1.0 INTRODUCTION2.0 BASIC GEOMETRY OF SPUR GEARS      2.1 Basic Spur Gear Geometry      2.2 The Law of Gearing      2.3 The Involute Curve      2.4  Pitch  Circles      2.5 Pitch            2.5.1 Circular Pitch            2.5.2 Diametral Pitch            2.5.3 Relation of Pitches3.0 GEAR TOOTH FORMS AND STANDARDS      3.1 Preferred Pitches      3.2 Design Tables      3.3 AGMA Standards4.0 INVOLUTOMETRY       4.1.1 Gear Nomenclature       4.1.2 Symbols       4.2 Pitch Diameter and Center Distance       4.3 Velocity Ratio       4.4 Pressure Angle       4.5 Tooth Thickness       4.6 Measurement Over-Pins       4.7 Contact Ratio       4.8 Undercutting       4.9 Enlarged Pinions       4.10 Backlash Calculation       4.11 Summary of Gear Mesh Fundamentals5.0 HELICAL GEARS       5.1 Generation of the Helical Tooth       5.2 Fundamental of Helical Teeth       5.3 Helical Gear Relationships       5.4  Equivalent Spur Gear       5.5  Pressure Angle       5.6  Importance of Normal Plane Geometry       5.7 Helical Tooth Proportions       5.8 Parallel Shaft Helical Gear Meshes               5.8.1 Helix Angle               5.8.2 Pitch Diameter               5.8.3 Center Distance               5.8.4 Contact Ratio               5.8.5 Involute Interference       5.9 Crossed Helical Gear Meshes               5.9.1 Helix Angle and Hands               5.9.2 Pitch

T25   T25T25T27T27T28T28T28T28     T29T29T29    T31T37T37T38T38138T39144144145145T48    T52T53T53T54T54T54T55T55155T55T55T55156156T56156

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                   5.9.3 Center Distance                   5.9.4 Velocity Ratio        5.10 Axial Thrust of Helical Gears     6.0 RACKS7.0 INTERNAL GEARS        7.1 Development of the Internal Gear        7.2 Tooth Parts of Internal Gear        7.3 Tooth Thickness Measurement        7.4 Features of Internal Gears8.0 WORM MESH        8.1 Worm Mesh Geometry        8.2 Worm Tooth Proportions        8.3 Number of Threads        8.4 Worm and Wormgear Calculations                  8.4.1 Pitch Diameters, Lead and Lead Angle                  8.4.2 Center Distance of Mesh        8.5 Velocity Ratio9.0 BEVEL GEARING        9.1 Development and Geometry of Bevel Gears        9.2 Bevel Gear Tooth Proportions        9.3 Velocity Ratio        9.4 Forms of Bevel Teeth10.0 GEAR TYPE EVALUATION11.0 CRITERIA OF GEAR QUALITY        11.1 Basic Gear Formats        11.2 Tooth Thickness and Backlash         11.3 Position Error (or Transmission Error)          11.4 AGMA Quality Classes        11.5 Comparison With Previous AGMA and International Standards12.0 CALCULATION OF GEAR PERFORMANCE CRITERIA        12.1 Backlash in a Single Mesh        12.2 Transmission Error        12.3 Integrated Position Error        12.4 Control of Backlash        12.5 Control of Transmission Error13.0 GEAR STRENGTH AND DURABILITY        13.1 Bending Tooth Strength        13.2 Dynamic Strength        13.3 Surface Durability        13.4 AGMA Strength and Durability Ratings

T57T57T57    T58     T58T59T60T61    T61T62T62T62T63T63T64    T64T66T66T67T68   T68T70T70T73T73   T76T77T77T78T78  T78T82T88T88

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Catalog D190

file:///C|/A3/D190/HTML/D190T22.htm [9/27/2000 4:11:52 PM]

14.0 GEAR MATERIALS        14.1 Ferrous Metals                 14.1.1 Cast Iron                14.1.2 Steel        14.2 Non Ferrous Metals                14.2.1 Aluminum                14.2.2 Bronzes        14.3 Die Cast Alloys        14.4 Sintered Powder Metal        14.5 Plastics        14.6 Applications and General Comments15.0 FINISH COATINGS        15.1 Anodize        15.2 Chromate Coatings        15.3 Passivation        15.4 Platings        15.5 Special Coatings        15.6 Application of Coatings16.0 LUBRICATION        16.1 Lubrication of Power Gears        16.2 Lubrication of Instrument Gears        16.3 Oil Lubricants        16.4 Grease        16.5 Solid Lubricants        16.6 Typical Lubricants17.0 GEAR FABRICATION        17.1 Generation of Gear Teeth                17.1.1 Rack Generation                17.1.2 Hob Generation                17.1.3 Gear Shaper Generation                17.1.4 Top Generating        17.2 Gear Grinding        17.3 Plastic Gears18.0 GEAR INSPECTION        18.1 Variable-Center-Distance Testers                18.1.1 Total Composite Error                18.1.2 Gear Size                18.1.3 Advantages and Limitations of Variable-Center-Distance Testers...        18.2 Over-Pins Gaging        18.3 Other Inspection Equipment        18.4 Inspection of Fine-Pitch Gears        18.5 Significance of Inspection and Its Implementation

   T91T91T91T92T92T92T92T92T92T99   T99T100T100T100T100T100    T101T101T101T103T103T103    T105T105T105T105T106T106T107     T107T107T107T107T108T108T108T108

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19.0 GEARS, METRIC        19.1 Basic Definitions        19.2 Metric Design Equations        19.3 Metric Tooth Standards        19.4 Use of Strength Formulas        19.5 Metric Gear Standards                19.5.1 USA Metric Gear Standards                19.5.2 Foreign Metric Gear Standards20.0 DESIGN OF PLASTIC MOLDED GEARS        20.1 General Characteristics of Plastic Gears        20.2 Properties of Plastic Gear Materials        20.3 Pressure Angles        20.4 Diametral Pitch        20.5 Design Equations for Plastic Spur, Bevel, Helical and Worm Gears                20.5.1 General Considerations                20.5.2 Bending Stress - Spur Gears                20.5.3 Surface Durability for Spur and Helical Gears                20.5.4 Design Procedure - Spur Gears                20.5.5 Design Procedure Helical Gears                20.5.6 Design Procedure - Bevel Gears                20.5.7 Design Procedure - Worm Gears       20.6 Operating Temperature       20.7 Eftect of Part Shrinkage on Gear Design       20.8 Design Specifications       20.9 Backlash       20.10 Environment and Tolerances       20.11 Avoiding Stress Concentration       20.12 Metal Inserts       20.13 Attachment of Plastic Gears to Shafts       20.14 Lubrication       20.15 Inspection       20.16 Molded vs Cut Plastic Gears       20.17 Elimination of Gear Noise       20.18 Mold Construction       20.19 Conclusion 

   T109T122T124T125T126T126T126    T131T132T139T139T139T139T140T141T143T146T146T147T147T147T150T150T150T150T151T151T152T152T152T153T153T158

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1.0 INTRODUCTION

This section presents a technical coverage of gear fundamentals. It is intended as a broad coverage written in a manner that iseasy to follow and to understand by anyone interested in knowing how gear systems function. Since gearing involves specialtycomponents it is expected that not all designers and engineers possess or have been exposed to all aspects of this subjectHowever, for proper use of gear components and design of gear systems it is essential to have a minimum understanding of gearbasics and a reference source for details.       For those to whom this is their first encounter with gear components, it is suggested this section be read in the orderpresented so as to obtain a logical development of the subject. Subsequently, and for those already familiar with gears, thismaterial can be used selectively in random access as a design reference.

2.0 BASIC GEOMETRY OF SPUR GEARS

The fundamentals of gearing are illustrated through the spur-gear tooth, both because it is the simplest, and hence mostcomprehensible, and because it is the form most widely used, particularly in instruments and control systems.

2.1 Basic Spur Gear Geometry

The basic geometry and nomenclature of a spur-gear mesh is shown in Figure 1.1. The essential features of a gear mesh are:

1. center distance2. the pitch circle diameters (or pitch diameters)3. size of teeth (or pitch)4. number of teeth5. pressure angle of the contacting involutes

Details of these items along with their interdependence and definitions are covered in subsequent paragraphs.

2.2 The Law of Gearing

A primary requirement of gears is the constancy of angular velocities or proportionality of position transmission, Precisioninstruments require positioning fidelity. High speed and/or high power gear trains also require transmission at constant angularvelocities in order to avoid severe dynamic problems.        Constant velocity (i.e. constant ratio) motion transmission is defined as “conjugate action” of the gear tooth profiles. Ageometric relationship can be derived (1,7)* for the form of the tooth profiles to provide cojugate action, which is summarized asthe Law of Gearing as follows:       “A common normal to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through afixed 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 conjugate Curves.

___________*Numbers in parenthesis refer to references at end of text.

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2.3 The Involute Curve

There are almost an infinite number of curves that can be developed to satisfy the law of gearing, and many different curve formshave been tried in the past. Modem gearing (except for clock gears) based on involute teeth. This is due to three majoradvantages of the involute curve:

1. Conjugate action is independent of changes in center distance.2. The form of the basic rack tooth is straight-sided, and therefore is relatively simple and can be  accurately made; as agenerating tool ft imparts high accuracy to the cut gear tooth.3. One cutter can generate all gear tooth numbers of the same pitch.

The involute curve is most easily understood as the trace of a point at the end of a taut string that unwinds from a cylinder. It isimagined that a point on a string, which is pulled taut in a fixed direction, projects its trace onto a plane that rotates with thebase circle. See Figure 1.2. The base cylinder, or base circle as referred to in gear literature, fully defines the form of the involuteand in a gear it is an inherent parameter, though invisible.          The development and action of mating teeth can be visualized by imagining the taut string as being unwound from onebase circle and wound on to the other, as shown in Figure 1.3a Thus, a single point on the string simultaneously traces aninvolute on each base circles rotating plane. This pair of involutes is conjugate, since at all points of contact the common normalis the common tangent which passes through a fixed point on the line-of-centers. It a second winding/unwinding taut string iswound around the base circles in the opposite direction, Figure 1 .3b, oppositely curved involutes are generted which canaccommodate motion reversal. When the involute pairs are properly spaced the result is the involute gear tooth, Figure 1.3c.

2.4 Pitch Circles

Referring to Figure 1.4 the tangent to the two base circles is the line of contact, or line-of-action in gear vernacular. Where thisline crosses the line-of-centers establishes the pitch point, P. This in turn sets the size of the pitch circles, or as commonly called,the pitch diameters. The ratio of the pitch diameters gives the velocity ratio:       Velocity ratio of gear 2 to gear 1 = Z =  D1                                                                        (1)                                                        D2

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2.5 Pitch

Essential to prescribing gear geometry is the size, or spacing of the teeth along the pitch circle. This is termed pitch and there aretwo basic forms.       2.5.1 Circular pitch — A naturally conceived linear measure along the pitch circle of the tooth  spacing. Referring to Figure1.5 it is the linear distance (measured along the pitch circle ar between corresponding points of adjacent teeth. it is equal to thepitch-circle circumference divided by the number of teeth:pc = circular pitch = pitch circle circumference = Dπ                                         (2)                                   number of teeth           N                 2.5.2 Diametral pitch — A more popularly used pitch measure, although geometrically much less evident, is one that is ameasure of the number of teeth per inch of pitch diameter. This is simply: expressed as:             Pd = diametral pitch = N                                                                       (3)                                              DDiametral pitch is so commonly used with fine pitch gears that it is usually contracted simply to "pitch" and that it is diametral isimplied.      2.5.3 Relation of pitches: From the geometry that defines the two pitches it can be shown that they are related by theproduct expression:

              Pd x Pe   =   π                                                                                     (4)This relationship is simple to remember and permits an easy transformation from one to the other.

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3.0 GEAR TOOTH FORMS AND STANDARDS

involute gear tooth forms and standard tooth proportions are specified in terms of a basic rack which has straight-sided teeth forinvolute systems. The American National Standards Institute (ANSI) and the American Gear Manufacturers Association (AGMA)have jointly established standards for the USA. Although a large number of tooth proportions and pressure angle standards havebeen formulated, only a few are currently active and widely used. Symbols for the basic rack are given in Figure 1.6 andpertinent standards for tooth proportions in Table 1.1.      Note that data in Table 1.1 is based upon diametral pitch equal to one. To convert to another pitch divide by diametral pitch.

3.1 Preferred Pitches

Although there are no standards for pitch choice a preference has developed among gear designers and producers. This is givenin Table 1.2. Adherence to these pitches is very common in the fine- pitch range but less so among the coarse pitches.

3.2 Design Tables

For the preferred pitches it is helpful in gear design to have basic data available as a function of the number of teeth on eachgear, Table 1.3 lists tooth proportions common to a given diametral pitch, as well as the diameter of a measuring wire. Table 1.6lists pitch diameters and the over-wires measurement as a function of tooth number (which ranges from 18 to 218) and variousdiametral pitches, including most of the preferred fine pitches. Both tables are for 20° pressure-angle gears.

3.3 AGMA Standards

In the United States most gear standards have been developed and sponsored by the AGMA. They range from general and basicstandards, such as those already mentioned for tooth form, to specialized standards. The list is very long and only a selected few,most pertinent to fine pitch gearing, are listed in Table 1.4. These and additional standards can be procured from the AGMA bycontacting the headquarters office at 1500 King Street; Suite 201; Alexandria, VA 22314 (Phone: 703-684-0211).

a = Addendumb = Dedendumc = Clearancehk = Working Depthht = Whole DepthPc = Circular Pitchrf = Fillet Radiust = circular Tooth Thicknessφ = Pressure Angle

Figure 1.6 Extract from AGMA 201.02 (ANSI B6.1 1968)

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TABLE 1.1 TOOTH PROPORTIONS OF BASIC RACK FORSTANDARD INVOLUTE GEAR SYSTEMS

Tooth Parameter

Symbolin

RackFig. 1.6

14-1/2ºFull DepthinvoluteSystem

20ºFull DepthinvoluteSystem

20ºCoarse-Pitch

involuteSpur Gears

20ºFine-PitchinvoluteSystem

1. System Sponsors2. Pressure Angle3. Addendum4. Dedendum5. Whole Depth6. Working Depth7. Clearance.8. Basic Circular Tooth    Thickness on Pitch Line9. Fillet Radius In    Basic Rack10. Diametral Pitch Range11. Governing Standard:     ANSI                     AGMA               

−−φabhthkCt  rf  --   ----

ANSI & AGMA14-1/2°

1/P1.157/P2.157/P

2/P0.157/P1 5708/P

    1-1/3 x

    not specified

    B6.1

201.02

ANSI20°1/P

1.157/P2.157/P

2/P0.157/P1.5708/P

   1-112 X

  not specified

    B6.1

--

AGMA20°

1.000/P1.250/P2.250/P2.000/P0250/P

π/2P  

0.300/P  

not specified   --

201.02

ANSI & AGMA20°

1.000/P1.200/P + 0.0022.200/P + 0.002

2.000/P0.200/P + 0.002

1.5708/P    

not standardized 

not specified  

B6.7207.06

TABLE 1.2 PREFERRED DIAMETRAL PITCHES

Class Pitch

Coarse

1/21246810

Class Pitch

Medium-Coarse

12141618

Class Pitch

Fine

2024324864728096120128

Class Pitch

Ultra-Fine150180200

TABLE 1.3 BASIC GEAR DATA FOR 20° P.A. FINE-PITCH GEARSDiameter Pitch 32 48 64 72 80 96 120 200Diameter ofMeasuring Wire* .0540 .0360 .0270 .0240 .0216 .0180 .0144 .0086

Circular PitchCircular ThicknessWhole DepthAddendumDedendumclearance

.09817

.04909

.0708

.0313

.0395

.0083

.06545

.03272

.0478

.0208

.0270

.0062

.04909

.02454

.0364

.0156

.0208

.0051

.04363

.02182

.0326

.0139

.0187

.0048

.03927

.01963

.0295

.0125

.0170

.0045

.03272

.01638

.0249

.0104

.0145

.0041

.02618

.01309

.0203

.0083

.0120

.0037

.01571

.00765

.0130

.0050

.0080

.0030Note: Outside Diameter for N number of teeth equals the Pitch Diameter far (N+2) number at teeth.*For 1.7290 wire diameter basic wire system.

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TABLE 1.4 SELECTED LIST OF AGMA STANDARDS

GeneralAGMA 390

AGMA2000-A88

Gear Classification HandbookGear Classification And Inspection Handbook

spurs AndHelicals

AGMA 201AGMA 207

Tooth portions For Coarse-Pitch Involute Spur GearsTooth Proportions For Fine-Pitch Involute Spur Gears And Helical Gears

Non-Spur

AGMA2005-B88AGMA 203AGMA 374

Design-Manual For Bevel Gears     Fine-Pitch On-Center Face Gears For 20° Involute Spur PinionsDesign For Fine-Pitch Worm Gearing

4.0 INVOLUTOMETRY

Basic calculations for gear systems are included in this section for ready reference in design. More  advanced calculations areavailable in the listed references.

4.1.1 GEAR NOMENCLATURE*

ACTIVE PROFILE is that part of the gear tooth profile which actually comes in contact with the profile of its mating tooth along theline of action.

ADDENDUM (a) is the height by which a tooth projects beyond the pitch circle or pitch line; also, the radial distance between thepitch circle and the addendum circle (Figure 1.1); addendum can be defined as either nominal or operating.

AXIAL PITCH (pa) is the circular pitch in the axial plane and in the pitch surface between corresponding sides of adjacent teeth, inhelical gears and worms. The term axial pitch is preferred to the term linear pitch. (Figure 1.7) 

AXIAL PLANE of a pair of gears is the plane that contains the two axes. In a single gear, an axial plane may be any planecontaining the axis and a given point.

BASE DIAMETER (Db = gear, and db = pinion) is the diameter of the base cylinder from which involute tooth surfaces, eitherstraight or helical, are derived. (Figure 1.1); base radius (Rb = gear, rb = pinion) is one half of the base diameter.

BASE PITCH (pb) in an involute gear is the pitch on the base circle or along the line-of-action. Correspcndng sides of involutegear teeth are parallel curves, and the base pitch is the constant and  fundamental distance between them along a commonnormal in a plane of rotation. (Figure 1.8)

BASIC RACK is a rack that is adopted as the basis for a system of interchangeable gears.

BACKLASH (B) is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth on the pitchcircles. As actually indicated by measuring devices, backlash may be______________*Portions of this section are repented with permission from the Barber-Colman Co., Rockford, Ml.

T31

determined variously in the transverse, normal, or axial planes, and either in the direction of the pit  circles or on theline-of-action. Such measurements should be corrected to corresponding values a transverse pitch circles for generalcomparisons. (Figure 1.9)

CENTER DISTANCE (C), Distance between axes of rotation of mating spur or helical gears.

CHORDAL ADDENDUM (ac) is the height from the top of the tooth to the chord subtending the circular-thickness arc. (Figure1.10)

CHORDAL THICKNESS (tc) is the length of the chord subtending a circular-thickness arc. (Figure 1.10)

CIRCULAR PITCH (pc) is the distance along the pitch circle or pitch line between corresponding profiles of adjacent teeth. (Figure1.1)

CIRCULAR THICKNESS (t) is the length of arc between the two sides of a gear tooth on the p4 circle, unless otherwise specified.(Figure 1.10)

CLEARANCE-OPERATING (c) is the amount by which the dedendum in a given gear exceeds addendum of its mating gear. (Figure1.1)

CONTACT RATIO (Spur) is the ratio of the length-of-action to the base pitch.

CONTACT RATIO (Helical) is the contact ratio in the plane of rotation plus a contact portion a tributted to the axial advance.

DEDENDUM (b) is the depth of a tooth space below the pitch line; also, the radial distance beta, the pitch circle and the rootcircle. (Figure 1.1); dedendum can be defined as either nominal or operating.

DIAMETRAL PITCH (Pd) is the ratio of the number of teeth to the number of inches in the pitch diameter. There is a fixed relationbetween diametral pitch (Pd) and circular pitch (pc): pc = π / Pd

FACE WIDTH (F) is the length of the teeth in an axial plane.

FILLET RADIUS (r,) is the radius of the fillet curve at the base of the gear tooth. In generated this radius is an approximate radiusof curvature. (Figure 1.13)

FULL DEPTH TEETH are those in which the working depth equals      2000"                                                                                                        diametral pitch

GENERATING RACK is a rack outline used to indicate tooth details and dimensions for the design of a hob to produce gears of abasic rack system.

HELIX ANGLE (ψ) is the angle between any helix and an element of its cylinder. In helical gears a worms, it is at the pitchdiameter unless otherwise specified. (Figure 1.7)

INVOLUTE TEETH of spur gears, helical gears, and worms are those in which the active portion of the profile in the transverseplane is the involute of a circle.

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LEAD (L) is the axial advance of a helix for one complete turn, as in the threads of cylindrical worms and teeth of helical gears.(Figure 1.11)

LENGTH-OF-ACTION (ZA) is the distance on an involute line of action through which the point of contact moves during the actionof the tooth profiles. (Figure 1.8)

LEWIS FORM FACTOR (Y, diametral pitch; yc, circular pitch). Factor in determination of beam strength of gears.

LINE-OF-ACTION is the path of contact in involute gears. It is the straight line passing through the pitch point and tangent to thebase circles. (Figure 1.12)

LONG- AND SHORT-ADDENDUM TEETH are those in which the addenda of two engaging gears are unequal.

MEASUREMENT OVER PINS (M). Distance over two pins placed in diametrically opposed tooth spaces (even number of teeth) ornearest to it (odd number of teeth). 

NORMAL CIRCULAR PITCH, Pcn, is the circular pitch in the normal plane, and also the length of the arc along the normal helixbetween helical teeth or threads. (Figure 1.7)

NORMAL CIRCULAR THICKNESS (tn) is the circular thickness in the normal plane. In helical gears. it is an arc of the normal helix,measured at the pitch radius.

NORMAL DIAMETRAL PITCH (Pdn) is the diametral pitch as calculated in the normal plane.

NORMAL PLANE is the plane normal to the tooth. For a helical gear this plane is inclined by the helix angle, ψ, to the plane ofrotation.

OUTSIDE DIAMETER (Do gear, and do = pinion) is the diameter of the addendum (outside) circle (Figure 1.1); the outside radius(Ro gear, ro pinion) is one half the outside diameter.

PITCH CIRCLE is the curve of intersection of a pitch surface of revolution and a plane of rotation. According to theory, it is theimaginary circle that rolls without slip with a pitch circle of a mating gear.   (Figure 1.1)

PITCH CYLINDER is the imaginary cylinder in a gear that rolls without slipping on a pitch cylinder or pitch plane of another gear.

PITCH DIAMETER (D = gear, d = pinion) is the diameter of the pitch circle. In parallel shaft gears, the pitch diameters can bedetermined directly from the center distance and the number of teeth by proportionality. Operating pitch diameter is the pitchdiameter at which the gears operate. (Figure 1.1) The pitch radius (R = gear, r pinion) is one half the pitch diameter (Figure 11).

PITCH POINT is the point of tangency of two pitch circles (or of a pitch circle and pitch line) and is on the line-of-centers. Also, forinvolute gears, it is at the intersection of the line-of-action and a straight line connecting the two gear centers. The pitch point ofa tooth profile is at its intersection with the pitch circle. (Figure 1.1) 

PLANE OF ROTATION is any plane perpendicular to a gear axis.

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PRESSURE ANGLE (φ), for involute teeth, is the angle between the line-of-action and a line tangent to the pitch circle at the pitchpoint. Standard pressure angles are established in connection with standard gear-tooth proportions. (Figure 1.1)

PRESSURE ANGLE — NORMAL (φn) is the pressure angle in the normal plane of a helical or spiral tooth

PRESSURE ANGLE — OPERATING (φr) is determined by the specific center distance at which the gears operate. It is the pressureangle at the operating pitch diameter.

STUB TEETH are those in which the working depth us less than     2.000”                                                                                                  diametral pitch

TIP RELIEF is an arbitrary modification of a tooth profile whereby a small amount of material is removed near the tip of the geartooth. (Figure 1.13)

TOOTH THICKNESS (T) Tooth thickness at pitch circle (circular or chordal — Figure 1.1).

TRANSVERSE CIRCULAR PITCH (Pt) is the circular pitch in the transverse plane. (Figure 1.7)

TRANSVERSE CIRCULAR THICKNESS (tt) is the circular thickness in the transverse plane.

TRANSVERSE PLANE is the plane of rotation and, therefore, is necessarily perpendicular to the go axis.

TRANSVERSE PRESSURE ANGLE (φt) is the pressure angle in the transverse plane.

UNDERCUT is the loss of profile in the vicinity of involute start at the base circle due to tool cutter action in generating teeth withlow numbers of teeth. Undercut may be deliberately introduced to facilitate finishing operations. (Figure 1.13)

WHOLE DEPTH (ht) is the total depth of a tooth space, equal to addendum plus dedendurn, also equal to working depth plusclearance. (Figure 1.1)

WORKING DEPTH (hk) is the depth of engagement of two gears; that is, the sum of their addenda.

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4.1.2 Symbols

The symbols used in this section are summarized below.This is consistent with most gear literature and the publications of AGMAand ANSI.

SYMBOL NOMENCLATURE & DEFINITION

B backlash, linear measure alongpitch circle a addendum

BLAbacklash, linear measurealong line-of-action b dedendum

aB backlash in arc minutes  c clearance

C center distance d pitch diameter, pinion

∆ change in center distance dwpin diameter, for over-pinsmeasurement

Co operating center distance e eccentricity

Cstd standard center distance  hk working depth  

D pitch diameter ht whole depth

Db base circle diameter mp contact ratio

Do outside diameter n number of teeth, pinion

DR root diameter nw number of threads in worm

F face width pa axialpitch

K factor; general pb base pitch

L length, general; also lead of worm pc circular pitch

M measurement over-pins pcn normal circular pitch

N number of teeth, usually gear r pitch radius, pinion

Nc critical number of teeth for no undercutting rb base circle radus, pinion

Nv virtual number of teeth for helical gear rt fillet radius

Pd diametral pitch ro outside radius, pinion

Pdn normal diametral pitch t tooth thickness, and forgeneral use for tolerance

pt horsepower, transmitted yc Lewis factor, circular pitch

R pitch radius, gear or general use γ pitch angle, bevel gear

Rb base circle radius, gear θ rotation angle, general

Ro outside radius, gear λ lead angle, worm gearing

RT testing radius µ mean value

T tooth thickness, gear v gear stage velocity ratio

Wb beam tooth strength φ pressure angle

Y Lewis factor, diametral pitch φο operating pressure angle

Z mesh velocity ratio ψ helix angle (Wb = base helix angle;operating helix angle)

ω angular velocity

invφ involute function

4.2 Pitch Diameter and Center Distance

As already mentioned in par. 2.4, the pitch diameters for a meshing gear pair are tangent at a point on the line-of-centers calledthe pitch point. See figure 1.4. The pitch point always divides the line of centers proportional to the number of teeth in each gear.

          Center distance = C  =  D1 + D2 = N1 + N2                                                                          (5)                                                   2            2Pd

and the pitch-circle dimensions are related as follows:D1 = R1 = N1                                                                      (6)D2    R2    N24.3 Velocity RatioThe gear ratio, or velocity ratio, can be obtained from several different parameters:Z = D1 = N1 = ω1                                                                                     (7)       D2    N2     ω2The ratio, Z, in this equation is the ratio of the angular velocity of gear 2 to that of gear 1.

4.4 Pressure Angle

The pressure angle is defined as the angle between the line- of-action (common tangent to the base circles in Figs. 1.3 and 1.4)and a perpendicular to the line-of-centers. See Figure 1.14. From the, geometry of these figures, it is obvious that the pressureangle varies (slightly) as the cen distance of a gear pair is altered. The base circle is related to the pressure angle and pitchdinmeter by the equation:

Db = D cos φ                               where D and φ are the standard values or alternately,    (8)Db = D cos φ                               where D and φ are the exact operating values.

This basic formula shows that the larger the pressure angle the smaller the base circle. Thus, for standard gears, 14½° pressureangle gears have base circles much nearer to the roots of teeth than 20° gears. It is for this reason that 14 ½° gears encountergreater undercutting problems than 20° gears. This is further elaborated on in section 4.8.

4.5 Tooth ThicknessThis is measured along the pitch circle. For this reason it is specifically called the circular tooth thickness. This is shown in Figure1.1. Tooth thickness is related to the pitch as follows:   T = Pc =  π                                                                                                  (9)            2         2Pd

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The tooth thickness (T2) at a given radius, R2, from the center can be found from a known value (T1) and knownpressure angle (θ1) at that radius (R1), as follows:

T2 = T1  R2 - 2R2 -2R2 (inv θ2 - inv θ1)                                                               (10)                   R1where:      inv θ =tan θ - θ = involute function.

      To save computing time involute-function tables have been computed and are available in the references. An abridged litingis given in Table 1.5.

4.6 Measurement Over-Pins

Often tooth thickness is measured indirectly by gaging over pins which are placed in diametrically opposed tooth spaces, or thenearest to it for odd numbered gear teeth. This is pictured in Figure 1.15.          For a specified tooth thickness the over-pins measurement, M, is calculated as follows:          For an even number of teeth:

  M =  D cos θ  + dw                                        (11)           cos θ1For an odd number of teeth  M = D cos θ cos 90º   + dw                       (12)          cos θ1where the value of θ1 is obtained frominv θ1 = T + invθ +      dw        - € π                    (13)               D                    D cos θ €€€€€€€€€€€€ Ν

       Tabulated values of over-pins measurements for standard gears are given in Table 1.6. This provides a rapid means forcalculating values of M, even for gears with slight departures trom standard tooth thicknesses.       When tooth thickness is to be calculated from a known over-pins measurement, M, the equations can be manipulated toyield:       T = D  ( π   +  inv θc - inv θ -     dw     )                                                                        (14)                    N                            D cos θ

    where:        cos θc = D cos θ                                                                                                          (15)                      2Rc

for an even number of teeth:        Rc  = M - dw                                                                                                                   (16)                     2and for an odd number of teeth:        Rc   =     M - dw                                                                                                                (17)                  2 cos 90º                            N

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TABLE 1.5  INVOLUTE FUNCTONSInv θ = tan θ - θ for values of θ from 10º  to 40º

Degreesθ

Minutes0 12 24 36 48

1011121314

0.001800.002390.003120.003980.00488

0.001910.002530.003280.004170.00520

0.002020.002670.003440.004360.00543

0.002140.002810.003620.004570.00566

0.002260.002960.003790.004760.00590

1516171819

0.006150.007500.009020.010760.01272

0.006400.007790.009350.011130.01314

0.006670.008090.009690.011420.01357

0.006940.008390.010040.011910.01400

0.007210.008700.010390.012310.01444

2021222324

0.014900.017340.020060.023040.02635

0.015370.017860.020630.023680.02705

0.015850.018400.021220.024330.02776

0.016340.018940.021820.024990.02849

0.016830.019490.022420.025660.02922

2526272829

0.029980.033940.038290.043020.04816

0.030740.034780.039200.044020.04924

0.031520.035630.040130.045030.05034

0.032320.036500.041080.046060.05146

0.033130.037390.042040.047100.05260

3031323334

0.053750.059810.066360.073450.08110

0.054920.061080.067730.074930.08270

0.056120.162370.069130.076440.08432

0.057330.063680.070550.077970.08597

0.058560.065020.071990.079520.08765

353637383940

0.089340.098220.107780.118060.129110.14097

0.091060.100080.109780.120200.121410.14344

0.092810.010960.111800.122380.133750.14595

0.094590.103880.113860.124590.136120.14850

0.096390.105820.115940.126830.138530.15108

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4.7 Contact Ratio

To assure smooth continuous tooth action, as one pair of teeth ceases contact a succeeding pair of teeth must already have comeinto engagement. It is desired to have as much overlap as possible. A measure of this overlapping action is the contact ratio. Thisis a ratio of the length of the line-of-action to the base pitch. Figure 1.16 shows the geometry. The length-of-action is determinedfrom the intersection of the length-of-action arid the outside radii. The ratio of the length-of-action to the base pitch isdetermined from:                   mp = (Ro² - Rb²) +(ro² - rb²) - Csin φ                                                                        (18)                                            Pc COS φ

       It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1,calculated for all tolerances at their worst-case values.        A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining timeone pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such as high contact ratiogenerally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pairor specially designed non-standard external spur gears.

4.8 Undercutting

From Figure 1.16 it can be seen that the maximum length of the line-of-contact is limited to the length of the common tangent.Any tooth addendum that extends beyond the tangent points (T and T') is not only useless, but interferes with the root fillet areaof the mating tooth. This results in the typical undercut tooth, shown in Figure 1.17. The undercut not only weakens the toothwith a wasp-like waist, but also removes some of the useful involute adjacent to the base circle.

From the geometry of the limiting length-of-contact (T-T', Figure 1.16) it is evident that interference is first encountered by theaddenda of the gear teeth digging into the mating-pinion tooth flanks. Since addenda are standardized by a fixed ratio (1/Pd) theinterference condition becomes more severe as the number of teeth on the gear increases. The limit is reached when the gearbecomes a rack. This is a realistic case since the hob is a rack-type cutter. The result is that standard gears with

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tooth numbers below a critical value are automatically undercut in the generating process. The limiting number of teeth in a gearmeshing with a rack is given by the expression:     Nc  =       2                                                                                                     (19)

                 sin²φ

     This indicates the minimum number of teeth free of undercutting decreases with increasing Pressure angle. For 14½º thevalue of Nc is 32, and for 20° it is 18. Thus, 200 pressure angle gears with low numbers of teeth have the advantage of muchless undercutting and, therefore, are both stronger and smoother acting.

4.9 Enlarged Pinions

Undercutting of pinion teeth is undesirable because of losses of strength, contact ratio and smoothness ofaction. The seventy of these faults depends upon how far below N, the tooth number is. Undercutting for the first few numbers issmall and in many applications its adverse effects can be neglected.

          For very small numbers of teeth, such as ten and smaller, and forhigh-precision applications, undercutting should be avoided. This is achieved bypinion enlargement (or correction as often termed), wherein the pinion teeth, stillgenerated with a standard cutter, are shifted radially ourward to form a fullinvolute tooth free of undercut The tooth is enlarged both radially andcircumferentially. Comparison of a tooth form before and after enlargement isshown in Figure 1.18.           The details of enlarged pinion design, mating gear design and, in general,profile-shifted gears is a large and involved subject beyond the scope of thiswriting. References 1, 3, 5 and 6 offer additional information. For measurementand inspection  Figure 1.18 Comparison of such gears, in particular, consultreference 5.

4.10  Backlash Calculation

Up to this point the discussion has implied that there is no backlash. If the gears are of standard tooth proportion design andoperate on standard center distance they would function ideally with neither backlash nor jamming.      Backlash is provided for a variety of reasons and cannot be designated without consideration of machining conditions. Thegeneral purpose of backlash is to prevent gears from jamming and making contact on both sides of their teeth simultaneously. Asmall amount of backlash is also desirable to provide for lubricant space and differential expansion between the gear componentsand the housing. Any error in machining which tends to increase the possibility of jamming makes it necessary to increase theamount of backlash by at least as much as the errors. Consequently, the smaller the amount of backlash, the more accuratemust be the machining of the gears. Runout of both gears, errors in profile, pitch, tooth thickness, helix angle and centerdistance — all are factors to consider in the specification of the amount of backlash. On the other hand, excessive backlash isobjectionable, particularly if the drive is frequently reversing or if there is an overrunning load. The amount of backlash must notbe excessive for the requirements of the job, but it should be sufficient so that machining costs are not higher than necessary.      In order to obtain the amount of backlash desired, it is necessary to decrease tooth thickness (see Figure 1.19). Thisdecrease must almost always be greater than the desired backlash because of

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the errors in manufacturing and assembling. Since the amount of the decrease in tooth thickness depends upon the accuracy ofmachining, the allowance for a specified backlash will vary according to the manufacturing conditions.      It is customary to make half of the allowance for backlash on the tooth thickness of each gear of a pair, although there areexceptions. For example, on pinions having very low numbers of teeth, it is desirable to provide all of the allowance on themating gear so as not to weaken the pinion teeth.

        In spur and helical gearing, backlash allowance is usually obtained by sinking the hob deeper into the blank than thetheoretically standard depth. Further, it is true that any increase or decrease in center distance of two gears in any mesh willcause an increase or decrease in backlash. Thus, this is an alternate way of designing backlash into the system.        In the following we give the fundamental equations for the determination of backlash in a single gear mesh. For thedetermination of backlash in gear trains, it is necessary to sum the backlash of each mated gear pair. However, to obtain thetotal backlash for a series of meshes it is necessary to take into account the gear ratio of each mesh relative to a chosenreference shaft in the gear train. For details see Reference 5.          Backlash is defined in Figure 1.20a as the excess thickness of tooth space over the thickness of the mating tooth. There aretwo basic ways in which backlash arises: Tooth thickness is below the zero-backlash value; and the operating center distance isgreater than the zero-backlash value.          If the tooth thickness of either or both mating gears is less than the zero-backlash value, the amount of backlashintroduced in the mesh is simply this numerical difference:                  B = Tstd - Tact = ∆T                                                             (20)

where:

B     = linear backlash measured along the pitch circle (Figure 1.20b)Tstd = no backlash tooth thickness on the operating-pitch circle, which is the standard teeth          thickness for ideal gearsTact = actual tooth thickness

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When the center distance is increased by a relatively small amount, ∆C, a backlash space developsbetween mating teeth, as in Figure 1.21. The relationship between center distance increase and linearbacklash, BLA, along the line of action, is:

BLA = 2(∆C)sin φ                                                                                    (21)

            This measure along the line-of-action is useful when inserting a feeler gage between teeth to measure backlash.            The equivalent linear backlash measured along the pitch circle is given by:

B = 2(∆C) tan φ                                                                                     (22a)

where:∆C = change in center distanceφ = pressure angle

Hence, an approximate relationship between center distance change and change in backlash is:

∆C= 1.933 ∆B for 14½° pressure-angle gears                                             (22b)∆C= 1.374 ∆B for 20° pressure-angle gears                                                (22c)

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        Although these are approximate relationships they are adequate for most uses. Their derivation, limitations, and correctionfactors are detailed in Reference 5.        Note that backlash due to center distance opening is dependent upon the tangent function of the pressure angle. Thus, 20°gears have 41% more backlash than 14½º gears, and this constitutes one of the few advantages of the lower pressure angle.        Equations 22 are a useful relationship, particularly for converting to angular backlash. Also for fine-pitch gears the use offeeler gages for measurement is impractical, whereas an indicator at the pitch line gives a direct measure. The two linearbacklashes are related by:

                       BLA                                                                       (23)  

        B  = _____                     cos φ       The angular backlash at the gear shaft is usually the critical factor in the gear application. As seenfrom Figure 1.20a this is related to the gear’s pitch radius as follows:                               B                                                     (24)

       aB = 3440   ____  (arc minutes)

                               R1

       Obviously, angular backlash is inversely proportional to gear radius. Also, since the two meshing gears are usually ofdifferent pitch diameters, the linear backlash of the measure converts to different angular values for each gear. Thus, an angularbacklash must be specified with reference to a particular shaft or gear center.

4.11 Summary of Gear Mesh Fundamentals

The basic geometric relationships of gears and meshed pairs given in the above sections are summarized in Table 1.7.

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TABLE 1.7 SUMMARY OF FUNDAMENTALSSPUR GEARS

To Obtain From Known Symbol and Formula

Pitch diameter Number of teeth and pitchD = N = N·Pc

    Pd      π

Circular Pitch Diametral pitch or number ofteeth and pitch diameter

Pc   = π = €€ πD          Pd       N

Diametral pitch Circular pitch or number ofteeth and pitch diameter

Pd  =   π = N           Pc   D

Number of teeth Pitch and pitch diameterN =DPd = D

                  Pc 

Outside diameter Pitch and pitch diameter orpitch and number of teeth

 Do =D + 2 = N+2              Pd   Pd

Root diameter Pitch diameter and dedendum  DR = D - 2b

Base circle diameter Pitch diameter and pressure angle  Db=D cos φ

Base pitch Circular pitch and pressure angle  Pb = Pc cos φ

Tooth thickness atstandard pitch diameter Circular pitch

 Tstd = Pc = πD           2     2N

Addendum Diametral pitch a =  1          Pd

Center distance Pitch diameters Or numberof teeth and pitch

 C=D1+D2=N1+N2=Pc(N1+N2)    2         2Pd         2π

Contact ratio Outside radii, base radii, centerdistance and pressure angle

 mp = (Ro²-Rb²)½+(ro²-rb²)½-C sin φPc cos φ

Backlash (linear) From change in center distance  B = 2 (∆C) tan φBacklash (linear) From change in tooth thickness  B = ∆TBacklash (linear)

along line of acvon Linear backlash along pitch cirde  BLA = B cos φ 

Backlash, angular Linear backlash aB = 6880 B (arc minutes)

D    Minimum number of

teeth for no undercutting Pressure angle  N =     2         sin² φ 

DedendumPitch diameter and

root diameter ( DR )  b = ½(D-DR)

Clearance Addendum and dedendum c = b - a  Working depth Addendum  hk = 2a

Pressure angle( standard )

Base circle diameter and pitchdiameter

φ =cos-1 Db/D 

Operating pressureangle

Actual operating pitch diameterand base circle diameter

 φ =cos-1 Db/D' 

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TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALSHELICAL GEARING

To Obtain From Known Symbol and Formula

Normal circular pitch Transverse circular pitch Pcn = Pc cos ψ

Normal diametral pitch Transverse diametral pitchPdn =   Pd 

           cos ψ

Axial pitch Circular pitchesPa = Pc cot ψ = Pcn

                        sin ψ

Normal pressure angle Transverse pressure angle tan φn = tan φ cos ψ

Pitch diameter Number of teeth and pitchD  =    N  =       N      

                Pd        Pdn cos ψ  

Center distance(parallel shafts) Number of teeth and pitch

C  =  N1 + N2               2 Pdn cos ψ  

Center distance(crossed shafts) Number of teeth and pitch

C =  1      (    N1   +  N2     )       2 Pdn     cos ψ1€    cos ψ2

Shaft angle(Crssed shafts) Helix angles of 2 mated gears θ = ψ1 +  ψ2

Addendum Pitch; or outside and pitchdiameters

a = 0.5 ( Do - D ) =  1                                Pd

DedendumPitch diameter and root

diameter (DR) b = 0.5 ( D - DR )

Clearance Addendum and dedendum c = b-a

Working depth Addendum hk = 2a

Transverse pressureangle

Base circle diameter andpitch circle diameter

cos φt = Db / D

Pitch helix angleNumber of teeth,

normal diametral pitch andpitch diameter

cos ψ =  N                Pn D

Lead Pitch diameter andpitch helix angle L = π D cos ψ

INVOLUTE GEAR PAIRS

To Obtain Symbols Spur or Helical Gears ( g gear; p = pinion)

Length of action ZAZA = (C² - (Rb+rb)²)½  (maximum)  

ZA = (Ro²-Rb²)½   (ro²-rb²-C sin φr)½

Start of active profile SAPSAPp = -(Ro²-Rb²)½

SAPg = Zmax-(ro²-rb²)½

Contact ratio Rc Rcg = ((SAP)² + Rb²)½; Rcp = ((SAP)² + rb²)½

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TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALSWORM MESHES

To Obtain From Known Symbol and Formula

Pitch diameter of worm Number of teeth and pitchdw = nw Pcn         p sin λ

Pitch diameter ofworm gear Number of teeth and pitch

Dg = Ng Pcn           π cos λ 

Lead angle Pitch, diameter, teethλ = tan-1    nw  =  sin-1 nw Pcn             Pddw                 pdw

Lead of worm Number of teeth and pitchL = nwpc = nw pcn                     cos λ

Normal circular pitch Transverse pitch and lead angle Pcn = Pc cos λ

Center distance Pitch diametersC = dw + Dg

       2

Center distance Pitch, lead angle, teethC = Pcn [   Ng  +    nw   ]       2π      cos λ      sin λ

Velocity ratio Number of teethZ = Ng         nw

BEVEL GEARINGTo Obtain From Known Symbol and Formula

Velocity ratio Number of teethZ = N1       N2

Velocity ratio Pitch diametersZ = D1      D2

Velocity ratio Pitch anglesZ = sin γ1    sin γ2

Shaft angle Pitch angles Σ = γ1 + γ2 

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5.0 HELICAL GEARS

The helical gear differs from the spur gear in that its teeth are twisted along a helical path in the axial direction. It resembles thespur gear in the plane of rotation, but in the axial direction it is as if there were a series of staggered spur gears. See Figure 1.22.This design brings forth a number of different features relative to the spur gear, two of the most important being as follows:

1. tooth strength is improved because of the elongated helical wrap aroundtooth base support. 2. contact ratio is increased due to the axial tooth overlap. Helical gears thustend to have greater load-carrying capactiy than spur gears of the same size.Spur gears, on the  other hand, have a somewhat higher efficiency.

Helical gears are used in two forms:

1. Parallel shaft applications, which is the largest usage.2. Crossed-helicals (or spiral gears) for connecting skew shafts, usually at tightangles.

5.1 Generation of the Helical Tooth

The helical tooth form is involute in the plane of rotation and can be developed in a manner similar to that of the spur gear.However, unlike the spur gear which can be viewed essentially as two dimensional, the helical gear must be portrayed in threedimensions to show changing axial features.         Referring to Figure 1.23, there is a base cylinder from which a taut plane is unwrapped, analogous to the unwinding tautstring of the spur gear in Figure 12. On the plane there is a straight line AB, which when wrapped on the base cylinder has ahelical trace AoBo. As the taut plane is unwrapped any point on the line AB can be visualized as tracing an involute from the basecylinder. Thus, there is an infinite series of involutes generated by line AB, all alike, but displaced in phase along a helix on thebase cylinder.         Again a concept analogous to the spur-gear tooth development is to imagine the taut plane being wound from one basecylinder on to another as the base cylinders rotate in opposite directions. The result is the generation of a pair of conjugatehelical involutes. If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, acomplete involute helicoid tooth is formed.

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5.2 Fundamental of Helical Teeth

In tho piano of rotation the helical gear tooth is involute and all of the relationships govorning spur gears apply to the helical.However, tho axial twist of the teeth introduces a holix anglo. Since the helix angle varies from the base of the tooth to theoutside radnjs, the helix angle, w~ is detned as the angle between the tangent to the helicoidal tooth at the intersection of thepitch cylinder and the tooth profile, and an element of the pitch cylinder. See Figure 1.24.      The direction of the helical twist is designated as either left or right. The direction is defined by the right-hand rule.

5.3 Helical Gear Relationships

For helical gears there are two related pitches: one in the plane of rotation and the other in a plane normal to the tooth. Inaddition there is an axial pitch. These are defined and related as follows: Referring to Figure 1.25, the two circular pitches arerelated as follows:

       Pcn = Pc cos ψ = normal circular pitch                                                  (25)

     The normal circular pitch is less than the transverse or circular pitch in the plane of rotation, the ratio between the two beingequal to the cosine of the helix angle. Consistent with this, the normal diametral pitch is greater than the transverse pitch:

Pdn =   Pd     = normal diametral pitch                                                           (26)          cos ψ

        The axial pitch of a helical gear is the distance between corresponding points of adjacent teeth measured parallel to thegears axis—see Figure 1.26. Axial pitch, p1. is related to circular pitch by the expressions: 

Pa = Pc cot ψ =  Pcn  = axial pitch                                                                 (27)                          sin ψ

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5.4 Equivalent Spur Gear

The true involute pitch and involute geometry of a helical gear is that in the plane of rotation. However, in the normal plane, lookingat one tooth, there is a resemblance to an involute tooth of a pitch corresponding to the normal pitch. However, the shape of thetooth corresponds to a spur gear of a larger number of teeth, the exact value depending on the magnitude of the helix angle.

The geometric basis of deriving the number of teeth in this equivalent toothform spur gear is given in Figure 1.27. The result of the transposed geometryis an equivalent number of teeth given as:      NV =    N                                                         (28)               cos³ψ

     This equivalent number is also called a virtual number because this spurgear is imaginary. The value of this number is its use in determining helicaltooth strength.

5.5 Pressure Angle

Although strictly speaking, pressure angle exists only for a gear pair, a nominal pressure angle can be considered for an individualgear. For the helical gear there is a normal pressure angle as well as the usual pressure angle in the plane of rotation. Figure 1.28shows their relationship, which is expressed as:        tan φ = tan φn                                                                                 (29)                   cos ψ

5.6 Importance of Normal Plane Geometry

Because of the nature of tooth generation with a rack-type hob, a single tool can generate helical gears at all helix angles as well asspur gears. However, this means the normal pitch is the common denominator, and usually is taken as a standard value. Since thetrue involute features are in the  transverse plane, they will differ from the standard normal values. Hence, there is a real need forrelating parameters in the two reference planes.

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5.7 Helical Tooth Proportions

These follow the same standards as those for spur gears. Addendum, dedendum, whole depth and clearance are the sameregardless of whothor measured in tho piano of rotation er the normal piano. Pressure angle and pitch are usually specified asstandard values in tho normal plane, but there are times when they are specified standard in the transverse plane.

5.8 Parallel Shaft Helical Gear Meshes

Fundamental information for the design of gear meshes is as follows:     5.8.1 Helix angle — Both gears of a meshed pair must have the same helix angle. However, thehelix directions must be opposite, i.e., a left-hand mates with a right-hand helix.     5.8.2 Pitch dIameter — This is given by the same expression as for spur gears, but if the normalpitch is involved it is a function of the helix angle. The expressions are:              D = N =         N                                                                       (30)                    Pd      Pdn cos ψ       5.8.3 Center distance — Utilizing equation 30, the center distance of a helical gear mesh is:              C = ( N1+N2 )                                                                     (31)                     2 Pdn cos ψ     

      Note that for standard parameters in the normal plane, the center distance will not be a standard value compared tostandard spur gears. Further, by manipulating the helix angle (ψ) the center distance can be adjusted over a wide range ofvalues. Conversely, it is possible

a. to compensate for significant center distance changes (or erçors) without changing the speed  ratio between parallel gearedshafts; andb. to alter the speed ratio between parallel geared shafts without changing center distance by manipulating helix angle along withtooth numbers.

5.8.4 Contact Ratio — The contact ratio of helical gears is enhanced by the axial overlap of the teeth. Thus, the contact ratio isthe sum of the transverse contact ratio, calculated in the same manner as for spur gears (equation 18), and a term involving theaxial pitch.           (mp)total = (mp)trans + (mp)axial                                                     (32)

where

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        (mp)trans = value per equation 18        (mp)axial =   F  = F tan ψ   =  F sin ψ                              Pa        Pc               Pcnand F = face width of gear.

5.8.5 Involute interference — Helical gears cut with standard normal pressure angles can have  considerably higher pressureangles in the plane of rotation (see equation 29), depending on the helix angle. Therefore, referring to equation 19, the minimumnumber of teeth without undercutting can be significantly reduced and helical gears having very low tooth numbers withoutundercutting are feasible.

5.9  Crossed Helical Gear Meshes

These are also known as spiral and screw gears. They are used for interconnecting skew shafts, such as in Figure 1.29. They canbe designed to connect shafts at any angle, but in most applications the shafts are at right angles.          5.9.1   Helix angle and hands — The helix angles need not be the same. However, their sum must  equal the shaftangle:                    ψ1 + ψ2 = θ                                                                                             (33)

where:

ψ1, ψ2 = the respective helix angles of the two gearsθ          = shaft angle (the acute angle between the two shafts when viewed in a direction parallel               ing a common perpendicular between the shafts)

Except for very small shaft angles, the helix hands are the same.

5.9.2 Pitch — Because of the possibility of ditferent helix angles for the gear pair, the transverse  pitches may not be the same.However, the normal pitches must always be identical.

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     5.9.3 Center Distance — The pitch diameter of a crossed-helical gear is given by equation 30, and the center distancebecomes:       C =    1   (   N1    +     N2  )                                                                         (34)              2Pdn     cos ψ1      cos ψ2    

      Again it is possible to adjust the center distance by manipulating the helix angle. However, both gear helix angles must bealtered consistently in accordance with equation 33.      5.9.4 Velocity ratio — Unlike spur and parallel shaft helical meshes the velocity ratio (gear ratio) cannot be determined fromthe ratio of pitch diameters, since these can be altered by juggling of helix angles. The speed ratio can be determined only fromthe number of teeth as follows:           velocity ratio Z = N1                                                                                 (35)                                     N2or if pitch diameters are introduced the relationship is:            Z =   D1 cos ψ1                                                                                        (36)                    D2 cos ψ2

5.10 Axial Thrust of Helical Gears   

In both parallel-shaft and crossed shaft applications helical gears develop an axial thrust load. This is a useless force that loadsgear teeth and bearings and must accordingly be considered in the housing and bearing design. In some special instrumentdesigns this thrust load can be utilized to actuate face clutches, provide a friction drag, or other special purpose. The magnitudeof the thrust load depends on the helix angle and is given by the expression:

WT =Wt tanψ                                                                                                       (37)

where:

WT = axial thrust load

Wt = transmitted load

     The direction of the thrust load is related to the hand of the gear and the direction of rotation. This is depicted in Figure 1.29.When the helix angle is larger than about 20°, the use of double helical gears with opposite hands (Figure 1 .30b) or herringbonegears (Figure 1.30a) is worth considering.

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6.0 RACKS

Gear racks (Figure 1.31) are important components in that they are a means of converting rotational motion into linear motion.In theory the rack is a gear with infinite pitch diameter, resulting in an involute profile that is essentially a straight line, and thetooth is of simple V form. Racks can be both spur and helical. A rack will mesh with all gears of the same pitch. Backlash iscomputed by the same formula as for gear pairs, equation 22. However, the pressure angle and the gears pitch radius remainconstant regardless of changes in the relative position of the gear and rack. Only the pitch line shifts accordingly as the gearcenter is altered. See Figure 1.32.

7.0 INTERNAL GEARS

A special feature of spur and helical gears is their capability of being made in an internal form, in which an internal gear mateswith an ordinary external gear. This offers considerable versatility in the design of planetary gear trains and miscellaneousinstrument packages.

7.1 Development of the Internal Gear

The gears considered so far can be imagined as equivalent pitch circle friction discs which roll on each other with external contactIf instead, one of the pitch circles rolls on the inside of the ether, it forms the basis of internal gearing. In addition, the largergear must have the material forming the teeth on the convex side of the involute profile, such that the internal gear is an inverseof the common external gear, see Figure 1.33a.     The base circles, line of action and development of the involute profiles and action are shown in Figure 1.33b. As with spurgears there is a taut generating string that winds and unwinds between the base circles. However, in this case the string does notcross the line of centers, and actual contact and involute development occurs on an extension of the common tangent. Otherwise,action parallels that for external spur gears. 

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7.2 Tooth Parts of Internal Gear

Because the internal gear is reversed relative to the external gear, the tooth parts are also reversed relative to the ordinary(external) gear. This is shown in Figure 1.34. Tooth proportions and standards are the same as for external gears except that theaddendum of the gear is reduced to avoid trimming of the teeth in the fabrication process.

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Tooth thickness of the internal gear can be calculated with equations 9 and 20, but one must remember that the tooth and spacethicknesses are reversed, (see Figure 1.35). Also, in using equation 10 to calculate tooth thickness at various radii, (see Figure1.36), it is the tooth space that is calculated and the internal gear tooth thickness is obtained by a subtraction from the circularpitch at that radius, Thus, applying equation 10 to Figure 1.36,

7.3 Teeth Thickness Measurement

In a procedure similar to that used for external gears, tooth thickness can be measured indirectly by gaging with pins, but thistime the measurement is "under" the pins, as shown in Figure 1.37. Equations 11 thru 13 are modified accordingly to yield:      For an even number of teeth:        M= 2 ( Rc - dw )                                                                         (38)                           2      For an odd number of teeth:        M = 2(Rc cos 90º - dw )                                                              (39)                            N      2      inv φ1=inv φ + π - T -    dw                             N      D     Dcos φ

where:        Rc = cos φ   R                cos φ1

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7.4 Features of Internal Gears

General advantages:      1. Lend to compact design since the center distance is less than for external gears.      2. A high contact ratio is possible.      3. Good surface endurance due to a convex profile surface working against a concave surface.General disadvantages:      1. Housing and bearing supports are more complicated, because the external gear nests          within the internal gear.      2. Low velocity ratios are unsuitable and in many cases impossible because of interferences.      3. Fabrication is limited to the shaper generating process, and usually special tooling is required.

8.0 WORM MESH

The worm mesh is another gear type used for connecting skew shafts, usually 90º, see Figure 1.38. Worm meshes arecharacterized by high velocity ratios. Also, they offer the advantage of the higher loadcapacity associated with their line contact in contrast to the point contact of the crossed-helical mesh

8.1 Worm Mesh Geometry

The worm is equivalent to a V-type screw thread, as evident from Figure 1.39. The mating worm-gear teeth have a helical lead. Acentral section of the mesh, taken through the worm’s axis and perpendicular to the wormgear’s axis, as shown in Figure 1.39,reveals a rack-type tooth for the worm, and a curved involute tooth form for the wormgear. However, the involute features areonly true for the central section. Sections on either side of the worm axis reveal non-symmetric and non-involute tooth profiles.Thus, a worm-gear mesh is not a true involute mesh. Also, for conjugate action the center distance of the mesh must be an exactduplicate of that used in generating the wormgear. To increase the length of action the wormgear is made of a throated shape towrap around the Worm.

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8.2 Worm Tooth Proportions

Worm tooth dimensions, such as addendum, dedendum, pressure angle, etc., follow the same standards as those for spur andhelical gears. The standard values apply to the central section of the mesh, (see Figure 1.40a). A high pressure angle is favoredand in some applications values as high as 25º and 30° are used. 

8.3 Number of Threads

The worm can be considered resembling a helical gear with a high helix angle. For extremely high helix angles, there is onecontinuous tooth or thread. For slightly smaller angles them can be two, three, or even more threads. Thus, a worm ischaracterized by the number of threads, nw.

8.4 Worm and Wormgear Calculations

Referring to Figure 1.40b and recalling the relationships established for normal and transverse pitches in Par.5, the followingdefines the geometry of worm mesh components.

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8.4.1 Pitch Diameters, Lead snd Lead Angle

Pitch diameter of worm = dw = nw Pcn                                  (40)                                              π sin λ 

Pitch diameter of wormgear = Dg = Ng Pcn                            (41)                                                    π cos λ

where:         nw = number of threads of worm         L   = lead of worm = nwpc = nw Pcn                                                       cos λ         λ   = lead angle = tan-1  nw                                                Pddw                                 = sin-1  nw Pcn                                                  πdw         Pcn = Pc cos λ  

8.4.2 Center Distance of Mesh              c = dw + Dg = Pcn [ Ng    +    nw  ]                               (42)

                         2        2π   cos λ    sin λ

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8.5 Velocity Ratio

The gear ratio of a worm mesh cannot be calculated from the ratio of the pitch diameters. It can be determined only from theratio of tooth numbers:

velocity ratio = Z = no. teeth in worm gear =     Ng                              (43)                              no. threads in worm

9.0 BEVEL GEARING

For intersecting shafts, bevel gears offer a good means of transmitting motion and power. Most transmissions occur at rightangles (Figure 1.41), but the shaft angle can be any value. Ratios up to 4:1 are common, although higher ratios are possible aswell.

9.1 Development and Geometry of Bevel Gears

Bevel gears have tapered elements because they can be generated by rolling cones, their pitch surfaces lying on the surface of asphere. Pitch diameters of mating bevel gears belong to frusta of cones, as shown in Figure 1.42. In the full development on thesurface of a sphere, a pair of meshed bevel gears and a crown gear are in conjugate engagement as shown in Figure 1.43.     The crown gear, which is a bevel gear having the largest possible pitch angle (defined in Figure 1.43), is analogous to the rackof spur gearing, and is the basic tool for generating bevel gears. However, for practical reasons the tooth form is not that of aspherical involute, and instead, the crown gear profile assumes a slightly simplified form. Although the deviation from a truespherical involute is minor, it results in a line of action having a figure-S trace in its extreme extension, see Figure 1.44. Thisshape gives rise to the name "octoid" for the tooth form of modem bevel gears.

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9.2 Bevel Gear Tooth Proportions

Bevel gear teeth are proportioned in accordance with the standard system of tooth proportions used for spur gears. However, thepressure angle of all standard design bevel gears is limited to 200. Pinions with a small number of teeth are enlargedautomatically when the design follows the Gleason system.        Since bevel-tooth elements are tapered, tooth dimensions and pitch diameter are referenced to the outer end (heel). Sincethe narrow end of the teeth (toe) vanishes at the pitch apex (center of reference generating sphere) there is a practical limit tothe length (face) of a bevel gear. The geometry and identification of bevel gear parts is given in Figure 1.45.

9.3 Velocity Ratio

The velocity ratio can be derived from the ratio of several parameters:

          velocity ratio = Z = N1 = D1   =   sin γ1                                                  (44)

                                                   N2       D2        sin γ2

where:

        γ = pitch angle (Figure 1.45)

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In the simplest design the tooth elements are straight radial, converging at the cone apex. However, it is possible to have theteeth curve along a spiral as they converge on the cone apex, resulting in greater tooth overlap, analogous to the overlappingaction of helical teeth. The result is a spiral bevel tooth. In addition, there are other possible variations. One is the zerol bevel,which is a curved tooth having elements that start and end on the same radial line.       Straight bevel gears come in two variations depending upon the fabrication equipment. All current Gleason straight bevelgenerators are of the Ceniflex form which gives an almost imperceptible convexity to the tooth surfaces. Older machines producetrue straight elements.. See Figure 1 .46a. Straight bevel gears are the simplest and most widely used type of bevel gear for thetransmission of power and/or motion between intersecting shafts. Straight-bevel gears are recommended:

         1. When speeds are less than 1000 fpm — at higher speeds, straight bevel gears may be noisy.         2. When loads are light, or for high static loads when surface wear is not a critical factor.         3. When space, gear weight, and mountings are a premium. This includes planetary gear             sets, where space does not permit the inclusion of rolling-element bearings. In this case             ground gears are a necessity.

Other forms of bevel gearing include the following:

• Conii1ex gears (Figure 1.46b) are made in special straight-bevel gear-cutting machines that  crown the sides of the teeth intheir lengthwise direction. The teeth, therefore, tolerate small amounts of misalignment in the assembly of the gears and somedisplacement of the gears under load without concentrating the tooth contact at the ends of the teeth. As a result, these gearsare capable of transmitting heavier loads than the straight bevel gears under the same operating conditions.• Spiral bevels (Figure 1.46c) have curved oblique teeth which contact each other gradually and smoothly from one end to theother. Imagine cutting a straight bevel into an infinite number of short face-width sections, angularly displace one relative to theother, and one has a spiral bevel gear. Well-designed spiral bevels have two or more teeth in contact at all times. Theoverlapping tooth action transmits motion more smoothly and quietly than with straight bevel gears.• Zerol bevels (Figure 1.46d) have curved teeth similar to these of the spiral bevels, but with zero spiral angle at the middle ofthe face width; and they have lithe end thrust.

Both spiral and Zerol gears can be cut on the same machines with the same circular face-mill cutters or ground on the samegrinding machines. Both are produced with localized tooth contact which can be controlled for length, width, and shape.       Functionally, however, Zerol bevels are similar to the straight bevels and thus carry the same ratings. In fact, Zerols can beused in the place of of straight bevels without mounting changes. Zerol bevels are widely employed in the aircraft industry, whereground-tooth precision gears are generally required. Most hypoid cutting machines can cut spiral bevel, Zerol or hypoid gears.

________“The material in this paragraph has been reprinted with the permission of McGraw Hill Book Co., Inc.,New York, N.Y. from “Design of Bevel Gears” by W. Coleman, Gear Design and Applications, N.Chironis, Editor, McGraw Hill, New York, N.Y. 1967, p.57.

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10.0  GEAR TYPE EVALUATION

The choice of gear type is dependent upon a number of considerations involving physical space and shaft arrangement, load, gearratio, and desired precision or quality level. A general guide is to choose the simplest gear type that can accomplish theobjectives.     Spur gears are the first choice if they can do the job, as they are the easiest to make. That means they are the leastexpensive and, if required, can be made to the highest precision.  Helical gears are slightly more complicated than straight spurs,but are the choice if loads and speeds are demanding. Helicals are superior to spurs in load capacity. Also, they offer avoidance ofundercutting in small tooth number pinions; and helicals can be designed to neatly span non-standard center distances. Crossedhelicals are an acceptable skew shaft drive only if the loads are small. Worm gearing and bevels offer right angle drives for skewand intersecting shafts respectively. Each offers special features and advantages if needed.     Internal gears can fill a real need nicely, but they should only be used when the application requires their unique feature.     Special gears such as spiroid, helicon beveleid and face should be avoided as much as possible because of limited features,complex forms to produce and inspect, limited fabrication sources, and relative high cost.     Table 1.8 summarizes comments and evaluations of the various gear types.

11.0 CRITERIA OF GEAR QUALITY

In addition to the sizing of gear parameters, it is necessary to ensure that their specifications and manufacture result in thedesired gear quality, This includes not only tolerances, but an understanding of what compromises gear quality.

11.1 Basic Gear Formats

Specification of a gear requires a drawing that shows details of the gear body, the mounting design, face width, any specialfeatures, and the fundamental and essential gear data. This gear data can be efficiently and consistently specified on the geardrawing in a standardized block format. The format varies in accordance with gear type. A typical data block for standardfine-pitch spur gears is given in Figure 1.47. Formats for coarse pitch gears, helical gears and other gear types are given in detailin the appendix of Ref. 5.

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TABLE 1.8 SUMMARY AND EVALUATION OF GEAR TYPES

Type PrecisionRating Features Applications Comments

Regarding PrecisionSpur excellent Parallel shafting

High speedsand loadsHighest efficiency

Applicable to alltypes of trains anda wide range ofvelocity ratios,

Simplest tooth elements offering max -imum precision. First choice, recom -mended for all gear meshes, exceptwhere very high speeds and loads orspecial features of other types, such asright - angle drive, cannot be avoided.

Helical good Parallel shaftingVery high speedsand loadsEfficiency slightlyless than spur mesh

Most applicable tohigh speeds andloads; also usedwherever spursare used.

  Equivalent quality to spurs except forcomplication of helix angle.Recommended for all high- speedand high-load meshes. Axial thrustcomponent must be accommodated.

Crossed- helical

poor Skewed shaftingPoint contactHigh slidinglow speedslight load.

Relatively lowvelocity ratio;low speeds andlight loads only.Any angleskew shafts.

To be avoided for precision meshes.Point contact limits capacity and precision.Suitable for right - angle drives ifA less expensive substitute for bevelgears. Good lubrication essential becauseof point contact and high sliding action.

Internalspur

fair Parallel shaftsHigh speedsHigh loads

Internal drivesrequiring highspeeds and highloads; offers lowsliding and highstress loading; goodfor high capacity,long tie. Used Inplanetary gears toproduce largereduction ratios.

Not recommended for precision meshesbecause of design, fabricabon, andinspection limitations. Should only beused when internal feature is necessary.

Bevel fair togood

IntersectingshaftsHigh speedsHigh loads

Suitable for 1:1 andhigher velocityratios and for right-angle mashes(and other angles)

Good choice for right-angle drive,particularly low ratios. However,comptcaled tooth form and fabricationlimits achievement of precision.Should be located at one of the lesscritical meshes of the train.

Wormmesh

fair togood

Right - angleskew shaftsHigh velocityratioHigh speedsand loadsLow efficiencyMost designsnonreversible

High velocity ratioAngular meshesHigh loads

Worm can be made to high precision,but worm gear has inherent limitations.To be considered for average precisionmeshes, but can be of high precision withcare. Best choice for combination highvelocity ratio and right- angle drive. Highsliding requires excellent lubrication.

Specials(face,

Spiroid,Helicon,Beveloid)

poor tofair

Intersecting andskew shaftsModest speedsand loads

Special cases To be avoided as precision meshes. Sig -nificant nonconjugate action with departure from nominal center distance andshaft angles. Fabrication requires specialequipment and inspection is limited.

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f

11.2 Tooth Thickness and Backlash

One of the most important criteria of gear quality is the specification and control of tooth thickness. As mentioned in Par. 4.10,the magnitude of tooth thickness and its tolerance is a direct measure of backlash when the gear is assembled with its mate.     Although it is possible to set the tooth thickness and tolerance to any value within a wide range, convenient quality classeshave been established by AGMA in Gear Classification and Inspection Handbook (ANSI/AGMA 2000 - A88 ). This information isreproduced in Table 1.9. The previous issue of this specification, (390.02), offered a more detailed table of backlash allowanceand tolerance which is still a useful design guide. See Table 1.10. Although no longer part of current AGMA standards, it isconsistent with Table 1.9.      Note that the data in Table 1.9 is for unassembled spur and helical gears; i.e. an individual gear. Backlash for a meshed gearpair due to tooth thickness tolerance will be the sum of two values from Table 1.9, Most often the same tolerance is applied toeach gear of a meshed pair.

11.3 Position Error (or Transmission Error)

In many precision gear applications the transmission of motion from shaft-to shaft must have a high degree of linearity. This isknown by several names: transmission linearity, angular transmission accuracy, and index accuracy. Theoretically, involute gearswill function perfectly. However, in practice there are deviations from ideal motion transmission due to involute profile variations,spacing errors, pitch line runout, and radial out-of-position. Combinations of all these errors cause a net position error, which istransmitted to the instrument or machine involved.

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The single most important criterion of the above position errors is the total composite error of the gear (TCE). This is definedsimply as the maximum variation in center distance as the gear is rolled, intimately meshed with a master gear, on avariable-center-distance fixture. The device has one floating center, and as the gears are rolled any eccentricity, tooth-to-toothvariation, and profile deviation results in center distance variation. This variation can be measured and plotted, as shown inFigure 1 .48.TheTCE parameter encompasses the combination of run out and tooth-to-tooth errors as indicated in Figure 1.48.The latter, which is essentially the variation over a tooth cycle, is known as tooth-to-tooth composite error (TTCE).         Control of TCE and TTCE is achieved by specifying maximum values. Since TCE includes TTCE it is only necessary to specifyboth when a finer control of the TFCE is desired.         The relationship between TCE and transmission error, ET, is adequately approximated by the expression:

            ET = Etc     sin θ , where θ = angular position of the gear                          (45)                          2  

      This relationship indicates that the position error fluctuates sinusoidally between maximum lead and lag values.

*TABLE1.9 TOOTH THICKNESS TOLERANCE, (tT) (ALL TOLERANCE VALUES IN INCHES)FOR UNASSEMBELED SPUR AND HELICAL GEARS

QualityNumber

DiametralPitch

Tolerance CodesA B C D

3and4

0.5 0.074      1.2 0.031      2.0 0.019      3.2 0.012      5.0 0.0075      

5

0.5 0.074      1.2 0.031      2.0 0.019 0.0093    3.2 0.012 0.006    5.0 0.0075 0.0037    8.0 0.005 0.0025    

6

0.5 0.074      1.2 0.031      2.0 0.019 0.0093    3.2 0.012 0.006    5.0 0.0075 0.0037    8.0 0.005 0.0037    12.0 0.003 0.0018    20.0 0.0024 0.0012 0.0006  32.0 0.0016 0.0008 0.00043  

7thru15

0.5 0.074      1.2 0.031      2.0 0.019 0.0093 0.0048  3.2 0.012 0.006 0.003  5.0 0.0075 0.0037 0.0019  8.0 0.005 0.0025 0.00125 0.0006312.0 0.003 0.0018 0.0009 0.0004420.0 0.0024 0.0012 0.0006 0.000332.0 0.0016 0.0008 0.00043 0.000250.0 0.0012 0.0006 0.0003 0.0001480.0 0.0008 0.00045 0.00022 0.00011120.0 0.00067 0.00034 0.00017 0.00009

*Extracted from AGMA Standard 2000-ABB, Gear Classification and Inspection Handbook Tolerances and Measuring Methods forUnassembled Spur and Helical Gears, with permmision of the publisher, American Gear Manufacturers Association, 1500 KingStreet, Alexenderia , Virginia 22314

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Equation 45 yields a linear position error measured in inches along the pitch circle. If an angular transmission error, aET, isdesired it is necessary to divide by the pitch radius of the gear. Thus:              aET = Etc sin q (radians) =    3440  Etc sin θ (arc minutes)                                 (46)                        2R                                D         

The above defines the error of a single gear. In practice, one is interested in the total error of a mesh arising from errors of bothgears. Concerning only the maximum error (in order to avoid the complexity of phase angles*), the peak total mesh error is:

                      maximum peak error = (aET)mesh = (Etc)1+(Etc)2 3440 (arc minutes)             (47)                                                                                R1,2    

where: subscripts 1 and 2 represent each of the meshing gears; and            R1 and R2 are the respective pitch radii. These yield the angular error for the respective            gear center of the particular pitch radius being used, as shown in equation 47.

**TABLE 1.10 AGMA BACKLASH ALLOWANCE AND TOLERANCECOARSE- PITCH GEARS

Center Distance( Inches )

Normal Dlametral Pitches0.5 - 1.99 2 - 3.49 3.5 - 5.99 6 - 9.99 10 - 19.99

Up to 5         0.005 - .015Over 5 to 10 .       0.010 - .020 0.010 - 020Over 10 to 20     0.020 -030 0.015 - .025 0.020 - .030Over 20 to 30   0.030-.040 0.025 -.030 0.020 - .030  Over 30 to 40 0.040 - .060 0.035 - .045 0.030 - .040 0.025- .040  Over 40 to 50 0.050-.070 0.040 - .055 0.035-.O50 0.030 - .040  Over 50 to 80 0.060 - .080 0.045 - .065 0.040 - .060    Over 80 to 100 0.070 - .095 0.050 - .080      Over 100 to 120 0.080 - .110        

FINE - PITCH GEARS

BacklashDesignation

Normal DiametralPitch Range

Tooth Thinning toObtain Backlash Resulting Approximate

Backlash (par mesh)Normal PlaneAllowance

( per gear)Tolerance

( per gear )

A

20 thru 4546 thru 7071 thru 9091 thru 200

.002.0015.001

. 00075

0 to .0020 to .002

0 to .001750 to .00075

  .004 to .008  .003 to .007

   .002 to .0055   . 0015 to .003

B20 thru 6061 thru 120121 thru 200

.001.00075.0005

0 to .0010 to .000750 to .0005

.002 to .004.0015 to .003001 to .002

C20 thru 6061 thru 120121 thru 200

.0005.00035. 0002

0 to .00050 to .00040 to .0003

.001 to .002.0007 to .00013.0004 to .001

D20 thru 6061 thru 120121 thru 200

.00025.0002. 0001

0 to .000250 to .00020 to .0001

.0005 to .001.0004 to .0008. 0002 to .0004

E20 thru 6061 thru 120121 thru 200

Zero0 to .000250 to .00020 to.0001

0 to.00050 to .00040 to.0002

                                 *See Reference 5 for the case of considering phase angles.                               **Extracted from AGMA Gear Classification Manual AGMA 390.02, September 1964

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11.4 AGMA Quality Classes

Using criteria that are indicators and measures of gear quality, the AGMA has established a convenient standardization that formsa continuous spectrum of quality classes ranging from the crudest to the most   precise gears. For all gears, coarse and finepitches, them am 13 classes numbered 3 through 15.         AGMA Gear Classification and Inspection Handbook (ANSI/AGMA 2000-A88) specifically defines various gear qualityparameters for these 13 classes. This includes tolerance ranges for runout, pitch, profile, lead, total composite error, andtooth-to-tooth composite error. These values are for spur and helical gearing. In addition, them are separate table values for rackand pinions, bevel and hypoid gears, and fine pitch worm gearing. Also presented are class tolerances of key parameters for spurand helical inspection master gears.

11.5 Comparison with previous AGMA and International Standards

It is assumed that the present AGMA Gear Classification and Inspection Handbook (ANSI/AGMA 2000- A88) is readily available toall those who wish to obtain additional information and tables related to this subject. Many designers, however, may not haveaccess to the tables published in previous AGMA 390.02 and AGMA 236.04 standards. For this reason, Tables 1.10A and 1 .10Bare presented.          Furthermore, as a result of increased international trade and the influx of metric gears, it is useful to compare differentnational gear standard values. Such a comparison giving approximate equivalence of values is given in Table 1.10 C.

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TABLE 1.10A FINE-PITCH GEAR TOLERANCES FOR AGMA QUALITY CLASSESAGMA

QualityNo.

No. of TeethAnd

Pitch Diameter

DiametralPitch

Range

Tooth-to-TeethCompositeTolerance

TotalCompositeTolerance

5

Up to 20 teeth inclusiveOver 20 teeth, up to 1.999”Over 20 teeth. 2” to 3.999”

Over 20 teeth, 4” & over

20 to 8020 to 3220 to 2420 to 24

0.00370.00270.00270.0027

0.00520.00520.00610.0072

6

Up to 20 teeth inclusiveOver 20 teeth,up to 1.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth, 4" & over

20 to 20020 to4820 to 3220 to 24

0.00270.00190.00190.0019

0.00370.00370.00440.0052

7

Up to 20 teeth InclusiveOver 20 teeth, up to l.999"Over 20 teeth, 2" to 3.999”

Over 20 teeth, 4” & over

20 to 20020 to20020 to 4820 to 40

0.00190.00140.00140.0014

0.00270.00270.00320.0037

8

Up to 20 teeth inclusiveOver 20 teeth, up to l.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth,4” & over

20 to 20020 to20020 to 10020 to64

0.00140.00100.00100.0010

0.00190.00190.00230.0027

9

Up to 20 teeth inclusiveOver 20 teeth, up to 1.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth, 4” & over

20 to 20020 to 20020 to 20020 to 120

0.00100.00070.00070.0007

0.00140.00140.00160.0019

10

Up to 20 teeth inclusiveOver 20 teeth, up to l.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth,4” &over

20 to 20020 to20020 to 20020 to200

0.00070.00050.00050.0005

0.00100.00100.00120.0014

11

Up to 20 teeth inclusiveOver 20 teeth, up to 1.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth,4” & over

20 to 20020 to 20020 to 20020 to200

0.00050.00040.00040.0004

0.00070.00070.00090.0010

12

Up to 20 teeth inclusiveOver 20 teeth, up to 1.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth, 4” &over

20 to 20020 to 20020 to 20020 to 200

0.00040.00030.00030.0003

0.00050.00050.00060.0007

13

Up to 20 teeth inclusiveOver 20 teeth,upto 1.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth, 4” & over

20 to 20020 to20020 to 20020 to 200

0.00030.00020.00020.0002

0.00040.00040.00040.0005

14

Up to 20 teeth inclusiveOver 20 teeth, up to 1.999”Over 20 teeth, 2” to 3.999”

Over20 teeth,4” & over

20 to 20020 to 20020 to 20020 to200

0.000190.000140.000140.00014

0.000270.000270.000320.00037

15

Up to 20 teeth inclusiveOver 20 teeth, up to1.999”Over 20 teeth, 2’ to 3.999”Over 20 teeth, 4” & over

20 to 20020 to 20020 to 20020 to 200

0.000140.000100.000100.00010

0.000190.000190.000230.00027

16

Up to 20 teeth inclusiveOver 20 teeth,upto 1.999”Over 20 teeth, 2” to 3.999”

Over 20 teeth, 4” & over

20 to 20020 to20020 to 20020 to 200

0.000100.000070.000070.00007

0.000140.000140.000160.00019

*From AGMA “Gear Classification Manual for Spur, Helical and Heningbone Gears.” AGMA 390.02, Sept. 1964.

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TABLE 1.10B COMPARISON OF NEW AND PREVIOUS FINE-PITCHAGMA QUALITY CLASSES*

PREVIOUS FINE-PITCH SYSTEM , AGMA 236.04 FINE-PITCH SYSTEM, AGMA 390.02AGMA

QualityNo.

Tooth-to-ToothComposite

Error

TotalComposite

Error

AGMAQuality

No.

Tooth-to-ToothComposite

(Error) Tolerance

TotalComposite

(Error) Tolerance

Commercial 1 0.0020 0.0060 5 or 6 0.0027 or0.0019    

0.0052 or0.0037   

Commercial 2 0.0015 0.0040 6 or 7 0.0019 or0.0014  

0.0037 or0.0027  

Commercial 3 0.0010 0.0020 8 0.0010 0.0019Commercial 4 0.0007 0.0015 9 0.0007 0.0014

Precision 1 0.0004 0.0010 10 or 11 0.0005 or0.0004  

0.0010 or0.0007  

Precision 2 0.0003 0.0005 12 0.0003 0.0005

Precision 3 0.0002 0.00025 13 or 14 0.0002 or0.00014 

0.0004 or0.00027 

* Extracted from AGMA Gear Classification Manual AGMA 390.02, Sept. 1964.   For more current standard, consult ANSI/AGMA 2000-A88, March 1988.

 TABLE 1.10C QUAUTY NUMBER COMPARISON OF

DIFFERENT NATIONAL GEAR STANDARDSInternational

ISOW. Germany

DINJapan

JISU.S.A.AGMA

4 4 0 135 5 1 126 6 2 117 7 3 108 8 4 99 9 5 8

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12.0 CALCULATION OF GEAR PERFORMANCE CRITERIA

Essential to proper application of gears is the derivation of values of performance criteria Most important are: backlash,transmission error and total position error. In evaluating a gear mesh, its performance depends not only on specific gearparameters, but also on many installation and design features such as bearings, shafting, and housing.

12.1 Backlash In a Single Mesh

All sources of backlash must be identified and combined in order to obtain the total backlash for the mesh. Sources can begrouped according to the following categories:

I.

       II.        III.                           IV.           V.      

Design backlash allowance1. Gear size allowance — any reduction of tooth thickness (or testing radius)     below nominal value2. Center distance — any increase in center distance above nominal value    Major tolerances1. Gear size tolerance (tooth thickness or testing radius)2. Center distance tolerance    Gear center shift due to secondary sources1. Fixed bearing eccentricities    a. Outer-race eccentricity of ball bearing    b. Inside-diameter and outside-diameter runout of sleeve bearing2. Racial clearances due to tolerances and allowances    a. Racial play of ball bearing    b. Fit between shaft and bearing bore    c. Fit between outside diameter of beating and housing bore

    Backlash sources which are functions of gear rotation1. Total composite error2. Clearance between gear bore and shaft3. Runout at point of gear mounting4. Eccentricity of rotating race of ball beating      Miscellaneous sources1. Dimensional changes due to thermal expansion or contraction2. Deflections: teeth, gear body, shaft, and housing

A more complete and detailed coverage of these backlash sources is given in Reference 5. From the above listing of backlashsources, those which contribute significantly can be evaluated and summed. Thus, the total backlash for a mesh is expressed as:

                 Bmesh = Σ B                                                                                      (48)

When using equation 48, it should be noted that all sources of radial backlash, such as center- distance tolerance and racial shiftdue to eccentricities, must be converted to backlash measured along the pitch circle in accordance with equation 22a prior toaddition of sources such as tooth- thickness tolerances, etc. Also, note that sources of backlash can be divided into twocategories: those of constant magnitude; and those the magnitude of which varies with gear rotation. The latter sources areassociated with runout. Thus, backlash can be expressed as follows:

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   B = Bc + Bv                                                                                                          (49)

where:

Bc = constant backlashBv = variable backlash

12.2 Transmission Error

The sources of transmission error originate both from the gears and their installation. Some of these are also sources of backlash.The list of usual sources is as follows:

I.

        II.             

         

         

Position error in the individual gears1. Total composite errora. Single-cycle errors (pitch-line runout)b. High-frequency tooth-to-tooth composite errors (TTCE)    Installation errors1. Runout sources    a. Clearance between gear bore and shaft    b. Runoutat point of gear mounting    c. Eccentricity of rotating race of ball bearing    d. Miscellaneous runouts:        component shaft        composite gear assembly2. Miscellaneous error sources     a. Shaft couplings     b. Material creep of shaft and bearings

The above errors are converted to angular-position error in the same manner as TCE is converted by equation 46. Thus, the totaltransmission error for each mesh is proportional to the sum of all eccentricity error sources:

a(ET)mesh = ± 3440 ΣEi (arc minutes)                                                                         (50)                            R

where:        Ei = eccentricity (one half runout value) of error contributors

A more detailed explanation and analysis of transmission error can be obtained from Reference 5.

12.3 Integrated Position Error

Backlash and transmission error should be distinguished from functional considerations which are not necessarily related to gearperformance. For example, in a servomotor gear train, backlash may be very important, whereas position error may beimmaterial. Alternatively, in a unidirectional position sensor gear-train, backlash may be of little concern, while transmission errormight be critical. Often however, positional accuracy is most important in the overall accuracy of gear trains. In such cases,backlash combines with transmission error to yield an integrated position error (IPE). In essence the

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errors. However, this combination is not necessarily simple since many of the transmission-error sources are identical to thoseassociated with variable backlash. In addition, transmission error varies between maximum lead and lag values. Details of theintegration are beyond the scope of this coverage, but can be found in Reference 5. The basic equation for the peak value is:

(peak) IPE = Ei = ±(ET + Bc )                                                                       (51)                                      2

where:Bc = backlash constant with rotationET = transmission error (± peak value)

12.4 Control of Backlash

In the many cases in which it is necessary to minimize backlash, a proper control must be chosen. The direct approach ofnarrowing all allowances and tolerances on sources is effective. Accordingly, precision gear qualities are specified, particularlywith regard to testing radius (tooth thickness) and TCE. However, there are practical limitations since cost increases exponentiallywith precision. Some method of circumventing extremes of precision must be used. An alternate means of controlling backlash isto use adjustable centers or to spring-load the gears by one of several different designs. In this regard, the spring-loaded scissorgear has particular merit since all backlash is continually eliminated. However, it is limited to low torque applications. ConsultReference 5 for an in-depth coverage of various types of backlash control and elimination schemes. 

12.5 Control of Transmission Error

The methods available for controlling transmission error are much more limited than the means for controlling backlash. Themost effective is the direct control of errors by specification of close tolerances. This means precision categories for TCE, TTCE,and for installation components such as shafting and ball bearings.     In special cases, such as when the gear ratio of the mesh is unity, it is possible to calibrate the gears to match pitchlinerunouts to provide cancellation of error. However, besides being costly and not foolproof, this method is very limited since itrequires not only a 1:1 gear ratio, but also identical runout errors for both gears.

13.0 GEAR STRENGTH AND DURABILITY

Gear failure can occur due to tooth breakage or surface failure in the form of fatigue and wear. The first is referred to as toothstrength and the latter as durability. Strength is determined in terms of tooth-beam stresses for both static and dynamicconditions, following well established formulas and procedures. Durability ratings are evaluated in terms of surface stressesincluding the influence not only of dynamics, but also of material combinations, lubrication and a considerable number ofempirically derived factors.

13.1 Bending Tooth Strength

Tooth loading produces stresses that can ultimately result in tooth breakage. This is not a prevalent

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type of failure because mechanical properties of gear materials are well known, and the design equations are sufficientlyaccurate. The analysis of bending stresses is as follows:        In transmitting power, the driving, force acts along the line-of-action, and the tooth senses a moving force acting from thetip to the base, as shown in Figure 1.49. The load can be resolved into a tangential force, W1, causing bending, and a normalforce, WN, causing compression. These are shown in Figure 1.50 along the corresponding net stresses.       Based upon the above static analysis, Wilfred Lewis, in 1892, presented his expression for tooth beam strength which is nowreknowned as the classic Lewis equatien:                   Wt   =  SFY                                                               Pd

As a static beam resisting a fixed load in position and magnitude, this equation is usually adequate. However, it does not takeinto account the dynamics of meshing teeth. In that regard, later investigators have modified and improved the original Lewisequation.

Beam Strength (Figure 1.51)Improved results can be obtained by use of Barth’s modified Lewis formula, which takes velocity into consideration but not wear.Impact and fatigue stresses become more pronounced as pitch-line velocity increases. The formula includes a velocity factor andis satisfactory for commercial gears at pitch-line velocities up to 1,500 fpm:

  Wt   =  SFY   (    600    )              Pd         600+V 

where: Wt = transmitted load                                                  (52)S = maximum bending tooth stress, at the root outer fibers.F = face width of gearY = Lewis factorPd= diametral pitchV = velocity of the pitch point in feet per minute.

For non-metallic gears, the velocity factor is changed from (   600  ) to  (    150     +  0.25 )                                                                                     600+V           200+V                  The Lewis factor is dimensionless and independent of tooth size, and a function only of shape. Lewis factors for standardteeth are given in Table 1.11.       A safe stress level depends upon the material and the number of stress cycles to which the teeth are subjected. This can beevaluated from an S-N curve, modified Goodman diagram, Soderberg line, or equivalent data. Reference 6 contains helpfulinformation on fatigue stress analysis.

Table 1.12 gives safe stresses for a number of engineering materials. An estimate for the maximum allowable bending stress, Sin equation 52, can then be obtained by multiplying the stress given in Table 1.12 by two factors: a service factor given in Table1.13 and a lubrication factor given in Table 1.14.       Use of a proper limiting stress value, Se in equation 52, results in a calculated tooth load, W0, based on beam strength. Foracceptable designs, Wb>= Wt       The tangentially transmitted load is calculated from the transmitted horsepower as follows:

    Wt = 126,000 Pt                 DNr

where: Pt = transmitted horsepower                                                (53)Nr= gear speed in revolutions per minuteD = gear pitch diameter

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TABLE 1.11 LEWIS Y FACTORS

No. ofTeeth

Full Depth Involute

14½o 20o

101112131415161718192022242628303234

0.1760.1920.2100.2230.2360.2450.2550.2640.2700.2770.2830.2920.3020.3080.3140.3180.3220.325

0.2010.2260.2450.2640.2760.2890.2950.3020.3080.3140.3200.3300.3370.3440.3520.3580.3640.370

No. ofTeeth

Full Depth Involute

14½o 20o

363840455055606570758090100150200300

Rack

0.3290.3320.3360.3400.3460.3520.3550.3580.3600.3610.3630.3660.3680.3750.3780.382

0390

0.377 0.383 0.389 0.399 0.408 0.415 0.421 0.425 0.429 0.433 0.436 0.442 0.446 0.458 0.463 0.471

 0.484

TABLE 1.12 SAFE STRESSES**Safe beam stress or static stress of materials for gears(values of sw for use in the modified Lewis equations)

Material*Safe Stress

swUltimate Strength

swYield Stress

swCast iron, ordinaryCast iron, good gradeSemisteelCast steel Forged carbon steel :SAE 1020 casehardenedSAE1030 not treated 1035 not treated 1040 nat treated 1045 not treated 1045 hardened 1050 hardened Alloy steels:Ni, SAE 2320, casehardenedCr-Ni, SAE 3245, heat treated Cr-Van. SAE 6145, heat treated Manganese bronze, SAE 43 Gear bronze. SAE 62 Phosphor bronze, SAE 65 Aluminum bronze, SAE 68 Rawhide Fabrnil BakeliteMicarta 

8,00010,00012,00020,000

18,000

20,00023,00025,00030.00030,00035.000

50,00065,00067,50020.00010,00012,00015,0006,0006,0006,0006,000  

24.00030,00036,00065,000

55,000

60,00070,00080,00090.00095,000100,000

100,000120,000130,00060.00030,00036,00065.000

18,000 bending18,000 bending 

36,000

30.000

33,00038,00045,00050.00060,00060,000

80,000100,000110,00030,00015,00020,00025,000

 

         * For materials not given in this table the safe stress can be taken as 1/3 of the ultimate               strength.          **Repinted with permission from: Doughtie, Valiance, Kreisle: Design of Machine Members,            McGraw Mill Co. 1964, p.268.

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D190 Catalog

TABLE 11.3 SERVICE FACTORS

Type of LoadType of Service

8-10 hr perday

24 hrper day

Intermittent,3 hr per day

SteadyLight shockmedium shockHeavy shock

1.000.800.650.55

0.800.650.550.50

1.251.000.800.65

TABLE 1.14 LUBRICATION FACTORS

Type ofLubrication Lubrication Factor

Submerged in oilOil dripGreaseIntermittentLubrication

1.00

0.800.650.50

The loading conditions assumed by the original Lewis equation are very conservative.A modification that results in a more realistic situation was made by Dudley(Reference 3), that takes into account multiple teeth sharing load. When the contact ratio factoris added as well, the modified Lewis equation becomes:

Wt   =  mpSFY   (    600    )            for steel gears                              (54)

             Pd                600+V 

where the contact ratio m takes into account the fact that when the load is at the tip of the tooth, it isshared by a second pair of teeth.

The following tables are useful in determining gear load ratings:Table 1.15                 :  Ratings for steel spur gearsTables 1.16 & 1.17   :  Ratings for small-pitch spur gearsTable 1.18                  : Ratings for hardened steel helical gearsTables 1.19 & 1.20    : Ratings of worms and worm gears.

13.2 Dynamic StrengthEquations 52 and 54 give adequate results for gear meshes that are in a static situation. Whengears are in action, however, tooth loading is greater than the static value due to dynamiceffects. In a gear system, dynamic forces arise from a combination of the masses involved, theirelasticity and the forcing function representing the prescribed motion. Inaccuracies in gear-toothprofiles cause accelerations and decelerations during gear action which reflect as inertia forces,and can greatly exceed static tooth loading. The severity of dynamic forces is a function ofpitch-line velocity and tooth errors.An accurate prediction of dynamic forces is very difficult.Various factors and formulas have beer, devised to increase the static tooth force to a value thatsafety represents the dynamic condition. A

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D190 Catalog

NOTE: All charts are based on 30,000 p.s.i. yield stress. For other yield stress values multiply gear by thickness matetial stressratio.Example: 72 pitch 140 teeth brass gear torque is 200 in. oz. Table for 72 pitch yields 0.062 face width for these conditions.Multiplied by 1.5 stress ratio, the final face width of 0.093 is obtained.

 

TABLE 1.17 STRESS RA11OS FOR VARIOUS GEAR MATERIALS** 

Gear Material Yield Stress, psi  Stress Ratio

3140 Steel Stainless Steel 416Aluminum Alloy 24 S-T4Stainless Steel 303Phosphor BronzeSAE 1020BrassPhenolicNylon

*70,000-150,000*50,000-115,000

40,00030,00020,00020,00020,0008,0006,000

*0.43-0.20*0.60-0.26

0.751.001.501.501.503.755.00

* depends upon heat treatment.**By permission, Product Engineering, October 1955

TABLE 1.18 RA11NGS FOR HARDENED STEEL HELICAL GEARS**

Numberof

Teeth 

Horsepower at Various R.P.M.*24 D.P. - 1/4" Face 20 D.P. - 3/8" Face

100  200  300  600   900  1200    1800 100  200  300  600   900   1200   1800810121516

1820242530

3236404850

6072

 .03  .05   .07  .14   .20    .26     .37  .04  .07   .10  .19   .27    .34     .47 .04  .08   .12  .23   .32    .41     .55 .05  .11   .16  .29   .41    .51     .68  -      -       -     -      -       -        - 

 .04  .09   .13  .23   .32    .40     .53 .05  .10   .14  .26   .35    .44     .57 .06  .12   .17  .30   .41    .50     .64  -      -       -     -      -       -        -  .08  .14   .21  .36   .48    .58     .73

  -      -       -     -      -       -        -  .09  .17   .25  .42   .55    .66     .81  -      -       -     -      -       -        -  .12  .22   .31  .52   .66    .77     .92  -      -       -     -      -       -        - 

 .15  .27   .37  .60   .75    .86    1.01 .17  .32   .43  .67   .82    .93    1.07

 .05  .11   .16  .30   .43    .55     .76 .07  .14   .21  .40   .56    .71     .47 .09  .18   .26  .48   .68    .85     .55 .12  .23   .33  .61   .85   1.05     .68  -      -       -     -      -       -        - 

  -      -       -     -      -       -        -  .11  .21   .30  .54   .73    .89    1.14  -      -       -     -      -       -        -  .14  .26   .37  .65   .86   1.04    1.30 .16  .31   .43  .75   .98   1.16    1.43

  -      -       -     -      -       -        -   -      -       -     -      -       -        -  .22  .40   .56  .93  1.19   1.38    1.65  -      -       -     -      -       -        -  .27  .49   .67 1.08  1.35   1.55    1.81

 .32  .57   .77  1.20  1.48   1.67    1.91   -      -       -     -      -       -        - 

*Above ratings are for gears used on PARALLEL SHAFTS. Perpendicular    shaft  applications are not recommended for transmission of power.**Reprinted by permission from Browning Manufacturing - Cat. No. 6.

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D190 Catalog

dynamic factor DF is used to modify static tooth strength equations 52 and 54, such that:

Wd=Wt * DF                                                                                             (55)

and for acceptable designs:

wb >=wd 

With the aid of empirical data. Buckingham established the dynamic increment of the transmittedforce as a function of: profile errors; acceleration forces; elasticity properties; forces required to deform the teeth an amountequivalent to the tooth errors; and pitch line velocity. His simplified equation is:

For spur gears:

wd = wt +  .05V(FC+Wt    )                  .05V +(FC+Wt)½                                                                         (56)

and for helical gears:

wd = wt +  .05V(FC Cos2¥+Wt)Cos¥                  .05V +(FC Cos2¥+Wt)½                                                             (57)

where:

V = pitch line velocity in feet per minuteF = active face width in inches C = deformation factor

Values of the factor C for common material combinations and a range of tooth error (action errors) is presented in Table 1.21. These errors can be equated to total composite and tooth-to-tooth composite errors.

TABLE 1.21 VALUES OF DEFORMA11ON FACTOR C

Materials, Pinion and GearToothForm

Error in Action, Inches0.0005 0.001 0.002 0.003 0.004 0.005

Cast Iron & Cast IronSteel & Cast IronSteel & Steel

14½o400550800

8001,1001,600

1,6002,2003,200

2,4003,3004,800

3,2004,4006,400

4,0005,5008,000

Cast Iron & Cast IronSteel & Cast IronSteel & Steel

20o

fulldepth

415570830

8301,1401,660

1,6602,2803,320

2,4903,4204,980

3,3204,5606,640

4,1505,7008,300

Cast Iron & Cast IronSteel & Cast IronSteel & Steel

20ostubtooth

430590860

8601,1801,720

1,7202,3603,440

2,5803,5405,160

3,4404,7206,880

4,3005,9008,600

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13.3 Surface Durability

The Lewis formula and its modification to Incorporate dynamic conditions is limited to beam-stress analysis. In addition, there arestresses generated in the surface layers of the teeth by the direct crushing action of the forces. These stresses can exceed thematerial limits and can result in pitting, scoring, scuffing, seizing and plastic deformation.

Pitting — This is the removal of small bits of metal from the surface, due to fatigue, thereby leaving small holes or pits. This iscaused by high tooth loads leading to excessive surface stress, a high local temperature due to high rubbing speeds, orinadequate lubrication. Minute cracking of the surface develops, spreads and ultimately results in small bits breaking out of thetooth surface.

Scoring — This is a heavy scratch pattern extending from tooth root to tip. It appears as if a heavily-loaded tooth pair hasdragged foreign matter between sliding teeth. It can be caused by lubricant failure, incompatible materials and overload.

Scuffing — This is a surface destruction composed of plastic material flow plus superimposed gouges and scratches caused byloose metallic particles acting as an abrasive between teeth. Both scoring and scuffing are associated with welding (or seizing)and plastic deformation. Frequently it is difficult to distinguish among the several types of failure as there is considerableintermingling.

There have been many attempts to derive expressions for calculating safe surface stress. The Buckingham durability equationsbased on Hertzian contact stresses and the work of others can be found in the references. All of the various design equations andprocedures are closely related to specific empirical data and experience. The AGMA equations are in wide use in the UnitedStates.

13.4 AGMA Strength and Durability Ratings

The AGMA rating formulas again represent a combinations of analysis, approximations, and empirical data. A complete treatmentof AGMA practices is too extensive for this discussion and only an introductory survey is offered. More details are available fromAGMA literature and Chapter 11 of Reference 6.

The AGMA formulas pertain to strength and surface durability, with dynamic and other effects induded. The equations are:

Tooth Strength (bending stress):

St = WtKo .  Pd   .   KsKm                                                           (58)         Kv         F        J Surface Durability:               Sc = Cp    WtCo .  Cs   .   CtCm                                                           (59)                Cv      dF           l        

These equations relate stress to load, size and stress parameters.The calculated stresses must be less than the allowable stress values of the material, which in turn depend on the nature of theapplication. The allowable stresses are as follows:

Allowable surface durability stress:

St =< Sat KL                                                             (60)

             KrKr

                                     T88

Allowable surface durability stress:

Sc =< Sac CLCH                                                                            (61)

              CRCTDefinition of terms in the above equations is given in Table 1.22.Tooth strength, equation 58, is essentially a modification of the Lewis formula. The extent of depar-ture and tie improved accommodation to actual performance is dependent upon the coefficients asso-ciated with each termThe surface durability equation is related to the well established Hertzian contact-stress formula.Again, coefficients in the above equations are intended to relate the theory more closely to actual gear-tooth behavior.The meaning of the coefficients in the above equations are as follows:

Load distribution factors — Cm&Km These factors concern phenomena that cause non-uniform load distribution across the gear width:profile errors, eccentricity of mounting, non-parallelism of shafts and defiections and distortions.The effect of these errors is to cause a load concentration.

Overload factors —  Km& Co

TABLE 1.22 DEFINITIONS OF SYMBOLS IN AGMA RATING FORMULASTerm Strength Durability

LOAD:Transmitted LoadDynamic FactorOverload Factor

WtKvKo

WtCvCo

SIZE:Pinion Pitch Diameter Net Face WidthTransverse Diameteral pitchSize Factor 

--F

 PdKo

dF--Co

STRESS DISTRIBUTION:Load Distribution FactorGeometry factorSurface Condition factor

 KmJ--

CmICf

STRESS:Caiculated StressAllowable StressElastic CoetficientHardness-Ratio FactorLife FactorTemperature FactorFactor of Safety 

StSat----KLKtKR

ScSacCPCHCLCTCR

                                                                     T89

Dynamic factors— Kv & Cv These relate to speed and gear errors which lead to dynamic loading. As pitch-line velocity increases, the dynamic load incrementincreases linearly. However, the dynamic effects of tooth errors is much more complex. Tooth-to-tooth errors, which arise in avariety of forms, have a different dynamic effect than runout errors. Also, elastic tooth deflections cause apparent errors.

Life factors — KL & CL These factors are primarily intended to take into account performance of gears the life of which canbe finite.

Factors of safety — KR & CR Although factors of safety are old in engineering practice, in this case they identify the degree of reliability sought in a clearfashion.

Temperature factors — KT & CT These factors modify the design in accordance with adverse temperature effects on lubricant performance. Usually this factordoes not become significant until temperature exceeds 200’F.

Surface factors—CPCH& CP The three durability factors, C,, C & C for surface condition, hardness ratio and elastic coefficient rates the resistance of thegear-tooth surface to wear.

Size factors — KS& CS These reflect the non-uniformity of material characteristics, such as hardness, and the dimensional parameters of the gear. Thelatter include: diameter, face width, tooth size and ratio of case depth to tooth size.

Geometry factors — J & I These relate to the tooth proportions, primarily concerning radii of curvature and parameters controlling load sharing. They aresomewhat akin the Lewis Y factors. For standard toothproportions, these have fixed values.

Allowable stress — Sat & Sac This is the rated stress value of the material as specified by the manufacturer or standards, or obtained from material testing.This value takes into account cyclic stressing and is the nominal endurance stress rating of the material.

Numerical values of factors — Specific factor values are available from AGMA publications, or duplicated extracted information.Procedures for determining these factors are given in the AGMA literature. When conditions are such that a given factor isunimportant or insufficient informationexists for its adequate evaluation it is usually safe to equate the factor to unity. In most cases, this results in a conservative ormid-value rating.

Evaluations of equations — The above information constitutes an outline of the procedures offered by AGMA for determiningstrength and durability ratings. As an outline it cannot include detail;and to apply the procedures the reader should refer to thereferences.

Additional design equations — The AGMA beam strength and durability equations have been custom modified and refined by anumber of gear designers and manufacturers, creating a variety of design techniques and equations. Often this may beproprietary information, but will be available for specific use with customers’ needs. In addition, there are a host of varied designequations used by

T90

foreign gear designers. This multiplicity of equations underlines that gear strength and durability is notan exact engineering  science, but rather is empirical and experience dependent. Also, the user shouldbe aware that most gear equations and empirical results pertain to coarse pitch gears. The literatureoffers much less about tine pitch instrument gearing.

Computer programs — The AGMA design equations involving various parameters are defined withspecific detail in the standard. Several of these equation terms are subject to design modification,but are complexly derived. Examples are geometry factors (I & J) which are alterable by profile modifications. Many computer programs have been generated which efficiently handle these complex calculations.

In addition  to strength and durabtity design, software exists for the entire gear and gear train design including the selection of gear type, pitch, geometry and materials. Programs are purchasable from a  number of universities and software houses.

14.0 GEAR MATERIALS

In order for gears to achieve their intended performance, life and reliability, the selection of a suitablegear material is very important. Often not all design requirements are compatible. High load capacityrequires a tough, hard material which is difficult to machine; whereas high precision favors materialsthat are easy to machine and, therefore, have lower strength and hardness ratings. Light weight andsmall size favors light non-ferrous materials, while high capacity requires the opposite. Thus, tradeoffs and compromise  arerequired to achieve an optimum design.Gear materials vary widely, ranging from ferrous metals, through the many non-ferrous and light-weight metals, to the variousplastics. The gear designer and user faces a myriad of choices. Thefinal slection should be based upon an understanding of material properties and application requirements.

14.1 Ferrous Metals

Despite the introduction of many new exotic metals and plastics with impressive characteristics, ferrous metals are still the mostwidely used far gears, because they offer high strength, responseto heat treatment and low cost. Cast iron and steel, carbon steels and alloy steels are in common use.

14.1.1 Cast Iron is widely used for large gears where it is advantageous to save machining costsby molding the gear blank. Cast steels also offer this advantage together with higher tensile and yieldstrengths, but cast iron is superior under dynamic conditions, providing excellent internal dampingproperties. 

14.1.2 Steels are divided into two main divisions: plain carbon and alloy. The carbon steels offerlow cost reasonably easy machining and ability to be hardened. A major disadvantage is the lack ofresistance to corrosion.When elements other than carbon are added to the iron, the steel is termed "alloy steel". Thesecover a wide range from low-grade types to special high alloys offering exceptionally high strengths.Stainless steels  are contained within this large category. Alloy steels offer a wide range of heattreatment properties that makes the category of alloy steels the most versatile.Stainless Steels are divided into two types: the so called 300 series true stainless steels, whichresist nearly all corrosive conditions; and the 400 series, which although not truly stainless, offer lesscorrosion resistance only in certain environments (such as certain acids and salt water) and areotherwise considered stainless. The further significant distinction between the two series is that the300 series generally are much more difficult to machine, non-magnetic and non-heat-treatable,although somewhat responsive to cold working. The 400 series are magnetic, almost every alloy is

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heat treatable and have a much better index of machinability corresponding to some of the carbon steels. Table 1.23 listsmechanical properties of typical gear steels. Table 1.24 presents relative machinability of various steels.

14.2 Non-Ferrous Metals

The commonly used non-ferrous materials are the aluminum alloys and bronzes. Zinc diecast alloys are used also. Non-ferrousmetals generally or selectively offer good machinability, light weight, corrosion resistance and are non-magnetic.14.2.1 Aluminum as a gear material has the special feature of light weight, and moderately good strength for the low weight Itis also corrosion resistant and easy to machine. A major disadvantage is the large coefficient of thermal expansion compared tosteels. Many aluminum alloys differ in ease of forming, machining and casting. Aluminum alloys respond to cold working and heattreatment Mechanical properties for several alloys are given in Table 1.25.14.2.2 Bronzes have long been used for gear materials. They possess favorable frictional and wear properties when mating withsteel gears. They are particularly advantageous in worm meshes and crossed-helical meshes because of the large amount ofsliding. Bronzes are extremely stable and offer excellent machinability. The material can be cast, but bar stock and forgings aresuperior. Chief disadvantages are the high specific weight (highest of the gear materials) and relatively high cost.There are many bronze alloys, but only a few are extensively used for gears. These are the four alloys listed in Table 1.25. Thistable also lists brasses that are used for low load fine pitch gears.

14.3 Die Cast Alloys

Many high-volume low-cost gears are produced by the die-cast process. Most are produced in alloys of aluminum and zinc, and afew in bronze and brass. Properties of alloys suitable fOr gears are given In Table 1.26.

14.4 Sintered Powder Metal

This is a process of molding fine metal powder and alloying ingredients under high pressure and then firing to fuse the mass. It isa high-production means of producing relatively high-strength gears at low cost. Metals used for gears are iron-based mixtures,bronzes and brasses. Powder metals are expensive, but offsetting this the scrap losses are very small. Properties of sinteredpowder alloys suitable for gears is presented in Table 127.

14.5 Plastics

Plastics gears offer quiet operation, wear resistance, damping, lightweight, non-corrosiveness, minimum or no lubrication and lowcost. On the debit side, they are difficult to machine to high precision and are subject to large temperature-induced dimensionalchanges and instability. Gears can be directly finish molded with teeth, entirely machined from bar and plate stock, or cut frommolded blanks.Phenolic laminates have bases of either paper, linen, or cotton cloth with relative strengths in that order. They offer relativelygood strength and in cotton-canvas base are suitable for large gears and high loads. Properties for gear phenolics are given inTable 1.28.

                                                             T92

TABLE 1.25 COMPARATIVE PROPERTIES OF MATERIALS

Material (ASTM No.)Tensile

Strength(psi)

YieldStrength

(psi)

 108 CyclesEnduranceLimit(psi)

Elongation(%in 2") Hardness

Aluminum Alloys:    Wrought:          2011-T8          2024-T4          7075-T6          5025-H34          6061-T6

     Cast:         195-T6         356-T6

5900068.00083,00038,00045,000 

36,00040,000 

4500047,00073,00031,00040,000 

24,00027,000 

18,00019,00023,00018,00014,000

8,00013,000

1218111312

55 

Brirtell 100Brinell 120Brinell 150Brinell 68Brinell 95

Brinell 75Brinell 90

Bronzes:Aluminum Bronze-B150-2 (annealed)Phosphor Bronze- B139CSilicon Bronze - B98B (Hard)Manganese Bronze - B138-A

100,00080,00065,00080,000

60,00045,00035,00065,000

28,00031,00025,00017,000

25331025

Rockwell B90Rockwell B80Rockwell B80

Brinell 80

Brasses:Free Cutting -B16Yellow B-36-8Naval - B124-3 (¼H)Cartridge - B134-6

55,00061,00070,00072,000

44,00050,00048,00052,000

20,000-

16,0001

22,0002

32232530

Rockwell B75Rockwell B70Rockwell B80Rockwell B80

Data for brasses end bronzes is for 1/2 hard temper condition unless noted otherwise.    1 Endurance limit at 3x108 cycles    2 Endurance limit at 5x107 cycles 

TABLE 1.26 PROPERTIES OF DIE-CASTING ALLOYS SUITABLE FOR GEARS

Material NominalComposition (%)

TensileStrength

(psi)

YieldStrength

(psi)

ShearStrength

(psi)

Comp-ressive

Strength(psi)

108 CyclesEndurance

Limit(psi)

Elonga-tion (%in 2")

Hardness(Brinell)

Aluminum Alloys:    13    85   380

12 Si5 Si , 4 Cu8.5 Si, 3.5 Cu

37,00040,00031,000

18,00024,00031,000

28,00023,00031,000

21,000

25,000

19,000

21,000

2

4

807580

MagnesiumAlloysASTM-AZ91 9 AI,0.2 Zn 0.13 Mn 33,000 22,000 20,000 22,000 10,000 -

14,000 3 60

Zinc:ASTM-xxiii(Zamak 3)ASTM-xxv(Zamak 5)

3.5 to 4.3Al,0.1 Cu(max), .03 to .08Mg3.5 to 4.3Al,0.75 Cu, .03 to .08Mg

41,000

47,000

  31,000

38,000

60,000

87,000

6,900

8,200

10

7

82

91

Taken from: Michalec, G.W., i’redsion Gearing, Wiley 1968

                                                                     T95

TABLE 1.27 TYPICAL SINTERED POWDER GEAR ALLOYS

Name Composition(%) SpecificationDesignation

Ultimate Tensile

Strength(psi)

Apparent Hardness

(Rockwell)Comments

Iron-Copper AlloyCopper 7-11Iron-remainder

SAE Type 3ASTM B222-58

40,000 H-95

Offers a controlled amount of porositysuitable for lubricant Impregnation

Copper-Steel AlloyIron 94,0 min.Copper 1.0 - 4.0Other 2.0 max.

ASTMB310-58T

Class A Type II60,000 B-56

Good for Gearapplications subject tohigh impact

Carbon-Steel Alloy

Iron 95.5 min.Silicon 0.3 max.Aluminum 0.2 minOther 3.0

SAE Type 6Class C

ASTM B310-58T50,000 A-40

Excellent wearresistance

Alloy SteelAISI 4630

Carbon 0.30Manganese 0.50Silicon 0.25Nitrogen 1.7Molybdenum 2.5Iron balance

Z2* 160,000 C-35The highest strengthsintered powdermaterial

Iron High Density

Iron 97.90Copper 0.15Silicon 0.20Aluminum 0.15Other 1.60

ASTMB309-58TClass A

52,000 A-60

Good gear material forimpact, strength andhardness. High densityallows it to be casehardened bycarburizing or nitriding

Phosphor Bronze

Copper 87.0 Min.Tin 9.5 - 10.5Phosphor 0.3 - 0.5Other 1.5 max.

SAE Type 1Class AASTM

B202-58TType 1 Class AMil B 5687A

Type 1 Comp.A

30,000 H-75One of the strongestsintered bronzes

Beryllium CopperBeryllium 1.5Cobalt 0.25Copper balance

150Pt 75,000 to100,000

B-85A maximum strengthberyllium alloy

 

* Designation of Keystone Carbon Co.t  Designation of The Brush Beryllium Co.

Taken from: Michalec. G.W., "Precisjon Gearing". Wiley 1966

T96

TABLE 1.28 PROPERTIES OF PLASTIC MATERIALS

Property

NYLONASTM Type 66 ASTM Type 6

0.2%Moisture

2.5 % Moisture

0.2% Moisture

 2.8%Moisture

Tensile StrengthYield StrengthCompressive Strength. psiWater Absorption % (24 hrs.)Saturation %Density- lbs/in3

Modules of Elasticity. psi (Flexural)Coefficient of Linear Thermal Expansion oF

11.800  13.800 13.000 

 1.5 8

 .0414.1 x 105

5 x 10-5

11.2008.500

7,200 ---

1.75 x 105

9,000 7.4008.000 

1.6 to 2.09.5.041

2.5 x 1054.6 to 

5.4x10-5

8.7005.900

----

1.1 x 105-

             Data at 70o F

PropertyDELRIN*

100 500Yield Strength - psiShear Strength - psiCompressive Stress - 1% deformation - psiWater Absorption %(24 hrs)Saturation %Density- lbs/in3

Modules of Elasticity. psi (Flexural)Coefficient of Linear Thermal Expansion oF

10.000 9.500 5,200 .25 .9 

0.514 4.1 x 105

5.5 x 10-5

10.000 9.500 5,200 .25 .9 

0.514 4.1 x 105

4.5 x 10-5

             Data at 70o - 75o F             Registered trade name of E.I. DuPont de Nemours & Co.

Properties

PHENOLIC LAMINATESNEMA Grade

X XXX C L

Base Kraft Paper Paper Cotton CanvasFabric

Fine WeaveCotton Linen

Fabric

Tensile Strength - psi   Lengthwise   CrosswiseFlexural Strength - psi   Lengthwise   CrosswiseCompressive Strength - psi   FlatwiseModules of Elasticity. psi (Flexural)   Lengthwise   CrosswiseWater Absorption %(24 hrs)Coefficient of Linear Thermal Expansion oF   Lengthwise   Crosswise

21,00017,000

26,00024,000

36,000

1.8 x 108

1.3 x 1080.9

1.1 x 10-5

1.4 x 10-5

16,00013,000

14,00012,000

32,000

1.3 x 108

1.0 x 1080.3

0.94 10-5

1.4 x 10-5

11,5009,500

22,00018,000

37,000

1.0 x 108

0.9 x 108

1.04 x 10-5

1.22 x 10-5

14,50011,000

23,00018,000

35,000

1.1 x 108

0.8 x 108

0.77 x 105

1.04 x 10-5

                                                                                      T97

Catalog D190

TABLE 1.29 SUMMARY OF MATERIAL FEATURES AND APPLICATIONS

Material Outstanding Features Applications Obtainable Precision   Rating

Ferrous:Cast Irons

Cast Steels

Plain-Carbon Steels 

Alloy Steels

Stainless Steels:    300 Series

    400 Series

Low cost, good machining,highInternal dampingLow cast, high strength

Good machining, heat- treatableHeat treatable, higheststrength durability

High corrosion resistance, nonmagnetic. nonhardenableHardenable, magnetic.moderate stainless steelproperties

Large-size, moderate powerrating, commercial gearsPower gears, mediumratings Power gears, mediumratingsSeverest powerrequirements

Extreme corrosion, lowpower ratings

Low to medium powerratings, moderate corrosion

Commercial quality

Commercial quality

Commercial to medium precisionPrecision and high  and precision

Precision

High precisionNonferrous:Aluminum Alloys

Brass Alloys

Bronze Alloys

Magnesium Alloys

Nickel Alloys

Titanium Alloys

Die-Cast Alloys

Sintered Powder Alloys

Light weight. noncorrosive,excellent machinabilityLow cost, noncorrosive,excellent machinability

Excellent machinability, lowfriction, and good compati-bility with steel mates

Extreme light weight,poorcorrosion resistance

Low coefficient of thermalexpansion, poor machin-ability

High strength for moderateweight, corrosion resistant

Low cost, no precision, lowstrength

Low cost, low quality, moderatestrength

Extremely light-dutyinstrument gearsLow-cost commercialequipment

Mates for steel power gears

Special lightweight,low-load uses

Special thermal cases

Special lightweight strength

High production, lowquality,commercial

High production, low qualitycommercial

High precision

Medium precision

High prectsion

Medium precision

Commercial grade

Medium precision  applications

Low-grade commercial Commercial

 Nonmetalic:Delrin

Phenolic Laminates

Nylons

Teflon (Fluorocarbon)

Wear resistant, long life, low waterabsorption

Quiet operation, higheststrength plastic

Low friction, no lubricant,high water absorption

Low friction, no lubricant

Long life, low noise, lowloads

Medium loads, low noise 

Long life, low noise, lowloads

Special low friction

Commercial

Commercial

 Commercial

Commercial

Taken from: Michalec, GW., "Precision Gearing", Wiley 1966

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Nylon has good wear resistance, even when operating without lubricant. A major disadvantage is instability in the presence ofmoisture and humidity. Delrin* is similar to nylon in many respects, but is super or with regard to rigidity, dimensional stability,and resistance to moisture. Properties are listed in Table 1.28.These comments and data apply in particular to gears machined from plastic stock. Alternately, a greater volume of plastic gearsare produced by molding. This subject is covered in detail in Par. 20.2.

14.6 Application, and General Comments For large gears and power applications, the ferrous materials are used. The greater the load and durability requirements, themore essential are the high-alloy steels. Plain carbon steels are in common use for low-quality commercial gears.An exception in the ferrous group are the stainless steels. These are predominantly used in the small-gear, fine-pitch instrumentfields because of their corrosion resistance. For fine-pitch precision applications, stainless steels are excellent. Although the 400series is easier to machine and can have superior properties as a result of heat treatment, the 303 type of stainless steel hasreasonable machinability and offers superior corrosion resistance. In addition, when used in conjunction with aluminum housings,its coefficient of thermal expansion matches that of aluminum much better than the 400 series.The aluminum alloys, particularly 2024-T4, are excellent instrument gear materials when used within their strength ratings.Aluminums have no value as a power gear material and should not be used beyond low-load instrument-type applications.Bronze is excellent for worm gears through the full range from light loads to power applications. It is also appropriate far use inspur and helical meshes that have high velocity and/or significant loading.

Plastic materials are best suited for small gears of the instrument and light commercial product variety. Their poorer machiningcharacteristics and greater instability make them undesirable for precision applications. Their quiet operation and minimallubrication requirements render them particularly attractive far consumer products.A summary of material features is presented in Table 1.29.

15.0 FINISH COATINGS

Thin finish coatings are often applied to metal gears for protection against the environment or for decorative purposes. The typeof finish chosen is related to the material, corrosive conditions, and level of gear quality and precision.

Finish coatings on the active surfaces of gear teeth must accomplish their objectives without altering dimensions, profile, orsurface finish. This limits coatings to thin coverings of oxides or a substance  that permanently adheres to the base, and not allare suited to extend over the active tooth surfaces.

15.1 Anodize

An excellent finish for aluminum gears is anodize. This is an artificially induced thin, but even and hard coating of oxide. The thickness of the coating can be varied by process control, and can be troublesome in the maintenance of close tolerances. Consequently, anodizing of precision aluminumgears is usually limited to the gear blank prior to tooth cutting.

* Registered trade name of E.I. duPont de Nemours and Co.

                                                        T99

Because the oxide film is somewhat porous. it can be impregnated with dyes of various colors. Anodized gears possess not onlyimproved appearance, but also other significant protection against many corrosive atmospheres and salt sprays.

15.2 Chromate Coatings Applicable to aluminum, bronze, zinc and magnesium, these are low-temperature dip-bath processes that produce a chemical filmof chromate which is extremely thin and does not alter dimensions. However, the thin film has little wear resistance and offerscorrosion protection only against non-abrasive environments. Coating color varies with the particular metal and alloy. Most oftenthere is an iridescent color, which generated the common trade name Iridite, Dyes can be added to produce a wide assortment ofcolors. Because there is no dimensional change, chromating can be applied to all gears, including precision, after tooth cutting.

15.3 PasslvatlonThis is not a coating, in the strict sense, but a conditioning of the surface. It is particularly applicable to stainless steels. Theprocess is essentially a low strength nitric acid dip. It results in an invisible oxide film that develops the "stainless" property,removes "tramp iron" and reduces the metal’s anodic potential in the galvanic series. Passivation causes no dimensional changesand does not discolor or otherwise alter the natural surface. If anything, it prevents random staining due to "free iron" particlesleft from machining. All quality stainless steel gears can be passivated after complete machining since dimensions and stabilityare unaffected.

15.4 Plating.The common electroplating materials, such as cadmium, chromium, nickel and copper, are not suitable for gear surfaces sincethey alter dimensions. Also, susceptibility to localized buildup precludes their use on any precision part. Use of these platingsshould be limited to the application of coatingt prior to cutting of the teeth and of any other gear dimensions requiring closetolerances.

15.5 Special CoatingsIn recent years, special extra thin precision coatings have been developed and are available under different commercial names,Some claim surface hardness, wear resistance, low coefficient of friction, anti-corrosive qualities, etc. There are many successfulapplications on record. Each case however should  be investigated and tested.

15.6 Application of Coating.It is advisable to finish coat all gears which operate in a corrosive environment or must meet the requirements of militaryequipment applications. In addition, appearance considerations may compel a  protective  finish.Aluminum gears are best protected when anodized in a natural color but not on the tooth surfaces. A chromate coating is adequate for many applications and is acceptable in many military equipment specifications.

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Passivation of stainless steels is a necessity for good practice and military equipment standards.Even for non-military applications, this is advisable to preclude discolorations from free iron particles and minimization of galvanicinteraction with other parts. Bronze gears could be chromate coated after cutting or cadium plated in the blank state, followed by chromating after toothgeneration. Table 1.30 summarizes features of the various coatings.

16.0 LUBRICATIONLubrication serves several purposes, but its basic and most important function is to protect the sliding and rolling tooth surfacesfrom seizing, wear, and other phenomena associated with surface failure by film separation. This is particularly pertinent to powergearing. In addition, lubrication aids all gearing in that it reduces friction and protects against corrosion.

16.1 Lubrication of Power Gear.Power gear trains require sealed housings with a lubricant bath. Depending on the magnitude of the transmitted power andspeed, it may be necessary to use a circulating system with lubricant cooling. Lubricant can be supplied as a liquid bath or finespray. Lubrication of small, low-power gear trains can be accomplished with a grease pack in some cases. Many consumer homeproducts are so lubricated.

16.2 Lubrication of instrument Gear.Because of their much smaller size and capacity, generally lower speeds, and small or negligible power transmission, instrumentgear lubrication is very different from that of power gears. Often, the lubricants main purpose is to reduce friction.Instrument gears that are relatively highly loaded and working near full capacity require equally good lubrication systems aspower gears. The difference is that, in these extremely low powers, the heat dissipation is not a problem, therefore the unit canbe packed and sealed without concern for lubricant circulation, filtering, etc.The lightly loaded gear trains can be of the open variety, in which a thin lubricant film is brushed on the teeth during assemblyand reapplied only as maintenance and usage dictate. In such applications, it is important that gear speeds are not so great thatthe lubricant is flung away by centrifugal force. Also, the lubricant should have a minimum "Spreading" rating. For this reason,greases are often favored.Open housing gear trains are subject to contamination and it is advisable to guard against excessive exposure. Instruments, theouter enclosures of which must often be removed far maintenance of other items, should be.worked on in clean and controlledenvironments. Where prolonged or uncontrollable exposure occurs, temporary or permanent inner dust covers for the gear trainare recommended. This is particularly advisable in hybrid electronic instrument boxes in which the danger of solder splatter andother debris is high.

16.3 Oil  LubricantsOils are the most common lubricants and come in various-types. The compounding of oils provides combinations and generationof various properties. The most basic lubricant is petroleum to which animal, vegitable and synthetic oils and additives arecombined to yield specific properties.

                                                                    T101

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Catalog D190

Oils offer a wider range of operating speeds than greases. Also, they are easier to handle and aremore effective because of their liquid nature.

16.4 GreaseGrease is a combination of liquid and solids, in which the latter serve as a reservoir for the liquid lubricant as well as impartingcertain of their own properties. Grease has the advantage of remaining in place and not spreading as oils, and has a much lowerevaporation rate. Also, it can provide a lubricant film at heavy loads and at low speeds.

16.5 Solid LubricantsIn recent years a number of "dry fllm" lubricants have been developed. These have the advantages of wide temperature range,no dispersion, and no evaporation. Hence, they are well suited for space and other vacuum applications; and they are easier touse in open gearing since they do not contaminate as rapidly as oils and grease. However, most solid films alter dimensionssignificantly - and somedrastically. The latter cannot be tolerated in quality gearing. Dry-film lubricants represent a one-shot application of lubricant thatmust last the life of the gears, despite a continual eroding and wearing away of the film from the start of its use.

16.6 Typical LubricantsThe choice of lubricants is very wide. Military specifications govern most types and classes of lubricants, to which manymanufacturers’ products qualify. Table 1.31 is a list of typical gear oils and grease lubricants and their applications.

17.0 GEAR FABRICATIONThe fabrication of a complete gear normally includes most or all of the following operations:

1. Blank fabrication2. Tooth generation3. Refining of tooth shape (shaving, grinding, honing)4. Heat treatment5. Deburring and cleaning6. Finish Coating

Although it is not necessary to apply all six operations to every gear, the basic operations 1, 2, and determine the quality level ofa gear.Blank fabrication involves all the general and special features of the gear body. Tooth generation involves only machine-cut oraround gears, as in other fabrication methods, the teeth and body are formed simultaneously. The refining operation (shaving,grinding, or honing) is a special means of improving quality, particularly in high-volume production. Heat treatment is limited togears requiring surface hardness and/or strength. Deburring and cleaning is essential for all gears irrespective of method ofmanufacture or quality. Finish coats are limited to certain materials and environments requiring corrosion protection or improvedappearance.Modern methods of producing gear teeth cover a wide variety:

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Table 1.31 Typical Gear Lubricants

Lubricant  Type

MilitarySpecification

Useful Temp.Range (o F)

Remarks Applications

Oils:Petroleum oil

Diester oil

Diester oil

Silicone oil

MIL-L-644B

MIL-L-6085A

MIL-L-7808C

MIL-L-7808C

-10 to 250

-67 to 350

-67 to 400

-100 to 600

Good general purpose lubricant.

General purpose, low startingtorque, stable over a widetemperature range.

Suitable for oil spray or mistsystem, at high temperature.

Best load carrier of silicone oils,widest temperature range.

All quality gears having a narrowrange of operating temperature.Precision instrument gears andsmall machinery gears.

High speed gears.

Power gears requiring widetemperature ranges.

Greases Diester Oil- lithiumsoap

Diester Oil- lithiumsoap

Petroleum oil- sodiumsoapSilicone oil- non-soap

Silicone oil- lithiumsoap

MIL-G-7421A

MIL-G-3278A

MIL-L-3545

MIL-G-2501 3B

MIL-G-1 5719 -A

-100 to 200

-67 to 250

-20 to 300

-65 to 400

0 to 350

Particularly suited for lowstarting torques, lowtemperatures.

General purpose light grease.

High temperature only.

Good high temperature features.

High temperature use only.

Moderately loaded gears.

Precision instrument gears, andgenerally lightly loaded gears.

High speed and high loads.

High temperature, moderatelyloaded gear trains.Light to moderately loaded gears,low speeds.

Solid LubricantsMolybdenumdisulfide(MoS2)powder

Graphite in resinbinder

MoS2 in resin binder

MIL-M7866A

-350 to 750;>20O0 in vacuum

-100 to 450

-100 to 450

Highly stable, radiationresistant, useable in vacuumover wide temp. range.Application by spray and bakingup to 3500 F. Film thickness.0003 to .001 in.Application by spray and bakingup to 350oF. Film thickness.0003 to .001 in. Stable invacuum.

 

Light duty precision gears.

Low precision and commercialquality gears. Light loads.

Space gear trains and vacuum.

                                                                             T104

Catalog D190

1. Machine cut2. Grinding3. Casting4. Molding5. Forming (drawing, extruding, rolling)6. Stamping

Each method offers special characteristics relating to quality, production quantity, cost, materialand application.

17.1 Generation of Gear TeethMachining constitutes the most important method of generating gear teeth. It is suitable for high precision gears in both smalland large quantities.

17.1.1 Rack generation — This is the basic method of producing involute teeth. The rack cutter forms conjugate tooth profileson the blank as the rack and blank are given proper relative motion by the drive mechanism of the generating machine. As therack traverses the gear blank, it is reciprocated across the blank face. Cutting edges on the rack teeth generate mating conjugateteeth on the blank. The chief disadvantage of this method is that the rack has a limited length which necessitates periodicindexing. This limits both operating speed and accuracy.

17.1.2 Hob generation — This is the most widely used method of cutting gear teeth. It is similar to rack generation except thatthe rack is in the form of a worm. Referring to Figure 1.39, the central section of the hob is identical to that of the worm andgear. The differences are that the thread of the hob is axially gashed or fluted in several places so as to form cutting edges, whilethe sides and top of these teeth are relieved behind the gash surface to permit proper cutting action. This arrangement, in eftect,gives an infinitely long rack so that cutting is both steady and continuous. To generate the full Width of the gear, the hob slowlytraverses the face of the gear as it rotates. Thus, the hob has a basic rotary motion and a unidirectional traverse at right angles.Both movements are relatively simple to effect, resulting in a very accurate process.A further advantage of hobbing is that the hob can be swiveled relative to the blank axis. This permits cutting helical gears of allangles with the same tooling.With regard to accuracy, hobbing is superior to the other cutting processes. Gears can be directly hobbed to ultra-precisiontolerances without resorting to any secondary refining processes.

17.1.3 Gear shaper generation — This process, unlike the other two, employs a gear-shaped cutter instead of a rack or theequivalent. Uke a rack cutter, a given gear-shaped cutter is conjugate to all tooth numbers of that pitch. Thus, a gear made as acutting tool can generate the teeth of a blank when the two are rotated at proper speeds. The cutting tool can be imagined as agear that axially traverses the blank with a reciprocating axial motion as it rotates. The teeth on the gear cutter are appropriatelyrelieved to form cutting edges on one face.Although the shaping process is not suitable for the direct cutting of ultra-precision gears and generally is not as highly rated ashobbing, it can produce precision quality gears. Usually it is a more rapid process than hobbing.Two outstanding features of shaping involve shouldered and internal gears. Compound gears and shaft gears frequently aredesigned so compactly that a hob cutter interferes with adjacent material.In such cases, shaping can be used since the stroke of the gear-shaped cutter requires very little round space on one side of thegear. For internal gears, the shaping process is the only basic methodof tooth generation.The shaping process can be used for the generation of helical gears. However, each helix angle requires special tooling.Therefore, with regard to helical gears, shaping is not as convenient and is

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Catalog D190

more expensive than hobbing.17.1.4 Top generating — This is a fabrication option utilizing cutters that finisb-.cut the outside diameter of the teethsimultaneously with the cutting of the tooth profiles.It can be used in both the hobbing and shaping processes, although more prevalent in hobbing and among the fine pitches. Themain advantages of topping are:

1. Liberal tolerances can be applied to the outside diameter of the blank.2. The deburring problem is reduced.3. The gear can be nested on its outside diameter for machining modifications of the       body should such a speaal need arise.

17.2 Gear GrindingAlthough grinding is often associated with quantity fabrication of high quality gears as a secondary refining operation, it is also abasic process for producing hardened gears. In addition, many high-precision fine-pitch gears have their teeth entirely groundfrom the blank state.Gear-tooth grinding can involve either form grinding or the generating process. The latter is basically more accurate because thedressing of the grinding wheel involves a straight-sided tooth.There are a number of distinct advantages to ground gears. These are listed as follows:

1. Achievement of high precision is possible because the process can remove very little    material in the final pass.2. Grinding results in a much finer surface finish than any machining process.3. Hardened steel alloys can be ground.4. Residual surface stresses are minimal.

Being able to use heat-treatable hard steel alloys raises the bending stress and surface endurancestress levels by very significant amounts. See Table 1.23. Often it is the difference between a reliable and unreliable gear. Inparticular, case hardened, carburized or nitrided gears offer outstanding strengths and performance. They are typically used forthe most demanding tasks, such as aircraft drives.Ground gears’ superior load carrying capacity is not only due to the hardened alloys higher mechanical properties, but alsobecause of the finer surface finish. A fine finish enables maintenance of a good continuous oil film versus boundary lubricationand breakthrough. The result is higher load capactry along with reduced wear and longer useful life.Although there are distinct advantages for ground gears, there are some limitations and disadvantages. These are:

1. Grinding is limited to ferrous materials.2. Hard metals grind better than soft ones.3. Grinding of helicals and worms has limitations that possibly involve profile deviations    and removal.4. Pro-grind hobbing requires special protruberance hobs to provide grind wheel    clearance at the root5. Gear grinding machinery is scarcer than hobbing machines.6. Grinding is a secondary operation which increases total gear cost

Despite involving higher cost and other limitations, ground gears are always superior in precisionand strength.                                                    T106

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17.3 Plastic GearsThese can be produced by the normal hobbing and shaping processes. In addition, they can be produced by various moldingtechniques. The latter methods are not  accurate as cut gears due to shrinkage, mold variations, and flow inconsistencies.Regardless of method, the fabrication of plastic gears suffers in comparison with metal gears dueto temperature instability, material flow, and generally poorer cutting qualities. Attainable quality is less than for metals andvaries with the particular plastic.See Section 20.10, which deals with plastic gears in greater detail.

18.0 GEAR INSPECTIONThe performance of a gear can be assured only by confirmation of its critical dimensions and parameters. With increasing gearprecision, adequate and proper inspection has become a paramount requirement.There are many aspects of gear inspection and the subject is too large for complete coverage in this discussion. However, two ofthe most basic and important inspection criteria, which will be discussed in the following paragraph, are total composite error(TCE) and tooth thickness.

18.1 Varlable-Center-Distance TestersBoth TCE and tooth thickness can be measured by means of roll testing with a variable-center-distance fixture. There are manyvarieties, but essentially all consist of a fixture having two parallel shafts (or precision centers), one fixed arid the other floatingon smooth, low-friction ways. The test gear is mounted on one shaft while an accurate known quality master gear is mounted onthe other shaft. The pair is held in intimate contact by spring loading or the equivalent. As the test gear is rotated, tooth-to-tootherrors and runout are revealed as a variation in the center distance of the pair. This variation can be sensed, amplified anddisplayed as a dial reading or recorded on a chart. See Figure 1.48. Sensitivity of the measurement is on the order of 50 to 100millionths of an inch.The unique feature of gear roll testing is that the inspection parallels the gear in its actual usage.Thus, roll testing is a functional inspection.

18.1.1 Total Composite Error — The TCE is dearly revealed in roll testing and its components can be identified. Referring toFigure 1.48, it is evident that the magnitude of runout and TTCE can be extracted. From this, the gear quality can be judged.Also, when parameters are out of tolerance, the fabricator can identify the source of the difficulty and take appropriate correctiveaction.

18.1.2 Gear Size — If the center distance setting of the roll tester is carefully established, the absolute readings are anindication of tooth thickness. Thus, in Figure 1.48, the mean line of the trace is a measure of tooth thickness. The high and lowreadings indicate the extreme variation of tooth thickness at the nominal pitch radius. Changes in center-distance are an indirectmeasure of tooth thickness and must be converted with the aid of equation 22.

18.1.3 Advantages and Limitations of Variable-Center-DIstance Testers — The functional test of a gear is desirable as itreveals characteristics that occur in the real application. Also, the method is rapid and, therefore, suitable for production gearinspection. Ability to obtain a hard copy record is also a distinct advantage.Rolling of the gears is not usually relied upon for the determination of. pitch radius. For the measurement of TCE and TTCE,however, roil testing gives excellent results. Repeatability arid absolute measure are usually good, being in the order of .0001inch. On the other hand, size measurement is not as reliable as an absolute measure. This is due to the nature of the fixture andthe integration of several error sources in the calibration process. A repeatability of .0002 inch is considered good, and often it iseven better.

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18.2 Over-Pin. GagingThe equations relating tooth thickness and a measurement over cylindrical pins or rolls inserted between the teeth were given inParagraph 4.6. This is a widely used method for gaging gears during fabrication (while they are still in the gear generatingmachine) and during final inspection. Accuracy of the over-pins measurement is on the order of .0001 inch.A major disadvantage of over-pins gaging is the inability to correlate precisely with variable-center-distance measurements. Thisis because over-pins gaging is insensitive to pitch-line runout. On the other hand, rolling a gear necessarily involves the TCE andits runout component. The best correlation is obtained by equating the over-pins measurement to the average value of centerdistance found in the roll test.Apart from the correlation problem, over-pins measurements by themselves are inadequate because the undetected runout canbe out-of-control causing interference with its mate. It is necessary, therefore, to control and to inspect runout.

18.3 Other Inspection EquipmentIn addition to the basic inspection methods and equipments described in Paragraphs 18.1 and 18.2, other special-purposeequipment is available. This includes involute-profile form checkers, tooth-spacing gages and runout checkers. Also, for highprecision gears, equipment is available for inspecting the position error of individual gears and the transmission error of a geartrain.

18.4 Inspection of Fine-Pitch Gear.Because of their small dimensions, fine-pitch gears do not easily lend themselves to the kind of detailed tooth measurementssuitable for large, coarse-pitch gears. Hence, fine-pitch gears are almost exclusively inspected by functional testing on avariable-center-distance fixture.Over-pins measurements are also used, but generally are restricted to a reference measurement, This is primarily used in thefabrication process as a set-up dimension, and in inspection departments which are not equipped to roll test gears.

18.5 Significance of Inspection and its ImplementationThe inspection operation is essential to obtaining a quality product. In effect, it is a policing operation that ensures conformanceto dimensional tolerances and other drawing specifications.The effort, care and cost of inspection are related to the quality level. Precision-gear inspection demands a much greater effortthan that for low quality gears. Equipment must be of the best grade, calibrated periodically and restricted to use by qualifiedpersonnel. Control of temperature environment is essential for measurements on the order of .0001 inches. The cleanliness ofequipment, gears and working area are also very important.

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GEAR DESIGN - METRIC19.0 GEARS, METRIC

19.1 Basic Definition.Metric gearing is distinguished not only by different units of length, but also by its own unique design standard. Historically,metric gears arose as a result of a different approach to the standardization of tooth proportions and this constitutes a majorobstacle to the adoption of the metric system by the American gear industry.In the inch system diametral pitch was created as a convenient means far relating pitch diameters to center distance. Thus,diametral pitch is defined as:

Pd = N = number of teeth per inch of pitch diameter               (62)

      D

where: N = number of teethD = pitch diameterPd = diametral pitch

From this relationship there are particular integer - values of diametral pitch that yield integer values for center distance ininches. Thus 8, 16, 32 and 64 diametral pitches, to mention only some, can be  associated with tooth numbers which can resultin center distances equal to an integral multiple of one inch and/or convenient fractions of an inch.In the metric system the module is analogous to pitch, and is defined as:

m = D = amount of pitch diameter per tooth, in millimeters.                (63)          N

This defines the module as analogous to the reciprocal of diametral pitch. However, the module is a dimension (length of pitchdiameter per tooth); whereas diametral pitch is the number of teeth to a unit length of pitch diameter. Again convenient centerdistances in metric measure are obtained by choosing integer module values and/or selected fractional values.One consequence is that each system (inch diametral pitch and metric module) has adopted preferred standard values which arenon - interchangeable.It should be noted that the term diametral pitch is associated with the inch system. In the metric system the nearest analogue topitch is termed "module", and the word pitch is reserved for tooth spacing along the pitch circle. In the inch system, the toothspacing measure is more accurately called "circular" pitch.

If the equations for diametral pitch and module are solved for pitch diameter and these values equated by introducing theconversion factor 25.4, we obtain:

             Pd * m=25.4                                       (64)

This shows that inch diametral pitch and the metric module are related by the decimal factor 25.4. It is obvious that conversionresults in decimal values, often awkward numbers, for one or the other measure. It follows that convenient values in one systemwill not be convenient values in the other. For this reason each system (inch diametral pitch and metric module) has adoptedpreferred standard values which are non-interchangeable. Table 1.32 lists the commonly used pitches/modules of both systems,with preferred values in bold-face type. Corresponding equivalent values are given, but these are of no help since odd valuedpitches and modules are usually not tooled for.

It becomes obvious, therefore, that direct replacement of conventional inch gearing with metric gearing is impossible. The bestthat can be done is to shift to the nearest standard module when converting from the inch system. One should keep in mind,however, the preferred module sizes which exist in different countries. The degree of non-correspondence between pitch andmodule is best measured by the circular pitch and the circular tooth thickness. These values are given in inches and millimeters inTable 1.32.

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As a consequence, metrification of gearing requires a completely new design with regard to gear dimensions and centerdistances. This in turn involves new gear cutting tools.Preferred module sizes in the United States are established only for the coarser gears by means of IS( recommendation R54 (seeTable 1.33). Judging by their acceptance by the industrialized metric countries, the following modules are expected to bepreferred for the finer gears:

0.3,0.4,0.5, 0.8, 1.0

To facilitate work with these modules we have computerized the basic relationship:

D = m * N

and created Table 1.34 for number of teeth, N ranging from 5 through 205.The pitch diameters are calculated in Table 1.34 both in millimeters and inches. We expect this tab to be of great help todesigners in developing a feel for metric gear sizes and for determining center distances.The subject of measurement over pins was dealt with in section 4.6. For inch-size gears Table 1.8 listed  the over-wiremeasurements. Similarly for module-type gears, computerized Table 1.34 was produced. This lists both pitch diameters as well asover-wire measurements in both millimeters and inches.

TABLE 1.33 Modules and Diametral Pitches of Cylindrical Gears forGeneral and Heavy Engineering*(ISO Recommendation R54 1977)

Modules m Diametral Pitches PI II III I II1

1.25

1.5

2

2.5

3

4

5

6

8

10

12

16

20

25

1.125

1.375

1.75

2.25

2.75

3.5

4.5

5.5

7

9

11

14

18

22

(3.25)(3.75)

(6.5)

20

16

12

10

8

6

5

4

3

2.5

2

1.5

1.25

1

0.75

18

14

11

9

7

5.5

4.5

3.5

2.75

2.25

1.75

0.875

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19.2 Metric Design EquationsSome of the gear design equations are dimensionless and are derived from geometric proportions and relationships. Theseequations will not be affected by the use of metric units as opposed to inch units since the units cancel. lt is important, however,that the same units are used consistently throughout.  When this is not the case the problem of metriflcation can be approachedin two steps. The first step is to express the present inch-base units in metrics and to modify the constants and coefficientsaccordingly. This procedure will yield results expressed in the form presently used in engineering practice in industrialized metriccountries.

The second step is to express these results in Si units which differ slightly from the conventional metric units. Thls is true forstress calculations but does not affect gear dimensioning.

Metrification in the U.S. is taking place at a time when the SI (International System of Units) has been adopted in most metriccountties, but its use has not spread to the practical design engineering profession. For. these countries, transition to the SIsystem represents a change which is accompanied by a degree of reluctance. The standardization related to transition to metricsin the U.S. is expected to introduce the SI units as well, in a single step.

lf we concentrate on the large number of equations which are independent of the system of measuring units, there will be noproblem with metrification. Most of the kinematic design equations that appear American gear texts. and are associated withinch-system gears, are suitable for use with metric gear dimensions, provided that a proper substitution of module (in) is madefor-pitch.

For equations involving diametral pitch:

Pd  is  replaced by 25.4                                 (65)

                    mRecalling that:

Pd * Pc   = π€€€€ m                                                           25.4

we find that for equations involving circular pitch:

Pc  is replaced by π€€€€ m                                      (66)                          25.4

Note: When converting between metric module and the inch diametral pitch, the conversion factor and relationship can beremembered from the simple product of the two pitch measures:

m * Pd  = 25.4

By this means, all geometric and all kinematic equations involving pitch parameters can be used. However, by the above,conversion results are still given in inch measurements. Thus, this is a way to adapt the metric module to kinematic designequations given in inch units.

Basic kinematic and geometric design equations for spur gears in both metric module and inch  diametral-pitch forms are given intable 1.35. These equations show the essence of using the modules versus inch diametral pitch.

Some equations which are identical in both systems are:

1. Over-pins measurements.2. Relationship between tooth thicknesses at different radii from gear center.3. Long and short addendum equations.4. Profile-shifted gear-design equations: i.e., enlarged gear teeth, non-standard center distance

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19.3 MetrIc Tooth Standmrds*

The metric module was developed in a number of versions that differ in minor ways. The German module, defined by the DINstandard, is widely used throughout Europe. However, the Japanese have their own version, defined in JIS standards. Thedeviations among these and other national metric standards are fortunately minor: the various metric standards, differ only withregard to dedendum size and root radii.Even these minor deviations are resolved by a new unified module standard sponsored and promoted by the InternationalStandards Organization (ISO). This unified version, shown in fIgure 1.52, conforms to the new SI system in all respects.Currently, Germany, Japan, Great Britain and other major industrial countries on the metric system, are shifting to this ISOstandard, which has been advocated as the basis for American metric gearing.

                    ISO standard metric gear tooth is defined by a rack of module m = 1. ISO gears share many features with inch-size Americangears: 200 pressure angle, plus similar addendum and dedendum ratios.

Tooth proportions for the standard, which applies to cylindrical gears of the spur and helical varieties, are given in terms of thebasic rack, as shown in the illustration. Dimensions, in millimeters, are normalized for module m = 1. Corresponding values forother modules are obtained by multiplying each dimension by the value of the specific module, m. Major tooth parameters aredescribed by the standard:

• Tooth form is straight-sided and full-depth, forming the basis of a family of full-depth interchangeable gears.• Pressure angle is 200, conforming to world-wide acceptance of 200 as the most versatile pressure angle.• Addendum is equal to the module, m, which conforms to the American practice of addendum equaling 1/P.• Dedendum is equal to 1.250 m, which corresponds to American practice for coarse pitch gears (see Table 1.1).• Root radius is slightly greater than current American standards specify.• Tip radius has a maximum tip-rounding specified. This rounding is a deviation from American standards, which do not specifyrounding. However, as a maximum or limit value, American gear makers are not prevented from specifying a tip radius as nearzero as possible.

Note that the basic racks for metric gears and for American inch gears are essentially identical. For metric gears, specific sizedimensions are obtained from multiplying by m (the module). Gears conforming to diametral pitch American standards are sizedby dividing the basic rack dimensions by the specific diametral pitch (P).___________________________Apart from minor changes in wording. this paragraph, including figure 1.52, is quoted or reproduced wilh the permissior ofMachine Design magazine from the following article: "Shifting to Metric", by G.W. Mchalec and F. Buchsbaum Machire Design,Vol. 45, August 9,1973, pp. 94-97.

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19.5 Metric Gear Standards

With recent increasing presence of metric geanng in the USA it is important that designers and gear users have knowledge of andready reference to various metric gear standards used throughout the world.

19.5.1 USA Metric Gear Standards — Metric gears designed and produced in the USA should conform to the ISO standard.This is the latest metric standard based upon SI units which have beendecreed as the most precise metric measurement for standardized international use.The latest (1989) ISO gear standards are listed in Table 1.36. They can be procured from ANSI. 1430 Broadway, New York, N.Y.10018

19.5.2 ForeIgn Metric Gear Standards — Several of the major industrialized countries that have been dedicated for a longtime to metric measurement countries have developed their own standards for metric gearing. In general they have similarstandards, and since the establishment of ISO and SI units have adopted these standards as theirs.With increasing international trade and worldwide manufacture of common products, availability and familiarity with appropriateforeign standards have become important. To serve that need Table 1.37 offered as a listing of key gear standards in use inseveral major countries and geographic areas.

TABLE 1.36 ISO METRIC GEARING STANDARDSISO 53:1974 Cylindrical gears for general and heavy engineering — Basic rack

ISO 54:1977 Cylindrical gears for general and heavy engineering — Modules and diametralpitches

ISO 677:1976 Straight bevel gears for general and heavy engineering — Basic rack

ISO 678:1978 Straight bevel gears for general and heavy engineering-. Modules and diarmetralpitches

ISO 701:1979 International gear notation — Symbols for geometrical dataISO 1122-1:1983 Glossary of gear terms — Part 1: Geometrical definitionsISO 1328:1975 Parallel involute gears -. ISO system of accuracy

ISO 1340:1976 Cylindrical gears.- Information to be given to the manufacturer by the purchaserIn order to obtain the gear required

ISO 1341:1976 Straight bevel gears - Information to be given to the manufacturer by thepurchaser in order to obtain the gear required

ISO 2490:1976 Single-start solid (monobloc) gear hobs with axial keyway, 1 to 20 module and 1to 20 diametral pitch - Nominal dimensions

ISO/TR 4407:1902 Addendum modification of the teeth of cylindrical gears for speed-reducing andspeed-increasing gear pairs

ISO 4468:1982 Gear hobs - Single start- Accuracy requirements

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TABLE 1.37 FOREIGN METRIC GEAR STANDARDS  AUSTRALIA

ASB62    1965AS B 66  1969 AS B 214 1966ASB2I7   1966AS 1637

 Bevel gears Worm gears (inch series) Geometrical dimensions for worm gears — Units Glossary for gearing International gear notation symbols for geometric data (similar to ISO 701)

 FRANCENF E 23-001 1972NF E 23-002 1972NF E 23-005 1965NF E 23-006 1967NF E 23-011 1972

NF E 23-012 1972

NF L 32-611 1955

 Glossary of gears (similar to ISO 1122)  Glossary of worm gears  Gearing — Symbols (similar to ISO 701)  Tolerances for spur gears with Involute teeth (similar to ISO 1328)  Cylindrical gears for general and heavy engineering — Basic rack and modules (similar to ISO 467 and ISO 53) Cylindrical gears — Information to be given to the manufacturer by the producerCalculating spur gears to NFL 32-610

GERMANY - DIN (Deutsches Institut für Normung) DIN 37 12.61DIN 780 Pt 1 05.77 DIN 780 P12 05.77

DIN 867 02.88

DIN 868 12.76

DIN 3961 08.78DIN 3962 Pt 1 08.78

DIN 3962 Pt 2 08.78

DIN 3962 Pt 3 08.78

DIN 3963 08.78

DIN 3964 11 .80

DIN 3965 Pt 1 08.86DIN 3965 Pt 2 08.86DIN 3965 Pt 3 08.86

DIN 3965 Pt 4• 08.86

DIN 3966 Pt 1 08.78

DIN 3966 Pt 2 08.78

DIN 3967 08.78

DIN 3970 Pt 1 11.74

DIN 3910 Pt 2 1114

 Conventional and simplified representation of gears and gear pairs [4]  Series of modules for gears — Modules for spur gears [4]  Series of modules for gears — Modules for cylindrical worm gear  transmissions [4]  Basic rack tooth profiles for involute teeth of cylindrical gears for general and heavy engineering [5]  General definitions and specification factors for gears, gear pairs and gear trains [11]  Tolerances for cylindrical gear teeth — Bases [8]  Tolerances for cylindrical gear teeth — Tolerances for deviations of individual parameters [11]  Tolerances for cylindrical gear teeth — Tolerances for tooth trace deviations (4]  Tolerances for cylindrical gear teeth — Tolerances for pitch-span deviations [4]  Tolerances for cylindrical gear teeth— Tolerances far working deviations [11]  Deviations of shaft center distances and shaft position tolerances of casings for cylindrical gears [4] Tolerancing of bevel gears — Basic concepts (5]  Tolerancing of bevel gears — Tolerances for individual parameters [11] Tolerancing of bevel gears — Tolerances for tangential composite errors [11] Tolerancing of bevel gears — Tolerances for shaft angle errors and axes intersection point deviations [5]  Information on gear teeth in drawings — Information on involute teeth for cylindrical gears [7]  Information on gear teeth in drawings — Information on straight bevel gear teeth [6] System of gear fits - Backlash, tooth thickness allowances, tooth thickness tolerances - Principles [12]  Master gears for checking spur gears - Gear blank and tooth system [8]  Master gears for ducking spur gears - Receiving arbors [4]

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TABLE 1.37 CONT. FOREIGN METRIC GEAR STANDARDS 

 GERMANY CONT. - DIN (Deutsches Institut für Normung) DIN 3971 07.80DIN 3972 02.52

DIN 3975 10.76DIN 3976 11.80

DIN 3977 02.81

DIN 3978 08.76 DIN 3979 07.79  DIN 3993 Pt 1 08.81

D1N3993 P12 08.81

D1N3993 Pt3 08.81

DIN 3993 P14 08.81

DIN 3998 09.76Suppl 1DIN 3998 Ptl 09.76DIN 3998 Pt2 09.76

DIN 3998 P13 09.76

DIN 3998 P14 09.76DIN 58405 Pt1 05.72

DIN 58405 P12 05.72

DIN 68405 P13 05.72DIN 68405 Pt 4 05.72

DIN ISO 2203 06.76

 Definitions and parameters for bevel gears and bevel gear pair [12] Reference profiles of gear-cutting tools for involute tooth systems according to DIN 887[4]Terms and definitions for cylindrical worm gears with shaft angle 90o[9]Cylindrical worms — Dimensions, correlation of shaft center distances and gear ratios ofworm gear drives [6]Measuring element diameters for the radial or diametral dimension for testing tooththickness of cylindrical gears [8]Helix angles for cylindrical gear teeth [5]Tooth damage on gear trains — Designation, characteristics, causes [11]Geometrical design of cylindrical Internal involute gear pairs — Basic r ules [17]Geometrical design of cylindrical internal involute gear pairs —Diagrams for geometricallimits of internal gear-pinion matings [15]Geometrical design of cylindrical internal involute gear pairs —Diagrams for thedetermination of addendum modification coefficients [15]Geometrical design of cylindrical internal involute gear pairs -. Diagrams for limits of internalgear-pinion type cutter matings [10] Denominations on gear and gear pairs — Alphabetical index of equivalent terms [10]Denominations on gears and gear pairs — General definitions [11]Denominations on gears and gear pairs — Cylindrical gears and gear pairs [11]Denominations on gears and gear pairs — Bevel and hypoid gears and gear pairs [9]Denominations on gears and gear pairs — Worm gear pairs [8]Spur gear drives for fine mechanics — Scope, definitions, principal design data, classification[7]Spur gear drives for fine mechanics — Gear fit selection, tolerances, allowances [9]Spur gear drives for fine mechanics — Indication in drawings, examples for calculation [12]spur gear drives for fine mechanics — Tables [15]Technical drawings — Conventional representation of gears

NOTE: Standards available in English from: ANSI, 1430 Broadway, New York, NY 10018: or Beuth          Verlag GmbH, Burggrafenstrasse 6. D-1000 Berlin 30, West Germany: or Global Engineering

  Documents, 2806 McGaw Avenue, P.O. Box 19539, Irvine, CA 92714, Telex 692 373.           Easylink 380 124; or I.S.L.I., 160 Old Derby Street, Hingham, MA 02018, Telex 948 658.

ITALYUNI 3521 1954UNI 3522 1954UNI 4430 1960UNI 4760 1961UNI 6586 1969

UNI 6587 1969

UNI 6588 1969

UNI 6773 1970

 Gearing - Module series  Gearing - Basic rack Spur gears — Order Information for straight and bevel gears  Gearing - Glossary and geometrical definitions  Modules and diametral pitches of cylindrical and straight bevel gears for general and heavy engineering (corresponds to ISO54 and 678) Basic rack of cylindrical gears for general engineering (corresponding to ISO 53)Basic rack of straight bevel gears for general and heavy engineering (corresponds to ISO677)International gear notation -. Symbols for geometrical data (Corresponding to ISO 701)

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TABLE 1.37 CONT. - FOREIGN METRIC GEAR STANDARDS  JAPAN - JIS (Japanese Industrial Standards)

B 0003 1989B 0102 1988B 1701 19738 1702 1976B 1703 1976B 1704 1978B 1705 1973B 1721 1973B 1722 1974B 1723 1977B 1741 1977B 1751 1976B 1752 1989B 1753 1976B 4350 1987B 4351 1986B 4354 1988B 4355 1988B 4356 1985B 4357 1988B 4358 1976

Drawing office practice for gears.Glossary of gear termsInvolute gear tooth profile and dimensionsAccuracy for spur and helical gearsBacklash for spur and helical gearsAccuracy for bevel gearsBacklash for bevel gearsShapes arid dimensions of spur gears for general engineeringShapes and dimensions of helical gears for general useDimensions of cylindrical worm gearsTooth contact marking of gearsMaster cylindrical gearsMethods of measurement of spur and helical gearsMeasuring method of noise of gearsGear cutter tooth profile and dimensionsStraight bevel gear generating cuttersSingle thread hobsSingle thread fine pitch hobsPinion type cuttersRotary gear shaving cuttersRack type cutters 

NOTE: Standards available in English from: ANSI, 1430 Broadway, New York, NY 10018; or International Standardization Cooperation Centre, Japanese Standards Association, 4-1-24 Akasaka, Minato-ku, Tokyo 107.

UNITED KINGDOM - BSI (BrItish Standards Institute)BS 235 1972BS 438 Pt1 1987

BS 436 Pt 2 1984

 BS436 Pt3 1986

BS721 Pt 1 1984BS721 Pt2 1983BS978 Pt1 1984BS978 Pt2 1984BS978 Pt3 1984BS978 Pt4 1965BS1807 1981

BS2007 1983

BS2062 Pt 1 1985

BS2082 Pt2 1986

BS2518 Pt 1 1983

BS2518 Pt2 1983

 Specification of gears for electric traction  Spur and helical gears — Basic rack form, pitches and accuracy (diametral pitch series)  Spur and helical gears — Basic rack form, modules and accuracy (1 to 50 metric module)  (Parts I & 2 related but not equivalent with ISO 53.54, 1328,1340 & 1341)  Spur gear and helical gears-Method for calculation of contact and root bending stresses, limitations for metallic involute gears (Related but not equivalent with ISO/ DIS 633611, 2 & 3)  Specification for worm gearing — Imperial units  Specification for worm gearing — Metric units  Specification for fine pitch gears — Involute spur and helical gears  Specification for fine pitch gears — Cydoidal type gears Specification for fine pitch gears - Bevel gears  Specification for fine pitch gears - Hobs and cutters  SpecifIcation for marine propulsion gears and similar drives: metric module Specification for circular gear shaving cutters, 1 to 8 metric module,accuracy requirements  Specification for gear hobs — Hobs for general purpose: 1 to 2O dp., inclusive  Specification for gear hobs — Hobs for gears for turbone reduction and similar drives  Specification for rotary form relieved gear cutters - Diametral pitch  Specification for rotary relieved gear cutters - Metric modules

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TABLE 1.37 CONTD. FOREIGN METRIC GEAR STANDARDS  UNITED KINGDOM CONT. — BSI (British Standards Institute)

BS 2519 Pt 1 1976BS 2519 Pt 2 1976

BS 2697 1976BS 3027 1968BS 3696 Pt 1 1984

BS 4517 1984BS 4582 Pt 1 1984BS 4582 Pt 2 1986BS 5221 1987BS 5246 1984

BS 6168 1987

 Glossary for gears — Geometrical definitions Glossary for gears — Notation (symbols for geometrical data for use in gear notation) Specification for rack type gear cutters Specification for dimensions of worm gear units  Specification for master gears — Spur and helical gears (metric module)  Dimensions of spur and helical geared motor units (metric series) Fine pitch gears (metric module) — Involute spur and helical gears  Fine pitch gears (metric module) — Hobs and cutters  Specification for general purpose, metric module gear hobs Specification for pinion type cutters for spur gears — 1 to 8 metric module  Specification for non-metallic spur gears

NOTE: Standards available from: ANSI. 1430 Broadway, New York, NY 10018; or BSI, Linford Wood, MiltonKeynes MK146LE, United Kingdom.

ADDITIONAL  GEAR DESIGNLITERATURE  AND SOFTWARE

From noted authoritiesIn the field of GEAR DESIGN.

such as:

Earl Buckingham

J W. Dudley

JE Shigley

Clifford E. Adams

and others is made available......

See complete listing with detailed descriptionand ordering informationon pages T159 and T160

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GEAR DESIGN  -  PLASTIC

20.0 DESIGN OF PLASTIC MOLDED GEARS

Plastic gears are continuing to displace metal gears in a widening arena of applications. Their unique characteristics are alsobeing enhanced with new developments, both in materials and processing. In this regard plastics contrast somewhat dramaticallyfrom metals, in that the latter materials and processes are essentially fully developed and, therefore, are in a relatively staticstate of development.Among the various methods of producing plastic gears, molding is unique in many respects. For that reason, it is singled out forin-depth treatment in this separate section.

20.1 General Characteristics of Plastic Gears

Among the characteristics responsible for the large increase in plastic gear usage the following are probably the most significant:

1. Cost effectiveness of the injection-molding process.2. Elimination of machining operations; capability of fabrication with inserts and integral   designs.3. Low density: light weight, low inertia.4. Uniformity of parts.5. Capability to absorb shock and vibration as a result of elastic compliance.6. Ability to operate with minimum or no lubrication, due to inherent lubricity.7. Relatively low coefficient of friction.8. Corrosion resistance; elimination of plating, or protective coatings.9. Quietness of operation.10. Tolerances often less critical than for metal gears, due in part to their greater     resilience.11. Consistency with trend to greater use of plastic housings and other components.12. One step production; no preliminary or secondary operations.

At the same time the design engineer should be familiar with the limitations of plastic gears relative to metal gears. The mostsignificant of these are as follows:

1. Less load-carrying capacity due to lower maximum allowable stress; the greater    compliance of plastic gears may also produce stress concentrations.2. Plastic gears cannot generally be molded to the same accuracy as high-precision    machined metal gears.3. Plastic gears are subject to greater dimensional instabilities due to their greater    coefficient of thermal expansion and moisture absorption.4. Reduced ability to operate at elevated temperatures; as an approximate figure,   operation is limited to less than 250 degreeso F. Also limited cold temperature   operations.5. Initial high mold cost in developing correct tooth form and dimensions.6. Can be negatively affected by certain chemicals and even some lubricants.7. Improper molding tools and process can produce residual internal stresses at the    tooth roots resulting in over stressing and/or distortion with aging.8. Cost of plastics track petrochemical pricing and thus are more volatile and increasing    in comparison to metals                                                        T131

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20.2 Properties of Plastic Gear Materials Popular materials for Plastic Gears are acetal resins such as DELRIN*, nylon resins such as ZYTEL*and NYLATRON** and acetal copolymers  such as CELCON***. The physical and mechanical properties of a these materials varywith regard to strength, rigidly, dimensional resistance, fabrication requirements, moisture absorption etc. Standardized tabulardata is available from various manufacturers catalogs. In general, the information and data is less simplified and fixed than for themetals. This is because plastics are subject to wider formulation variations and are often regarded as proprietary compounds andmixtures. Tables 1.38 through 1.43A are representative listings of physical and mechanical properties of gear plastics taken from avariety of sources.

It is common practice to use plastics in combination with different metals and materials other than plastics.  Such is the case whengears have metal hubs, inserts, rims, spokes, etc. In these cases one must be cognizant of the fact that plastics have an order ofmagnitude different Coefficients of Thermal Expansion as well as Density and Modulus of Elasticity. For this reason TABLE 1 .43A ispresented.Other properties and features that enter into considerations for gearing are given in Table 1.44 (Wear) and Table 1.45 (Poisson’sRatio).Moisture has a significant impact on plastic properties as can be seen in Tables 1.38 thru 1.43. Ranking of plastics is given in Table1.46. In this table, rate refers to expansion from dry to full moist condition. Thus, a 0.20% rating means a dimensional increase of0.002 inch per inch. Note that this is only a rough guide as exact values depend upon factors of composition and processing, boththe raw material and gear molding. For example, it can be seen that the various types and grades of nylon can range from 0.07%to 2.0%.

Table 1.47 lists safe stress values for a few basic plastics and the effect of glass fiber reinforcement

TABLE 1.38 PHYSICAL PROPERTIES OF PLASTICS USED IN GEARS

Material

TensileStrength(psi x103)

FlexuralStrength(psi x 103)

CompressiveModulus(psi x 103)

HeatDistortionTemperature(oF @ 264 psi)

WaterAbsorption(% 24 hr)

RockwellHardness

MoldShrinkage(in./in.)

AcetalABSNylon 6/6Nylon6/1OPolycarbonateHigh ImpactPolystyrenePolyurethanePolyvinylChloridePolysulflonMoS2Filled Nylon

8.8-1.04.5-8.5

11.2-13.17-8.58-9.5

1.9-4

4.5-86-9

10.210.2

13-145-13.514.610.5 11-13

5.5-12.5

7.18-15

16.410

410120-200

400400350

300-580 

85300-400

370330

230-255180-245

200145 

285-290

160-205

180-205140-175

345140

0.250.2-0.5

1.3.4

0.15

.05-.10

.60-.8007-.40

0.220.4

M94 R120R80-120R115-123R111 M70

R112

M25-69

M29 R90 R100-120

M69-R120D785

0.022/0.0030.007/ 0.007

0.0150.015/0.0050.007/ 0.003

0.005

0.009/0.0020.004

0.00760.012

Reprinted with the permission of Plastic Design and Processing Magazine; see Raf. 11.

________________*Registered trademark. E.I. du Pont de Nemours and Co., Wilmington, Delaware 19898.**Registered trademark, The Polymer Corporation, P.O. Box 422, Reading Pennsylvania, 19603 ***Registered trademark, CelaneseCorporation, 26 Msin St., Chaitham, N.J.  07928

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TABLE 1.39 PROPERTY CHART FOR BASIC POLYMERS FOR GEARINGWater

Absorption 24 hr

MoldShrinkage

TensileStrength*Yield#Break

FlexuralModulus

IzodImpact

StrengthNotched

deflect.Temp.

@264 psi

Coeff.Linear

ThermalExpan.

SpecificGravity

Units % In./In. psi psi ft-lb/in. oF 10-5 oFASTM D570 D955 D638 D790 D256 D648 D696 D7921.Nylon 6/62.Nylon63.Acetal4.Polycarbonate    30% G.F    15% PTFE5.Polyester (thermoplastic)6.Polyphenylene   Sulfide    30% Sulfide   15% PTFE7.Polyester   elastomer8.Phenolic  (molded)  

1.51.60.2

0.06

0.08

0.03

0.3

0.45

.015/.030

.013/.025

.016/.030.0035

.020

.002

.012

.007

*11,200*11,800*10,000*17,500

*8,000#12,000*19,000

*3,780#5,500#7,000

175,000395,000410,000

1,200,000

340,000

1,300,000

--

340,000

2.11.1

1.4/2.32

1.2

1.10

--

.29

220150255290

130

500

122

270

4.5varies

4.65.81.50

5.3

1.50

10.00

3.75

1.13/1.151.131.421.55

1.3

1.69

1.25

1.42

These are average values for comparison purposes only.Source: Clifford E. Adams, "Plastic Gearng", Marcel Dekker Inc. N.Y.1986. Ref.13

TABLE 1.40 PHYSICAL PROPERTIES OF OELRIN ACETAL RESIN AND ZYTEL NYLON RESIN

Properties - Units ASTM

"DELRIN"

500          100

"ZYTEL" 100

.2% Moisture 2.5% Moisture

Yield Strength, psiShear Strength, psi Impact Strength, (Izod)Elongation at Yield,%Modulus of Elasticity, psiHardness, RockwellCoefficient of Linear ThermalExpansion, in/in. oFWater Absorption      24hrs.%    Saturation, %

Specific Gravity

D638D732D256D638D790D785D696

D570

D792

         10,000           9,510  1.4                  2.3  15                   75         410,000        M94 R120        4.5 x 10-5

                      0.25             0.9             1.425

11,8009,600

0.95

410,000M79 R1184.5 x 10-5

1.58.01.14

8,500

2.025

175,000M59 R108

 1.14

Test conducted at 73o FReprinted with the permission of E.l. DuPont de nemours and Co.; see Ref. 8

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TABLE 1.46 MATERIAL RANKING BY WATER ABSORPTION RATE

Material Rate of change %PolytetrafluoroethylenePoly ethylene: medium density                     high density                      high molecular weight

                     low density

Polyphenylene sulfldes (40% glass filled) 

Polyester: thermosetting and alkyds                     low shrink                     glass-preformed chopping roving

Polyester: linear aromaticPolyphenylene sulfide: unfilledPolyester: thermoplastic(18% glass)Polyurethane: cast liquid methane

Polyester synthetic: fiber filled - alkyd                             glass filled - alkyd                             mineral filled - alkyd                             glass-woven cloth                             glass premixed, chopped

Nylon 12 (30% glass)Polycarbonate (10-40% glass)Styrene acrylonitrite copolymer(20-33% glass filled)

Polyester thermoplastic:              thermoplastic PTMT (20% asbestos)              glass sheet molding  

Polycarbonate < 10% glassPhenolic cast - mineral filledPolyester alkyd - asbestos filledPolycarbonate - unfilledPolyester cast - rigidAcetal: TFE

Nylon 6/12 (30-35% glass)         6/10 (30-35% glass)

Polyester alkyd vinyl ester thermosetStyrene acrylonitrite copolymer: unfilledPolycarbonate ABS alloyPhenolic cast : unfilled

Acetal copolymer          homopolymer

Nylon 12 (unmodified)Acetal (20% glass)Poly(ancide-imide)Acetal (25% glass)Nylon 11 (unmodified)

0.0<0.01<0.01<0.01<0.015

0.01

0.0 1-0.250.01-1.0

0.020.02

0.02-0.070.02-1.5

0.05-0.200.05-0.250.05-0.500.05-0.500.06-0.28

0.070.07-0.200.08-0.22

0.090.10

0.10-0.15

0.120.12-0.36

0.140.15-0.180.15-0.60

0.20

0.200.20

0.200.20-0.300.20-0.350.20-0.40

0.220.25

0.250.25-0.29

0280.29030

Source: Clifford E. Adams, "Plastic Gearing", Marcel Oekker, Inc. New York, 1986. Ref 13

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TABLE 1.46 (CONTINUED)

Material Rate of change %Polyester elastomerPoly imideNylon 6/12 (unmodified)          6/10 (unmodified)Polyester-thermosetting and alkyds (cast flexible)Nylon 6 (cast)Poly urethane elastomer thermoplastic

Nylon 6/6: MOS2      30 - 35% glass          unmodified          nucleated

Nylon 6 (30-35% glass)       unmodified

       nucleated

Nylon 6/6-6(copolymer)         

0.30-0.600.320.400.40

0.50-2.50.60-1.20.70-0.90

0.80-1.10.90

1.1-1.51.1-1.5

1.31.3-1.91.3-1.9

1.5-2.0

Source: Clifford E. Adams, "Plastic Gearing", Marcel Dekker, Inc. New York, 1986. Ref. 13

20.3 Pressure Angles

Pressure angles of 14½o, 20o and 25o are used in plastic gears. The 20o pressure angle is usually preferred due to its strongertooth shape and reduced undercutting compared to the 14½o pressure-angle system. The 25o pressure angle has the highestload-carring ability, but is more sensitive to center-distance variation and hence runs less quietly. The choice is dependent on theapplication.

20.4 Diametral Pitch

The determination of the appropriate diametral pitch is a compromise between a number of different design requirements. Asmaller pitch number is associated with larger and stronger teeth. For a given pitch diameter, however, this also means a smallernumber of teeth with a correspondingly greater likelihood of undercut at very low tooth numbers. Larger teeth are generallyassociated with more sliding than smaller teeth.On the other side of the coin, larger pitch numbers, which are associated with smaller teeth, tend to provide greater load sharingdue to the compliance of plastic gears. However, a limiting condition would eventually be reached when mechanical interferenceoccurs as a result of too much compliance. Smaller teeth are also more sensitive to tooth errors and may be more highlystressed.A good procedure is probably to size the pinion first, since it is the more highly loaded member. It should be proportioned tosupport the required loads, but should not be overdesigned.

20.5 Design Equations For Plastic Spur, Bevel, Helical and Worm Gears

20.5.1 General Considerations—The load-carrying capacity of a plastic spur gear is reached under a limiting load, which isdetermined either by bending fatigue strength or wear(surface durability). The latter is measured by contact stress. The characterof the limiting load depends on the presence or absence of lubrication and its nature, as shown in Table 1.48

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TABLE 1.4 LOAD-CARRYING UNIT VS. LUBRICATION OF SPUR GEARS FOR ACETAL COPOLYMER (CELCON), ACEIALRESIN (DELRIN) AND NYLON RESIN (NYLON ETC.).

Natureof Lubricant  Limiting Load Determined by:Continuous lubrication

Initial lubrication only

No lubrication

Bending fatigue strength

Celcon and Zytel: Bending fatigue strengthDelrin: Wear (contact strength)

Wear (contact stress)

Like metal gearing, plastic spur gears are the most prevalent type and also the easiest to design inregard to stress level. The design procedure for helical and bevel gears is analagous to those for spur, differing mainly in thevalues of the various equation modification parameters. We first consider the design for bending fatigue strength and follow thiswith a discussion of design for surface durability (contact stress), For a given spur gear Table 1.48 determines which of theseshould be used.

In regard to standards it should be noted that the AGMA standards have been generated around metal gears. An exception isAGMA 141.01 which is devoted to plastic gearing.

20.5.2 Bending Stress -Spur Gears—The basic Lewis Formula, the use of which for metal gears has been described in Section 13,is used also for plastic gears with suitable modifying factors. It can be expressed in the form:

       Wt = SFY  KLKvKt                                                       (70)

                 Pd

where

KL = lubrication factor

Kv = velocity-dependent dynamic factor

Kt = temperature factor

As already explained in Section 13.1, the remaining notation is as follows:

          Wt  = tangentially transmitted Load lbs

           S   = maximum tooth bending stress , lbs/in2           F   = face width of gears, in.

       Y = Lewis form factor f (Table 1.49)

ln the event torque or horsepower (H.P.) are specified, these are related to the tangent.ially transmitted load as follows:

Torque (in.-lbs.) = (1/2)WtD                                            (71)and H.P. = WtDn                                                             (72)                 126050  where            D = pitch diameter of gear, inches                  n  = gear speed in revolutions per minute

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TABLE 1.49 LEWIS FORM FACTOR, YTOOTH FORM FACTOR LOAD NEAR THE PITCH POINT

Number of Teeth 14½o 20o Full Depth 20o Stub141516171819202224262830343843506075100150300

Rack

--------

0.5090.5220.5350.5400.5530.5660.5750.5880.6040.6130.6220.6350.6500.660

---

0.5120.5210.5340.5440.5590.5720.5880.5970.6060.6280.6510.6720.6940.7130.7350.7570.7790.8010.823

0.5400.5660.5780.5870.6030.6160.6280.6480.6640.6780.6880.6980.7140.7290.7390.7580.7740.7920.8080.8300.8550.881

             Reprinted with the permission of E. I. DuPont de Nemours and Co.; see Ref. 8.

20.5.3 Surface Durability for Spur and Helical Gears —Excessive contact stresses can cause wear of the tooth surface andcan be a limiting factor in the performance of a plastic gear. For spur gears which are not lubricated or (in some cases) onlyinitially lubricated, both contact stress and bending stress should be checked, the smaller load being the limiting load.For spur gears the Hertzian contact stress, 5,, in the elastic range is given by:

  

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For helical gears, the contact stress is given by the equation:

 The other symbols have the same meaning as in the case of spur gears.

The allowable contact stress is a function of lubrication, gear and pinion material, speed of operation, ambient temperature andcycles of operation during the life of the gear. The most reliable information on -this point is likely to be that obtainable from theplastics manufacturer.In applying eqs. (73) and (74) it is not always easy to predict the values of the elastic moduli. The values given in the literaturesuch as figures 1.53 and 1.54 are functions of temperature. If the temperature does not vary too much, the ambient temperaturecan be used for a first estimate.For bevel gears the determination of contact stresses is beyond the scope of this treatment and the cited references at the end ofthis section can be consulted for further information.

  

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 2. ln determining the Lewis form factor from Table 1.49 the number of teeth should be the formative Numberr of teeth, Nt   where    Nt = N                                                       (77)           cosý 

   and  ý = pitch angle

3. Both bending stress and contact stress should be checked for all bevel gears regardless of lubrication, as either type of stressmay be limiting. The subject of contact stresses in bevel gears is an Involved subject for which the reader is referred to thereferences.

20.5.7 Design Procedure- Worm Gears—The design of worm-gear drives involves consideration of a rumber of factors, whichdo not arise in the design of spur and helical gears. These include the stresses associated with the theoretical line contactbetween the teeth of the worm and gear - and the wear associated with the relatively high sliding velocities at the toothinterface. Plastic worm gears meshing with either with  a plastic worm or a metal worm have been used. In either case theload-carrying capacity of the combination is substantially less than that of metal gears and worms. For design calculations, whichare the scope of this discussion, the reader is referred to the references.

3.6 Operating Temperature

As a general guideline plastic gears should be used only for temperatures below 250oF, as their load carrying capacity decreaseswith temperature. The actual recommended maximum temperature can be considerably below 2500o F, depending upon theapplication. Limiting factors include the nature of the lubrication, loads, speeds, thermal expansion, nature of operation(continuous or intermittent) and the material properties of the plastics involved.The combination of a plastic and a metal gear improves heat dissipation. If space permits, the plasticcan be proportioned so as to maximize the rate of heat transferred to its surroundings.In order to  estimate the operating temperature limit it is recommended that the bending stress andcontact stress can  be calculated, taking into account the reduction in tensile strength and elastic moduli with temperature(Figures 1.53, 1.54).

20.7 Effect of Part Shrinkage on Gear Design

The nature of the part and the molding operation have a significant effect on the molded gear. From the design point of view themost important effect is the shrinkage of the gear relative to the size of the mold cavity.Gear shrinkage depends upon mold proportions, gear geometry, material, ambient temperature andis  usually expressed in inches per inch. For example, if a plastic gear with a shrinkage rate of 0.022 in./in. has a pitch diameterof 2 inches while in the mold, the pitch diameter after molding will be reduced by  (2)(0.22)"  or 0.44" and becomes 1.956" afterit leaves the mold. Depending upon thematerial and the molding process shrinkage rates ranging from about 0.OO1in./in. to 0.030 in/in, occur in (see Tablet 1.38 andFig.1.61). Sometimes shrinkage rates are expressed as a percentage. For example a shrinkage rate of 0.0025 in/in, can be stated as a 0.25% shrinkage rate.

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20.8 Design Specifications

Basic gear formals have been discussed in Section 11.1, which also included a typical data block for spur gears (Figure 1.47). Theinformation required for plastic injection-molded gears is very similar. Additional information, which would be useful, wouldinclude fillet radius and whole depth. Other data, which is pertinent to the function of the gear and which the design engineermay wish to add to the data block, includes the following: material and pitch diameter of mating gear, operating temperature,lubrication, moisture/humidity data, and annealing, if required.

20.9 Backlash

Due to the thermal expansion of plastic gears, which is significantly greater than that of metal gears (see Tables 1 .40 thru1.43A) and the effects of tolerances, one should make sure that meshing gears do not bind in the course of service. Severalmeans are available for introducing backlash into the system. Perhaps the simplest is to enlarge center distance. This procedureis described in Section 4.10. Care must be taken, however, to ensure that the contact ratio remains adequate.It is possible also to thin out the tooth profile during manufacturing, but this adds to the manufacturing cost and requires carefulconsideration of the tooth geometry.To some extent the flexibility of the bearings and clearances can compensate for thermal expansion. It a small change in centerdistance is necessary and feasible, it probably represents the best and least expensive compromise.

20.10 Environment and Tolerances

In any discussion of tolerances for plastic gears it is necessary to distinguish between manufacturingtolerances and dimensional changes due to environmental conditions.As far an manufacturing is concerned, plastic gears can be made to high accuracy, if desired. For injection-molded gears TCE canreadily be held within a range of roughly 0.003" - 0.005", with a corresponding TTCE of about 0.OO1 - 0.002". Higher accuraciescan be obtained if the more expensive filled materials, mold design, tooling and quality control are warranted.In addition to thermal expansion changes there are permanent dimensional changes as the result of moisture absorption. Inaddition, there are dimensional changes due to compliance under load. The coefficient of thermal expansion of plastics is on theorder of four to ten times those of metals (see Tables 1.40, 1.43A). In addition, most plastics are hygroscopic (i.e., absorbmoisture) and dimensional changes on the order of 0.1% or more can develop in the course of time, if the humidity is sufficientAs a result, one should attempt to make sure that a tolerance which is specified is not smaller than the inevitable dimensionalchanges which arise as a result of environmental conditions. At the same time, the greater compliance of plastic gears ascompared to metal gears suggests that the necessity for close tolerances need not always be as high as those required for metalgears.

20.11 Avoiding Stress Concentration

In order to minimize stress concentration and maximize the life of a plastic gear the root fillet radius should be as large aspossible, consistent with conjugate gear actiotn. Sudden changes in section and sharp corners should be avoided, especially inview of the possibility of additional residual stresses, which may have occurred in the course of the molding operation.

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20.12 Metal Inserts

Injection-molded metal inserts are used in plastic gears for a variety of reasons:

(a) To avoid an extra finishing operation(b) To achieve greater dimensional stability, because the metal will shrink less and is not sensitive      to moisture, is also, a better heat sink.(c) To provide greater load-carting capacity(d) To provide increased rigidity(e) To permit repeated assembly and disassembly(f) To provide a more precise bore to shaft fit

Inserts can be molded into the part or subsequently assembled. By subsequent insertion of inserts stress concentrations may bepresent which can result in cracking of the parts. The interference limits for press fits must be obeyed depending on the materialused, also proper minimum wall thicknesses around the inserts must be left. The insertion of inserts can be accomplished byultrasonically driving in the insert. In this case the material actually melts into the knurling at the insert periphery.Inserts are usually produced by screw-machines and made of aluminum or brass, It is advantageous to attempt to match thecoefficient of thermal expansion of the plastic to the materials used for inserts. This will reduce the residual stresses in the plasticpart of the gear during contraction while cooling after molding.When metal inserts are used generous radii and fillets in the plastic gear are recommended to avoid stress concentration, It isalso possible to use other types of metal inserts, such as self-threading, self-tapping screws, press fits and knurled inserts. Oneadvantage of the first two of these is that they permit repeated assembly and disassembly without part failure of fatigue.

20.13 Attachment of Plastic Gears To Shafts

Several methods of attaching gears to shafts are in common use. These include splines, keys, integral shafts, set screws, andplain and knurled press fits. Table 1.53 lists some of the basic characteristics of each of these fastening methods:

TABLE 1.53 CHARACTERISTICS OF VARIOUS SHAFT-ATTACHMENT METHODSNature of

Gear ShaftConnection

TorqueCapacity

Cost Disassembly Comments

Set Screw

Press fit

Knurled ShaftConnection

Spindle

Key

Integral Shaft

Limited

Limited

Fair

Good

Good

Good

Low

Low

Low

High

ReasonablyLowLow

Not good unlessthreaded metalinsert is used

Not Possible

Not Possible

Good

Good

Not Possible

Questionable reliability,particularly under

vibration or reversingdrive

Residual stresses need tobe considered

A permanent assembly

Suited for closetolerances

Requires good fits

Bending load on shaftneeds to be watched

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20.14 Lubrication

Depending on the application, plaslic gears can operate with continuous lubrication, initial lubrication  or no lubrication. Accordingto LD. Martin lnjection Molded Plastic Gears, Plastics Design and Processing, 1968; Pert 1, August,pp 38-45; Part 2.September,pp.33-35);

(a) all gears function more effectively with lubrication and will have a longer service life; (b) a light spindle oil (SAE 10) is generally recommended as are the usual lubricants; these include silicone and hydrocarbon oils,and in some cases cold water is acceptable as well ; and (c) under certain conditions dry lubricants, such as molybdenum disulfide, can be used to reduce tooth friction.

Ample experience and evidence exists that substantiates plastic gears can operate with a metal mate without the need of alubricant so long as the stress levels are not exceeded. It is also true that in the case of a moderate stress level, relative to thematerials rating plastic gears can be meshed together without a lubricant However, as the stress level is increased there is atendency for localized plastic to plastic welding which increases friction and wear. The level of this problem varies with theparticular plastic type.A key advantage of plastic gearing is that for many applications running dry is adequate. When asituation of stress and shock level is uncertain, using the proper lubricant will provide a safety margin and certainly will cause noharm. The chief consideration in choosing a lubricants chemical compatability with the particular plastic. Least likely to encounterproblems with typical gear oils and greases are: nylons, delrins, phenolics, polyethylene and polypropylene. Materials requiringcaution are: polystyrene, polycarbonates, polyvinyl chloride and ABS resins.An alternate to external lubrication is to use plastics fortified with a solid state lubricant. Molybdenum disulfide in nylon and Delrinare commonly used. Also, graphite, coloidal carbon and silicone are used as fillers.In no event should there be need of an elaborate sophisticated lubrication system such as for metal gearing. If such a system iscontemplated then the choice of plastic gearing is in question. Simplicity is the plastic gears inherent feature.

20.15 Inspection

In view of the compliance of injection-molded gears, the dimensional accuracy of such gears is determined by avariable-center-distance fixture (see Section 18.1). This type of gear testing is both functional and utilizes a much moreconsistent measuring contact force than an over-wires measurement. In view of the hygroscopic nature of plastic gears, careshould be taken to minimize dimensional changes between inspection and use of the gear.

20.15 Molded vs. Cut Plastic Gears

Although not nearly as common as the injection-molding process, both thermosetting and thermoplastic plastic gears can bereadily machined. The machining of plastic gears can be considered for high-precision parts with close tolerances and for thedevelopment of prototypes for which the investment in a mold may not be justified.Standard stock gears of reasonable precision are produced by molding blanks with brass inserts, which are subsequently hobbedto close tolerances.When to use molded gears vs. cut plastic gears is usually determined on the basis at production quantity, body features that mayfavor molding, quality level and unit cost Often the initial prototype quantity will be machine cut, and investment in molding toolsis deferred until the product and market is assured. However, with some plastics this approach can encounter problems.

The performance of molded vs. cut plastic gears is not always identical. Differences occur due to subtle causes. Bar stock andmolding stock may not be precisely the same. Molding temperature can have an effect. Also, surface finishes will be different forcut vs. molded gears. And finally there is the impact of shrinkage with molding which may not have been adequatelycompensated.An example is  Dupont's. Vespel SP3, a good high temperature polyamide plastic available  in rod form. However  Vespel formolding is not available in SP3 formulation another  similar Vespel resin must be substituted.                                                              T152

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20.17 Elimination of Gear Noise

In complete conjugate action and/or excessive backlash are  the source of noise. Plastic-molded gears are generally less accuratethan their metal counterparts. Furthermore, due to the presence of a larger TCE there is more backlash built into the gear train.To avoid noise, more resilient material, such  as Urethane can be used. Figure 1.63 shows several gears made of Urethane whichin mesh with Delrin gears produce a practically noiseless gear train. The face width of the Urethane gears must be increasedcorrespondingly to compensate for lower load carrying ability of this material.

20.18 Mold Construction

Depending on the quantity of gears to be produced a decision has to be made to make one single cavity or a multiplicity ofidentical cavities. If more than one cavity is involved these are used as "family molds" inserted in mold bases which canaccommodate a number of cavities for identical or different parts. Since special terminology will be used we shall first describethe elements shown in Figure 1.64.

1 — LOCATING RING is the element which assures the proper location of the mold on the platen with respect to the nozzlewhich injects the molten plastic.2— SPRUE BUSHING is the element which mates with the nozzle. It has a spherical or flat receptacle which accurately mateswith the surface of the nozzle.3— SPRUE is the channel in the sprue bushing through which the molten plastic is injected.4— RUNNER is the channel which distributes material to different cavities within the same mold base.5-CORE PIN is the element which by its presence restricts the flow of plastic; hence, a hole or void will be created in the moldedpart.6-EJECTOR PINS are pins-operated by the molding machine. These have a relative motion with respect to the cavity in thedirection which will cause ejection of the part from the mold.7-FRONT SIDE is considered the side on which the sprue bushing and the Nozzle are located.8- GATE is the orifice through which the molten plastic enters the cavity.* - VENT is a miniscule opening through which the air can be evacuated from the cavity as the molten plastic fills it. The Vent isconfigured to let air escape, but does not fill up with plastic.____________ *Not visible due to its small size.

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Furthermore, the shrinkage of the material in the direction of the flow will be different from that perpendicular to the flow As aresult, a slide-gated gear or rotating part will be somewhat elliptical rather than round.In order to eliminate this problem, "diaphragm gating" can be used which will cause the injection of material in all directions atthe same time (see Figure 1.66). The disadvantage of this method is the presence of a burr at the hub and no means of supportof the core pin because of the presence of the sprue.The best, but most elaborate way is "multiple-pin-gating" (Figure 1.67). In this case the plastic injected at several placessymmetrically located. This will assure reasonable viscosity of plastic when the material welds, as well as create uniformshrinkage in all directions. The problem is the elaborate nature of the mold arrangement- so called 3-plate molds (Figure 1.68)accompanied by high costs. If precision is a requirement, this way of molding is a must, particularly it the gears are of a largerdiameter.To compare the complexity of a  3-plate mold with a 2plate mold which is used for edge gating, Figure 1.69 can serve as anillustration.

20.19 Conclusion

In this section we have attempted to highlight the procedure for proper design of plastic gears as well as illustrate the difficultiesand complexities involved in the production of molded plastic gears for technical applications.It is a fact that many gears are produced by molders for whom a gear is not any different from a "fancy door-knob". In many instances taking this position is justified - such as toys etc. However, if technical applications andstringent requirements are involved, it is imperative that a thorough knowledge of disciplines such as: gear design, mold design,tool-making, molding and machining (for secondary operation) is used in order to produce a superior or even an acceptableproduct.

ACKNOWLEDGEMENT OF REPRINTSReferences:

1. Earle Buckingham, Manual of Gear Design",3 Vols., Industrial Press, New York, 19352. Chironis, N.P.(Editor): "Gear Design and Application", McGraw-Hill Book Co., Inc. New York, NY., 19673. D.W. Dudley, "Gear Handbook", McGraw-Hill, NewYork,19624. Knut 0. Kverneland (Editor), "World Metric Standards for Engineering", Industrial Press, New York, NY. 19785. G.W. Michalec,"Precision Gearing: Theory and Practice", John Wiley & Sons, New York, 19666. J.E. Shigley,"Mechanical Engineering Design", McGraw-Hill, New York, 19637. W. Steeds,"Involute Gears, Longmans, Green and Co., London, 19488. El. DuPont de Nemeurs and Co., "Gears of DELRIN and ZYTEL’", Wilmington, Delaware.9. Celanese Plastics and Specialties Co., "Design and Production of Gears in CELCON Acetal Copolymer", Chatham, New Jersey,197910. The Polymer Corp, "Nylatron Nylon Gear Design Manual’, Reading, Pa.11. L.D. Martin, "Injection Molded Plastic Gears",Plastic Design  and Processing Magazine , Pt.1 ,pp. 38-45. August 1968.12. E.l. DuPont de Nemours and Co.. DELRIN Design Handbook", Willinggton. Delaware, 196713. Clifford E. Adams, "Plastie Gearing" Marcel Dekker, Inc.  New YorK 1986 -

Literature of general interest:

R.W.Woodbury: "History of Gear Cutting Machines"  M.I.T. Technology PressCambridge, Mass, 1958D.W.Dudley: "The Evolution of the Gear Art" AGMA Paper No. 990.14(Published in book form  by AGMA Jan. 1969)

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