11. Gear Design
11. Gears
Objectives •
Understand basic principles of gearing. gearing.
•
Understand gear trains and how to calculate ratios.
•
•
Understand geometry of different gears and their dimensional pro perties.
Recognize different principles of gearing.
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Recognize different gearing systems and relative advantages and disadvantages between them.
Recognize the unorthodox ways gears can be used in different mot differentmotion motion systems.
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Introduction
Introduction
Gears are the most common means used for power transmission
They can be applied between two shafts which are
Parallel
Collinear
Perpendicular and intersecting
Perpendicular and nonintersecting
Inclined at any arbitrary angle
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Gears are made to high precision precision Purchased from gear manufacturers rather than made in house However it is necessary to design for a specific application so that proper selection can can be be made Used to be called toothed wheels dating back to 2600 b.c.
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An 18th Century Application Application of Gears for Powering Textile Machinery http://www.efunda.com/DesignStandards/gears/gears_history.cfm August 15, 2007
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11.2 Types of Gears
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Gear Types
Gear Parameters
Spur gears
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Internal gears
Most common form Used for parallel shafts
Number of teeth
Form of teeth
Size of teeth
Face Width of teeth
Style and dimensions of gear blank
Suitable for low to medium speed application
Design of the hub of the gear
Relatively high ratios can be achieved (< 7)
Degree of precision required
Steel, brass, bronze, cast iron, and plastics
Means of attaching the gear to the shaft
Can also be made from sheet metal
Means of locating the gear axially on the shaft
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Gear Types
Spur gear nomenclature
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Kalpakjian• Schmid Manufacturing Engineering and 11 Technology, Prentice Hall
Helical gears
Teeth are at an angle
Used for parallel shafts
Teeth engage gradually reducing shocks
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2
Helical Gears
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Helical Gear
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Helical Gear Characteristics
Helix angle 7 to 23 degrees
More power
Larger speeds
More smooth and quiet operation
Used in automobiles
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Herringbone Gears
Two helical gears with opposing helical angles sideside- by by-side Axial thrust gets cancelled
Helix angle must be the same for both the mating gears Produces axial thrust which is a disadvantage
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Herringbone Gears
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Herringbone Gear
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Herringbone Gear Machining
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Gear Types
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Bevel gears
They have conical shape
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Bevel Gears (Miter gears)
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Bevel Gears
For one-to-one ratio Used to change the direction
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Bevel Gears
Gear Types
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Worm gears For large speed reductions between two perpendicular and nonnon-intersecting shafts Driver called worm looks like a thread
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Rack and pinion
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A rack is a gear whose pitch diameter is infinite, resulting in a straight line pitch circle. Involute of a very large base circle approaches a straight line Used to convert rotary motion to straight line motion Used in machine tools
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Fig. 11.7 Rack and pinion
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Rack and pinion
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Internal spur gear
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Internal spur gear
Provides more compact drives compared to external gears They provide large contact ratio Relatively less sliding and hence less wear compare to external gears
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Internal spur gear
Internally Meshing Spur Gears Figure 14.14 Intern ally meshing spur gears.
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Gear Assemblies
©1998 McGraw-Hill, Hamrock, Jacobson and Schmid
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Fig. 11-9 Velocity Ratio
Identified based on the input and output shaft positions Parallel shaft
Perpendicular shaft
Spur gears
Bevel and Miter Rack -andand- pinion gears
Helical gears
Other types
Vr =
CrossCross-helix Worm gears August 15, 2007
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Velocity Ratio
Vr =
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N p
=
N p
=
Dg D p 34
Velocity Ratio
Velocity ratio is defined as the ratio of rotational speed of the input gear to that of the output gear N g
N g
Dg D p
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Vr = Velocity ratio
Vr =
N g
=
N p
Dg D p
N p = Number of teeth on pinion
Ng = Number of teeth on gear
D p = Pitch diameter of pinion
Dg = Pitch diameter of gear
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Example Problem 11-1: Velocity Ratios and Gear Trains
Example Problem 11-1: Velocity Ratios and Gear Trains (cont’d.) (11-1)
• For the set of four gears shown below, calculate output speed, output torque, and horsepower for both input and output conditions and overall velocity ratio:
N 2 N 4 V r = • N 1 N 3 V r =
60 60 9 • = 20 20 1
Output speed:
−
n4 =
3600 rpm •
n1 V r
1 = 400 rpm 9
Output torque:
−
T 4 = T 1 V r T 4 = 200 in-lb •
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Example Problem 11-1: Velocity Ratios and Gear Trains (cont’d.)
−
9 = 1800 in-lb 1
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Example Problem 11-2: Velocity Ratios and Gear Trains • For the gear train shown below, determine the train value, output speed, output direction, output torque, and output power.
Input horsepower: (2-6) Tn hp = 63,000 200 in-lb 3600 rpm 63,000
hp =
hp = 11.4 −
Output horsepower: 1800 in-lb 400 rpm 63,000
hp =
hp = 11.4
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Example Problem 11-2: Velocity Ratios and Gear Trains
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Example Problem 11-2: Velocity Ratios and Gear Trains (cont’d.)
(cont’d.) −
– Train value:
Vr Vr
=
N B
•
N A =
65 20
N D
•
N C •
60
=
20
N E
•
(11-1)
N D
I f:
G ea r A – cl oc kw is e Gear B – counterclockwise
9.75 / 1
Gear C – counterclockwise Gear D – clockwise
Idler cancels out and has no effect on overall train value.
Gear E – counterclockwise
– Output speed:
N A n E
Direction:
=
=
3000 rpm 9.75 / 1
V r
=
307.7 rpm
−
Output power: P = Tn E = 97.5 Nm 307.5
rev min min 60 sec
π
– Output torque:
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T E
=
T E
=
T A V r
10 Nm (9.75 / 1)
P = 1571 Nm/sec or =
J or W s
97.5 Nm 41
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P = 1.57 kW
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Spur Gears
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Pinion Gears
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Internal Gears
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Spur gear geometries
Pitch circle: is the imaginary circle on which most gear calculations are made. When two gears meet their pitch circles are tangent to each other Pitch diameter (D These are (Dp) and pitch radius (r): These the diameter diameter and radius radius of the pitch pitch circle. circle. Pitch point: The point on the the imaginary line joining the centers of the two meshing gears where the pitch circle circle touch
D-d 2
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Pitch circle
Figure 14.1 Spur gear drive. August 15, 2007
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Text Reference: Figure 14.1, page 616
©1998 McGraw-Hill, Hamrock, 48 Jacobson and Schmid
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Spur gear geometries
Spur gear geometries
Addendum circle : It is the circle that bounds the outer ends of the teeth and whose center is at the center of the gear (Fig. 7.2). 7.2). Dedendum circle : It is the circle that bounds the bottoms of the teeth and whose center is at the center of the gear (Fig. 7.2). 7.2). distance from the Addendum (a): is the radial distance pitch circle to the outer end of the teeth. (Fig. 7.2). 7.2). distance from the Dedendum (b): is the radial distance pitch circle to the bottom of the teeth. (Fig. 7.2). 7.2).
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Circular pitch (P c): is the distance between corresponding points on adjacent teeth measured measured along the pitch circle (Fig. 7.2). 7.2). Diametral pitch (Pd): specifies specifies the the number number of of teeth teeth per inch of of pitch pitch diameter. Tooth space: is the space between the adjacent teeth measured along the pitch circle (Fig. 7.2). 7.2). Tooth thickness: is the thickness of the tooth measured along the pitch circle (Fig. 7.2). 7.2).
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Spur gear geometries
Face width (W): is the length of the tooth measured parallel to the gear (Fig. 7.2). 7.2). Face: is the surface between the pitch circle and the top of the tooth (Fig. 7.2). 7.2). Flank: is the surface between the pitch circle and the bottom of the tooth (Fig. 7.2). 7.2). Pressure angle ( ): is the angle between the line of action and a line tangent to the two pitch circles at the pitch point. (Fig. (Fig. 14.8 Hamrock ). ).
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Figure 14.8 Pitch and base circles for pinion and gear as well as line of action and pressure angle.
r b = r cosϕ D b = D p cosϕ August 15, 2007
Spur gear geometries
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Gear terminology
Line of action: is the locus of all the points of contact between two meshing teeth from the time the teeth go into contact until they lose contact. Pinion: is the smaller of the two meshing gears. Backlash: is the difference (clearance) between the tooth thickness of one gear and the tooth space of the meshing gear measured along the pitch circle (Fig. 7.5) 7.5).
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©1998 McGraw-Hill, Hamrock, Jacobson and Schmid
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Clearance (c): is the addendum minus dedendum. dedendum. Working depth: is the distance that one tooth of a meshing gear penetrates into the the tooth space. Base circle: is an imaginary circle about which the tooth involute profile involute profile is developed. Fillet: is the radius that occurs where the flank flank of of the tooth meets the dedendum circle. circle. Module: diametral pitch in metric Module: replaces diametral pitch system. 54
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Specifications for standard gear teeth
Basic formulas for spur gears
Diametral pitch, Diametral pitch, Pd =
Circular pitch, Pc =
Addendum, a =
N P
Item
DP
π
DP
1 Pd
Dedendum, Dedendum, b = 1.250 Pd
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Clearance, c = b – a= b – a
Where
20°
14½ 14½°
Addendum (in.)
1.0/ Pd
1.0/ Pd
1.0/ P d
1/ Pd
Dedendum (in.)
1.250/ Pd
1.250/ P d
1.2/ Pd + 0.002
1.157/ Pd
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Basic formulas for spur gears
0.250
Pd
D p = pitch pitch diameter diameter of pinion pinion
Center to center distance D + D pp N g + N p CtoC = pg = 2 Pd 2 Where
N p = number of teeth on the pinion
It can be shown that
Pd × Pc = π
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D pp = pitch diameter of pinion N p = number of teeth on the pinion D pg = pitch diameter of gear Ng = number of teeth on the gear Pd = Diametral pitch Diametral pitch
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Metric System
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Metric System
1
Module (m) =
See Table Table 11.1 11.1 for for equivalents equivalents
Pd
Normally they are not converted
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14½ 14½° full depth
25°
Basic formulas for spur gears
Full depth & pitches finer than 20
20°
Pressure angle
N P
Full depth & pitches coarser than 20
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1
Diametral pitch, Diametral pitch, Pd =
m
Circular pitch, Pc =
m
Addendum, a = m
Dedendum, Dedendum, b = 1.25 m
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π
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Inch units
Metric units
A spur gear of the 14 ½ degree degree involute system system has 32 teeth of diametral pitch 8. Find of diametral pitch
A spur gear of the 14 ½ degree involute system has a module of 8 mm and 35 teeth. Find
The pitch diameter
The pitch diameter
The circular pitch
The circular pitch
The outside diameter (addendum diameter)
The outside diameter (addendum diameter)
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Example Problem 11-3: Pressure Angle Example Problem 11-3: Pressure Angle
• For the set of gears shown in Figure 11-17, if diametral pitch is 8, find the pitch diameter, circular pitch, and shaft center-to-center distance. • The pinion has 16 teeth and the gear has 32 teeth. − −
Pitch diameter:
Centerline distance:
(11-4) D p = −
−
−
(11-2)
N p N g or Pd Pd
Pinion: D p =
16 = 2 inches 8
D p =
32 = 4 inches 8
(cont’d.)
C-C=
Gear:
D p Dg N p + N g + or C - C = 2 2 2Pd C-C=
16 + 32 2 (8)
C – C = 3 inches
Circular pitch: (11-3) Pc =
Pc =
π 2
π D p
N p
in = .393 inch 16
• Circular pitch would be the same for both pinion and gear.
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Summary To understand the gears one should be familiar with the gear terminology.
Spur gears are most commonly used for transmission of power.
Mating gears should have the same diametral pitch. diametral pitch.
A number of gear manufacturing methods are available.
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Speed of mating gears is inversely proportional to the number of teeth.
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Good gear design should take care of the power, speed, life and material properties. 68
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