Fundamentals of Machine Elements, 2nd Ed.

Chapter 1: Introduction

The invention all admir’d, and each, how he To be th’ inventor missed; so easy it seem’d Once found, which yet unfound most would have thought Impossible John Milton

Hamrock • Fundamentals of Machine Elements

Product Development Approaches

Figure 1.1 Approaches to product development. (a) Classic approach, with large design iterations typical of the over-thewall engineering approach. (b) A more modern approach, showing a main design flow with minor iterations representing concurrent engineering inputs. Hamrock • Fundamentals of Machine Elements

Pugsley Method for Safety Factor Characteristica

A = vg

A=g

A=f

A=p

B= vg

g

f

p

C=

vg g f p

1.1 1.2 1.3 1.4

1.3 1.45 1.6 1.75

1.5 1.7 1.9 2.1

1.7 1.95 2.2 2.45

C=

vg g f p

1.3 1.45 1.6 1.75

1.55 1.75 1.95 2.15

1.8 2.05 2.3 2.55

2.05 2.35 2.65 2.95

C=

vg g f p

1.5 1.7 1.9 2.1

1.8 2.05 2.3 2.55

2.1 2.4 2.7 3.0

2.4 2.75 3.1 3.45

C=

vg g f p

1.7 1.95 2.2 2.45

2.15 2.35 2.65 2.95

2.4 2.75 3.1 3.45

2.75 3.15 3.55 3.95

!all Safety Factor: ns = ! d

Pugsley Equation: ns = nsxnsy

Characteristica

E=

ns s vs

D= ns

s

vs

1.0 1.0 1.2

1.2 1.3 1.4

1.4 1.5 1.6

a vg

= very good, g = good, f = fair, and p = poor. A = quality of materials, workmanship, maintenance, and inspection. B = control over load applied to part. C = accuracy of stress analysis, experimental data, or experience with similar parts.

Figure 1.2 Safety factor characteristics A, B, and C.

a vs

= very serious, s = serious, and ns = not serious D = danger to personnel. E = economic impact.

Figure 1.3 Safety factor characteristics D and E.


Design for Assembly Screw Screw Screw

Screw Cover Cover Screw Connecting rod

Bearing

Pin Bush

Seal

Seating

Pin

Needle bearing Piston

Rivet Screw

Piston Washer

Roll pin Pin

Bush Washer

Counterweight

Pin

Blade clamp Screw

Connecting rod

Blade clamp

Gear

Screw

Gear

Bearing Seating

Plug

Screw Housing

Bearing

Seating Housing Guard

Washer Screw Guard

Set screw Set screw

Screw

(a)

(b)

Figure 1.2 Effect of manufacturing and assembly considerations on design of a reciprocating power saw. (a) Original design, with 41 parts and 6.37-min assembly time; (b) modified design, with 29 parts and 2.58-min assembly time. Hamrock • Fundamentals of Machine Elements

(a) SI units Quantity Unit SI base units Length meter Mass kilogram Time second Temperature kelvin SI supplementary unit Plane angle radian SI derived units Energy joule Force newton Power watt Pressure pascal Work joule

SI symbol

Formula

m kg s K

-

rad

-

J N W Pa J

N-m kg-m/s2 J/s N/m2 N-m

SI Units

(b) SI prefixes Multiplication factor 1,000,000,000,000 = 1012 1,000,000,000 = 109 1,000,000 = 106 1000 = 103 100 = 102 10 = 101 0.1 = 10−1 0.01 = 10−2 0.001 = 10−3 0.000 001 = 10−6 0.000 000 001 = 10−9 0.000 000 000 001 = 10−12

Prefix tera giga mega kilo hecto deka deci centi milli micro nano pico

SI symbol for prefix T G M k h da d c m µ n p

Table 1.3 SI units and prefixes.


(a) Fundamental conversion factors English Exact SI unit value Length 1 in. 0.0254 m Mass 1 lbm 0.453 592 37 kg Temperature 1 deg R 5/9 K (b) Definitions Acceleration of gravity Energy

Length Power Pressure Temperature

Kinematic viscosity Volume

Approximate SI value 0.4536 kg -

Conversion Factors

1 g = 9.8066 m/s2 (32.174 ft/s2 ) Btu (British thermal unit) = amount of energy required to raise 1 lbm of water 1 deg F (1 Btu = 778.2 ft-lb) kilocalorie = amount of energy required to raise 1 kg of water 1K (1 kcal = 4187 J) 1 mile = 5280 ft; 1 nautical mile =6076.1 ft 1 horsepower = 550 ft-lb/s 1 bar = 105 Pa 9 degree Fahrenheit tF = tC + 32 (where tC is degrees 5 Celsius) degree Rankine tR = tF + 459.67 Kelvin tK = tC + 273.15 (exact) 1 poise = 0.1 kg/m-s 1 stoke = 0.0001 m2 /s 1 cubic foot = 7.48 gal

(c) Useful conversion factors 1 ft = 0.3048 m 1 lb = 4.448 N 1 lb = 386.1 lbm-in./s2 1 kgf = 9.807 N 1 lb/in.2 = 6895 Pa 1 ksi = 6.895 MPa 1 Btu = 1055 J 1 ft-lb = 1.356 J 1 hp = 746 W = 2545 Btu/hra 1 kW = 3413 Btu/hr 1 quart = 0.000946 m3 = 0.946 liter 1 kcal =3.968 Btu

Table 1.4 Conversion factors and definitions.


Invisalign Product

Figure 1.3 The Invisalign® product. (a) An example of an Aligner; (b) a comparison of conventional orthodontic braces and a transparent Aligner. Hamrock • Fundamentals of Machine Elements

Invisalign Part 1

Figure 1.4 The process used in application of Invisalign orthodontic treatment. (a) Impressions are made of the patient's teeth by the orthontist, and shipped to Align technology, Inc. These are used to make plaster models of the patient's teeth. (b) High-resolution threedimensional representations of the teeth are produced from the plaster models. The correction plan is then developed using computer tools. Hamrock • Fundamentals of Machine Elements

Invisalign Part 2

Figure 1.4 (cont.) (c) Rapid-prototyped molds of the teeth at incremental positions are produced through stereolithography. (d) An Aligner, produced by molding a transparent plastic over the stereolithography part. Each Aligner is work approximately two weeks. The patient is left with a healthy bite and a beautiful smile. Hamrock • Fundamentals of Machine Elements

Chapter 2: Load, Stress and Strain The careful text-books measure (Let all who build beware!) The load, the shock, the pressure Material can bear. So when the buckled girder Lets down the grinding span The blame of loss, or murder is laid upon the man. Not on the stuff - The Man! Rudyard Kipling, “Hymn of Breaking Strain” Hamrock • Fundamentals of Machine Elements

Establishing Critical Section

To establish the critical section and the critical loading, the designer: 1. Considers the external loads applied to a machine (e.g. an automobile). 2. Considers the external loads applied to an element within the machine (e.g. a cylindrical rolling-element bearing. 3. Located the critical section within the machine element (e.g., the inner race). 4. Determines the loading at the critical section (e.g., contact stress).


Example 2.1

y Pin

0.25 m

x Roller 0.75 m P = 10,000 N (a)

Wp

0.25 m Wr

0.75 m P = 10,000 N (b)

Figure 2.1 A simple crane and forces acting on it. (a) Assembly drawing; (b) free-body diagram of forces acting on the beam.


P

P

(a)

Types of Loads

P

P (b) P

y

P

V x P

P

V

(c) M y M

x (d) T

y T x (e) y V

M x T

z

Figure 2.2 Load classified as to location and method of application. (a) Normal, tensile; (b) normal, compressive; (c) shear; (d) bending; (e) torsion; (f) combined.

(f)


Sign Convention in Bending

y

y y'' < 0

y'' > 0

y'' < 0

y'' > 0

M>0

M<0

M<0

M>0

x (a)

x (b)

Figure 2.3 Sign convention used in bending. (a) Positive moment leads to tensile stress in the positive y direction; (b) positive moment acts in a positive direction on a positive face. The sign convention shown in (b) will be used in this book. Hamrock • Fundamentals of Machine Elements

Example 2.2 (1)

Normal, tensile Px = 0

Px

Py

(2)

y

B

Shear Py = –P

b a

z

B B

x

(3) M z

Bending Mz = - bP, My = 0

P B My (a)

(4)

Torsion T = –aP T

B (b)

Figure 2.4 Lever assembly and results. (a) Lever assembly; (b) results showing (1) normal, tensile, (2) shear, (3) bending, and (40 torsion on section B of lever assembly. Hamrock • Fundamentals of Machine Elements

Type of support

Reaction

θ

Support Types

θ

P

Cable

P

Roller

Py

Px Pin

Py Px

Table 2.1 Four types of support with their corresponding reactions.

M Fixed support


Examples 2.4 and 2.5 5

5

16

W 4

y µP2

(Rotation)

P2

12

30∞ 30∞ 130∞ 10 R. 130∞ 20∞ 20∞

20°

A

12

B 3 3 (a) W

4W 4W W 4W

(ml + mp)g

µdP

P1

dP

µP1 0

x

4W

4W

W 4W

µdP

dP

dP µdP

dP

µdP

(Rotation)

Figure 2.5 Ladder in contact Figure 2.6 External rim brake and forces with the house and the acting on it. (a) External rim brake; (b) ground while a painter is on external rim brake with forces acting on the ladder. each part. Hamrock • Fundamentals of Machine Elements (b)

60°

60°

Example 2.6 (a)

P

(b)

60°

60°

P

ma g

150 N

Figure 2.7 Sphere and forces acting on it. (a) Sphere supported with wires from top and spring at bottom; (b) free-body diagram of forces acting on sphere.


Beam Support Types

(a)

(b)

(c)

Figure 2.8 Three types of beam support. (a) Simply supported; (b) cantilevered; (c) overhanging.


Shear and Moment Diagrams Procedure for Drawing Shear and Moment Diagrams: 1. Draw a free-body diagram, and determine all the support reactions. Resolve the forces into components acting perpendicular and parallel to the beam’s axis, which is assumed to be the x axis. 2. Choose a position x between the origin and the length of the beam l, thus dividing the beam into two segments. The origin is chosen at the beam’s left end to ensure that any x chosen will be positive. 3. Draw a free-body diagram of the two segments, and use the equilibrium equations to determine the transverse shear force V and the moment M 4. Plot the shear and moment functions versus x. Note the location of the maximum moment. Generally, it is convenient to show the shear and moment diagrams directly below the free-body diagram of the beam. Hamrock • Fundamentals of Machine Elements

P

Example 2.7

A

C B

P __ 2

P

l __ 2

P __ 2

l __ 2

x P __ 2

(a) y

P __ 2

M

P __ 2

V

x

x P V = – __ 2

(b)

y

M

P l __ 2

A P __ 2

P V = __ 2

V

x

A

y

Pl Mmax = __ 4

P M = __ x 2

P M = __ (l – x) 2

M

x

x (d) V

x

(c)

Figure 2.9 Simply supported bar. (a) Midlength load and reactions; (b) freebody diagram for 0 < x < l/2; (c) free-body diagram l/2 ≤ x < l; (d) shear and moment diagrams. Hamrock • Fundamentals of Machine Elements

Singularity Concentrated moment

Graph of q(x)

Expression for q(x)

y q(x) = M 〈x – a〈–2

M x

Singularity Functions

a Concentrated force

y

P q(x) = P 〈x – a〈–1 x a

Unit step

y w0

q(x) = w0 〈x – a〈 0 x

a Ramp

y w0

w0 q(x) = __ 〈x – a〈1 b

x a Inverse ramp

b

y w q(x) = w0 〈x – a〈 0– __0 〈x – a〈1 b

w0 x a Parabolic shape

b

y q(x) = 〈x – a〈 2

Table 2.2 Singularity and load intensity functions with corresponding graphs and expressions.

x a


Using Singularity Functions Procedure for Drawing the Shear and Moment Diagrams by Making Use of Singularity Functions: 1. Draw a free-body diagram with all the singularities acting on the beam, and determine all support reactions. Resolve the forces into components acting perpendicular and parallel to the beam’s axis. 2. Referring to Table 2.2, write an expression for the load intensity function q(x) that describes all the singularities acting on the beam. 3. Integrate the negative load intensity function over the beam to get the shear force. Integrate the negative shear force over the beam length to get the moment (see Section 5.2). 4. Draw shear and moment diagrams from the expressions developed.


Example 2.8 M V V1

P __ 2

M1 V3

0

l __ 2

l

l __ 2

–P

M2

l

x

V2 V

M

P __ 2 P – __ 2

M3

x

l l __ 2

(a)

x l __ 2

l

x

(b)

Figure 2.10 (a) Shear and (b) moment diagrams for Example 2.8.


Example 2.9

10.5 8.5 7.5 w0

5.0

x l __ 4

l __ 4 P1

l __ 4

l __ 4

Shear, kN

y

1.5 3

6

9

12

x, m

–2.5

P2

–5.0

(a)

–8.5 (c)

Parabolic 40.0

34.53 = Maximum moment (at x 6.9 m)

l __ 6

w __0l 8

w__ 0l 2 x

l __ 4 R1

l __ 4 P1

l __ 2 P2

R2

Moment, kN-m

33.27 33.0 30.0

Parabolic

25.5

19.5 12.0 10.0 Cubic

(b) 0

3

6 (d)

9

12

x, m

Figure 2.11 Simply supported beam. (a) Forces acting on beam when P1=8 kN, P2=5 kN; w0=4 kN/m, l=12 m; (b) free-body diagram showing resulting forces; (c) shear and (d) moment diagrams for Example 2.9.


Example 2.10 A ma1g

RB ma1 ma2

B

ma2 g

C 1m

1.5 m 3m (a)

1m

0.5 m

RC 1.5 m

(b)

Figure 2.12 Figures used in Example 2.10. (a) Load assembly drawing; (b) free-body diagram.


Stress Elements y

y

σy

τzy

z

σz

σy

σy

τxy

0

0

τzx τxz

τyx

τyx τxy

τ yx τxy

τ yz

y

σx

σx

x

x

σx

0

z

Figure 2.13 Stress element showing general state of threedimensional stress with origin placed in center of element.

(a)

(b)

Figure 2.14 Stress element showing twodimensional state of stress. (a) Threedimensional view; (b) plane view.


x

Stresses in Arbitrary Directions

(a)

(b)

Figure 2.15 Illustration of equivalent stress states. (a) Stress element oriented in the direction of applied stress. (b) stress element oriented in different (arbitrary) direction.

y A cos φ

τφ

φ

σφ

σx

φ

Figure 2.16 Stresses in an oblique plane at an angle φ.

A

τ xy x

τ yx σy

A sin φ


τ

σx + σ y σc = ______ 2

Mohr’s Circle

σx

σy τ1

σy , τ xy τxy σ2 0

σy

2φ τ

σc

σx

σ1

σ

2φ σ

τxy σx ,–τ xy

τ2

Figure 2.17 Mohr’s circle diagram of Equations (2.13) and (2.14).


Constructing Mohr’s Circle The steps in constructing and using Mohr’s circle in two dimensions are as follows: 1. Calculate the plane stress state for any x-y coordinate system so that σx , σy , and τxy are known. 2. The center of the Mohr’s circle can be placed at ! " σ x + σy ,0 2 3. Two points diametrically opposite to each other on the circle correspond to the points (σx , −τxy ) and (σy , τxy ). Using the center and either point allows one to draw the circle. 4. The radius of the circle can be calculated from stress transformation equations or through trigonometry by using the center and one point on the circle. For example, the radius is the distance between points (σx , −τxy ) and the center, which directly leads to #! "2 σx − σy r= + τx2 2 5. The principal stresses have the values σ1,2 = center ± radius. 6. The maximum shear stress is equal to the radius. 7. The principal axes can be found by calculating the angle between the x axis in the Mohr’s circle plane and the point (σx , −τxy ). The principal axes in the real plane are rotated one-half this angle in the same direction relative to the x axis in the real plane. 8. The stresses in an orientation rotated φ from the x axis in the real plane can be read by traversing an arc of 2φ in the same direction on the Mohr’s circle from the reference points(σx , −τxy ) and (σy , τxy ). The new points on the circle correspond to the new stresses (σx! , −τx! y! ) and (σy! , τx! y! ), respectively.


τ (cw)

τmax = 9.43 ksi

Example 2.12

B

5 2φ σ=58°

σc=14 ksi

σy

σ 1=23.43 ksi

A

9.4 3

σ 2=4.57 ksi 0 C

8

2 φ τ=32°

(a) (σx, -τxy) = (9 ksi, -8 ksi)

–τ (ccw)

(a)

y

σ 1=23.43 ksi σ2

y σc = 14 ksi

τ = τmax = 9.43 ksi

A

D x

C σ2 =4.57 ksi

σ1

(b)

29°

σc

B

τ2 τ1

(c)

σc

σc = 14 ksi 16°

x

Figure 2.18 Results from Example 2.12. (a) Mohr’s circle diagram; (b) stress element for proncipal normal stress shown in xy coordinates; (c) stress element for principal shear stresses shown in xy coordinates.


Three Dimensional Mohr’s Circle τ τ1/3 τ1/2 τ 2/3

σ3

σ

σ2

σ1

σ σ1

τ1/2

σ2

(a)

(b)

Figure 2.19 Mohr’s circle for triaxial stress state. (a) Mohr’s circle representation; (b) principal stresses on two planes. Hamrock • Fundamentals of Machine Elements

Shear stress, τ

Example 2.13

Shear stress, τ

τ 1/3 τ1/3

τ 2/3 σ2 σ3

τ1/2

τ 2/3

τ1/2 σ1

10 20 Normal stress, σ

σ3 30

σ1 0


30

(a) (c)

Shear stress, τ

τ1

σ2

y (σy , τxy)

σc 0 2φ σ


2φ τ (σx , – τxy) x

τ2

σ1 30

Figure 2.20 Mohr’s circle diagrams for Example 2.13. (a) Triaxial stress state when σ1=23.43 ksi, σ2 = 4.57 ksi, and σ3 = 0; (b) biaxial stress state when σ1=30.76 ksi, σ2 = -2.76 ksi; (c) triaxial stress state when σ1=30.76 ksi, σ2 = 0, and σ3 = -2.76 ksi.

(b)


Octahedral Stresses σ2

σ1

σ3

∫

σoct

σoct

+ σoct

σoct

(a)

(b)

τoct τoct

τoct τ oct

(c)

Figure 2.21 Stresses acting on octahedral planes. (a) General state of stress; (b) normal stress; (c) octahedral stress. " 2 #$1/2 1! 2 2 2 2 2 !oct = ("x − "y) + ("y − "z) + ("z − "x) + 6 !xy + !yz + !xz 3 !x + !y + !z !oct = 3


Strain in Cubic Elements y

y – δy

x

y

δx

y

δx

y

z

z – δz

(a)

θ yx

x

δx θ yx

y

x

0

x

(b)

z (a)

Figure 2.22 Normal strain of a cubic element subjected to uniform tension in the x direction. (a) Threedimensional view; (b) twodimensional (or plane) view.

x

0 (b)

Figure 2.23 Shear strain of cubic element subjected to shear stress. (a) Three-dimensional view; (b) two-dimensional (or plane) view.


Plain Strain y

y

y

ε x dx ε y dy

dx

γ xy ___ 2

A

dy x (a)

x (b)

γ___ xy 2 0

B

x

(c)

Figure 2.24 Graphical depiction of plane strain element. (a) Normal strain εx; (b) normal strain εy; (c) shear strain γxy


Strain Gage Rosette in Example 2.16

90°

45° 0°

x

Figure 2.25 Strain gage rosette used in Example 2.16.


Glue Spreader Shaft Case Study

Adhesive

(a)

y

Slice

RA

RB x

Sheet 75

Block Roll

70

70

70

75

Shear, N

400 N 400 N 400 N 400 N (b)

Figure 2.26 Expansion process used in honeycomb materials.

Moment, N-m

Expanded panel

800 400 0 –400 –800

x (c) x

0 –40 –80 (d)

Figure 2.27 Glue spreader case study. (a) Machine; (b) free-body diagram; (c) shear diagram; (d) moment diagram. Hamrock • Fundamentals of Machine Elements

Chapter 3

Give me matter, and I will construct a world out of it. Immanuel Kant Hamrock • Fundamentals of machine Elements

Centroid of Area

(a)

(b)

Figure 3.1 Ductile material Figure 3.2 Failure of a from a standard tensile test brittle material from a apparatus. (a) Necking; (b) standard tensile test failure. apparatus. Hamrock • Fundamentals of machine Elements

Fiber Reinforced Composites

Lead Epoxy

σ2

Material

Pure aluminum Wood

σ1

Steel

2

τ 12

1

Nylon Graphite 0

100

200 300 400 500 Ratio of strength to density, N-m/kg

Figure 3.3 Strength/density ratio for various materials.

600 × 10 4

Fiber

Figure 3.4 Cross section of fiber-reinforced composite material.

Hamrock • Fundamentals of machine Elements

Stress-Strain Diagram for Ductile Material Elastic Plastic Y

Sy

U E

R P

Y

Sy P

0

Stress, σ = _P_ A

Stress, σ = _P_ A

Su

__ Slope, E = σ ε 0′ 0.002

Strain, ε = _δ_ l

Figure 3.5 Typical stressstrain curve for a ductile material.

0

0′ 0.002

Strain, ε = _δ_ l

Figure 3.6 Typical stress-strain behavior for ductile metal showing elastic and plastic deformations and yield strength Sy. Hamrock • Fundamentals of machine Elements

Comparison of Brittle and Ductile Materials

Brittle B

c

Sf

B′

Compression

Stress, σ = _P_ A

Stress, σ = _P_ A

Ductile

t

Sf

0

x

C

Strain, ε = _δ_ l

C′

Figure 3.7 Typical tensile stressstrain diagrams for brittle and ductile metals loaded to fracture. Hamrock • Fundamentals of

Tension Strain, ε = _δ_ l

Figure 3.8 Stress-strain diagram for ceramic in tension and in compression. machine Elements

Example 3.6

Magnesia Steel

Figure 3.9 Bending strength of bar used in Example 3.6.


Behavior of Polymers Brittle (T << Tg)

Stress, σ = _P_ A

Limited plasticity (T ≈ 0.8 Tg)

Extensive cold drawing (T ≈ Tg)

Sy

Viscous flow (T >> Tg) 0.01

Strain, ε = _δ_ l

Figure 3.10 Stress-strain diagram for polymer below, at, and above its glass transition behavior. Hamrock • Fundamentals of machine Elements

10 4

Metals

Polymers

Ceramics

Lead Copper

Density of Materials

Steels Cast iron

Density, ρ, kg/m3

Zinc alloys Sintered iron

Alumina Aluminum tin

Silicon nitride Silicon carbide

Aluminum

Magnesium Silicone rubber Acetal Phenol formaldehyde Nylon 10 3 8 × 10 2

Natural rubber Polyethylene

Graphite

Figure 3.11 Density for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


Material Metals Aluminum and its alloysa Aluminum tin Babbitt, lead-based white metal Babbitt, tin-based white metal Brasses Bronze, aluminum Bronze, leaded Bronze, phosphor (cast)b Bronze, porous Copper Copper lead Iron, cast Iron, porous Iron, wrought Magnesium alloys Steelsc Zinc alloys Polymers Acetal (polyformaldehyde) Nylons (polyamides) Polyethylene, high density Phenol formaldehyde Rubber, naturald Rubber, silicone Ceramics Alumina (Al2 O3 ) Graphite, high strength Silicon carbide (SiC) Silicon nitride (Si2 N4 ) a Structural alloys b Bar stock, typically 8.8 ×103 kg/m3 c Excluding “refractory” steels. d “Mechanical” rubber.

Density, ρ kg/m3 lbm/in.3 2.7 × 103 3.1× 103 10.1 × 103 7.4 × 103 8.6 × 103 7.5 × 103 8.9 × 103 8.7 × 103 6.4 × 103 8.9 × 103 9.5 × 103 7.4 × 103 6.1 × 103 7.8 × 103 1.8 × 103 7.8 × 103 6.7 × 103

0.097 0.11 0.36 0.27 0.31 0.27 0.32 0.31 0.23 0.32 0.34 0.27 0.22 0.28 0.065 0.28 0.24

1.4× 103 1.14× 103 .95× 103 1.3× 103 1.0× 103 1.8× 103

.051 .041 .034 .047 .036 .065

103 103 103 103

0.14 0.061 0.10 0.12

3.9× 1.7× 2.9× 3.2×

(0.03 lbm/in.3 ).

Density of Materials

Table 3.1 Density for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


10 12

10 11

Metals

Polymers

Ceramics

Carbides Alumina

Steels Cast iron Brass, bronze Aluminum Zinc alloys Magnesium alloys Babbitts

Modulus of Elasticity

Graphite

Modulus of elasticity, E, Pa

10 10 Phenol formaldehyde Acetal Nylon 10 9

Polyethylene

10 8

10 7

10 6

Figure 3.12 Modulus of elasticity for various metals, Natural rubber polymers, and ceramics at room temperature (20°C, 68°F). Hamrock • Fundamentals of machine Elements

Modulus of Elasticity, E Material GPa Mpsi Metals Aluminum 62 9.0 Aluminum alloysa 70 10.2 Aluminum tin 63 9.1 Babbitt, lead-based white metal 29 4.2 Babbitt, tin-based white metal 52 7.5 Brasses 100 14.5 Bronze, aluminum 117 17.0 Bronze, leaded 97 14.1 Bronze, phosphor (cast)b 110 16.0 Bronze, porous 60 8.7 Copper 124 18.0 Iron, gray cast 109 15.8 Iron, malleable cast 170 24.7 b Iron, spheroidal graphite 159 23.1 Iron, porous 80 11.6 Iron, wrought 170 24.7 Magnesium alloys 41 5.9 Steel, low alloys 196 28.4 Steel, medium and high alloys 200 29.0 Steel, stainlessc 193 28.0 Steel, high speed 212 30.7 Zinc alloys d 50 7.3 Polymers Acetal (polyformaldehyde) 2.7 0.39 Nylons (polyamides) 1.9 0.28 Polyethylene, high density .9 0.13 Phenol formaldehyde e 7.0 1.02 Rubber, naturalf .004 0.0006 Ceramics Alumina (Al2 O3 ) 390 56.6 Graphite 27 3.9 Cemented carbides 450 65.3 Silicon carbide (SiC) 450 65.3 Silicon nitride (Si2 N4 ) 314 45.5 a Structural alloys. b For bearings. c Precipitation-hardened alloys up to 211 GPa (30 Mpsi). d Some alloys up to 96 GPa (14 Mpsi). e Filled. f 25%-carbon-black “mechanical” rubber.

Modulus of Elasticity

Table 3.2 Modulus of elasticity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


Material Metals Aluminum and its alloysa Aluminum tin Babbitt, lead-based white metal Babbitt, tin-based white metal Brasses Bronze Bronze, porous Copper Copper lead Iron, cast Iron, porous Iron, wrought Magnesium alloys Steels, Zinc alloys d Polymers Acetal (polyformaldehyde) Nylons (polyamides) Polyethylene, high density Phenol formaldehyde Rubber Ceramics Alumina (Al2 O3 ) Graphite, high strength Cemented carbides Silicon carbide (SiC) Silicon nitride (Si2 N4 ) a Structural alloys.

Poisson’s ratio, ν 0.33 0.33 0.33 0.22 0.33 0.26 0.20 0.30 0.33 0.30 0.27 0.40 0.35 0.50 0.28 0.19 0.19 0.26

Poisson’s Ratio

Table 3.3 Poisson’s ratio for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


3 × 10 2 2

10 2

Metals

Polymers

Ceramics

Aluminum Copper Brass Magnesium alloys

Thermal Conductivity Graphite

Cast iron Bronze Steel

Thermal conductivity, Kt, W/m-°C

Alumina Stainless steel

10

Silicon carbide

Natural rubber 1

Figure 3.13 Thermal conductivity for various Acetal, nylon metals, polymers, and ceramics at room temperature (20°C, 68°F). Hamrock • Fundamentals of machine Elements Polyethylene

10 –1

Thermal conductivity, Kt Material W/m·◦ C Btu/ft·hr·◦ F Metals Aluminum 209 120 Aluminum alloys, casta 146 84 Aluminum alloys, siliconb 170 98 c Aluminum alloys, wrought 151 87 Aluminum tin 180 100 Babbitt, lead-based white metal 24 14 Babbitt, tin-based white metal 56 32 Brassesa 120 69 Bronze, aluminuma 50 29 Bronze, leaded 47 27 Bronze, phosphor (cast)d 50 29 Bronze, porous 30 17 Coppera 170 98 Copper lead 30 17 Iron, gray cast 50 29 Iron, spheroidal graphite 30 17 Iron, porous 28 16 Iron, wrought 70 40 Magnesium alloys 110 64 Steel, low alloyse 35 20 Steel, medium alloys 30 17 Steel, stainlessf 15 8.7 Zinc alloys 110 64 Polymers Acetal (polyformaldehyde) 0.24 0.14 Nylons (polyamides) 0.25 0.14 Polyethylene, high density 0.5 0.29 Rubber, natural 1.6 0.92 Ceramics Alumina (Al2 O3 )g 25 14 Graphite, high strength 125 72 Silicon carbide (SiC) 15 8.6 a At 100◦ C. b At 100◦ C (∼ 150 W/m·◦ C at 25◦ C). c 20 to 100◦ C. d Bar stock, typically 69 W/m·◦ C. e 20 to 200◦ C. f Typically 22 W/m·◦ C at 200◦ C. g Typically 12 W/m·◦ C at 400◦ C.

Thermal Conductivity

Table 3.4 Thermal conductivity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


2 × 10 – 4

10

Metals

Polymers

Ceramics

Linear Thermal Expansion Coefficient

Polyethylene Silicone rubber Natural rubber Acetal, nylon

–4

_ Linear thermal expansion coefficient, a , (°C)–1

Nitrile rubber

Zinc Magnesium Aluminum Brass, copper Most bronzes

Phenol formaldehyde

Babbitts Steel 10 – 5

Leaded bronze Cast irons Sintered iron Alumina Silicon carbide Silicon nitride Graphite

10 – 6

Figure 3.14 Linear thermal expansion coefficient for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


Linear thermal expansion coefficient, a ¯ Material (◦ C)−1 (◦ F)−1 Metals Aluminum 23 ×10−6 12.8×10−6 a −6 Aluminum alloys 24 ×10 13.3×10−6 −6 Aluminum tin 24 ×10 13.3×10−6 −6 Babbitt, lead-based white metal 20 ×10 11×10−6 Babbitt, tin-based white metal 23 ×10−6 13×10−6 −6 Brasses 19×10 10.6×10−6 −6 Bronzes 18 ×10 10.0×10−6 −6 Copper 18×10 10.0×10−6 −6 Copper lead 18×10 10.0×10−6 −6 Iron, cast 11 ×10 6.1×10−6 −6 Iron, porous 12×10 6.7×10−6 −6 Iron, wrought 12 ×10 6.7×10−6 −6 Magnesium alloys 27×10 15×10−6 Steel, alloysb 11 ×10−6 6.1×10−6 −6 Steel, stainless 17 ×10 9.5×10−6 −6 Steel, high speed 11 ×10 6.1×10−6 −6 Zinc alloys 27 ×10 15×10−6 Polymers Thermoplasticsc (60-100)×10−6 (33-56)×10−6 d −6 Thermosets (10-80)×10 (6-44)×10−6 −6 Acetal (polyformaldehyde) 90×10 50×10−6 Nylons (polyamides) 100×10−6 56×10−6 −6 Polyethylene, high density 126×10 70×10−6 e −6 Phenol formaldehyde (25-40)×10 (14-22) ×10−6 f −6 Rubber, natural (80-120)×10 (44-67)×10−6 g −6 Rubber, nitrile 34×10 62×10−6 −6 Rubber, silicone 57×10 103×10−6 Ceramics Alumina (Al2 O3 )h 5.0×10−6 2.8×10−6 Graphite, high strength 1.4-4.0×10−6 0.8-2.2×10−6 −6 Silicon carbide (SiC) 4.3×10 2.4×10−6 −6 Silicon nitride (Si3 N4 ) 3.2×10 1.8×10−6 a Structural alloys. b Cast alloys can be up to 15 ×10−6 (◦ C)−1 . c Typical bearing materials. d 25 × 10−6 (◦ C)−1 to 80 × 10−6 (◦ C)−1 when reinforced. e Mineral filled. f Fillers can reduce coefficients. g Varies with composition. h 0 to 200◦ C.

Thermal Expansion Coefficient

Table 3.5 Linear thermal expansion coefficient for various metals, polymers, and ceramics at room temperature (20°C, 68°F). Hamrock • Fundamentals of machine Elements

2.0

Metals

Polymers

Ceramics

Natural rubber

Specific Heat Capacity

1.8

Specific heat capacity, Cp, kJ/ kg-°C

1.6

Thermoplastics

1.4

1.2

1.0 Magnesium Aluminum Graphite

0.8

Carbides, alumina 0.6

0.4

Steel Cast iron Copper

Figure 3.15 Specific heat capacity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).

0.2


Specific heat capacity, Cp kJ/kg·◦ C Btu/lbm·◦ F

Material Metals Aluminum and its alloys 0.9 0.22 Aluminum tin 0.96 0.23 Babbitt, lead-based white metal 0.15 0.036 Babbitt, tin-based white metal 0.21 0.05 Brasses 0.39 0.093 Bronzes 0.38 0.091 a Copper 0.38 0.091 Copper lead 0.32 0.076 Iron, cast 0.42 0.10 Iron, porous 0.46 0.11 Iron, wrought 0.46 0.11 Magnesium alloys 1.0 0.24 b Steels 0.45 0.11 Zinc alloys 0.4 0.096 Polymers Thermoplastics 1.4 0.33 Rubber, natural 2.0 0.48 Ceramics Graphite 0.8 0.2 Cemented Carbides 0.7 0.17 a Aluminum bronze up to 0.48 kJ/kg·◦ C (0.12 Btu/lbm·◦ F. b Rising up to 0.55 kJ/kg·◦ C (0.13 Btu/lbm·◦ F) at 200 ◦ C (392◦ F).

Specific Heat Capacity

Table 3.6 Specific heat capacity for various metals, polymers, and ceramics at room temperature (20°C, 68°F).


Example 3.12 l1 = 1 m l 2 = 0.3 m

l3 = 0.5 m

P1 = 3 kN

d = 0.1 m

Figure 3.16 Rigid beam assembly used in Example 3.12.


1000 Diamond SiC Si3N4 B

Be

Aluminas Sialons BeO ZrO2 Si

CFRP Uniply

100

() E __ ρ

(m/s)

Modulus of elasticity, E, GPa

104 10

Fir Parallel to grain Balsa

Laminates GFRP KFRP

Ash Oak Pine

Cu alloys

Ti alloys Al alloys Rock, stone Cement, concrete

CFRP

Engineering composites

Porous ceramics

Mg alloys

Specific Elastic Modulus

Mo W alloys alloys Ni alloys

Glasses Pottery

KFRP GFRP 1/2

WC-Co

Engineering ceramics

Zn alloys Tin alloys Lead alloys

Engineering alloys

MEL PC Epoxies PS PMMA

Wood products

PVC Woods 3 × 103

1.0

Ash Lower E limit for true solids Oak Pine Fir Perpendicular to grain Spruce Balsa

Nylon

HDPE

Guidelines for minimumweight design

PTFE

E __ =C ρ

LDPE

Figure 3.17 Modulus of elasticity plotted against density.

Plasticized PVC

103 Polymers, foams

0.1

Cork

0.01 100

Engineering polymers

Polyesters

PP

300

Elastomers

3 × 10 2

Hard butyl

Polymers, foams

Soft butyl

PU

1/2 E __ =C ρ

Silicone 1/ 3 E __ =C ρ

1000

3000 Density, ρ, kg/m 3

(G ≈ 3E/8; K ≈ E)

10 000

30 000


10 000

SiC Diamond Si3N4 Sialons

Engineering ceramics B

Glasses

100 Strength, S, MPa

Fir

Balsa

MEL PVC Epoxies PS Polyesters HDPE

Wood products

Ash Oak Pine Fir Perpendicular to grain LDPE

Woods 10

Soft butyl

Balsa

PU

Silicone

Mo alloys Ni alloys Cu alloys

Lead alloys

Cement, concrete Porous ceramics


Guidelines for minimumweight design

Figure 3.18 Strength plotted against density.

Polymers foams

Cork

300

Specific Strength

W alloys

Cast irons

Engineering alloys

Elastomers

σf __ =C ρ

0.1 100

PTFE

Steels Ti alloys

Zn Stone, alloys rock

Ash PP Oak Pine

Parallel to grain

Engineering alloys

Al alloys

Mg alloys

Nylons PMMA

Cermets

Al2O3 Ge MgO

Si

CFRP GFRP Uniply KFRP Pottery Engineering CFRP composites Be Laminates GFRP KFRP

1000

1

ZrO2

1/ σf 2 __ =C ρ

2/ σf 3 __ =C ρ

1000

3000

10 000

30 000

Density, ρ, kg/m 3


1000 Diamond Minimum energy storage per unit volume

Engineering alloys

Yield before buckling

Cermets Mo alloys

W

Beryllium

WC SiC Boron Si3N4 MgO Al2O3

Steels ZrO2 Ni alloys CFRP BeO Uniply Silicon Cu alloys Zn alloys Ti alloys Ge Common rocks GFRP Engineering Al alloys Brick, etc. Glasses ceramics Mg alloys Cement Sn + Engineering Laminates Concrete composites CFRP GFRP Lead Ash Oak Pine Porous Mel ceramics ll to grain Epoxies PMMA Cast irons


100

10

–4 S __ = 10 E

Balsa Woods Wood products Polyester Ash Oak

1.0

–2 S __ = 10 E

–3 S __ = 10 E

Pine ⊥ to grain

PS S __ = 0.1 E

PVC Nylons PP

Design guidelines

HDPE PTFE

Balsa


LDPE

Polymers, foams

Figure 3.19 Modulus of elasticity plotted against strength.

S __ =C E

0.1

3/ 2 S __ =C E

Hard butyl

PU

Elastomers

Cork

2

S __ =C E

Maximum energy storage per unit volume Buckling before yield

Silicone 0.01 0.1

Modulus of Elasticity vs. Strength

Soft butyl 1

10

100

1000

10 000

Strength, S, MPa


10 –10

Wear Constant

10 –11 Al Alloys

10 –12

Copper

PTFE

10 –13 Archard wear constant, KA, m2/ N

Mild steel

Engineering alloys

W __ = 10 –3 A

Stainless steels Medium-carbon steels High-carbon Tools steels steels


10 – 4

10 –14 LDPE Unfilled thermoplastics HDPE 10 –15 Filled PTFE

Cast irons

Nylons

Nitrided steels

Filled thermosets

Filled thermoplastics Filled polymides

Cemented carbides Al2O3 Si3N4 SiC Sailons

Bronzes

10 –5 Engineering ceramics

Diamond

10 –16

Engineering composites

Maximum bearing pressure, Pmax

10 –6

10 –7 10 –17 Range of K A for p << pmax

10 –18

1

10 –8

Rising K A as p nears pmax

Figure 3.20 Archard wear constant plotted against limiting pressure.

W __ = 10 –9 A 10

100 Limiting pressure, pl , MPa

1000

10 000


1000 Engineering ceramics Si C AL7 O3 Mild steel Porous ceramics

Cement, concrete Brick, stone, concrete

Stone, brick, and pottery

AL alloys alloys Mg alloys GFRP

Glasses

Engineering composites Engineering alloys

Ash, oak Parallel to grain

Pines

Balsa wood products

Woods

Ash, oak

1.0

Ti alloys KFRP

Pb alloys

10


Zn

Modulus of Elasticity vs. Cost

S alloys Zr O3

CFRP

Cu alloys

100

Cermets

Si3 N4

Ni alloys

Cast irons

W alloys

PF

PS

PMMA

PC

Polymides


PVC Epoxies Nylons Polyesters PP HDPE

Perpendicular to grain Pines

Design guidelines

PTFE LDPE

Balsa

PVC (plasticised)

Figure 3.21 Modulus of elasticity plotted against cost times density.

E ––– =C CRρ

0.1 Polymer foams

Hard butyl

Elastomers 1/2

E ––– = C CRρ Soft butyl 0.01 0.1

1

Silicones

10 100 Relative cost times density, Mg/m3

1/3

E ––– = C CRρ 1000

10 000


Sand Casting

Figure 3.22 Schematic illustration of the sand casting process.


Forging and Extrusion Figure 3.23 An example of the steps in forging a connecting rod for an internal combustion engine, and the die used.

Container liner

Container

Billet Die

Pressing stem

Die backer

Dummy block

Extrusion

(a)

(b)

Figure 3.24 The extrusion process. (a) Schematic illustration of the forward or direct extrusion process; (b) Examples of cross-sections commonly extruded. Hamrock • Fundamentals of machine Elements

Polymer Extruder Barrel heater/cooler Thrust bearing

Hopper

Barrel liner Throat Screw

Thermocouples

Melt Filter Breaker screen plate thermocouple

Barrel

Gear reducer box

Throat-cooling channel Feed zone Melting zone

Adapter

Die

Melt-pumping zone

Motor

Figure 3.25 Schematic illustration of a typical extruder. Hamrock • Fundamentals of machine Elements

formsa

Material Available Aluminum B, F, I, P, S, T, W Ceramics B, p, s, T Copper and brass B, f, I, P, s, T, W Elastomers b, P, T Glass B, P, s, T, W Graphite B, P, s, T, W Magnesium B, I, P, S, T, w Plastics B, f, P, T, w Precious metals B, F, I P, t, W Steels and stainless steels B, I, P, S, T, W Zinc F, I, P, W a B=bar and rod; F = foil; I = ingot; P = plate and sheet; S = structural shapes; T = tubing; W=wire. Lowercase letters indicate limited availability. Most of the metals are also available in powder form, including prealloyed powders.

Available Forms of Materials

Table 3.8 Commercially available forms of materials.


Tolerance vs. Surface Roughness

Figure 3.26 A plot of achievable tolerance versus surface roughness for assorted manufacturing operations. Hamrock • Fundamentals of machine Elements

Chapter 4: Stresses and Strains

I am never content until I have constructed a mechanical model of the subject I am studying. If I succeed in making one, I understand; otherwise I do not. William Thomson (Lord Kelvin)


Centroid of Area y

y¯ =

R

x¯ =

R

dA A C _ y

y x _ x x

A y dA

A A x dA

A

A1y¯1 + A2y¯2 + · · · = A1 + A2 + · · · =

A1x¯1 + A2x¯2 + · · · A1 + A2 + · · ·

Figure 4.1 Centroid of area.


Example 4.1 y

y a dA

d b

r

y

c f e

Figure 4.2 Rectangular hole within a rectangular section used in Example 4.1.

x

A

0

x x

Figure 4.3 Area with coordinates used in describing area moment of inertia.


Example 4.2 y

r dy

φ

x

Figure 4.4 Centroid of area.


Parallel-Axis Theorem y

dA

A

Ix! = Ix + Ady2 y

0

C

x

Iy! = Iy + Adx2

dy 0′

x′

Figure 4.5 Coordinates and distance used in describing parallel-axis theorem. Hamrock • Fundamentals of Machine Elements

Examples 4.3 and 4.4 y

y′ y r x

y′c = 4r

a b

dy = 4r

2 cm 60°

60°

1 cm

6 cm

x

x′ dx = 3r x′c = 3r

Figure 4.6 Triangular cross section with circular hole in it, used in Example 4.3.

Figure 4.7 Circular crosssectional area relative to x’ - y’ coordinates, used in Example 4.4.


Cross section Circular area y

Centroid _ x =0 _ y=0

x

r Hollow circular area y

_ x =0

ri r

x

_ y=0

Triangular area a y _ x _ y

C

_ a+b x = _____ 3 _ __ h y= 3

h x

b Rectangular area y C

h

x b Area of circular sector y r α C x α _ x Quarter-circular area y _ x r C

_ y

_ y

C a

_ r sin α x = _2_ ______ 3 α

x

Area

Ix = Ix_ = _π_ r 4 4 π _ _ Iy = Iy = _ r 4 4 π 4 _ _ J = r 2

A = πr 2

Ix = Ix_ = _π_ r 4 – ri4 4 π _ _ Iy = Iy = _ r 4 – ri4 4 π 4 _ _ J = r – ri4 2

( (

(

) )

(

bh A = ___ 2

bh_3, I _ = ___ bh_3 Ix = ___ x 3 12 hb_3, I _ = ___ hb_3 Iy = ___ y 3 12 _ bh J = ___ b 2 + h2 12

A = bh

( (

) )

(

)

_1_ sin 2α) 2 _1_ sin 2α) 2

A = r 2α

_ _ __ x = y = _4r 3π

πr 2 A = ___ 4

_ 4a_ x = __ 3π _ __ 4b y= _ 3π

π – ___ 4 ab3 ____3 , I _ = ___ Ix = _πab x 16 16 9π 3b π – ___ 4 a 3b ____ Iy = _πa , Iy_ = ___ 16 16 9π πab_ a 2 + b 2 J = ___ 16

πab A = ____ 4

(

)

( (

(

)

)

bh_3, I _ = ___ bh_3 Ix = ___ x 12 36 bh____________ b2 + ab + a2 Iy = ___ 12 bh____________ b2 – ab + a2 Iy_ = ___ 36 _ bh 2 + h2 + a2 + ab) ___ (b J = 36

4 Ix = _r__ (α – 4 4 Iy = _r__ (α + 4 J = _1_ r 4α 2

Properties of Cross Sections

A = π r 2 – ri 2

___4 Ix = Iy = _πr 16 π – ___ 4 r4 _ _ Ix = Iy = ___ 16 9π 4 ___ J = _πr 8

x

Area of elliptical quadrant y x_ b

_ b x = __ 2 _ __ y= h 2

Area moment of inertia

) )

Table 4.1 Centroid, area moment of inertia, and area for seven cross sections

)


Mass Element y

y

x2 + z2

dma x

dma x2

+

y2

z2 + y2 r

y

y

0

z

x

z

Figure 4.8 Mass element in three-dimensional coordinates and distance from the three axes.

x

x

Figure 4.9 Mass element in two-dimensional coordinates and distance from the two axes.


Shape

Equations

Rod y d x

l

z Disk

y

2t ρ h ______ ma = πd 4 2 ____ ad Imx = _m 8 2 ____ ad Imy = Imz = _m 16

th x z Rectangular prism

y

x z Cylinder

a

y d x

z l Hollow cylinder

y di do x

z l

2lρ _____ ma = _πd 4 m d2 _ ____ a Imx = 8 2 + 4l 2) ___ a_(3d _______ _____ Imy = Imz = m 48

2 2 o – di ) ___________ ma = πlρ(d 4 2 – d 2) m (d _ __________ a o i Imx = 8 m (3d 2 + 3d 2 + 4l 2) Imy = Imz = ____a ____o________i__________ 48

y

Sphere

πd 3ρ ma = _____ 6

d x

Mass and Mass Moment of Inertia

ma = abcρ 2 ma_(a b_2_) ____+___ Imx = ___ 12 2 ma_(a c_2_) ____+___ Imy = ___ 12 2 + c2) ma_(b ___ ___ ______ Imz = 12

b c

2 lρ _____ ma = πd 4 2 ____ al Imy = Imz = _m 12

Table 4.2 Mass and mass moment of inertia of six solids.

ma d 2 Imx = Imy = Imz = ____ 10

z


Circular Bar with Tensile Load P

P Internal load

External load P

P

Figure 4.10 Circular bar with tensile load applied. Hamrock • Fundamentals of Machine Elements

z

Twist Due to Torque ro

TL != GJ

γ θz

!=

l

r T

Tc J

Figure 4.11 Twisting of member due to applied torque.

θ


Deformation in Bending M

y Longitudinal lines become curved Transverse lines remain straight, yet rotate (a)

M z

(b)

Figure 4.12 Bar made of elastomeric material to illustrate effect of bending. (a) Undeformed bar; (b) deformed bar.


x

Neutral Surface and Deformation in Bending 0

y

l

r

Axis of symmetry

r

∆θ

z Bending axis

Neutral surface

∆s = ∆ x ∆x

M

x

Neutral axis

y

y

∆s′ ∆x

Neutral axis

∆x (a)

Figure 4.13 Bending occurring in cantilevered bar, showing neutral surface

(b)

Figure 4.14 Undeformed and deformed elements in bending.


Stress in Bending

σ max c

σ

M x

Mc !=− I

y

Figure 4.15 Profile view of bending stress variation.


Example 4.10 8 mm

C

Neutral axis 120 mm

y 8 mm B 80 mm

A

x

Figure 4.16 U-shaped cross section experiencing bending moment, used in Example 4.10


Deformation of Member in Bending Center of initial curvature

M

ri

_ r

φ

y

e

c b (a)

dφ

c′

Neutral axis

r

y

co

Centroidal axis

rn

_ r

_ c

d′ d ci

M

rn

a ro

b

Neutral surface Centroidal surface

Centroidal axis

e CG

ro

_ c

dr Neutral axis r

_ r ri

rn

(b)

Figure 4.17 Curved member in bending. (a) Circumferential view; (b) cross-sectional view

Figure 4.18 Rectangular cross section of curved member


Transverse Shear P

(a) P

(b)

Figure 4.19 How transverse shear is developed. (a) Boards not bonded together; (b) boards bonded together. Hamrock • Fundamentals of Machine Elements

Deformation Due to Transverse Shear

V (a)

(b)

Figure 4.20 Cantilevered bar made of highly deformable material and marked with horizontal and vertical grid lines to show deformation due to transverse shear. (a) Undeformed; (b) deformed. Hamrock • Fundamentals of Machine Elements

Moments and Stresses on Element

A′ σ M

σ′

τ wt dx

(a)

σ M M + dM

σ′ τ

y′

M + dM

(b)

Figure 4.21 Three-dimensional and profile views of moments and stresses associated with shaded top segment of element that has been sectioned at y’ about neutral axis. Shear stresses have been omitted for clarity. (a) three-dimensional view; (b) profile view. Hamrock • Fundamentals of Machine Elements

Cross section

Maximum shear stress 3V τmax = ___ 2A

Rectangular

Maximum Shear Stress for Different Cross Sections

4V τmax = ___ 3A Circular

2V τmax = ___ A Round tube

V τmax = ___ Aweb

Table 4.3 Maximum shear stress for different beam cross sections.

I-beam


Example 4.13 150 lb

300 lb

T = 1000 in-lb

T = 1000 in-lb 25 lb

75 lb

5 in.

50 lb 5 in.

Figure 4.22 Shaft with loading considered in Example 4.13.

5 in. 200 lb

250 lb

P

V 50 lb

M

75 lb

1000 in-lb 1250 in-lb

200 lb

x

x

x –250 lb -25 lb (a)

(b)

(c)

Figure 4.23 (a) Shear force; (b) normal force and (c) bending moment diagrams for the shaft in Fig. 4.22. Hamrock • Fundamentals of Machine Elements

Stress Elements in Example 4.13.

A

D

B

C

Figure 4.24 Cross section of shaft at x=5 in., with identification of stress elements considered in Example 4.13.


Shear Stress Distributions

Figure 4.25 Shear stress distributions. (a) Shear stress due to a vertical shear force; (b) Shear stress due to torsion. Hamrock • Fundamentals of Machine Elements

Coil Slitter

Recoiler

Two-roll tension stand

Slitter Operator's console

Entry pinch rolls and guide table

Pay-off reel (uncoiler)

Peeler

Coil-loading car

(a)

Slitter blade Set screw Collar

Rubber roller Steel spacers Key Driveshaft

500 lbf

180 lbf

RB

Lower driveshaft

540 lbf

Rubber rollers

le tab jus Ad eight h

10 in.

10 in.

5 in.

RA Collar Lower slitter blade Steel spacers (behind slitter blade)

(b)

(c)

6 in.

Figure 4.26 Design of shaft for coil slitting line. (a) Illustration of coil slitting line; (b) knife and shaft detail; (c) free-body diagram of simplified shaft for case study.


Coil Slitter Results

Center at (21,900/d3, 0) V

M x 290 lbf

720 lb x

4300 in.-lb

–430 lb

Radius =

Figure 4.27 Shear diagram (a) and moment diagram (b) for idealized coil slitter shaft.

(

2

) (

2

)

21,900 + _______ –11,000 = ______ 24,510 ______ d3 d3 d3

(σx, – τxy) = (43,800/d 3, 11,000/d 3 )

Figure 4.28 Mohr's circle for location of maximum bending stress.


Chapter 5: Deformation

Knowing is not enough; we must apply. Willing is not enough; we must do. Johann Wolfgang von Goethe


Example 5.1

P y x l

P

V

M

Figure 5.1 Cantilevered beam with concentrated force applied at free end. Used in Example 5.1. Hamrock • Fundamentals of Machine Elements

Deflection by Singularity Functions 1. Draw a free-body diagram showing the forces acting on the system. 2. Use force and moment equilibrium to establish reaction forces acting on the system. 3. Write an expression for the load-intensity function for all the loads acting on the system while making use of Table 2.2. 4. Integrate the negative load-intensity function to give the shear force and then integrate the negative shear force to obtain the moment. 5. Make use of Eq. (5.9) to describe the deflection at any value. 6. Plot the following as a function of x: (a) Shear (b) Moment (c) Slope (d) Deflection


Force on Simply Supported Beam y l x a

b

Pb_ RA = __ l

Pa_ RB = __ l

P (a)

y x

V

x a Pb_ RA = __ l

x–a P

M

Figure 5.2 Free-body diagram of force anywhere between simplysupported ends. (a) Complete bar; (b) portion of bar.

(b)


y l a

Cantilever with Step Load

b w0

A

–yl

B

x

C (a) a

b w0

(

MA = w0b a + _b 2

) RA = w0b (b) a

(

w0b a + _b 2

x–a w0 V

) w0b x (c)

M

Figure 5.3 Cantilevered bar with unit step distribution over part of bar. (a) Loads and deflection acting on cantilevered bar; (b) free-body diagram of forces and moments acting on entire bar; (c) free-body diagram of forces and moments acting on portion of bar.


y l a

b P

A

C

B

x

Cantilever with Concentrated Force

(a) a

b P MC

RA

RC (b) a

x–a P

V M

RA x

Figure 5.4 Cantilevered bar with other end simply-supported and with concentrated force acting anywhere along bar. (a) Sketch of assembly; (b) free-body diagram of entire bar; (c) free-body diagram of part of bar.

(c)


Type of loading

Deflection for any x

Concentrated load at any x y P a b x yl

l Unit step distribution over part or all of bar y a b w 0

x yl

l

Beam Deflections

P ( x – a 3 – x3 + 3x2a) y = – –— 6EI

w0 bx 3 bx 2 y = –— –— – –— a + b– EI 6 2 2 1 –— x–a4 24

Moment applied to free end y M x yl

l

Mx 2 y = – –—— 2EI

(a)

y

P a

b x

P b– x y = –— 6EI l

3–

a3x x – a 3 + 3a2x – 2alx – ––– l

l

– Pb l

– Pa l

y

b w0 a

c x

x–a 4– x–a–b

4



+ x b3 + 6bc2 + 4b2c + 4c3 – 4l 2 c + b–  2 

l

w0b – ––– c + b 2 l

w0b  y = – –—–– 4 c + b– x 3 – –l 24lEI  2 b 

Table 5.1 Deflection for three different situations when one end is fixed and one end is free and two different situations of simply supported ends.

w0b – ––– a + b 2 l

(b)


y

Cantilever with Moment

a P A

B yl

C

x

Mo l (a) y a P yl, 1

x

(b) y Mo yl, 2 x

l (c)

Figure 5.5 Bar fixed at one end and free at other with moment applied to free end and concentrated force at any distance from free end. (a) Complete assembly; (b) free-body diagram showing effect of concentrated force; (c) free-body diagram showing effect of moment.


Stress Elements

σz

dx dx

dy dz

Figure 5.6 Element subjected to normal stress.

γdz γ

dy τ dz

Figure 5.7 Element subjected to shear stress.


Strain Energy

Loading type Axial Bending Torson Transverse shear (rectangular section)

Factors involved P, E, A

Strain energy for special case where all three factors are constant with x P 2l U= 2EA

M, E, I

M 2l U= 2EI

T, G, J

T 2l U= 2GJ

V, G, A

3V 2 l U= 5GA

General expression for strain energy !l P 2 U= dx 2EA 0 !l M 2 U= dx 2EI 0 !l T 2 U= dx 2GJ 0 !l 3V 2 U= dx 5GA 0

Table 5.2 Strain energy for four types of loading. Hamrock • Fundamentals of Machine Elements

Castigliano’s Theorem ∂Qi

The load Qi is applied to a particular point of deformation and therefore is not a function of x. T !U yi = to Qi before integrating for the general expressio permissible to take the derivative with respect !Q i

The following procedure is to be employed in using Castigliano’s theorem: 1. Obtain an expression for the total strain energy including (a) Loads (P , M , T , V ) acting on the object (use Table 5.2) (b) A fictitious force Q acting at the point and in the direction of the desired deflection 2. Obtain deflection from y = ∂U/∂Q. 3. If Q is fictitious, set Q = 0 and solve the resulting equation.


Cantilever with Concentrated Force

y P

b A

x

B

C

l (a) Q b

P

x l (b)

Figure 5.8 Cantilevered bar with concentrated force acting distance b from free end. (a) Coordinate system and important points shown; (b) fictitious force shown along with concentrated force.


Linkage System

l, A1, E1 θ

P1 θ

A

θ l, A2, E 2

(a)

Q

A

θ P

P2

P

(b)

Figure 5.9 Linkage system arrangement. (a) Entire assembly; (b) free-body diagram of forces acting at point A. Hamrock • Fundamentals of Machine Elements

Example 5.10 x

C

B h

y l

A

P

Q Figure 5.10 Cantilevered bar with 90° bend acted upon by horizontal force at free end. Hamrock • Fundamentals of Machine Elements

Chapter 6: Failure Prediction for Static Loading The concept of failure is central to the design process, and it is by thinking in terms of obviating failure that successful designs are achieved. Henry Petroski Design Paradigms


Rectangular Plate with Hole Kc =

actual maximum stress average stress

a

P

P a (a)

σ σmax

σ

a b

P (c) (b)

Kc = 1 + 2

!a" b

Figure 6.1 Rectangular plate with hole subjected to axial load. (a) Plate with cross-sectional plane; (b) one-half of plate with stress distribution; (c) plate with elliptical hole subjected to axial load. Hamrock • Fundamentals of Machine Elements

Rectangular Plate with Hole 3.0

Stress concentration factor, Kc

2.8 2.6 2.4 2.2 2.0 1.8 P

1.6

h

1.4

P

P = ––––––– P Snom= –– A (b – d)h

1.2 1.0

b

d

0

0.1

0.2

0.3

0.4

0.5

0.6

Diameter-to-width ratio, d/b

Figure 6.1 Stress concentration factors for rectangular plate with central hole. (a) Uniform tension. Hamrock • Fundamentals of Machine Elements

Rectangular Plate with Pin-Loaded Hole 11.0 d


10.0

c/b = 0.35

9.0

P

b

c

P/2 h

P/2 P = ––––––– P σnom= –– A (b – d)h

8.0 7.0 6.0 5.0 4.0

c/b = 0.50

3.0

c/b ≥ 1.0

2.0 1.0

0

0.1

0.2 0.3 0.4 Diameter-to-width ratio, d/b

0.5

0.6

Figure 6.2 Stress concentration factors for rectangular plate with central hole. (b) pin-loaded hole. Hamrock • Fundamentals of Machine Elements

Rectangular Plate with Hole in Bending 3.0


2.8

M

2.6 2.4

0.25

2.0

0.5

1.8

1.0

1.6

2.0

1.4

∞

1.2 1.0

M

Mc 6M σnom= –––– = –––––––2 I (b – d)h

d/h = 0

2.2

b

d

h

0

0.1

0.2 0.3 0.4 Diameter-to-width ratio, d/b

0.5

0.6

Figure 6.2 Stress concentration factors for rectangular plate with central hole. (c) Bending. Hamrock • Fundamentals of Machine Elements

Axially Loaded Rectangular Plate with Fillet 3.0


2.8

P

2.6

b

r

2.4

P

h

H

P P Snom= –– = –– A bh

2.2 2.0 1.8

H/h = 3 2 1.5 1.15 1.05 1.01

1.6 1.4 1.2 1.0

0

0.05

0.10 0.15 0.20 Radius-to-height ratio, r/h

0.25

0.30

Figure 6.3 Stress concentration factors for rectangular plate with fillet. (a) Axial load. Hamrock • Fundamentals of Machine Elements

Rectangular Plate with Fillet in Bending 3.0 2.8 M


2.6

M

h

H

b

r

2.4

6M ––– = ––– σnom= Mc I bh2

2.2 2.0 1.8 1.6

H/h = 6 2 1.2 1.05 1.01

1.4 1.2 1.0

0

0.05


0.25

0.30

Figure 6.3 Stress concentration factors for rectangular plate with fillet. (b) Bending. Hamrock • Fundamentals of Machine Elements

Axially Loaded Rectangular Plate with Groove 3.0


2.8

P

2.6

P

h

H b

r

2.4

P P Snom = –– = –– A bh

2.2 2.0

H/h = ! 1.5 1.15

1.8 1.6

1.05

1.4

1.01

1.2 1.0

0

0.05

0.10

0.15

0.20

0.25

0.30

Radius-to-height ratio, r/h

Figure 6.4 Stress concentration factors for rectangular plate with groove. (a) Axial load. Hamrock • Fundamentals of Machine Elements

Rectangular Plate with Groove in Bending 3.0


2.8

M

2.6

h

b

r

2.4

M

H

6M ––– = ––– σnom= Mc I bh2

2.2 2.0 1.8

H/h = ∞ 1.5 1.15 1.05 1.01

1.6 1.4 1.2 1.0

0

0.05


0.25

0.30

(b)

Figure 6.4 Stress concentration factors for rectangular plate with groove. (b) Bending. Hamrock • Fundamentals of Machine Elements

Axially Loaded Round Bar with Fillet 2.6 r P


2.4

D

d

P

2.2 4P Snom = –P– = ––– A Pd2

2.0 1.8 1.6

D/d = 2 1.5 1.2 1.05

1.4 1.2 1.0

1.01 0

0.1

0.2

0.3

Radius-to-diameter ratio, r/d (a)

Figure 6.5 Stress concentration factors for round bar with fillet. (a) Axial load. Hamrock • Fundamentals of Machine Elements

Round Bar with Fillet in Bending 3.0 r

2.8

M

D

d

M


2.6 2.4

Mc 32M Snom = ––– = –––– I Pd3

2.2 2.0 1.8 1.6 D/d = 6 3 1.5 1.1 1.03 1.01

1.4 1.2 1.0

0

0.1 0.2 Radius-to-diameter ratio, r/d

0.3

Figure 6.5 Stress concentration factors for round bar with fillet. (b) Bending. Hamrock • Fundamentals of Machine Elements

Round Bar with Fillet in Torsion 2.6 r


2.4

T

D

d

T

2.2 –– = 16T ––– τnom = Tc J πd 3

2.0 1.8 1.6 1.4

D/d = 2 1.2 1.09

1.2 1.0

0

0.1 0.2 Radius-to-diameter ratio, r/d (c)

0.3

Figure 6.5 Stress concentration factors for round bar with fillet. (c) Torsion. Hamrock • Fundamentals of Machine Elements

Axially Loaded Round Bar with Groove 3.0

r

2.8

P

D

d

P


2.6 P 4P Snom = –– = –––2 A Pd

2.4 2.2 2.0 1.8 1.6 1.4

D/d 2 1.1 1.03

1.2

1.01

1.0

0


0.3

Figure 6.6 Stress concentration factors for round bar with groove. (a) Axial load. Hamrock • Fundamentals of Machine Elements

Round Bar with Groove in Bending 3.0

r

2.8 M

D

d

M


2.6 Mc 32M σnom = ––– = –––– I πd 3

2.4 2.2 2.0 1.8 1.6

D/d > 2 1.1 1.03 1.01

1.4 1.2 1.0

0


0.3

Figure 6.6 Stress concentration factors for round bar with groove. (b) Bending. Hamrock • Fundamentals of Machine Elements

Round Bar with Groove in Torsion r

2.6


2.4

T

D

d

T

2.2 Tc 16T τnom = –– = –––3 J πd

2.0 1.8 1.6 1.4

D/d ≥ 2 1.1 1.01

1.2 1.0

0

0.1

0.2

0.3

Radius-to-diameter ratio, r/d

Figure 6.6 Stress concentration factors for round bar with groove. (c) Torsion. Hamrock • Fundamentals of Machine Elements

Round Bar with Hole 3.0 A xia


2.8 2.6

Be

2.4

l

T

M

P

P

D

M

Nominal stresses:

d n di

Axial load:

ng

P P σnom = –– = 2 A (πD /4) - Dd

2.2

Bending (plane shown is critical):

2.0 T or

1.8

M Mc σnom= ––– = 3 I (πD /32) - (dD2/6)

sio n

1.6

Torsion:

1.4

Tc T τnom = –– = J (πD3/16) - (dD2/6)

1.2 1.0

T

0

0.1 0.2 Hole diameter-to-bar diameter ratio, d/D

0.3

Figure 6.7 Stress concentration factors for round bar with hole. Hamrock • Fundamentals of Machine Elements

P

P

Stress Contours

(a)

P

P

(b)

P

P

(c)

P

P

(d)

Figure 6.8 Flat plate with fillet axially loaded showing stress contours for (a) square corners; (b) rounded corners; (c) small grooves; and (d) small holes.


Modes of Fracture

A

(a)

(b)

(c)

Figure 6.9 Three modes of crack displacement. (a) Mode I, opening; (b) mode II, sliding; (c) Mode III, tearing. Hamrock • Fundamentals of Machine Elements

Fracture Toughness √

Kci = Y !nom "a Material Metals Aluminum alloy 2024-T351 Aluminum alloy 7075-T651 Alloy steel 4340 tempered at 260◦ C Alloy steel 4340 tempered at 425◦ C Titanium alloy Ti-6Al-4V

Yield stress, Sy ksi MPa

Fracture toughness, Kci √ √ ksi in. MPa m

47

325

33

36

73

505

26

29

238

1640

45.8

50.0

206

1420

80.0

87.4

130

910

40-60

44-66

Ceramics Aluminum oxide Soda-lime glass Concrete

— — —

— — —

2.7-4.8 0.64-0.73 0.18-1.27

3.0-5.3 0.7-0.8 0.2-1.4

Polymers Polymethyl methacrylate Polystyrene

— —

— —

0.9 0.73-1.0

1.0 0.8-1.1

Table 6.1 Yield stress and fracture toughness data for selected engineering materials at room temperature. Hamrock • Fundamentals of Machine Elements

Failure Prediction for Multiaxial Stresses I. Ductile Materials

Maximum Shear Stress Theory (MSST): Sy !1 − !2 = ns

Distortion-Energy Theory (DET) " Sy 1 ! 2 2 2 1/2 √ (!2 − !1) + (!3 − !1) + (!3 − !2) = ns 2


Failure Prediction for Multiaxial Stresses II. Brittle Materials Maximum Normal Stress Theory (MNST) Sut !1 ≥ ns

Suc or !3 ≤ ns

Internal Friction Theory (IFT)

!1 !3 1 If !1 > 0 and !3 < 0, − = Sut Suc ns Sut If !3 > 0, !1 = n Sucs If !1 < 0, !3 = ns

Modified Mohr Theory If !1 > 0 and !3 < −Sut , If !3 > −Sut If !1 < 0,

Sut !3 SucSut = Suc − Sut nsSuc − Sut Sut !1 = ns Suc !3 = ns

!1 −


Three-Dimensional Yield Locus σ2 Centerline of cylinder and hexagon

σ1 DET

σ2

MSST

σ3 σ1

σ3

Figure 6.10 Three-dimensional yield locus for MSST and DET.

View along axis of cylinder


MSST and DET for Biaxial Stress State + S2

+ S2 –0.577 Syt

Sy A

Syt

G

45o

0.577 Syt

Sy

E B

– S1

Syt

D

Syt

+ S1

–Syt

– S1

45o

Shear diagonal

H

C Sy – S2

+ S1

–Syt

F –Sy

Syt

–0.577 Syt

–Syt –Syt

0.577 Syt

– S2 Shear diagonal

Figure 6.12 Graphical Figure 6.11 Graphical representation of distortion energy representation of maximum-sheartheory for biaxial stress state. stress theory for biaxial stress state. Hamrock • Fundamentals of Machine Elements

Example 6.6

2500 N

Arm Torsion bar

300 mm

Bearing

100 mm

Figure 6.13 Rear wheel suspension used in Example 6.6


Examples 6.7 and 6.8

T

y

T1 T1

S3

0 S2

z T

x

S

S1

(b) T2

(a)

(c)

Figure 6.14 Cantilevered round bar with torsion applied to free end used in Example 6.7. T

y

T1 = Tmax Element Sx

z d

T

l

dx dz

x

Sx Txz

S3 S2 –4

S1 4

8 12 16 20 24 28

(b) P

T2

(a)

(c)

Figure 6.15 Cantilevered round bar with torsion and transverse force applied to free end used in Example 6.8. Hamrock • Fundamentals of Machine Elements

S

Maximum Normal Stress Theory + σ2

Most suitable for fibrous brittle materials, glasses, and brittle materials in general.

Sut

Suc

Sut 0

+ σ1

Sut !1 ≥ ns Suc

Sut !3 ≤ ns

Figure 6.16 Graphical representation of maximum-normal-stress theory (MNST) for biaxial stress state. Hamrock • Fundamentals of Machine Elements

Internal Friction and Modified Mohr Theories σ2

Pure shear: σ1 = –σ2 MMT

Sut

IFT

Sut σ1

0 Sut

Suc

Figure 6.17 Internal friction theory and modified Mohr theory for failure prediction of brittle materials. Hamrock • Fundamentals of Machine Elements

Experimental Verification

σ2 Maximum normal stress

Steel Cast iron σ1

Cast iron Steel Copper Aluminum –1.0

σ2 σult Maximum distortion energy 1.0

Maximum shearing stress 0

1.0

σ1 σult

–1.0

(a)

(b)

Figure 6.18 Experimental verification of yield and fracture criteria for several materials. (a) Brittle fracture; (b) ductile yielding. Hamrock • Fundamentals of Machine Elements

Stress Analysis of Artificial Hip 0.462

0.38 r

0.59 r 0.90

5∞ 1.452

A

1.06 B

0.500 diam 0.25 r

A

C

B

0.115 0.06 r

C

0.931 4.54 r 1.255

36.5∞

2.858

5.39

3∞taper

Figure 6.19 Inserted total hip replacement.

Figure 6.20 Dimension of femoral implant (in inches).

Figure 6.21 Sections of femoral stem analyzed for static failure.


Chapter 7: Failure Prediction for Cyclic and Impact Loading All machines and structural designs are problems in fatigue because the forces of Nature are always at work and each object must respond in some fashion. Carl Osgood, Fatigue Design


On the Bridge!

Figure 7.1 “On the Bridge”, an illustration from Punch magazine in 1891 warning the populace that death was waiting for them on the next bridge. Note the cracks in the iron bridge. Hamrock • Fundamentals of Machine Elements

Methods to Maximize Design Life 1.

2.

3.

4.

By minimizing initial flaws, especially surface flaws. Great care is taken to produce fatigueinsusceptible surfaces through processes, such as grinding or polishing, that leave exceptionally smooth surfaces. These surfaces are then carefully protected before being placed into service. By maximizing crack initiation time. Surface residual stresses are imparted (or at least tensile residual stresses are relieved) through manufacturing processes, such as shot peening or burnishing, or by a number of surface treatments. By maximizing crack propagation time. Substrate properties, especially those that retard crack growth, are also important. For example, fatigue cracks propagate more quickly along grain boundaries than through grains (because grains have much more efficient atomic packing). Thus, using a material that does not present elongated grains in the direction of fatigue crack growth can extend fatigue life (e.g., by using cold-worked components instead of castings). By maximizing the critical crack length. Fracture toughness is an essential ingredient. (The material properties that allow for larger internal flaws are discussed in Chapter 6.)


Stress Cycle and Test Specimen

1 cycle 7 __

Tension +

σa 0

Compression –

Stress

σmax

3 16

σr

σm

σmin

0.30

Time

Figure 7.2 Variation in nonzero cyclic mean stress.

9 7–8 R

Figure 7.3 R.R. Moore machine fatigue test specimen. Dimensions in inches.


Cyclic Properties of Metals

Material

Condition

Yield strength Sy , MPa

1015 4340 1045 1045 1045 1045 4142 4142 4142 4142 4142

Normalized Tempered Q&Ta 80◦ F Q&T 306◦ F Q&T 500◦ F Q&T 600◦ F Q&T 80◦ F Q&T 400◦ F Q&T 600◦ F Q&T 700◦ F Q&T 840◦ F

228 1172 — 1720 1275 965 2070 1720 1340 1070 900

1100 2014 2024 5456 7075

Annealed T6 T351 H311 T6

97 462 379 234 469

Fracture Fatigue strength ductility ! σf coefficient, MPa "!f Steel 827 0.95 1655 0.73 2140 — 2720 0.07 2275 0.25 1790 0.35 2585 — 2650 0.07 2170 0.09 2000 0.40 1550 0.45 Aluminum 193 1.80 848 0.42 1103 0.22 724 0.46 1317 0.19

Fatigue strength exponent, a

Fatigue ductility exponent, α

-0.110 -0.076 -0.065 -0.055 -0.080 -0.070 -0.075 -0.076 -0.081 -0.080 -0.080

-0.64 -0.62 -1.00 -0.60 -0.68 -0.69 -1.00 -0.76 -0.66 -0.73 -0.75

-0.106 -0.106 -0.124 -0.110 -0.126

-0.69 -0.65 -0.59 -0.67 -0.52

Table 7.1 Cyclic properties of some metals. Hamrock • Fundamentals of Machine Elements

Fatigue Crack Growth

10-2

∆σ1

∆σ2 da dN

10

Regime B 1 mm/min

-4

da = C(∆K)m dN m 1

10-6 one lattice spacing per cycle

Number of cylces, N (a)

10-8

1 mm/hour

1 mm/day Regime C

1 mm/week

log ∆K (b)

Figure 7.4 Illustration of fatigue crack growth. (a) Size of a fatigue crack for two different stress ratios as a function of the number of cycles; (b) rate of crack growth, illustrating three regimes. Hamrock • Fundamentals of Machine Elements

Crack growth rate at 50 Hz

Regime A Crack growth rate da/dN (mm/cycle)

Crack length, a

∆σ2 > ∆σ1

Kc

Fatigued Part Cross-Section

Figure 7.5 Cross-section of a fatigued section, showing fatigue striations or beachmarks originating from a fatigue crack at B.


High Nominal Stress Mild stress concentration

Severe stress concentration

No stress concentration

Mild stress concentration

Tension-tension or tension-compression

No stress concentration

Low Nominal Stress Severe stress concentration

Rotational bending

Reversed bending

Unidirectional bending

Fatigue Fracture Surfaces

Beachmarks

Fracture surface

Figure 7.6 Typical fatigue fracture surfaces of smooth and notched cross-sections under different loading conditions and stress levels.


Fatigue stress ratio, Sf /Sut

Fatigue Strength of Ferrous Metals

1.0 0.9 0.8 Not broken

0.7 0.6 0.5 0.4 103

104

105 Number of cycles to failure, N′

106

107

Figure 7.7 Fatigue strengths as a function of number of loading cycles. (a) Ferrous alloys, showing clear endurance limit.


Fatigue Strength of Aluminum Alloys

Alternating stress, sa, ksi

80 70 60 50 40 35 30 25

Wr ou Per

20 18 16 14 12 10 8 7 6 5 103

ght

ma nen tm old c

ast

San dc ast

104

105 106 107 Number of cycles to failure, N′

108

109

Figure 7.7 Fatigue strengths as a function of number of loading cycles. (b) Aluminum alloys, with less pronounced knee and no endurance limit. Hamrock • Fundamentals of Machine Elements

Fatigue Strengths of Polymers 60 83103

40

Phenolic Epoxy

30

Diallyl-phthalate Alkyd

4

Nylon (dry)

2

6

20 10 0 103

PTFE

Alternating stress, sa, psi

Alternating stress, sa, MPa

50

Polycarbonate Polysulfone 0 104

105

106

107

Number of cycles to failure, N′

Figure 7.7 Fatigue strengths as a function of number of loading cycles. (c) Selected properties of assorted polymer classes. Hamrock • Fundamentals of Machine Elements

Endurance Limit vs. Ultimate Strength 1603103 Carbon steels Alloy steels Wrought irons

140 120 Endurance limit, S′e , psi

0.5

0.6 S_′_e_ = Su

0.4 100 x103 psi

100 80 60 40 20 0 0

20

40

60

80

100

120 140 160 180 Tensile strength, Sut, psi

200

220

240

260

280 3003103

Figure 7.8 Endurance limit as function of ultimate strength for wrought steels. Hamrock • Fundamentals of Machine Elements

Endurance Limit

Material Magnesium alloys Copper alloys Nickel alloys Titanium Aluminum alloys

Number of cycles 108 108 108 107 5 × 108

Relation Se! = 0.35Su 0.25Su < Se! < 0.5Su 0.35Su < Se! < 0.5Su 0.45Su < Se! < 0.65Su Se! = 0.40Su (Su < 48 ksi Se! = 19 ksi (Su ≥ 48 ksi

Table 7.2 Approximate endurance limit for various materials.


Notch Sensitivity Use these values with bending and axial loads  Steel,  Su, ksi (MPa) Use these values with torsion  as marked 1.0

180 (1241) 379) 1 ( 0 ) 20 120 (827 ) 965 2) 0( 80 (55 4 1 ) 4) 689 60 (41 0( 0 1 2) (55 4) 0 8 (41 ) 60 (345 50

Notch sensitivity, qn

0.8

0.6

Usage: K f = 1 + (Kc − 1)qn

Aluminum alloy (based on 2024–T6 data)

0.4

0.2

0

0

0.5

1.0

1.5 2.0 2.5 Notch radius, r, mm

3.0

3.5

4.0

0

0.02

0.04

0.06 0.08 0.10 Notch radius, r, in.

0.12

0.14

0.16

Figure 7.9 Notch sensitivity as function of notch radius for several materials and types of loading. Hamrock • Fundamentals of Machine Elements

Surface Finish Factor Ultimate strength in tension, Sut , ksi 60 1.0

80

100

140

160

Polished

0.8 Surface finish factor, kf

120

180

200

220

240

Ground

k f = eSutf

Machined or cold drawn

0.6

Manufacturing process Grinding Machining or cold drawing Hot rolling As forged

Hot rolled 0.4

As forged

0.2

0 0.4

Factor e MPa ksi 1.58 1.34 4.51 2.70 57.7 272.0

14.4 39.9

Exponent f -0.085 -0.265 -0.718 -0.995

Table 7.3 Surface finish factor. 0.6

0.8

1.0

1.2

1.4

1.6

Ultimate strength in tension, Sut , GPa

Figure 7.10 Surface finish factors for steel. (a) As function of ultimate strength in tension for different manufacturing processes. Hamrock • Fundamentals of Machine Elements

Surface finish factor, kf

Roughness Effect on Surface Finish Factor 1.0 0.9 0.8 0.7 0.6

Surface finish Ra, µin.

0.5 0.4

40

60

80

2000

1000 500

250

125

63

32

100 120 140 160 180 Ultimate strength in tension, Sut, psi

16 8

200

1 4 2

220

(b)

Figure 7.10 Surface finish factors for steel. (b) As function of ultimate strength and surface roughness as measured with a stylus profilometer. Hamrock • Fundamentals of Machine Elements

240x103

Reliability Factor

Probability of survival, percent 50 90 95 99 99.9 99.99

Relaibility factor, kr 1.00 0.90 0.87 0.82 0.75 0.70

Table 7.4 Reliability factor for six probabilities of survival.


Shot Peening 200

300

200

150

Peened - smooth or notched

100

690 Not peened - smooth

345

50

70

483 Al 7050-T7651 Ti-6Al-4V

414 Shot peened 345

50

276

40

207

Polished

Machined

0

690 1380 2170 Ultimate tensile strength, Sut, (MPa) (a)

0

30 20

138 Not peened - notched (typical machined surface)

60

104

105 106 107 Number of cycles to failure, N' (b)

Figure 7.11 The use of shot peening to improve fatigue properties. (a) Fatigue strength at two million cycles for high strength steel as a function of ultimate strength; (b) typical S-N curves for nonferrous metals. Hamrock • Fundamentals of Machine Elements

108

ksi

1035

100

Alternating stress, σa, MPa

1380

ksi

Fatigue strength at two million cycles (MPa)

MPa

Example 7.4

r = 2.5 mm

P

25 mm

P

25 mm 25 mm

30 mm 25 mm r = 2.5 mm

P

P (a)

(b)

Figure 7.12 Tensile loaded bar. (a) Unnotched; (b) notched.


Influence of Non-Zero Mean Stress Gerber Line ! "2 K f ns!a ns!m + =1 Se Sut

Syt Alternating stress, σa

Yield line

Se

Gerber line Goodman line Soderberg line

0 Mean stress, σm

Syt

Sut

Goodman Line ! "2 K f !a !m 1 + = Se Sut ns Soderberg Line !

"2

K f !a !m 1 Figure 7.13 Influence of nonzero + = mean stress on fatigue life for tensile Se Syt ns loading as estimated by four empirical relationships. Hamrock • Fundamentals of Machine Elements

Modified Goodman Diagram Su

Su +σ B

Sy

C

Sy σmax

N Se /Kf σmax

σm

L

A

D 45∞

σmin

M

H 0

–σm

Sy

Su

σm

σmin E

– Se /Kf –σ

45∞

F

G a

–Sy b

c

d

Figure 7.14 Complete modified Goodman diagram, plotting stress as ordinate and mean stress as abscissa.


Modified Goodman Criterion Line

Equation ! " Se Se = + σm 1 − Kf S u Kf

AB

σmax

BC

σmax = Sy

CD

σmin = 2σm − Sy

DE

σmin =

!

EF

σmin = σm −

FG

σmin = −Sy

GH

σmax = 2σm + Sy

HA

σmax = σm +

Se 1+ Kf Su Se Kf

Se Kf

"

σm −

Se Kf

Range Sy − Se /Kf 0 ≤ σm ≤ Se 1− Kf S u Se Sy − Kf ≤ σm ≤ Sy Se 1− Kf S u Se Sy − Kf ≤ σm ≤ Sy Se 1− Kf S u Se Sy − Kf 0 ≤ σm ≤ Se 1− Kf S u Se − S y ≤ σm ≤ 0 Kf Se −Sy ≤ σm ≤ − Sy Kf Se −Sy ≤ σm ≤ − Sy Kf Se − S y ≤ σm ≤ 0 Kf

Table 7.5 Equations and range of applicability for construction of complete modified Goodman diagram.


Modified Goodman Criterion Region in Fig. 7.14

Failure equation

a

σmax − 2σm = Sy /ns

b

σmax − σm =

c

σmax + σm

d

σmax =

Sy ns

!

Se ns Kf

Se Kf Su

" −1 =

Se ns Kf

Validity limits of equation Se −Sy ≤ σm ≤ − Sy Kf Se − Sy ≤ σ m ≤ 0 Kf Se Sy − Kf 0 ≤ σm ≤ Se 1− Kf Su Se Sy − Kf ≤ σm ≤ Sy Se 1− Kf Su

Table 7.6 Failure equations and validity limits of equations for four regions of complete modified Goodman diagram.


Example 7.7

F

90 B

Stress, ksi

60

30

C 45°

A D

120

0

–30

Mean stress, σm, ksi E

45

90

Figure 7.15 Modified Goodman diagram for Example 7.7. Hamrock • Fundamentals of Machine Elements

Alternating Stress Ratio for Cast Iron

Alternating stress ratio, σa/Su

1.5 1.0 0.5

S —e ≈ (0.4)(0.9) = 0.36 Su

0 –4.0 –3.5 –3.0 –2.5 –2.0 –1.5 –1.0 –0.5 Mean stress ratio, σm/Su

0

0.5

1.0

Figure 7.16 Alternating stress ratio as function of mean stress ratio for axially loaded cast iron.


Properties of Mild Steel

80

80 /S u Ratio S y gth S u n e r t s e U lti m at

60

60

Total elongation

40

Yield strength S y

20

0 –6 10

40

20

10–5

10–4

10–3

10–2

10–1

1

10

102

Sy/Su, percent

100

0

Elongation, percent

Ultimate and yield stresses, Su and Sy, ksi

100

103

Average strain rate, s–1

Figure 7.17 Mechanical properties of mild steel at room temperature as a function of strain rate. Hamrock • Fundamentals of Machine Elements

Example 7.10

y V 2 ft 1.5 in. 5 ft (a)

M P

x

18 in. (b)

(c)

Figure 7.18 Diver impacting diving board, used in Example 7.10. (a) Side view; (b) front view; (c) side view showing forces and coordinates. Hamrock • Fundamentals of Machine Elements

V Sheer force, N

Brake Stud Design Analysis

P

Machine frame

A–A

Moment, N–m

M Px

(a)

R = 0.375 in.

1.375 in.

P

1 in.

2.25 in.

Shoulder

x

Stress, Pa

3.0625 in.

σa σm Time (b)

Figure 7.19 Dimensions of existing brake drum design.

Figure 7.20 Press brake stud loading. (a) Shear and bending-moment diagrams for applied load; (b) stress cycle. Hamrock • Fundamentals of Machine Elements

Chapter 8: Lubrication, Friction and Wear

“...among all those who have written on the subject of moving forces, probably not a single one has given sufficient attention to the effect of friction in machines...” Guillaume Amontons (1699)


Conformal and Nonconformal Surfaces Journal

Fluid film

Outer ring

Sleeve

Figure 8.1 Conformal surfaces.

Rolling element

Inner ring

Figure 8.2 Nonconformal surfaces.


Contacting Solids

W

x

Solid a ray

rax

y x rby

rbx

Solid b

Figure 8.3 Geometry of contacting solids.

y W


Radii of Curvature Sphere r

rax = r ray = r

Cylinder rax

Conic frustum rax

ray

ray

Barrel shape rax

Concave shape rax

ray

–ray

(a) Thrust –rby

Ri

Radial inner

Radial outer

–rby

–rby

rbx

–rbx

rbx (b) Thrust –rby Ri β α−β rbx (c)

Cylindrical inner –rby

Cylindrical outer rby

rbx

–rbx

Figure 8.4 Sign designations for radii of curvature of various machine elements. (a) Rolling elements; (b) ball bearing races; (c) rolling bearing races.


Ellipsoidal Contact

D ––x 2 x

pmax

pH = pmax

p y

!

"

2x 1− Dx

Dy –– 2

pmax =

#2

"

2y − Dy

6W !DxDy

Figure 8.5 Pressure distribution in an ellipsoidal contact.


#2$1/2

Elliptic Integrals

y

10

Elliptic integrals, E and F

Elliptic integral E Elliptic integral F Ellipticity parameter ke

4

8

3

6

2

4

1

2

0

4

8

12 16 20 24 Radius ratio, αr

28

32

Figure 8.6 Variation of ellipticity parameter and elliptic integrals of first and second kinds as function of radius ratio.

Ellipticity parameter, ke

5

Dy –– 2 x

y Dy –– 2 x

Dx –– 2

1 ≤ αr ≤ 100 2/π ke = αr F = π2 + qa ln αr where qa = π2 − 1 E = 1 + αqar

Dx –– 2

0.01 ≤ αr ≤ 1.0 2/π ke = αr F = π2 − qa ln αr where qa = π2 − 1 E = 1 + qa αr

Table 8.1 Simplified equations.


Hydrodynamic Lubrication

W

W

W

ub

W

wa pa ub

h min

ps (a)

Conformal surfaces pmax ≈ 5 MPa h min = f (W, ub, η0, Rx , Ry) > 1 µm No elastic effect

Figure 8.7 Characteristics of hydrodynamic lubrication.

(b)

(c)

Figure 8.8 Mechanism of pressure development for hydrodynamic lubrication. (a) Slider bearing; (b) squeeze film bearing; (c) externally pressurized bearing.


Elastohydrodynamic Lubrication W

W

ub

ub

h min Nonconformal surfaces High-elastic-modulus material (e.g., steel) pmax ≈ 0.5 to 4 GPa h min = f (W, ub, η0, Rx , Ry , E′, ξ) > 0.1 µm Elastic and viscous effects both important

Figure 8.9 Characteristics of hard elastohydrodynamic lubrication.

h min Nonconformal surfaces (e.g., nitrite rubber) pmax ≈ 0.5 to 4 MPa h min = f (W, ub, η0 , Rx , Ry , E ′) ≈ 1 µm Elastic effects predominate

Figure 8.10 Characteristics of soft elastohydrodynamic lubrication.


Regimes of Lubrication Boundary film Bulk lubricant

(a)

(b)

(c)

Figure 8.11 Regimes of lubrication. (a) Fluid film lubrication surfaces separated by bulk lubricant film. This regime is sometimes further classified as thick or thin film lubrication; (b) partial lubrication - both bulk lubricant and boundary film play a role; (c) boundary lubrication - performance depends essentially on boundary film. Hamrock • Fundamentals of Machine Elements

Friction and Lubrication Condition 10

Unlubricated

Coefficient of friction, µ

1

Boundary Elastohydrodynamic

10 –1

10 –2 Hydrodynamic 10 –3

10 –4

Figure 8.12 Bar diagram showing coefficient of friction for various lubrication conditions.


Wear Rate and Lubrication

Wear rate

Seizure Severe wear

Hydrodynamic Unlubricated Elastohydrodynamic Boundary

Figure 8.13 Wear rate for various lubrication regimes.

Relative load


Surface Roughness

z

Mean reference line

Centerline average roughness Ra =

1 N|zi| ! N i=1

Root-mean-square roughness Figure 8.14 Surface profile showing surface height variation relative to mean reference line.

Rq =

!

1 Nz2i ! N i=1


"1/2

Typical Surface Roughness Arithmetic average surface roughness, Ra µm µin. Processes Sand casting; hot rolling Sawing Planing and shaping Forging Drilling Milling Boring; turning Broaching; reaming; cold rolling; drawing Die casting Grinding, coarse Grinding, fine Honing Polishing Lapping Components Gears Plain bearings – journal (runner) Plain bearings – bearing (pad) Rolling bearings - rolling elements Rolling bearings - tracks

12.5-25 3.2-25 0.8-25 3.2-12.5 1.6-6.3 0.8-6.3 0.4-6.3 0.8-3.2 0.8-1.6 0.4-1.6 0.1-0.4 0.03-0.4 0.02-0.2 0.005-0.1

500-1000 128-1000 32-1000 128-500 64-250 32-250 16-250 32-128 32-64 16-64 4-16 1.2-16 0.8-8 0.2-4

0.25-10 0.12-0.5 0.25-1.2 0.015-.12 0.1-0.3

10-400 5-20 10-50 0.6-5 4-12

Table 8.2 Typical arithmetic average surface roughness for various manufacturing processes and machine components. Hamrock • Fundamentals of Machine Elements

Viscosity ub

Friction force F

ub

u

Area A

h

F/A shear stress != = ub/h shear strain rate

z

Figure 8.15 Slider bearing illustrating absolute viscosity.


Viscosity Conversion Factors To convert from cP kgf·s/m2 N·s/m2 reyn, or lb·s/in.2

To cP 1 9.807 × 103 103 6.9 × 106

kgf·s/m2

N·s/m2 Multiply by 1.02 × 10−4 10−3 1 9.807 −1 1.02 × 10 1 7.03 × 102 6.9 × 103

lb·s/in.2 1.45 × 10−7 1.422 × 10−3 1.45 × 10−4 1

Table 8.3 Absolute viscosity conversion factors.


–50

0

100

200

400

600 800 1000

LB 550 X 103

SAE 10

Viscosity Data

SAE 70 10–4 Polypropylene glycol derivatives

LB 100 X Polymethyl siloxanes (silicones)

DC 500 A 102

Residuum (specific gravity, 0.968)

10–5

Fluorolube light grease Di(n-butyl) sebacate

101

100

Crude oil (specific gravity, 0.855) Kerosene

DC 200 E Halocarbons Fluorolube FCD–331

Mercury

Gasoline (specific gravity, 0.748)

10–6

10–7

Octane

Water plus 23% NaCl

Gasoline (specific gravity, 0.680)

Navy Symbol 2135

Water 20.7 MPa (3000 psi)

10–1

10–8

Superheated steam (14.7 psig) Air 10–2 –50

Saturated steam Hydrogen 0

Absolute viscosity, η, lb-s/in.2

Absolute viscosity, η, cP

Temperature, tm, °F

100 Temperature, tm, °C

200

6.9 MPa (1000 psi) 300 400 500

Figure 8.16 Absolute viscosities of a number of fluids for a wide range of temperatures.

10–9


Viscosity of Selected Fluids Fluid

Advanced ester Formulated advanced ester Polyalkyl aromatic Synthetic paraffinic oil (lot 3) Synthetic paraffinic oil (lot 4) Synthetic paraffinic oil (lot 2) plus antiwear additive Synthetic paraffinic oil (lot 4) plus antiwear additive C-ether Superrefined napthenic mineral oil Synthetic hydrocarbon (traction fluid) Fluorinated polyether

Temperature, tm , ◦ C 38 99 149 38 99 149 Absolute viscosity Kinematic viscosity at p = 0 at p = 0, ηk , η0 , cP m2 /s 25.3 4.75 2.06 2.58 × 10−5 0.51 × 10−5 0.23 × 10−5 27.6 4.96 2.15 2.82 × 10−5 0.53 × 10−5 0.24 × 10−5 25.5 4.08 1.80 3.0 × 10−5 0.50 × 10−5 0.23 × 10−5 414 34.3 10.9 49.3 × 10−5 4.26 × 10−5 1.4 × 10−5 375 34.7 10.1 44.7 × 10−5 4.04 × 10−5 1.3 × 10−5 −5 −5 370 32.0 9.93 44.2 × 10 4.00 × 10 1.29 × 10−5 375

34.7

10.1

44.7 × 10−5

4.04 × 10−5

1.3 × 10−5

29.5 68.1 34.3 181

4.67 6.86 3.53 20.2

2.20 2.74 1.62 6.68

2.5 × 10−5 7.8 × 10−5 3.72 × 10−5 9.66 × 10−5

0.41 × 10−5 0.82 × 10−5 0.40 × 10−5 1.15 × 10−5

0.20 × 10−5 0.33 × 10−5 0.19 × 10−5 0.4 × 10−5

Table 8.4 Absolute and kinematic viscosities of various fluids at atmospheric pressure and different temperatures.


Piezoviscous Properties of Fluids Fluid

Advanced ester Formulated advanced ester Polyalkyl aromatic Synthetic paraffinic oil (lot 3) Synthetic paraffinic oil (lot 4) Synthetic paraffinic oil (lot 2) plus antiwear additive Synthetic paraffinic oil (lot 4) plus antiwear additive C-ether Superrefined napthenic mineral oil Synthetic hydrocarbon (traction fluid) Fluorinated polyether

Temperature, tm , ◦ C 38 99 149 Pressure-viscosity coefficient, ξ, m2 /N 1.28 × 10−8 0.987 × 10−8 0.851 × 10−8 1.37 × 10−8 1.00 × 10−8 0.874 × 10−8 1.58 × 10−8 1.25 × 10−8 1.01 × 10−8 1.77 × 10−8 1.51 × 10−8 1.09 × 10−8 1.99 × 10−8 1.51 × 10−8 1.29 × 10−8 1.81 × 10−8 1.37 × 10−8 1.13 × 10−8 1.96 × 10−8

1.55 × 10−8

1.25 × 10−8

1.80 × 10−8 2.51 × 10−8 3.12 × 10−8 4.17 × 10−8

0.980 × 10−8 1.54 × 10−8 1.71 × 10−8 3.24 × 10−8

0.795 × 10−8 1.27 × 10−8 0.939 × 10−8 3.02 × 10−8

Table 8.5 Pressure-viscosity coefficients of various fluids at different temperatures. Hamrock • Fundamentals of Machine Elements

Temperature, °F 32 50 10 54 3 2

75

100

125

150

175

200

225

250

275 10-3

1

10-4

.5 .4 .3 .2 .1

10-5

.06 .04 .03 .02 SAE 70 .01

10-6

60 50

.005 .004

40 30

.003

3×10-7 20 10

.002

Absolute viscosity, η, reyn

Absolute viscosity, η, N-s/m2

Viscosity of Single Grade SAE Oils

0

20

30

40

50

60 70 80 Temperature, °C

90 100 110 120 130 140

Figure 8.17 Absolute viscosities of SAE lubricating oils at atmospheric pressure. (a) Single grade oils.


Temperature, °F 32 50 10 54 3 2

75

100

125

150

175

200

225

250

275 10-3

1

10-4

.5 .4 .3 .2 .1

10-5

.06

Absolute viscosity, η, reyn

Absolute viscosity, η, N-s/m2

Viscosity of Multigrade SAE Oils

20W-50

.04 .03

10W-30

.02

20W-40 .01 10-6 20W

.005

5W-30

.004

10W .003

.002

Figure 8.17 Absolute viscosities of SAE lubricating oils at atmospheric pressure. (b) Multigrade oils.

3×10-7

0

20

30

40

50

60 70 80 Temperature, °C

90 100 110 120 130 140


Viscosity of SAE Single-Grade Oils C2 ! = C1 exp tF + 95 C2 ! = C1 exp 1.8tC + 127 SAE Grade 10 20 30 40 50 60

(English units) (SI units)

Constant C1 reyn N-s/m2 1.58 × 10−8 1.09 × 10−4 1.36 × 10−8 9.38 × 10−5 1.41 × 10−8 9.73 × 10−5 1.21 × 10−8 8.35 × 10−5 1.70 × 10−8 1.17 × 10−4 1.87 × 10−8 1.29 × 10−4

Constant C2 1157.5 1271.6 1360.0 1474.4 1509.6 1564.0

Table 8.6 Curve fit data for SAE single-grade oils. Hamrock • Fundamentals of Machine Elements

Friction

W W

Coulomb Friction Law: F

F µ= W

F

(a)

(b)

Figure 8.18 Friction force in (a) sliding and (b) rolling.


Coefficient of Friction Data Coefficient of friction, µ Self-mated metals in air Gold Silver Tin Aluminum Copper Indium Magnesium Lead Cadmium Chromium Pure metals and alloys sliding on steel (0.13% carbon) in air Silver Aluminum Cadmium Copper Chromium Indium Lead Copper - 20% lead Whitemetal (tin based) Whitemetal (lead based) α-brass (copper - 30% zinc) Leaded α/β brass (copper - 40% zinc) Gray cast iron Mid steel (0.13% carbon)

2.5 0.8-1 1 0.8-1.2 0.7-1.4 2 0.5 1.5 0.5 0.4 0.5 0.5 0.4 0.8 0.5 2 1.2 0.2 0.8 0.5 0.5 0.2 0.4 0.8

Table 8.7 Typical coefficients of friction for combinations of unlubricated metals in air. Hamrock • Fundamentals of Machine Elements

Abrasive and Adhesive Wear Archard Wear Law:

W

k1W L v= 3H

Small spot welds Transferred surface of softer metal

θ F

Asperities

L

Figure 8.19 Conical asperity having mean angle θ plowing through a softer material. This action simulates abrasive wear.

Intimate contact between metals of two opposing surfaces

Figure 8.20 Adhesive wear model.


Wear Coefficient Data

Rubbing materials Gold on gold Copper on copper Mild steel on mild steel Brass on hard steel Lead on steel Polytetrafluoroethylene (teflon) on steel Stainless steel on hard steel Tungsten carbide on tungsten carbide Polyethylene on steel

Coefficient of friction, µ 2.5 1.2 0.6 0.3 0.2 0.2 0.5 0.35 0.5

Adhesive wear coefficient, k1 0.1-1 0.01-0.1 10−2 10−3 2 × 10−5 2 × 10−5 2 × 10−5 10−6 10−8 − 10−7

Table 8.8 Coefficients of rubbing friction and adhesive wear constant for selected rubbing materials.


(a)

Fatigue Wear

(b) Periodic applied load

(c)

Subsurface defects

(d) Coalescence of defects

Fatigue spall

Wear particle

Figure 8.21 Fatigue wear simulation. (a) Machine element surface is subjected to cyclic loading; (b) defects and cracks develop near the surface; (c) the cracks grow and coalesce, eventually extending to the surface until (d) a wear particle is produced, leaving a fatigue spall in the material


Chapter 9: Columns

And as imagination bodies forth the forms of things unknown, The poet’s pen turns them to shapes And gives to airy nothingness a local habitation and a name. William Shakespeare A Midsummer Night’s Dream Hamrock • Fundamentals of Machine Elements

Equilibrium Regimes

P P

P Re

0

0

0 We

(a)

Re

We

We (b)

(c)

Figure 9.1 Depiction of equilibrium regimes. (a) Stable; (b) neutral; (c) unstable.


Example 9.1 θ l

P θ

mag

Figure 9.2 Pendulum used in Example 9.1.


Column with Pinned Ends P x P

y P′

y

l

M

dy

dx

V

ds

y P

P

P (a)

(b)

(c)

Figure 9.3 Column with pinned ends. (a) Assembly; (b) deformation shape; (c) load acting.


P

Buckling of Columns

y x

Euler Equation:

n2!2EI Pcr = l2

P

Figure 9.4 Buckling of rectangular section. Hamrock • Fundamentals of Machine Elements

Column Effective Lengths End condition description

Both ends pinned

One end pinned and one end fixed

Both ends fixed

One end fixed and one end free P

P

P

P

l

le = 0.7l Illustration of end condition

l = le

le = 0.5l

l

l

le = 2l

P

P

P

P

Theoretical effective column length

le = l

le = 0.7l

le = 0.5l

le = 2l

AISC (1989)– recommended effective column length

le = l

le = 0.8l

le = 0.65l

le = 2.1l

Table 9.1 Effective length for four end conditions. Hamrock • Fundamentals of Machine Elements

Buckling for Different Slenderness Ratio Critical Slenderness Ratio Yielding Yield strength

A

Normal stress, σ

Buckling Johnson equation T

# ! " le 2E!2 Cc = = rg E Sy

Euler Equation Euler equation

Safe

(Pcr )E "2 E (!cr )E = = A (le/rg)2

AISC Equations (le /rg) T Slenderness ratio, le /rg

Figure 9.5 Normal stress as a function of slenderness ratio.

Johnson Parabola (Pcr )J (!cr )J = = A

Sy2 Sy − 2 4" E


! "2 le rg

AISC Buckling Criterion

Elastic Buckling Inelastic Buckling

!all =

12"2E 23 (le/rg)2

!all =

#$ ! " (le /rg )2 1 − 2C2 Sy

Allowable Stress Reduction

c

n!

5 3 (le/rg) (le/rg)3 n! = + − 3 8Cc 8Cc3


50 mm

Example 9.3

27.6 mm

41.7 mm

Pcr = 738.6 N Pcr = 4094 N (a)

(b)

50 mm 24.5 mm

43.6 mm

Pcr = 773.5 N Pcr = 5672 N (c)

(d)

Figure 9.6 Cross-sectional areas, drawn to scale, from results of Exampe 9.3, aas well as critical buckling load for each cross-sectional area.


P

P

M′ = Pe

e A

e

y

x

x

x

y l

Eccentrically Loaded Column

P

y

M y

l

P (c)

B

Secant Equation: !

" # $ % le P ymax = e sec −1 2 EI

M′ = Pe

e P

P x (a)

x (b)

Figure 9.7 Eccentrically loaded column. (a) Eccentricity; (b) statically equivalent bending moment; (c) free-body diagram through arbitrary section.


Chapter 10: Stresses and Deformations in Cylinders

In all things, success depends on previous preparation. And without such preparation there is sure to be failure. Confucius, Analects


Classes of Fit Class 1

Description Loose

Type Clearance

2

Free

Clearance

3

Medium

Clearance

4

Snug

Clearance

5

Wringing

Interference

6

Tight

Interference

7

Medium

Interference

8

Heavy force or shrink

Interference

Applications Where accuracy is not essential, such as in building and mining equipment In rotating journals with speeds of 600 rpm or greater, such as in engines and some automotive parts In rotating journals with speeds under 600 rpm, such as in accurate machine tools and precise automotive parts Where small clearance is permissible and where moving parts are not intended to move freely under load Where light tapping with a hammer is necessary to assemble the parts In semipermanent assemblies suitable for drive or shring fits on light sections Where considerable pressure is needed to assemble and for shrink fits of medium sections; suitable for press fits on generator and motor armatures and for car wheels Where considerable bonding between surfaces is required, such as locomotive wheels and heavy crankshaft disks of large engines

Table 10.1 Classes of fit. Hamrock • Fundamentals of Machine Elements

Recommended Tolerences for Classes of Fit

Class 1 2 3 4 5 6 7 8

Allowance, a 0.0025d2/3 0.0014d2/3 0.0009d2/3 0.000 — — — —

Interference, δ — — — — 0.000 0.00025d 0.0005d 0.0010d

Hub tolerance, tlh 0.0025d1/3 0.0013d1/3 0.0008d1/3 0.0006d1/3 0.0006d1/3 0.0006d1/3 0.0006d1/3 0.0006d1/3

Shaft tolerance, tls 0.0025d1/3 0.0013d1/3 0.0008d1/3 0.0004d1/3 0.0004d1/3 0.0006d1/3 0.0006d1/3 0.0006d1/3

Table 10.2 Recommended tolerances in inches for classes of fit.


Recommended Tolerances for Classes of Fit

Class 1 2 3 4 5 6 7 8

Allowance, a 0.0073d2/3 0.0041d2/3 0.0026d2/3 0.000 — — — —

Interference, δ — — — — 0.000 0.00025d 0.0005d 0.0010d

Hub tolerance, tlh 0.0216d1/3 0.0112d1/3 0.0069d1/3 0.0052d1/3 0.0052d1/3 0.0052d1/3 0.0052d1/3 0.0052d1/3

Shaft tolerance, tls 0.0216d1/3 0.0112d1/3 0.0069d1/3 0.0035d1/3 0.0035d1/3 0.0052d1/3 0.0052d1/3 0.0052d1/3

Table 10.3 Recommended tolerances in millimeters for classes of fit.


Maximum and Minimum Shaft and Hub Diameters

Type of fit Clearance Interference

Hub diameter Maximum, Minimum dh.max dh,min d + tlh d d + tlh d

Shaft diameter Maximum Minimum ds,max ds,min d−a d − a − tls d + δ + tls d+δ

Table 10.4 Maximum and minimum diameters of shaft and hub for two types of fit.


Internally Pressurized Thin-Walled Cylinder Stresses in ThinWalled Cylinders: S1

r

z Q l

z

di pi

r (a)

!r = 0

S2

th

Q

pi r !" = th

(b)

Figure 10.1 Internally pressurized thinwalled cylinder. (a) Stress element on cylinder; (b) stresses acting on element.

pi r !z = 2th


Thin-Walled Cylinder - Front View SQthdl dQ piri dQdl pi

dQ/2

ri ro

th

Figure 10.2 Front view of internally pressurized, thin-walled cylinder. Hamrock • Fundamentals of Machine Elements

Thick-Walled Cylinder - Front View

SQ dQ/2 dQ

Sr + dSr

Sr

SQ r

ro

SQ dQ/2 dr

dQ/2

dQ/2

SQ sin dQ/2 SQ dQ/2

ri pi (b) po

(a)

Figure 10.3 Complete front view of thick-walled cylinder internally and externally pressurized. (a) With stresses acting on cylinder; (b) with stresses acting on element. Hamrock • Fundamentals of Machine Elements

dDQ ––– dr

1 $Dr – ––– r $Q

Cylindrical Element $Dr Dr + ––– $r dr

rdQ DQ + $D –––Q dQ $Q

DQ dr Dr

r dQ

dQ

Q

Initial element Deformed element

Figure 10.4 Cylindrical polar element before and after deformation.


Pressurized Cylinders Normal stress, S

Normal stress, S

po

SQ Sr

ri Tension

ri

Radius, r

Compression

Tension Compression

ro

Radius, r

pi

ro SQ

Sr

Figure 10.5 Internally pressurized, thick-walled cylinder showing circumferential (hoop) and radial stresses for various radii.

po

Figure 10.6 Externally pressurized, thick-walled cylinder showing circumferential (hoop) and radial stresses for various radii.


Rotating Cylinders Normal stress, S

Normal stress, S

SQ SQ Tension

Sr

Sr Radius, r

Radius, r

ri

Compression

ro

Figure 10.7 Stresses in rotating cylinder with central hole and no pressurization.

ro

Figure 10.8 Stresses in rotating solid cylinder and no pressurization.


Press Fit - Side View

ri δrs

δrs

l

δrh

δrh rf ro

Figure 10.9 Side view showing interference of press fit of hollow shaft to hub. Hamrock • Fundamentals of Machine Elements

rf

Press Fit - Front View

ro

ri

(a)

pf

rf

rf

ri

pf ro

(b)

Figure 10.10 Front view showing (a) cylinder assembled with an interference fit and (b) hub and hollow shaft disassembled (also showing interference pressure).


Chapter 11: Shafts

When a man has a vision, he cannot obtain the power from that vision until he has performed it on the Earth for the people to see. Black Elk Hamrock • Fundamentals of Machine Elements

Shaft Design Procedure 1.

2.

Develop a free-body diagram by replacing the various machine elements mounted on the shaft by their statically equivalent load or torque components. To illustrate this, Fig. 11.1(a) shows two gears exerting forces on a shaft, and Fig. 11.1(b) then shows a free-body diagram of the gears acting on the shaft. Draw a bending moment diagram in the x-y and x-z planes as shown in Fig. 11.1(c). The resultant internal moment at any section along the shaft may be expressed as

Mx = 3.

4. 5. 6.

!

2 + M2 Mxy xz

Develop a torque diagram as shown in Fig. 11.1(d). Torque developed from one power-transmitting element must balance the torque from other power-transmitting elements. Establish the location of the critical cross-section, or the x location where the torque and moment are the largest. For ductile materials use the maximum-shear-stress theory (MSST) or the distortion-energy theory (DET) covered in Sec. 6.7.1. For brittle materials use the maximum-normal-stress theory (MNST), the internal friction theory (IFT), or the modified Mohr theory (MMT), covered in Sec. 6.7.2.


z Az A

Shaft Assembly

y P1

T P1

Bz

Ay

B

x T

By P2

P2 (a)

(b)

My

Mz

x

x

Moment diagram caused by loads in x-z plane

Moment diagram caused by loads in x-y plane (c)

Tx

T x (d)

Figure 11.1 Shaft assembly. (a) Shaft with two bearings at A and B and two gears with resulting forces P1 and P2; (b) free-body diagram of torque and forces resulting from assembly drawing; (c) moment diagram in xz and xy planes; (d) torque diagram.


40 mm

160 mm

Example 11.1 Chain

(a) y

RA

40 mm

160 mm

M x

RB P (b)

M

x

Figure 11.2 Illustration for Example 11.1. (a) Chain drive assembly; (b) free-body diagram; (c) bending moment diagram.

–80 N-m (c)


z

z 0.250 m 0.250 m

A 0.050 m

y

C

0.250 m 0.150 m 300 N B

150 N

550 N

500 N

200 N 475 N 950 N

0.075 m D

Example 11.2

x

650 N 475 N

(a) A

0.150 m

7.5 N-m

x 400 N

0.250 m

y

7.5 N-m

(b)

C

B

D

A

0.250 m 0.250 m 0.150 m 475 N 950 N 475 N

C

0.250 m 150 N

My(N-m)

B

D

0.250 m 0.150 m 650 N 500 N

Mz (N-m)

My = 118.75 N-m Mz = 37.5 N-m

My = 75 N-m

x(m)

x(m)

(c)

(d) C 7.5 N-m

A

B

D 7.5 N-m

0.250 m

0.250 m

0.150 m

Tx(N-m)

x(m) –7.5

Figure 11.3 Illustration for Example 11.2. (a) Assembly drawing; (b) free-body diagram; (c) moment diagram in xz plane; (d) moment diagram in xy plane; (e) torque diagram.

(e)


Fluctuating Stresses on Shafts TFa Tm pKfsTa

Sm pKfSa

Tm pKfsTa

Sm pKfSa

Se/2ns D

(a)

G

y

SF

A

0 Sm +KfSa

TF

Tm +KfsTa

F

A cos F

A sin F

x

Tm +KfsTa

H

F Sy/2ns

Figure 11.5 Soderberg line for shear stresses.

(b)

Figure 11.4 Fluctuating normal and shear stresses acting on shaft. (a) Stresses acting on rectangular element; (b) stresses acting on oblique plane at angle ϕ. Hamrock • Fundamentals of Machine Elements

TFm

4( ~ B) 2

Derivation in Eq. (11.29)

(A ~ )2 +

sin 2! A˜ = tan 2! = cos 2! 2B˜

2φ ~ 2B

Figure 11.6 Illustration of relationship given in Eq. (11.29). Hamrock • Fundamentals of Machine Elements

~ A

Shaft Design Equations Using DET and Soderberg Criteria !d 3Sy ns = !" #2 " #2 Sy Sy 3 32 Mm + K f Ma + Tm + K f sTa Se 4 Se 

32ns  d= !Sy

#$

Sy Mm + K f Ma Se

%2

$

Sy 3 + Tm + K f sTa 4 Se

%2

1/3 


Example 11.4

d3

d2 d1

Figure 11.7 Section of shaft in Example 11.4.


Critical Frequency of Shafts

!=

!

Single Mass k = ma

k

"

W /y = W /g

!

g "

Multiple Mass - Raleigh Equation ma y W(t)

Figure 11.8 Simple singlemass system.

!cr =

!

g "i=1,...,n Wi#i,max "i=1,...,n Wi#2i,max

Multiple Mass - Dunkerly Equation 1 1 1 1 = 2 + 2 +···+ 2 2 !cr !1 !2 !n


Example 11.5

PA

y x1

x2 A

R1

PB x3

x

B R2

Figure 11.9 Simply supported shaft arrangement for Example 11.5.


Keys and Pins

Figure 11.10 Illustration of keys and pins. (a) Dimensions of shaft with keyway in shaft and hub; (b) square parallel key; (c) flat parallel key; (d) tapered key; (e) tapered key with Gib head, or Gib-head key. The Gib head assists in removal of the key; (f) round key; (g) Woodruff key with illustration of mountingl (h) pin, which is often grooved. The pin is slightly larger than the hole so that friction holds the pin in place; (i) roll pin. Elastic deformation of the pin in the smaller hole leads to friction forces that keep the pin in place. Hamrock • Fundamentals of Machine Elements

w w

Plain Parallel Keys

w – 2 w

l w

d

Shaft diameter in.

Key width in.

0.500 0.625 0.750 0.875 1.000 1.125 1.250 1.375 1.500 1.675 1.750 1.875

0.125 0.1875 0.1875 0.1875 0.25 0.25 0.25 0.3125 0.375 0.375 0.375 0.50

Distance from keyseat to opposite side of shaft, in. 0.430 0.517 0.644 0.771 0.859 0.956 1.112 1.201 1.289 1.416 1.542 1.591

Shaft diameter in.

Key width in.

2.000 2.250 2.500 2.750 3.000 3.250 3.500 3.750 4.000 4.500 5.000 6.000

0.50 0.50 0.625 0.625 0.75 0.75 0.875 0.875 1.00 1.00 1.25 1.50

Distance from keyseat to opposite side of shaft, in. 1.718 1.972 2.148 2.402 2.577 2.831 3.007 3.261 3.437 3.944 4.296 5.155

Table 11.1 Dimensions of selected square plain parallel stock keys.


Tapered Keys

Shaft diameter in.

0.5-0.5625 0.625-0.875 0.9375-1.25 1.3125-1.375 1.4375-1.75 1.8125-2.25 2.3125-2.75 2.875-3.25 3.375-3.75 3.875-4.5 4.75-5.5 5.75-6

Square type Width Heighta w h in. in. 0.125 0.125 0.1875 0.1875 0.25 0.25 0.3125 0.3125 0375 0.375 0.5 0.5 0.625 0.625 0.75 0.75 0.875 0.875 1.00 1.00 1.25 1.25 1.50 1.50

Flat type Width Heighta w h in. in. 0.125 0.09375 0.1875 0.125 0.25 0.1875 0.3125 0.25 0.375 0.25 0.5 0.375 0.625 0.4375 0.75 0.50 0.875 0.625 1.00 0.75 1.25 0.875 1.50 1.00

Available lengths, l Minimum Maximum Available in. in. increments in. 0.50 2.00 025 0.75 3.00 0.375 1.00 4.00 0.50 1.25 5.25 0.625 1.50 6.00 0.75 2.00 8.00 1.00 2.50 10.00 1.25 3.00 12.00 1.50 3.50 14.00 1.75 4.00 16.00 2.00 5.00 20.00 2.50 6.00 24.00 3.00

Table 11.2 Dimensions of square and flat taper stock keys.


Woodruff Keys Key No.

Suggested shaft sizes, in.

204 305 405 506 507 608 807 809 810 812 1012 1212

0.3125-0.375 0.4375-0.50 0.6875-0.75 0.8125-0.9375 0.875-0.9375 1.00-1.1875 1.25-1.3125 1.25-1.75 1.25-1.75 1.25-1.75 1.8125-2.5 1.875-2.5

Nominal key sizea , in. w×l 0.062 × 0.500 0.094 × 0.625 0.125 × 0.625 0.156 × 0.750 0.156 × 0.875 0.188 × 1.000 0.250 × 0.875 0.250 × 1.125 0.250 × 1.250 0.250 × 1.500 0.312 × 1.500 0.375 × 1.500

Height of key, in. h 0.203 0.250 0.250 0.313 0.375 0.438 0.375 0.484 0.547 0.641 0.641 0.641

Shearing Area, in.2 0.030 0.052 0.072 0.109 0.129 0.178 0.198 0.262 0.296 0.356 0.438 0.517

Table 11.3 Dimensions of selected Woodruff keys.


Set Screws

Screw Diameter (in.) 0.25 0.375 0.50 0.75 1.0

Holding Force (lb) 100 250 500 1300 2500

Table 11.4 Holding force generated by setscrews.


Flywheel

Tl

Tm

Figure 11.11 Flywheel with driving (mean) torque Tm and load torque Tl.


Coefficient of Fluctuation Type of equipment Crushing machinery Electrical machinery Electrical machinery, direct driven Engines with belt transmissions Flour milling machinery Gear wheel transmission Hammering machinery Machine tools Paper-making machinery Pumping machinery Shearing machinery Spinning machinery Textile machinery

Coefficient of fluctuation, Cf 0.200 0.003 0.002 0.030 0.020 0.020 0.200 0.030 0.025 0.030-0.050 0.030-0.050 0.010-0.020 0.025

Table 11.5 Coefficient of fluctuation for various types of equipment. !max − !min 2 (!max − !min) Cf = = !avg !max + !min


Design Procedure for Sizing a Flywheel 1. Plot the load torque Tl versus θ for one cycle. 2. Determine Tl,avg over one cycle. 3. Find the locations θωmax and θωmin. 4. Determine kinetic energy by integrating the torque curve. 5. Determine ωavg. 6. Determine Im from Eq. (11.72). Ke Im = C f !2ave

7. Find the dimensions of the flywheel.


Example 11.7

Torque, T, N-m

160

144 N-m

120 80 40 0

12 N-m 0

π –

2

12 N-m π

3––π 2

2π

θ

Figure 11.12 Load or output torque variation for one cycle used in Example 11.7.


Materials for Flywheels

Material Ceramics Composites: Ceramic-fiber-reinforced polymer Graphite-fiber-reinforced polymer Berylium High-strength steel High-strength aluminum (Al) alloys High-strength magnesium (Mg) alloys Titanium alloys Lead alloys Cast iron

Performance index, Mf kJ/kg 200-2000 (compression only) 200-500 100-400 300 100-200 100-200 100-200 100-200 3 8-10

Comment Brittle and weak in tension. Use is usually discouraged The best performance; a good choice. Almost as good as CFRP and cheaper; an excellent choice. Good but expensive, difficult to work, and toxic All about equal in performance; steel and Al-alloys less expensive than Mg and Ti alloys High density makes these a good (and traditional) selection when performance is velocity limited, not strength limited.

Table 11.6 Materials for flywheels.


Chapter 12: Hydrodynamic and Hydrostatic Bearings A cup of tea, standing in a dry saucer, is apt to slip about in an awkward manner, for which a remedy is found in introduction of a few drops of water, or tea, wetting the parts in contact. Lord Rayleigh (1918)

A Kingsbury Bearing.


Density Wedge and Stretch p

p

ρ

ub

ub(x)

Figure 12.1 Density wedge mechanism.

Figure 12.2 Stretch mechanism.


Physical Wedge and Normal Squeeze

p

p

wa

ub

wb

Figure 12.3 Physical wedge mechanism.

Figure 12.4 Normal squeexe mechanism.


Translation Squeeze and Local Expansion p

p ua

ub

Figure 12.5 Translation squeeze mechanism.

Heat

Figure 12.6 Local expansion mechanism.


ua Aa

Moving

ua Da

Ca

E

Velocity Profiles in Slider Bearings

Ba

B A Stationary

A

(a)

Aa

Ba B

A (b) N

ua A′

B′

z

ua

Aa

h

A

Wz

L

ua

Ka Ba

ua Da

H

B

x

I B

Stationary

Figure 12.7 Velocity profiles in a parallel-surface slider bearing.

Ca Ja

M

J

A

z

A (c)

Figure 12.8 Flow within a fixed-incline slider bearing (a) Couette flow; (b) Poiseuille flow; (c) resulting velocity profile. Hamrock • Fundamentals of Machine Elements

Thrust Slider Bearing ω Bearing pad A

Wt

A′

ro ri Thrust bearing Lubricant

Bearing pad

Figure 12.9 Thrust slider bearing geometry. Hamrock • Fundamentals of Machine Elements

Force and Pressure Profiles for Slider Bearing Film pressure distribution

& sh

Wz

Wa ub

Wxa

ho

q

z

Wza Wzb

Fa

q qs

sh

ho Fb

qs (both sides) l

x

ub

Figure 12.10 Force components and oil film geometry in a hydrodynamically lubricated thrust slider bearing.

Figure 12.11 Side view of fixedincline slider bearing.


Design Procedure for Fixed-Incline Thrust Bearing 1.

Choose a pad length-to-width ratio. A square pad (λ = 1) is generally thought to give good performance. If it is known whether maximum load or minimum power loss is more important in a particular application, the outlet film thickness ratio Ho can be determined

2.

from Fig. 12.13. Once λ and Ho are known, Fig. 12.14 can be used to obtain the bearing number Bt.

3.

From Fig. 12.15 determine the temperature rise due to shear heating for a given λ and Bt. The volumetric specific heat Cs = ρCp, which is the dimensionless temperature rise

4.

6 2 parameter, is relatively constant for mineral oils and is equivalent to 1.36 × 10 N/(m °C). Determine lubricant temperature. Mean temperature can be expressed as

!tm t˜m = tmi + 2

where tmi = inlet temperature, °C. The inlet temperature is usually known beforehand. Once the mean temperature tm is known, it can be used in Fig. 8.17 to determine the viscosity of SAE oils, or Fig. 8.16 or Table 8.4 can be used. In using Table 8.4 if the temperature is different from the three temperatures given, a linear interpolation can be used. (continued)


Design Procedure for Fixed-Incline Thrust Bearing 5. Make use of Eqs. (12.34) and (12.68) to get the outlet (minimum) film thickness h0 as

h0 = Hol

!

!0ubwt WzBt

Once the outlet film thickness is known, the shoulder height sh can be directly obtained from sh = ho/Ho. If in some applications the outlet film thickness is specified and either the velocity ub or the normal applied load Wz is not known, Eq. (12.72) can be rewritten to establish ub or Wz. 6. Check Table 12.1 to see if the outlet (minimum) film thickness is sufficient for the pressurized surface finish. If ho from Eq. (12.72) ≥ ho from Table 12.1, go to step 7. If ho from Eq. (12.72) < ho from Table 12.1, consider one or both of the following steps: a. b.

Increase the bearing speed. Decrease the load, the surface finish, or the inlet temperature. Upon making this change return to step 3. 7. Evaluate the other performance parameters. Once an adequate minimum film thickness and a proper lubricant temperature have been determined, the performance parameters can be evaluated. Specifically, from Fig. 12.16 the power loss, the coefficient of friction, and the total and side flows can be determined.


Slider Bearings: Configuration and Film Thickness Sliding surface or runner

ro ri

Na

Dimensionless minimum film thickness, Ho = ho /sh

1.0

.8 Maximum normal load

.6

.4

.2

0

l

Minimum power consumed

1

2

3

4

Length-to-width ratio, λ = l/wt

Figure 12.13 Chart for determining Pads minimum film thickness corresponding to maximum load or Figure 12.12 Configuration of minimum power loss for various multiple fixed-incline thrust slider pad proportions in fixed-incline bearing bearings. Hamrock • Fundamentals of Machine Elements

Film Thickness for Hydrodynamic Bearings Surface finish (centerline average), Ra µm µin. 0.1-0.2 4-8

Examples of manufacturing methods Grind, lap, and superfinish Grind and lap

Approximate relative costs 17-20

Allowable outlet (minimum) film thicknessa , ho µm µin. 2.5 100

Description of surface Mirror-like surface without toolmarks; close tolerances .2-.4 8-16 Smooth surface with17-20 6.2 250 out scratches; close tolerances .4-.8 16-32 Smooth surfaces; close Grind, file, 10 12.5 500 tolderances and lap .8-1.6 32-63 Accurate bearing surGrind, precision 7 25 1000 face without toolmarks mill, and file 1.6-3.2 63-125 Smooth surface withShape, mill, 5 50 2000 out objectionable toolgrind and marks; moderate turn tolerances a The values of film thickness are given only for guidance. They indicate the film thickness required to avoid metal-to-metal contact under clean oil conditions with no misalignment. It may be necessary to take a larger film thickness than that indicated (e.g., to obtain an acceptable temperature rise). It has been assumed that the average surface finish of the pads is the same as that of the runner.

Table 12.1 Allowable outlet (minimum) film thickness for a given surface finish. Hamrock • Fundamentals of Machine Elements

Thrust Bearing - Film Thickness 10

Dimensionless minimum film thickness, Ho = ho /sh

6

Length-towidth ratio, λ

4

0

2

1/2

1 1

2 3 4

.6 .4 .2 .1

0

1

2

4

6

10

20

40

60

100

200

400

1000

Bearing number, Bt

Figure 12.14 Chart for determining minimum film thickness for fixed-incline thrust bearings. Hamrock • Fundamentals of Machine Elements

Thrust Bearings - Temperature Rise 1000

Dimensionless temperature rise, 0.9 Cs wt l ∆t m /Wz

600 400


200

100

4 3

60

2

40

1 20

1/2

0 10 6 4 0

1

2

4

6

10

20

40

60

100

200

400

1000

Bearing number, Bt

Figure 12.14 Chart for determining dimensionless temperature rise due to viscous shear heating of lubricant for fixed-incline thrust bearings. Hamrock • Fundamentals of Machine Elements

Thrust Bearings - Friction Coefficient 1000

Dimensionless coefficient of friction, µl / sh

600


400 200 100

4

60 40

1/2

1

3

2

0

20 10 6 4 2 1

0

1

2

4

6

10

20

40

60

100

200

400

1000

Bearing number, Bt

Figure 12.16a Chart for determining friction coefficient for fixedincline thrust bearings. Hamrock • Fundamentals of Machine Elements

Thrust Bearings - Power Loss 1000

Power loss variable, 1.5 hpl / Wz ub sh, W-s/N-m

600 Length-towidth ratio, L

400 200

4 100 1

60 40

3

2

1/2

0

20 10 6 4 2 1

0

1

2

4

6

10

20

40 60

100

200

400

1000

Bearing number, Bt (b)

Figure 12.16b Chart for determining power loss for fixed-incline thrust bearings. Hamrock • Fundamentals of Machine Elements

Thrust Bearings - Lubricant Flow 4

q

q – qs

0

Dimensionless flow, q/wt ub sh

qs (both sides)

Length-towidth ratio, L

1/2

1

3

2

4 3

1

0

2

1

2

4

6

10

20

40

60

100

200

400

1000

Bearing number, Bt (c) Figure 12.16c Chart for determining lubricant flow for fixedincline thrust bearings.


Thrust Bearings - Side Flow 1.0

Volumetric flow ratio, qs /q

.8

Length-towidth ratio, λ .6

4 3 .4

2

.2

1 1/2

0 0

1

2

4

6

10

20

40

60

100

200

400

1000

Bearing number, Bt

Figure 12.16d Chart for determining lubricant side flow for fixedincline thrust bearings. Hamrock • Fundamentals of Machine Elements

Wr

Journal Bearing Pressure Distribution e 0a 0 Wb hmin

F0

& Film pressure, p Fmax

Figure 12.17 Pressure distribution around a journal bearing.

pmax


Concentric Journal Bearing

u Na

Figure 12.18 Concentric Journal Bearing

r

c wt

2πr u

u Bearing

h=c Journal

Figure 12.19 Developed journal bearing surfaces for a concentric journal bearing. Hamrock • Fundamentals of Machine Elements

Typical Radial Load for Journal Bearings

Application Automotive engines: Main bearings Connecting rod bearing Diesel engines: Main bearings Connecting rod bearing Electric motors Steam turbines Gear reducers Centrifugal pumps Air compressors: Main bearings Crankpin Centrifugal pumps

Average radial load per area, Wr∗ psi MPa 600-750 1700-2300

4-5 10-15

900-1700 1150-2300 120-250 150-300 120-250 100-180

6-12 8-15 0.8-1.5 1.0-2.0 0.8-1.5 0.6-1.2

140-280 280-500 100-180

1-2 2-4 0.6-1.2

Table 12.2 Typical radial load per area Wr* in use for journal bearings. Hamrock • Fundamentals of Machine Elements

Journal Bearing - Film Thickness

Figure 12.20 Effect of bearing number on minimum film thickness for four diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements

Journal Bearings - Attitude Angle

Attitude angle, Φ, deg

100 Diameter-towidth ratio, λj

80

0 1

60

2 4

40

20

0

10 –2

10 –1

100

101

Bearing number, Bj

Figure 12.21 Effect of bearing number on attitude angle for four different diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements

Journal Bearing - Friction Coefficient 2 × 102

Dimensionless coefficient of friction variable, rbµ /c

102

Diameter-towidth ratio, λj 2

1

10

0

4

1

100 0

10 –2

10 –1 Bearing number, Bj

100

101

Figure 12.21 Effect of bearing number on coefficient of friction for four different diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements

Journal Bearing - Flow Rate

Dimensionless volumetric flow rate, Q = 2πq/rbcwtω b

8

Diameter-towidth ratio, λj

6

4 2 1

4

0

2

0

10 – 2

10 –1

100

101

Bearing number, Bj

Figure 12.23 Effect of bearing number on dimensionless volume flow rate for four different diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements

Journal Bearing - Side Flow Side-leakage flow ratio, qs /q

1.0 4

.8


2

.6 .4

1 .2

0

10 –2

10 –1

100

101

Bearing number, Bj

Figure 12.21 Effect of bearing number on side-flow leakage for four different diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements

Journal Bearing - Maximum Pressure

Dimensionless maximum film pressure, Pmax = Wr /2rb wt pmax

1.0 .8


0

.6 1 .4

2 4

.2

0

10 – 2

10 –1

100

101

Bearing number, Bj

Figure 12.25 Effect of bearing number on dimensionless maximum film pressure for four different diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements


φ0 φmax

100

25 0

1

2

4

80

20 4

60 2

40

15 10

1 20

5

0

0

0 10 – 2

10 –1

100

Location of maximum pressure, φmax, deg

Location of terminating pressure, φ0, deg

Journal Bearing - Maximum and Terminating Pressure Location

101

Bearing number, Bj

Figure 12.21 Effect of bearing number on location of terminating and maximum pressure for four different diameter-to-width ratios. Hamrock • Fundamentals of Machine Elements

Minimum film thickness, h min; power loss, hp; outlet temperature, tmo; volumetric flow rate, q

Journal Bearing - Effect of Radial Clearance

hmin hp

t mo

q

0

.5

1.0

1.5

2.0

2.5

3.0 × 10–3

Radial clearance, c, in.

Figure 12.27 Effect of radial clearance on some performance parameters for a particular case. Hamrock • Fundamentals of Machine Elements

Squeeze Film Bearing w = – ∂h ∂t

l z

ho

Surface b x

Surface a Figure 12.21 Parallel-surface squeeze film bearing Hamrock • Fundamentals of Machine Elements

Wz

Wz Bearing runner

Bearing pad Bearing recess

Restrictor

Recess pressure, pr = 0

pr > 0

Flow, q=0

Supply pressure, ps = 0

ps > 0

Manifold

Fluid Film in Hydrostatic Bearing

(b)

(a)

Wz

Wz

ho

q

p = pl

q=0

p = ps

p = pl

(d)

(c) Wz + ∆Wz

Wz – ∆Wz ho + ∆ ho

ho – ∆ho

q

q

p = pr + ∆ pr p = ps

(e)

p = pr

p = pr – ∆ pr p = ps

Figure 12.29 Formation of fluid film in hydrostatic bearing system. (a) Pump off; (b) pressure buildup; (c) pressure times recess area equals normal applied load; (d) bearing operation; (e) increased load; (f) decreased load.

(f)


Wz

Radial Flow Hydrostatic Bearing ro ri

p=0

sh

pr

q

ho

Figure 12.30 Radial flow hydrostratic thrust bearing with circular step pad.


Chapter 13: Rolling Element Bearings Since there is no model in nature for guiding wheels on axles or axle journals, man faced a great task in designing bearings - a task which has not lost its importance and attraction to this day.

Hamrock • Fundamentlas of Machine Elements

Radial Ball Bearings Type

Conrad or deep groove

Approximate range of bore sizes, mm Mini- Maximum mum 3 1060

Relative capacity Radial Thrust

Limiting speed factor

Tolerance to misalignment

1.00

a 0.7

1.0

±0◦ 15"

Maximum capacity or filling notch

10

130

1.2-1.4

a 0.2

1.0

±0◦ 3"

Self-aligning, internal

5

120

0.7

b 0.2

1.0

±2◦ 30"

Self-aligning, external

—

—

1.0

a 0.7

1.0

High

Double row, maximum

6

110

1.5

a 0.2

1.0

±0◦ 3"

Double row, deep groove

6

110

1.5

a 1.4

1.0

0◦

Table 13.1 Characteristics of representative radial ball bearings. Hamrock • Fundamentlas of Machine Elements

Angular-Contact Ball Bearings Type

Approximate maximum bore size, mm 320



b 1.00-1.15

a,b 1.5-2.3

b 1.1-3.0

±0◦ 2"

Duplex, back to back

320

1.85

c 1.5

3.0

0◦

Duplex, face to face

320

1.85

c 1.5

3.0

0◦

Duplex, tandem

320

1.85

a 2.4

3.0

0◦

Two directional or split ring

110

1.15

c

1.5

3.0

±0◦ 2"

Double row

140

1.5

1.85

0.8

0◦

One-directional thrust


c

Table 13.2 Characteristics of representative angular-contact ball bearings. Hamrock • Fundamentlas of Machine Elements

Thrust Ball Bearings

Type

One directional, flat race

Approximate range of bore sizes, mm Minimum Maximum 6.45 88.9

Relative thrust

Limiting speed

Tolerance to mis-

a 0.7

0.10

b 0◦

One directional, grooved race

6.45

1180

a 1.5

0.30

0◦

Two directional, grooved race

15

220

c 1.5

0.30

0◦

Table 13.1 Characteristics of representative thrust ball bearings.


Cylindrical Roller Bearings Type

Separable outer ring, nonlocating (N)

Approximate range of bore sizes, mm MiniMaximum mum 10 320

Relative capacity Radial thrust



1.55

0

1.20

±0◦ 5"

Separable inner ring, nonlocating (NU)

12

500

1.55

0

1.20

±0◦ 5"

Separable inner ring, one-direction locating (NJ)

12

320

1.55

a Locating

1.15

±0◦ 5"

Separable inner ring, two-direction locating

20

320

1.55

b Locating

1.15

±0◦ 5"

Table 13.1 Characteristics of representative cylindrical roller bearings. Hamrock • Fundamentlas of Machine Elements

Spherical Roller Bearings Type

Single row, barrel or convex

Approximate range of bore sizes, mm MiniMaximum mum 20 320




2.10

0.20

0.50

±2◦

Double row, barrel or convex

25

1250

2.40

0.70

0.50

±1◦ 30"

Thrust

85

360

a 0.10

a 1.80

0.35-0.50

b 0.10

b 2.40

±3◦

2.40

0.70

0.50

±1◦ 30"

Double row, concave a Symmetric

50

130

rollers. rollers.

b Asymmetric

Table 13.5 Characteristics of representative spherical roller bearings. Hamrock • Fundamentlas of Machine Elements

Radial Single-Row Ball Bearings bw cd

da de di

db

do

d

Figure 13.1 (a) Cross section through radial single-row ball bearings; (b) examples of radial single-row ball bearings. Hamrock • Fundamentlas of Machine Elements

Race Conformity and Axial Shift βf cd – 4

r d

x ro

ri

ro

d – 2

cr ri

cd cr – — 2

cd – 4

Figure 13.2 Cross section of ball and outer race, showing race conformity.

Axis of rotation (a)

c —e 2 Axis of rotation (b)

Figure 13.1 Cross section of radial bearing, showing ball-race contact due to axial shift of inner and outer races. (a) Initial position; (b) shifted position.


48 44 40

Total conformity ratio, B 0.02

Free Contact Angle and Endplay

Free contact angle Dimensionless endplay

32 28

.04

24 .06 20

.08

16

.040 .032

12

.08 .06 .04

.024

8

.02

.016

4 0

.008 0 .001 .002 .003 .004 .005 .006 .007 .008 .009 Dimensionless diametral clearance, cd /2d

Dimensionless endplay, ce/2d

Free contact angle, βf

36

Figure 13.4 (a) Free contact angle and endplay as function of cd/2d for four values of total conformity ratio.


Ball Bearing β

d

r d

de + d cos β ————— 2

θs

sh

de – d cos β ————— 2 C L

Figure 13.5 Shoulder height in a ball bearing.

ωi

de /2 ωo

Figure 13.6 Cross section of ball bearing.


Spherical, Cylindrical and Toroidal Bearings

lt

le

ll

ll d

d rr

rr

(a)

(b)

Figure 13.7 (a) Spherical roller (fully crowned) and (b) cylindrical roller; (c) section of toroidal roller bearing. Hamrock • Fundamentlas of Machine Elements

Spherical Roller Bearing

Outer ring

d

rr

ri

c —d 2

c —e 2

rr

ro – cd /2

de

β

β

ro

γd ro

Axis of rotation

Figure 13.8 Geometry of spherical roller bearing.

Figure 13.9 Schematic diagram of spherical roller bearing, showing diametral play and endplay. Hamrock • Fundamentlas of Machine Elements

Contact Angle and Angular Velocity of Ball

βo

Pzo

ωro

βo ωso

Centrifugal force

βi ωri

ωsi

Pzi βi

Figure 13.10 Contact angles in ball bearing at appreciable speeds.

φs

ωi

ωo

Figure 13.11 Angular velocities of ball


ωb

Ball Spin Orientations ωb

ωb

(a)

(b)

Figure 13.12 Ball-spin orientations for (a) outer-race and (b) inner race control. Hamrock • Fundamentlas of Machine Elements

Tapered Roller Bearing

d

β de /2

Figure 13.13 Tapered-roller bearing. (a) Tapered-roller bearing with outer race removed to show rolling elements; (b) simplified geometry for tapered-roller bearing. Hamrock • Fundamentlas of Machine Elements

di

di + 2d

do

Radially Loaded Rolling Element Bearing

cd /2

c

ψ

cd /2 (a)

(b)

cd /2 δm δmax

δψ ψl

ψ δmax

Figure 13.14 Radial loaded rollingelement bearing. (a) Concentric arrangement; (b) initial contact; (c) interference.

(c)


Contact Ellipse and Angular Contact Ball Bearing

Outer race

P β βf

sh

r

ro

θs

βf β

δt δm cr

cr – cd /2 δt

ce /2

ri Inner race

β cr – cd /2

βf

Dy D x

cr cr + δm

δt P

Figure 13.15 Contact ellipse in bearing race under load.

C L

Figure 13.16 Angular-contact ball bearing under thrust load.


Back-to-Back Arrangement

Stickout

Stickout (a) Pa

Bearing a

Bearing b

Pa

Figure 13.17 Angular-contact bearings in back-to-back arrangement, shown (a) individually as manufactured and (b) as mounted with preload.

(b)


Load-Deflection Curve

Applied thrust load, Pa

Bearing b load System thrust load, Pt Preload

Bearing a load

∆a

∆b

Stickout Axial deflection, δa

Figure 13.18 Thrust load-axial deflection curve for typical ball bearing. Hamrock • Fundamentlas of Machine Elements

Principal dimensions db mm in. 15 0.5906

20 0.7874

25 0.9843

30 1.1811

35 1.3780

40 1.5748

da

32 1.2598 32 1.2598 35 1.3780 35 1.3780 42 16535 42 1.6535 47 1.8504 52 2.0472 72 2.8346 47 1.8504 52 2.0472 62 2.4409 80 3.1496 55 2.1654 62 2.4409 72 2.8346 90 3.5433 62 2.4409 72 2.8346 80 3.1496 100 3.9370 68 2.6672 80 3.1496 90 3.5433 110 4.3307

Basic load ratings Dynamic Static bw

8 0.3510 8 0.3543 11 0.4331 13 0.5118 8 0.3150 12 0.4724 14 0.5512 15 0.5906 19 0.7480 12 0.4724 15 0.5906 17 0.6693 21 0.8268 15 0.5118 16 0.6299 19 0.7480 23 0.9055 14 0.5512 17 0.6693 21 0.8268 25 0.9843 15 0.5906 18 0.7087 23 0.9055 27 1.0630

C N lb 5590 1260 5590 1260 7800 1750 11,400 2560 6890 1550 9360 2100 12,700 2860 15,900 3570 30,700 6900 11,200 5520 14,000 3150 22,500 5060 35,800 8050 13,300 2990 19,500 4380 28,100 6320 43,600 9800 15,900 3570 25,500 5370 33,200 7460 55,300 12,400 16,800 3780 30,700 6900 41,000 9220 63,700 14,300

C0

2850 641 2850 641 3750 843 5400 1210 4050 910 5000 1120 6550 1470 7800 1750 15,000 3370 6550 1470 7800 1750 11,600 2610 19,300 4340 8300 1870 11,200 2520 16,000 3600 23,600 5310 10,200 2290 15,300 3440 19,000 4270 31,000 6970 11,600 2610 19,000 4270 24,000 5400 36,500 8210

Allowable load limit wall N lb 120 27.0 120 27.0 160 36.0 228 51.3 173 38.9 212 47.7 280 62.9 335 75.3 640 144 275 61.8 335 75.3 490 110 815 183 355 79.8 475 107 670 151 1000 225 440 98.9 655 147 815 183 1290 290 490 110 800 147 1020 229 1530 344

Speed ratings Grease Oil

rpm 22,000

28,000

22,000

28,000

19,000

24,000

17,000

20,000

17,000

20,000

17,000

20,000

15,000

18,000

13,000

16,000

10,000

13,000

15,000

18,000

12,000

15,000

11,000

14,000

9000

11,000

12,000

15,000

10,000

13,000

9000

11,000

8500

10,000

10,000

13,000

9000

11,000

8500

10,000

7000

8500

9500

12,000

8500

10,000

7500

9000

6700

8000

Mass

kg lbm 0.025 0.055 0.030 0.066 0.045 0.099 0.082 0.18 0.050 0.11 0.090 0.15 0.11 0.15 0.14 0.31 0.40 0.88 0.080 0.18 0.13 0.29 0.23 0.51 0.53 0.51 0.12 0.26 0.20 0.44 0.35 0.77 0.74 1653 0.16 0.35 0.29 0.64 0.46 1.00 0.95 2.10 0.19 0.42 0.37 0.82 0.63 1.40 1.25 2.75

Designation

Bearing Ratings

— 16002 6002 6202 6302 16004 6004 6204 6304 6406 6005 6205 6305 6405 6006 6206 6306 6406 6007 6207 6307 6407 6008 6208 6308 6408

Table 13.1 Single-row, deepgroove ball bearings.


Principal dimensions db mm in. 15 0.5906

20 0.7874

25 0.9843

30 1.811

35 1.3780

40 1.5748

45 1.7717

50 1.9685

da

35 1.3780 42 1.6535 47 1,8504 52 2.0472 52 2.0472 62 2.4409 62 2.4409 72 2.8346 72 2.8346 80 3.1496 80 3.1496 90 3.5433 85 3.3465 100 3.9370 90 3.5433 110 4.3307

Basic load ratings Dynamic Static bw

11 0.4331 13 0.5118 14 0.5512 15 0.5906 15 0.5906 17 0.6693 16 0.6299 19 0.7480 17 0.693 21 0.8268 18 0.7087 23 0.9055 19 0.7480 25 0.9843 20 0.7874 27 1.0630

C N lb 12,500 2810 19,400 4360 25,100 5640 30,800 6920 28,600 6430 40,200 9040 38,000 8540 51,200 11,500 48,400 10,900 64,400 14,500 53,900 12,100 80,900 18,200 60,500 13,600 99,000 22,300 64,400 14,500 110,000 24,700

C0

10,200 2290 15,300 3440 22,000 4950 26,000 5850 27,000 6070 36,500 8210 36,500 8210 48,000 10,800 48,000 10,800 63,000 14,200 53,000 11,900 78,000 17,500 64,000 14,400 100,000 22,500 69,500 15,600 112,000 25,200

Allowable load limit wall N lb 1200 274 1860 418 2750 618 3250 731 3350 753 4550 1020 4450 1020 6200 1390 6100 1390 8150 1830 6700 1510 10,200 2290 8150 1830 12,900 2900 8800 1980 15,000 3370

Speed ratings Grease Oil

rpm 18,000

22,000

16,000

19,000

13,000

16,000

12,000

15,000

11,000

14,000

9500

12,000

9500

12000

9000

11,000

8500

10,000

8000

9500

7500

9000

6700

8000

6700

8000

6300

7500

6300

7500

5000

6000

Mass

kg lbm 0.047 0.10 0.086 0.19 0.11 0.24 0.15 0.33 0.13 0.29 0.24 0.53 0.20 0.44 0.36 0.79 0.30 0.66 0.48 1.05 0.37 0.82 0.65 1.05 0.43 0.95 0.90 2.00 0.48 1.05 1.15 2.55

Designation

Bearing Ratings

— NU 202 EC NU 302 EC NU 204 EC NU 304 EC NU 205 EC NU 305 EC NU 206 EC NU 306 EC NU 207 EC NU 307 EC NU 208 EC NU 308 EC NU 209 EC NU 309 EC NU 210 EC

Table 13.7 Singlerow cylindrical roller bearings.

NU 310 EC


Combined Load Pr = Pcosαp

P

αp Pa = Psinαp

P0 = X0 Pr +Y0 Pa Bearing type Radial deep-groove ball Radial angular-contact ball

Radial self-aligning ball Radial spherical roller Radial tapered roller

β β β β β

= 20◦ = 25◦ = 30◦ = 35◦ = 40◦

Single row X0 Y0 0.6 0.5 0.5 0.42 0.5 0.38 0.5 0.33 0.5 0.29 0.5 0.26 0.5 0.22 cot β 0.5 0.22 cot β 0.5 0.22 cot β

Double row X0 Y0 0.6 0.5 1 0.84 1 0.76 1 0.66 1 0.58 1 0.52 1 0.44 cot β 1 0.44 cot β 1 0.44 cot β

Table 13.8 Radial factor X0 and thrust factor Figure 13.19 Combined load acting Y0 for statically stressed radial bearings. on radial deep-groove ball bearing. Hamrock • Fundamentlas of Machine Elements

Fatigue Spall

(a)

(b)

Figure 13.20 Typical fatigue spall (a) Spall on tapered roller bearing; (b) detail of fatigue spall. Hamrock • Fundamentlas of Machine Elements

Fatigue Failure 4

Life

3

2 S1 = 1 – M1

1

~ L ~ L = 0.2

0

20

40 60 Bearings failed, percent

80

100

S2 = 1 – M2

S3 = 1 – M3

…

Sm = 1 – Mm

Figure 13.22 Representation of m similar stressed volumes.

Figure 13.21 Distribution of bearing fatigue failures. Hamrock • Fundamentlas of Machine Elements

Specimens tested, percent

Bearing Fatigue Failures 95 90 80 70 60 50 40 30 20 10 5

2

1

2

4 6 10 20 40 60 100 Specimen life, millions of stress cycles

1000

Figure 13.23 Typical Weibull plot of bearing fatigue failures. Hamrock • Fundamentlas of Machine Elements

Capacity Formulas Bearing type e Deep-groove ball bearings

Pa /C0 = 0.025 Pa /C0 = 0.04 Pa /C0 = 0.07 Pa /C0 = 0.13 Pa /C0 = 0.25 Pa /C0 = 0.50 Angularβ = 20◦ contact ball β = 25◦ bearings β = 30◦ β = 35◦ β = 40◦ β = 45◦ Self-aligning ball bearings Spherical roller bearings Tapered-roller bearings

0.22 0.24 0.27 0.31 0.37 0.44 0.57 0.68 0.80 0.95 1.14 1.33 1.5 tan β 1.5 tan β 1.5 tan β

Single-row bearings Pa Pa ≤e >e Pr Pr X Y X Y 1 0 0.56 2.0 1 0 0.56 1.8 1 0 0.56 1.6 1 0 0.56 1.4 1 0 0.56 1.2 1 0 0.56 1 1 0 0.43 1 1 0 0.41 0.87 1 0 0.39 0.76 1 0 0.37 0.66 1 0 0.35 0.57 1 0 0.33 0.50

1

0

0.40

0.4 cot β

Double-row bearings Pa Pa ≤e >e Pr Pr X Y X Y

1 1 1 1 1 1 1 1 1

1.09 0.92 0.78 0.66 0.55 0.47 0.42 cot β 0.45 cot β 0.42 cot β

0.70 0.67 0.63 0.60 0.57 0.54 0.65 0.67 0.67

Table 13.9 Capacity formulas for rectangular and elliptical conjunctions for radial and angular bearings. Hamrock • Fundamentlas of Machine Elements

1.63 1.41 1.24 1.07 0.93 0.81 0.65 cot β 0.67 cot β 0.67 cot β

Radial and Thrust Factor for Thrust Bearings

Bearing type e β = 45◦ β = 60◦ β = 75◦ Spherical roller thrust Tapered roller Thrust ball

1.25 2.17 4.67 1.5 tan β 1.5 tan β

Single acting Pa >e Pr X Y 0.66 1 0.92 1 1.66 1 tan β 1 tan β 1

Double acting Pa Pa ≤e >e Pr Pr X Y X Y 1.18 0.59 0.66 1 1.90 0.55 0.92 1 3.89 0.52 1.66 1 1.5 tan β 0.67 tan β 1 1.5 tan β 0.67 tan β 1

Table 13.10 Radial factor X and thrust factor Y for thrust bearings.


Material Factors Material 52100 M-1 M-2 M-10 M-50 T-1 Halmo M-42 WB 49 440C

¯ Material factor, D 2.0 0.6 0.6 2.0 2.0 0.6 2.0 0.2 0.6 0.6-0.8

Table 13.11 Material factors for through-hardened bearing materials.


Lubrication and Bearing Life 3.5 300

250

Region of lubrication-related surface distress

From Skurka (1970)

3.0

Mean curve recommended for use

2.5 Region of possible surface distress

150

Lubrication factor, Fl

~ L10 life, percent

200

~ L10 life

100

2.0 From Tallian (1967)

1.5 1.0

Calculated from AFBMA

50

.5 0 .6

.8

1

2 4 6 Dimensionless film parameter, Λ

8

10

Figure 13.24 Group fatigue life L10 as a function of dimensionless film parameter.

0 .6

.8

1

2 4 6 Dimensionless film parameter, Λ

8

10

Figure 13.25 Lubrication factor as a function of dimensionless film parameter.


Chapter 14 Just stare at the machine. There is nothing wrong with that. Just live with it for a while. Watch it the way you watch a line when fishing and before long, as sure as you live, you’ll get a little nibble, a little fact asking in a timid, humble way if you’re interested in it. That’s the way the world keeps on happening. Be interested in it. Robert Pirsig, Zen and the Art of Motorcycle Maintenance


Spur Gears

Figure 14.1 Spur gear drive. (a) Schematic illustration of meshing spur gears; (b) a collection of spur gears.


Helical Gears

Figure 14.2 Helical gear drive. (a) Schematic illustration of meshing helical gears; (b) a collection of helical gears. Hamrock • Fundamentals of Machine Elements

Bevel Gears

Figure 14.3 Bevel gear drive. (a) Schematic illustration of meshing bevel gears; (b) a collection of bevel gears. Hamrock • Fundamentals of Machine Elements

Worm Gears

(a)

(b)

Figure 14.4 Worm gear drive. (a) Cylindrical teeth; (b) double enveloping; (c) a collection of worm gears. Hamrock • Fundamentals of Machine Elements

Spur Gear Geometry

Pinion Line of action

r bp rp r op

Ou tsPii dtec h(

pdd) iam

ete

Base circle

Pitch circle

r, d

op

Pressure angle, F

Tooth profile Pitch circle Whole depth, ht

Center distance, cd

Addendum, a Working depth, hk Clearance, cr Base diameter, dbg Ro ot dia me ter Circular tooth thickness Chordal tooth thickness

Dedendum, b

Root (tooth) Fillet Top land

r bg rg

rog

Pit ch

dia

me te

r, d

g

Circular pitch, pc

Gear

Figure 14.5 Basic spur gear geometry.

Pitch point


To p

la

nd

Fa c

ew

id th

Gear Teeth

Outside c

Width of space

Pitch circl e

Clearance

Fillet radius

Dedendum circle

Bo tto

m

la

nd

Tooth thickness

nk

Circular pitch Fl a

Dedendum

Fa c

Addendum

e

ircle

Clearance circle

Figure 14.6 Nomenclature of gear teeth. Hamrock • Fundamentals of Machine Elements

Standard Tooth Size

2 12

2

3

4

Class Coarse Medium coarse Fine

5

Ultrafine 6

7

8

9

10

12

14

Figure 14.7 Standard diametral pitches compared with tooth size.

16

Diametral pitch, pd , in−1 1/2, 1, 2, 4, 6, 8, 10 12, 14, 16, 18 20, 24, 32, 48, 64 72, 80, 96, 120, 128 150, 180, 200

Table 14.1 Preferred diametral pitches for four tooth classes


d

=8

n (m .00 i

= 8; d

= 20 0

Data for all curves: gr = 4, NP=24 Ka = 1.0 20° full depth teeth

3p

it c

h;

200

100

Power vs. Pinion Speed

150 100

70 60 =4.00 in (m=4; d=100 mm) 50 6 pi tc h ; d 40

70 50

30 20

20

15

10.0

12 p

i tc h ;

(m=2; d 00 in . 2 = d

) =50 mm

10.0 7.0 6.0 5.0 4.0

7.0 5.0

Power transmitted, kW

Power transmitted, hp

30

3.0 3.0 2.0 24

;d p i t ch

= 1. 0 0

m) d=25 m ; 1 = m in (

2.0 1.5 1.0

1.0

0.7

0.7

0.5

0.5 0

600

1200

1800

2400

3000

Figure 14.8 Transmitted power as a function of pinion speed for a number of diametral pitches.

3600

Pinion speed, rpm


Gear Geometry Formulas

Parameter Addendum Dedendum Clearance

Symbol a b c

Coarse pitch (pd < 20 in.−1 ) 1/pd 1.25/pd 0.25/pd

Fine pitch (pd ≥ 20 in.−1 ) 1/pd 1.200/pd + 0.002 0.200/pd + 0.002

Metric module system 1.00 m 1.25 m 0.25 m

Table 14.2 Formulas for addendum, dedendum, and clearance (pressure angle, 20°; full-depth involute).


Pitch and Base Circles Pinion

Base circle Pitch circle

rbp

ωp

rp

φ a Pitch point, pp

ωg

Pitch circle b

Base circle φ

rg

rbg Gear

Figure 14.9 Pitch and base circles for pinion and gear as well as line of action and pressure angle.


Involute Curve

Base circle Involute

A4 A3

C4 B4

A2

C3 A1

C2 C1 A0

B1

B3 B2

0

Figure 14.10 Construction of the involute curve.


Construction of the Involute Curve 1. Divide the base circle into a number of equal distances, thus constructing A0, A1, A2,... 2. Beginning at A1, construct the straight line A1B1, perpendicular with 0A1, and likewise beginning at A2 and A3. 3. Along A1B1, lay off the distance A1A0, thus establishing C1. Along A2B2, lay off twice A1A0, thus establishing C2, etc. 4. Establish the involute curve by using points A0, C1, C2, C3,... Gears made from the involute curve have at least one pair of teeth in contact with each other.


Contact Parameters Arc of approach qa

Arc of recess qr

Line o f actio n

Outs

a

P

A

Outside circle

ide c ircle

B b

Pitc h ci rcle

Motion Lab

Figure 14.11 Illustration of parameters important in defining contact. Hamrock • Fundamentals of Machine Elements

Line of Action

Length of line of action: Lab =

!

2 − r2 + rop bp

!

2 − r 2 − c sin ! rog d bg

Contact ratio:

" # c tan ! 1 !" 2 2 2 − r2 − d Cr = rop − rbp + rog bg pc cos ! pc

Figure 14.12 Details of line of action, showing angles of approach and recess for both pinion and gear. Hamrock • Fundamentals of Machine Elements

Backlash

P itc

h ci

B

ci ase

r cle

rcle

0 φ

Pitch

Base circle

Backlash

circl e Pressure line

Backlash

Figure 14.13 Illustration of backlash in gears.

Diametral pitch pd , in.− 1 18 12 8 5 3 2 1.25

Center distance, cd , in. 2 4 8 16 32 0.005 0.006 — — — 0.006 0.007 0.009 — — 0.007 0.008 0.010 0.014 — — 0.010 0.012 0.016 — — 0.014 0.016 0.020 0.028 — — 0.021 0.025 0.033 — — — 0.034 0.042

Table 14.3 Recommended minimum backlash for coarsepitched gears.


Meshing Gears Gear 2 (N2 teeth)

Gear 1 (N1 teeth)

W2

r2 W1

Gear 1 (N1)

r1 r2

W1

(+)

(–)

r1

W2

Figure 14.14 Externally meshing gears.

Figure 14.15 Internally meshing gears.


Gear 2 (N2)

Gear Trains N2

Figure 14.16 Simple gear train. N1

N2

N5

N1

Figure 14.17 Compound gear train.

N6

N3

N4

N7


N8

Example 14.7 Input Shaft 1

A

NA = 20

A

B NB = 70 Shaft 2

B

NC = 18 C D

ND = 22

Shaft 3 E N = 54 E

Shaft 4

Only pitch circles of gears shown

C D

E

Figure 14.18 Gear train used in Example 14.7.

Output


Planetary Gear Trains Important planet gear equations: Ring Planet Arm

R P A

P

!ring − !arm Nsun =− !sun − !arm Nring

R P

Sun S

S

P

! planet − !arm Nsun =− !sun − !arm Nplanet Nring = Nsun + 2Nplanet

(a)

(b)

Figure 14.19 Illustration of planetary gear train. (a) With three planets; (b) with one planet (for analysis only).

Zp =

!L − !A !F − !A


Gear Quality AGMA quality index (16) (6) 14 (17) 9 8 (7) 15 13 12 11 10

100

(4)(5)

Gear shaper-hobbing

1 2

0.010 3

4 5

6 7

0.0010

0.0005

0.5

0.00010

1

0.00005 in.

Relative cost

10

0.015

Shaving Production grinding Special methods

8 9 10 11 12

DIN quality number

Figure 14.20 Gear cost as a function of gear quality. The numbers along the vertical lines indicate tolerances.

Application Cement mixer drum driver Cement kiln Steel mill drives Corn pickers Punch press Mining conveyor Clothes washing machine Printing press Automotive transmission Marine propulsion drive Aircraft engine drive Gyroscope Pitch velocity ft/min m/s 0-800 0-4 800-2000 4-10 2000-4000 10-20 > 4000 > 20

Quality index, Qv 3-5 5-6 5-6 5-7 5-7 5-7 8-10 9-11 10-11 10-12 10-13 12-14 Quality index, Qv 6-8 8-10 10-12 12-14

Table 14.4 Quality index Qv for various applications.


Form Cutting

Form cutter

Gear blank

(a)

(b)

(c)

Figure 14.21 Form cutting of teeth. (a) A form cutter. Notice that the tooth profile is defined by the cutter profile. (b) Schematic illustration of the form cutting process. (c) Form cutting of teeth on a bevel gear. Hamrock • Fundamentals of Machine Elements

Pinion-Shaped Cutter

Figure 14.22 Production of gear teeth with a pinion-shaped cutter. (a) Schematic illustration of the process; (b) photograph of the process with gear and cutter motions indicated. Hamrock • Fundamentals of Machine Elements

Gear Hobbing Top view

Gear blank

(b) Helical gear

Hob rotation Hob Hob

Gear blank

(a)

(b)

Figure 14.23 Production of gears through the hobbing process. (a) A hob, along with a schematic illustration of the process; (b) production of a worm gear through hobbing. Hamrock • Fundamentals of Machine Elements

Grade 2

350 300 250 200

60

400

r Th

ou

Th

gh

ro

rd -h a

-h u gh

150

e ne

ed trid Ni

50

ided Nitr

40

Material

d

d ene ard

Grade 1

30

Grade

Allowable bending stress number MPa

ksi

Through-hardened steels

1 2

0.703 HB + 113 0.533 HB + 88.3

0.0773 HB + 12.8 0.102 HB + 16.4

Nitriding throughhardened steels

1 2

0.0823 HB + 12.15 0.1086 HB + 15.89

0.568 HB + 83.8 0.749 HB + 110

ksi

Allowable bending stress number, Sb, MPa

Allowable Bending Stress

20

100 120 150

200

250 300 350 Brinell hardness, HB

400

10 450

Figure 14.24 Effect of Brinell hardness on allowable bending stress number for steel gears. (a) Through-hardened steels. Note that the Brinell hardness refers to the case hardness for these gears. Hamrock • Fundamentals of Machine Elements

Allowable Bending and Contact Stress Material designation Steel Through-hardened Carburized & hardened

Nitrided and throughhardened Nitralloy 135M and Nitralloy N, nitrided 2.5% Chrome, nitrided

Cast Iron ASTM A48 gray cast iron, as-cast ASTM A536 ductile (nodular) iron

Bronze Sut > 40, 000 psi (Sut > 275GP a) Sut > 90, 000 psi (Sut > 620GP a)

Grade

Typical Hardnessa

1 2 1 2 3 1 2 1 2 1 2 3

— — 55-64 HRC 58-64 HRC 58-64 HRC 83.5 HR15N — 87.5 HR15N 87.5 HR15N 87.5 HR15N 87.5 HR15N 87.5 HR15N

Class 20 Class 30 Class 40 60-40-18 80-55-06 100-70-03 120-90-02

— 174 HB 201 HB 140 HB 179 HB 229 HB 269 HB

Allowable bending stress, σall,b lb/in.2 MPa See Fig. 14.24a See Fig. 14.24a 55,000 380 65,000b 450b 75,000 515 See Fig. 14.24b See Fig. 14.24b See Fig. 14.24b See Fig. 14.24b See Fig. 14.24b See Fig. 14.24b See Fig. 14.24b

Allowable contact stress, σall,b lb/in.2 MPa See Fig. 14.25 See Fig. 14.25 180,000 1240 225,000 1550 275,000 1895 150,000 1035 163,000 1125 170,000 1170 183,000 1260 155,000 1070 172,000 1185 189,000 1305

5000 8500 13,000 22,000-33,000 22,000-33,000 27,000-40,000 31,000-44,000

34.5 59 90 150-230 150-230 185-275 215-305

50,000-60,000 65,000-75,000 75,000-85,000 77,000-92,000 77,000-92,000 92,000-112,000 103,000-126,000

345-415 450-520 520-585 530-635 530-635 635-770 710-870

5700

39.5

30,000

205

23,600

165

65,000

450

Table 14.5 Allowable bending and contact stresses for selected gear materials. Hamrock • Fundamentals of Machine Elements

Allowable Bending Stress

70 me 2.5% Chro Grade 3 -

400

2 Grade

Grade lloy - Nitra

Grade 1 Grade 1

50

300 Nitrallo

Material

60

hrome 2 - 2.5% C

2.5% Chro

me

40

y

Grade

Allowable bending stress number MPa

ksi

Niralloy

1 2

0.594 HB + 87.76 0.784 HB + 114.81

0.0862 HB + 12.73 0.1138 HB + 16.65

2.5% Chrome

1 2 3

0.7255 HB + 63.89 0.7255 HB + 153.63 0.7255 HB + 201.91

0.1052 HB + 9.28 0.1052 HB + 22.28 0.1052 HB + 29.28

ksi

Allowable bending stress number, Sb, MPa

500

30

200

20 100 250

275

300

325

350

Brinell hardness, HB

Figure 14.24 Effect of Brinell hardness on allowable bending stress number for steel gears. (b) Flame or induction-hardened nitriding steels. Note that the Brinell hardness refers to the case hardness for these gears. Hamrock • Fundamentals of Machine Elements

Allowable Contact Stress

1300 1200 1100

Grade 1: Sc = 2.41 HB +237 (MPa) 0.349 HB + 34.3 (ksi) Grade 2: Sc = 2.22 HB +200 (MPa) 0.322 HB + 29.1 (ksi)

{ {

e2 d ra

175

G

1000

e1 d a Gr

900

150 ksi

Allowable contact stress number, Sc, MPa

1400

125

800 100

700 600 150

200

250 300 350 Brinell hardness, HB

400

450

75

Figure 14.25 Effect of Brinell hardness on allowable contact stress number for two grades of through-hardened steel. Hamrock • Fundamentals of Machine Elements

Stress Cycle Factor 4.0

400 HB: YN = 9.4518 N-0.148 Case Carb.: YN = 6.1514N-0.1192

Stress cycle factor, Yn

3.0

250 HB: YN = 4.9404 N-0.1045 2.0

Nitrided: YN = 3.517 N-0.0817

1.0 160 HB: YN = 2.3194 0.9 0.8 0.7

YN = 1.3558 N-0.0178

N-0.0538

YN = 1.6831 N-0.0323

0.6 0.5 102

103

104

105

106

107

108

109

1010

Number of load cycles, N

Figure 14.26 Stress cycle factor. (a) Bending stress cycle factor YN. Hamrock • Fundamentals of Machine Elements

Stress Cycle Factor 2.0

1.5

Stress cycle factor, Zn

Zn = 2.466 N-0.056 Zn = 1.4488 N-0.023

1.1 1.0 0.9

Nitrided Zn = 1.249 N-0.0138

0.8 0.7 0.6 0.5 102

103

104

105

106

107

108

109

1010

Number of load cycles, N

Figure 14.26 Stress cycle factor. (a) pitting resistance cycle factor ZN. Hamrock • Fundamentals of Machine Elements

Reliability Factor

Probability of Reliability factora , survival, percent KR 50 0.70b 90 0.85b 99 1.00 99.9 1.25 99.99 1.50 a Based on surface pitting. If tooth breakage is considered a greater hazard, a larger value may be required. b At this value plastic flow may occur rather than pitting.

Table 14.6 Reliability factor, KR.


Hardness Ratio Factor 1.16

R

a, p

1.14

R

Hardness ratio factor,CH

a,p

1.12 1.10

=

R

ap

1.08

=

0. 8

=1

0. 4

µm

.6 µ

µm

(1 6

(3 2µ

m

µi n. )

in. )

(64

µin

.)

1.06 1.04 1.02 1.00 180 200

For Rap > 1.6, use CH = 1.0 250

300

350

Brinell hardness of gear, HB

400

Figure 14.27 Hardness ratio factor CH for surface hardened pinions and through-hardened gears.


Loads on Gear Tooth Wr

W φ Wt Pitch circle

Figure 14.24 Loads acting on an individual gear tooth.


Loads and Dimensions of Gear Tooth W

Wr φ Wt

Wt l

bw

t

rf a x l t (a)

(b)

Figure 14.29 Loads and length dimensions used in determining tooth bending stress. (a) Tooth; (b) cantilevered beam. Hamrock • Fundamentals of Machine Elements

Bending and Contact Stress Equations Lewis Equation AGMA Bending Stress Equation Hertz Stress AGMA Contact Stress Equation

Wt pd !t = bwY Wt pd KaKsKmKvKiKb !t = bwY j

pH = E

!

!

!

W 2!

"1/2

!c = pH (KaKsKmKv)1/2


Lewis Form Factor Number of teeth 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32

Lewis form factor 0.176 0.192 0.210 0.223 0.236 0.245 0.256 0.264 0.270 0.277 0.283 0.292 0.302 0.308 0.314 0.318 0.322

Number of teeth 34 36 38 40 45 50 55 60 65 70 75 80 90 100 150 200 300

Lewis form factor 0.325 0.329 0.332 0.336 0.340 0.346 0.352 0.355 0.358 0.360 0.361 0.363 0.366 0.368 0.375 0.378 0.382

Table 14.7 Lewis form factor for various numbers of teeth (pressure angle, 20°; full-depth involute). Hamrock • Fundamentals of Machine Elements

Spur Gear Geometry Factors 1000  170  85  50  35  25  17 

.50

Geometry factor, Yj

.40

Number of teeth in mating gear. Load considered applied at highest point of single-tooth contact.

.30 Load applied at tip of tooth .20

.10

0 12

125 15

20

25

30

40

60 80

∞

275

Number of teeth, N

Figure 14.30 Spur gear geometry factors for pressure angle of 20° and full-depth involute profile. Hamrock • Fundamentals of Machine Elements

Application and Size Factors Power source

Uniform

Uniform Light shock Moderate shock

1.00 1.20 1.30

Driven Machines Light shock Moderate shock Application factor, Ka 1.25 1.50 1.40 1.75 1.70 2.00

heavy shock 1.75 2.25 2.75

Table 14.8 Application factor as function of driving power source and driven machine.

Diametral pitch, pd , in.−1 ≥5 4 3 3 1.25

Module, m, mm ≤5 6 8 12 20

Size factor, Ks 1.00 1.05 1.15 1.25 1.40

Table 14.9 Size factor as a function of diametral pitch or module. Hamrock • Fundamentals of Machine Elements

Load Distribution Factor Km = 1.0 +Cmc(Cp f Cpm +CmaCe)

where ! 1.0 for uncrowned teeth Cmc = 0.8 for crowned teeth

 0.80 when gearing is adjusted at assembly Ce = 0.80 when compatability between gear teeth is improved by lapping  1.0 for all other conditions


Face width, bw (in.) 0.70

0

10

20

30

40

Pinion Proportion Factor

Pinion proportion factor, Cpf

0.60 0.50 0.40 .00 2 = 0 /d p 1.5 0 bw 1 .0 0 0.5 For bw/dp < 0.5 use curve for bw/dp = 0.5

0.30 0.20

Figure 14.31 Pinion proportion factor Cpf.

0.10 0.00 0

200

400

600

800

1000

Face width, b (mm)

Cp f

w  bw   − 0.025 bw ≤ 25 mm    10d   b p w − 0.0375 + 0.000492bw 25 mm < bw ≤ 432 mm = 10d p     bw   − 0.1109 + 0.000815bw − (3.53 × 10−7)b2w432 mm < bw ≤ 1020 mm  10d p


Pinion Proportion Modifier

! 1.0,(S1/S) < 0.175 Cpm = 1.1,(S1/S) ≥ 0.175 S1

S/2 S

Figure 14.32 Evaluation of S and S1.


Mesh Alignment Factor Face width, bw (in) 0

10

20

30

40 2

0.90

Cma = A + Bbw + Cbw

Mesh alignment factor, Cma

0.80

If bw is in inches:

0.70

Condition Open gearing Commercial enclosed gears Precision enclosed gears Extraprecision enclosed gears

0.60 0.50 0.40

e Op

ng

o ncl al e i c r

se

ea dg

r un

me

0.20 0.10 0

200

0.247 0.127 0.0675 0.000380

B

C

0.0167 0.0158 0.0128 0.0102

-0.765 s 10-4 -1.093 s 10-4 -0.926 s 10-4 -0.822 s 10-4

its

If bw is in mm:

i ts r un gea m d it s e o C los ar un enc ed g e s n o l o c n cisi ion e Pr e recis p a E x tr

0.30

0.00

i ear ng

A

400 600 Face width, bw (mm)

Condition Open gearing Commercial enclosed gears Precision enclosed gears Extraprecision enclosed gears 800

A

B

C

0.247 0.127 0.0675 0.000360

6.57 s 6.22 s 10-4 5.04 s 10-4 4.02 s 10-4 10-4

1000

Figure 14.33 Mesh alignment factor.


-1.186 s 10-7 -1.69 s 10-7 -1.44 s 10-7 -1.27 s 10-7

Dynamic Factor Pitch line velocity, ft/min 0 1.8

2500 Qv = 5

5000 Qv = 6 Qv = 7

1.7 1.6 Dynamic factor, Kv

7500

Qv = 8

1.5

Qv = 9

1.4 Qv = 10 1.3 Qv = 11

1.2 1.1 "Very accurate" gearing 1.0

0

10

20

30

40

50

Pitch line velocity, m/s

Figure 14.34 Dynamic factor as a function of pitch-line velocity and transmission accuracy level number. Hamrock • Fundamentals of Machine Elements

Chapter 15: Helical, Bevel and Worm Gears

The main object of science is the freedom and happiness of man.

Thomas Jefferson


Summary of Gear Design

Gear type Spur

Helical

Bevel

Worm

Advantages Inexpensive, simple to design, no thrust load is developed by the gearing, wide variety of manufacturing options. Useful for high speed and high power applications, quiet at high speeds. Often used in lieu of spur gears for high speed applications. High efficiency (can be 98% or higher), can transfer power across nonintersecting shafts. Spiral bevel gears transmit loads evenly and are quieter than straight bevel. Compact designs for large gear ratios. Efficiency can be 90% or higher.

Disadvantages Can generate significant noise, especially at high speeds, and are usually restricted to pitch-line speeds below 20 m/s (4000 ft/min). Generate a thrust load on a single face, more expensive than spur gears. Shaft alignment is critical, rolling element bearings are therefore often used with bevel gears. This limits power transfer for high speed applications (where a journal bearing is preferable). Can be expensive. Wear by abrasion is of higher concern than other gear types, can be expensive. Generate very high thrust loads. Worm cannot be driven by gear; worm must drive gear.

Table 15.1 Design considerations for gears.


Helical Gear

Tangent to helical tooth

Element of pitch cylinder (or gear's axis)

Pitch cylinder

Helix angle, ψ

(a)

(b)

Figure 13.1 Helical gear. (a) Front view; (b) side view.


p

cn

Helical Gear Pitches

ψ

ψ

pc

pa (a)

(b)

Figure 15.2 Pitches of helical gears. (a) Circular; (b) axial.


AGMA Equations for Helical Gears

Bending Stress: Wt pd KaKsKmKvKiKb !t = bwYh

Pitting Resistance: !c = pH

!

KaKsKmKv Ih

" 12


Geometry Factors for Helical Gears ! = 20◦, " = 10◦ Gear teeth 12 14 17 21 Ih Yh 26 Ih Yh 35 Ih Yh 55 Ih Yh 135 Ih Yh

12 P

14 G

P

17 G

P

21 G

P

G

Pinion Teeth 26 35 P G P G

55 P

135 G

P

G

Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua

Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua

Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua

0.127 0.46 0.46 0.143 0.47 0.49 0.164 0.48 0.52 0.195 0.49 0.55 0.241 0.50 0.60

0.131 0.49 0.49 0.153 0.50 0.53 0.186 0.52 0.56 0.237 0.53 0.61

0.136 0.54 0.54 0.170 0.55 0.57 0.228 0.57 0.62

0.143 0.59 0.59 0.209 0.60 0.63

0.151 0.65 0.65

Table 15.2 Geometry factors Yh and Ih for helical gears loaded at tooth tip.


Geometry Factors for Helical Gears ! = 20◦, " = 20◦ Gear teeth 12 14 17 Ih Yh 21 Ih Yh 26 Ih Yh 35 Ih Yh 55 Ih Yh 135 Ih Yh

12 P

14 G

P

17 G

P

21 G

P

Pinion Teeth 26 G P G

35 P

55 G

P

135 G

P

G

Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua

Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua

0.125 0.44 0.44 0.140 0.45 0.46 0.156 0.45 0.49 0.177 0.46 0.51 0.205 0.47 0.54 0.245 0.48 0.58

0.129 0.47 0.47 0.145 0.48 0.49 0.167 0.49 0.52 0.197 0.50 0.55 0.242 0.51 0.59

0.133 0.50 0.50 0.155 0.51 0.53 0.188 0.52 0.56 0.238 0.54 0.60

0.138 0.54 0.54 0.172 0.55 0.57 0.229 0.57 0.61

0.144 0.58 0.58 0.209 0.60 0.62

0.151 0.64 0.64



Geometry Factors for Helical Gears ! = 20◦, " = 30◦ Pinion Teeth 21 26 P G P G

Gear 12 14 17 teeth P G P G P G a 12 U 14 Ih Ua 0.125 a Yh U 0.39 0.39 a 17 Ih U 0.139 0.128 Yh Ua 0.39 0.41 0.41 0.41 21 Ih Ua 0.154 0.144 0.132 Yh Ua 0.40 0.42 0.42 0.43 0.44 0.44 26 Ih Ua 0.169 0.159 0.148 Yh Ua 0.41 0.44 0.43 0.45 0.45 0.46 35 Ih Ua 0.189 0.180 0.170 a Yh U 0.41 0.46 0.43 0.47 0.45 0.48 55 Ih Ua 0.215 0.208 0.200 a Yh U 0.42 0.49 0.44 0.49 0.46 0.50 135 Ih Ua 0.250 0.248 0.245 a Yh U 0.43 0.51 0.45 0.52 0.47 0.53 a A ‘U’ indicates that this geometry would produce an undercut

35 P

55 G

P

135 G

0.135 0.46 0.46 0.158 0.139 0.47 0.48 0.49 0.49 0.190 0.174 0.145 0.48 0.50 0.50 0.51 0.52 0.52 0.240 0.231 0.210 0.49 0.53 0.51 0.54 0.53 0.55 tooth form and should be avoided.

P

0.151 0.56 0.56



G

Geometry Factors for Helical Gears ! = 25◦, " = 10◦ Gear teeth 12 14 Ih Yh 17 Ih Yh 21 Ih Yh 26 Ih Yh 35 Ih Yh 55 Ih Yh 135 Ih Yh

12 P

14 G

P

17 G

P

G


35 P

55 G

P

135 G

P

G

Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua Ua

0.129 0.47 0.47 0.144 0.48 0.51 0.159 0.48 0.55 0.175 0.49 0.58 Ua 0.50 0.61 0.221 Ua 0.257 0.52 0.70

0.133 0.52 0.52 0.149 0.52 0.55 0.165 0.53 0.58 0.195 0.54 0.62 0.215 0.51 0.65 0.255 0.56 0.71

0.136 0.56 0.56 0.152 0.57 0.59 0.186 0.57 0.63 0.206 0.55 0.66 0.251 0.60 0.72

0.139 0.60 0.60 0.175 0.61 0.64 0.195 0.58 0.67 0.246 0.63 0.73

0.162 0.64 0.64 0.178 0.62 0.68 0.236 0.67 0.74

0.148 0.65 0.69 0.215 0.71 0.75

0.70 0.70 0.154 0.76 0.76



Geometry Factors for Helical Gears ! = 25◦, " = 20◦ Gear teeth 12 Ih Yh 14 Ih Yh 17 Ih Yh 21 Ih Yh 26 Ih Yh 35 Ih Yh 55 Ih Yh 135 Ih Yh

12 P

G 0.128 0.47 0.47 0.140 0.47 0.50 0.154 0.48 0.53 0.169 0.48 0.56 0.184 0.49 0.59 0.202 0.49 0.62 0.227 0.50 0.66 0.258 0.51 0.70

14 P

17 G

0.131 0.50 0.50 0.145 0.51 0.54 0.161 0.51 0.57 0.176 0.52 0.60 0.196 0.53 0.63 0.222 0.53 0.67 0.257 0.54 0.71

P

G

0.134 0.54 0.54 0.150 0.55 0.58 0.166 0.55 0.60 0.187 0.56 0.64 0.215 0.57 0.67 0.255 0.58 0.72


0.137 0.58 0.58 0.153 0.59 0.61 0.176 0.60 0.64 0.206 0.60 0.68 0.251 0.62 0.72

0.140 0.62 0.62 0.163 0.62 0.65 0.196 0.63 0.69 0.246 0.65 0.73

35 P

55 G

0.144 0.66 0.66 0.178 0.67 0.70 0.236 0.68 0.74

P

135 G

0.148 0.71 0.71 0.214 0.72 0.75

P

0.153 0.76 0.76



G

Geometry Factors for Helical Gears ! = 25◦, " = 30◦ Gear teeth 12 Ih Yh 14 Ih Yh 17 Ih Yh 21 Ih Yh 26 Ih Yh 35 Ih Yh 55 Ih Yh 135 Ih Yh

12 P

14 G

0.130 0.46 0.46 0.142 0.47 0.49 0.156 0.47 0.51 0.171 0.48 0.54 0.186 0.48 0.56 0.204 0.49 0.58 0.228 0.49 0.61 0.259 0.50 0.64

P

17 G

0.133 0.49 0.49 0.147 0.50 0.52 0.163 0.50 0.54 0.178 0.51 0.56 0.198 0.51 0.59 0.223 0.52 0.61 0.257 0.53 0.64

P

G

0.136 0.52 0.52 0.151 0.53 0.55 0.167 0.53 0.57 0.188 0.54 0.59 0.216 0.54 0.62 0.255 0.55 0.65


0.138 0.55 0.55 0.154 0.56 0.57 0.176 0.56 0.60 0.207 0.57 0.62 0.251 0.58 0.66

0.141 0.58 0.58 0.163 0.58 0.60 0.196 0.59 0.63 0.245 0.60 0.66

35 P

55 G

0.144 0.61 0.61 0.178 0.62 0.64 0.234 0.62 0.67

P

135 G

0.147 0.64 0.64 0.212 0.65 0.68

P

0.151 0.68 0.68



G

Bevel Gears Pitch apex to back Crown to back Pitch apex to crown Crown

Pitch apex Face width

Face angle

Shaft angle Pitch angle

Back angle

Pinion

Dedendum angle

Root angle Uniform clearance

Front angle

Gear

Pitch diameter

ist

an

ce

Outside diameter

Ba

ck c

on

ed

Back cone

Figure 15.3 Terminology for bevel gears.


Bevel Gears

Figure 15.4 Bevel gears with curved teeth. (a) Spiral bevel gears; (b) Zerol®. Hamrock • Fundamentals of Machine Elements

Bevel Gear Forces

Wt T W= = cos ! r cos ! Wa = Wt tan ! sin " Wr = Wt tan ! cos "

Figure 15.5 Forces acting on a bevel gear Hamrock • Fundamentals of Machine Elements

Design of Bevel Gears

Bending Stress:

 2Tp pd KaKvKsKm   English units  bwd p KxYb !t = 2T K K K K p a v s m   SI units  bwd p m pKxYb

Pitting Resistance:

% !  T E p     "bwd 2Ib KaKvKmKsKx English units % p !c = !  2T E  p   Cp "bwd 2Ib KaKvKmKsKx SI units p


Load Distribution Factor for Bevel Gears ! Kmb + 0.0036b2w English units Km = Kmb + 5.6b2w SI units

where Kmb = 1.00 for both gear and pinion straddle mounted (bearings on both sides of gear) = 1.10 for only one member straddle mounted = 1.25 for neither member straddle mounted


Size Factor for Bevel Gears

1.0

1.6 5

10

Outer transverse module, met 20 30

40

25

50

Face width, bw, mm 50 75 100

125

Ks = 1.0 for bw > 114.3 mm (4.5 in.)

1.00 Ks = 0.4867 + 0.2133/pd = 0.4867 + 0.008399 met

0.8

Size factor, Ks

Size factor, Ks

0.9

0.7 0.6

0.75

Ks = 0.00492 bw + 0.4375 (bw in mm) = 0.125 bw + 0.4375 (bw in in.)

0.50

Ks = 0.5 for bw < 12.7 mm (0.5 in.)

0.5 -1

Ks = 0.5 for met < 1.6 (pd < 16 in. ) 0.4

16

5

2.5

1.25 0.8 0.6 -1 Outer transverse pitch, pd, in. (a)

0.25 0.5

0

1.0

2.0 3.0 Face width, bw, in. (b)

Figure 15.6 Size factor for bevel gears. (a) Size factor for bending stress; (b) size factor for contact stress or pitting resistance. Hamrock • Fundamentals of Machine Elements

4.0

5.0

Straight Bevel Gear Geometry Factor for Contact Stess

Figure 15.7 Geometry factor for straight bevel gears with pressure angle = 20° and shaft angle 90°. (a) Geometry factor for contact stress, Ib. Hamrock • Fundamentals of Machine Elements

Straight Bevel Gear Geometry Factor for Bending Number of teeth on gear for which geometry factor is desired

100

13 15

20

Number of teeth in mate 25 30 35 40 45 50

100 90

90 80

80 70

70 60

60 50 40 30 20 10 0.16

0.20

0.24 0.28 Geometry factor, Yb

0.32

0.36

0.40

Figure 15.7 Geometry factor for straight bevel gears with pressure angle = 20° and shaft angle 90°. (b) Geometry factor for bending, Yb. Hamrock • Fundamentals of Machine Elements

Spiral Bevel Gears Geometry Factor for Contact Stress Number of teeth in gear 50

50

60

70 80 90 100

Number of pinion teeth

45 40

40

35 30

30

25 20

20 15

10 0.04

0.06

0.08 0.10 0.12 Geometry factor, Ib

0.14

0.16

Figure 15.8 Geometry factor for spiral bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for contact stress, Ib. Hamrock • Fundamentals of Machine Elements

Spiral Bevel Gears Geometry Factor for Bending Number of teeth in mate 12

100

20

30 40 50

Number of teeth on gear for which geometry factor is desired

90 80 70 60 60708090 100

50 40 30 20 10 0.12

0.16

0.20 0.24 0.28 Geometry factor, Yb

0.32

0.36

Figure 15.8 Geometry factor for spiral bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for bending, Yb. Hamrock • Fundamentals of Machine Elements

Zerol Bevel Gears Geometry Factor for Contact Stress Number of gear teeth 50

50

60

70

80

90 100

Number of pinion teeth

45 40

40

35 30

30 25 20

20 15

10

0.04

0.05

0.06 0.07 0.08 Geometry factor, Ib

0.09

0.10

0.11

Figure 15.9 Geometry factor for Zerol bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for contact stress, Ib. Hamrock • Fundamentals of Machine Elements

Zerol Bevel Gears Geometry Factor for Bending Number of teeth in mate

Number of teeth in gear for which geometry factor is desired

100

13 15

20 25 30 35 40 45 50

90 80 70 60 60

50

70

80 90 100

40 30 20 10 0.16

0.20

0.32 0.28 0.24 Geometry factor, Yb

0.36

0.40

Figure 15.9 Geometry factor for Zerol bevel gears with pressure angle = 20°, spiral angle = 25°, and shaft angle 90°. (a) Geometry factor for bending, Yb. Hamrock • Fundamentals of Machine Elements

Worm Gear Contact

Figure 15.10 Illustration of worm contact with a worm gear, showing multiple teeth in contact. Hamrock • Fundamentals of Machine Elements

Minimum Number of Worm Gear Teeth

Pressure angle, φ deg 14.5 17.5 20 22.5 25 27.5 30

Minimum number of wormgear teeth 40 27 21 17 14 12 10

Table 15.3 Suggested minimum number of worm gear teeth for customary designs.


Forces on a Worm Gear

Figure 15.11 Forces acting on a worm. (a) Side view, showing forces acting on worm and worm gear. (b) Three-dimensional view of worm, showing worm forces. The worm gear has been removed for clarity. Hamrock • Fundamentals of Machine Elements

AGMA Equations for Worm Gears The rated input power is   

vtW f NWt dg + English units h pi = (126, 000)Z 33, 000   !Wt dg + vtW f SI units 2Z

where    !Ndwm English units cos " vt = 12 #d   wm SI units 2 cos "


AGMA Equations for Worm Gears (cont.) Tangential Force

where

 0.8 Csdgm bwCmCv English units 0.8 Wt = Csdgm bwCmCv  SI units 75.948

 % 2 &0.5  + 0.463 ≤ Z < 20 0.0200 −Z + 40Z − 76 % 2 &0.5 Cm = 0.0107 −Z + 56Z + 5145 20 ≤ Z < 76  1.1483 − 0.00658Z 76 ≤ Z   0.659 exp (−0.0011vt )0 < vt ≤ 700 ft/min (−0.571) 700 ft/min < vt ≤ 3000 ft/min Cv = 13.31vt  65.52v(−0.774) 3000 ft/min < vt t


AGMA Equations for Worm Gears (cont.) Friction Force

µWt Wf = cos ! cos "n

where  vt = 0 ft/min 0.150 $ % 0 < vt ≤ 10 ft/min µ = 0.124 exp $−0.074vt0.645%  0.103 exp −0.110vt0.450 + 0.01210 ft/min < vt


Materials Parameter for Worm Gears Mean gear pitch diameter, d (mm) 1000II

70 100

111

I 200 I

I 500 I I I Ill 1000 I Ce n tr ifu gal ly

I

S ta

900

ti c

cas t

Materials factor, Cs

ill

c as ed org rf to

t cas

700

12

1IO00 1000

ch

nd

800 800

Sa

Materials factor, Cs

I 2500 I

20 20

Center distance, cd (mm) 30 30

50 50

40 40

60 60

70 70

7575

i i i 3 i in

I i i i i 1i i i i i i i i i 900

IIIIII II

Y

I i i i i i i

800 800 1-i

/

I I I CALlbN A. .-..a. r*al m-r A ..IlT

C;HtC;K I-ItiUHt

1 HIWJ

USE THE LOWER OF THE TWO VALUES

3

600 600 í FACTORFOR CENTER t DISTANCESc 3.00 IN (76 mm) I I 500 500 1 í I I I I III

2.5 3 2.5

! I

I Y I I

I I

I

I

I I

90 20 25 30 30 40 40 50 60 60 70 160 90 4 55 6 7 891010 15 20 MEAN GEAR DIAMETER, Dm -INCHES Mean gear pitch diameter, d (in.)

(a)

700 0.5 0.5

1.0 1.0

1.5 2.0 DISTANCET:NCHES Center C::TER distance, cd (in.)

2.5 2.5

3.0 3.0

(b)

Figure 15.12 Materials parameter Cs for bronze worm gears and worm minimum surface hardness of 58 Rc. (a) Materials factor for center distances cd greater than 76 mm (3 in); (b) Materials factor for center distances cd less than 76 mm (3 in). When using the figure in (b), the value from part (a) should be checked and the lower value used. See also Table 15.4. Hamrock • Fundamentals of Machine Elements

Materials Factor for Worm Gear

Manufacturing Process Sand casting Static chill cast or forged Centrifugally cast

Pitch diameter d ≤ 64 mm (2.5 in.) d ≥ 64 mm d ≤ 200 mm (8 in.) d > 200 mm d ≤ 625 mm (25 in.) d > 625 mm

Units for pitch diameter in. mm 1000 1000 1189.6365 − 476.5454 log d 1859.104 − 476.5454 log d 1000 1000 1411.6518 − 455.8259 log d 2052.012 − 455.8259 log d 1000 1000 1251.2913 − 179.7503 log d 1503.811 − 179.7503 log d

Table 15.4 Materials factor Cs for bronze worm gears with worm having surface hardness of 58 Rc.


Food Mixer Gear Train Case Study

Speed

Torque

Tmax= 20 ft-lbf Maximum current

Motor current, A

Table 15.13 The gears used to transmit power from an electric motor to the agitators of a commercial mixer. Hamrock • Fundamentals

Table 15.14 Torque and speed of motor as a function of current for the industrial mixer used in the Case Study. of Machine Elements

Chapter 16: Fasteners and Power Screws

Engineers need to be continually reminded that nearly all engineering failures result from faulty judgments rather than faulty calculations. Eugene S. Ferguson, Engineering and the Mind’s Eye


Thread Geometry p

l p

Crest β

d

β

α

l p

l p

ht

dc dp

dr

Root (a)

Figure 16.1 Parameters used in defining terminology of thread profile.

(b)

(c)

Figure 16.2 (a) Single-, (b) double-, and (c) triple-threaded screws.


Acme, UN, and M Threads

29∞

60∞

(a)

Figure 16.3 Thread profiles. (a) Acme; (b) UN.

(b)

0.125 ht 0.125 p

0.5 p

β

0.625 ht 0.375 ht

0.5 p

Figure 16.4 Details of M and UN thread profiles.

0.25 ht 0.25 p


Pitch diameter

Equivalent Threads and Acme Profile Inch series Bolts Nuts 1A 1B 2A 2B 3A 3B

Metric series Bolts Nuts 8g 7H 6g 6H 8h 5H

Table 16.1 Inch and metric equivalent thread classifications.

p p 2.7

Figure 16.5 Details of Acme thread profile. (All dimensions are in inches.)

0.5p + 0.01

β = 29∞

dc

dp

dr

p – 0.052 2.7


Crest diameter, dc , in. 1/4 5/16 3/8 7/16 1/2 5/8 3/4 7/8 1 1 1/8 1 1/4 1 3/8 1 1/2 1 3/4 2 2 1/4 2 1/2 2 3/4 3 3 1/2 4 4 1/2 5

Number of threads per inch, n 16 14 12 12 10 8 6 6 5 5 5 4 4 4 4 3 3 3 2 2 2 2 2

Acme Threads Tensile stress area, At , in.2 0.02632 0.04438 0.06589 0.09720 0.1225 0.1955 0.2732 0.4003 0.5175 0.6881 0.8831 1.030 1.266 1.811 2.454 2.982 3.802 4.711 5.181 7.338 9.985 12.972 16.351

Shear stress area, As , in.2 0.3355 0.4344 0.5276 0.6396 0.7278 0.9180 1.084 1.313 1.493 1.722 1.952 2.110 2.341 2.803 3.262 3.610 4.075 4.538 4.757 5.700 6.640 7.577 8.511

Table 16.2 Crest diameters, threads per inch, and stress areas for Acme threads.


Power Screws Load, W (Screw is threaded into W) dp/2 β/2 β/2 Pitch, p

α Thrust collar

!

(d p/2)(cos !n tan " + µ) Tr = W + rcµc cos !n − µtan " !

"

(d p/2)(µ− cos !n tan ") Tl = −W + rcµc cos !n + µtan "

Equal rc


"

C B

Forces on Power Screw

A

Axis of screw

A

D B

Pn cosθn cosα θn α β/2

Pn

β/2

H 0

E

0 dp/2

Pn cosθn cosα tan(β/2)

E (a)

(b)

C

A

Pn cosθn cosα Pn cosθn

µPn sinα µP n

α Pn cosθn sinα H

α

µPn cosα

0

µcW

W

P

Figure 16.7 Forces acting in raising load of power screw. (a) Forces acting on parallelepiped; (b) forces acting on axial section; (c) forces acting on tangential plane.

(c)


Types of Threaded Fasteners

(a)

(b)

(c)

Figure 16.8 Three types of threaded fastener. (a) Bolt and nut; (b) cap screw; (c) stud.


Illustration

Type Hex-head bolt

Description An externally threaded fastener with a trimmed hex head, often with a washer face on the bearing side.

Application notes Used in a variety of general purpose applications in different grades depending on the required loads and material being joined.

Carriage bolt

A round head bolt with a square neck under the head and a standard thread.

Used in slots where the square neck keeps the bolt from turning when being tightened.

Elevator bolt or belt bolt

A bolt with a wide, countersunk flat head, a shallow conical bearing surface, an integrally-formed square neck under the head and a standard thread.

Used in belting and elevator applications where head clearances must be minimal.

Serrated flange bolt

A hex bolt with integrated washer, but wider than standard washers and incorporating serrations on the bearing surface side.

Used in applications where loosening hazard exists, such as vibration applications. The serrations grip the surface so that more torque is needed to loosen than tighten the bolt.

Flat cap screw (slotted head shown)

A flat, countersunk screw with a flat top surface and conical bearing surface.

A common fastener for assembling joints where head clearance is critical.

Buttunhead cap screw (socket head shown)

Dome shaped head that is wider and has a lower profile than a flat cap screw.

Designed for light fastening applications where their appearance is desired. Not recommended for high-strength applications.

Lag screw

A screw with spaced threads, a hex head, and a gimlet point. (Can also be made with a square head.)

Used to fasten metal to wood or with expansion fittings in masonry.

Step bolt

A plain, circular, oval head bolt with a square neck. The head diameter is about three times the bolt diameter.

Used to join resilient materials or sheet metal to supporting structures, or for joining wood since the large head will not pull through.

Types of Bolts and Screws

Table 16.3 Common types of bolts and screws.


Types of Nuts Illustration Nuts

Type Hex nut

Nylon insert stop

Cap nut

Castle nut

Description

Application notes

A six-sided internally threaded fastener. Specific dimensions are prescribed in industry standards. A nut with a hex profile and an integral nylon insert.

The most commonly used generalpurpose nut.

Similar to a hex nut with a dome top.

Used to cover exposed, dangerous bolt threads or for aesthetic reasons.

A type of slotted nut.

Used for general purpose fastening and locking. A cotter pin or wire can be inserted through the slots and the drilled shank of the fastener. Used to join two externally threaded parts of equal thread diameter and pitch.

The nylon insert exerts friction on the threads and prevents loosening due to vibration or corrosion.

Coupling nut

A six-sided double chamfered nut.

Hex jam nut

A six-sided internally threaded fastener, thinner than a normal hex nut. A hex nut preassembled with a free spinning external tool lock washer. When tightened, the teeth bite into the member to achieve locking. An internally threaded nut with integral pronounced flat tabs.

Used in combination with a hex nut to keep the nut from loosening. A popular lock nut because of ease of use and low cost.

A hex nut with integrated washer, but wider than standard washers and incorporating serrations on the bearing surface side.

Used in applications where loosening hazard exists, such as vibration applications. The serrations grip the surface so that more torque is needed to loosen than tighten the bolt.

K-lock or keplock nut

Wing nut

Serrated nut

Used for applications where repetitive hand tightening is required.

Table 16.4 Common nuts and washers for use with threaded fasteners. Hamrock • Fundamentals of Machine Elements

Types of Washers Washers Flat washer

A circular disk with circular hole, produced in accordance with industry standards. Fender washers have larger surface area than conventional flat washer.

Designed for general-purpose mechanical and structural use.

Belleville washer

A conical disk spring.

Split lock washer

A coiled, hardened, split circular washer with a slightly trapezoidal cross-section

Used to maintain load in bolted connections. Preferred for use with hardened bearing surfaces. Applies high bolt tension per torque, resists loosening caused by vibration and corrosion.

Tooth lock washer

A hardened circular washer twisted teeth or prongs.

with

Internal teeth are preferred fro aesthstics since the teeth are hidden under the bolt head. External teeth give greater locking efficiency. Combination teeth are used for oversized or out-ofround holes or for electrical connections.

Table 16.4 (cont.) Common nuts and washers for use with threaded fasteners. Hamrock • Fundamentals of Machine Elements

Pb (tension)

Pj (compression)

Bolt and Nut Forces

0b

0j

δ b (extension)

kb

δ j (contraction) (a)

kj Pi Pb

0b

Figure 16.9 Bolt-andnut assembly modeled as bolt-and-joint spring.

Pj

Extension (b)

0j

Figure 16.10 Force versus deflection of bolt and member. (a) Separated bolt and joint; (b) assembled bolt and joint. Hamrock • Fundamentals of Machine Elements

Force vs. Deflection Pj

Pb Pi + kbek Pi Load

P = increase in Pb plus decrease in Pj Pi – kjek

0b

0j Deflection

ek (extension of bolt = reduction in contraction of joint)

Figure 16.11 Forces versus deflection of bolt and joint when external load is applied. Hamrock • Fundamentals of Machine Elements

Bolt Stiffness

dc Lse = Ls + 0.4dc

Ls

1 4 = kb !E Lt

dr

!

Ls + 0.4dc Lt + 0.4dr + 2 dc dr2

Lte = Lt + 0.4dr

Figure 16.12 Bolt and nut. (a) Assembled; (b) stepped-shaft representation of shank and threaded section. Hamrock • Fundamentals of Machine Elements

"

Member Stiffness as Conical Frustum dw

αf

L

dc

Figure 16.13 Bolt and nut assembly with conical frustum stress representation of joint. Hamrock • Fundamentals of Machine Elements

Member Stiffness Equations Decription Single member, general case

Single member, αf = 30◦

Two members, same Young’s modulus, E, back-to-back frustaa Two members, same Young’s modulus, back-to-back frusta, α = 30◦ , di = dw = 1.5dc a

Member stiffness, km πEj dc tan αf ! " kji = (2Li tan αf + di − dc )(di + dc ) ln (2Li tan αf + di + dc )(di − dc ) 1.813Ej dc ! " kji = (1.15Li + di − dc )(di + dc ) ln (1.15Li + di + dc )(di − dc ) πEj dc tan αf ! " kj = (2Li tan αf + di − dc )(di + dc ) 2 ln (2Li tan αf + di + dc )(di − dc ) 1.813Ej dc " ! kj = (2.885Li + 2.5dc ) 2 ln (0.577Li + 2.5dc )

Two members, same material, Wileman kj = Ei dc Ai eBi dc /Li methoda,b a Note that this is stiffness for the complete joint, not a member in the joint. b See Table 16.6 for values of A and B for various materials. i i

Table 16.5 Member stiffness equations for common bolted joint configurations. Hamrock • Fundamentals of Machine Elements

Wileman Member Stiffness k j = EidcAieB j dc/L

Material Steel Aluminum Copper Gray cast iron

Poisson’s ratio, ν 0.291 0.334 0.326 0.211

Modulus of elasticity, E, GPa 206.8 71.0 118.6 100.0

Numerical Ai 0.78715 0.79670 0.79568 0.77871

constants, Bi 0.62873 0.63816 0.63553 0.61616

Table 16.6 Constants used in joint stiffness formula.


Example 16.6 3/2 dc

30∞

15

dc

1

3 10

8

10

25 2 12.5

d2

(a)

(b)

Figure 16.14 Hexagonal bolt-and-nut assembly used in Example 16.6. (a) Assembly and dimensions; (b) dimensions of frustrun cone. Hamrock • Fundamentals of Machine Elements

Strength of Steel Bolts SAE grade 1

Head marking

Range of crest diameters, in. 1/4 – 1 1/2

Ultimate tensile strength Su , ksi 60

Yield strength, Sy , ksi 36

Proof strength, Sp , ksi 33

2

1/4 – 3/4 > 3/4 – 1 1/2

74 60

57 36

55 33

4

1/4 – 1 1/2

115

100

65

5

1/4 – 1 >1 - 1 1/2

120 105

92 81

85 74

5.2

1/4-1

120

92

85

7

1/4 – 1 1/2

133

115

105

8

1/4 – 1 1/2

150

130

120

8.2

1/4 -1

150

130

120

Table 16.7 Strength of steel bolts for various sizes in inches.


Strength of Steel Bolts Metric grade 4.6 4.8 5.8 8.8 9.8 10.9 12.9

Head marking 4.6 4.8 5.8 8.8 9.8 10.9 12.9

Crest diameter, dc , mm M5 – M36

Ultimate tensile strength, Su , MPa 400

Yield strength, Sy , MPa 240

Proof strength, Sp , MPa 225

M1.6 – M16

420

340a

310

M5 – M24

520

415a

380

M17 – M36

830

660

600

M1.6 – M16

900

720a

650

M6 – M36

1040

940

830

M1.6 – M36

1220

1100

970

Table 16.8 Strength of steel bolts for various sizes in millimeters.


UN Coarse and Fine Threads Crest diameter, dc , in. 0.0600 0.0730 0.0860 0.0990 0.1120 0.1250 0.1380 0.1640 0.1900 0.2160 0.2500 0.3125 0.3750 0.4750 0.5000 0.5625 0.6250 0.7500 0.8750 1.000 1.125 1.250 1.375 0.500 1.750 2.000

Coarse threads (UNC) Number of Root Tensile stress threads per, diameter, area, At , inch, n dr , in. in.2 — — — 64 0.05609 0.00263 56 0.06667 0.00370 48 0.07645 0.00487 40 0.08494 0.00604 40 0.09794 0.00796 32 0.1042 0.00909 32 0.1302 0.0140 24 0.1449 0.0175 24 0.1709 0.0242 20 0.1959 0.0318 18 0.2523 0.0524 16 0.3073 0.0775 14 0.3962 0.1063 13 0.4167 0.1419 12 0.4723 0.182 11 0.5266 0.226 10 0.6417 0.334 9 0.7547 0.462 8 0.8647 0.606 7 0.9703 0.763 7 1.095 0.969 6 1.195 1.155 6 1.320 1.405 5 1.533 1.90 4.5 1.759 2.5

Fine threads (UNF) Number of Root Tensile stress threads per, diameter, area, At , inch, n dr , in. in.2 80 0.04647 0.00180 72 0.05796 0.00278 64 0.06909 0.00394 56 0.07967 0.00523 48 0.08945 0.00661 44 0.1004 0.00830 40 0.1109 0.01015 36 0.1339 0.01474 32 0.1562 0.0200 28 0.1773 0.0258 28 0.2113 0.0364 24 0.2674 0.0580 24 0.3299 0.0878 20 0.4194 0.1187 20 0.4459 0.1599 18 0.5023 0.203 18 0.5648 0.256 16 0.6823 0.373 14 0.7977 0.509 12 0.9098 0.663 12 1.035 0.856 12 1.160 1.073 12 1.285 1.315 12 1.140 1.581 — — — — — —

Table 16.9 Dimensions and tensile stress areas for UN coarse and fine threads. Root diameter is calculated from Eq. (16.2) and Fig. 16.4. Hamrock • Fundamentals of Machine Elements

M Coarse and Fine Threads Crest diameter, dc , mm 1 1.6 2 2.5 3 4 5 6 8 10 12 16 20 24 30 36 42 48

Coarse threads (MC) Root Tensile Pitch, diameter, stress area, p, mm dr , mm At , mm2 0.25 0.7294 0.460 0.35 1.221 1.27 0.4 1.567 2.07 0.45 2.013 3.39 0.5 2.459 5.03 0.7 3.242 8.78 0.8 4.134 14.2 1.0 4.917 20.1 1.25 6.647 36.6 1.5 8.376 58.0 1.75 10.11 84.3 2.0 13.83 157 2.5 17.29 245 3.0 20.75 353 3.5 26.21 561 4.0 31.67 817 4.5 37.13 1121 5.0 42.59 1473

Fine threads (MF) Root Tensile Pitch, diameter, stress area, p, mm dr , mm At , mm2 — — — 0.20 1.383 1.57 0.25 1.729 2.45 0.35 2.121 3.70 0.35 2.621 5.61 0.5 3.459 9.79 0.5 4.459 16.1 0.75 5.188 22 1.0 6.917 39.2 1.25 8.647 61.2 1.25 10.65 92.1 1.5 14.38 167 1.5 18.38 272 2.0 21.83 384 2.0 27.83 621 3.0 32.75 865 — — — — — —

Table 16.10 Dimensions and tensile stress areas for M coarse and fine threads. Root diameter is calculated from Eq. (16.2) and Fig. 16.4. Hamrock • Fundamentals of Machine Elements

Pj

Pj

Separation of Joint

Figure 16.15 Separation of joint.


Pj

Pb, max Pi

Pi

Load on bolt

Pb

Pj, min

0b

Deflection

SAE grade 0-2 4-8

∆δ

Metric grade 3.6-5.8 6.6-10.9

Load on joint

Threads in Fatigue Loading

Figure 16.16 Forces versus deflection of bolt and joint as function of time.

0j

Rolled threads 2.2 3.0

Cut threads 2.8 3.8

Fillet 2.1 2.3

Table 16.11 Fatigue stress concentration factors for threaded elements.


Gaskets

Gasket

Figure 16.17 Threaded fastener with unconfined gasket and two other members.


Failure Modes for Fasteners in Shear

(a)

(b)

(c)

(d)

Figure 16.18 Failure modes due to shear loading of riveted fasteners. (a) Bending of member; (b) shear of rivet; (c) tensile failure of member; (d) bearing of member on rivet. Hamrock • Fundamentals of Machine Elements

Example 16.9 P = 1000 lb 7

8

A

B

2 A

3

B

y

3 2

rB

rA

rC

x

C

rD

D D

C

(a)

(b) τtA

rC

A

3

B τtB

α

τd

4.635

τd

5

7– 8

3 3

rD

3

β 2.365 (c)

τtC

5 C

τtD

τd (d)

7– 8

D 1– 2

τd (e)

Figure 16.19 Group of riveted fasteners used in Example 16.9. (a) Assembly of rivet group; (b) radii from centroid to center of rivets; (c) resulting triangles; (d) direct and torsional shear acting on each rivet; (e) side view of member. (All dimensions in inches.)


Weld Symbols Bead

Basic arc and gas weld symbols Plug Groove Fillet or Bevel V slot Square

Basic resistance weld symbols U

J

Finish symbol

Root opening, depth of filling for plug and slot welds Effective throat

Flash or upset

Seam

Length of weld in inches Pitch (center-to-center spacing) of welds in inches

F A

Depth of preparation or size in inches Reference line

Field weld symbol

S(E) T

(Both sides) (Other (Arrow side) side)

R

Specification, process or other reference

Basic weld symbol or detail reference

Projection

Groove angle or included angle of countersink for plug welds

Contour symbol

Tail (omitted when reference is not used)

Spot

Weld-all-around symbol L@P A

B

Arrow connects reference line to arrow side of joint. Use break as at A or B to signify that arrow is pointing to the grooved member in bevel or J-grooved joints.

Figure 16.20 Basic weld symbols. Hamrock • Fundamentals of Machine Elements

Fillet Weld 1 — 16

in. Actual weld configuration

he

Assumed weld configuration te

Shear plane of weld at throat he (a)

Load 1 — 16

1

in. clear for plates –4 in. thick

Shear stress Shear stress L

te te Shear planes

L

Figure 16.21 Fillet weld. (a) Cross-section of weld showing throat and legs; (b) shear planes.

(b)


Geometry of Welds Dimensions of weld

Bending

Torsion P a

Weld

a x

d x

x A=d

P

Weld Ju = d 3/12

Iu = d 2/6

x

T = Pa

M = Pa

c = d/2

b a

Weld d

x

x

P

Weld

P

x x

Iu = d 2/3

A = 2d

P

a

d(3b2 + d 2) Ju = ––––––––– 6

b a

Weld d x

x

P

Weld

P

x x A = 2b

Iu = bd P

a

b3 + 3bd2 Ju = –––––––– 6

Table 16.12 Geometry of welds and parameters used when considering various types of loading. Hamrock • Fundamentals of Machine Elements

Geometry of Welds

–y d

b

a

–x

x

a

Weld

x

x b2

Weld

x

–x = –––––– 2(b + d)

A = b+ d

P

d2

4bd + At top Iu = ––––––– 6 d 2(4b + d) At bottom Iu = –––––––– 6(2b + d)

b

d x A = d + 2b

(b + d)4 – 6b2d2 Ju = ––––––––––––– 12(b + d)

x

x

d2 –y = –––––– 2(b + d)

a

–x

a

Weld

x

x b2 –x = –––––– 2(b + d)

P

x

P

P

Weld x

Iu = bd + d 2/6 x

Weld

Weld (2b + d)3 b2(b + d)2 Ju = –––––– – ––––––––– 12 (2 b + d)

Table 16.12 (cont.) Geometry of welds and parameters used when considering various types of loading. Hamrock • Fundamentals of Machine Elements

Geometry of Welds Dimensions of weld –y

Bending

Torsion

b a

Weld d x

x

x

d2 – y = –––––– (b + 2d)

A = b + 2d

b d x A = 2b + 2d

x

x

x Weld

2bd + d2 At top Iu = ––––––– 3 2 d (2b + d) At bottom Iu = –––––––– 3(b + d)

a

Weld all around

P

Weld

P

x

P

Weld a

P

(b + 2d)3 d2(b + d)2 Ju = –––––– – ––––––––– 12 (b + 2d)

Weld all around

P

x x

Iu = bd + d 2/3 P

a

(b + d)3 Ju = –––––––– 6


Geometry of Welds

b

a

Weld x

d x

Weld

P

P

x

x x

x

Iu = bd + d 2/3

A = 2b + 2d Weld

Weld P

Weld all around

d x

x

a

Weld all around

P

a

b 3 + 3bd 2+ d 3 Ju = ––––––––—— 6

P

x

x x A = πb

x Iu = π(d 2/4) P

a

Ju = π(d 3/4)


Electrode Properties

Electrode number E60XX E70XX E80XX E90XX E100XX E120XX

Ultimate tensile strength, Su , ksi 62 70 80 90 100 120

Yield strength, Sy , ksi 50 57 67 77 87 107

Elongation, ek , percent 17-25 22 19 14-17 13-16 14

Table 16.13 Minimum strength properties of electrode classes.


Example 16.10 y 100

τtx τty

300

l2

A

A 45

l1 y

80

150 20 kN x

τtx

B

B (a)

τty

x

(b)

Figure 16.22 Welded bracket used in Example 16.10. (a) Dimensions, load and coordinates; (b) torsional shear stress components at points A and B. (All dimensions in millemeters.) Hamrock • Fundamentals of Machine Elements

Welds in Fatigue

Type of weld Reinforced butt weld Tow of transverse fillet weld End of parallel fillet weld T-butt joint with sharp corners

Fatigue stress concentration factor, Kf 1.2 1.5 2.7 2.0

Table 16.14 Fatigue strength reduction factors for welds.


Adhesive Joints (a)

b

L

(b)

(c)

(d)

Figure 16.23 Four methods of applying adhesive bonding. (a) Lap; (b) butt; (c) scarf; (d) double lap.


P

θ

Scarf Joint

τ

P

θ

σx

tm

σn

(a)

θ

A tm

M

M

A (b)

y

x l

O

ro ri

T

T

Figure 16.24 Scarf joint. (a) Axial loading; (b) bending; (c) torsion.

(c)


Integrated Snap Fasteners

h 2 h

Deflected (a)

Rigid (b)

(c)

Figure 16.25 Common examples of integrated fasteners. (a) Module with four cantilever lugs; (b) cover with two cantilever and two rigid lugs; (c) separable snap joint for chassis cover. Hamrock • Fundamentals of Machine Elements

Snap Joint Design Shape of cross section

A

B a

c

h

c

c2

b Type of design

1

P α

h h _ 2

Figure 16.26 Cantilever snap joint.

(Permissible) deflection

y

b

Trapezoid

P

l

l

Rectangle

h

c1

h

y

εl2 y = 0.67 ___ h

a + b(1) ___ εl2 y = _______ 2a + b h

εl2 y = 1.09 ___ h

a + b(1) ___ εl2 y = 1.64 _______ 2a + b h

εl2 y = 0.86 ___ h

a + b(1) ___ εl2 y = 1.28 _______ 2a + b h

Cross section constant over length h _ 2

2

h

y

All dimensions in direction y (e.g., h) decrease to one-half b _ 4

3

b

z

All dimensions in direction z (e.g., b and a) decrease to one-quarter

Figure 16.27 Permissible deflection of different snap fastener cantilever shapes. Hamrock • Fundamentals of Machine Elements

Friction for Integrated Snap Fastener Mateirals

Material Polytetrafluoroethylene PTFE (teflon) Polyethylene (rigid) Polyethylene (flexible) Polypropylene Polymethylmethacrylate (PMMA) Acrylonitrile-butadiene-styrene (ABS) Polyvinylchloride (PVC) Polystyrene Polycarbonate

Coefficient of friction On self-mated On steel polymer 0.12-0.22 — 0.20-0.25 0.40-0.50 0.55-0.60 0.66-0.72 0.25-0.30 0.38-0.45 0.50-0.60 0.60-0.72 0.50-0.65 0.60-0.78 0.55-0.60 0.55-0.60 0.40-0.50 0.48-0.60 0.45-0.55 0.54-0.66

Table 16.15 Coefficients of friction for common snap fastener polymers


Hydraulic Baler Case Study

End cap Seal

Cylinder flange

Figure 16.28 End cap and cylinder flange.


Chapter 17: Springs

Entia non multiplicantor sunt prater necessitatum. (Do not complicate matters more than necessary.) Galileo Gallilei


Stress

Stress Cycle

∆U

U

Strain

Figure 17.1 Stress-strain curve for one loading cycle.


Spring Materials

Common name Specification High-carbon steels Music wire ASTM A228

Modulus of elasticity, E, psi

Shear modulus of elasticity, G psi

Density, ρ, lb/in.3

Maximum service temperature, ◦F

30 × 106

11.5 × 106

0.283

250

Hard drawn

ASTM A227

20 × 106

11.5 × 106

0.283

250

Stainless steels Martensitic

AISI 410, 420

29 × 106

11 × 106

0.280

500

Austenitic

AISI 301, 302

28 × 106

10 × 106

0.282

600

16 × 106

6 × 106

0.308

200

Copper-based alloys Spring brass ASTM B134 Phosphor bronze

ASTM B159

15 × 106

6.3 × 106

0.320

200

Beryllium copper

ASTM B197

19 × 106

6.5 × 106

0.297

400

Nickel-based alloys Inconel 600

—

31 × 106

11 × 106

0.307

600

Inconel X-750

—

31 × 106

11 × 106

0.298

1100

Ni-Span C

—

27 × 106

9.6 × 106

0.294

200

Principal characteristics High strength; excellent fatigue life General purpose use; poor fatigue life Unsatisfactory for subzero applications Good strength at moderate temperatures; low stress relaxation Low cost; high conductivity; poor mechanical properties Ability to withstand repeated flexures; popular alloy High elastic and fatigue strength; hardenable Good strength; high corrosion resistance Precipitation hardening; for high temperatures Constant modulus over a wide temperature range

Table 17.1 Typical properties of common spring materials. Hamrock • Fundamentals of Machine Elements

Strength of Spring Materials Ap Sut = m d

Material Music wirea Oil-tempered wireb Hard-drawn wirec Chromium vanadiumd Chromium siliconee

Size range in. mm 0.004-0.250 0.10-6.5 0.020-0.500 0.50-12 0.028-0.500 0.70-12 0.032-0.437 0.80-12 0.063-0.375 1.6-10

Exponent, m 0.146 0.186 0.192 0.167 0.112

Constant, Ap ksi MPa 196 2170 149 1880 136 1750 169 2000 202 2000

Table 17.2 Coefficients used in Equation (17.2) for five spring materials.


Helical Coil P

R

P d

l = 2π RN

P R

D R

(a)

P

P

P

T = PR P

(b)

R (c)

P

Figure 17.2 Helical coil. (a) Straight wire before coiling; (b) coiled wire showing transverse (or direct) shear; (c) coiled wire showing torsional shear.


Shear Stresses on Wire and Coil Spring axis

d

(a)

d

(c)

D/2 Spring axis

d

(b)

d

(d)

D/2

Figure 17.3 Shear stresses acting on wire and coil. (a) Pure torsional loading; (b) transverse loading; (c) torsional and transverse loading with no curvature effects; (d) torsional and transverse loading with curvature effects. Hamrock • Fundamentals of Machine Elements

Compression Spring End Types

(a)

(b)

(c)

(d)

Figure 17.4 Four end types commonly used in compression springs. (a) Plain; (b) plain and ground; (c) squared; (d) squared and ground. Hamrock • Fundamentals of Machine Elements

Compression Spring Formulas

Term Number of end coils, Ne Total number of coils, Nt Free length, lf Solid length, ls Pitch, p

Plain 0 Na pNa + d d(Nt + 1) (lf − d)/Na

Type of spring end Plain and ground Squared or closed 1 2 Na + 1 Na + 2 p(Na + 1) pNa + 3d dNt d(Nt + 1) lf /(Na + 1) (lf − 3d)/Na

Squared and ground 2 Na + 2 pNa + 2d dNt (lf − 2d)/Na

Table 17.3 Useful formulas for compression springs with four end conditions.


Lengths and Forces in Helical Springs (P = 0) Pr

Po Ps lf li lo

ga

(a)

(b)

(c)

ls

(d)

Figure 17.5 Various lengths and forces applicable to helical compression springs. (a) Unloaded; (b) under initial load; (c) under solid load.


Force vs. Deflection lf li lo ls Length, l

0

Spring force, P

Ps

Po

Pi 0 0

δi

δo

Deflection, δ

δs

Figure 17.6 Graphical representation of deflection, force and length for four spring positions.


Buckling Conditions Ratio of deflection to free length, δ/l f

0.80 Stable

Unstable

0.60

0.40

Stable

Unstable Parallel ends

0.20 Nonparallel ends 0

3

4 5 6 7 8 9 Ratio of free length to mean coil diameter, lf /D

10

Figure 17.7 Critical buckling conditions for parallel and nonparallel ends of compression springs. Hamrock • Fundamentals of Machine Elements

P

Extension Spring Ends

P

d r3

d

A

r1

r2 r4 B

(a)

(b) P

P

d d

r3 r1

A r2 r4 B

(c)

Figure 17.8 Ends for extension springs. (a) Conventional design; (b) Side view of Fig. 17.8(a); (c) improved design; (d) side view of Fig. 17.8(c).

(d)


Helical Extension Springs do

30

200

28 26

175

24

lf

lb

20 125

ga

18

Preferred range

16

100

14 12

75

10 8

50 25

lh

22

150

6 4 4

6

8

10 12 Spring index

14

16

Figure 17.10 Preferred range Figure 17.9 Dimensions of of preload stress for various helical extension spring. spring indexes. Hamrock • Fundamentals of Machine Elements

Preload stress, ksi

ll

Preload stress, MPa

di

Helical Torsion Spring

P

P d

a D

Figure 17.11 Helical torsion spring.


Leaf Spring b 2 b 2

nb b b 2 b 2

P l

P

t

x

(a) l

P

t

P

x

(b) b

Figure 17.12 Leaf spring. (a) Triangular plate, cantilever spring; (b) equivalent multiple-leaf spring. Hamrock • Fundamentals of Machine Elements

Belleville Springs 200

Percent force to flat

160

Di

2.275

1.000

120 1.414 80

40

0.400

h 0

t

Height-tothickness ratio, 2.828

0

20

40

60

80 100 120 Percent deflection to flat

140

160

180

Do

Figure 17.13 Typical Belleville Spring.

Figure 17.14 Force-deflection response of Belleville spring.


200

Stacking of Belleville Springs

(a)

(b)

Figure 17.15 Stacking of Belleville springs. (a) in parallel; (b) in series.


Dickerman Feed Case Study

Gripping unit (sliding)

500 400 300 200 100

Gripping unit (fixed)

S-ring

0

0.04

0.08 0.12 0.16 Wire diameter, d, in.

0.20

0

0.04

0.08 0.12 0.16 Wire diameter, d, in.

0.20

1.4

Fixed rear guide

1.2 Safety factor, ns

Cam

Maximum force, Pmax, lbf

600

Figure 17.16 Dickerman feed unit.

1.0 0.8 0.6 0.4 0.2

Figure 17.17 Performance of the spring in the case study. Hamrock • Fundamentals of Machine Elements

Chapter 18: Brakes and Clutches

Nothing has such power to broaden the mind as the ability to investigate systematically and truly all that comes under thy observation in life. Marcus Aurelius, Roman Emperor


Brake and Clutch Types P

(a)

P P

P

P

P

(d)

(b)

P

P

P (e) (c)

P

P

Figure 18.1 Five types of brake and clutch. (a) internal, expanding rim type; (b) external contracting rim type; (c) band brake; (d) thrust disk; (e) cone disk. Hamrock • Fundamentals of Machine Elements

Thrust Disk

θ

ro

dr r

ri

–– Dimensionless torque, T = T/2µPro

0.6

0.5

0.4

0.3 –– Tw 0.2

Figure 18.2 Thrust disk clutch surface with various radii.

–– Tp

0

0.2

0.4 0.6 0.8 Radius ratio, β = ri /ro

1.0

Figure 18.3 Effect of radius ratio on dimensionless torque for uniform pressure and uniform wear models.


Properties of Common Friction Materials

Friction materiala Molded Woven Sintered metal Cork Wood Cast iron; hard steel

Coefficient of friction, µ 0.25-0.45 0.25-0.45 0.15-0.45 0.30-0.50 0.20-0.30 0.15-0.25

Maximum contact pressure,b pmax psi kPa 150-300 1030-2070 50-100 345-690 150-300 1030-2070 8-14 55-95 50-90 345-620 100-250 690-1720

Maximum bulk temperature, tm,max ◦F ◦C 400-500 204-260 400-500 204-260 400-1250 204-677 180 82 200 93 500 260

Table 18.1 Representative properties of contacting materials operating dry.


Friction for Wet Clutch Materials

Friction materiala Coefficient of friction, µ Molded 0.06-0.09 Woven 0.08-0.10 Sintered metal 0.05-0.08 Paper 0.10-0.14 Graphitic 0.12 (avg.) Polymeric 0.11 (avg.) Cork 0.15-0.25 Wood 0.12-0.16 Cast iron; hard steel 0.03-0.16 a When rubbing against smooth cas iron or seel.

Table 18.2 Coefficient of friction for contacting materials operating in oil.


Cone Clutch dr —— sin α

dA

rdθ dP

α

dr

dw r

dθ

D

θ

d

b

Figure 18.4 Forces acting on elements of cone clutch. Hamrock • Fundamentals of Machine Elements

Short-Shoe Brake d4 d3

W

C D

d1

B µP P

d2 ω

r

Figure 18.5 Block, or short-shoe brake, with two configurations. Hamrock • Fundamentals of Machine Elements

Example 18.3

14 in.

P

1.5 in.

W

14 in. 36 in.

Figure 18.6 Short-shoe brake used in Example 18.3.


Long-Shoe, Internal Expanding Rim Brake W

Rotation

W

d6

θ2

d7

θ θ1 r

d5

Drum Lining

A

d5

Figure 18.7 Long-shoe, internal, expanding rim brake with two shoes.


Forces and Dimensions y

d7 sin θ dP

µdP cos θ

dP sin θ

µdP

θ

θ

Wx

dP cos θ µdP sin θ W

θ2 Wy

θ θ1

A

x

Rx

d6 d7 Rotation

r – d7 cos θ

Ry

r

Figure 18.8 Forces and dimensions of long-shoe, internal, expanding rim brake. Hamrock • Fundamentals of Machine Elements

Example 18.4 y 15°

15°

W W d

d b

a

10°

a

10°

A

10°

B r

x

10°

b d

d

W W ω 15°

15°

Figure 18.9 Four-long-shoe, internal expanding rim brakes used in Example 18.4.


Long-Shoe, External, Contracting Rim Brake Wx Wy

W y µdP sin θ

θ µdP cos θ

µdP

dP θ2

d6

dP sin θ

θ dP cos θ θ1

Rx

θ

A d7

x

Ry

r Rotation

Figure 18.10 Forces and dimensions of long-shoe, external, contracting rim brake. Hamrock • Fundamentals of Machine Elements

Pivot-Shoe Brake y

µdP sin θ

Rotation

dP r

θ

µdP cos θ

dP sin θ

µdP

d7 cos θ – r

θ2 dP cos θ

Rx

x

θ1 r cos θ Ry

d7

Figure 18.11 Symmetrically loaded pivot-shoe brake.


Band Brake

dθ φ

θ 0

dθ (F + dF) cos — 2 F + dF dθ (F + dF) sin — 2

Drum rotation

dθ F cos — 2

r dθ

dθ — 2

dP µdP

dθ — 2

F

dθ F sin — 2

r

dθ F1

F2 0 (a)

(b)

Figure 18.12 Band brake. (a) Forces acting on band; (b) forces acting on element. Hamrock • Fundamentals of Machine Elements

Example 18.7

Rotation φ

Cutting plane for free-body diagram W F2 F1

d10 d8

Figure 18.13 Band brake used in Example 18.7.

d9


Capacities of Brake and Clutch Materials

Operating condition Continuous: poor heat dissipation Occasional: poor heat dissipation Continuous: good heat dissipation as in oil bath

pu (kPa)(m/s) (psi)(ft/min) 1050 30,000 2100 60,000 3000 85,000

Table 18.3 Product of contact pressure and sliding velocity for brakes and clutches.


Hydraulic Crane Brake Case Study W

17.5 in.

30°

30°

18 in.

Figure 18.14 Hoist line brake for mobile hydraulic crane. The crosssection of brake with relevant dimensions is given.


Chapter 19: Flexible Machine Elements

Scientists study the world as it is; engineers create the world that has never been. Theodore von Karmen


Flat Belt

B α α

A

D

α D1

φ1 01

φ2

D2 02

α

α

cd

Figure 19.1 Dimensions, angles of contact, and center distance of open flat belt. Hamrock • Fundamentals of Machine Elements

Weighted Idler

Idler Pivot Weight Tight side

Figure 19.2 Weighted idler used to maintain desired belt tension. Hamrock • Fundamentals of Machine Elements

Synchronous Belt Tooth included angle

Circular pitch

Backing

A Neoprene-encased tension member

Section A–A

Facing A

Neoprene tooth cord

Pulley face radius Pulley pitch radius

Figure 19.3 Synchronous, or timing, belt. Hamrock • Fundamentals of Machine Elements

V-Belt in Groove wt dN/2 –––– sin β ht

dN –––– 2

2β ≈36∞

Figure 19.4 V-belt in sheave groove.


Driven unit Agitators Liquid Semisolid Compressor Centrifugal Reciprocating Conveyors and elevators Package and oven Belt Fans and blowers Centrifugal, calculating Exhausters Food machinery Slicers Grinders and mixers Generators Farm lighting and exciters Heating and Ventilating Fans and oil burners Stokers Laundry machinery Dryers and ironers Washers Machine tools Home workshop and woodworking Pumps Centrifugal Reciprocating Refrigeration Centrifugal Reciprocating Worm gear speed reducers, input side

Overload factor 1.2 1.4 1.2 1.4

Overload Service Factors

1.2 1.4 1.2 1.4 1.2 1.4 1.2 1.2 1.4 1.2 1.4 1.4 1.2 1.4 1.2 1.4 1.0

Table 19.1 Overload service factors f1 for various types of driven unit.


Pulley Design Considerations

Belt type 2L 3L 4L

Size of belt, in. 1 × 18 4 3 8 1 2

× ×

7 32 5 16

Minimum pitch diameter, in. Recommended Absolute 1.0 1.0 1.5

1.5

2.5

1.8

Table 19.3 Recommended pulley dimensions in inches for three electric motor sizes.

Motor horsepower, hp 0.50 0.75 1.00

Table 19.2 Recommended minimum pitch diameters of pulley for three belt sizes.

Motor speed, rpm 575 695 870 1160 1750 Recommended pulley diameter, in. 2.50 2.50 2.50 — — 3.00 2.50 2.50 2.50 — 3.00 3.00 2.50 2.50 2.25


Loss in arc of contact, deg. 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90

Correction factor 1.00 0.99 0.98 0.96 0.95 0.93 0.92 0.89 0.89 0.87 0.86 0.84 0.83 0.81 0.79 0.76 0.74 0.71 0.69

Arc Correction Factor

Table 19.4 Arc correction factor for various angle of loss in arc of contact.


Power Ratings for V-Belts Speed of faster shaft rpm 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Pulley effective outside diameter, in. 1.00 2.00 3.00 4.00 5.00 6.00 Rated horsepower, hp 0.04 0.05 0.06 0.08 0.10 0.12 0.05 0.08 0.12 0.16 0.19 0.21 0.06 0.12 0.17 0.22 0.25 0.30 0.08 0.15 0.23 0.28 0.32 0.39 0.10 0.18 0.27 0.34 0.39 0.44 0.12 0.14 0.16 0.18 0.20

0.21 0.24 0.28 0.31 0.35

0.31 0.35 0.38 0.42

0.40 0.44 0.46

0.31 0.44

0.47

(a)

Table 19.5 Power ratings for light-duty V-belts. (a) 2L section with wt = 1/4 in and ht=1/8 in. Hamrock • Fundamentals of Machine Elements

Power Ratings for V-Belts (cont.) Speed of faster shaft rpm 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Pulley effective outside diameter, in. 1.50 1.75 2.00 2.25 2.50 2.7 3.00 Rated horsepower, hp 0.04 0.07 0.09 0.12 0.14 0.17 0.19 0.07 0.12 0.16 0.21 0.25 0.30 0.34 0.09 0.15 0.22 0.29 0.35 0.41 0.47 0.10 0.19 0.27 0.35 0.43 0.51 0.59 0.11 0.21 0.31 0.41 0.51 0.60 0.69 0.11 0.11 0.11 0.10 0.09

0.23 0.25 0.26 0.25 0.26

0.35 0.38 0.40 0.42 0.42

0.45 0.50 0.54 0.56 0.57

0.57 0.62 0.66 0.68 0.69

0.68 0.74 0.78 0.80 0.80

0.78 0.84 0.88 0.90 0.89

(b)

Table 19.5 Power ratings for light-duty V-belts. (b) 3L section with wt = 3/8 in and ht=1/4 in. Hamrock • Fundamentals of Machine Elements

Power Ratings for V-Belts (cont.)

Speed of faster shaft rpm 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

0.08 0.11 0.12 0.11 0.09

Pulley effective outside diameter, in. 2.25 2.50 2.75 3.00 3.25 3.50 3.75 Rated horsepower, hp 0.14 0.19 0.24 0.29 0.34 0.39 0.44 0.21 0.31 0.41 0.50 0.60 0.69 0.78 0.26 0.40 0.54 0.67 0.81 0.94 1.07 0.30 0.47 0.65 0.82 0.99 1.15 1.31 0.31 0.53 0.73 0.94 1.13 1.32 1.51

0.49 0.87 1.20 1.47 1.69

0.06 0.02 — — —

0.31 0.30 0.27 0.22 0.15

1.84 1.92 1.92 1.84 1.65

2.00

0.56 0.57 0.56 0.54 0.47

0.79 0.83 0.83 0.81 0.75

1.02 1.07 1.09 1.07 1.01

1.24 1.31 1.33 1.30 1.23

1.45 1.53 1.55 1.51 1.41

1.65 1.73 1.75 1.69 1.66

4.00

(c)

Table 19.5 Power ratings for light-duty V-belts. (c) 4L section with wt = 1/2 in and ht=9/32 in. Hamrock • Fundamentals of Machine Elements

Center Distance for V-Belts Pulley combination Driver Driven pitch pitch diameter, diameter, in. in. 2.0 2.0 3.0 3.0 2.0 2.5 2.0 3.0 3.0 4.5

Nominal center distance Short center Medium center Center Center Belt distance, Belt distance, number in. number in. 3L200 6.4 3L250 9.4 3L250 7.4 3L310 10.4 3L210 6.6 3L270 9.6 3L220 6.7 3L280 9.7 3L290 8.2 3L350 11.2

2.0 2.0 2.25 2.5 3.0

3.5 4.0 4.5 5.0 6.0

3L240 3L250 3L270 3L290 3L330

7.3 7.2 7.7 8.1 8.9

3L300 3L310 3L330 3L350 3L390

10.3 10.3 10.7 11.1 11.9

2.0 2.0 3.0 2.0 2.0

5.0 6.0 9.0 7.0 9.0

3L250 3L310 3L410 3L340 3L390

8.0 8.6 10.3 9.2 9.9

3L340 3L370 3L470 3L400 3L450

11.0 11.6 13.4 12.2 13.0

2.0 1.5

10.0 9.0

3L420 3L390

10.4 10.1

3L480 3L450

13.6 13.3

(a)

Table 19.6 Center distances for various pitch diameters of driver and driven pulleys. (a) 3L type of V-Belt. Hamrock • Fundamentals of Machine Elements

Center Distance for V-Belts (cont.) Pulley combination Driver Driven pitch pitch diameter, diameter, in. in. 2.5 2.5 3.0 3.0 3.0 4.5 4.0 6.0 3.0 6.0 3.5 7.0 3.0 7.5 4.0 10.0 3.0 9.0 3.5 10.5 4.0 12.0 3.0 10.5 4.0 14.0 3.0 12.0 3.5 14.0 2.0 9.0 4.0 18.0 2.4 12.0 2.5 18.0 2.8 14.0 2.0 11.0 2.0 12.0 2.5 15.0 3.0 13.0 2.0 14.0

Minimum center Center Belt distance, number in. 4L170 4.0 4L200 4.8 4L240 5.5 4L300 6.5 4L280 6.2 4L320 7.0 4L320 6.8 4L410 8.5 4L360 7.5 4L420 8.8 4L470 9.6 4L410 8.6 4L530 10.5 4L450 9.1 4L520 10.4 4L350 7.5 4L650 12.8 4L440 8.9 4L480 9.3 4L510 10.3 4L400 8.0 4L430 8.5 4L530 10.3 4L630 12.2 4L490 9.5

Nominal center distance Short center Medium center Center Center Belt distance, Belt distance, number in. number in. 4L150 8.0 4L330 12.0 4L280 8.8 4L360 12.8 4L320 9.6 4L400 13.6 4L380 10.6 4L460 14.5 4L360 10.3 4L440 14.3 4L400 11.1 4L480 15.1 4L400 11.0 4L480 15.0 4L490 12.5 4L570 16.7 4L440 11.7 4L520 15.8 4L500 13.0 4L580 17.1 4L550 13.8 4L630 18.0 4L490 12.9 4L570 17.0 4L610 15.0 4L690 19.1 4L530 13.4 4L610 17.6 4L600 14.8 4L680 19.0 4L430 11.8 4L510 15.9 4L730 17.3 4L810 21.6 4L520 13.3 4L600 17.5 4L560 14.3 4L640 18.5 4L590 14.7 4L670 19.0 4L480 12.5 4L560 16.7 4L510 13.0 4L590 17.3 4L610 14.9 4L690 19.2 4L710 16.8 4L790 21.2 4L570 14.1 4L650 18.4

(b)

Table 19.6 Center distances for various pitch diameters of driver and driven pulleys. (b) 4L type of V-Belt. Hamrock • Fundamentals of Machine Elements

Wire Rope

Figure 19.5 Cross-section of wire rope.

(a)

(b)

Figure 19.6 Two lays of wire rope. (a) Lang; (b) regular. Hamrock • Fundamentals of Machine Elements

Wire Rope Data Rope 6 × 7 Haulage

Weight per height, lb/ft 1.50d2

Minimum sheave diameter in. 42d

Rope diameter, d, in. 1 − 1 12 4

Size of outer wires d/9 d/9 d/9 d/13 − d/16 d/13 − d/16 d/13 − d/16 d/22 d/22 d/15 − d/19 d/15 − d/19 —

Modulus of elasticity,a psi 14 × 106 14 × 106 14 × 106 12 × 106 12 × 106 12 × 106 11 × 106 11 × 106 10 × 106 10 × 106 —

Strength,b psi 100 × 103 88 × 103 76 × 103 106 × 103 93 × 103 80 × 103 100 × 103 88 × 103 92 × 103 80 × 103 124 × 103

Material Monior steel Plow steel Mild plow steel 1 3 2 6 × 19 Standard 1.60d 26d − 34d − 24 Monior steel 4 hoisting Plow steel Mild plow steel 1 6 × 37 Special 1.55d2 18d − 3 12 Monior steel 4 flexible Plow steel 1 1 2 8 × 19 Extra 1.45d 21d − 26d − 12 Monior steel 4 flexible Plow steel 1 3 2 7 × 7 Aircraft 1.70d — −8 Corrosion-resistant 16 steel Carbon steel — — 124 × 103 1 3 2 7 × 9 Aircraft 1.75d — − 18 Corrosion-resistant — — 135 × 103 8 steel Carbon steel — — 143 × 103 1 5 2 19-Wire aircraft 2.15d — − 16 Corrosion-resistant — — 165 × 103 32 steel Carbon steel — — 165 × 103 a The modulus of elasticity is only approximate; it is affected by the loads on the rope and, in general, increases with the life of the rope. b The strength is based on the nominal area of the rope. The figures given are only approximate and are based on 1-in. rope sizes and 1/4-in. aircraft cable sizes.

Table 19.7 Wire rope data. Hamrock • Fundamentals of Machine Elements

Application Safety factor,a ns Track cables 3.2 Guys 3.5 Mine shafts, ft Up tp 500 8.0 1000-2000 7.0 2000-3000 6.0 Over 3000 5.0 Hoisting 5.0 Haulage 6.0 Cranes and derricks 6.0 Electric hoists 7.0 Hand elevators 5.0 Private elevators 7.5 Hand dumbwaiters 4.5 Grain elevators 7.5 Passenger elevators, ft/min 50 7.60 300 9.20 800 11.25 1200 11.80 1500 11.90 Freight elevators, ft/min 50 6.65 300 8.20 800 10.00 1200 10.50 1500 10.55 Powered dumbwaiters, ft/min 50 4.8 300 6.6 800 8.0 a Use of these factors does not preclude a fatigue failure

Minimum Safety Factors

Table 19.8 Minimum safety factors for a variety of wire rope applications.


Effect of D/d Ratio Relative service life, percent

100

Strength loss, percent

50 40 30 20

80 60 40 20

10

0 0

10

20

30

D/d ratio

Figure 19.7 Percent strength loss in wire rope for different D/d ratios.

40

10

20

30

40

50

D/d ratio

Figure 19.8 Service life for different D/d ratios.


60

Allowable Bearing Pressures

Rope Regular lay 6×7 6 × 19 6 × 37 8 × 19 Lang lay 6×7 6 × 19 6 × 37

Material Cast Cast Chilled Manganese Wooda iron b steelc cast irond steele Allowable bearing pressure, pall , psi 150 250 300 350

300 480 585 680

550 900 1075 1260

650 1100 1325 1550

1470 2400 3000 3500

165 275 330

350 550 660

600 1000 1180

715 1210 1450

1650 2750 3300

Table 19.9 Maximum allowable bearing pressures for various sleeve materials and types of rope. Hamrock • Fundamentals of Machine Elements

Rolling Chains pt /2

Link plate

Roller link

Pin link

Roller

B

Pin

C

pt A

θr

Pitch circle

0

Bushing Pitch

Pitch

Figure 19.9 Various parts of rolling chain.

Figure 19.10 Chordal rise in rolling chains.


rc

Strength of Rolling Chains

Chain number a 25 a 35 a 41 40 50 60 80 100 120 140 160 180 200 240

Pitch, pt , in. 1/4 3/8 1/2 1/2 5/8 3/4 1 1 14 1 12 1 34 2 2 14 2 12 3

Roller Diameter, Width, in. in. 0.130 1/8 0.200 3/16 0.306 1/4 5/16 5/16 2/5 3/8 15/32 1/2 5/8 5/8 3/4 3/4 7/8 1 1 1 1 18 1 14 1 13 1 13 32 32 9 1 16 1 12 1 78 1 78

Pin diameter, d, in. 0.0905 0.141 0.141 0.156 0.200 0.234 0.312 0.375 0.437 0.500 0.562 0.687 0.781 0.937

Link plate thickness, a, in. 0.030 0.050 0.050 0.060 0.080 0.094 0.125 0.156 0.187 0.219 0.250 0.281 0.312 0.375

Average ultimate strength, Su , lb 875 2100 2000 3700 6100 8500 14,500 24,000 34,000 46,000 58,000 76,000 95,000 130,000

Weight per foot, lb 0.084 0.21 0.28 0.41 0.68 1.00 1.69 2.49 3.67 4.93 6.43 8.70 10.51 16.90

Table 19.10 Standard sizes and strengths of rolling chains.


Chain Power Ratings Table 19.11: Transmitted power of single-strand, No. 25 rolling chain. Number of teeth in small sprocket 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 28 30 32 35 40 45 50 55 60

100 0.054 0.059 0.064 0.070 0.075 0.081 0.086 0.097 0.097 0.103 0.108 0.114 0.119 0.125 0.131 0.148 0.159 0.170 0.188 0.217 0.246 0.276 0.306 0.336

500 900 Type I 0.23 0.39 0.25 0.43 0.27 0.47 0.30 0.50 0.32 0.54 0.34 0.58 0.37 0.62 0.39 0.66 0.41 0.70 0.44 0.74 0.46 0.78 0.48 0.82 0.51 0.86 0.53 0.90 0.56 0.94 0.63 1.07 0.68 1.15 0.73 1.23 0.80 1.36 0.92 1.57 1.05 1.78 1.18 1.99 1.30 2.21 1.43 2.43 Type II

1200

1800

2500

3000

3500

0.50 0.55 0.60 0.65 0.70 0.75 0.81 0.86 0.91 0.96 1.01 1.06 1.12 1.17 1.22 1.38 1.49 1.60 2.76 2.03 2.31 2.58 2.96 3.15

0.73 0.80 0.87 0.94 1.01 1.09 1.16 1.23 1.31 1.38 1.46 1.53 1.61 1.69 1.76 1.99 2.14 2.30 2.53 2.93 3.32 3.72 4.12 4.53 III

0.98 1.07 1.17 1.27 1.36 1.46 1.56 1.66 1.76 1.86 1.96 2.06 2.16 2.26 2.37 2.67 2.88 3.09 3.40 3.93 4.46 5.00 5.54 6.09

1.15 1.26 1.38 1.49 1.61 1.72 1.84 1.95 2.07 2.19 2.31 2.43 2.55 2.67 2.79 3.15 3.39 3.64 4.31 4.63 5.26 5.89 6.53 7.18

1.32 1.45 1.58 1.71 1.85 1.98 2.11 2.25 2.38 2.52 2.65 2.79 2.93 3.07 3.20 3.62 3.90 4.18 4.62 5.32 6.04 6.77 7.51 8.25

Small sprocket speed, rpm 4000 4500 5000 5500 6000 6500 Type II lubrication 1.42 1.19 1.01 0.88 0.77 0.68 1.62 1.36 1.16 1.00 0.88 0.78 1.78 1.53 1.30 1.13 0.99 0.88 1.93 1.71 1.46 1.26 1.11 0.94 2.06 1.89 1.62 1.40 1.23 1.09 2.23 2.08 1.78 1.54 1.35 1.20 2.38 2.28 1.95 1.69 1.48 1.31 2.53 2.49 2.12 1.84 1.52 1.43 2.69 2.70 2.30 2.00 1.75 1.55 2.84 2.91 2.49 2.16 1.89 1.68 2.99 3.13 2.68 2.32 2.04 1.80 3.15 3.36 2.87 2.49 2.18 1.93 3.30 3.59 3.07 2.66 2.33 2.07 3.46 3.83 3.27 2.83 2.48 2.20 3.61 4.07 3.48 3.01 2.64 2.34 4.28 4.54 4.12 3.57 3.13 2.78 4.40 4.89 4.57 3.96 3.47 3.08 4.71 5.24 5.03 4.36 3.83 3.39 5.19 5.78 5.76 4.99 4.38 3.88 6.00 6.67 7.04 6.10 5.35 4.75 6.81 7.58 8.33 7.28 6.39 5.66 7.64 8.49 9.33 8.52 7.48 6.63 8.46 9.41 10.3 9.83 8.63 7.65 9.30 10.3 11.3 11.2 9.83 8.72 Type IV lubrication

7000

7500

8000

8500

9000

10000

0.61 0.70 0.79 0.88 0.96 1.07 1.18 1.28 1.39 1.50 1.61 1.73 1.85 1.97 2.10 2.49 2.76 3.04 3.48 4.25 5.07 5.93 6.85 7.80

0.55 0.63 0.71 0.79 0.88 0.97 1.06 1.16 1.25 1.35 1.46 1.56 1.67 1.78 1.89 2.24 2.49 2.74 3.13 3.83 4.57 5.35 6.17 7.03

0.50 0.57 0.64 0.72 0.80 0.88 0.96 1.05 1.14 1.23 1.32 1.42 1.51 1.61 1.72 2.04 2.26 2.49 2.85 3.48 4.15 4.96 5.60 6.38

0.46 0.52 0.59 0.66 0.73 0.80 0.88 0.96 1.04 1.12 1.21 1.29 1.38 1.47 1.52 1.86 2.06 2.27 2.60 3.17 3.79 4.44 5.12 5.83

0.42 0.48 1.54 0.60 0.67 0.74 0.81 0.88 0.96 1.03 1.11 1.19 1.27 1.35 1.44 1.71 1.89 2.06 2.38 2.91 3.48 4.07 4.76 5.35

0.36 0.41 0.46 0.51 0.57 0.63 0.69 0.75 0.81 0.88 0.95 1.01 1.06 1.16 1.23 1.46 1.62 1.78 2.04 2.49 2.97 3.48 4.01 4.57


Rolling Chain Data

Table 19.12 Service factors for rolling chains.

Type of driven load Smooth Moderate shock Heavy shock

Type of input power Internal combustion Electric motor Internal combustion engine with or engine with hydraulic drive turbine mechanical drive 1.0 1.0 1.2 1.2 1.3 1.4 1.4 1.5 1.7

Table 19.13: Multiple-strand factors for rolling chains. Number of strands 2 3 4

Multiple-strand factor, a2 1.7 2.5 3.3


Dragline φ

Hoist rope Gantry line system

Mast (20 ft) Multi-part line

Drag rope θ

5 ft

15 ft

Figure 19.11 Typical dragline.


Chapter 20: Elements of Microelectromechanical Systems (MEMS)

There is plenty of room at the bottom. Richard Feynman


(a)

(b)

Mirror

Mirror support post Landing tips

Mirror -10°

Mirror +10°

Torsion hinge

DPT Device

Address electrode Electrode Yoke support post Hinge support post Metal 3 address pads

Hinge

Landing sites

Yoke Landing tip

Bias/Reset Bus (c)

CMOS substrate

To SRAM (d)

Figure 20.1 The Texas Instruments digital pixel technology (DPT) device. Hamrock • Fundamentals of Machine Elements

Lithography

Figure 20.2 Pattern transfer by lithography. Note that the mask in step 3 can be a positive or negative image of the pattern. Hamrock • Fundamentals of Machine Elements

Etching Directionality

(a)

undercut

(b)

(c)

Mask layer {111} face 54.7°

Etch front

Etch front Etch material (e.g. silicon)

Final shape

Etch front

Figure 20.3 Etching directionality. (a) Isotropic etching: etch proceeds vertically and horizontally at approximately the same rate, with significant mask undercut. (b) Orientation-dependant etching (ODE): etch proceeds vertically, terminating on {111} crystal planes with little mask undercut. (c) Vertical etching: etch proceeds vertically with little mask undercut. Hamrock • Fundamentals of Machine Elements

(a)

Bulk Micromachining

Diffused layer (e.g. p-type Si)

Substrate (e.g. n-type Si) (b) Non-etching mask (e.g. silicon nitride)

(c) Freestanding cantilever

(111) planes

Figure 20.4 Schematic illustration of bulk micromachining. (a) Diffuse dopant in desired pattern. (b) Deposit and pattern masking film. (c) Orientation-dependant etch (ODE), leaving behind a freestanding structure.


Surface Micromachining

Figure 20.5 Schematic illustration of the steps in surface micromachining. (a) deposition of a phosphosilicate glass (PSG) spacer layer; (b) etching of spacer layer; (c) deposition of polysilicon; (d) etching of polysilicon; (e) selective wet etching of PSG, leaving the silicon substrate and deposited polysilicon unaffected. Hamrock • Fundamentals of Machine Elements

Micromirror (a)

(b)

Figure 20.6 (a) SEM image of a deployed micromirror. (b) Detail of the micromirror hinge. (Source: Sandia National Laboratories.)


Micro-Hinge Manufacture (a)

(b)

(c)

Poly1

Spacer layer 1

Spacer Layer 2

Silicon

(e)

(d) Poly2

Figure 20.7 Schematic illustration of the steps required to manufacture a hinge. (a) Deposition of a phosphosilicate glass (PSG) spacer layer and polysilicon layer. (b) deposition of a second spacer layer; (c) Selective etching of the PSG; (d) depostion of polysilicon to form a staple for the hinge; (e) After selective wet etching of the PSG, the hinge can rotate. Hamrock • Fundamentals of Machine Elements

LIGA

Figure 20.8 The LIGA (lithography, electrodeposition and molding) technique. (a) Primary production of a metal final product or mold insert. (b) Use of the primary part for secondary operations, or replication Hamrock • Fundamentals of Machine Elements

Beams in MEMS

Table 20.1 Summary of important beam situations for MEMS devices. Hamrock • Fundamentals of Machine Elements

Atomic Force Microscope Probe

(a)

(b)

Figure 20.9 Scanning electron microscope images of a diamond-tipped cantilever probe used in atomic force microscopy. (a) Side view with detail of diamond; (b) bottom view of entire cantilever. Hamrock • Fundamentals of Machine Elements

Rectangular Plate a/b α β

1.0 0.0138 0.3078

1.2 0.0188 0.3834

1.4 0.0226 0.4356

1.6 0.0251 0.4680

1.8 0.0267 0.4872

2.0 0.0277 0.4974

∞ 0.0284 0.5000

Table 20.2 Coefficients α and β for analysis of rectangular plate pressure sensor. pb4 !max = −" 3 Et pb2 !max = " 2 t


Electrostatic Actuation Stationary comb

Fy

Moving comb a

Beam spring suspension V (volts)

y h

x

Fy (a)

a y

Anchors

h

x

V (volts)

(c) (b)

Figure 20.10 Illustration of electrostatic actuatuation. (a) Attractive forces between charged plates; (b) forces resulting from eccentric charged plate between two other plates; (c) schematic illustration of a comb drive. Hamrock • Fundamentals of Machine Elements

Comb Drive

Figure 20.11 A comb drive. Note the springs in the center provide a restoring force to return the electrostatic comb teeth to their original position. From Sandia National Laboratories. Hamrock • Fundamentals of Machine Elements

Rotary Electrostatic Motor

Rotor Stator

12° 18°

27°

(a)

(b)

Figure 20.12 (a) Schematic illustration of a rotary electrostatic motor, sometimes called a slide motor; (b) scanning electron microscope image of a rotary micromotor. Hamrock • Fundamentals of Machine Elements

Capilary Tube for Microflow

Piezoelectric coating with transducer

Flow Traveling wave direction

Flexible tube wall

Figure 20.13 Capilary tube for microflow. (a) Schamitic illustration of tube construction; (b) induced traveing wave and fluid flow. Hamrock • Fundamentals of Machine Elements

Thermal Inkjet Printer a) Actuation Bubble Ink Heating element b) Droplet formation

c) Droplet ejection

d) Liquid refills

Satellite droplets

Figure 20.14 (a) Sequence of operation of a thermal inkjet printer. (a) Resistive heating element is turned on, rapidly vaporizing ink and forming a bubble. (b) Within five microseconds, the bubble has expanded and displaced liquid ink from the nozzle. (c) Surface tension breaks the ink stream into a bubble, which is discharged at high velocity. The heating element is turned off at this time, so that the bubble collapses as heat is transferred to the surrounding ink. (d) Within 24 microseconds, an ink droplet (and undesirable satellite droplets) are ejected, and surface tension of the ink draws more liquid from the reservoir.


Piezoelectric Inkjet Mechanism

PZT actuator Teflon coating

Ink reservoir

Nozzle

Ink droplet Paper

Ink dot

Figure 20.15 Schematic illustration of a piezoelectric driven inkjet printer head. Hamrock • Fundamentals of Machine Elements

Metal Oxide Sensors

Metal Coating BaTiO3 /CuO SNO2 SNO2 SNO2 SNO2 WO3 Fe2 O3 Ga2 O3 MoO3 In2 O3

Catalyst La2 O3 , CaCO3 Pt + Sb Pt Sb2 O3 CuO Pt Ti-doped Au Au — —

Detected gas CO2 CO Alcohols H2 , O2 , H2 S H2 S NH3 CO CO NO2 , CO O3

Table 20.4 Common metal oxide sensors.


Accellerometer Stationary polysilicon fingers Direction of acceleration

Spring (beam)

Suspended inertial mass

C1 C2

Anchor to substrate

(a)

(b)

Figure 20.16 (a) Schematic illustration of accellerometer; (b) photograph of Analog Devices’ ADXL-50 accelerometer with a surface micromachined capacitive sensor (center), on-chip excitation, self-test and signal conditioning circuitry. The entire chip measures 0.500 by 0.625 mm. Hamrock • Fundamentals of Machine Elements

Fundamentals of Machine Elements, 2nd Ed.

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