. The key idea is that the lateral friction of the tires prevents direct lateral motion (skidding sideways) but does not restrict the bicycle from moving right or left while it rolls. The non-holonomic constraints on the wheels are actually quite important for bicycle stability. A bicycle with no lateral constraints at all—imagine a bicycle on a frictionless, icy surface—cannot be made stable. But a bicycle with ordinary (holonomic) lateral constraints preventing all lateral motion also cannot be made stable—think of how quickly a bicycle will fall over if the wheels fall into crack in the pavement. In other words, a stable bicycle requires lateral friction at the wheels that does not prevent the bicycle from moving left or right on the road. Imagine a bicycle attached to a rail. The wheels roll without slipping along the rail, and the entire bike can rotate about the rail. (The contact patch must stay on the rail, but the wheel can pivot about the track as if on a linear bearing.) A bicycle launched on this track with spinning wheels would not wobble, but would simply fall over shortly after launch. As will be shown, the same would happen to a bicycle with both wheels attached to a single rigid frame (no headset bearing between the frame and fork). A bicycle becomes stable above low speeds primarily due to the coupling between the front fork/wheel assembly and the rear main frame and rear wheel. For this coupling to occur, the wheels must be able to move laterally on the road surface. (This is why longitudinal road cracks pose such a hazard to bicyclists—if the front wheel is caught in a crack, it cannot move laterally.) A better understanding of this coupling requires an investigation of the geometry of a typical bicycle. The front fork assembly (which of course includes the front wheel) is able to pivot, or steer, relative to the main frame of the bike. Almost invariably, the axis of the pivot is angled backwards, as shown in Figure 10-2. The angle between the ground and the steering axis is known as the headtubeangle, f. Most bicycles have

170


Figure 10-2 Some important bicycle parameters

the front wheel offset somewhat from the steer axis. This is known as fork offset or rake. (Fork offset is obtained by bending the fork blades, mounting the wheel dropouts off the fork center, welding the fork blades at an angle to the steering axis, or a combination of these.) Other parameters of interest include the location of the center of gravity location of the bicycle3 and the wheelbase. Figure 10-3 depicts the front wheel parameters in more detail. Trail (T) is the distance, measured along the road surface, from the intersection of the steering axis with the ground to the center of the tire contact patch. Trail is widely considered an important parameter related to bicycle handling and stability, and is sometimes provided in bicycle specifications. It is related to the more important parameter (from a dynamics standpoint) called mechanical trail (Tm). Mechanical trail is the perpendicular distance from the steering axis to the center of the tire contact patch. All road forces acting on the tire do so at the contact patch. Lateral forces produce moments about the steering axis, and mechanical trail is just the moment arm producing these torques. Trail (and mechanical trail) depends on the wheel radius R, the headtubeangle f, and the fork offset, S, and can be found with the following formulas: R cos φ − S T

=

sin φ

Tm = Rcosf – S

(10-1) (10-2)

3

The mass properties of the rear frame with rider and the front assembly are actually quite important and include the locations of the centers of gravity, the mass of each system, and the inertia tensors for each system.

171


Figure 10-3 Front wheel parameters

Any lateral force on the front wheel produces a steering torque as long as trail is not equal to zero. Due to the coupling between the front fork assembly and the rest of the bicycle, that torque not only changes steer angle, but also causes the bike to lean, or roll, to one side. This begins to get to the heart of bicycle stability. Consider again the bicycle launched across a flat, level surface such as a large parking lot. Initially, the bike is upright and balanced. Shortly after launch, the bike starts to fall to one side or the other, just as the rolling wheel did. If there was no lateral friction, the wheels would simply slide out from under the bike, and the CG would fall vertically until the bike crashed to the ground. But since there is lateral friction, a lateral force is developed at the wheels acting in the direction of the lean. (If the bike leans left, the force is directed to the left.) What happens next is complex, and depends on gravity, mass and mass distribution for the front fork assembly and the main frame, ground forces at the wheels, inertial and gyroscopic effects, and centrifugal force. It is also strongly dependent on speed. The common and crucial element is that vehicle roll (leaning to the side) and steering are coupled: if the bike leans, the fork turns. Conversely, if the fork is turned, the bike leans. When a bicycle with conventional geometry leans to the left, the handlebar and fork rotate so as to steer left. This is due primarily to the ground forces acting on the tire contact patch (which is located behind the steering axis) and partially to gyroscopic effects of the front wheel. On the other hand, if the bicycle, initially straight and vertical, is momentarily steered to the left, (by bumping one end the handlebar in the forward direction for instance) the bike will initially lean to the right—again due to the tire forces.

172


Acting together, these tendencies cause a disturbed bike to weave back and forth, oscillating between the effects of steer and lean. If the speed is within a certain range, the oscillations are stable, and the bike eventually returns to vertical, travelling in a straight line. The interaction between all the various forces acting on the bicycle, in conjunction with the coupling between the steering and rolling, lead to the following theoretical conclusions: • When the bicycle is stopped or moving at very low speeds, it is unstable and will simply fall over. • When the bicycle is moving at low speeds it is still unstable, but will oscillate back and forth prior to falling. The oscillations become progressively larger until the bike falls. • Above a certain speed, the oscillations get progressively smaller until the bicycle has returned to rolling straight in the upright position. Note that the new heading (yaw angle) may not be the same as the srcinal heading. • At very high speeds, the bicycle may not be able to recover. Instead of overcompensation and oscillatory behavior, the bicycle never quite makes it back to vertical. It will begin a gradual arc and eventually fall over. It is worthwhile to note that this instability is usually quite mild, and easily overcome by a rider. • A bicycle with negative trail can never be stable. • Increasing the mass moment of inertia of the wheels slightly improves the low-speed weave instability, but the high-speed capsize instability is made worse. These conclusions apply to a bicycle with an inert rider, as if the rider were replaced with a sack of concrete of the same weight. They are interesting, and help understand bicycle stability, but do not accurately predict the behavior of a bicycle with a real rider. The rider is constantly interacting with the bicycle, subtly and often unconsciously shifting her weight or providing steering inputs to achieve the desired response. Stability is less important than how the bicycle responds to these subtle inputs, or how the bicycle handles.

Bicycle Handling The handling characteristics of a bicycle are quite important to the rider, and must be compatible with the mission of the vehicle. Handling is much more than stability—few, if any, riders would want the most stable bicycle possible. (The

173


possible exception are bicycles built purely for high-speed sprints, such as those held annually at Battle Mountain, Nevada. These vehicles are designed to go as fast as possible in a straight line. Perhaps extreme high-speed stability is an advantage in this situation.) While stability affects handling, it is only one factor among several that affect the response of the bicycle. Handling qualities relate the rider’s actions to the response of the vehicle. Indirectly, it also includes the response of the bike to ‘noise’ inputs, such as rough pavement, road debris, or wind gusts. When the handlebar is pushed or moved, the quickness, magnitude, and nature of the response, collectively, are what is known as the handling characteristics, or handling qualities, of the bicycle. Handling qualities of a particular bicycle depend on the operational environment. Speed has a strong effect on handling, as every cyclist has learned through experience. Handling is also strongly dependent on surface condition and, to a much smaller extent, on the ambient weather conditions including wind and temperature. Consider a bicycle ridden at 5 m/s on smooth, dry pavement on a summer afternoon. A small push on the handlebar will result in a very different response than on the same bicycle at 20 m/s. The same small push riding on a frozen lake in January would result in yet a different response, even at the same speed. Changes in any number of variables can affect handling performance. For example, changing tires can have a significant effect. Perceived handling performance is quite personal and subjective. A bicycle that has very light controls might be perceived by one rider as difficult to control, while a second rider might find it nimble and practical. Each rider’s abilities, riding experience, and expectations affect perceptions of acceptable handling—so how does an engineer design for handling? Precise predictions of handling during the design stage of a new bicycle are very difficult to make. The number of variables involved—both design variables that can be controlled and environmental variables that cannot—make the problem very complex. Analytical models are very complex and incomplete with regard to handling. The traditional approach has been to start with a known design, modify it, and check the result. This approach works well for a new design based on similar bicycles for which data is readily available. Large bicycle manufacturers with years of experience producing similar bicycles can probably design a new model with high confidence in the handling characteristics. This is particularly true as upright bicycles have been manufactured for so long that there is a large body of comparative data available. The sport builder designing her first recumbent bicycle has quite a different problem. Not only is data scarce, but cost in time and money to build, test, modify and repeat may be unaffordable. Fortunately, it is generally adequate to be able to only roughly predict the handling qualities during

174


the design of a new vehicle. As long as the handling characteristics of a new design are generally as desired, the design is a success. Experimental programs can then fine-tune the handling to greater precision. General handling characteristics may be described as “easy to balance,” “responsive and sporty,” “stable during cruise.” These might correspond to an entry-level recumbent; a sport or racing bike; or a touring recumbent, respectively. Patterson’s method provides the designer with a tool that can predict the general handling characteristics of a new design in the early design stage, even before complete mass properties are available.

Patterson’s Method Bill Patterson developed a method for predicting bicycle handling for a bicycle design course he taught at California Polytechnic Institute. 4 The method is based on the premises that (1) the method should be an easy tool for the bicycle designer to use, and (2) handling qualities can be defined mathematically by considering the motion and force of the rider’s hands on the handlebar. In order to make the method easy to use, the dynamic equations have been extremely simplified. The mathematical accuracy of the result is questionable, but the method does work. The pragmatic engineer is willing to forgo mathematical rigor for a design tool that works! The second premise allows the intentions of the rider to be considered, if only in a limited way. Most bicycle riders use more than their hands to control a bicycle. Subtle weight shifts and body movements contribute significantly to controlling a bicycle. These are the only methods of controlling a bicycle when riding without hands. Nonetheless, being able to predict the response to hand inputs provides a good insight into the handling qualities of the bicycle. Patterson’s method will allow designers to determine bicycle geometry that produces desired handling qualities during the early stages of design. The required input data is minimal, and the equations are fairly simple to implement. However, the method does not predict handling with a high level of precision, and precision should not be an expectation. As noted, there are many variables involved, so a simple method that predicts general handling characteristics is actually quite a valuable tool. It is quite possible for a novice to design a bicycle that meets design goals such as: easy to ride; very stable at speed; light controls; nimble and sporty; or good low-speed maneuverability. While the method is simple to use, it does require accurate interpretation of the results. Initially, it is often useful to compare a new design with known bicycles. Two examples are included 4

Patterson, W. B., 2001, The Chronicles of the Lords of the Chainring, W. B. Patterson, Santa Maria, CA, USA.

175


at the end of this section to provide a basis for comparison. Analyzing a bicycle that is ridden frequently also assists with interpretation of the results. Patterson uses a slightly different set of parameters than is used throughout this book. The wheel radius is measured to the center of the tire, rather than the tire patch. This affects the value obtained when calculating trail. Patterson’s value of trail will be slightly smaller than that calculated by Equation 10-1. Mechanical trail is physically identical to that previously defined, but must be calculated differently to account for the different definition of wheel radius. Rather than use the head tube angle, Patterson measures the declination of the head tube from the vertical, and denotes this angle as b. Beta is the complement of the head tube angle f. These definitions are illustrated in Figure 10-4. Nomenclature for Patterson’s Method L b h If Ir kx K K1–K5 m

Wheelbase Horizontal distance from rear axle to center of gravity Vertical distance from ground to center of gravity Mass moment of inertia of the front tire about the wheel axis Mass moment of inertia of the rear tire about the wheel axis Radius of gyration about the centroidal x-axis (longitudinal axis) Roll Control Authority constant Patterson’s constants (see text for explanation and interpretation) Combined mass of bicycle and rider

Figure 10-4 Patterson’s steering parameter definitions

176


Q Rf Rr RH S T

Moment about the head tube axis Front wheel radius, measured from axle to mid-point of tire Rear wheel radius, measured from axle to mid-point of tire Handlebar radius, measured from steering axis to center of hand Fork offset Trail (see Figure 10-4)

Tm Mechanical trail (see Figure 10-4) V Velocity of the bicycle Wx Roll rate (rate of rotation about the longitudinal axis) Wz Yaw rate (rate of rotation about the vertical axis) b Compliment of the head tube angle, measured from the vertical d Steering angle The change in bicycle roll rate with respect to control inputs is considered to be the most important handling characteristic. At low speeds, the fork is rotated through a relatively large angle during maneuvers, and the bicycle is controlled by actually moving the handlebars. Roll control authority is the change in roll rate with respect to the distance the handlebar is moved, and is very important for low-speed handling. At higher speeds, the fork rotates through only a very small angle, and the force applied to the handlebar becomes more important than the distance through which it moves. Patterson assumed that the intentions of the rider could be measured by a weighted sum of the movement and the force applied to the handlebar through the rider’s hands. The change in the roll rate with respect to the intention function is the control sensitivity of the bicycle, another very important handling characteristic. Roll authority increases linearly with speed, but control sensitivity increases non-linearly with speed to a maximum, and then decreases. The value of the maximum sensitivity and the speed at which it occurs are both significant. A third speed-dependent function also provides important information regarding handling. The forces acting at the tire contact patch produce moments about the steering axis. Conceptually, the total moment can be thought of as a torsional spring with a stiffness that is dependent on speed. This torsional spring constant is the third important characteristic that the designer should consider. It is non-linear with speed, decreasing at an increasing rate. The five “Patterson constants,” K, K1, K2, K3, K4, and K5 are used in the formulas for roll control authority, control sensitivity, and the torsional spring constant. Once a user has some experience with the method, the values of these constants can provide insight into the handling qualities of a new design. The values of K3 and K5 are set by the user, and used to define acceptable handling. The remaining constants are calculated based on bicycle parameters.

177


Trail is calculated as usual with Patterson’s method. The formula is repeated here for convenience. (Recall that the wheel radius is measured to the center of the tire, rather than the ground.)

T

=

R sin( β)

cos(β)

−

S

cos( β)

(10-3)

The equation for mechanical trail must be modified to adjust for the radius of the tire. Physically, it is the moment arm about the steering axis relative to the tire contact patch. All lateral forces developed at the tire produce a torque about the steer axis. Mechanical trail is the torque arm.

Tm = Tcos(b) + rtsin(b)

(10-4)

Roll Control Authority is defined as the change in the roll rate of the bicycle with respect to the distance the handlebar is moved. It is most applicable to lowspeed maneuvering, in which the handlebar is deflected through relatively large angles. It is a linear function of speed, and the slope K provides insight into lowspeed control. Roll control authority is given by

 b = δ hL 

dWx Rd H

 cos(β)   V R 

(10-5)

= KV

(10-6)

H

Or dWx RH d δ

Where

K

b  =  hL   cos(β)    R 

(10-7)

H

For bicycles designed for low-speed maneuvering, such as short-wheelbase bikes designed for urban use, the value of K should be relatively large, greater than 3.0 m–2. Greater values of K provide better low-speed control authority, but may result in bicycles that are overly responsive at high speeds. Touring bicycles,

178


or bicycles designed for high-speed cruise, should have lower values. A long wheelbase touring bike may have a value of K as low as 0.35 m –2. Figure 10-5 illustrates the roll control authority plot for a typical bicycle. The mid-range value of K indicates this is a good all-around bicycle. The torsional spring constant is the torque acting on the steering column. It is a decreasing quadratic function of speed, and is quite important for handling qualities. It is given by the change in torque on the steering column with respect to the change in steer angle: dQ dδ

= K 1 − K 2V

2

(10-8)

Where the coefficients are Patterson’s constants K1 and K2, given by: (10-9) And

K

T 2

=

m

 mb cos(β)   k 2  L2   k 2 − h 2   x

(10-10)

x

ROLL CONTROL AUTHORITY AS A FUNCTION OF SPEED 45 K = 2.03 40

) -s m / 35 1 ( Y 30 IT R O H 25 T U A L 20 O R T N 15 O C 10

5 0 0

2

4

6

8 10 12 SPEED (m/s)

14

16

18

20

Figure 10-5 Roll control authority

179


The torsional spring constant typically has a positive value at zero speed and decreases quadratically as speed increases, as illustrated in Figure 10-6. When the spring constant is positive, the bicycle is unstable. The rider must move the handlebar one way, while applying force in the opposite direction! This is typical of most bicycles at very low speeds. As speed increases, the spring constant decreases, and at some speed becomes zero. This is the threshold at which the bicycle becomes easy to ride. As the speed increases, the torsional spring constant becomes much more negative, providing a torque that tends to oppose disturbances and stabilize the bicycle. Figure 10-6 shows the torsional spring constant curve for a long-wheelbase bicycle with a 63 kg rider. The bicycle is a Gold Rush, manufactured by Easy Racers, Inc. in California. This is a well-known bicycle with very nice handling characteristics and a very light feel to the controls. The plot indicates the initial value of 8.21 N-m/rad. For a long-wheelbase fast cruiser, this is a reasonable value. It should be lower for a bike designed to spend much time at low speeds. K1 should not exceed 10 N-m/rad, and should normally be smaller for low-speed bikes. Sometimes it may be difficult to achieve this value. Values up to 25 N-m/rad can be used if necessary, but these bicycles may be difficult to control at low speeds. A more vertical head tube (smaller b) will reduce the value of K1.

BICYLE HANDLING CURVES FOR EASY RACERS GOLD RUSH 20

2

SPRING RATE = 8.21 - 0.27*V 0

–20

) d r/a m - –40 (N E T A R –60 G IN R P S –80

–100 ROOT = 5.47 m/s –120 0

2

4

6

8 10 12 SPEED (m/s)

14

16

18

Figure 10-6 Torsional spring constant as function of speed

180

20


The curve in the figure crosses zero at about 5.5 m/s, again an acceptable value for a bike designed for the open road. For city bikes, a lower value would be better. The curve drops off rather slowly, reaching –100 N-m/rad at about 20 m/s. This indicates a bicycle with a very light feel, a characteristic of the Gold Rush. The value for K2, 0.27 kg, is toward the low end of the acceptable range. Bicycles with greater control authority may require higher values of K 2. Normally these bicycles will have higher values for both K 2 and K4. Long-wheelbase recumbents such as the Gold Rush have low control authority and can have very light controls. The constant K2 given by Equation 10-10 does not include terms due to angular momentum of the wheels. For lightweight bicycle wheels, the error is not large, but for heavy or wide tires, there may be a significant discrepancy. Two additional terms must be added to K2 if angular momentum is to be included. Motorcycle wheels have substantially more inertia than bicycle wheels, so the additional terms should be used for motorcycle design. Bicycles designed with large, heavy wheels should also include these terms. The inertia for light road wheels is small enough that Equations 10-9 and 10-10 yield acceptable results. If heavy wheels are used or enhanced accuracy is required, the inertia of the wheels should be measured and included. The two additional terms, together calledDK2, are given by

∆K= 2

If

cos(β) LR f

b cos( )β   β +sin( )  h 

(10-11)

The torsional spring constant with the angular momentum terms included is then given as dQ dδ

=K −1

(K + 2∆

K 2 )V 2

A key factor in bicycle handling and stability is the coupling that occurs between steering and rolling. If the bicycle is leaned away from the vertical position, the front wheel and fork assembly will rotate. The moment on the fork assembly that produces this rotation is known as fork flop. More rigorously, fork flop is the change in moment on the steering column with respect to the change in roll angle, or:

dQ dθ

mgb  = −T cos(β) *    L 

181


Patterson divides both sides by the handlebar radius in order to obtain the force on the rider’s hands rather than the moment. Patterson’s fork flop in terms of force is given by:

FF

mgb  T βcos( )*  LR 

dQ −= dRH θ

=

  

1 H

  

(10-12)

As with most of the parameters, there is a range of acceptable values for fork flop. Riders need some minimum amount of fork flop to provide feedback, or ‘feel’ for the bicycle. Bicycles with no fork flop are very difficult to ride. A disturbance that causes the bike to lean can not be felt at the handlebar, and there will be minimal stabilizing effect on the bicycle. The only practical ways to achieve this undesirable condition are to reduce trail to zero (T = 0), or reduce the front wheel load b  to zero   mg = 0. This emphasizes the importance of trail and a good front/rear L weight distribution. Patterson suggests that 225 N/rad is a reasonable maximum for a recumbent; above that value, riders have more difficulty. For upright bicycles, lower values are typical. For example, a downhill mountain bike with a 64° headtube angle will have around 60 N/rad fork flop, while a road bike with a 73° headtube may have as little as 40 N/rad. Recumbents usually require somewhat higher values. The Easy Racers Gold Rush Replica, which has very light steering, is at the lower end of the range with around 80 N/rad. Some special-purpose bicycles can exceed 225 N/rad, but the handling becomes more sluggish and control forces larger. For bicycles that spend very little time maneuvering, such as speed bikes, this could be an advantage. Some speed bikes do exceed the recommended maximum, trading maneuverability for straight-line, high-speed stability. Since fork flop is dependent on trail, the maximum allowable fork flop can be used to set a maximum acceptable value for trail. Substituting the maximum value of fork flop into Equation 10-12 and solving for T gives an expression for the maximum trail:

TMAX

=

FFMAX  b mg cos(β)     L   RH

(10-13)

 

If trail exceeds this value, fork flop will be excessive and the control forces may be excessive for comfortable riding. This is a very useful design tool. A bicycle should not be overly responsive to control inputs at any speed. Patterson considers handling to be the bicycle response to the rider’s intentions,

182


where intentions are defined as a weighted sum of handlebar motion and handlebar force. This is perhaps a bit ad-hoc, but it does appear to work in practice. Note that this definition of intent does not include lateral weight shift, which most bicycle riders do automatically while riding. A mathematical expression for rider’s intent is given by

INT =R

Hδ

K +

3

Q RH

(10-14)

The first term is just the distance the rider’s hands move the handlebar during maneuvering, while the second term is the force acting on the rider’s hands, weighted by the proportionality constant K3. Patterson recommends a K3 value of 1/1500 m/N, based on experiments taken with a variable geometry bicycle. (The value turned out to be very similar to the US Air Force recommendations for the control spring for jet fighter aircraft, which may provide some additional validation for this number, at least for healthy men.) This value appears satisfactory for most bicycle design problems; although it should perhaps be increased when designing bicycles for small children or people with reduced arm strength. Control sensitivity is the change in the roll rate of the bicycle with respect to the intentions of the rider, or

dWx dINT

. If a bicycle has a large and quick response to a rid-

er’s handlebar inputs, it may be difficult to control. The rider will tend to over control the bike. That is, the rider will push the handlebars harder than was needed, and the bike responds too much. This is similar to what often happens when a child is learning to ride a bicycle for the first time. The child over controls, and has difficulty controlling the bicycle. As control sensitivity increases, even experienced riders have difficulty with over controlling the bike. Some people will not be able to ride a bike with excessive control sensitivity, while others may be uncomfortable on it. Riding such a bicycle on roads shared with motor traffic is potent ially dangerous, even for experienced riders. It is thus important to keep control sensitivity at reasonable levels. However, control sensitivity limit is subjective, and varies from person to person. The recommended values provided here seem to work for most people, and permit the design of a bicycle with desired handling characteristics. Since we have an expression for rider’s intention, we can get the following expression for control sensitivity. dWx dINT

K V

= RH

K + 3  RH

4

 ( −V K1 + K 

(10-15) 2 2

)

183


The constant K4 is just the control authority constant K multiplied by the handlebar radius. K4 = KR H

(10-16)

Control sensitivity, like control authority and the torsional spring constant, is a function of velocity. It is non-linear, increasing from zero when the bike is not moving up to some maximum, and then decreasing. Both the maximum value, and the speed at which it occurs, are important considerations for the designer. Patterson recommends a limit of 12 rad/m-s for touring bicycles and 18 rad/m-s for nimble bicycles that are required to do a lot of maneuvering. Experience indicates that the 12 rad/m-s recommendation may be a bit low, and difficult to achieve for many bike designs. The higher limit seems more practical for most bicycles, and may be taken as a working upper limit unless design requirements specifically dictate otherwise. Bicycles with values up to 25 rad/m-s can be ridden, but take more skill and are more difficult to ride. Figure 10-7 shows a typical control sensitivity plot. The shape of the curve—gradual curvature with smoothly arcing trajectory that peaks below 18 rad/m-s—is typical of good-handling bicycles. The peak value is a

ROLL CONTROL SENSITIVITY AS A FUNCTION OF SPEED 18 16 ) -s m / 14 (1 Y 12 T I IV IT10 S N E S 8 L O R T 6 N O C 4

2 MAX SENS = 16.3 rad/m-s AT 14.7 m/s

0 0

2

4

6

8 10 12 SPEED (m/s)

14

16

18

Figure 10-7 Roll control sensitivity for typical bicycle

184

20


reasonable 16.3 rad/m-s. This bicycle has been ridden at speeds exceeding 22 m/s (48.4 mph) with no handling problems. Taking the derivative of Equation 10-15, setting it to zero and solving for V gives the velocity of the maximum sensitivity. Substituting back into Equation 10-15 gives the value for the maximum control sensitivity. Simplifying the result and eliminating second order terms yields the following approximation for maximum control sensitivity: K4

dWx dINT

=

MAX

(10-17)

2 K 3K 2

Once an upper limit for the maximum control sensitivity is set, Equation 10-17 can be used to determine the minimum acceptable trail. Designating the control sensitivity upper limit as CSMAX, expanding K1, K2, and K4, and simplifying gives

TMIN

K    b  1 =  2 32 +   2 2  CS ⋅ 2  MAX    m  h x

1

k

 

(10-18)

The first term consists of constants, since CSMAX and K3 are determined a-priori. These can be grouped into a new constant K 5. The expression for minimum trail then becomes (10-19)

For a maximum sensitivity limit of 18 rad/m-s and K 3

1 =

1500

, K5 = 1.2. This

value appears to work very well for most bicycle designs. A value of 2.6 corresponds to the lower sensitivity limit of 12 rad/m-s. Higher K5 values should be used to design bicycles that have very low control sensitivity, such as high-speed sprint bikes. The resulting bikes will require significant control input for maneuvering, and would be inappropriate in urban riding requiring high maneuverability. Lower values of K5 produce bikes with more sensitivity requiring less handlebar input for maneuvering. These bikes can be effective for everyday transportation in a variety of environments.

185


Summary Table 10-1 Summary of Parameters Used with Patterson’s Method Parameter K

Units (SI) 1/m2

TypicalValues 1 to 3 m–2

C o mme nt s • K is the slope of the control authority line. • Higher values give better low speed control authority. (Often for SWB bikes) • Lower values best for road/ touring bikes (often LWB) • Increase K by: • Shortening wheelbase (decrease A) • Moving seat Forward (Increase B) • Lower seat (Decrease h) • More vertical headtube (decrease b) • Higher values may make bike overly responsive (‘twitchy’) at high speeds.

K1

N-m/rad Less than 10 Nm/ rad best. Avoid large values greater than 25 Nm/rad

• “Trim” force on handlebars during a steady turn • Minimize K1 for bikes that spend much time at slow speeds. • K1 is reduced by more vertical headtube (smaller b)

K2

186

N-s2/m

.2 to .3 Typical

Torsional spring constant: TSC = K1 – K2V2


K3

m/N

1/1500

• Patterson’s recommendation. • May need to increase forchildren or smaller and weaker riders.

K4

1/m

.15–.5

Maxallowablesensitivitydepends on K4. Lower K4 reduces max sensitivity.

K5

Kg-m2

1.2 Kg-m2

• Selected by User • Determines the responsiveness of the bicycle based on limiting max sensitivity. • Patterson recommends: • 2.6 for “Normal” Bike • May be high in practice • 1.2 for responsive bike • Works well for many bikes • 1.8 works well for all-around bike • High-speed racing bikes can have larger values—pilot can pedal without concern for controlling the bike.

Range: 1.2 to 2.6 Kg-m2

Trail

m

Minvaluedepends • Important design parameter on max allowable for bicycles. sensitivity. • High speed bikes must have adequate trail to avoid exMax value depends cess sensitivity. on limiting fork • As front wheel diameter flop.

decreases, fork offset must decrease to maintain trail. • Seat back angle and CG height affect the minimum required trail.

187


Table 10-1 (Continued) Parameter

ForkFlop

Units (SI) N

TypicalValues

50–225N/rad

Minimum value: >200 N/rad Low CG >50 N/rad High CG 225 N/rad Max

Max Sensitivity

rad/ (m-s)

C o mme nt s

• Adequate fork flop makes the bike easy to control—the rider can sense roll error and make corrections without conscious effort. • Excessive fork flop makes bike sluggish. • Minimum recommended value depends on CG height. • Max value can be exceeded for bikes requiring exceptional high-speed stability.

Less than 18 rad/ • It may be difficult to achieve (m-s) for smooth 12 rad/(m-s) for some bikes. control at high Very satisfactory handling can speed. often result even if this limit is Patterson recomexceeded. mends less than • K5 establishes the maximum 12 rad/(m-s) for sensitivity limit. normal bikes and • 18 rad/(m-s) corresponds less than 18 rad/ to K5 = 1.2 Kg-m2 (m-s) for more • 12 rad/(m-s) corresponds responsivebikes. to K5 = 2.6 Kg-m2 • Max Sensitivity depends on K 2 and K4 • Bikes with higher K4 and better low speed capability should have larger K2 and give more ‘feel’ at the handlebar. • LWB bikes have low K4 and can have light feel.

188


Function Control Authority

Comments • Control authority is the responsiveness (agility) of the bike, and varies linearly with speed. • Control authority is enhanced by: • Decreasing wheelbase (reduce L) • • • •

Moving CG forward (increase b) Reducing CG height (reduce h)larger effect at high speeds More vertical headtube (decrease b) Decrease handlebar width (decrease Rh)

• Steep line (high K) gives better low speed handling, but the bike may be prone to overcontrol at high speeds. • SWB bikes tend to have higher K than LWB bikes. • Shallow line (low K) has poorer low speed handling, but higher max speed without over control problems. Torsional Spring • Positive values are unstable, negative values provide Constant stable control. (Most bikes are unstable at very low

speeds.) • Initial value should be low, less than 10 N-m/rad for easy starting • Root should be small value for easy starting and lowspeed riding. • More vertical headtube enhances low-speed stability. Control Sensitivity

• The ‘feel’ of the bike. • Light feel bikes have higher control sensitivity. • Heavier feel bikes have lower control sensitivity. Should be smooth, somewhat flat curve with a maximum less than 18 rad/(m-s)

Additional Comments

As B increases (forward CG) roll control is enhanced. This facilitates hill climbing and low-speed handling. More vertical headtube increases responsiveness and reduces low-speed instability. Wider handlebars increase the speed at which maximum sensitivity occurs. This may solve ‘twitchy’ handling problems.

189


Example 10-1 A recumbent bicycle design is proposed for a vehicle to be used for touring on paved surfaces. It should have good high-speed handling qualities. Most riding will occur on open roads, although some may also be in urban areas. A medium feel is desired. Based on the design mission for the v ehicle, the standard recommendations for Patterson’s K 3, and max fork flop are a ppropriate, giving values of K3 = 1/1500 m/N and max fork flop = 225 N/rad. Since this bike is a touring bike that will spend some time in urban riding, a mid-range value of K 5 = 1.8 kg m 2 is chosen. (Note, this value seems good for many all-around bicycles.) The initial design, completed by a novice designer, included the following specifications:

Parameter Totaldesignmasswithrider DesignCGlocationforwardofrearwheel Design CG height Radius of gyration about centroidal x-axis

Value 75

Kg

.62

Fork offset (rake) Handlebarradiusfromsteeraxistohands

m

.65

m

.25

Wheelbase Headtubeinclinationfromvertical

Units

m

1.1 19 .030

m Degrees m

.2

m

Fronttiresize

28-406

ERTOsize

Rear tire size

28-622

ERTO size

It is assumed that the inertia of the tires can be neglected. This bicycle was analyzed with Patterson’s method. The results indicated an unsatisfactory design. The program output indicated: Bicycle Handling Program Analysis Based on Patterson’s Method Bicycle: QB Bike – First Iteration

190


Trail = 0.043 m Trail is not within Limits of 0.273 to 0.115 m Patterson’ Constants: K1 = 4.912 N/rad K2 = 0.212 N-s^2/m-rad K4 = 0.820 1/m Fork Flop = 84.3 N/rad Maximum Sensitivity = 36.0 AT 16.1 m/s MAXIMUM SENSITIVITY = 36.0 AT 16.1 m/s

BICYLE HANDLING CURVES FOR QB BIKE -- FIRST ITERATION ) 40 20 s 2

120 ) s 100 /m (1 y ti 80 r o th u 60 a l o tr 40 n o C

) 0 d a r/ -20 m (N

K = 4.10

E T -60 A R G-80 IN -100 R P S -120

0 5

10

15

20

SPEED (m/s)

25

-140 0

ROOT = 4.82 m/s

5

10 15 20 SPEED (m/s)

m / 35 (1 Y30 IT IV25 T I S20 N E S15 L O 10 R T N5 O C0

-40

20

0

SPRING RATE = (4.91 - 0.21*V ) N-m/rad

25

0

MAX SENS = 36.0 rad/m-s AT 16.1 m/s

5

10 15 20 25 SPEED (m/s)

The control authority K is a high for bike designed to spend much time on the open road. The spring rate plot indicates a moderate feel, with both K1 and K2 within the recommended ranges. The maximum sensitivity is very high,exceeding the limit by more than a factor of two, and the trail is below the minimum trail requirement. Fork flop, at 84 N-m, is low, but within limits. This bicycle has excessive control sensitivity at speed, and should be re-designed. Long wheelbase bicycles tend to have good cruising characteristics, so significantly increasing the wheelbase might help. The rider is quite laid back in a very aerodynamic position. Placing him more upright would not only make the vehicle more comfortable for long rides, but also increase the radius of gyration, thus reducing the high sensitivity of the bicycle. It might improve visibility for the rider, which could be important for touring. Increasing trail would also help reduce the high sensitivity. For a bike designed for long hours cruising, a more inclined headtube might be a good means of increasing trail. The handlebars on a long wheelbase bike can be wider, which will improve

191


control authority. Finally, a touring bicycle might be more comfortable with larger tires, perhaps 40 mm section width. Incorporating these changes, and moving the rider aft to fit the crank behind the rear wheel gives the following revised parameters:

Parameter Totaldesignmasswithrider

Value 75

Design CG location forward of rear wheel Design CG height Radius of gyration about centroidal x-axis Wheelbase Headtube inclination from vertical Fork offset (rake) Handlebar radius from steer axis to hands

Units Kg

.5 .7

m m

.4

m

1.64 31 .025 .29

m Degrees m m

Fronttiresize

40-406

ERTOsize

Reartiresize

40-622

ERTOsize

The revised design was analyzed with Patterson’s method, giving the following results. Bicycle Handling Program Analysis based on Patterson’s Method Bicycle: QB Long Wheelbase Bike Trail = 0.105 m Trail is within Limits of 0.099 to 0.339 m Patterson’ Constants: K1 = 9.718 N/rad K2 = 0.295 N-s^2/m-rad K4 = 0.373 1/m Fork Flop = 69.5 N/rad Maximum Sensitivity = 13.9 at 19.9 m/s

192


The revised design is a long wheelbase recumbent that will handle well during high-speed cruising, yet it still retains adequate maneuverability for in town. The revised design is compatible with the mission of the vehicle. Control authority is reduced, but the value for K is within the expected range for this type of bike. The torsional spring constant indicates a fairly light feel, although K1 is near the upper limit and the root is a bit on the high side. This can be acceptable for a touring bike, however, as most of the time it will be operated at speeds above 6 m/s. The design could be improved by further reducing K1. The maximum control sensitivity is below the maximum value, indicating that the bike will not be overly sensitive at speed. Trail is high, but within the acceptable range. Fork flop is a bit low, but within the acceptable range. The controls should feel light to the rider.

193

CHAPTER

MULTI-TRACK VEHICLE HANDLING

11

PERFORMANCE

Multi-Track Vehicle Handling A multi-track vehicle generally has at least one axle with more than one wheel. Imagine a vehicle riding in a straight line over a few inches of new snow. Unlike a bicycle, which would leave a single track in the snow, a multi-track vehicle will leave two or more tracks. Hence the name. Multi-track vehicles include tricycles (three wheels), quadricycles (four wheels) and other, less common wheel numbers and arrangements. In general, multi-track vehicles exhibit zero-speed stability. That is, they do not fall over when stopped. At speed, both stability and performance can vary significantly, depending on configuration, design, and environmental variables. This chapter introduces basic handling parameters and stability indices for multi-track vehicles. Stability and handling are two different, but not unrelated, topics. Stability is the tendency for a vehicle, under specified conditions, to return to its current state after a disturbance. For example, a vehicle in a steady-state turn experiences a transient steering input into the direction of the turn. The steering input is a disturbance that decreases the turn radius and increases the lateral acceleration. If the vehicle is stable, all wheels remain in contact with the ground and the vehicle continues the turn. If the increase in lateral acceleration due to the transient input is excessive, the vehicle can roll over toward the outside of the turn, a form of instability called rollover. Handling is a general term used to describe the way a vehicle responds to driver inputs. It includes a diverse set of parameters related to steering, suspension, weight distribution, and tire/road interaction. There is no single definition of what constitutes “good” handling: desirable characteristics depend on the mission and nature of the vehicle. The important handling characteristics should be

195


considered early in the design stage, and incorporated into the vehicle design specifications. Even so, the handling characteristics of a vehicle are often a compromise with other design constraints. A tricycle designed to be sporty and fun to ride may require very responsive steering, good tactile feedback (good “road feel”), and a high resistance to rollover. In contrast, steering stability and low rolling resistance may be more important on a touring trike. It is generally not possible for a single vehicle to optimize all aspects of handling. This chapter is devoted to handling characteristics for rigid multi-track vehicles. It includes low- and high-speed cornering, tire stiffness, understeer gradient, and rollover and pitchover thresholds. Rigid vehicles do not include suspension other than that provided by tires. All concepts presented here are valid for suspended vehicles, but the formulas become more complicated, as suspension effects must be included. The interested reader is encouraged to read more in the references.

Definitions and Nomenclature Figure 11-1 illustrates several vehicle parameters applicable to handling and stability. Three wheels of a tadpole tricycle are shown. The wheelbase L is the longitudinal distance from the rear axle to the front axle. The center of gravity is located a distance b forward of the rear axle and a heighth above the ground. The castor angle is the projected angle between the steering axis and a vertical plane, as seen in the side view. The castor angle is positive when the steering axis is inclined rearward when moving along the axis upwards from the ground. The kingpin inclination angle is the projected angle between the steering axis and a vertical plane as seen in the front view of the vehicle. The inclination of the plane

Figure 11-1 Definition of multi-track terms

196

Multi-Track Vehicle Handling Performance

of the wheel from a vertical plane is the camber angle. Camber angle is positive when the top of the wheel leans away from the vehicle. The scrub radius, also known as the kingpin offset, is the horizontal distance, measured in the front projection, from the center of the tire contact patch to the intersection of the steering axis and the ground. Wheel track is the horizontal distance between the tire contact patches, and should not be confused with the kinematic track, which is the horizontal distance between the intersection of the wheel axes and the steering axes. Kinematic track is significant in the design of steering mechanisms, but is only applicable when two side-by-side wheels are steered.

Low-Speed Cornering Ackerman Steering At low speeds, no lateral force is developed, and the wheels should roll without slipping over the pavement. For this to occur, the extended centerlines of the axles for all wheels, as seen from above, must intersect at a common point, as shown in Figure 11-2. The figure uses a bicycle model to define the Ackerman steering angle. Multitrack vehicles can be modeled as equivalent single-track vehicles that do not lean while turning. This model is quite different from an actual bicycle, and is used for illustrative purposes only.

Figure 11-2 Ackerman steer angle and turn radius

197


The turn radius is the distance from the center of the turn to the center of gravity of the vehicle. For the bicycle model shown, the steer angle of the front wheel is related to the turn radius and wheelbase by

L δ A = tan −1   R

(11-1)

This angle is known as the Ackerman angle, and is the correct steer angle for a tricycle with only one steered wheel. For multitrack vehicles with pairs of steered wheels—such as most tadpole trikes and all quads—the inside and outside wheel angles must differ in order to prevent slip. This is apparent from Figure 11-3. For true rolling to occur, lines passing through the axles of each wheel must intersect at a common point. Due to the offset of the front wheels, the inside wheel must have a larger steer angle than the outside wheel for this to occur. The angles for the inside and outside wheels depend on wheelbase and kinematic track as well as turn radius. They are given by L  δi = tan −1  T  R−  2  

δo = tan

−1

 L T R+ 2 

(11-2)

  

Figure 11-3 Ackerman steering angle and turn radius for tadpole tricycle

198


Where T is the kinematic track, or the distance between the kingpin axes at the location of the steering arms. Note that kinematic track differs from wheel track for nearly all practical vehicles. This is due to both the kingpin inclination angle and the wheel offset. The correct steering angles are achieved by the steering mechanism. In practice, there are only a few mechanisms that are used for steering. They include the track rod (Figure 11-4), the symmetric six-bar linkage (Figures 11-8 and 119), and rack and pinion (Figure 11-10). None of these mechanisms can achieve the angles in Equation 11-2 exactly. In general, the angles are exact for only two steering angles—one for neutral steer (wheels straight) and one at a small turn radius. At all other turn radii, the actual steering angles deviate from the ideal. The amount of deviation is known as steering error, and can be minimized with well-designed mechanisms. The largest deviations occur at large steer angles, corresponding to small turn radii. The minimum turn radius should thus be specified before optimizing the steering mechanism for minimum deviations. Tadpole and quad HPVs require close attention to optimizing steer angle over the full range of steering. Human-powered vehicles are power-limited, and poor steering accuracy can lead to increased resistance during maneuvering. The relative cost of this resistance is higher than for motorized vehicles. Exact steering

Figure 11-4 Track rod steering mechanism, showing neutral steer and 30° steer

angles for a tadpole trike

199


mechanisms that do not deviate from the true Ackerman angle over the entire range of steering angles are possible, but the complexity outweighs the advantage. The mechanisms described here offer a very good compromise between complexity and accuracy. Properly designed, the steering errors are quite small. Three different mechanisms commonly used for Ackerman steering are described in this chapter. The mechanisms can be compared on the basis of complexity, accuracy (or the design effort required to obtain a specified level of accuracy), and sensitivity. Steering sensitivity is the change in steering angle of the wheels with respect to the change in position of the handlebar. The rider of a vehicle with very sensitive steering need only move the handlebar a very short distance to have a significant change in the steering angle of the wheels. Overly sensitive steering produces vehicles that are difficult to control, while vehicles with less sensitive steering may be sluggish to respond. Track Rod Track rod steering consists of steering arms attached to each steering knuckle which are connected by the track rod, as shown in Figure 11-5. Track rod steering is simple, provides good Ackerman compensation, and makes toe-in adjustment easy and straightforward. The track rod simply ensures that the two wheels steer together to achieve Ackerman steering angles. It requires a separate linkage for actual steering control. This could be a tie rod from the steering column to one of the steering arms. Direct steering, in which separate handlebars attach directly to each steering knuckle is also used with track rod steering. Catrike uses this method, for example. A variety of other methods with more complexity may also be used. The track rod is usually placed behind the axle. For the aft mounted track rod, the steering arms angle inward, and toe-in—used to preload the steering mechanism and eliminate looseness—will place the track rod in tension. Kinematically, it can be placed forward of the axle, but the track arms must then angle outward,

Figure 11-5 Track rod steering parameters

200


toward the wheels. Interference with the outside wheel usually precludes the forward arrangement. Toe-in—a slight angling inwards of the wheels—is used to pre-load the steering linkage. With a track rod behind the axle, toe-in places the track rod in compression. The track rod must be made larger and stronger to prevent buckling. (Toe-out—angling the wheels outward slightly—can be used in this case.) Figure 11-4 shows a track rod steering mechanism for a tadpole tricycle. Note that when the wheels are turned, the linkage ensures that the inside wheel turns more than the outside wheel. The important parameters for designing a track rod mechanism are defined in Figure 11-5. An old rule-of-thumb for determining the correct neutral steer angle and track rod length is to choose the neutral steer angle such that the steering arm centerlines intersect the vehicle centerline at two-thirds of the wheelbase aft of the front axle. This generally produces a mechanism with small steer errors. The neutral steer angle, qo, obtained by this method is:

=θo

2  L  −1 tan +=3   1 t   2 

π 2

+tan

−1

 4L   3t   

π 2

(11-3)

This angle usually provides a good starting point for steering design. However, it is not optimal for most vehicles. With the ability to model a mechanism in a computer program, it becomes relatively easy to optimize the steering kinematics in order to minimize steer error, and thus obtain a much more accurate mechanism than that obtained by the rule-of-thumb. Track rod steering is perhaps the easiest to design and optimize. If the kinematic track and kingpin location and orientation are known, then only two parameters—the steering arm length and neutral steer angle—are required. The track rod length is then fully defined. As noted, a good start point is to draw lines from the steering axis to a point on the vehicle centerline two-thirds back from the front axle. Then draw a line parallel with the front axle that intersects these lines a short distance back from the steering axes. This generally provides a good initial guess for the mechanism. A computer program can evaluate, improve, and optimize the kinematics. The sensitivity of track rod steering depends on how the handlebars are connected to the system. A very simple method is to attach the handlebars directly to the kingpins. This usually produces sensitive steering, although sensitivity can be decreased by increasing the length of the handlebars. Other mechanisms are possible, and may result in lower sensitivity.

201


Kinematic Analysis of Steering Mechanisms An optimal steering design involves a kinematic analysis of the mechanism and a comparison to the theoretical Ackerman angles. Link lengths and angles can be modified until the difference from Ackerman is minimized over the entire range of steering. The simplest kinematic analyses assume that the entire steering mechanism is planar—that is, all the links lie in one plane. However, most real steering mechanisms are not actually planar, but spatial, meaning that the links move in a more complex way that must be described in three dimensions. For example, a non-zero kingpin inclination angle always results in a spatial, non-planar mechanism. Fortunately, the mechanisms described here can be designed using the simpler planar kinematic equations with little loss of accuracy. (In some cases, the steering error in the resulting mechanism is less than predicted by the planar assumption under which it was designed. This cannot be counted on, however.) Several solution techniques are available for solving plane kinematic problems. As an illustration, a position analysis of a track rod steering mechanism using the Chase equations is presented in Appendix 11-1 of this chapter. Most mechanism textbooks can provide more details on kinematic position analyses. Many computer-aided design software packages include motion analysis, making a complete three-dimensional spatial solution relatively easy. This may take more time and effort to model the steering assembly and set up a kinematic analysis than to use the simpler planar algorithms, although it will likely be more accurate. One approach is to use programs that solve the simpler algorithms early in the design, and verify the result with a CAD model and kinematic simulation. In either case, a procedure to evaluate, improve, and optimize the linkage is useful. The following outline presents such a procedure: 1. Identify any parameters whose values are known a-priori (such as kinematic track, for example) 2. Assume or estimate the remaining kinematic parameter values(track arm length, steering arm length, neutral steer angle, etc.) 3. Define the minimum turn radius for the analysis 4. Calculate the Ackerman angle fora range of turn radii from the minimum to neutral steer (infinite turn radius, straight line motion) a. Also calculate the corresponding inside outside wheel angle b. Calculate the corresponding outside wheel angle 5. For each radius in the range, set the inside wheel angle to coincide with that of Step 4a, and use a kinematic analysis to find the angle of the outside wheel. a. Methods such as that in Appendix 11-1 may be used for the solution

202


6. Compare the angle obtained in Step 5 with the theoretically correct outside wheel angle given in Step 4b. The difference is the steering error. a. A plot of the error as a function of turn radius may be helpful 7. If desired, calculate the mean square error to obtain a single estimate of the steering error over the entire range. a. The max error at any turn radius may also be of interest 8. If the error is too large, adjust the values in Step 2and repeat a. Iterate until the error is sufficiently small A Matlab program, Ackerman_TR.m, is available with this text for analyzing track rod steering mechanisms. It uses the method outlined above to evaluate the kinematics and plot the steer error as a function of turn radius. The correct inside wheel steer angles di and the corresponding steering arm angleqi is calculated for a range of turn radii, starting with the minimum turning radius of the vehicle. The actual outside wheel steering arm angle for each position is then evaluated based on the mechanism kinematics. The actual values are compared with the correct angles corresponding to di of Equation 11-2. Two figures of merit for the steering accuracy are provided—the mean squared steering error and the maximum deviation of the inside wheel from that required for true Akerman steering. The mean square steering error is given by MSE =

n

1

∑θ( n

−θ o,true

2 o,actual )

(11-4)

i =1

where qo is the outside wheel steering arm angle and the summation is over all turn radii analyzed (the default number of positions is 50). qo,true is the theoretically required angle given by Equation 11-2, whileqo,actual is the angle that is actually achieved by the mechanism. The maximum deviation is given by MD =

max θ( −oθ,true i

)

o,actual

(11-5)

The mean square error provides a global index of merit for the entire range of turn radii. It is generally the criteria to use for optimization. The maximum deviation provides an index of the worst possible deviation from true steering. Both should be minimized. With due care, the max deviation can usually be well under one degree. The program reports the maximum deviation with the turn radius at which it occurs. Two version of the program are provided. One is used interactively, allowing the user to vary the track arm length or angle and see how the steering error

203


STEERING ANGLE FOR OUTSIDE WHEEL 210

205

) g e d ( E 200 L

DESIRED FUNCTION ACTUAL FUNCTION

N G A L E 195 E H W E ID 190 S T U O 185

180

0

20

40

60 80 TURN RADIUS (ft)

100

120

Figure 11-6 Track rod steering angle example plot

STEERING ANGLE ERROR FOR OUTSIDE WHEEL 1

0.8

0.6 ) g e d ( R O R R E

0.4

0.2

0

-0.2

-0.4

0

20

40

60 80 TURN RADIUS (ft)

100

Figure 11-7 Track rod steering angle error example

204

120


changes. The second version is a function, which can be used for automatic optimization of the mechanism. Similar programs are provided for the other steering mechanisms. Sample Output ACKERMAN STEERING ERROR VEHICLE: TADPOLE Mk 1 WHEELBASE: TRACK:

40.00 in 32.00 in

TYPE OF STEERING MECHANISM: TRACK ROD TRACK ARM LENGTH: STEERING ARM CENTERS: TRACK ROD LENGTH:

2.100 in 32.000 in 29.964 in

NEUTRAL STEER ANGLE:

151.0 deg

MEAN SQUARE ERROR = 0.1186 deg^2 MAX POS DEVIATION = MAX NEG DEVIATION =

0.68 deg AT RADIUS = 8.83 ft –0.33 deg AT RADIUS = 5.50 ft

Six-Bar Mechanism Six-bar mechanisms are used on many high-performance tricycles. The handlebar is often a part of the linkage, so no additional parts are required for steering control. It is possible to achieve good accuracy with this type of mechanism, and steering sensitivity can be controlled. Design of an accurate six-bar mechanism is more difficult, however, because there are

five independent variables involved in the design. The lengths of the steering arm, control rods, and input rocker, along with the distance along the vehicle X-axis from the front axle to the handlebar pivot must be determined. Many design options are feasible, and several different geometries are used in practice. Two examples are illustrated. Figure 11-8 shows an uncrossed mechanism. This configuration can produce very sensitive steering. Care should be exercised to ensure that the sensitivity is not excessive. The crossed linkage is shown in Figure 11-9. Many classic Greenspeed trikes use this mechanism,

205


Figure 11-8 Six-bar steering linkage

Figure 11-9 Six-bar crossed steering linkage

which generally provides accurate steering that is not overly sensitive. These mechanisms are analyzed with the Matlab program Ackerman_6b.m . Rack and Pinion With rack and pinion steering the steering column turns a pinion which drives a rack left or right. Tie rods connect the rack with the steering arms. The steering sensitivity can easily be altered by changing the number of teeth on the pinion. This in no way affects the steering angles, and is an advanta ge of the rack and pinion mechanism. Rack and pinion steering is not often used on human-powered vehicles. However, the steering analysis is valid for any type of steering mechanism in which the inboard tie-rod joint translates left and right, rather than moving in an arc. It is of intermediate complexity, having four independent parameters. The steering error can be comparable to that of the track rod, but may be somewhat more difficult to obtain. The mechanism is illustrated in Figure 11-10, and can be analyzed with the program Ackerman_RP.m.

Figure 11-10 Rack-and-pinion steering

206


Figure 11-11 High-speed cornering parameters

High-Speed Cornering During high-speed cornering, the tires must develop a lateral force. “High speed” is simply any speed at which the lateral, or sideways, accelerationay and the lateral tire force Fy are not negligibly small. Any vehicle in a high-speed turn is subject to centripetal accelerationay

V =

2

R

and the associated forceFy

= −

m

V

2

R

.

If the tires do not have enough traction to resist this force, the vehicle will slide out of the turn—imagine a high-speed turn on ice for example. The tire forces affect all aspects of vehicle handling and stability, from the feel of the handlebar in a turn to the ability to negotiate a turn without rolling over or sliding. Of course, turns are not the only situation requiring lateral tire forces. A vehicle traveling in a straight line along a road with a cross slope in a strong cross wind will also require forces to resist the sideways push. The mechanisms by which tires develop lateral forces are the same in each case, either by steering or leaning the tires. In a turn, vehicle weight and the inertial force are balanced by the vertical and lateral forces at the tires. The relative locations of these forces determine what happens to the vehicle as the speed (and lateral load) increase. Weight and inertial forces are body loads, considered to act through the center of gravity (CG) of the vehicle, while the tire forces act through the contact patch of each tire. This produces moments that cause the vehicle to turn, but also tend to roll the vehicle over if speed is too high. Narrow vehicles with high centers of gravity are more

207


Example 11-1 A rigid tadpole tricycle has a wheelbase of 1.01 meters and a mass, including the driver, of 85 kg. The center of gravity is located .62 meters forward of the rear axle and .65 meters above the ground. The trike is making a turn with a 7 meter radius at a speed of 4.0 m/s. Find the lateral acceleration and the vertical and lateral tire forces for each axle. If the tricycle will roll over at 4.9 m/s in this turn, what is the maximum lateral acceleration it could sustain? Solution: The lateral acceleration (in dimensionless G units) is Ay

=

V2 Rg

4

2

=

=

.233 G’s

7 × 9.81

The vertical forces on each axle are .62  b =  × × = 85 9.81 511.9 N Wf =   mg 1.01 L    .62  b   = − 1 Wr  =mg− 1 × × = 85 9.81 322.0  L  1.01 

N

The lateral forces on each axle are just the vertical forces multiplied by the lateral acceleration, or Fyf

=

Wf A =y

511.9 × =

.233

119.3 N

Fyr

=

Wr A =y

322.0 × =

.233

75.0 N

The maximum lateral acceleration, in G’s, that the tricycle could sustain before incipient rollover is known as the rollover threshold, and is Ay ,RT

208

=

2 Vmax

Rg

=

4.92 7 × 9.81

=

0.350 G’s


likely to roll over in a turn. Less obvious is the effects of tire forces on steering. Both rollover and steering effects are described in this chapter. The individual tire forces depend on the weight and lateral acceleration of the vehicle, the wheelbase and wheel track, and the location and height of the center of gravity. The vertical forces are distributed between the front and rear axles in proportion to the CG location, as described in Chapter 7. The lateral loads are distributed between the axles in a like manner—that is, in proportion to the vertical axle loads. How Tires Develop Forces: Tires develop a lateral force through either of two mechanisms: slipping or leaning. When a side force is applied to the vehicle, such as in a turn, it causes the tire to slip in the direction of the force. This causes an angle, known as the slip anglea, to develop between the direction the tire is actually moving (considering both the forward rolling motion and the slipping motion) and the direction the tire is heading (the rolling direction). The slip angle is illustrated in Figure 11-12. Friction between the pavement and the slipping

Figure 11-12 Slip angle as viewed from above looking down on the wheel

209


wheel resists the lateral force in proportion to the slip angle—a greater lateral force requires larger slip angles. During a turn, all tires contribute to the lateral force, and hence all tires develop a slip angle. (Do not confuse the slip angle with the steering angle—the two are quite different. The steering angle is the angle the steered wheels are rotated from the neutral steer position, measured relative to the vehicle frame. It is caused by turning the handlebar.) Camber is the inclination of the wheel relative to the plane of the road, as in Figure 11-13 (viewed from behind the vehicle). Camber also produces a lateral force, sometimes called camber thrust, acting on the rolling wheel. Camber thrust is proportional to the camber angle, but is much smaller than the force produced by a slipping. Therefore it has less effect on high-speed handling for multitrack vehicles than does slip angle. Tire Stiffness The lateral force developed by the tire increases in proportion to the slip angle a. The constant of proportionality between the force and the slip angle is known as the cornering stiffness—a direct analogy to the stiffness of a torsional spring. Cornering stiffness is a key parameter in vehicle dynamics, and is defined so as to satisfy

Fy = –Caa

Figure 11-13 Camber angle, viewed from behind the vehicle

210

(11-6)


where

Ca = Cornering Coefficient The negative sign in Equation 11-6 is based on SAE sign conventions: A positive slip angle results in a negative lateral force. The tire shown in Figure 11-12 has a positive slip angle. The force would be to the left in the figure, or along the negative y-axis of the vehicle. The relationship between slip angle and lateral force is linear for small angles of a. Figure 11-14 shows the non-linearity of the cornering force at large slip angles. For this reason, the cornering coefficient is defined as the initial slope of the dFy curve, that is, C = . dα 0 α

α=

Figure 11-15 shows experimental cornering force data for Ritchy Tom Slick 26X1.4 bicycle tire. The cornering force was measured with 860 Newtons vertical load on the tire. Note the good linearity, particularly for low slip angles. This is typical of many bicycle tires. The cornering stiffness was determined from a linear regression to be 168 N/deg.

SLIP FORCE AS A FUNCTION OF SLIP ANGLE 1800 1600 1400 1200 ) N ( E 1000 C R O F P 800 I L S

600 400 200 0

CORNERING STIFFNESS = 150.0 N/deg

0

5

10

15

SLIP ANGLE (deg)

Figure 11-14 Slip force as a function of slip angle

211


SLIP FORCE AS A FUNCTION OF SLIP ANGLE 250

CORNERING STIFFNESS = 168.2 N/deg

200

) (N 150 E C R O F IP L 100 S

50

0

0

0.2

0.4

0.6 0.8 SLIP ANGLE (deg)

1

1.2

1.4

Figure 11-15 Cornering stiffness for Ritchey Tom Slick tire with 860 Newtons

vertical load Tire cornering stiffness depends on tire design parameters such as tire type, construction, cross-sectional shape, tread, and material. In addition, operational factors such as tire load and inflation pressure have significant effects on stiffness. As inflation pressure increases, Ca also increases, but at a decreasing rate. Likewise, as the vertical load on the tire increases, cornering stiffness increases some maximum value, beyond which it decreases. This is illustrated in Figure 11-16 for the same Ritchey Tom Slick tire shown in Figure 11-15. Note how the curve flattens out at high vertical loads. The non-linear relationship between cornering stiffness and vertical load is particularly significant for tricycles due to the lateral load transfer occurring on one, but not both, axles. This will be discussed further in the context of the understeer gradient. A distinction must be made between the cornering stiffness of a wheel and the axle stiffness. The relationship F = –Caa, as presented in Equation 11-6, applies to a single wheel. However, within the context of multi-track vehicle handling, the cornering stiffness of the axle is more important. In this case, the stiffness contribution from each tire on the axle must be combined. In this chapter, the term “tire stiffness” refers to the cornering stiffness of a single tire, while either of the terms “axle stiffness” or “stiffness” refer to the cornering stiffness of the axle. The same symbol, Ca, is used for both cases, except where confusion is likely.

212


CORNERING STIFFNESS FOR RITCHEY TOM SLICK TIRE 180 160 ) g 140 e d / (N S S E

120

N 100 F F I T S 80 G N I R E 60 N R O C 40

20 0 0

100

200

300 400 500 VERTICAL FORCE (N)

600

700

800

Figure 11-16 Effect of vertical load on cornering stiffness for a Ritchey Tom

Slick tire

Camber Stiffness If a rolling wheel is inclined with respect to the ground plane so as to give a camber angle g, a side force acting at the contact patch is developed. As with slip, the relationship between the lateral force and the camber angle is non-linear, but is approximately linear at small camber angles. The Camber Stiffness is defined as the slope of the curve ofFy versus camber angle g at g = 0, such that if no side slip is present Fy = C g g

(11-7)

For most tires, camber stiffness is approximately one tenth the magnitude of cornering stiffness. For example, a given tire produces a normalized lateral force

Fy F z

=

1.0

at 5˚ of side slip, but requires nearly 50˚ of camber to achieve the

1

same force. Most multi-track vehicles have very small camber change during maneuvering, and hence camber stiffness is a secondary effect. For these vehicles, cornering stiffness is the primary means of achieving the forces required for high-speed turns. In contrast, bicycles lean into turns, and can achieve very large

1

Cossalter, V. et al., “Dynamic Properties of Motorcycle and Scooter Tires: Measurement and Comparison,” Vehicle System Dynamics, Vol. 39, No. 5, pp. 329–352, 2003.

213


camber angles and commensurately greater camber thrust. Camber stiffness is thus more significant for bicycles.

Lateral Load Transfer and Rollover Threshold The lateral forces acting on a vehicle in a high-speed turn produce a moment that tends to roll the vehicle towards the outside of the turn. The moment is caused by the fact that while the inertial force acts through the center of gravity, which is at some height above the ground, the tire forces act at ground level. The moment is the product of the lateral force times the height of the CG, and causes the vertical load on the outside wheel to increase and the load on the inside wheel to decrease. This is known as lateral load transfer, and the amount of load transferred from to the outside wheel is proportional to the ratio of CG height to wheel track,

h

. At some point, the weight on the inside wheel is reduced to zero. With-

t

out corrective action, any further increase in lateral acceleration results in the vehicle rolling over, or capsizing. The lateral acceleration, expressed in G’s, that just makes the inside wheel load go to zero is called the rollover threshold. A derivation of the rollover threshold for a tadpole tricycle is provided in the appendix to this chapter. Equations 11-8 give the rollover thresholds for tadpole and delta tricycles and quadricycles. t     2h  b t  RTDELTA =  1 −    L 2 h  t  RTQUAD =    2h  RTTADPOLE

=

b

L

(11-8)

In some situations, a vehicle may roll over at speeds well below the rollover threshold computed with Equation 11-8. Hazards on the road surface can trip a vehicle, causing an early rollover. During braking, the rollover threshold is decreased for delta tricycles, but increased for tadpole tricycles. The opposite is true for acceleration, but since human-powered vehicles generally are able to achieve higher braking decelerations than accelerations, the delta is usually somewhat more susceptible to this problem. Quads are relatively insensitive to longitudinal acceleration effects on rollover. Sloped road surfaces affect the rollover threshold, particularly for tricycles. A cross slope on the road surface reduces the rollover threshold for turns made

214


Example 11-2 Compare the rollover thresholds of the following three vehicles: A) A rigid four-wheeled vehicle with a wheelbase of 1.0 meters, wheel track of 0.65 m and a mass, including the driver, of 90 kg. The center of gravity is located .50 meters forward of the rear axle and .70 m above the ground. B) A rigid tadpole tricycle with a wheelbase of 1.0 m, wheel track of 0.65 m and a mass, including the driver, of 90 kg. The center of gravity is located .65 m forward of the rear axle and .70 m above the ground. C) A rigid delta tricycle with a wheelbase of 1.0 m, wheel track of 0.65 m and a mass, including the driver, of 90 kg. The center of gravity is located .35 m forward of the rear axle and .70 m above the ground. Solution: Using Equation 11.8 we get:

   = 0.30 1.0  2 ×0.70  0.65  0.35   =  1 −  =  .70 .30 × 1.0 2 0   

RTTADPOLE

RTDELTA

RTQUAD

=

0.65  0.65

 0.65  =  = .46  2 × 0.70 

If the CG location of the tadpole tricycle was moved to 0.55 meters forward of the rear axle, the rollover threshold would drop to 0.25.

toward the downhill side, and increases it for turns made to the uphill side. This applies to all vehicles. When the slope is in the direction of travel, quads are not affected as significantly as tricycles. A tadpole trike heading straight downhill while making a turn will have a slightly increased rollover threshold. As the trike continues to turn toward the uphill side, the rollover threshold will decrease to that on level ground, and then to the worst case when the trike is headed slightly uphill. Conversely, a delta trike heading straight downhill will experience a slight reduction in rollover threshold. As the trike turns, the threshold will continue to decrease to a worst case condition on a slightly downhill cross slope.

215


Pitchover Threshold Longitudinal load transfer, or shifting vertical load from one axle to the other, occurs during braking and hard acceleration. It was described in Chapter 7. The pitchover threshold is defined similarly to the rollover threshold—the condition in which the vertical load on one axle just becomes zero. During acceleration, weight is transferred from the front axle to the rear axle, while braking transfers weight to the front axle. As noted in Chapter 7, only pitchover due to braking is significant for human-powered vehicles, where it is quite possible to lift the real wheel off the ground while braking hard. This can be a real hazard on vehicles with high and/or forward centers of gravity such as upright bicycles and tadpole tricycles. On a level road, the pitchover threshold due to braking can be found by summing moments about the front wheel patch. The result is given by: PTb

=

a x braking ,

g

L =

−

b

(11-9)

h

The pitchover threshold for acceleration is similarly found by summing moments about the rear tire patch. The result is PTa

=

a x accel ,

b =

(11-10)

g h Understeer Gradient During high-speed turns, the steering angle often varies with lateral acceleration. For example, a delta tricycle that requires 8 degrees of handlebar at 1 m/s may only require 4 degrees of rotation for the same turn at 7 m/s. For four-wheeled human-powered vehicles, the difference is usually small, but for tricycles it can be significant. Good design, including CG location, track width, and tire selection can ensure adequate handling for any type of vehicle, including quads, delta trikes, and tadpole trikes. Recall that in a steady-state high-speed turn, the tires must sustain a force equal to the vehicle mass times the lateral acceleration, ormax. This is accomplished through a combination of slip and camber forces. For most multi-track vehicles, the camber change during cornering is slight, and slip is dominant. (There are exceptions, however , including leaning trikes.) The following discussion neglects the effects of camber. Consider a vehicle slowly increasing speed while in a constant radius turn on level pavement. At low speeds, the lateral acceleration, and hence the slip angles,

are negligible. As speed increases, the lateral acceleration increases asα y

=

V2 R

,

so the tires must sustain a steadily increasing side load. This is accomplished by increasing the slip angle on both the front and rear axles. If the slip angle on the

216


front and rear axles increase in exact proportion to the static weight fraction on each axle, no steering angle change is required, that is, the driver will not need to turn the handlebars as the speed increases. This is known as neutral steer. However, if the slip angle on either axle increases either more or less than this amount, the change in lateral load on each axle is unbalanced, and a steering angle correction is required. The handlebar may need to be turned into or out of the turn. If the handlebar must be turned into the turn as speed increases, the vehicle is said to have understeer. Conversely, if the handlebars must be steered out of the turn, the vehicle exhibits oversteer. The understeer gradient is the amount by which the steering angle must change as speed increases in a constant radius turn. The steering angle required for low speed turns is the Ackerman angle dA, given by Equation 11-1. During high-speed turns,the actual steering angled deviates from the Ackerman angle, and the deviation depends strongly on the lateral acceleration. The lateral force on the vehicle is shared by both the front and the rear axles. This implies that a slip angle must exist on each axle in order for the vehicle to sustain a steady-state turn. Figure 8 illustrates the slip angles on the front and rear axles of a tadpole tricycle in a high-speed turn to the right. Notice that in order to produce the slip angle on the front wheels, the handlebar must be turned further into the turn, but the slip on the real axle requires a reduction in steering angle. The steering angle thus depends on the Ackerman angle and the front and rear slip angles:

d = dA + af – ar

(11-11)

It is more convenient to relate the steering angle to the lateral acceleration. This can be done by noting that the lateral force on a given axle is related to the slip angle by the axle corning stiffness, Fy = Caa. In a steady-state turn, the lateral forces on the front and rear axles are simply the axle weight fraction times the lateral acceleration, or Fyf

=

Wf A y

Fyr

=

Wr A y

(11-12)

Substituting and solving for the front and rear slip angles gives W 

αf =  f  C αf W αr =  r  C αr

 Ay    Ay 

(11-13)

217


Substituting Equation 11-13 into the steer angle equation (Equation 11-11) gives an expression for the steering angle in terms of the Ackerman angle and a correction for lateral acceleration.

δ=δ +A

 Wf  C−  αf

Wr 

 Ay

C αr 

(11-14)

The term in the brackets is known as theundersteer gradient, K. An alternate, and simpler, form of Equation 11-14 is

d = dA + KAy

(11-15)

W W  = f − r  C C αr   αf

(11-16)

Where K is given by: K

If K is positive, the vehicle exhibits understeer, if negative, it has oversteer. In either Equation 11-15 or 11-14, the steering angles are given in degrees, cornering

Figure 11-17 High-speed cornering parameters

218


stiffness in units of force per degree (N/degree or lb/degree), and the understeer gradient is given in degrees per G. For tricycles, the understeer gradient varies greatly with vehicle speed.

Example 11-3 A rigid four-wheeled vehicle has a wheelbase of 1.2 meters and a mass, including the driver, of 100 kg. The center of gravity is located .55 meters forward of the rear axle and .70 meters above the ground. The axle cornering stiffness for each axle is 12°/G. Find the understeer gradient K, and state whether this vehicle exhibits understeer, oversteer or neutral steer. Also find the steering angle required to maintain a level 18 meter radius turn at 0.5, 4, 8, and 12 m/s. The kinematic track is 0.72 m. Solution: To find the understeer gradient, the front/rear weight distribution must be determined using the methods of Chapter 12. b

.55

Wf

=

= LW

Wr

 b   .55 = −1 = = W− 1× × 100 L   1.2 

×1.2×

100 = 9.81

450 N 9.81 531

N

The understeer gradient is obtained from Equation 11-6 as: K

W

=  −f  C αf

450 531   =  − −= °  12 12   

Wr

6.8 /G

C αr

The understeer gradient is negative, implying that the vehicle experiences oversteer during cornering. The value is small. Nonetheless, handling could be improved by increasing the understeer gradient to a small, positive value. This could be done either by shifting the center of gravity forward or replacing tires on one axle. For example, if the rear tires were replaced with a pair that had a cornering stiffness of Ca = 15N/°, the understeer gradient would be positive 2.1°/G. The Ackerman steering angle is given by:

= δA

180 

   π

−1

tan = 

  

  R L

  °= tan  π

180



−1.2 1

3.8

18

219


The steering angle at 8 m/s is given by:

δ =+

A

δ+KA y =

V 2 =A − K    rg 





3.8 6.8 = °

8

2

 1.3  18 × 9.81 

The steering angles for 0.5, 4.0, and 12.0 m/s can be computed in a similar manner. The result is Speed (m/s) 0.1 4.0 8.0 12.0

Ay (G) 0.00 0.09 0.36 0.82

Steer Angle (deg) 3.81 3.20 1.34 –1.74

The figure shows a plot of the steer angle with respect to the vehicle speed around the test circle. Note the decreasing steering angle as lateral acceleration increases. This is oversteer. Also notice that the steering angle deceases to zero at about 10 m/s. This represents an instability known as the critical speed.

220


Critical Speed: As lateral forces increase on a vehicle with oversteer, the steering angle must be reduced. At some speed, the steering angle will decrease to zero, that is, the wheels will be in the neutral steer position, although the vehicle is still turning. Loss of control of the vehicle is imminent. This represents an instability known as the critical speed, which only exists for vehicles that experience oversteer. Recalling that oversteer corresponds to a negative value of K, the critical speed is given by: v crit =

 180 Lg  −    π  K 

(11-17)

In design, the critical speed should be greater than any normal operating speed of the vehicle. This is particularly important with delta tricycles, which have a tendency towards increased oversteer with lateral acceleration. In practice, good handling for tadpole tricycles is usually obtained when the characteristic speed is only achieved well above the rollover threshold. Effect of Lateral Load Transfer: Recall that the cornering stiffness of a tire is dependent on the vertical load on that tire. The relationship is non-linear: as the vertical load increases, the cornering stiffness also increases, but at a decreasing rate. Notice that the curve in Figure 11-18, which illustrates this relationship, is concave down. A very significant consequence of this non-linear relationship

Figure 11-18 Cornering stiffness of a tire as a function of vertical load

221


is the effect of lateral load transfer on axle stiffness, and hence the understeer gradient K. As lateral acceleration increases, load is transferred from the inside wheels to the outside wheels. This has a net effect of decreasing the stiffness of an axle with multiple wheels. For four-wheeled vehicles, the change of stiffness is typically similar on the front and rear axles and the effect on K is usually minor. For tricycles, this is not the case. Lateral load transfer occurs only on the axle with two wheels. Thus, there is a potentially significant change in stiffness on one axle and none on the axle with only one wheel. It is clear from Equation 11-6 that this can have a significant effect on the understeer gradient. This condition is exacerbated in practice, because the CG fraction is often greatest for the axle with two wheels in order to reduce the rollover threshold and balance the static load on each wheel. To quantify the effects of lateral load transfer on the understeer gradient, the cornering stiffness must be modeled as a function of the vertical load. A quadratic model may be used,2 so that C

α

AF ztire ,tire = ,

BF −

2

(11-18)

ztire ,

Table 11-1 includes some values of the polynomial coefficients for several tires. Note that Equation 11.18 is written for a single tire. For an axle with only one wheel, the tire stiffness is the same as the axle stiffness, and the vertical load is just the weight fraction on that axle. Without considering the effect of lateral load transfer, the stiffness of an axle with two identical wheels is twice the stiffness of each tire:

C α,axle

  W  = 2  A  axle    2

  Waxle  − B 2  

2

  

(11-19)

During high-speed cornering, some axle weight is transferred to the outside wheel. If the amount of load transferred from the inside to the outside wheel is

DFz, the vertical load on the outside wheel becomes wise, the load on the inside wheel is

2

Fz ,i

W =  axle  2

Fz ,o

W =  axle  2

  + ∆Fz . Like

  − ∆Fz . Substituting these values 

Gillespie, Thomas D., 1992, Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA, USA. 3 Cole, D.J. and Khoo, Y.H., 2001, “Prediction of vehicle stability using a ‘back to back’ tyre test method,” Int. J. Vehicle Design, 26(5), pp. 573–582.

222


Table 11-1 Cornering Tire Properties for Several Bicycle Tires Tire

Schwalbe Durano 28-4061 Ritchey Tom Slick 26 × 1.41 Tioga Comp Pool 20 × 1.751 Unspecified bicycle tire 2

A

B

0.2718195

0.0000702

0.4214691

0.0002715

0.4468100

0.0002787

.2532

.000211

1

Measured by the author 2 Based on data from Cole and Khoo 3

into Equation 11-19 and simplifying gives the formula for the multi-track axle stiffness of a rigid vehicle, Equation 11-20.

  Waxle    B−    2 

C α,axle =A2 

 Waxle B F− ∆

  2 

2

(

z



)2 

(11-20)



As the transferred load increases, the axle stiffness decreases by 2B(DFz)2. The lateral load transfer DFz can be found using the methods of Chapter 12, and is given by ∆Fz = W

h t

(11-21)

Ay

For vehicles with two wheels on each axle, the stiffnessecrease d on the front and rear axles are at least of similar magnitude, and the effect on K may be small. This is not so with tricycles, in which only one axle experiences a stiffness decrease. The axle with one wheel is not affected by lateral load transfer, and the axle stiffness is simply the stiffness of the tire, Equation 11-18. The axle with two wheels does experience lateral load transfer, and the stiffness is given by Equation 11-20. Substituting each expression into the general formula for understeer gradient, Equation 11-16 gives equations for the understeer gradient of delta and tadpole tricycles. K TADPOLE

W

f =     Wf   Wf  2  Af  2 f −B  f2z      

2

−B

 ( ∆F )2  

−

W

ArW r

r (11-22) −r rB (W )2   

223


K DELTA

Wf Wr  = − 2 A W B W  − ( )   W   W  f f ff 2  Ar  r r −B  r zr    2   2 

2

−B

    ( ∆F )2    

(11-23)

Because the lateral load transfer depends on the lateral acceleration, the understeer gradient of a rigid tricycle cannot be conveniently made independent of Ay as implied by Equation 11-16. For tricycles, the understeer gradient K is dependent on the CG fraction

b L

, the CG height ratio

h t

, and the lateral acceleration

Ay, as well as the front and rear stiffness coefficients Af , Bf , Ar, and Br. The understeer gradient for tadpole and delta trikes are plotted in Figure 1119 and Figure 11-20. In each figure, the lines terminate at the rollover threshold condition. The height ratio was set at a constant value, .67 for the tadpole trike data and .78 for delta trikes. These values are typical. Increasing the height ratio shifts all curves to the left, while decreasing it shifts them right. Both figures assume the same stiffness coefficients for all tires. Using a stiffer tire on the axle with one wheel will shift the curves vertically. A stiffer rear wheel on a tadpole trike will shift the curves upward, while a stiffer tire on the front of a delta trike will shift the curves downward. The curvature in all cases is dependent on the magnitude of B, the second coefficient in Equation 11-18.

Figure 11-19 Understeer gradient K for tadpole trikes with height ratio of .67

224


Figure 11-20 Understeer gradient K for delta trikes with height ratio of .78

It is clear that proper tire selection is critical for optimal tricycle handling. Ideally, tires with a more linear dependence on vertical force (e.g., very small values of B) should be used on the axle with two wheels. The CG fraction can then be adjusted to provide near-neutral steer over a wider range of lateral accelerations along with an acceptable rollover threshold. Unfortunately, data for tire coefficients is not readily available to the individual vehicle designer. Other factors also affect the understeer gradient. Most important is the effect of camber. If the wheels have significant camber, or if the camber changes significantly due to suspension or steering, the lateral force produced should be considered when calculating the understeer gradient.

Summary of Multi-Track Handling Characteristics Tricycles exhibit greater static stability than bicycles but less dynamic stability than four-wheeled vehicles. Quads resist vehicle rollover much better than either type of tricycles. Of the rigid tricycles, stability differences exist between delta and tadpole configurations. •

•

For equal wheel track, wheelbase, and CG height, four-wheeled vehicles are less likely to roll over than either delta or tadpole tricycles. On level ground, otherwise identical delta and tadpole trikes are equally likely to roll over if the centers of gravity are the same height and equally distant from the axle with two wheels.

225


•

•

•

•

•

•

•

•

While turning on an inclined surface, a trike is most likely to roll over when the slope is angled down in a direction normal to the rollover axis: forward of the lateral plane for delta trikes, aft of the lateral plane for tadpole trikes. Braking during cornering will make a delta trike more prone to rollover, and a tadpole trike less likely to rollover. (Assuming equal brake forces on right and left wheels.) Hard acceleration during cornering will have the opposite effect (but generally less so, as the deceleration from hard braking is usually greater than most riders can accelerate). For a tadpole trike, a high-speed turn toward the uphill side of a cross slope while accelerating is the worst case for rollover. (However, in practice a turn on a down slope may be more hazardous, due to the enhanced ability to accelerate. This is seen in many tadpole rollover accidents.) For a delta trike, a high-speed turn toward the downhill side of a cross slope while braking is the worst case. Tadpole trikes are generally more prone to braking pitchover, because the center of gravity is usually further forward. During hard cornering, a tadpole trike will have a significant tendency toward understeer because of lateral load transfer. During hard cornering, a delta trike will have a significant tendency toward oversteer, with the possibility of exceeding the critical speed for poorly designed trikes.

The understeer gradient is strongly dependent on the tire stiffness. The charts presented in this chapter assumed that the same tires were used on all three wheels. This is probably not the optimum solution. Tire choice is quite important for all multi-track vehicles, and in particular for tricycles. Optimal tricycle design requires proper choice of both CG location and tires. Unfortunately, essentially no tricycle (or bicycle) tire rate data is available in the public domain, other than that presented here. A good testing program may be required to optimize performan ce of multi-track vehicles.

Appendix 11-1 Kinematic Solution of the Track Rod Steering Mechanism The track rod steering mechanism, such as that illustrated in Figure 11-5, is a four-bar linkage, with the first “bar” being the frame of the vehicle, the second and fourth bars the steering arms, and the third bar the track rod. Each bar can be represented as a plane vector, with length and magnitude, as shown in Figure 11-21. In the following analysis, vectors are designated with a superscript arrow,

226


Figure 11-21 Kinematic vector diagram for track rod steering



such as R1, while scalers are shown without the arrow R1, for example. Angles are given by ∠R1. Thus R1 = R1∠θR1. If the kinematic track is known, both the magnitude and the angle of vector R1 are known. The length of the steering arms R2 can be assumed or estimated. The neutral steer angle, or the value of the steering arm angle qo corresponding to riding in a straight line, must be estimated or initially calculated from Equation 11-3. The length of the track rod can be calculated using trigonometry and symmetry: 



R3 = R1 – 2R2 sin (qo)

(11-24)

The loop closure equations are used to represent the linkage illustrated in Figure 11-21. In this case, they are written as 





R1R 2= 3R 4 R +



(11-25)

+

The goal is to calculate the angle of one steering arm (and hence the wheel angle) for a given the angle of the opposite steering arm. In this way, the angles of both wheels can be compared to the Ackerman angles as described in the chapter. With the angle q2 given, the unknowns becomeq3 and q4, with the latter being the one of most interest. It represents the rotation angle of the opposite steering arm and wheel. The loop closure equation can be rearranged to place all the fully known vectors on the left-hand side: 

(R1 − R2





)R3= R 4



+

227








For simplicity, let (R1 R 2 ) C and note that any vector can be represented by the product of its magnitude and a unit vector in the same direction, such that C CC . Also, the unit vector k is normal to the plane of the mechanism. Thus, −

=



=

ˆ

ˆ

CCˆ = R

R4 ∠θ ∠θ3 + 4

3

With these simplifications, the loop closure equation is in a convenient format for solution using the Chase equations.4 The solution is 2



R3±R=−

3



= ∓− R4 R

3

2  R324R−2+C2   ˆR 34  k C  ×+  ( C 2C  

 R324R−2+C2   k C  +× 2C 

2

2  ˆR 43  (C 

+C R ˆ− 2 2

) 

ˆ 2C

(11-26)

+C R ˆ− 2 2

) 

ˆ 2C 

The angle q4 can be easily determined once the vectorR4 is known.

Appendix 11-2 Derivation of Rollover Threshold for Tadpole Trike A quasi-static rollover analysis provides a first-order estimate of the lateral acceleration required to produce a rollover. The inertial force due to lateral acceleration acts through the center of gravity towards the outside of the turn. Since the center of gravity is located above ground level, but the resisting forces at the tires are at ground level, a rolling moment is induced. In the case of a tricycle, the line about which the vehicle will roll extends from the contact patch ofthe outside tire to the contact patch of the center tire, shown by axis OO in Figure 1. This discussion will assume a tadpole configuration, but delta configuration analysis is similar. The vehicle is assumed to be in a steady-state turn at constant speed and constant radius. The rollover axis is inclined with respect to the vehicle centerline by an angle g, as shown by axis OO in Figure 11-22. It depends on the wheel track

4

See Shigley, J.E. and John Joseph Uicker, Jr., Theory of Machines and Mechanisms, 2nd ed, McGraw-Hill, 1995, Chapter 2 for derivation and examples of the Chase equation approach to kinematics.

228


Figure 11-22 Tadpole tricycle rollover definitions

and wheelbase. For the tadpole trike illustrated, the rollover inclination angle is given by t  γ = tan −1   2  L

(11-27)

For a delta trike, the equation is similar except that the angle is inclined in the opposite direction. For quads with equal track widths on both axles, g is zero. The inertial force may acts in the radial direction of the turn, which differs from the lateral axis of the vehicle by a small angleq. The effects of this angle are small when considering only lateral accelerations, but are significant when both lateral and longitudinal accelerations act on the trike. Thus, the terms related to this angle are retained initially in the derivation. Angle q depends on the slip angles of the rear wheels, the CG location, and the turn radius. It is given by

θ = α−

R

b R

 

sin −1  α cos( )R 

(11-28)

229


Where ar is the slip angle on the rear tire, b is the longitudinal distance from the rear axle to the center of gravity, andR is the turn radius. The angle ar is considered positive if the turn center is forward of the lateral centroidal axis of the vehicle. Thus, ar is negative as shown in Figure 11-22. Only the component of the inertial forcemay perpendicular to the axis OO contributes to rollover. For quadricycles with equal wheel track front and rear, axis OO is parallel to the longitudinal axis of the vehicle and the angle l is zero. For tricycles, the rollover axis is inclined at an angle l to the vehicle centerline. The component of the lateral force perpendicular to axis OO is given by λ TADPOLE=γ− θ λ DELTA=γ+ θ

(11-29)

λ QUAD = θ or, substituting for q,

λTADPOLE=γ +

R

b

λDELTA=γ−

sin −1  αcos( α + )R

R −1  b sin  cos( α α+ R

λQUAD =

  

b

sin −1  αcos( α − )R

)R

  

  

R

R

(11-30)

R

Summing moments about the rollover axis, and assuming a level road gives

may(h cos(l)) – mg(b sin(g)) + Fzit cos(g) = 0

(11-31)

Equation 11-31 is valid for all vehicle types—tadpoles, deltas and quads. Solving for ay and dividing through by mg gives the acceleration, in g’s, required to produce a given vertical loadFzi on the inside wheel

ay g

F zi tcos( ) γ mg cos(λ)h

b sin( γ) −

=

(11-32)

Incipient rollover occurs whenFzi equals zero. This does not imply that a rollover is guaranteed at this lateral acceleration. It is the point at which the inside wheel will begin to lift off the ground. A skillful rider can maintain control of a trike on two wheels by leaning into the turn. However, the vehicle is unstable.

230


Without corrective action, it will proceed to the point where rollover is unavoidable. Setting Fzi equal to zero in Equation 11-32 gives the acceleration required to unload the inside wheel: ay

=

g

b sin( γ) cos(λ)h

(11-33)

The rollover threshold is a common metric used to estimate the propensity of a vehicle to roll over. It is simply the lateral acceleration required to just unload the inside wheel on a level surface, and is given by Equati on 11-33. The angle q is generally small, so cos( l) » cos( g). With this assumption and substituting for tan( g) from Equation 11-27 the rollover threshold for a tadpole tricycle is expressed as ay

=

g

b



t 



h  2L 

(11-34)

This provides an easy way to approximate the rollover threshold. In most cases, the difference between Equations 11-33 and 11-34 is very small. Rollover thresholds for delta tricycles and quadricycles are derived in a similar manner. The angles l and g are zero for quadricycles with equal wheel track front and rear. Delta tricycles have a rollover axis inclined in the opposite direction as tadpole trikes—the axis slopes toward the centerline moving from rear axle to front axle. The rollover threshold for each type of vehicle is then given by Equation 11-8, repeated here for completeness. t   2h    b t  = 1 −    L   2h 

RTTADPOLE

RTDELTA

RTQUAD

=

b

L

(11-35)

 t  =   2h 

231

CHAPTER

DRIVE TRAIN DESIGN

12

T

he human rider provides the power. The drive wheels develop the force to push the vehicle forward. In between is the drive train—the set of components that transmit power from human to wheels. While many aspects of a vehicle affect performance, comfort, and effectiveness, few do so as much as the drive train. Cadence, force, and stroke must be set correctly to prevent fatigue, discomfort, and even injury. However, the torque and speed of the drive wheels generally differs substantially from the optimal values for the rider, described in Chapter 5. Good drive train design bridges this gap, and does so with very little loss of energy. Simple drive trains—known as single speeds—have only a single velocity ratio between cranks and wheels. That is, the speed of wheel rotation relative to crank rotation is constant. This is inefficient for most practical vehicles that must operate at a range of speeds and under a variety of conditions. Single-speed drive trains are selected for simplicity, reliability, and low cost, rather than performance. For some riders, these attributes are most important, and single speeds are very desirable. Most riders prefer the efficiency of multi-speed drives that allow the rider to maintain an optimal cadence over a wide range of vehicle speeds. Going uphill or into a stiff headwind, a low gear is chosen, permitting a lower vehicle speed without slowing to an uncomfortably low cadence. Conversely, high-speed descents are possible by shifting to a higher gear. Riding comfort and efficiency are improved, and the risk of muscle injury is reduced by using a multi-speed drive.

Gearing Gearing is a general term that refers to the ratio of the crank, or power input, speed to vehicle speed. Humans have a fairly limited range of cadence—cranking speed—for which power is effectively generated. Effectiveness in this context includes high efficiency, high power, and minimal damage to joints and muscles.

233


The effective cadence range is much smaller than the useful speed range of most land vehicles, which is why most vehicles have drivetrains with multiple speeds (commonly, but incorrectly, called “gears”). Single-speed bicycles are not uncommon, and are often simple, light, and inexpensive. They can only match a rider’s optimal cadence at one speed, leading to reduced performance and potential physiological damage. Vehicles with multiple speeds provide improved performance, an extended useful speed range, and reduced likelihood of injury. This is particularly true for recumbent vehicles, due to the inability to stand up on the pedals and to the high pedal forces that can be obtained with a firm seat back. Bicycle gearing is traditionally denoted by the development, measured in meters, or in gear-inches—the unit used more commonly in the United States. The two measurements are defined differently, and are not simply conversions between meters and inches. Gear development is the distance, measured in meters, that the bike rolls forward when the crank is turned one complete revolution. For a simple chain-driven bicycle with NCHAINRING teeth on the chainring, NFREEWHEEL teeth on the freewheel or rear cog, and a drive wheel diameter in meters of NDRIVE, the gear development is N CHAINRING GD

=

N FREEWHEEL DDRIVE × π

If the drivetrain includes an internal gear hub, the internal velocity ratio of the hub must be included. If VRINTERNAL is the velocity ratio of the hub in a particular gear, the development is given by

GVR D =

INTERNAL

N CHAINRING D N FREEWHEEL

DRIVE × π

Velocity ratio is just the ratio of drive wheel speed to crank speed, or

VR

=

ω DRIVE_WHEEL ωCRANK

For a simple chain drive, the velocity ratio is equal to the ratio of driving cog tooth numbers to driven cog tooth number,

VR

N CHAINRING =

. Regardless of the

N FREEWHEEL

complexity of the drive train, the development can be expressed as

GD = VR ´ DDW ´ p

234

(12-1)

Drive Train Design

In the United States, gears are measured in gear-inches, defined as the product of the velocity ratio and the drive wheel diameter, measured in inches.

G = VR ´ DDW

(12-2)

Historically, gear-inches were initially used for ordinary bicycles in the late nineteenth century. These bicycles had direct drive: the pedals were connected directly to the driving wheel. The only means of changing the gearing was to change the diameter of the drive wheel. In order to achieve reasonable speeds, the drive wheels were enlarged to as much as 60 inches in diameter. Since the gearing was directly proportional to the wheel diameter, it made sense to use the diameter as a measure of the gearing. As chain drives begin to see more use, the equivalent direct drive wheel diameter was used so that gearing could be compared directly with the older direct-drive bikes. This convention stuck, and gives us the rather peculiar unit of gear inches. Determining the appropriate range of gearing is an important first step towards drivetrain design. Single speed bicycles built for adult riders typically have gearing around 5 m or 60 inches. This provides a good gear for riding on flat roads at moderate speeds. At low speeds, the cadence can become uncomfortably low. This is particularly true when climbing steep grades. Maximum speed is limited to the rider’s maximum cadence. Cyclists that like these bicycles tend to pedal with low cadence and must have strong legs to climb hills or accelerate. Table 12-1 shows the low and high gear, along with the gear range, for several types of production cycles. The values are for specific models, but are representative of their type.

Table 12-1 Typical Gear Ranges for Various Types of Cycles Meters Development Use RoadRacing Urban Touring

Low

Gear Inches

Hi

Lo w

Hi

Range

2.9 2.0

10.1 8.4

36.6 25.2

126.7 105.8

3.5 4.2

1.8

8.7

22.1

108.5

4.9

RecumbentBike

2.0

10.0

24.8

125.1

5.0

RecumbentTrike

1.8

10.0

22.0

125.1

5.7

XCMountainRacing

1.4

7.9

17.2

99.0

5.8

235


For most general-purpose riding, gearing should range between 3 and 7 meters development (40 and 90 gear inches). For most people, this roughly corresponds to a comfortable speed range of 3 to 11 m/s (7 to 25 mph), although certainly lower and higher speeds are possible. Many experienced cyclists are adept at pedaling at high cadences, and can comfortably pedal at somewhat higher speeds. Greater gears should be used for vehicles intended for high-speed operation, and many racing bicycles having top gears around 10 meters (125 gear inches) or a little more. Speeds up to 17 m/s (37 mph) can be achieved at a moderately brisk cadence of 100 RPM, and maximum speeds of around 23 m/s are possible for cyclists capable of pedaling at high cadence of 135 RPM. (It is important to note that being able to pedal at a high cadence does not imply the ability to ride on level ground at 23 m/s. This speed in still wind on level ground requires an extremely high power output. See Chapter 13) Streamliners designed to obtain very high speeds often have even higher gears. Vehicles operating in hilly environments, touring vehicles and vehicles designed to carry heavy loads often have lower gearing, particularly tricycles or quads that can ride extremely slowly with no balance problems. Table 12-2 summarizes gear ranges and typical applications. In selecting the gearing range for a new vehicle, keep in mind the target rider. Unless the rider is a trained cyclist or a very strong rider, gears above around 9 meters (113 inches) will not result in higher speeds because of the large pedal forces required. The exception is downhill or downwind runs, where the rider receives assistance from nature. Novice designers frequently specify gearing based solely on desired speeds. This is poor practice, as rider capability and power requirements should be considered as well.

Recumbent Drivetrains The difference between upright and recumbent riding positions requires additional gearing design considerations. In the upright position, the rider is able to stand up on the pedals, and even pull against the handlebar. In this position, it is easier to apply large forces at low cadences. Some upright cyclists, preparing for a hard attack on a hill, will actually shift to a higher gear and then rise to a standing position. In essence, the gearing range can be extended by standing. This is impossible with recumbent vehicles. Hill climbing on a recumbent requires either hard pushing against the cranks or spinning fast in alow gear. If the bottom bracket is positioned high relative to the seat, spinning at high cadences can be difficult with standard length crank arms. Many riders find that it is easier to spin to high cadences on vehicles with high bottom brackets if short crank arms are used. As a result, designers should consider the following additional factors when designing gearing for a recumbent vehicle:

236

Drive Train Design

Table 12-2 Gearing Ranges and Applications Gear (inches) <20

Development (meters) <1.6

Application

Extremely low gearsfor steep hills and heavy loads. Used on loaded trikes and quads (vehicles without low-speed balance problems)

20

1.6

30

2.4

40

3.2

50

4.0

60

4.8

70

5.6

80

6.4

90

7.2

100

8.0

110

8.8

120

9.6

130

10.4

>130

•

•

•

>10.4

Steephills,usuallywithloads

NormalRidingformostconditions

Highspeedsordownhillruns

Very high gears for streamliners and speed records. Only used if power simulation indicates speeds in excess of 23 m/s (50 mph) are achievable.

The gearing range on a recumbent vehicle should be greate r than that for a comparable upright vehicle. Usually, increasing the range by 40–60% is sufficient. Recumbent vehicles using standard length cranks (around 170 mm) should shift the gear range up somewhat relative to comparable upright vehicles. Recumbent vehicles, particularly with high crank positions, using short crank lengths (115–140 mm) should not shift the gear range up, and might require a slight downward shift in the range.

237


Any special features or functions of the vehicle should also be considered. Freight haulers and pedicabs require extra low gearing. In addition, these vehicles require strong, robust drivetrains. Light 10 or 11-speed drive systems would be inappropriate for these types of vehicles. Acceleration performance depends on both gearing and quality of shift components. A range of gears with evenly spaced increments is required for good acceleration performance. The two goals are to maintain a cadence as close as possible to that which provides maximum power, while simultaneously minimizing the number of shifts. Shifts should be minimized because each one interrupts the power required for acceleration. The two goals are in opposition to each other: small steps are best for maintaining the ideal cadence, but require more frequent shifts. For upright bicycles, the optimum cadence is reduced when the rider stands on the pedals. This in effect gives at least some acceleration advantage, as a critical shift may be postponed by transitioning from a standing to a sitting position. (This technique requires practice to be effective, however.) A good drivetrain also minimizes overlap of gears, reducing the number of gears required and extra weight and inertia. Many triple cranks have substantial overlap, with many essentially equivalent gears. See the triple drivetrain graphed in Figure 12-1, for example. On the small chainring, only three out of the nine sprockets extend the gearing range, while the large chainring extends the range with only two sprockets. Thus, this 27-speed drivetrain only has 14 effectively independent gears. (Some overlap may be desirable, to minimize double shifting on both front and rear derailleurs, but not to the extent shown in the example.) The Matlab program bikegearf.m computes and plots gearing and speed ranges for each gear. This is a useful program in the early stages of drivetrain design. Several sample data files for various types of vehicles are also included for comparison. The program requires a Matlab script file (.m file) to be prepared containing drivetrain information. Data required includes the teeth numbers for all chainrings, tooth numbers for intermediate drives, if any, cassette tooth numbers, type of hub, drive wheel size, and low, medium, and high preferred cadences. (For data file format, see the bikegearf.m help or one of the sample data files.)

Drive Train Configurations The type, number, and layout of drivetrain components define the drivetrain configuration. Drive trains can be extremely simple, as on a child’s direct-drive tricycle, or quite complex, as on a back-to-back tandem tricycle. Deciding on a

238

Drive Train Design

Example 12-1 Matlabgear analysis was completed for a tandem bicycle that uses a 700c drive wheel, 9 speed cassette, and a triple chainring. The crankset is a compact triple, with a small “granny” gear, or smallest sprocket. A wide-range mountain bike cassette is specified, with tooth numbers ranging from 11 to 34. No intermediate drive is used. Program output provides a summary of gearing information:

Bicycle Gear Calculator Vehicle: Tandem Mk3 Drive Wheel: 40-622 ISO Low Gear: 19.5 inches High Gear: 115.6 inches Range: 5.92

Chainring Tooth Numbers: 46 34 24 Cassette Tooth Numbers: 11 13 15 17 20 Gears (inches) High Mid 115.6 85.4 97.8 72.3 84.8 62.6 74.8 55.3 63.6 47.0 55.3 40.9 48.9 36.1 42.4 37.4

31.3 27.6

23

26

30

34

Low 60.3 51.0 44.2 39.0 33.2 28.8 25.5 22.1 19.5

This information is presented graphically on two plots. The first plot provides a logarithmic plot of the gears corresponding to each chainring. A logarithmic plot is used to better illustrate gear spacing. Ideally, the fractional

239


increase from one gear to the next should be constant. This means that each shift feels like the same amount of change. On a logarithmic plot, constant fractional increases are plotted with equal spacing, so it is quite easy to see if the gears are evenly spaced. It is also easy to see if different combinations of gears are redundant. That is, on a 27 speed drivetrain such as this example, does more than one combination of chainring and cassette cog provide nearly the same gear? (In this example, the answer is yes.) The second plot provides an indication of comfortable speed ranges corresponding to each gear. The use specifies low, medium, and high cadences that are used for generating this plot. In this case, 70, 90, and 120 RPM was selected. This represents a comfortable range for most experienced cyclists.

Figure 12-1 Gear plot for triple drivetrain

Figure 12-2 Speed ranges corresponding to gears in example bike

240

Drive Train Design

It might be a bit high for a recumbent, particularly with full-length crank arms and/or a high crank. For this bicycle, the 19.5 low gear permits loaded climbing on steep hills. The bicycle should have good low-speed stability to ensure it is easily balanced a speeds less than two meter per second. On downhill descents, this vehicle should be capable of exceeding 18 meters per second. suitable configuration for a given vehicle is an important, but sometimes difficult task. Examples of configurations include: •

•

•

•

Simple chain with centerline rear-wheel drive: This is the most common drive configuration for upright bicycles, upright tadpole tricycles, and some long-wheel base recumbents. A single chain runs directly from a crank to the drive hub. This configuration may be used with single or multi-speed drives. A belt drive can be used in this configuration, but the vehicle must have a frame designed to accommodate belts. In some cases a tensioner is used on the slack side of the chain. Complex chain drive with centerline rear-wheel drive: This is a common drivetrain for recumbent bicycles and tadpole tricycles. Power-side idlers or intermediate drives are used to re-route the chainline from the crank to the hub (which are usually distant from each other). Front wheel drive with fixed crank: This is the most common configuration for front-wheel drive recumbent bicycles. The crank is attached to the main frame of the bicycle, and does not swing with the front fork. A single chain may be routed from the crank, around an idler, and then around the front hub. Guides or pivoting idlers direct the slack side of the chain back to the crank. The chain twists during steering, so the guides must be well designed to prevent chain derailments. Alternatively, two chains are used with an intermediate drive. The output chain—running from the mid-drive to the hub—will still twist during steering. This twist is entirely eliminated in some designs by using a universal joint in the mid-drive, positioned accurately on the steering axis. In this case one side of the mid-drive is fixed to the bicycle frame and the other pivots with the fork. Front wheel drive with pivoting crank: This configuration is used on some front-wheel drive vehicles to simplify the drivetrain. The Cruzbike production bicycles and the srcinal Flevobike used this configuration. In the United States, Tom Traylor has designed numerous pivoting crank front-wheel drive bikes. All of these have the crank attached to the fork (which is usually

241


•

a larger, more complex structure that for other drive configurations). The entire drivetrain pivots during steering, so a simple chain drive is sufficient. Simplicity is achieved, but the vehicles have unique handling, and may require a bit of experience to ride comfortably. Dual wheel rear-wheel drives: Several configurations are used for rear-wheel drive delta tricycles and quadricycles, which have dual rear wheels. A rigid axle driving both wheels will always cause some amount of slip during maneuvering, as the wheels are constrained to rotate at the same speed. This should be avoided in most cases. The simplest viable option isto drive only one wheel. This can produce a turning moment onthe vehicle during heavy pedaling, which can lead to impaired handling under some conditions. For example, climbing steep grades on wet, slick, or loose surfaces may cause control problems. It may also be less efficient at high torques due to additional tire slip. For most situations, however, this design workssatisfactorily. (Rigid quads, should avoid single-wheel drive, as there is a possibility that the drive wheel may have insufficient vertical load for good traction. Suspension systems can alleviate this problem.) Using a differential todrive both wheels while permitting differing speeds during cornering is an alternative to driving onlyone wheel. Yawing moments are avoided, and excellent handling is possible. However, this configuration is expensive, and suitable differentials are not readily available. A compromise is to use overrunning clutches on each wheel, and drive both wheels. During maneuvering, power is diverted to the inside wheel and the outside wheel overruns. (Using a split chain drive with freewheels on each drive wheel is one way to realize this configuration.) This does not really eliminate the yaw moment in turns, and usually adds weight and complexity to thedesign.

Drive Train Des ign Drive train design includes configuration, layout, and component specifications. This should be done after body position and gross vehicle geometry have been established, but prior to frame design, although there is overlap between these three design stages. Drive train design depends on the body orientation of the rider or riders, and on the gross geometry of the vehicle. During the vehicle configuration design stage— when gross geometry and position are determined—the designer should consider the drivetrain. Locations for the driving wheel(s) and crank (or crank equivalent if a non-traditional drive system is used) are determined in this early design stage. The drivetrain design follows, and includes layout of drivetrain components, chain routing, gearing requirements and implementation, and consideration for

242

Drive Train Design

adjusting the drivetrain for different size riders. Safety, drive train maintenance, and operating environment should also be considered at this time. As with most design, strive for simplicity. For chain drives, the chain line is important for optimal function. Chainline, in terms of components, refers to the offset of the chain from the vehicle centerline. Cranks and hubs require specific offsets for best operation. The correct chainline can be obtained from the component manufacturer for the specific models specified. Component manufacturers use chainlines ranging from 43.5 to 51 mm. For example, Shimano uses either 47.5 or 50 mm for mountain cranks, and either 43.5 or 45 mm for road cranks. When specifying a crank, the correct bottom bracket must be selected to obtain the desired chainline. For single-speeds and hub gears, ensure the chainline for the cog matches that of the crank. Refer to the manufacturer’s technical specifications or Sutherland’s Handbook for Bicycle Mechanics 1 for specific compatibility information.

Drive Train Technologies The current state of the art in drive train technology is very advanced. Most drive train component development in recent years has been devoted to making derailleur shift systems faster, lighter, and more accurate. There has also been a revival of interest in internally geared hubs, with several new models appearing on the market in the last decade. Improvements include improved range and efficiency, smaller step sizes, and quieter operation. Noticeable improvement in belt drives has also occurred, and bicycle-specific belt drives are now commercially available. •

Direct drive: The crank axis is collocated with the driving wheel axis. This option was used on ordinary, or high-wheeled bicycles, and is often used today for children’s tricycles and unicycles. Schlumpf makes a two-speed direct drive hub, marketed for unicycles. It would certainly be possible to make a direct drive recumbent bicycle, and several designers have pointed out advantages to this design. However, with current hubs the gearing is too low for most applications.2

1

Sutherland, Howard and William Horner, 2004, Sutherland’s Handbook for Bicycle Mechanics, 7th Ed., Sutherland Publications, Berkeley, CA. 2 Several designers have proposed multi-speed direct-drive hubs, and claim significant advantages to this configuration. See Kretschmer [1] and Garnet [2], for example. To the author’ knowledge, however, the two-speed hub made by Schlumpf for unicycles is the only commercially available multi-speed direct-drive hub.

243


•

•

Simple chain drive: The simple chain drive is familiar to all cyclists as the most conventional drive train used for upright bicycles. It consists of a crank and drive hub, and may have multiple speeds obtained either by derailleurs or hub gears. (Schlumpf and Pinion make multi-speed geared cranksets that can replace the front derailleur and multiple chainrings.) Some mechanism must be provided to properly tension the chain. For drivetrains with rear derailleurs, the spring-loaded derailleur cage functions as a tensioner. Single-speed and hub-gear drive trains do not have a derailleur: if not equipped with a separate tensioner, these vehicles maintain chain tension either by sliding the hub within horizontally slotted dropouts or rotating an eccentric bottom bracket. Tandems often use an eccentric bottom bracket to tension the timing chain running between the two cranks. If the distance between the bottom bracket and the hub is large, additional tensioning capability may be required, such as a second tensioner. Complex chain drive: Many recumbent vehicles have long and complex chain lines with additional components such as idlers and intermediate drives. Idler sprockets re-direct the chain to a new direction. This may be required to route a chain under a seat, for example. Idlers on the power-side chain must be securely mounted and durable, as considerable force may be applied from the chain.Intermediate drives, or mid-drives, are used with multiple chains. They not only re-route the chain, but can alter the velocity ratio, and sometimes add additional speeds as well.The chain from the crank is routed to the input sprocket on an intermediate drive. An output sprocket is rigidly attached to the same shaft as the input sprocket, but a different chain runs from it to the hub or another downstream component. If the two sprockets have different tooth numbers, the velocity ratio will be changed. In this case, the overall velocity ratio is given by: VR

=

N CHAINRING N MID_INPUT

•

244

×

N MID_OUTPUT N FREEWHEEL

It is possible to add multiple cogs to the input of a mid-drive unit in order to obtain additional speeds. This is usually accomplished using a multiplecog freewheel. Figure 12-3 illustrates an example of a bicycle with a complex chain drive that includes an intermediate drive with a velocity ratio change and a slack-side idler to direct the chain above the front wheel. Belt drives: Belt drives are similar to chain drives, but use a toothed belt in lieu of a roller chain. Chains are very efficient, but require regular maintenance (particularly in wet conditions), can be noisy, and tend to produce oily

Drive Train Design

Figure 12-3 Example of complex chain drive

stains on clothing or skin. Belts are only slightly less efficient than chains— 98% for belts compared to 99.6% for single-speed or 98–99% for derailleur chain drives3—but are quieter, cleaner, and require very little maintenance. Belts cannot be used with derailleurs, essentially limiting multi-speed drives to hub gears and geared cranks such as those produced by Schlumpf or Pinion. The wide range of components used with chain drives is not available for belt drives, reducing component selection significantly. Bicycle-specific belts •

and components are available from Gates and Schlumpf. Other drives: A variety of different drive systems have been used throughout cycling history, including shaft drives, cable drives, levers, hydraulic transmissions, and electric drives. Some, such as shaft drives and at least one cable drive, are currently used by at least some manufacturers. Almost all are less efficient and heavier than chain drives. Shaft drives are clean and low-maintenance, but require internally geared hubs for multiple speeds. Several shaft-driven bicycles are currently on the market. Cable drives often require somewhat complex mechanismsfor operation, and generally continuous rotary motion is not possible. Hydraulic and electric transmissions convert human power into fluid or electric power, which is then converted back to mechanical power at the drive wheels. These systems usually have low efficiencies and are often heavy, making them impractical for most vehicles. They do provide an integral means of energy storage, however. Several designers have attempted to exploit this through regenerative braking. The weight and efficiency penalties of such systems generally override the

3

Casteel, Elizabeth A. and Archibald, Mark, “A study on the Efficiency of Bicycle Hub Gears,” ASME 2013 International Mechanical Engineering Congress & Exposition, San Diego, CA, USA, November 2013.

245


benefits. Chester Kyle evaluated the feasibility of regenerative braking. 4 His conclusion—that the penalty is greater than the benefit—is still valid today. Hybrid electric vehicles use a more conventional chain drive with an auxiliary electric power supply. The goal is power assist, rather than power transmission. Since all elements required for regenerative braking can be included in the auxiliary system, regenerative braking may be feasible. In essence, the penalty has already been incurred, so any benefit is a bonus. Chain drives are the most prevalent technology used for land human-powered vehicles. The low weight, high efficiency, and advanced state of commercially available components make chain high-performance systems easily obtainable and affordable. There is certainly opportunity for proponents of alternate drives, but a clear advantage must be shown over chain drives. Belts and shafts offer the best alternatives, primarily due to low maintenance. This is particularly true for commuters and those that ride frequently in dirty environments.

Efficiency of Chain Drives The performance of human-powered vehicles is limited to the power capabilities of the human body. It is thus quite important to maximize the fraction of power generated by the rider that goes into actual propulsion of the vehicle, that is, the drivetrain efficiency. Competitive cyclists require high efficiency drivetrains for winning races, particularly time trials. (In time trials, the speed of the vehicle is paramount, as opposed to road races where strategy, technique, and teamwork are also important factors.) A reduction in drivetrain efficiency of 2% would have been more than enough to cost Chris Boardman his 1996 world hour record, or the 1996 US Olympic 4000 meter team pursuit their gold medal. 5 Efficient drivetrains benefit all riders—commuters, competitors, and recreational riders—by using more of the body’s power for propulsion. Drivetrain efficiency is defined in Equation 13-2, repeated here for convenience: DRIVE η= P

WHEEL

PCRANK

4

(12-3)

Kyle, Chester R. 1988, “The Mechanics and Aerodynamics of Cycling,” in Medical and Scientific Aspects of Cycling, Edmund R. Burke and Mary M. newsom, Eds., Human Kinetics. 5 Kyle, Chester R. and Berto, Frank, 2001, “The Mechanical Efficiency of Bicycle Derailleur and Hub-Gear Transmissions,” Human Power, 52, 3–11.

246

Drive Train Design

Figure 12-4 Typical torque-efficiency curve for a chain drive

Efficiency for chain and belt drives is strongly dependent on output torque. At low torques these drives are very inefficient, but as torque increases, the efficiency rises and stabilizes at a relatively high value. The relationship between torque and efficiency can be modeled as an exponential curve of the form 6

η = A − Be −λ TO

(12-4)

A typical curve is shown in Figure 12-4. Note that efficiency initially rises fairly rapidly but does not approach its final value until torques are quite large. Typically, output torque for casual riders is fairly low, often below 20 N-m. Hard sprints and uphill drives can result in higher torques, particularly for strong riders. In actual riding, the output torque is not constant, but exhibits high variability due to variable pedal forces throughout the crank cycle and changing ride conditions. This all contributes to the difficulty in measuring and reporting drive efficiencies, and probably explains discrepancies between different studies. Casteel and Archibald measured the efficiency of several drivetrains, including derailleur, belt drive, and gear hubs.7 Efficiency of each gear was measured at a variety of power levels, and the data was fit to the exponential model of Equation 12-4. Table 12-2 summarizes the results. The efficiency shown is the average over all the gears in each drive. (There are typically notable variations between gears in a given drivetrain.) Chain drives are exceptionally efficient. Geared hubs are

6

Casteel, Elizabeth A. and Archibald, Mark, “A study on the Efficiency of Bicycle Hub Gears,” ASME 2013 International Mechanical Engineering Congress & Exposition, San Diego, CA, USA, November 2013. 7 Ibid.

247


Table 12-3 Average Efficiencies for Various Drive Trains

Drivetrain

Num. Speeds

Average Efficiency

Single-SpeedChain

1

99.6%

Single-Speed Belt

1

98.1%

7

98.9%

7-Speedderailleur Roholoff500/14Speedhub

14

LowRange,Speeds1–7

96.5%

High Range, Speeds 8–14

98.5%

Overallaverage

97.5%

Sturmy-ArcherX-RK8(W) SRAM Dual-Drive Shimano Alfine 11

8 3 11

91.1% 99.0% 93.2%

Summarized from Casteel, Elizabeth A., and Archibald, Mark, “A study on the Efficiency of Bicycle Hub Gears,” ASME 2013

generally less efficient than derailleur systems, although there is a wide spread between models. There are many factors that affect efficiency in addition to the type of drivetrain. Increased chain tension improves efficiency. Larger cogs improve efficiency relative to smaller cogs at the same speed and power. Minimizing the number of friction points—tensioners, idlers, etc. in the drivetrain will improve efficiency, although this can be difficult in some recumbent vehicles. A clean and properly lubricated system will also improve efficiency. Several researchers have investigated bicycle chain drive efficiency, including Kyle and Berto, Spicer et al.,8 and Rohloff and Greb.9 Unfortunately, none of these papers report standard deviations or include a statistical analysis. Only the Rohloff paper indicated ranges of data. This lack of statistical analysis calls into question at least some of the conclusions, particularly as it is very likely that variability in

8

Spicer, James B. et al., On the efficiency of bicycle chain drives, Human Power, 50, Spring 2000, 3–9. 9 Rohloff, Bernhard, andGreb, Peter, 2004, “Efficiency Measurement of Bicycle Transmission— A Neverending Story?,” Human Power, 55, Winter 2003/2004, 11–15.

248

Drive Train Design

the data is on the order of some of the differences between drivetrains. 10 Spicer et al. investigated the efficiency of bicycle transmissions both analytically and experimentally with the goal of determining the factors that affect efficiency. 11 They found that efficiency is strongly dependent on the tension in the chain. As chain tension increases, so does the efficiency. Specifically, efficiency decreases linearly in proportion to the inverse of chain tension, or

η∝

1

. This relationship

T holds true regardless of power or speed. For a specified chainring/cog combination at a constant speed, this implies that as power increases, so does drive train efficiency. They also found that larger sprockets are more efficient than smaller sprockets with efficiency improvements of 2 to 5% when changing from a 52-11 chainring/cog combination to a 52-21 chainring/cog. The effects of three different lubricants and of chainring/cog axial offset were negligible in this study. A large number of chain lubricants are available on the market. It is probable that maintaining a clean and properly lubricated chain is much more important for good efficiency than the particular lubricant selected. (The susceptibility of the lubricant to environmental factors such as water, mud, dirt and road salt may vary significantly between products, however.) Modern narrow multi-speed chains are designed to be laterally flexible. This probably accounts for the insignificant efC

fect due to chain offset. More rigid single-speed chains would probably show a reduction in efficiency with offset. The overall efficiencies measured in the Spicer study ranged from 80.9% to 98.6%. Most measured efficiencies were above 90%. All efficiencies below 90% were at low chain tensions. Spicer concludes that the primary factors affecting drivetrain efficiency are the size of the sprockets and the tension in the chain. Kyle and Berto12 tested several derailleur and hub gear models using a custom dynamometer. Each transmission system was tested in all or most available speeds 10

A lively debate between Chester Kyle and Bernhard Rohloff was published Human Power number 55. The disagreement over proper experimental methods was sharpened by a discrepancy in efficiency of about 2% between the two studies. Although differing methods and equipment likely played a role, the argument might have been avoided by inclusion of a statistical analysis. Rohloff reported a range ofvalues of about 2%, and Kyle claimed reproducibility within 1%. The 2% difference in efficiency might well be statistically insignificant. This should be a reminder to experimentalists to include relevant statistics with published data. 11 Spicer, James B. et al., On the efficiency of bicycle chain drives, Human Power, 50, Spring 2000, 3–9. 12 Kyle, Chester R. and Berto, Frank, 2001, “The Mechanical Efficiency of Bicycle Derailleur and Hub-Gear Transmissions,” Human Power, 52, 3–11.

249


and at different power levels up to 200 Watts. The crank speed for all tests was set at 75 RPM. The general trends agreed with the Spicer study: efficiency increased with increased power and larger sprockets. For all drivetrains, efficiency varied from one gear to the next. It is interesting to note that the peak efficiency of some of the hub gears tested did not correspond to the gear with a 1:1 velocity ratio (which should have the highest efficiency in theory). Specific efficiency values ranged from 87 to 97%, although the apparatus used included an extra chain and set of bearings estimated to decrease efficiency by about 2%. Hub gears were generally less efficient than derailleur gears, although the difference was small— on the order of 2%—for some hub gears. All hub gears were lubricated with oil, rather than grease, which probably improved the efficiency. Rohloff and Greb presented data that, in general, corresponded with Kyle and Berto. However, they tested at a higher power level of 314 Watts and used a broken-in Rohloff Speedhub 500/14 in comparison with a Shimano 24-speed mountain bike drivetrain. They justified the higher power test on the basis of field tests of chain and chainring wear. Results indicated that the derailleur system and the Rohloff hub gear had comparable efficiencies. Rohloff also tested both systems with new chains and sprockets, and with chains and sprockets that had experienced 1000 km of field use and were not cleaned prior to testing. The derailleur system showed a 1% reduction in efficiency, while no measurable efficiency reduction was noted for the Speedhub 500/14. Specific effects of idlers and chain tensioners are not known, and were not well addressed in these studies. It can be assumed that any additional component in the drivetrain incurs losses, and therefore reduces efficiency. Components on the tension side of the drive system are likely to incur greater losses. This is supported indirectly in the Spicer paper. A part of their experimental work included infrared imaging of the rear cog and derailleur. Bright areas on these images revealed hot areas, produced by frictional losses. The derailleur pulleys are quite bright in the images—indicating significant energy losses. (If the pulleys are plastic, as on many derailleurs, the high temperatures may be due in part to low thermal conductivity coupled with the small pulley mass.) It is very likely that other tensioners would incur similar losses. Tension-side idlers are likely to have even larger losses. Guidelines for designers based on these studies include: • •

•

250

Avoid small sprockets for gears that are frequently used. Avoid small sprockets on intermediate drives, idlers and tensioners if possible. If chainline offsets must occur, use only multi-speed chains.

Drive Train Design

•

•

Hub gears are usually somewhat less efficient than derailleur gears, although the difference is slight in some models, particularly at higher power levels. A relatively smaller difference exits between hub gears and derailleur systems if the drive system is frequently exposed to harsh and dirty conditions.

Chain Drive Design

Design of chain drive systems require identification of gearing requirements, componentselection, and establishment of satisfactory chain routing. For some vehicles, such as many upright bicycles, this is fairly straightforward. Other vehicles, such as recumbents, delta trikes, and quads, are frequently more challenging. Determination of overall gear range requirement is generally the first step. Table 12-1 may be helpful in determining gear ranges. For example, a touring vehicle may need to carry heavy loads up steep grades, but also should be capable of high-speed mountain descents. From Table 12-1, the low gear should be approximately 2.0 meters development and the high gear should be around 10 meters. The range factor is the ratio of high to low gear, in this case about 5.0, a value easily obtained using standard drive components. Most mountain bicycles have gear ranges equal to or slightly greater than five. It is important to check the gearing range, as that determines the type of components that are required. If an extremely large range of gearing is necessary, greater than about 5.5, special components or additional components may be required in the drive train. Once the overall range is established, the drive train configuration can be laid out and components selected. Be sure to consider: • • •

Mechanism for tensioning each chain Means of guiding chain onto each cog or chainwheel (preventing derailments) Means to retain aderailed chain

Some means of tensioning each chain is required. Tensioning can be accomplished with spring-loaded tensioners, eccentric bottom brackets, or slides such as used on many single-speed and hub gear bikes. Spring-loaded tensioners may reduce the drive train efficiency somewhat, but often offer benefits of convenience. For multi-speed derailleur drivetrains, spring-loaded tensioners are required to take up the slack as the chain moves to smaller sprockets. Derailleurs have built-in tensio ners to accommodate this. Vehicles with adjustable cranks, such as sliding booms, can use the adjustment to tension the chain. This simplifies the drive train, but requires a different length chain for each position of the crank. If the vehicle is always ridden by the same person, this

251


may not be a factor, but it is certainly inconvenient if riders with different leg lengths use the vehicle. If this method is used, it is imperative to include a chain keeper to prevent loss of the chain in the event of a derailment. (If one end of a chain can come off of a sprocket entirely, a dangerous accident can occur.) Chain keepers retain the chain in the event of derailment, preventing the chain from becoming entangled in wheels or other components. They can be important safety items. Power-side idlers should be avoided if possible. For some vehicles, a straight chain line is impossible, and tension-side idlers or intermediate drives are necessary. Ensure they are mounted securely, as tension-side loads can be quite high. Also use good bearings to minimize friction in the tensioner. Identifying a chain line that avoids interference with the frame or other components can be a difficult challenge. Components such as seats and accessories may change location. Forks and front wheels move during operation. All of these factors must be taken into account to avoid chain interference problems. If derailleurs are used, the chain location moves, and the entire range of chain motion must be kept clear. Chains also vibrate, which takes up additional space requirements. Longer chains need greater clearance than short chains due to larger amplitudes of vibration. The chainline should be checked for interference at each step in the design process, including design changes. Be sure to consider guards for chains and sprockets that are exposed. If a rider can reach a sprocket, his or her hand can get caught between the sprocket and chain—a very painful and dangerous accident. Chain guards can prevent such accidents, and also prevent grease marks on clothing or skin. Some recumbent vehicles use tubes to prevent the latter. Tubes may be incorporated with guards over sprockets to provide both safety and cleanliness. In summary, drive train design consists of the following steps: 1. Determine the gearing requirements required for the vehicle, including low, high, and range. 2. Determine the drive train configuration. 3. Select component types and sprocket tooth numbers to ensure gearing requirements are met. 4. Lay out the chain line, ensuring that chain interference is avoided. 5. Determine and select secondary components such as tensioners, chain keepers, and chain guards.

252

CHAPTER

LAND VEHICLE FRAMES AND

13

STRUCTURES

T

he frame is the essence of a human-powered vehicle. A well-designed and fabricated frame will be strong enough to withstand all normal service loads, stiff enough to ensure good handling response and efficient power transfer, yet light and comfortable for the rider. These are stringent requirements, demanding much attention during design.

Functional Requirements The vehicle frame must support the weight of the rider and any cargo, provide mounting locations for fairings, components and accessories, and resist drive train forces and forces induced by maneuvering. In many cases, it must also provide at least some degree of safety in the event of an accident. For efficient power transfer, the frame should be longitudinally stiff along the chain line. Chain forces can be quite large, and often the chain is offset laterally from the supporting frame member. This induces bending as well as compression in the frame, so resistance to flexure in the lateral direction is required as well. Lateral stiffness is also required for good, crisp handling. Excessive lateral flexibility can result in poor control during hard cornering. In the vertical direction, compliance is sometimes included by design to provide a more comfortable ride. This is particularly applicable to recumbent vehicles since the rider cannot stand up and use legs as shock absorbers as the upright cyclist can. Some advanced upright bicycles tune the stiffness of the chainstays to provide high longitudinal and lateral stiffness, with lower stiffness in the vertical direction to provide a smoother ride. Tapered forks also provide a bit of vertical compliance, which can be good for bicycle handling as well as comfort.

253


Frame design should start with known loads and reactions. This generally means that vehicle gross geometry and rider position should be determined prior to designing the frame. Preliminary vehicle handling and stability analyses should be completed first, as these dictate gross geometry of the vehicle. The chainline should also be established—at least tentatively—prior to frame design. Chain loads can be large, and must be considered in the frame design process. With known gross geometry and drive train, the loads acting on the frame can be estimated for a variety of scenarios, called load cases. Tentative geometry is then developed to resist the loads in each load case. Failure modes associated with the different load cases are identified, and the frame is analyzed to predict performance and weight. Usually, the finite element method is used for the analyses. Initially, the goal is to establish a frame geometry that meets the structural requirements for the vehicle—that is, it has the required life and willnot fail under expected or design operating conditions. The ultimate goal is to design a frame that meets the structural requirements with minimum weight, yet is easy to manufacture—quite a challenge.

Yielding Frame failures are due to several different mechanisms. Most metal frames are made of ductile materials such as steel or aluminum, which deform plastically, or yield, prior to fracture. The resulting permanent deformation is noticeable, and provides warning that the frame is damaged prior to catastrophic failure. Yielding is essentially a static failure mode, in which the single application of a large external load deforms the frame. Normal operation of a vehicle should never result in permanent deformation, which is considered a failure mode. The frame should be designed such that all stresses are below the yield strength of the frame material under all design operating conditions. Given the material yield strength Sy and the required factor of safety n, the maximum allowable Von Mises stress is given by: σ′ <

Sy n

(13-1)

(Von Mises stress is a scaler stress equivalent, and is reported by many FEA programs. For more information see any good strength of materials text.)

Fatigue Fatigue failures are caused by repeated cyclical loads applied to the frame. Design for fatigue is an important—and difficult—task for the frame engineer. After many cycles of a load that produces relatively low stresses, a member can abruptly

254

Land Vehicle Frames and Structures

fracture. The failure usually occurs rapidly and with little warning, and can occur even if the stresses are well below the yield strength of the material. The fatigue life of a material is the number of cycles of a specified alternating stress that produces a fatigue failure. The fatigue strength is the stress level corresponding to failure at a specified number of cycles. Unlike yield strength or tensile strength, fatigue strength cannot be stated as a simple, single number. The number of cycles must be specified. For most materials, including aluminum, there is no lower limit to fatigue life—any cyclical stress can eventually result in a fatigue failure. In practice, the fatigue strength at a large number of cycles is used, say 500 million cycles. Steel and titanium exhibit an unusual but fortuitous property known as an endurance limit. This is the lowest stress level that will produce fatigue failures—cyclical stresses less than this limit will not produce failure no matter how many cycles are applied. (Assuming the absence of extraneous factors such as corrosion, wear, etc. that can combine with stress loading to produce failures at lower than expected stress levels.) There can be two stress components in fatigue analyses—alternating and mean stress. The mean stress is the average stress magnitude, while the alternating stress is the variable component. The modified Goodman formula is often used to determine the factor of safety for such problems. 1

n

σ

σ m +

=

Sut

a

(13-2)

Se

Where sm is the mean stress component, sa is the alternating stress component, Sut is the ultimate tensile strength, and Se is the endurance limit or fatigue strength of the material. For example, the weight of the rider produces a mean stress in the frame. Pedal loads produces alternating stresses that are superimposed over the stress due to weight. (Road vibrations also produce alternating stress, although they are more difficult to predict and quantify.) Figure 13-1 MEASURED AXIAL STRESS -- TOP OF DOWNTUBE

10 5 ) a P

0 -5

(M-10 S S E-15 R T S-20

-25 -30 -35 0

0.5

1

1.5

2

2.5 3 TIME (s)

3.5

4

4.5

5

Figure 13-1 Measured stress data, showing fluctuations due to pedaling

255


shows actual stress data taken from the downtube of an upright diamond frame bicycle while climbing a hill. The stress fluctuations with each pedal stroke are clearly seen. Note also the compressive (negative) mean stress. The stress is due to the combined effects of pedaling and the weight of the rider.

Inadequate Stiffness

Failure modes are not necessarily based on strength. Inadequate stiffness should also be considered a failure mode. Frame stiffness is not a global property, but varies with locations on the frame and the directions of measurement. Think of the frame members—chainstays, main tube, etc.—as springs. The stiffness of each spring affects the deformation of theframe under a given load. Unfortunately, unlike strength-based criteria, it is difficult to determine specific deflection limits appropriate for a given frame design. A better approach is to use the total strain energy in the frame due to a given load. Strain energy is a global measure that provides an overall indication of frame stiffness. Once the frame strength requirements are met, the strain energy should generally be reduced. For a given load case, a frame with lower strain energy will be stifferthan one with greater strain energy. The operational stresses in a vehicle frame are usually complex and fluctuating. Forces due to pedaling can be large, and fluctuate with each pedal cycle. Road roughness imposes random fluctuations with magnitudes that vary significantly depending on road surface, tires, speed, and suspension system. Except for manufacturers with experience and plenty of field data, the resulting stresses are difficult or impossible to accurately predict a-priori. The methods described in this chapter are useful when there is no prior experimental data to guide the analysis. Properly applied, a safe, relatively lightweight frame can be designed. More effective optimization is possible if stress testing under actual field conditions is completed on one or more prototype vehicles.

Finite Element Modeling of Frames Finite element analysis (FEA) enables structural verification and optimization of vehicle frames and components during vehicle design. A broad selection of commercial FEA software products is available to the vehicle designer. Most companies that produce computer-aided design software also offer FEA packages. In addition, there are several stand-alone FEA programs, spanning a broad spectrum of prices and features. There are many benefits to using FEA, and with the widespread availability of programs, there is no reason the HPV designer should avoid it.

256


A detailed explanation of FEA is beyond the scope of this book. There are many good books1 on the topic for those interested in learning FEA in greater depth. A brief, qualitative overview must suffice here. The key concept behind FEA is that a structure, such as a bicycle frame, can be theoretically divided into a finite number of small elements. The elements are connected to each other at nodes. Elements have elastic properties based on the type of material. In essence, each element can be thought of as a small spring. When a load is applied to the structure, the elements (springs) deform due to the load that is transmitted internally through the structure. As the elements deform, the nodes are displaced, that is, they change position. All finite element programs first determine the deformation of the elements and the corresponding displacements of the nodes. Stresses can then be computed for each element based on Hooke’s law, which relates stress to strain. FEA programs usually provide graphical output for a variety of measures, such as stress, deflection, and strain energy. Data summaries in text format are also provided. Generally, it is relatively easy to set up and run a simple FEA analysis. Virtually all programs available today will produce colorful images depicting stresses and deflections. However, if good modeling practice is not used, the output is meaningless. New FEA users may fall into the trap of blindly trusting that the results of an FEA analysis accurately reflect the behavior of the real structure. Of course, correct simulations of the structure is the goal of all finite element analyses, but in order to have confidence that the results are meaningful, the analysis must carefully complete each of the following steps: •

•

• •

• • •

Identify the applicable load cases and corresponding failure modes and failure criteria. Clearly define the loads to be applied to the structure, including location, direction, type, and magnitude. Determine how the structure should be constrained for each load case. Correctly construct a geometric model of the structure, recognizing the limitations and advantages of any idealizations used. Correctly apply loads and constraints. Define and run the analyses. Assess the results: consider convergence and deflection.

1

Knight, Charles E., 1993 The Finite Element Method in Mechanical Design , PWS-Kent Publishing Co., Boston, MA, USA.

257


With any analysis, there is a compromise between accuracy and cost. Cost may be actual pecuniary cost or may be cost in terms of time and effort. The most accurate predictions of structural behavior can be made bybuilding a vehicle, instrumenting it with strain gauges, accelerometers, etc., and conducting extensive testing in the actual operating environment. However, this is extremely costly in terms of time, effort and equipment. Physical testing can be very valuable, particularly near the end of the design cycle, but it is generally not affordable during the early design stages. Early in the design process, FEA is used in lieu of physical testing because it is much faster and less expensive. This contrast illustrates the proper use of FEA in the design process. Begin with models that are easy, quick, and inexpensive, but perhaps not as accurate. As the design matures, shift to more detailed models and fewer idealizations. This requires more time and effort, butprovides more accurate results. At some point, physical testing may be required to verify the FEA models, but this should only be done once the FEA has reached its limits. Note that the limiting factor for FEA is often (but not always) uncertainty in loads.

FEA Modeling—Idealizations, Loads, Constraints, and Validatio n Correct modeling and set-up is required for accurate FEA analyses. Modeling includes geometry, which is usually established in a CAD program, but may be constructed directly in the FEA package. Model geometry must reflect the geometry of the physical frame, but may be simplified to decrease modeling time and speed up the analysis and design cycles. The level of detail should be consistent with the goals of the analysis, the stage of the design, and idealizations used. Earl y models should use simple geometry that can exploit simple element types and produce rapid design iterations. In all models, loads must be correctly and realistically applied to the structure, which must also be constrained in such a way as to reflect actual conditions. Failure to correctly define the loads, constraints, and idealizations can lead to grossly inaccurate results. For this reason, each model should be carefully evaluated prior to accepting the results. Commercial FEA programs offer users a selection of element types. This simply refers to the mathematical model used to relate forces and displacements across the element. The size of the mathematical model and the resulting computational effort can vary significantly. Choosing appropriate element types for a given analysis helps the analyst optimize the balance between modeling time, analysis run time, and model accuracy. In the early design stages, models using element types that are quick and easy to implement, modify, and evaluate help establish gross frame geometry. Subsequent more detailed analyses can evaluate or optimize details such as joint design and stress concentration effects. A brief

258


overview of three element types and their application to design of HPV frames may be useful for the designer unfamiliar with FEA.

Beam Elements Beam idealizations are used to model prismatic structural elements that are long relative to their cross-section dimensions. Beam elements can support transverse loads and flexural moments. The beam elements in most commercial FEA software also include torsional loads, which are usually significant in HPV design. Frame tubes on traditional bicycle frames are examples of structural elements that may be idealized as beams. Beam idealizations are particularly useful in the early design stages, where the primary concern is establishing viable oroptimal gross geometry and many gross design alternatives must be evaluated quickly. Frames idealized with beam elements are usually very easy to model and quick to analyze, permitting rapid assessment and improvement. However, beam idealizations are subject to several limitations. If these are violated, the results will be incorrect, sometimes significantly so. The three significant limitations of the beam idealization are: • •

•

The beam element has a constant cross-section along its length. The transverse dimensions (cross-section dimensions) are small compared to the length. The stress and strain vary linearly across any cross-section of the element.

The consequences of these limitations should be well understood by the analyst. Frame members made of tapered tubes cannot be accurately modeled with beam elements. A common example is commercially available unicrown fork blades, in which the diameter can taper from 31.8 mm (1.25 in) to 20 mm (.787 in) over a length of 432 mm (17 in).2 Butted frame tubes—where the wall thickness gradually changes along the length of the tube—also violate the constant cross-section requirement. If the objective is an accurate estimate of stress and deflection at any location in the frame, b eam elements would be inap propriate for these members. if the is to establish gross geometry, idealizations may beHowever, used with the objective understanding that some error will be beam incurred. Tapered or butted tubes can be modeled as prismatic (constant cross-section) sections as a rough approximation to the actual geometry. By selecting appropriate dimensions, the analysis can either be correct on average or be conservative. For butted tubes, using the smallest wall thickness for the entire length of the 2

True-Temper Product Guide, 2004.

259


segment will result in overestimation of the peak stress. In the early design stages, this may be acceptable. Tapered tubes modeled using the minimum diameter may significantly overestimate the peak stress and deflection. It is often advisable to run multiple analyses using the smallest, largest and mean diameters. This places bounds on the estimates of stress and deflection. In the event the range between these bounds is excessive, beam idealizations should not be used. For accurate results with beam idealizations, the transverse dimensions of the structural member should be small compared to the length of the member. Beam elements should have a length at least eight to fifteen times the maximum transverse dimension.3 For example, a round tube with a 25 mm (1 inch) diameter should have a minimum length of 200 mm (8 in) for accurate estimation of stress and deflection. Sometimes short beam elements are used—knowing the stress and strain estimates will be inaccurate—in order to provide convenient loading locations on a structure modeled with beam elements. A common example is the bottom bracket shell, which may be 63 mm long with a 38 mm diameter. This clearly violates the minimum length-to-diameter ratio. However, the stresses and deflections at locations well away from the bottom bracket shell will be accurate. The model can be quite appropriate for validating gross geometry if stresses in the bottom bracket shell are disregarded, which may be acceptable during the initial design. This clearly would be inappropriate for an analysis to determine stress concentrations in the region of the bottom bracket shell itself. Beam elements may be curved or straight. Curved frame members with solid or thick-walled cross-sections can usually be modeled accurately with beamelements provided the length-to-transverse dimension ratio is sufficient. However, curved tubes with thin walls can posesignificant problems if idealized with beam elements. When a thin-walled curved tube is subject to flexure, the cross-section distorts. The cross-section of a curved, circular tube will be flattened into an approximate ellipse, for example. This distortion affects the stress/strain distribution across the cross-section of the member, and can violate the requirement of linear distribution of stress and strain. The result is that stresses will be under estimated, sometimes by a significant amount. Figure 13-2 shows the stress distributionin a curved, thinwalled tube subject to flexure. The peak stress shown in the figure is a result of the distortion in the tube cross-sectional shape, and would be missed if beam idealizations had been used to model this part. Fortubes with thick walls, the distortion is minimal, and curved beam elements canprovide good estimates of stress.

3

Peterson, Leisha A. and Kelly J. Londry, 1986, “Finite-Element Structural Analysis: A New Tool for Bicycle Frame Design,” Bike Tech, 5(2), Summer 1986.

260


Figure 13-2 Example of beam idealization for a tube

Figure 13-3 Von Mises stress in a bent tube modeled with shell idealizations. The

peak stress would be underestimated with beam idealizations Consider a rather extreme example, a 50.8 mm (2.0 in) diameter tube with a 0.9 mm (.035 in) wall thickness bent into a 305 mm (12 in) centerline bend radius and lightly loaded in pure flexure. A beam element predicts a maximum Von Mises stress of 6.6 MPa (958 psi), which underestimates the true maximum by nearly 70%. The same frame with shellpsi). elements provides a much more accurate stress member estimate analyzed of 19.2 MPa (2,792 This illustrates the dangers of modeling thin-walled curved tubes with beam idealizations. Summary of Frame Analysis Using Mainly Beam Idealizations •

Objective: Determine gross geometry and tube sizes

•

261


•

Advantages: Fast solution time Minimal time investment for modeling and set-up Rapid design cycle/optimization Limitations: Accurate only for long structural members

• • • •

• •

Will not capture: Stress concentrations Localized stress in dropouts, joints, etc. Stresses in curved thin-wall tubes will be underestimated by a significant amount Application Requirements: MUST have node wherever: A Load or Constraint is to be applied Two or more frame members joined An accurate displacement is required Tube wall section changes (butted tubes) • •

•

•

•

• • • •

Shell Elements Structures that are very thin relative to theirtransverse dimensions can be modeled with shell elements. Examples include thin-walled tubes and formed sheetmetal components. The shell idealization collapses the thickness of the member, reducing the element dimension from three to two. The elements are triangles or

Figure 13-4 Example of beam idealizations and point mass idealizations

262


quadrilaterals. Compared to beam idealizations, Shell elements require more modeling time and effort, but produce more accurate results. Tapered tubes, frame members with non-constant cross-sections, and some stress concentrations such as holes can be accurately modeled. Stresses near frame joints can be more accurately modeled with shell idealizations. Figure 13-4 shows a tube idealized as a shell. Note that the tube has no thickness. The thickness must be specified when the shell elements are defined, and is included in the definition of the shell. However, it is considered constant, and thus is not required in the CAD model. Figure 13-5 shows shell idealizations used in the analysis of a long-wheelbase recumbent bicycle frame under heavy pedaling loads. Much more comprehensive information is available with shell models.

Figure 13-5 Example of shell idealizations of a tube

Figure 13-6 Shell idealization used to model LWB bicycle frame

263


Summary of Frame Analysis Using Shell Idealizations •

Objective: Determine stresses, deflections, and strain energy in frame Advantages: More accurate than beams for some geometry

• •

•

Ability to identify stress concentrations More realistic loads, constraints and geometry Much quicker run time than solid elements, particularly for thin-walled tubes Limitations: Applicable only to thin members relative to cross-sectional dimensions Increased modeling time required Slower design iteration rate relative to beams

• • •

•

• • •

Solid Elements Solid elements are used when the part geometry is not compatible with beam or shell idealizations. Solid elements are often used with existing CAD solid models, so there may be little overhead in preparing the model for analysis. Geometrically, solid elements are tetrahedral, prismatic, or hexahedral in shape. Analyses using solid elements typically take much longer to run than the simpler beam or shell idealizations, but there are essentially no geometric limitations to their use. Typically, run times are reduced by removing non-critical features such as rounds and fillets, or by exploiting symmetry.

Figure 13-7 Example of solid model of tube

264


Solid elements are not typically used to model thin-walled tubular frames, but are frequently used for components and geometry where shell idealizations are inappropriate. Solid elements and shells and beams can be used together in the same model.

Boundary Conditions: Loads and Constraints

Static FEA analyses require the model to be constrained against rigid body movement, meaning the structure cannot translate or rotate other than that caused by deformations. Imagine pushing a structure that is sitting on the ground. Unless the structure is constrained—say by bolting it to the ground—it will slide as a unit. This is an example of a rigid body mode that would prevent a static FEA analysis from running. The constraints must be sufficient to prevent all six rigid body modes—translation in X, Y, and Z, and rotation about X, Y, and Z. Likewise, analyses for models without any loads may fail. Unconstrained or unloaded models may not run, while incorrectly or inappropriately constrained models may run, but give erroneous results. Correct and appropriate application of loads and constraints is critical for accurate estimation of stress and deflection. Recommendations are included with the descriptions of specific load cases.

Load Cases A human-powered vehicle encounters several types of service loads that could potentially lead to structural failure. Predictable accidents or improper use can also impose loads sufficient to cause structural failure. The designer must produce a frame that does not fail due to service loads over the life of the vehicle, but must also make a frame that is as light as possible. These requirements are in direct conflict, and the degree to which both are satisfied is a measure the engineer’s skill and knowledge. A machine component that operates at a known constant load and speed is easy to analyze, and factors of safety are easy to estimate. Human-powered vehicles are far more challenging. Loads and speeds vary considerably, depending on the rider, road conditions, speed, pedaling effort, and other factors. The uncertainty of loading, coupled with a requirement for minimum weight, make frame design a challenging task. In describing specific load cases, recommendations for constraints and loads will be made relative to the vehicle coordinate system described in Chapter 7. The longitudinal coordinate is represented by X, and is positive towards the front of the vehicle. The vertical coordinate is Z, positive downward, and the lateral coordinate is Y, positive to the right.

265


Vertical Drop The vertical drop load case is a simplified way to simulate the long-term effects of road vibration in the absence of test data. It is a good starting point for new designs due to its ease of implementation and the relative value of the results. It can be used as a go/no-go test for new frame concepts—if a new frame design fails to meet the safety criterion for this test, it should be revised or re-designed prior to proceeding with other tests. Objective: Constraints:

Modeling:

Loads: Assessment:

266

To simulate the effects of long-term road vibration on the vehicle frame when no knowledge of actual loads is available. For bicycles, both rear dropouts should be constrained against translation in the vertical, longitudinal, and lateral directions, and the lower end of the headtube should be constrained in the vertical direction only. Improved accuracy is obtained if the stiffness of the fork is taken into consideration. This can be done by using beam elements attached to the headset bearings and fixed in the vertical direction at the front wheel dropouts. For multitrack vehicles with two rear wheels, one rear wheel mount should be constrained against translation in X, Y, and Z, while the other side should only be constrained against translation in X and Z (longitudinal and vertical directions). The front dropouts or kingpins should be constrained against vertical translation only. Tadpole trikes should have rear wheel constraints similar to a bicycle, and front kingpins constrained against vertical translation only. Masses should be added to the model to represent the rider’s weight and the mass of un-modeled components or equipment. The location of the masses should be selected to accurately represent the mass distribution of the vehicle/rider system. (Note, some FEA programs include special element types to help with load distribution. For example, Pro/Mechanica by PTC uses elements called weighted links.) The only load applied in this load case is a vertical gravity load equal to three times the acceleration due to gravity, or 3G’s. Qualitatively assess the frame deflections to verify constraints and loads. Assess convergence, as with any finite element analysis.


Figure 13-8 Vertical 3G drop loads and constraints

Compare the peak stress with the fatigue strength of the frame material. Note that if beam idealizations are used, stress concentrations may not be available or accurate. Also consider total strain energy and maximum displacement.

Horizontal Impact (CPSC or ISO Frame Test) Bicycle manufacturers must demonstrate that their frames meet physical safety tests. While the standards specify a physical test, it is prudent to ensure that a proposed frame design will meet the test. Hence, it is included as one of the recommended load cases. Several standards address bicycle frame testing. In the United States, the Consumer Product Safety Commission Requirements for Fork and Frame Assembly (CFR Part 16, 1512.14) is applicable. The CPSC standard is available on-line, and can be found at: http://www.access.gpo.gov/nara/cfr/waisidx_04/16cfr1512_04.html Internationally, ISO and JIS standards are used. These standards must be purchased. The standards differ somewhat on how the load is applied. The loads and constraints described comply with the CPSC test. The frame test involves a load applied to the front fork directed aft toward the rear wheel. It is generally similar to a low-speed frontal collision with a wall. The physical test requires the fork to be installed. The FEA analysis should include the fork, if feasible. If not, a statically equivalent load set should be applied at the

267


headtube bearing locations. The equivalent loads can be determined from a static analysis of the fork. However, the strain energy requirement cannot be evaluated without considering the contribution of the fork. Delta trikes can be analyzed similarly to bicycles. However, the analysis is not directly applicable to tadpole trikes and other vehicles with two front wheels. For these vehicles, consider the frontal collision possibilities and make appropriate adjustments. It may be prudent to include a single-wheel collision scenario, in which only one front wheel strikes an object. Objectives:

Constraints:

Loads:

Assessment:

(1) To assess ability of frame to pass the CPSC fork/frame test for bicycles. (2) To simulate the effects of a low-speed frontal collision. Both rear dropouts should be constrained against translation in the vertical and longitudinal directions, and one dropout should also be constrained in the lateral direction. The front dropout should be constrained in the vertical direction only. NOTE: This assumes the fork is included in the FEA model. A force of 890 N (200 lbf) shall be applied at the front axle in a direction directed toward the rear wheel axle. If this force does not produce 39.5 J (350 in-lb) strain energy, increase the magnitude of the force until strain energy of 39.5 J is obtained. Qualitatively assess the frame deflections to verify constraints and loads. Assess convergence, as with any finite element analysis. Compare the peak stress with the yield strength of the frame material. The CPSC Fork and Frame Test require that there be

Figure 13-9 Loads and constraints for CPSC frontal impact

268


no fracture in the frame or no deformation that would interfere with steering. Interpret this to mean that stress should not exceed the yield strength of the frame material. Also consider total strain energy (particularly if the load is force-limited to 890 N, rather than energy limited) and maximum displacement.

Maximum Acceleration This is a severe load case, simulating a strong rider accelerating hard from a dead stop. Chain tension is very high, and inertial loads are significant. The seat back sustains a significant load due to reaction against the hard pedaling. (For upright bicycles, the rider is assumed to be out of the saddle, and the reaction force is directed against the handlebar.) Loads for the maximum acceleration must be derived from the equations of equilibrium, first for the entire vehicle, and subsequently applied to the frame. The procedure can be tedious, particularly with drivetrains that include tension-side idlers or mid-drives. Usually some simplifying assumptions are necessary. Since this load case represents a significant source of stress in the frame, it is incumbent to carefully analyze all loads acting on the frame. Measurements indicate that strong amateur riders can generate 60 to 80 N-m of torque at the rear wheel during a hard startup effort on an upright bicycle. Usually recumbent riders produce less start-up torque than upright cyclists, with strong amateur riders achieving 40 N-m. Certainly thesemits li can be exceeded by very strong riders in the right circumstances. Depending on the crank length and gearing, this corresponds to pedal forces from 500 to 1400 N (110 to 320 lb). Chain tension is often significantly more than pedal loads. For example, a pedal force of 1400 N using standard 170 mm crank arms and a 48 tooth chainring produces a chain tension of 2.5 kN (560 lb)! The forces during hard acceleration can indeed be quite large. Objective:

To simulate the effects of hard start-up pedaling on the vehicle frame.

Constraints:

For bicycles, rear dropouts be constrained a similar mannerboth as the vertical dropshould test, except it is even in more important to include the front fork stiffness. Also, a lateral constraint at the dropout should be used to simulate the lateral resistance of the tire acting on the pavement. For multitrack vehicles, all wheels should be constrained in the vertical direction. The drive wheel(s) mounting locations should be constrained in the longitudinal (X) direction, and at

269


Modeling:

Loads:

270

least one wheel, usually the drive wheel—should be constrained in the lateral (Y) direction. The mass of the rider is significant in this analysis. Masses should be added to the model to represent the rider’s weight and the mass of un-modeled components or equipment. The location of the masses should be selected to accurately represent the mass distribution of the vehicle/rider system. Correct application of loads is critical—and non-trivial—for this load case. A gravity load equal to one G should be applied to the model. Loads representing pedaling must be applied to the bottom bracket shell at the location of the crank bearings. For tandems, apply loads to each bottom bracket shell. Loads must also be applied to the frame at the location of any tension-side idler sprockets. These loads should be applied at the mounting location for the sprockets. Idlers on the slack side of the chain may be neglected. If the vehicle has a mid-drive consisting of a bottom bracket, loads should be applied at the bearing locations similar to a crankset. For recumbents, there is usually a reaction load on the seat back due to the hard pedaling. This load must be applied to the frame at the location of the seat mounts and stays. For uprights and some recumbents, the pedaling force is partially reacted by pulling on the handlebar. For these vehicles, the appropriate handlebar loads should be applied.


Figure 13-10 Maximum acceleration loads and constraints

Assessment:

Qualitatively assess the frame deflections to verify constraints and loads. Assess convergence, as with any finite element analysis. Compare the peak stress with the yield strength of the frame material. Also consider total strain energy and maximum displacement. It may be particularly helpful to compare the strain energy of different frame members with the goal of minimizing weight while maintaining sufficient stiffness.

Hill Climb This load case is very similar to the max acceleration case, and usually yields similar results. It may not be necessary to run both load cases, particularly when resources are limited. Loads are similar to that of the max acceleration, but the force of gravity is acting at an angle from the vertical.

Maximum Front Braking In a rapid stop maneuver, the front wheel provides the greatest stopping force. The forces are large, and inertial loading is significant. The limiting factor during braking must be determined using the methods of Chapter 7—either imminent pitchover or skidding. For vehicles with high or forward centers of gravity, the limit is likely to be pitchover. In this case, the vertical force on the

271


rear wheel(s) is zero, and all weight is concentrated on the front wheel(s). Vehicles that are limited by skidding will have a reduced—but non-zero—vertical load component. In either case, the reaction forces at the headtube bearings or kingpin bearings can be easily computed by equilibrium of the front wheel/fork assembly.

Objective:

Constraints:

Modeling:

Loads:

Assessment:

To simulate the effects of hard front braking on the vehicle frame. (NOTE: The analysis described here is a simple analysis for the bicycle frame, and not the front fork. The fork may be highly stressed. A fork analysis requires braking forces to be identified and correctly modeled. See the qualitative description of rear braking for items to consider.) Constrain the head tube (or fork if included) against translation in X, Y, and Z. Constrain the rear dropouts in the Y and Z (lateral and vertical directions). (Only one dropout needs to be constrained in the lateral direction.) The mass of the rider is significant in this analysis, since inertial forces are important. Masses should be added to the model to represent the rider’s weight and the mass of un-modeled components or equipment. The location of the masses should be selected to accurately represent the mass distribution of the vehicle/rider system. Apply a 1G vertical gravity load directed downward in conjunction with a gravity load acting in the positive X direction. The magnitude of the latter should be the accelerationAx at the braking limit, given by Equation 7-34. Verify that the vertical reaction load at the rear axle is not negative, indicating the pitchover threshold has been exceeded. If so, reduce the longitudinal acceleration value. Qualitatively assess the frame deflections to verify constraints and loads. Assess convergence, as with any finite element analysis. Verify reaction loads are correct by comparing with calculated headtube reaction forces. Compare the peak stress with the yield strength of the frame material. Also consider total strain energy and maximum displacement. It may be particularly helpful to compare the strain energy of different frame members with the goal of minimizing weight while maintaining sufficient stiffness.

272


Maximum Rear Braking This load case is much less severe overall than the maximum fron t braking, due to the reduced stopping force and deceleration possible with rear braking. Local frame stresses at the brake mount locations may be severe, however, and should be investigated. After the limiting braking deceleration has been determined using the methods of Chapter 7, a free-body diagram of the rear wheel indicates the force at the brake mount. Note the frame location that reacts against the braking load. For caliper brakes, this will occur at the caliper mount bolt location. For cantilever or linear pull brakes, the force is reacted at the brake mounting posts. For disc brakes, the reaction location is the caliper mount, while for drum, coaster or roller brakes there may be a torque arm. In all cases, the magnitude of the force on the frame is calculated from the force on the wheel, adjusting for the offset between the wheel and frame locations. This is simply an equilibrium calculation, but the brake geometry must be well understood to avoid errors. Objective: Constraints:

Modeling:

Loads:

Assessment:

To simulate the effects of hard rear-only braking on the vehicle frame. Constrain the head tube (or fork if included) against translation in the vertical direction only. Constrain the rear dropouts in the X, Y, and Z directions. (Only one dropout needs to be constrained in the lateral direction.) The mass of the rider is significant in this analysis, since inertial forces are important. Masses should be added to the model to represent the rider’s weight and the mass of un-modeled components or equipment. The location of the masses should be selected to accurately represent the mass distribution of the vehicle/rider system. Include gravity loads as described in the maximum front braking case. In addition, include the local forces acting at the brake mount location. Qualitatively assess the frame deflections to verify constraints and loads. Assess convergence, as with any finite element analysis. Verify reaction loads are correct by comparing with calculated headtube reaction forces. Compare the peak stress with the yield strength of the frame material. Also consider total strain energy and maximum displacement. It may be particularly helpful to compare the strain energy of different frame members with the goal of minimizing weight while maintaining sufficient stiffness.

273


Steady-State Pedaling Steady-state pedaling can usually be modeled in a similar manner to maximum acceleration. The primary difference is the magnitude of the pedal forces and handlebar reaction forces.

Other Load Cases Additional load cases may be used as needed. For example, a test to simulate the Rinard frame deflection test4 may be appropriate. Multi-track vehicles should include a lateral load case simulating maximum cornering loads. Special-purpose vehicles may require special, additional load cases.

FEA Load Verification The load cases described work well for initial design of a new vehicle. Properly applied, they will result in a safe vehicle that is not overly heavy. In order to achieve the minimum weight possible, actual in-service loads should be measured experimentally. The measured loads can be applied to FEA models in order to obtain more accurate results. If appropriated testing equipment is available, the loads can be reproduced in the lab, perhaps after scaling, in order to physically test vehicle frames for fatigue failure. These tests can be expensive and time consuming, but should be considered before bringing a new vehicle on the market.

Initial Design Using Beam Idealizations The focus of this chapter is on initial development of a new frame design using simple methods without prior field data. Beam idealizations are an appropriate starting point for such cases. The general procedure for an initial frame analysis using finite element beam idealizations is as follows: 1. Create the idealized geometryand apply beam idealizations 2. Apply loads appropriate to each load case atnodes a. Apply loads to points (beam nodes) only 3. Apply constraints appropriate for eachload case a. Apply constraints to points (beam nodes) only 4

Rinard, Damon, Sheldon Brown’s web site, http://www.sheldonbrown.com/rinard/rinard_ frametest.html, accessed January 24, 2012.

274


4. Apply point masses at required locations a. Use weighted links or rigid links as required 5. Specify material for the model, or foreach beam 6. Run the analysis 7. Evaluate the results a. Check reaction forces at constraints. The expected values can usually

b.

c.

d.

e. f.

be computed with little effort. Discrepancies may indicate modeling errors. Check displaced shape. Unexpected or unreasonable displacements in shape may indicate modeling errors. i. Only if items a and b are satisfactory should stress and deflection results be accepted. Compare peak stress with allowable values i. Beware of artificial stress concentrations due to application of point forces or constraints. Stresses immediately adjacent to point loads and constraints should be considered suspect. ii. If stress exceeds allowable values: 1. Increase the moment of inertia of the beam by increasing the diameter or, less effectively, the wall thickness 2. Adjust gross frame geometryto better accommodate loads iii. If stress in any member is well below the allowable values for all load cases, reduce the size of the member or eliminate it. Check deflections i. If excessive, adjust member size or gross geometry to stiffen the frame Check strain energy and compare with other frame geometries. The stiffest frame will have the lowest strain energy under the same loading. Check the total mass of the frame by subtracting all point masses from the total mass of the model. Refine the design until all criteria are met and the frame has the lowest weight.

Detailed Frame Design Frame design is not complete once gross geometry and tube sizes are determined. Frame details are crucial for functionality of the frame. The frame must be designed such that components can be properly installed—the bottom bracket must fit into its shell; the headset must fit into the headtube; the wheels must fit into the dropouts. Brakes usually require special frame fittings that depend on the type of brake. Fittings for water bottles, derailleurs, and cables are usually

275


Figure 13-11 Bottom bracket shell dimensions

required on a finished frame. These items are best addressed in the design stage. Ensuring proper fit with components is perhaps the most crucial these steps.

Frame Compatibility with Components A new frame design must be compatible with the components used. Wheels must fit in the dropouts, the headset must fit both the fork and the headtube, and the crankset must fit in the bottom-bracket shell. There are many components on the market, and unfortunately for the builder, there is no single standard for any of the components that attaché to the frame. However, in most cases standards do exist that are applicable to many or most commercially available components. The following information has been compiled from several sources.5–7 It is not comprehensive—some commercially available components may differ from the requirements shown here. When in doubt, always check with the component manufacturer for more information.

Bottom Bracket Shells Cranksets, or more specifically the bottom bracket assembly containing the crank spindle and bearings, attach to the frame at the bottom bracket shell. These may be made from tubing or purchased through vendors at relatively low cost. Be sure to install correctly, with right-hand threads on the left side of the bike. The dimensions listed are the most common, and do not include oversize bottom 5

Sutherland, Howard and Willem Horner, 2004, Sutherland’s Handbook for Bicycle Mechanics, 7th Ed., Sutherland Publications, Berkeley, CA, USA. 6 Shimano, 2002, Shimano Technical Information STI 2002. 7 SRAM, 2010, SRAM Technical Manual 2010, online at http://www.sram.com/sites/default/ files/techdocs/my10-sram-tech-manual-rev-a.pdf, accessed January 24, 2012.

276


Table 13-1 Bottom Bracket Shell Dimensions Length

68mmor73mm

Thread

LeftSideofBike Right Side of Bike

1.370-24NS RH 1.370-24NS LH

brackets used for some cranksets. Also, eccentric bottom brackets require a special, larger shell. See the manufacturer’s instructions.

Head Tubes The head tube supports the fork. The headset bearings are usually press-fit into the head tube, requiring a close tolerance on the inside diameter of the head tube. Reamers are available from bicycle tool manufacturers to achieve the required tolerances. There are several standards for conventional headsets, as well as semi-integrated and integrated headsets. The following tolerances are for the inside diameter of the head tube, where the bearing cups are pressed in. Only tolerances for conventional headsets are included. The nominal size listed for each standard corresponds to the fork steerer tube outside diameter. Note that many new headset designs are now available.

Table 13-2 Headtube Tolerances Steerer Size

Headtube ID Tolerance

1in

30.0to30.1mm

11/8in

33.7to33.9mm

1inJIS

29.85to29.90mm

11/4in 11/2in

36.9to36.95mm 49.57to49.61mm

Description

1inch“Professional”or “European” size Typical road bike size This is recommended 1 inch size Typicalmountainbikesize Japanese standard Common on lower-priced Asian bicycles Oversize Oversize

277


Steerer Tubes The steerer tube is the extension of the fork that fits into the frame. The handlebar stem is attached to the steerer tube as well. The outside diameter of the steerer must have close tolerances to fit the headset bearings. The headset includes a crown race, on which the lower bearing rests. The crown race is pressed onto the crown race seat, a required part of the fork. The listed tolerance must

Figure 13-12 Fork steerer and crown race seat

Table 13-3 Steerer and Crown Race Seat Tolerances Standard

Steerer Tube OD

Steerer Thread

Crown Seat OD

1 in Threaded 1 in Threaded JIS 1 1/8 Threaded 1 1/4 Threaded

25.1–25.3 mm 25.1–25.3 mm 28.3–28.5 mm 31.5–31.7 mm

1–24tpi* 1–24tpi* 1 1/8–26tpi* 1 1/4–26tpi

26.43–26.49 mm 27.03–27.09 mm 30.015–30.075 mm 33.03–33.09 mm

1 in Threadless 1 1/8 in Threadless

25.1–25.3 mm 28.3–28.5 mm

None None

26.43–26.49 mm 30.015–30.075 mm

278


Figure 13-13 Over locknut spacing dimension

Table 13-4 Over-Locknut Dimensions for Common Hubs Wheel

Front Rear Rear

Use

Road/Mountain Road Mountain

Over-Locknut

100mm 130 mm 135 mm

Axle Diameter

9mm mm 10 10 mm

be maintained for at least 12.5 mm (.5 inch) into each end of the headtube. Fit between the crown race and the crown seat is important, and also requires a close tolerance. Steerer tubes can be threaded or threadless, depending on the type of headset used.

Dropout Locknut-to-Locknut Dimensions Wheel hubs fit into dropouts on the frame and fork. The distance along the hub axle between the lock nuts is the over locknut dimension. The frame and fork must be compatible with hubs. As before, not all hubs follow these standards, but these are currently the most common dimensions. In particular, the axle diameter

Figure 13-14 Derailleur hangar dimension definitions

279


Table 13-5 Selected Derailleur Hangar Dimensions SRAM Mountain Derailleurs L

X

28 8–10 30 7.5–10 SRAM Road Derailleurs L

X

A

R

25˚–30˚ 25˚–30˚

8.5Max 8.5Max

A

R

26 6–10 30˚–35˚ 28 8–10 30˚–35˚ Shimano Mountain Derailleurs L

X

8.5Max 8.5Max

A

28 6–10 25˚–30˚ 30 7.5–10 25˚–30˚ Shimano Road Derailleurs

B

11.5–13.5 11.5–13.5

B

11.5–12.5 11.5–12.55

T

7–8 7–8

T

7–8 7–8

T

7–8 7–8

L

X

A

T

24 26

4–10 6–10

30˚–35˚ 30˚–35˚

7–8 7–8

varies between hub manufacturers. Unless the hub uses an oversize axle, frame compatibility should not be a problem.

Integral Derailleur Hangar Dimensions Derailleur hangars provide an attachment location for rear derailleurs. They can be integrated with the frame in what is known as an integral derailleur hangar, or attached as separate piece know as a replaceable hangar. Either way, the dimensions locating the derailleur are important for good shifting performance.

280

CHAPTER

BICYCLE COMPONENTS

14

M

any of the component parts for human-powered vehicles are available as stock bicycle parts. There are literally thousands of off-the-shelf components available for bicycles and HPVs. Competition among bicycle component manufacturers drives rapid development of new products with improved performance, functionality, and weight. While the latest top tier products, usually designed for racing, are expensive, the rapid development cycle means that new technologies rapidly trickle down to low and mid-range components with much more moderate pricing. It is usually possible to purchase mid-range components that offer very good performance at a moderate price. It makes sense to use offthe-shelf bicycle components on an HPV in many cases. Certainly there are times when bespoke components are needed or desired, but with the quality, availability, and broad selection of standard parts, the designer should have a good reason to do so. This chapter will provide general information about some of the more common types of components and a few less common but interesting alternatives. The intent is to familiarize the reader with component types, and how the types of components relate to the overall vehicle design. Components must be compatible with frames as well as related components. Decision such as steerer diameter, type of bottom bracket, and hub type affect both the frame and the components that can be used. The very large number of variations means that the designer must be diligent in specifying components, or at least the type of components, concurrently with frame design. This chapter should provide guidance for many of the more common—and a few unusual—components. It is not comprehensive, so some time spent with catalogs, cycling magazine product reviews, or just visiting your local bike shop can be very advantageous.

281


Wheels and Tires The bicycle wheel is a beautiful example of structural and functional optimization. The wheels support the entire weight of a bicycle or tricycle, and dramatically affect HPV handling and performance. They must resist vertical loads due to the weight of the bicycle and rider, lateral loads during cornering, and torsional loads caused by the drivetrain or hub brakes. They provide suspension, ensure traction, roll with surprising ease, and weigh very little. This efficiency is due to an evolution of more than a century, leading to one of the most elegant and efficient structures ever invented. Structural efficiency refers to the ratio of sustained external loads to the structural weight. It is axiomatic that the most efficient structures minimize the number of compression members and maximize the number of tension members. The rim is the only compression member and wire spokes are loaded only in tension, so the wheel is extremely efficient. Under load, tension is reduced for spokes below the hub and increased for those above the hub. The change in tension should be much smaller than the actual tension in the spoke, improving fatigue strength for the spokes. A well-built wheel is quite light and strong, and can support a vertical load well over one hundred times its own weight. It is also efficient at transmitting torque through the drive train or hub brakes. Wheels with three or four large spokes and disk wheels—both usually made of composite materials—are also available. These wheels have an aerodynamic advantage over wire spoked wheels. Full disk wheels on unfaired vehicles can have very low drag, but are susceptible to cross winds. For this reason, they are not recommended for front wheels. Wheels with three or four spoke streamlined spokes comparable drag to the full disk wheel, but are significantly less susceptible to cross winds. Wheel properties can significantly affect vehicle performance, as described in Chapters 8, 9, and 11. These properties include vertical, lateral, and torsional strength and stiffness, wheel weight and mass moment of inertia, aerodynamic drag, rolling resistance, cornering and camber stiffnesses of the tire, and the shock absorbing ability of the tire. Most of these properties result from the wheel as a whole—hub, spokes, rim, and tire. Thus, the wheel assemblies are among the most important for a high-performance vehicle. The traditional wire-spoked bicycle wheel is a very sophisticated assembly, despite its structural elegance. Typical components of a wheel include the hub, spokes, spoke nipples, rim, rim tape, tube, and tire. Thehub contains the axle, bearings and hub body, and is discussed in more detail below. Spokes are typically made of round wire, which may be butted, or reduced in diameter, for most of the middle

282

Bicycle Components

length. Flat, aerodynamic wire spokes are available to reduce drag caused by the rotating spoke.

Tire and Rim Sizes Bicycle tire sizes can be quite confusing. For example, most 20 inch tires are not actually 20 inches in diameter, and there are at least two different—and incompatible—20 inch rim sizes. Several systems are used for marking tires, and only one uses an approach that is logical and meaningful. The ISO standard for wired edge tires specifies tires by two numbers, such as 28-622. The first is the tire section width (SW), illustrated in Figure 14-1. (Note: Tread on the sides of the tire sometimes makes the overall width of the tire greater than the section

Figure 14-1 ISO Tire Size Interpretation

283


width. The tire illustrated has little or no tread.) It is essentially the diameter of a cross-section of the tire, and is given in millimeters. The second number is the bead seat diameter (BSD), or the diameter of the rim at the location h were the tire bead is seated. The bead seat diameter must be the same for both rim and tire. The ISO system is the least ambiguous, but is not always used by cyclists and bicycle mechanics. For example, a 23-622 tire might be referred to as a 700c by 23 tire, a size common on upright road bicycles. Beaded seat tires are designated by the nominal wheel diameter, followed by the section width, both in inches. For example, a mountain bike tire might be designated as 26 × 2.0 inches. This would correspond to an ISO size of 50-557. Many tires are marked with both designations. The actual outside diameter of a wheel depends on the section width and the bead seat diameter (and to a lesser extent, the tire design). A good estimate of the outside diameter in mm is given by:

Table 14-1 Selected Tire Sizes and Typical Applications Common Size Designation 27inch 700c

26inch 650c

ISOSizes 630 622

559 571

Application Olderroadbicycles Roadbicycles;highracerrecumbents;rear wheel for some recumbents; 29er mountain bikes Mountainbikes,cruisers Smallsizeroadbikes Some triathalon and time trial bikes

24inchroad

520

TerryBicyclesFront Children’s road bikes

24inchBMX

507

BMXbikes, Children’s Mountain Bikes

20×11/8road 20inchBMX

451 406

BMXRoad Recumbent BMXandChildren’sbikes Some recumbent front wheels

16 inch

349

Some small-wheel folders, some recumbent and tadpole trikes (front)

16 inch

305

Folders; small children’s bikes; occasional recumbent (front)

284

Bicycle Components

Outside Diameter

»

BSD + 2 ´ SW

(14-1)

For most tires, this is fairly accurate. If this value is used to program cycle computers, the error is often less than 2% for road tires (but may be more for tires with thick or knobby tread or thick puncture-resistant belts). To obtain more accuracy for a given tire, the diameter must be measured with the tire at the correct inflation pressure and load. (Generally the circumference of the wheel is measured by rolling the vehicle with rider along a tape measure laid flat on the ground.) As an example, compare the actual outside diameter of a 26 × 2.0 tire with a 26 × 1.5 tire. The first has an actual diameter of 25.87 inches, very close to the nominal size. The second tire is 25.08 inches, nearly an inch smaller than the nominal 26 inches. For any wheel, the tire and rim must be compatible. For compatibility, the bead seat diameter must be the same for both tire and rim. Secondly, the tire section width should be between 1.3 and 3 times the inside rim width (as shown in Figure 14-2). This range is the maximum that will work for vehicles that do not use road brakes (caliper brakes). Caliper brakes do not open far enough to fit larger tires, so a better maximum section width would be twice the inside rim width.

Rims Rims are arguably the most critical component on an HPV. They are very highly loaded, resisting loads from the spokes, the inflated tire, radial and lateral loads

Figure 14-2 Rim Dimensions

285


incurred during riding, and often the force of brakes. Rims are most often made by extruding aluminum sections, then bending them into circles and welding the ends together. (A very few steel rims are still available. These should generally be avoided—not only are they heavier, but in wet conditions rim brake effectiveness is severely reduced with steel rims. This is a potentially severe safety concern.) Rim quality can vary greatly, in terms of weight, rigidity, strength, trueness, and aerodynamics. A fairly deep, hollow cross-section, such as in Figure 14-2, is stiffer and stronger for the same weight than a shallow or open section. It will better resist both the normal operational loads and the occasional extreme load due to a pothole or curb. Open cross sections should be avoided for quality wheels. Deep, aerodynamic sections can reduce the drag on a wheel, and are discussed in Chapter 14. Spoke hole eyelets—small grommets or sleeves that fit into the spoke holes—are often used with higher quality rims. Although they add some weight, eyelets strengthen the rim around the spoke hole and also make truing the wheel easier. Rims can fail because of wear, fatigue, excessive loads, or overheating. Most quality rims today have a very long fatigue life, on the order of 50,000 miles. Generally, wear or damage will require rim replacement long before fatigue becomes a problem. Impacts due to hitting a pothole, rock or curb can damage a rim. A noticeably warped rim should be replaced, as attempting to true—even if successful— can excessively stress the rim. Rim brakes cause wear on the side walls of the rim and can also cause rim heating. This wear is exacerbated by frequent riding in adverse conditions such as rain, snow, mud, or sand. Winter riding in northern latitudes where roads are frequently salted is particularly hard on rims, especially when used with rim brakes, where both corrosion and wear shorten rim life. When the rim side wall becomes excessively worn, tire pressure can cause the rim to rupture. A long section of rim can split off, with a high likelihood of loss of control and a crash (particularly for front wheel failures). Rim brakes can also produce very high temperatures in the rim, particularly on long descents with a heavy load. Although new rims can fail due to excessive heat, most rim failures not caused by impacts are probably due to a combination of wear and overheating.

Figure 14-3 Example of open section rim

286

Bicycle Components

Some rims designed for rim brakes have wear indicators, or small notches cut into the inside wall of the rim. When the wear indicators become visible as holes on the outside wall, the rim should be replaced.

Spokes and Spoke Nipples Spokes carry the wheel load from the rim to the hub. Most spokes have a head at the elbow, or bent end and threads at the other end. Some hubs are designed to use spokes without an elbow, but these are not common. Although there are many types and variations of spokes available, only a few are in common use: straight gauge wire spokes, butted spokes, and aero or bladed spokes. Straight gauge spokes are made from wire that is a constant diameter from the elbow to the threaded end. They are general-purpose spokes and are relatively inexpensive. However, they are slightly heavier and tend to have reduced fatigue strength when compared to butted spokes. For very small wheels that require short spoke lengths, they may be the only available option. Butted spokes have a reduced diameter in the center section of the spoke. If both the nipple and elbow ends have the same diameter, the spoke is double-butted. If the elbow end is larger than the nipple end, and both are larger than the center, the spoke is triple-butted. Butted spokes are somewhat lighter and more aerodynamic than straight-gauge spokes, and—importantly—they are much stronger at resisting fatigue. The increased fatigue strength is due to improvements in strength from cold-working done while reducing the diameter of the center section and from the reduced stiffness of the spoke. Aero, or bladed spokes, are used on track bicycles, time trial and some road bicycles to reduce drag caused by the spokes. They are made from flattened wire,

Table 14-2 Spoke Diameter and Gauge Sizes Type Straight gauge Aero (bladed) Double Butted

Siz( emm) 2.0

14

1.8

15

1.8

15

2.0/1.8

14/15

2.0/1.5 TripleButted

Siz( egauge)

2.34/1.8/2.0 2.0/1.7/1.8

14/17 13/15/14 14/16/15

287


and require either a special hub or a hub modified with slots for the spoke blade. Aero spokes have less torsional strength than round spokes. Wire spokes screw into spoke nipples, which fit into holes in the rim. Spoke nipples are commonly available in either brass or aluminum. Brass nipples have high strength and excellent friction properties, and seldom damage rims. Alloy aluminum nipples are about one third the weight of brass, but require more care when building a wheel. They are more prone to damage, and may also damage the rim if sufficient lubrication is not used. Hex head nipples are used specifically for deep-V rims. If a spoke nipple is over tightened, the nipple usually fails before the spoke, particularly for butted spokes. This is good, as spokes are much more expensive than nipples.

Hubs Literally hundreds of different hub models are available on the market. Hub features include the number of spokes, the drive mechanism, bearing type, axle type, and brake mount. In addition, important hub dimensions such as Over-Locknut (OLN) distance and spoke flange height can vary significantly. See Figure 14-1 for illustrations of hub parameters. Drive Mechanism: Rear hubs have some means to drive the wheel, while front hubs do not. (Although some HPVs use front-wheel-drive, common terminology refers to drive hubs as rear hubs.) Drive mechanisms include threads for cogs or freewheels, freehubs, or internal gears. Freehubs, currently the most common drive type, incorporate a clutch into the hub. A cassette containing a cluster of sprockets slides onto splines on the freehub body. Cassettes are available with 7 to 11 cogs, but in general a given freehub can only use the number of cogs for which it was designed. (Seven sprockets can be used on eight-speed freehubs if a spacer is used.) Freewheels are common on single-speed bicycles and some low-cost multispeed bicycles. A freewheel is a unit that contains both the clutch and the sprockets, and screws onto threads on the hub body. Hubs for BMX and single speed bicycles are designed for freewheels with only a single sprocket, while multispeed freewheels may have 5, 6, 7, or 8 sprockets. Over the years a number of manufacturers have made rear hubs with internal gears, ranging from two to 14 speeds. Most of these internally geared hubs use epicyclic gearing. Advantages usually include ease of shifting, less maintenance, and the ability to guard or enclose the drive train. Bearings: Hubs are available with either loose ball or cartridge bearings. Loose ball bearings use a cup that is permanently pressed into the hub body and cones mounted on the axle. The ball bearings are, as the name implies, loose.

288

Bicycle Components

If the axle is removed, the balls will fall out. For long life and minimum friction, it is imperative to keep these bearings properly lubricated and adjusted. Although there are exceptions, many front bearings use 10 balls with 3/16 inch diameter per side. (Some Campagnolo front hubs use the very similar looking 7/32 inch balls.) Most rear hubs use 9 balls per side with 1/4 inch diameters. (Campagnolo uses 10 balls per side.) Cartridge bearings are units that contain the balls and races. Sealed cartridge bearings are lubricated at the factory and contain seals to retain the grease. They are seldom used on inexpensive hubs, because cartridge bearings require tight tolerances on the fit of the bearings. Sealed cartridge bearings may have somewhat higher friction when new, but as the seals wear in, bearing friction decreases. Axle Type: Hubs may have solid axles or hollow axles used for quick release levers. The later are known as quick release or QR hubs. Quick release hubs require a minimum thickness for the dropouts on the frame—if the end of the axle extends beyond the dropout, the wheel cannot be properly secured. The consequences are potentially severe. Most road and mountain bicycles use a M9x1 thread front and M10x1 thread for rear axles rear, but there are many exceptions. Another common thread is 3/8 × 26TPI. This is often found on children’s bikes and BMX bikes. Oversize axles up to15mm are used on some BMX, mountain, and downhill bicycles. Some hubs used with front suspension forks use a 20 mm diameter through axle. Brake Mount: Hub brakes require special hubs designed for mounting the brake. Both disc brakes and drum brakes are used, but disc brakes are far more common today. Disc hubs have mounts for the brake rotor. The international standard disc brake mount specifies that the rotor attach to the hub with six M5x.8 screws located on a 44 mm diameter bolt circle. However, not all manufacturers adhere to this standard. Shimano has a proprietary Centerlock mount that uses splines and a lockring. Wheelbuilding: All components comprising a wheel must be compatible. The hub and rim must have the same number of spoke holes and the spokes must be the correct length. Brake discs must be compatible with the hub brake mount, and the over locknut distance must match that of the fork or frame. Spoke length is critical for a well-built wheel. The length of the spokes depends on the hub spoke flange dimensions, the effective rim diameter (ERD) and the lacing pattern. Important dimensions on the hub include the spoke hole circle diameter, the distance of the flange from the hub center plane (which may be different for left and right sides of the hub), and the diameter of the spoke holes. The ERD is the diameter of the rim as measured from the seat of a spoke nipple. Because of the many rim designs, the ERD will differ for rims with the same bead seat diameter. Lacing

289


patterns are based on the number of times a spoke crosses over another spoke. Radial patterns have no crosses, and can not transmit significant torque between the hub and the rim. Radial patterns should only be used on non-driving wheels without hub brakes, and are usually not used on wheels with more than 32 spokes. More commonly, tangential spoke patterns—with oneto three or more crosses (1X, 2X, 3X or 4X)—are used. Three cross patterns make good general-purpose wheels, with sufficient strength and rigidity for mostusage. One and two cross patterns are used primarily for road wheels, with one cross more common on deep section rims. Some mountain bikes also use a two cross pattern. Four cross patterns are used for hightorque applications, such as tandems. Drive wheels may use mixed spoke patterns on the left and right sides of the wheel. A variety of other patterns such as crow’s foot, three spoke, and others are also used occasionally. Spoke length calculators are available commercially and on the world wide web. Many have databases of hubs and rims, so actual measurements are not necessary. Two examples of on-line calculators as of this writing are: DT Swiss United Bicycle Institute

(https://spokes-calculator.dtswiss.com/en/) (http://www.bikeschool.com/spokes/index.cgi)

The Matlab program spoke.m, available for download with this text, will calculate spoke length given the requisite dimensions. Note that if the hub or lacing pattern is not symmetrical left and right, the program must be run twice, once for each side of the wheel. Also, unlike the

Drivetrain Components Virtually all current production bicycles and HPVs use leg cranks, chains, and sprockets for drivetrains. Most multi-speed vehicles use derailleurs, although internal hub gears have been used for many years, and are currently having somewhat of a renaissance. All drivetrain and shifting components must be compatible. For example, a 10 speed drivetrain (ten sprockets on the cassette) must use a chain, chainrings, derailleurs, and shifters designed for a 10 speed system. Otherwise performance—and in some cases functionality—will suffer. Bottom Brackets: The bottom bracket contains the bearings that support the crank. Currently, most bottom brackets are sealed units that thread into the bicycle frame and are replaced as a unit. Older bicycles and many children’s or lowcost bicycles use open bearings and one-piece cranks. Bottom brackets are specified by the spindle type, spindle length, and bottom bracket shell length. A square taper cartridge-style bottom bracket is illustrated

290

Bicycle Components

Figure 14-4 Hub parameters

in Figure 14-4. The spindle length is specified to obtain the correct chainline. A shorter or longer spindle will shift the chainrings inboard or outboard respectively. Bottom bracket shells—the part of the frame in which the bottom bracket fits—are typically either 68 or 73 mm wide. Most bottom brackets will on fit one size shell. Most bicycles use a 1.37 × 24 thread, with left-hand threads on the right side of the bicycle and right-hand threads on the left side of the bicycle, although several oversize bottom brackets are now on the market. Many spindle types are in common usage. Square taper spindles have been used for many years, and are still quite common on lower-cost cranksets. Two similar— but incompatible—tapers are used, Campagnolo and JIS. They differ in the width of the small end of the taper. Hollow spindles reduce weight without a corresponding loss of strength. Most use splines to engage the crank arms, but many proprietary designs are in use. The crank arm spline must be compatible with the bottom bracket for the parts to fit together. The international standard ISIS bottom brackets use 10 splines with a 21 mm spindle. Many Shimano bottom brackets use an eight spline design called Octolink. Two-piece cranks use a large diameter hollow spindle, often with external bearings, to further reduce weight while retaining strength and stiffness. Some newer, high-performance cranksets require oversize bottom bracket shells on the frame. These include the BB30, PF30, and other standards, with cups that press into the large bottom bracket shell.

Figure 14-5 Square taper cartridge-style bottom bracket

291


Cranksets: Cranksets consist of the crank arms and the chainrings. Some BMX bicycles and children’s bikes use one-piece cranks, but most bicycles use crank arms that attach to a separate bottom bracket spindle. Cranksets are selected based on number of chainrings, bottom bracket spindle type, and crank arm length. (If crank arms and chainrings are specified separately, rather than as a set, the chainring bolt pattern must also be specified.) Generally cranks come with one, two, or three chainrings, known as single, double, and triple cranksets respectively. Crank arms are only offered in a limited number of lengths by most manufacturers. Lengths of 170 mm for road bikes and 175 mm for mountain bikes are typical, with available lengths ranging from about 165 to 180 mm. (Many recumbent riders prefer much shorter crankarms—as short as 110 mm or so. Some BMX cranks are available in lengths down to about 140 mm, but only a few manufacturers make shorter crank arms.) Derailleurs and Shift Systems: The vast majority of multi-speed bicycles on the market today use derailleurs to shift the chain from one sprocket to the next. Rear derailleurs shift the chain between the cogs on the cassette or freewheel. (In current usage, the drivetrain is described by the number of cogs on the cassette, rather than the total number of combinations. Thus a 10 speed drivetrain has ten sprockets on the cassette, regardless of how many chainrings (1, 2, or 3) are used on the crank.) Front derailleurs shift between chainrings. Rear derailleurs are mounted either to the frame via an integral derailleur hangar or to a replaceable derailleur hangar that is attached to the rear dropout. (Correct dimensions for derailleur mounting is described in Chapter 13.) Generally the mounting screw has an M10x1 thread (10 mm by 26TPI for Campagnolo). Rear derailleurs are selected based on the maximum cassette cog size, the maximum total capacity (to take up chain slack) and the number of cassette cogs. The maximum cog size is given in number of teeth, and assumes a standard length for the derailleur hangar. A longer hanger may permit a derailleur to work with a larger rear cog. The maximum total capacity represents how much slack the derailleur can take up when the chain is on the smallest front and rear cogs. Total capacity represents the difference in the cassette tooth sizes added to the difference in the chainring tooth sizes. For example, if the cassette cogs range from 11 to 32 teeth, and the chainrings are 52-42-30 for the outer, middle, and inner chainring, respectively, then the total capacity required of the rear derailleur is (32 − 11) + (52 − 30) = 21 + 22 = 43 teeth A rear derailleur should be selected with a maximum cog size of at least 32 teeth and a maximum total capacity of at least 43 teeth.

292

Bicycle Components

Rear derailleur specifications include the number of cogs on the cassette—a 9-speed derailleur for example. However, in many cases a derailleur will work with other speed cassettes. (Shifters must be designed for the correct number of cogs, however.) Most rear derailleurs will shift toward the largest cog as cable tension is increased. In recent years Shimano has made a number of low-normal rear derailleurs that shift toward the largest cog as tension is increased. These derailleurs, formerly known as rapid-rise, can provide smooth, quiet shifts. Because they rely on spring tension to shift to lower gears (larger cogs) the shift gates on the cogs are crucial. At low cadence, a downshift may be delayed until the gates have rotated into the correct position. Front derailleurs are selected by the capacity, type of mount, and number of speeds. Front derailleur capacity is the difference between the number of teeth on the smallest chainring and the largest chainring. Typical capacities are 15 teeth for double cranksets and 22 teeth for triple cranksets. They can be mounted to the seat tube by a clamp or a brazed-on fitting. The seat tube diameter must be specified to ensure the correct size mount. The number of speeds on the rear cassette will affect the spacing of the front chainrings, and hence the front derailleur. The direction of shift cable routing is also a factor. Top-pull derailleurs use a cable that comes from above, while bottom-pull derailleurs use a cable that comes from below. Chains: Bicycles have used chain drives for well over a century. Chains are mechanically efficient, simple, and reliable. They continue to work well, albeit with poorer shifting and reduced efficiency, under very adverse conditions including dirt, salt, and inadequate lubrication. (Of course, extended operation under these conditions will significantly reduce the life of the chain.) Over time, bicycle chains have evolved into highly specialized pieces of equipment, flexible, light, and efficient. Chains are made of pairs of links held together by pins. Inner and outer links alternate, and pivot on the pins. All bicycle chains have a pitch, or distance from one pin to the next, of 0.5 inches. (Although this is the same pitch as an ANSI number 40 and 41 industrial chain, these are not compatible with bicycle chains and should not be used.) The chain width varies with the application. Bikes with derailleurs have chains with an inside width of 3/32 inches. The outside width depends on the number of sprockets. Bicycles without derailleurs use 1/8 inch wide chains. Table 14-4 shows sizes for various applications. Miscellaneous Drive Components: All chains require some means of tensioning, which can be as simple as a horizontal slotted dropout for the rear hub. This approach is used on many single-speed bicycles and some internal hub drives.

293


Table 14-3 Overview of Bicycle Chain Types and Applications ChainType

Fits

InsideWidth

OutsideWidth

Non-Derailleur

Single speed Internal gears

1/8 in

6-SpeedDerailleur 8-Speed Derailleur

Old5or6speed 5, 6, 7, and 8 speed

9-Speed Derailleur

9 speed 5, 6, 7, and 8 speed (front shifting may be affected)

10-SpeedDerailleur

10speed

3/16in

5.9–6.1mm

11-SpeedDerailleur

11speed

3/16in

5.5–5.7mm

3/16in 3/16 in 3/16in

7.6mmMIN 7.2–7.4 mm 6.6–6.8mm

With derailleur systems, the rear derailleur functions as a tensioner. Springloaded tensioners are available from several manufacturers when no derailleur is used. Another alternative is an eccentric bottom bracket such as used on many tandems to tension the timing chain. Idlers are used to re-route chains, and may be required on recumbents for which a straight chainline is impossible. Since the chain tension while pedaling can be large, idlers on the power side of the chain should be securely mounted to the frame, and care taken to correctly align them. Idler pulleys are available as stock items from some recumbent dealers. On some HPVs, multiple chains are used, and the idler is replaced with an intermediate drive, usually a bespoke design. Chain retention devices and chain guards are also available from several manufacturers. Unusual Drive Components: Several companies produce unusual drive train components or parts that may have special interest to human-powered vehicle designers. Several manufacturers make multi-speed cranksets, including Schlumpf, Truvativ, Suntour and Pinion. Schlumpf makes two-speed geared cranks that can be quite useful for extending the gearing range without using a front derailleur. They produce several models with different gear ratios, and are compatible with standard bottom bracket shells, although some shells require special reaming in order to use the Schlumpf drive. (Schlumpf also makes geared direct-drive hubs for unicycles) Truvativ’s HammerSchmidt is also a two-speed crank, capable of replacing multiple chainrings. Pinion produces several multi-speed bicycle transmissions incorporated into the crank, ranging from 9 to 18 gears. Unlike

294

Bicycle Components

Table 14-4 Drivetrain Component Manufacturers Manufacturer SRAM

Information Products

Brands URL

Shimano

SR Suntour

hub gears, etc. SRAM, Truvative http://www.sram.com/

Technical

http://www.sram.com/service (Documentation is available on-line for most SRAM products.)

Products

Full range of road/mountain/city bicycle components: shifters, derailleurs, cassettes, cranks, hub gears, etc.

URL

Camagnolo

Full range of road/mountain/city bicycle components: shifters, derailleurs, cassettes, cranks,

http://www.shimano.com/# (World site) http://bike.shimano.com/ (North America Cycling)

Technical

http://techdocs.shimano.com/techdocs/index. jsp

Products

Wide range of performance racing and road bike components: shifters, derailleurs, cassettes, cranks, etc.

URL

http://www.campagnolo.com/jsp/en/index/index.jsp

Technical

http://www.campagnolo.com/jsp/en/doc/doccatid_1.jsp

Products

Internally geared crank transmission Cranksets, front derailleurs, bottom brackets

URL Sturmey-Archer

Products

Full Speed Ahead (FSA)

Products

URL

URL

http://www.srsuntour-cycling.com/SID=si9af8dfb48477a447a236121236ac2b/index.php Internally geared hubs Cranks and shifters http://www.sturmey-archer.com/ Cranksets and chainrings; also wide range of non-drive parts http://www.fullspeedahead.com/ (Continued)

295


Table 14-4 (Continued) Manufacturer Schlumpf Innovations

Information Products

URL Rohloff

Products URL

Rotor Bike Components

Products

Gates Carbon Drive

Products

Fallbrook Technologies

Products

Sinz Racing

Products

URL URL URL

URL

Internally geared crank sets; Internally geared unicycle hubs; belt drive systems http://www.schlumpf.ch/hp/schlumpf/schlumpf_ engl.htm 14-Speed, wide-range, internally geared hub; Chains, tensioners, some tools http://www.rohloff.de/en/contact/ rohloff_world_wide/ Rotor Cranksets—reported to increase power/ reduce knee strain; Q-rings (non-circular chainrings); Other cranksets, components http://www.rotorbikeusa.com/ Carbon belt drive systems http://www.carbondrivesystems.com/ NuVinci Continuously-variable drive hub http://www.fallbrooktech.com/ BMX parts, but produce cranks with short arms down to 115 mm http://www.sinz-racing.com/

Schlumpf, these do require a specially designed frame to accept the transmission. SR Suntour’s V-Boxx is a 9-speed crank transmission that also requires a custom frame design. Table 14-4 lists several manufacturers of drivetrain components, including both major group manufacturers and some special component manufacturers. Be sure to read the technical application notes or instructions provided with each component to ensure compatibility and correct installation. Some require special frame fittings or even custom frame designs, and may not be compatible with the standards described in Chapter 13.

Headsets Headsets are assemblies that contain the bearings for the fork and steering system. Several basic types are currently used, each in several sizes. Threaded

296

Bicycle Components

headsets use a fork with threads on the steerer tube of the fork. Threadless headsets use a stem that clamps onto an un-threaded steerer tube. Threaded headsets, once used on all bicycles, are now used on bicycles that might require frequent handlebar adjustment, such as children’s bicycles, and many lower-priced bicycles. A nut is used to adjust the headset bearings and secure the assembly. A locknut prevents the adjustment nut from turning. The stem (which connects the handlebar to the fork) slides down into the steerer tube and is locked in place with an internal wedge. The wedge is tightened by a bolt extending down from the top of the stem. The stem and handlebar may be re-positioned or removed without readjusting the headset bearings. Threadless headsets are lighter, and are now used on most quality bicycles. Generally, they provide only a limited adjustment for handlebar position as spacers must be added or removed to change handlebar height. Some threadless stems have adjustable angles, which provide more adjustability at the cost of additional weight. Bearings are adjusted by a screw passing through the cap on top of the headset, which screws into a star nut pressed into the steerer tube. Once the correct adjustment is achieved, the stem is clamped tightly to the steerer tube to lock the adjustment. With threadless headsets, if the stem is loosened, the bearings must be readjusted. Variations on the standard threadless headset include low profile and integrated (bearings are inside headtube) and zero stack. The basic size of a headset is given by the outside diameter of the steerer tube. For example, a 1 inch headset has a steerer tube diameter of 1 inch. The fork, frame, and headset must all adhere to the same size and standard to be compatible. Standard threaded headsets are available in 1, 1 1/8, and 1 1/4 inch sizes. There are two different standards for 1 inch headsets, and they are not compatible, so caution is in order in this case. Threadless headsets are available in 1, 1 1/8, 1 1/4, and 1.5 inch sizes. There are many different options, including external cup (shown in the illustration) integrated, where the bearings are pressed directly into machined heatubes without a separate cup, and zero stack, with cups that are almost entirely inside the headtube. The Standardized Headset Identification System, or SHIS, standardizes the nomenclature of the very many headset options available It consists of a sixpart number, defining the type and size of the upper headset, the type and size of the lower headset, the stem clamp diameter and the crown race seat diameter. For example EC34/28.6|EC34/30 represents an external cup headset for a 1 1/8 steerer. Breaking it down, EC is the upper headset type. Types include EC for external cup, IS for integrated, and ZS for zero stack. The number following the type is the bore size for the upper headset. The third number, following a /, is the stem clamp diameter, in millimeters. The last three symbols following the | are for

297


Table 14-5 Typical Values Used in SHIS Codes Bore Size

Description

Common Size Description

30

1 inch External

1 inch

34 44

11/8inchExternal 11/8ZeroStack

11/8inch 1.5inch

49

1.5 inch External

56

1.5 inch Zero Stack

Stem Diameters

Crown Diameters

25.4

26

28.6 38.1

30 40

the lower headset, with type and bore as for the upper headset followed by the crown race seat diameter. Common values for bores, stem diameters, and crown race seat diameters are given in Table 14-5. Both threaded and threadless headsets are available with either cartridge bearings or loose ball bearings (sometimes the balls are held together with a retainer). Higher quality headsets often use sealed cartridge bearings.

Stems Stems connect the steerer tube to the handlebar. Stems for threaded headsets slip down into the fork steerer tube, and are locked in place by a wedge which is tightend by a screw extending up through the stem. The height of the stem can be adjusted by loosening the screw and raising or lowering the stem. Bearing adjustment is not affected. Stems for threadless headsets clamp onto the outside of the steerer tube. Vertical adjustment requires headset spacers to be added or removed. If the stem clamp is loosened for any reason, the headset bearings must be readjusted. Stems are produced in a variety of lengths and angles to ensure a good fit to the rider. Four parameters are required to specify a stem: size, length, angle, and handlebar size. The size is the diameter of the steerer tube, generally either 1 or 1 1/8 inches. Handlebar size refers to the size of the clamp at the handlebar end. These two parameters must be correct, or the stem will not fit the vehicle. The length and angle parameters are used to adjust vehicle fit to the rider. Some stems are available with adjustable angles. These provide flexibility if the vehicle will be used by multiple riders of differing sizes, but are generally heavier than non-adjustable stems. Some designers and mechanics use adjustable stems to determine a comfortable angle, then purchase a fixed angle stem for

298

Bicycle Components

better performance. Adjustable stems are available for both threaded and threadless headsets.

Handlebars Many styles of handlebars are in use for different types of bicycles. Some of the more common types used on upright bicycles include drop, or road bars, flat bars, and riser bars. Recumbent bicycles use a wide variety of handlebar styles including under-seat steering bars, very high riser bars, and bars mounted either between the legs or on the outside of the legs. A diverse variety of handlebars is used for HPVs. Long wheelbase recumbents often use either extremely high riser bars or under-seat steering bars. Other types use a tall mast to mount the handlebar between the rider’s legs. Some recumbent handlebars and masts are available from recumbent bike dealers. However, bespoke handlebars are more common with HPVs due to the diverse nature of these vehicles. With upright bicycles, the arms and hands support part of the rider’s weight. Drop handlebars are commonly used on upright road bicycles. These bars provide multiple hand positions—on top of the bar, on the brake hoods, or on the drops. With hands on the drops, the rider is generally in a good aerodynamic position. Riding with hands on the brake hoods or top bar gives a more upright position, comfortable for easy riding and providing deeper breathing for steep climbs. The ability to switch hand positions during a long ride prevents hand and arm soreness and discomfort. Flat or low riser bars are often used on mountain and city bicycles, and provide an upright, but not aerodynamic position. Bar ends can be added to provide additional hand positions, and are recommended for long rides. Aero bars are used on time trial and triathalon bicycles to give the rider the best possible aerodynamic advantage. These bars place the rider in a low aerodynamic position.

Brakes Brakes are used to slow or stop a vehicle, usually by converting kinetic energy to heat. Two broad categories of brakes are defined by the location at which the braking force is applied to the wheel. Rim brakes apply the force to the rim, while hub brakes apply the force to the hub. Since the wheel contact patch is located on the rim rather than the hub, the braking torque for rim brakes is not transmitted through the spokes. The braking torque for hub brakes is transmitted through the spokes, however. This is why wheels using hub brakes should never be built with

299


radial spokes (which are relatively poor at transmitting torque). There are several different type of both rim and hub brakes, and a large number of products on the market for each category. Rim brakes are usually mounted on the frame or fork, with pads that press against either side of the rim. Generally, this requires a traditional bicycle type mount for the wheel, with fork blades or frame members on either side of the wheel. It may be more difficult to use rim brakes with cantilevered wheels used on many multitrack (and some singletrack) vehicles. Caliper brakes usually mount to a single hole on a bridge in the rear triangle or the lower end of the steerer tube for forks. Cantilever, linear-pull, and U-brakes require mounting posts on the seatstays or fork blades. Rim brakes generally provide good braking torque, but can wear and eventually damage rims. Correct choice of pads can affect both effectiveness and rim wear. Most manufacturers produce pads designed for wet and muddy conditions. While rim brakes generally lose effectiveness when wet, using an appropriate pad will minimize the effect. The correct pad can also significantly affect the life of the rim under these conditions. Caliper brakes have been used for many years, but are now most often found on road bicycles. Sidepull calipers were quite common on road bicycles in the past. Today they are mostly used on BMX bicycles. They are still available in a fairly wide range of sizes. Centerpull calipers were widely used on road bicycles the 1970s. They provided some clearance for fenders or larger tires, but are seldom used today. By far the most common type of brake used on current road bicycles is the dual-pivot caliper. This is essentially a combination of side-pull and centerpull types, with one arm pivoting on the central pivot bolt (similar to a centerpull) and one arm pivoting on a second, off-center pivot (similar to a sidepull). The central pivot and second pivot are connected by a half-bridge. Dual-pivot calipers are lightweight and provide good modulation of the braking force. They have superior centering characteristics relative to the side-pull, and also a bit more mechanical advantage. However, it cannot normally be used with medium or wide tire widths, as there is insufficient clearance for the tire to clear the caliper arms. Cantilever brakes are currently found on many cyclocross and touring bicycles with drop-style handlebars. Two arms pivot on frame or fork mounted bosses, and are actuated by a straddle cable. They provide ample clearance for wide tires, and have about twice the mechanical advantage of dual-pivot brakes. These are both desirable characteristics for touring cycles, which may be heavily loaded and use wide tires. Unlike linear-pull brakes, cantilevers can use the same levers as caliper brakes, which are widely available for both flat bars and drop bars. As the pads wear, they move downward on the rim toward the spokes, and may require

300

Bicycle Components

re-adjustment. The mechanical advantage of cantilever brakes, along with centerpull and U-brakes, can be adjusted by changing the length of the straddle cable. Linear-pull brakes, sometimes generically called by Shimano’s trade name V-brakes, provide good mechanical advantage, ample clearance for fenders or wide tires, and avoid the problem of pads moving down the rim as they wear. They are frequently found on mountain bikes and touring bikes with flat bars. Linear-pull brakes have high mechanical advantage, and must use levers designed specifically for them to avoid overly powerful braking. They provide effective braking force with good modulation, and are relatively easy to set up and adjust. The mounting posts are identical with those used for cantilever brakes. Prior to the introduction of linear-pull brakes, U-brakes were used on many mountain bikes. The U-brake uses a center-pulled straddle cable, but pivots are mounted on the frame. The location of the bosses is incompatible with linear-pull or cantilever brakes however. Hub brakes apply torque directly to the hub of the wheel. Hub brakes include mechanically or hydraulically actuated disc brakes, drum brakes, coaster brakes and roller brakes. They are frequently used on bicycles designed for all-weather operation, such as mountain bikes and city bikes. Tandems sometimes use hub brakes to provide additional braking capacity for long downhill descents. (The extra weight on tandems demands more braking force, and hence more heat must be dissipated.) Hub brakes are usually more suitable for cantilevered wheels, such as often used on tricycles and quads, than are rim brakes. Disc brakes use a rotor attached directly to the hub. Unlike rim brakes, the stopping power of hub brakes depends on wheel diameter. Braking torque is the product of brake force and radial distance from the wheel center or T = F ´ r. For rim brakes, the brake force Fb is applied at the rim at a radius Rb, while the stopping forceFs at the contact patch has a lever arm Rs (see Figure 14-6). The stopping force is obtained by summing moments about the axle: Fs

R  = Fb  b   Rs 

(14-2)

Since both Rb and RS change by the same amount as the wheel diameter changes (assuming the tire size and rim type remain constant) the stopping force is only slightly dependent on wheel diameter, increasing slightly as rim diameter increases. For hub brakes, the moment arm for the braking force Rd remains constant regardless of wheel diameter. The stopping force is derived as before Fs

R  = Fd  d   Rs 

(14-3)

301


Figure 14-6 Threaded headset with loose bearings

 Rd   becomes significantly larger as the wheel diame Rs 

In this case, the ratio 

ter decreases. In effect, hub brakes are more effective for small wheels than for large ones, while rim brakes are slightly more effective for large wheels. Since many recumbent vehicles use small wheels, braking can be enhanced by using hub brakes. This can be an advantage for heavily laden vehicles such as tandems and freight haulers. Of course, rim brakes are still quite adequate for many recumbent applications. The disc brake rotor included in Figure 14-6 illustrates a fairly common type of hub brake, particularly for mountain bikes. Recently disc brakes have become more popular on city and touring bikes as well. Many tricycles, particularly tadpoles with cantilevered front wheels, also use disc brakes. Disc brakes have a caliper with pads mounted on either side of a rotor. The rotor is attached directly to the hub, while the caliper is mounted to the fork or frame. When actuated, the caliper pads squeeze the rotor to provide braking torque. Disc brakes mounts should be included in frame/fork design if these brakes are to be used. Disc brakes are available with either mechanical or hydraulic actuation.

302

Bicycle Components

Figure 14-7 Threadless headset with cartridge bearings

303


Figure 14-8 Forces for rim and hub brakes

Drum brakes are available from a few manufacturers. These brakes use shoes that expand to contact the inner cylindrical surface of a drum attached to the hub. Since they are enclosed, they offer good weather protection. However, braking effectiveness with currently available drum brakes is usually poor for full-size wheels. (Braking effectiveness with drum brakes mounted on small wheels may be quite adequate.) Roller brakes are also enclosed, but use rollers that are wedged between an internal actuator and an external shell. They are often found on internally geared hubs, but are available for front hubs as well. Braking effectiveness is reportedly adequate, although not as good as rim brakes on large wheels. This effectiveness should increase with small wheels. Both drum and roller brakes are usually actuated with cables. Coaster brakes are found on some single-speed or internally geared rear hubs. The brake is actuated by pedaling backwards. In most applications, coaster brakes produce modest braking effectiveness, and are not widely used on recumbents.

Compatibility of Levers and Brakes Brake levers must be matched with the brake for proper operation and effectiveness. The mechanical advantage of the lever is designed to work with each particular brake type, and some combinations will provide either inadequate brake force or overly powerful braking. Either situation poses a safety hazard. Levers are available for different handlebar types as well as brake types, and may vary in reach and size. Riders with smaller hands may need short-reach levers.

304

Bicycle Components

Drop-bar levers are designed to be mounted onto drop-style handlebars. The brake cable is sometimes routed along the bar, under the handlebar tape. Some brake levers are integrated with the shift lever. These are known as integrated shifters. Almost all drop-bar levers are compatible with caliper brakes, cantilever brakes, and roller, drum and U-brakes. The brake feel may differ depending on the type of brake, however. Cantilevers can have a soft or “mushy” feel with some of these levers. Linear pull brakes and disc brakes are not compatible with drop-bar levers with very few exceptions, and should not be used. Flat bar levers are available for all brake types. Dual-pivot, front drum, U-brake and cantilevers use normal levers. Cantilever brakes have a higher mechanical advantage, so they will require less force for a giving stopping power than dual-pivot brakes using the same levers. Disc and linear-pull brakes have higher mechanical advantage and require special levers. Most low-mechanical advantage levers are compatible with both disc and linear pull brakes. Some levers have adjustable mechanical advantage, and are compatible with any brake type.

Summary Components are specified based on intended vehicle use, performance level, and cost. Compatibility between components may be crucial for optimum performance, so manufacturer’s technical data and OEM instructions should be followed. As noted in Chapter 13, components must also be compatible with the frame. This requires that the vehicle designer specify at least the type of components to be used prior to final frame design. The rather intense competition between bicycle component manufacturers is a boon for human-powered vehicle designers, as it has produced a plethora of high-quality components spanning a wide price range. The many options ensure that components should not limit the design of a new human-powered vehicle. Nonetheless, it is sure that innovative designers and engineers will continue to develop and improve special-purpose components in order to achieve better performance, increased efficiency, reduced fatigue or lower cost.

305

INDEX A

Ackerman angle, 198, 217 Ackerman steering, 197 Adenosine triphosphate (ATP), 44 Adhesive bonding, 99 Aerobic, 44 Aerodynamic drag, 113, 135 Aerodynamic drag force, 151 Age, 48 Air pollution, 9 Aluminum, 89 Aluminum alloys, 80 Anaerobic, 44 Anaerobic threshold, 55, 58 Antioxidants, 52 Aramid fiber, 95 ATP, see Adenosine triphosphate (ATP)

Brakes, 299 Braking performance, 128 Brazing, 99 C

Cadence, 70 Camber, 210 Camber angle, 213 Camber stiffness, 213 Carbohydrates, 49

Bamboo, 85, 97 Bead seat diameter, 284 Beam idealizations, 259

Carbon fiber, 94 Cardiac output, 54 Cast metals, 81 Castor angle, 196 CFRP, 95 Chain line, 35 Chainline, 242 Chains, 293 Chromium-molybdenum steels, 87 Comfort, 61 Competition, 2 Control sensitivity, 177, 183, 184

Bearing friction, 139 Bearings, 288 Body configuration angle, 65 Body Position, 64 Bonding, 99 Bottom brackets, 290

Cornering stiffness, 210 Crank arm length, 67, 70 Cranksets, 292 Critical power, 47 Critical speed, 221 Crosswind component, 136

B

307


D

Delta, 17, 22 Derailleurs, 292 Design parameters, 28 Design process, 25 Design specifications, 27 Detail items, 36 Development, 234 Diet, 51 Drag, 113 Drag area, 159 Draisenne, 14 Drive train, 35, 233 Drivetrain efficiency, 246

Frame stiffness, 256 Freehubs, 288 Freewheels, 288 G

Gas-tungsten arc welding, 97 Gear-inches, 234 Gearing, 233 Gender, 48 GHGs, see Greenhouse gasses (GHGs) Glycogen, 44 Grade, 110 Greenhouse gas emissions, 9 Greenhouse gasses (GHGs), 8 Gross geometry, 34

E

Economics, 4 Efficiency, 42, 134

H

Elliptical drives, 71 Endurance limit, 255 Environment, 8 Exercise, 53

Handling, 195 Hazards, 73 Headsets, 296 Head-wind component, 135 Heart rate, 53 High-speed cornering, 207 Hip orientation angles, 64, 65 Human engine, 41 Hyponatremia, 50

F

Fairings, 101 Fast glycolytic, 43 Fatigue, 46, 254 Fats, 45, 49, 52 FEA, see Finite element analysis (FEA) Fiber reinforced-polymer composite, 92 Finite element analysis (FEA), 256 Fitness and health, 5 Fork flop, 181 Form drag, 151 Frame, 253 Frame design, 275

308

Hand cranks, 69

I

Induced drag, 152, 158 Inertia coefficient, 121, 123 Interference drag, 152 ISO 4210, 72 J

JIS D9414, 72 Joint angles, 64

Index

K

Kevlar, 95 Kinematic track, 197, 199 Kinetic energy, 140 Kingpin inclination angle, 196 Kingpin offset, 197 L

Lactic acid, 45 Laminar boundary layer, 156 Lateral acceleration, 115 Lateral forces, 115, 214 Lateral load transfer, 214, 221 Levers, 72 Lift, 158 Linear drives, 71 Lipids, 45 Load transfer, 116 Longitudinal load transfer, 124 Long-wheelbase, 20 Low-traction, 119

Non-metals, 81 Nutrition, 49 O

Onset of blood lactate accumulation (OBLA), 58 Ordinary, 14 Oversteer, 217 P

Patterson’s method, 175 Peak power, 70 Penny-farthing, 14 Pitchover, 125 Pitchover threshold, 127, 216 Planners, 15 Power, 48 Power losses, 134 Propulsive forces, 113 Proteins, 49 Prototype, 37

M

Magnesium, 96 Mechanical trail, 171, 178 Metabolic energy production, 57 MIL-T-6736B, 87 Mission statement, 27 Mobility, 8, 14 Monocoque, 101 Multi-track vehicle, 195 Muscle fibers, 46 Muscle structure, 42 Muscles, 42 N

Neurons, 42 Newton’s second law, 106

Q

Quadricycles, 22 R

Race Across America, 17 Recreation, 2 Recumbent, 14, 17, 18 Resins, 96 Respiratory exchange ratio, 56 Retarding forces, 113 Rims, 286 Roll control authority, 177, 178 Rolling resistance, 113, 138 Rollover threshold, 214, 228 Rowbike, 71

309


S

16CFR1512A, 72 SAE J670e, 106 Safety, 72 Scrub radius, 197 Seat, 62 Section width, 283 Separation point, 156 Shell, 101 Shell elements, 262 Shift systems, 292 Short-wheelbase, 20 Single-speed, 234 Skeletal muscles, 43 Skin friction drag, 151 Slip, 209 Slope, 110 Slow oxidative, 43 Solid elements, 264 Spokes, 287 Stability, 195 Stainless steel, 89 Static weight fraction, 107 Steel, 80, 86 Steering error, 199 Steering sensitivity, 200 Stems, 298 Streamliner, 20 Streamlines, 153 Sustainable transportation, 11 T

Tadpole, 17, 20 Tensioning, 251 Testing, 37 Tire stiffness, 210 Titanium alloys, 91

310

Toe-in, 201 Tolerances, 37 Torsional spring constant, 177, 180 Track rod, 200 Track rod steering, 201, 226 Tractive force, 113 Trail, 171, 178 Transportation, 16 Tricycles, 20 Trireme, 13 U

Understeer, 217 Understeer gradient, 216, 218, 223, 224 Upright bicycle, 18 V

Ventilation, 55 Vitamins and minerals, 49 W

Wheel bearings, 139 Wheelchairs, 69 Wheel track, 197 Wood, 85, 96 World Human-Powered Speed Challenge, 17

About the Author

Mark Archibald is a professor of Mechanical Engineering at Grove City College in western Pennsylvania, where he teaches a course on the design of human-powered vehicles. Hehas advised many student projects for both land and water vehicles. For several years, he was very involved with the American Society of Mechanical Engineer’s Human-Powered Vehicle Challenge, where he served a term as chief judge and a term as chairman. Much of the inspiration for this book came from his efforts to correct common design mistakes made by student competitors. He has also designed and built several successful bicycles that he and his family use on their summer bicycle tours.

311

Design of Human Powered Vehicles (Bicycle Drive Train)

Recommend Documents