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Rev: 12- 11-98 c.d.
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Table 1-5 (Cont.) Spur Gear Design Formulas
*All linear dimensions in millimetersSymbols per Table 1-4
SECTION 2 INTRODUCTION TO GEAR TECHNOLOGY
This section presents a technical coverage of gear fundamentals. It is intended as a broadcoverage written in a manner that is easy to follow and to understand by anyone interested inknowing how gear systems function. Since gearing involves specialty components, it is expected thatnot all designers and engineers possess or have been exposed to every aspect of this subject.However, for proper use of gear components and design of gear systems it is essential to have aminimum understanding of gear basics and a reference source for details.
For those to whom this is their first encounter with gear components, it is suggested thistechnical treatise be read in the order presented so as to obtain a logical development of the subject.Subsequently, and for those already familiar with gears, this material can be used selectively inrandom access as a design reference.
2.1 Basic Geometry Of Spur Gears
The fundamentals of gearing are illustrated through the spur gear tooth, both because it is thesimplest, and hence most comprehensible, and because it is the form most widely used, particularlyfor instruments and control systems.
The basic geometry and nomenclature of a spur gear mesh is shown in Figure 2-1 . Theessential features of a gear mesh are:
1. Center distance.
To Obtain
Dedendum
Outside Diameter
Root Diameter
Base Circle Diameter
Base Pitch
Tooth Thickness atStandard Pitch Diameter
Center Distance
Contact Ratio
Backlash (linear)
Backlash (linear)
Backlash (linear) AlongLine-of-action
Backlash, Angular
Min. No. of Teethfor No Undercutting
Use This Formula*
b = 1.25m
Do = D + 2m = m (N + 2)
DR = D 2.5m
Db = D cos
pb = m cos
Tstd = m2
m (N1 + N2)C = 2
1Ro 1Rb + 2Ro 2Rb C sinmp = m cosB = 2(C)tan
B = T
BLA = B cos
BaB = 6880 (arc minutes) D
2Nc = sin2
From Known
Module
Module and Pitch Diameter orNumber of Teeth
Pitch Diameter and Module
Pitch Diameter and Pressure Angle
Module and Pressure Angle
Module
Module and Number of Teeth
Outside Radii, Base Circle Radii,Center Distance, Pressure Angle
Change in Center Distance
Change in Tooth Thickness
Linear Backlash Along Pitch Circle
Linear Backlash
Pressure Angle
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2. The pitch circle diameters (or pitch diameters).3. Size of teeth (or module).4. Number of teeth.5. Pressure angle of the contacting involutes.
Details of these items along with their interdependence and definitions are covered in subsequentparagraphs.
Fig. 2-1 Basic Gear Geometry
Generally:
Larger Gear Diameter orRadius Symbols capitalletters
Smaller Gear Diameteror Radius Symbols lowercase letters
PINION
GEAR
Outside Diameter (da )
rb rra
Pitch Circle
Tooth Profile
Pitch CircleWhole Depth (h)
Addendum(ha)
CenterDistance(a)
Root(Tooth)Fillet
TopLand
Pitch Point
Rb
R
RaRoot Diameter (D
f )
Pitc
h D
iam
eter
(D)
Circular Pitch (p)
Chordal ToothThickness
Circular ToothThickness
Base Diameter (Db)Clearance
WorkingDepth (hw)
Line-of-Centers
Dedendum (hf)
Pressureangle ()
Line-of-Action
Base Circle
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2.2 The Law Of Gearing
A primary requirement of gears is the constancy of angular velocities or proportionality ofposition transmission. Precision instruments require positioning fidelity. High-speed and/or high-powergear trains also require transmission at constant angular velocities in order to avoid severe dynamicproblems.
Constant velocity (i.e., constant ratio) motion transmission is defined as "conjugate action" of thegear tooth profiles. A geometric relationship can be derived (2, 12)* for the form of the tooth profiles toprovide conjugate action, which is summarized as the Law of Gearing as follows:
"A common normal to the tooth profiles at their point of contact must, in all positions of thecontacting teeth, pass through a fixed point on the line-of-centers called the pitch point."
Any two curves or profiles engaging each other and satisfying the law of gearing are conjugatecurves.
2.3 The Involute Curve
There is almost an infinite number of curves that can be developed to satisfy the law of gearing,and many different curve forms have been tried in the past. Modern gearing (except for clock gears)is based on involute teeth. This is due to three major advantages of the involute curve:
Conjugate action is independent of changes in center distance.The form of the basic rack tooth is straight-sided, and therefore is relatively simple and canbe accurately made; as a generating tool it imparts high accuracy to the cut gear tooth.One cutter can generate all gear teeth numbers of the same pitch.
The involute curve is most easily understood as the trace of a point at the end of a taut stringthat unwinds from a cylinder. It is imagined that a point on a string, which is pulled taut in a fixeddirection, projects its trace onto a plane that rotates with the base circle. See Figure 2-2 . The basecylinder, or base circle as referred to in gear literature, fully defines the form of the involute and in agear it is an inherent parameter, though invisible.
The development and action of mating teeth can be visualized by imagining the taut string asbeing unwound from one base circle and wound on to the other, as shown in Figure 2-3a . Thus, asingle point on the string simultaneously traces an involute on each base circle's rotating plane. Thispair of involutes is conjugate, since at all points of contact the common normal is the common tangentwhich passes through a fixed point on the line-of-centers. If a second winding/unwinding taut string iswound around the base circles in the opposite direction, Figure 2-3b , oppositely curved involutes aregenerated which can accommodate motion reversal. When the involute pairs are properly spaced, theresult is the involute gear tooth, Figure 2-3c .
Fig. 2-2 Generation of an Fig. 2-3 Generation and Involute by a Taut String Action of Gear Teeth
* Numbers in parenthesis refer to references at end of text.
1.2.
3.
Trace Point
InvoluteCurve
Base Cylinder
UnwindingTaut String
InvoluteGeneratingPoint onTaut String
Base Circle
BaseCircle
Taut String
(a) Left-HandInvolutes
(b) Right-HandInvolutes
(c) Complete Teeth Generatedby Two Crossed GeneratingTaut Strings
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2.4 Pitch Circles
Referring to Figure 2-4 , the tangent to the two base circles is the line of contact, or line-of-actionin gear vernacular. Where this line crosses the line-of-centers establishes the pitch point, P. This inturn sets the size of the pitch circles, or as commonly called, the pitch diameters. The ratio of thepitch diameters gives the velocity ratio:
Velocity ratio of gear 2 to gear 1 is:
d1i = (2-1) d2
2.5 Pitch And Module
Essential to prescribing gear geometry is the size, or spacing of the teeth along the pitch circle.This is termed pitch, and there are two basic forms.
Circular pitch A naturally conceived linear measure along the pitch circle of the toothspacing. Referring to Figure 2-5 , it is the linear distance (measured along the pitch circle arc)between corresponding points of adjacent teeth. It is equal to the pitch-circle circumference dividedby the number of teeth:
pitch circle circumference dp = circular pitch = = (2-2) number of teeth z
Module Metric gearing uses the quantity module (m) in place of the American inch unit,diametral pitch. The module is the length of pitch diameter per tooth. Thus:
dm = (2-3)
z
Relation of pitches : From the geometry that defines the two pitches, it can be shown thatmodule and circular pitch are related by the expression:
p = (2-4)m
This relationship is simple to remember and permits an easy transformation from one to theother.
d1
d2
PitchPoint (P) Base Circle, Gear #1
Base Circle, Gear #2
PitchCircles
Fig. 2-4 Definition of Pitch Circle andPitch Point
Line ofcontact
p
Fig. 2-5 Definition of Circular Pitch
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Diametral pitch (Pd) is widely used in England and America to represent the tooth size.The relation between diametral pitch and module is as follows:
25.4m = (2-5)
Pd
2.6 Module Sizes And Standards
Module m represents the size of involute gear tooth. The unit of module is mm.Module is converted to circular pitch p, by the factor .
p = m (2-6)
Table 2-1 is extracted from JIS B 1701-1973 which defines the tooth profile anddimensions of involute gears. It divides the standard module into three series. Figure 2-6shows the comparative size of various rack teeth.
Table 2-1 Standard Values of Module unit: mm
Note: The preferred choices are in the series order beginning with 1.
Series 3
0.65
3.25
Series 1
0.1
0.2
0.3
0.4
0.5
0.6
0.8
1 1.25 1.5
2
2.5
3
Series 2
0.15
0.25
0.35
0.45
0.55
0.7 0.75
0.9
1.75
2.25
2.75
Series 1
4
5
6
8
10
12
16
20
25
32
40
50
Series 3
3.75
6.5
Series 2
3.5
4.5
5.5
7
9
11
14
18
22
28
36
45
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Fig. 2-6 Comparative Size of Various Rack Teeth
M1
M1.5
M2
M2.5
M3
M4
M5
M6
M10
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Circular pitch, p, is also used to represent tooth size when a special desired spacingis wanted, such as to get an integral feed in a mechanism. In this case, a circular pitch ischosen that is an integer or a special fractional value. This is often the choice in designingposition control systems. Another particular usage is the drive of printing plates to providea given feed.
Most involute gear teeth have the standard whole depth and a standard pressureangle = 20. Figure 2-7 shows the tooth profile of a whole depth standard rack toothand mating gear. It has an addendum of ha = 1m and dedendum hf 1.25m. If tooth depthis shorter than whole depth it is called a stub tooth; and if deeper than whole depth it isa high depth tooth.
The most widely used stub tooth has an addendum ha = 0.8m and dedendum hf = 1m.Stub teeth have more strength than a whole depth gear, but contact ratio is reduced. Onthe other hand, a high depth tooth can increase contact ratio, but weakens the tooth.
In the standard involute gear, pitch (p) times the number of teeth becomes the lengthof pitch circle:
d = mz
Pitch diameter (d) is then: (2-7)
d = mz
Fig. 2-7 The Tooth Profile and Dimension of Standard Rack
Metric Module and Inch Gear Preferences: Because there is no direct equivalence betweenthe pitches in metric and inch systems, it is not possible to make direct substitutions. Further, thereare preferred modules in the metric system. As an aid in using metric gears, Table 2-2 presentsnearest equivalents for both systems, with the preferred sizes in bold type.
Module mPressure Angle = 20Addendum ha = mDedendum hf 1.25mWhole Depth h 2.25mWorking Depth hw = 2.00mTop Clearance c = 0.25mCircular Pitch p = mPitch Perpendicular to Tooth pn = p cosPitch Diameter d = mzBase Diameter db = d cos
p
pn
p2
db
hf
d
hah
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Table 2-2 Metric/American Gear Equivalents
NOTE: Bold face diametral pitches and modules designate preferred values.
Continued on following page
203.2000200180
169.333150
127.000125120
101.60096
92.363684.6667
8078.153872.5714
7267.733
6463.50050.800
504844
42.33340
36.285736
33.866732
31.750030
28.222228
25.40002422
20.32002018
16.93331615
14.51431413
12.700012
11.288911
10.160010
0.01550.01570.01750.01860.02090.02470.02510.02620.03090.03270.03400.03710.03930.04020.04330.04360.04640.04910.04950.06180.06280.06550.07140.07420.07850.08660.08730.09280.09820.09890.10470.11130.11220.12370.13090.14280.15460.15710.17450.18550.19630.20940.21640.22440.24170.24740.26180.27830.28560.30920.3142
0.1250.127000.14111
0.150.16933
0.20.203200.21167
0.250.26458
0.2750.3
0.317500.3250.35
0.352780.375
0.396880.40.5
0.508000.529170.57727
0.60.63500
0.70.70556
0.750.79375
0.80.84667
0.90.90714
11.05831.15451.25
1.27001.4111
1.51.58751.69331.75
1.81431.9538
22.11672.25
2.30912.50
2.5400
0.3930.3990.4430.4710.5320.6280.6380.6650.7850.8310.8640.9420.9971.0211.1001.1081.1781.2471.2571.5711.5961.6621.8141.8851.9952.1992.2172.3562.4942.5132.6602.8272.8503.1423.3253.6273.9273.9904.4334.7124.9875.3205.4985.7006.1386.2836.6507.0697.2547.8547.980
0.00770.00790.00870.00930.01050.01240.01260.01310.01550.01640.01700.01860.01960.02010.02160.02180.02320.02450.02470.03090.03140.03270.03570.03710.03930.04330.04360.04640.04910.04950.05240.05570.05610.06180.06540.07140.07730.07850.08730.09280.09820.10470.10820.11220.12080.12370.13090.13910.14280.15460.1571
0.1960.1990.2220.2360.2660.3140.3190.3320.3930.4160.4320.4710.4990.5110.5500.5540.5890.6230.6280.7850.7980.8310.9070.9420.9971.1001.1081.1781.2471.2571.3301.4141.4251.5711.6621.8131.9631.9952.2172.3562.4942.6602.7492.8503.0693.1423.3253.5343.6273.9273.990
0.00490.00500.00560.00590.00670.00790.00800.00830.00980.01040.01080.01180.01250.01280.01380.01390.01480.01560.01570.01970.02000.02080.02270.02360.02500.02760.02780.02950.03130.03150.03330.03540.03570.03940.04170.04550.04920.05000.05560.05910.06250.06670.06890.07140.07690.07870.08330.08860.09090.09840.1000
0.1250.1270.1410.1500.1690.2000.2030.2120.2500.2650.2750.3000.3180.3250.3500.3530.3750.3970.4000.5000.5080.5290.5770.6000.6350.7000.7060.7500.7940.8000.8470.9000.9071.0001.0581.1551.2501.2701.4111.5001.5881.6931.7501.8141.9542.0002.1172.2502.3092.5002.540
DiametralPitch
P
Modulem
Circular Pitch Circular Tooth Thickness Addendum
inches millimeters inches millimeters inches millimeters
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Table 2-2 (Cont.) Metric/American Gear Equivalents
NOTE: Bold face diametral pitches and modules designate preferred values.
0.10830.11110.11810.12500.12800.13780.14290.14760.15750.16670.17720.18700.19690.20000.21650.23620.25000.25590.27560.28570.31500.31830.33330.35430.39370.40000.43310.47240.50000.55120.62990.66670.70870.78740.86610.94490.98431.00001.06301.10241.18111.25981.29921.33331.41731.53541.57481.65351.77171.96852.0000
2.7502.8223.0003.1753.2503.5003.6293.7504.0004.2334.5004.7505.0005.0805.5006.0006.3506.5007.0007.2578.0008.0858.4679.00010.00010.16011.00012.00012.70014.00016.00016.93318.00020.00022.00024.00025.00025.40027.00028.00030.00032.00033.00033.86736.00039.00040.00042.00045.00050.00050.800
0.17010.17450.18550.19630.20100.21640.22440.23190.24740.26180.27830.29380.30920.31420.34010.37110.39270.40200.43290.44880.49470.50000.52360.55660.61840.62830.68030.74210.78540.86580.98951.04721.11321.23681.36051.48421.54611.57081.66971.73161.85531.97902.04082.09442.22632.41192.47372.59742.78293.09213.1416
4.3204.4334.7124.9875.1055.4985.7005.8906.2836.6507.0697.4617.8547.9808.6399.4259.975
10.21010.99611.39912.56612.70013.29914.13715.70815.95917.27918.85019.94921.99125.13326.59928.27431.41634.55837.69939.27039.89842.41243.98247.12450.26551.83653.19856.54961.26162.83265.97370.68678.54079.796
Modulem
9.23649
8.46678
7.81547.2571
76.77336.3500
65.64445.34745.0800
54.61824.2333
43.90773.62863.50003.17503.1416
32.82222.54002.50002.30912.1167
21.81431.58751.50001.41111.27001.15451.05831.0160
10.94070.90710.84670.79380.76970.75000.70560.65130.63500.60480.56440.50800.5000
2.752.8222
33.1750
3.253.5
3.62863.75
44.2333
4.54.75
55.08005.5000
66.35006.5000
77.2571
88.08518.4667
910
10.1601112
12.7001416
16.9331820222425
25.4002728303233
33.867363940424550
50.800
0.34010.34910.37110.39270.40200.43290.44880.46380.49470.52360.55660.58750.61840.62830.68030.74210.78540.80400.86580.89760.98951.00001.04721.11321.23681.25661.36051.48421.57081.73161.97902.09442.22632.47372.72112.96843.09213.14163.33953.46323.71053.95794.08164.18884.45274.82374.94745.19485.56586.18426.2832
8.6398.8669.4259.97510.21010.99611.40011.78112.56613.29914.13714.92315.70815.95917.27918.85019.94920.42021.99122.79925.13325.40026.59928.27431.41631.91934.55837.69939.89843.98250.26553.19856.54962.83269.11575.39878.54079.79684.82387.96594.248
100.531103.673106.395113.097122.522125.664131.947141.372157.080159.593
DiametralPitch
P
AddendumCircular Tooth ThicknessCircular Pitch
inches millimetersinches millimetersinches millimeters
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2.7.2 Intersecting Axes Gears
1. Straight Bevel Gear
This is a gear in which the teeth have taperedconical elements that have the same direction as thepitch cone base line (generatrix). The straight bevelgear is both the simplest to produce and the mostwidely applied in the bevel gear family.
2. Spiral Bevel Gear
This is a bevel gear with a helical angle of spiralteeth. It is much more complex to manufacture, butoffers a higher strength and lower noise.
3. Zerol Gear
Zerol gear is a special case of spiral bevel gear.It is a spiral bevel with zero degree of spiral angletooth advance. It has the characteristics of both thestraight and spiral bevel gears. The forces acting uponthe tooth are the same as for a straight bevel gear.
2.7.3 Nonparallel and Nonintersecting Axes Gears
1. Worm and Worm Gear
Worm set is the name for a meshed worm andworm gear. The worm resembles a screw thread; andthe mating worm gear a helical gear, except that it ismade to envelope the worm as seen along the worm'saxis. The outstanding feature is that the worm offersa very large gear ratio in a single mesh. However,transmission efficiency is very poor due to a greatamount of sliding as the worm tooth engages with itsmating worm gear tooth and forces rotation by pushingand sliding. With proper choices of materials andlubrication, wear is contained and noise is low.
Fig. 2-14 Straight Bevel Gear
Fig. 2-15 Spiral Bevel Gear
Fig. 2-16 Zerol Gear
Fig. 2-17 Worm Gear
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