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780 Technical 015-12-95

Rev: 12- 11-98 c.d.

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D780 - Technical Section P-17D777 - Technical Section P-17

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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Rev: 12-9-98 c.d.D780 - Technical Section P-19D777 - Technical Section P-19

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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780 Technical 025-12-95


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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Rev: 12-9-98 c.d.

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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