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T25

1.0 INTRODUCTION 2.0 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 Pitches 3.0 GEAR TOOTH FORMS AND STANDARDS 3.1 Preferred Pitches 3.2 Design Tables 3.3 AGMA Standards 4.0 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 Fundamentals 5.0 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 T25 T27 T27 T28 T28 T28 T28 T29 T29 T29 T31 T37 T37 T38 T38 138 T39 144 144 145 145 T48 T52 T53 T53 T54 T54 T54 T55 T55 155 T55 T55 T55 156 156 T56 156 T21

Catalog D190

5.9.3 Center Distance 5.9.4 Velocity Ratio 5.10 Axial Thrust of Helical Gears

T57 T57 T57

6.0 RACKS 7.0 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 Gears 8.0 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 Ratio 9.0 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 Teeth 10.0 GEAR TYPE EVALUATION 11.0 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 Standards 12.0 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 Error 13.0 GEAR STRENGTH AND DURABILITY 13.1 Bending Tooth Strength 13.2 Dynamic Strength 13.3 Surface Durability 13.4 AGMA Strength and Durability Ratings T22

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

T58 T58 T59 T60 T61 T61 T62 T62 T62 T63 T63 T64 T64 T66 T66 T67 T68 T68 T70 T70 T73 T73 T76 T77 T77 T78 T78 T78 T82 T88 T88

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 Comments 15.0 FINISH COATINGS 15.1 Anodize 15.2 Chromate Coatings 15.3 Passivation 15.4 Platings 15.5 Special Coatings 15.6 Application of Coatings 16.0 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 Lubricants 17.0 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 Gears 18.0 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 T23

T91 T91 T91 T92 T92 T92 T92 T92 T92 T99 T99 T100 T100 T100 T100 T100 T101 T101 T101 T103 T103 T103 T105 T105 T105 T105 T106 T106 T107 T107 T107 T107 T107 T108 T108 T108 T108

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 Standards 20.0 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 T24

T109 T122 T124 T125 T126 T126 T126 T131 T132 T139 T139 T139 T139 T140 T141 T143 T146 T146 T147 T147 T147 T150 T150 T150 T150 T151 T151 T152 T152 T152 T153 T153 T158

1.0 INTRODUCTION This section presents a technical coverage of gear fundamentals. It is intended as a broad coverage written in a manner that is easy to follow and to understand by anyone interested in knowing how gear systems function. Since gearing involves specialty components it is expected that not all designers and engineers possess or have been exposed to all aspects of this subject However, for proper use of gear components and design of gear systems it is essential to have a minimum understanding of gear basics and a reference source for details. For those to whom this is their first encounter with gear components, it is suggested this section be read in the order presented so as to obtain a logical development of the subject. Subsequently, and for those already familiar with gears, this material can be used selectively 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 most comprehensible, 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. 2. 3. 4. 5.

center distance the pitch circle diameters (or pitch diameters) size of teeth (or pitch) number of teeth 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, Precision instruments require positioning fidelity. High speed and/or high power gear trains also require transmission at constant angular velocities 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. A geometric relationship can be derived (1,7)* for the form of the tooth profiles to provide cojugate action, which is summarized as the Law of Gearing as follows: “A common normal to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point.” Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate Curves.

___________

*Numbers in parenthesis refer to references at end of text. T25

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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 forms have been tried in the past. Modem gearing (except for clock gears) based on involute teeth. This is due to three major advantages 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 a generating 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 is imagined that a point on a string, which is pulled taut in a fixed direction, projects its trace onto a plane that rotates with the base circle. See Figure 1.2. The base cylinder, or base circle as referred to in gear literature, fully defines the form of the involute and 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 one base circle and wound on to the other, as shown in Figure 1.3a Thus, a single point on the string simultaneously traces an involute on each base circles rotating plane. This pair of involutes is conjugate, since at all points of contact the common normal is the common tangent which passes through a fixed point on the line-of-centers. It a second winding/unwinding taut string is wound around the base circles in the opposite direction, Figure 1 .3b, oppositely curved involutes are generted which can accommodate 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 this line 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 D2

(1)

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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 are two basic forms. 2.5.1 Circular pitch — A naturally conceived linear measure along the pitch circle of the tooth spacing. Referring to Figure 1.5 it is the linear distance (measured along the pitch circle ar between corresponding points of adjacent teeth. it is equal to the pitch-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 a measure of the number of teeth per inch of pitch diameter. This is simply: expressed as: Pd = diametral pitch = N (3) D Diametral pitch is so commonly used with fine pitch gears that it is usually contracted simply to "pitch" and that it is diametral is implied. 2.5.3 Relation of pitches: From the geometry that defines the two pitches it can be shown that they are related by the product 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 for involute 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 have been formulated, only a few are currently active and widely used. Symbols for the basic rack are given in Figure 1.6 and pertinent 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 given in 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 each gear, Table 1.3 lists tooth proportions common to a given diametral pitch, as well as the diameter of a measuring wire. Table 1.6 lists pitch diameters and the over-wires measurement as a function of tooth number (which ranges from 18 to 218) and various diametral 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 basic standards, 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 by contacting the headquarters office at 1500 King Street; Suite 201; Alexandria, VA 22314 (Phone: 703-684-0211).

a = Addendum b = Dedendum c = Clearance hk = Working Depth ht = Whole Depth Pc = Circular Pitch rf = Fillet Radius t = circular Tooth Thickness φ = Pressure Angle

Figure 1.6 Extract from AGMA 201.02 (ANSI B6.1 1968) T29

TABLE 1.1 TOOTH PROPORTIONS OF BASIC RACK FOR STANDARD INVOLUTE GEAR SYSTEMS Tooth Parameter 1. 2. 3. 4. 5. 6. 7. 8.

System Sponsors Pressure Angle Addendum Dedendum Whole Depth Working Depth Clearance. Basic Circular Tooth Thickness on Pitch Line 9. Fillet Radius In Basic Rack 10. Diametral Pitch Range 11. Governing Standard: ANSI AGMA

Symbol in Rack Fig. 1.6 −− φ a b ht hk C t

14-1/2º Full Depth involute System

20º Full Depth involute System

20º Coarse-Pitch involute Spur Gears

20º Fine-Pitch involute System

ANSI & AGMA 14-1/2° 1/P 1.157/P 2.157/P 2/P 0.157/P 1 5708/P

ANSI 20° 1/P 1.157/P 2.157/P 2/P 0.157/P 1.5708/P

AGMA 20° 1.000/P 1.250/P 2.250/P 2.000/P 0250/P π/2P

ANSI & AGMA 20° 1.000/P 1.200/P + 0.002 2.200/P + 0.002 2.000/P 0.200/P + 0.002 1.5708/P

rf

1-1/3 x

1-112 X

0.300/P

not standardized

--

not specified

not specified

not specified

not specified

---

B6.1 201.02

B6.1 --

-201.02

B6.7 207.06

TABLE 1.2 PREFERRED DIAMETRAL PITCHES Class

Coarse

Pitch

1/2 1 2 4 6 8 10

Class

MediumCoarse

Pitch

12 14 16 18

Class

Pitch

Class

Pitch

Fine

20 24 32 48 64 72 80 96 120 128

Ultra-Fine

150 180 200

TABLE 1.3 BASIC GEAR DATA FOR 20° P.A. FINE-PITCH GEARS Diameter Pitch 32 48 64 72 80 96 Diameter of .0540 .0360 .0270 .0240 .0216 .0180 Measuring Wire* Circular Pitch .09817 .06545 .04909 .04363 .03927 .03272 Circular Thickness .04909 .03272 .02454 .02182 .01963 .01638 Whole Depth .0708 .0478 .0364 .0326 .0295 .0249 Addendum .0313 .0208 .0156 .0139 .0125 .0104 Dedendum .0395 .0270 .0208 .0187 .0170 .0145 clearance .0083 .0062 .0051 .0048 .0045 .0041 Note: 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. T30

120

200

.0144

.0086

.02618 .01309 .0203 .0083 .0120 .0037

.01571 .00765 .0130 .0050 .0080 .0030

TABLE 1.4 SELECTED LIST OF AGMA STANDARDS General spurs And Helicals Non-Spur

AGMA 390 AGMA 2000-A88 AGMA 201 AGMA 207 AGMA 2005-B88 AGMA 203 AGMA 374

Gear Classification Handbook Gear Classification And Inspection Handbook Tooth portions For Coarse-Pitch Involute Spur Gears Tooth Proportions For Fine-Pitch Involute Spur Gears And Helical Gears Design-Manual For Bevel Gears Fine-Pitch On-Center Face Gears For 20° Involute Spur Pinions Design 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 are available 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 the line of action. ADDENDUM (a) is the height by which a tooth projects beyond the pitch circle or pitch line; also, the radial distance between the pitch 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, in helical 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 plane containing 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, either straight 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 involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a common normal 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 pitch circles. 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 the line-of-action. Such measurements should be corrected to corresponding values a transverse pitch circles for general comparisons. (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. (Figure 1.10) 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. (Figure 1.1) 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. (Figure 1.1) 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 root circle. (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 relation between 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 radius of 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 a basic 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 pitch diameter 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 transverse plane is the involute of a circle. T32

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 action of 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 the base 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) or nearest 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 helix between 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 of rotation. 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 the imaginary 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 be determined directly from the center distance and the number of teeth by proportionality. Operating pitch diameter is the pitch diameter 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, for involute gears, it is at the intersection of the line-of-action and a straight line connecting the two gear centers. The pitch point of a tooth profile is at its intersection with the pitch circle. (Figure 1.1) PLANE OF ROTATION is any plane perpendicular to a gear axis. T33

PRESSURE ANGLE (φ), for involute teeth, is the angle between the line-of-action and a line tangent to the pitch circle at the pitch point. 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 pressure angle 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 gear tooth. (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 with low 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 plus clearance. (Figure 1.1) WORKING DEPTH (hk) is the depth of engagement of two gears; that is, the sum of their addenda. T34

T35

T36

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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 AGMA and ANSI. SYMBOL NOMENCLATURE & DEFINITION backlash, linear measure along pitch circle

a

addendum

backlash, linear measure along line-of-action

b

dedendum

aB

backlash in arc minutes

c

clearance

C

center distance

d

∆

change in center distance

Co

operating center distance

Cstd

standard center distance

hk working depth

pitch diameter

ht whole depth

Db

base circle diameter

mp contact ratio

Do

outside diameter

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

N

number of teeth, usually gear

Nc

critical number of teeth for no undercutting

rb base circle radus, pinion

Nv

virtual number of teeth for helical gear

rt

Pd

diametral pitch

ro outside radius, pinion

B BLA

D

Pdn

normal diametral pitch

pitch diameter, pinion pin diameter, for over-pins dw measurement e

n

eccentricity

number of teeth, pinion

pcn normal circular pitch r

t

pitch radius, pinion

fillet radius

tooth thickness, and for general 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

Z

mesh velocity ratio

φο operating pressure angle helix angle (Wb = base helix angle; ψ 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 called the 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 2 2Pd

(5)

and the pitch-circle dimensions are related as follows: D1 = R1 = N1 (6) D2 R2 N2 4.3 Velocity Ratio The gear ratio, or velocity ratio, can be obtained from several different parameters: Z = D1 = N1 = ω1 (7) D2 N2 ω2 The 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 pressure angle varies (slightly) as the cen distance of a gear pair is altered. The base circle is related to the pressure angle and pitch dinmeter by the equation: Db = D cos φ Db = D cos φ

where D and φ are the standard values or alternately, where D and φ are the exact operating values.

(8)

This basic formula shows that the larger the pressure angle the smaller the base circle. Thus, for standard gears, 14½° pressure angle gears have base circles much nearer to the roots of teeth than 20° gears. It is for this reason that 14 ½° gears encounter greater undercutting problems than 20° gears. This is further elaborated on in section 4.8. 4.5 Tooth Thickness This is measured along the pitch circle. For this reason it is specifically called the circular tooth thickness. This is shown in Figure 1.1. Tooth thickness is related to the pitch as follows: T = Pc = π (9) 2 2Pd

T38

The tooth thickness (T2) at a given radius, R2, from the center can be found from a known value (T1) and known pressure angle (θ1) at that radius (R1), as follows: T2 = T1 R2 - 2R2 -2R2 (inv θ2 - inv θ1) R1 where: inv θ =tan θ - θ = involute function.

(10)

To save computing time involute-function tables have been computed and are available in the references. An abridged liting is 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 the nearest 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 cos θ1 For an odd number of teeth M = D cos θ cos 90º + dw cos θ1 where the value of θ1 is obtained from inv θ1 = T + invθ + dw - π D D cos θ Ν

(11)

(12)

(13)

Tabulated values of over-pins measurements for standard gears are given in Table 1.6. This provides a rapid means for calculating 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 to yield: T = D ( π + inv θc - inv θ dw ) (14) N D cos θ where: cos θc = D cos θ 2Rc

(15)

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

(16) (17)

T39

TABLE 1.5 INVOLUTE FUNCTONS Inv θ = tan θ - θ for values of θ from 10º to 40º Degrees θ 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0 0.00180 0.00239 0.00312 0.00398 0.00488 0.00615 0.00750 0.00902 0.01076 0.01272 0.01490 0.01734 0.02006 0.02304 0.02635 0.02998 0.03394 0.03829 0.04302 0.04816 0.05375 0.05981 0.06636 0.07345 0.08110 0.08934 0.09822 0.10778 0.11806 0.12911 0.14097

Minutes 24 0.00202 0.00267 0.00344 0.00436 0.00543 0.00667 0.00809 0.00969 0.01142 0.01357 0.01585 0.01840 0.02122 0.02433 0.02776 0.03152 0.03563 0.04013 0.04503 0.05034 0.05612 0.16237 0.06913 0.07644 0.08432 0.09281 0.01096 0.11180 0.12238 0.13375 0.14595

12 0.00191 0.00253 0.00328 0.00417 0.00520 0.00640 0.00779 0.00935 0.01113 0.01314 0.01537 0.01786 0.02063 0.02368 0.02705 0.03074 0.03478 0.03920 0.04402 0.04924 0.05492 0.06108 0.06773 0.07493 0.08270 0.09106 0.10008 0.10978 0.12020 0.12141 0.14344 T40

36 0.00214 0.00281 0.00362 0.00457 0.00566 0.00694 0.00839 0.01004 0.01191 0.01400 0.01634 0.01894 0.02182 0.02499 0.02849 0.03232 0.03650 0.04108 0.04606 0.05146 0.05733 0.06368 0.07055 0.07797 0.08597 0.09459 0.10388 0.11386 0.12459 0.13612 0.14850

48 0.00226 0.00296 0.00379 0.00476 0.00590 0.00721 0.00870 0.01039 0.01231 0.01444 0.01683 0.01949 0.02242 0.02566 0.02922 0.03313 0.03739 0.04204 0.04710 0.05260 0.05856 0.06502 0.07199 0.07952 0.08765 0.09639 0.10582 0.11594 0.12683 0.13853 0.15108

T41

T42

f

T43

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 come into engagement. It is desired to have as much overlap as possible. A measure of this overlapping action is the contact ratio. This is 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 determined from the intersection of the length-of-action arid the outside radii. The ratio of the length-of-action to the base pitch is determined 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 time one 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 ratio generally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pair or 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 area of the mating tooth. This results in the typical undercut tooth, shown in Figure 1.17. The undercut not only weakens the tooth with 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 the addenda of the gear teeth digging into the mating-pinion tooth flanks. Since addenda are standardized by a fixed ratio (1/Pd) the interference condition becomes more severe as the number of teeth on the gear increases. The limit is reached when the gear becomes a rack. This is a realistic case since the hob is a rack-type cutter. The result is that standard gears with T44

tooth numbers below a critical value are automatically undercut in the generating process. The limiting number of teeth in a gear meshing 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 the value of Nc is 32, and for 20° it is 18. Thus, 200 pressure angle gears with low numbers of teeth have the advantage of much less 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 of action. The seventy of these faults depends upon how far below N, the tooth number is. Undercutting for the first few numbers is small and in many applications its adverse effects can be neglected.

For very small numbers of teeth, such as ten and smaller, and for high-precision applications, undercutting should be avoided. This is achieved by pinion enlargement (or correction as often termed), wherein the pinion teeth, still generated with a standard cutter, are shifted radially ourward to form a full involute tooth free of undercut The tooth is enlarged both radially and circumferentially. Comparison of a tooth form before and after enlargement is shown 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 this writing. References 1, 3, 5 and 6 offer additional information. For measurement and inspection Figure 1.18 Comparison of such gears, in particular, consult reference 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 and operate 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. The general purpose of backlash is to prevent gears from jamming and making contact on both sides of their teeth simultaneously. A small amount of backlash is also desirable to provide for lubricant space and differential expansion between the gear components and the housing. Any error in machining which tends to increase the possibility of jamming makes it necessary to increase the amount of backlash by at least as much as the errors. Consequently, the smaller the amount of backlash, the more accurate must be the machining of the gears. Runout of both gears, errors in profile, pitch, tooth thickness, helix angle and center distance — all are factors to consider in the specification of the amount of backlash. On the other hand, excessive backlash is objectionable, particularly if the drive is frequently reversing or if there is an overrunning load. The amount of backlash must not be 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). This decrease must almost always be greater than the desired backlash because of T45

the errors in manufacturing and assembling. Since the amount of the decrease in tooth thickness depends upon the accuracy of machining, 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 are exceptions. For example, on pinions having very low numbers of teeth, it is desirable to provide all of the allowance on the mating 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 the theoretically standard depth. Further, it is true that any increase or decrease in center distance of two gears in any mesh will cause 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 the determination of backlash in gear trains, it is necessary to sum the backlash of each mated gear pair. However, to obtain the total backlash for a series of meshes it is necessary to take into account the gear ratio of each mesh relative to a chosen reference 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 are two basic ways in which backlash arises: Tooth thickness is below the zero-backlash value; and the operating center distance is greater 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 backlash introduced 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 gears Tact = actual tooth thickness T46

When the center distance is increased by a relatively small amount, ∆C, a backlash space develops between mating teeth, as in Figure 1.21. The relationship between center distance increase and linear backlash, 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 distance φ = pressure angle Hence, an approximate relationship between center distance change and change in backlash is:

∆C= 1.933 ∆B for 14½° pressure-angle gears ∆C= 1.374 ∆B for 20° pressure-angle gears

(22b) (22c) T47

Although these are approximate relationships they are adequate for most uses. Their derivation, limitations, and correction factors 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 of feeler gages for measurement is impractical, whereas an indicator at the pitch line gives a direct measure. The two linear backlashes 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 seen from 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 of different pitch diameters, the linear backlash of the measure converts to different angular values for each gear. Thus, an angular backlash 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. T48

To Obtain

TABLE 1.7 SUMMARY OF FUNDAMENTALS SPUR GEARS From Known

Pitch diameter

Number of teeth and pitch

Circular Pitch

Diametral pitch or number of teeth and pitch diameter

Diametral pitch

Circular pitch or number of teeth and pitch diameter

Number of teeth

Pitch and pitch diameter

Outside diameter

Pitch and pitch diameter or pitch and number of teeth

Root diameter

Pitch diameter and dedendum

Base circle diameter

Pitch diameter and pressure angle

Base pitch

Circular pitch and pressure angle

Tooth thickness at standard pitch diameter

Circular pitch

Addendum

Diametral pitch

Center distance

Pitch diameters Or number of teeth and pitch

Contact ratio

Outside radii, base radii, center distance and pressure angle

Backlash (linear) Backlash (linear) Backlash (linear) along line of acvon

=π= π Pd

D N

π=N Pc D N =DPd = D Pc

Pd =

Do =D + 2 = N+2 Pd Pd DR = D - 2b Db=D cos φ Pb = Pc cos φ Tstd = Pc = πD 2 2N a= 1 Pd C=D1+D2=N1+N2=Pc(N1+N2) 2 2Pd 2π

From change in center distance From change in tooth thickness Linear backlash along pitch cirde

BLA = B cos φ

Linear backlash

Minimum number of teeth for no undercutting

Pressure angle

Clearance Working depth Pressure angle ( standard ) Operating pressure angle

Pc

mp = (Ro²-Rb²)½+(ro²-rb²)½-C sin φ Pc cos φ B = 2 (∆C) tan φ B = ∆T

Backlash, angular

Dedendum

Symbol and Formula D = N = N·Pc Pd π

Pitch diameter and root diameter ( DR ) Addendum and dedendum Addendum Base circle diameter and pitch diameter Actual operating pitch diameter and base circle diameter T49

aB = 6880 B (arc minutes) D N= 2 sin² φ b = ½(D-DR) c=b-a hk = 2a φ =cos-1 Db/D φ =cos-1 Db/D'

TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALS HELICAL GEARING To Obtain

From Known

Symbol and Formula

Normal circular pitch

Transverse circular pitch

Pcn = Pc cos ψ

Normal diametral pitch

Transverse diametral pitch

Axial pitch

Circular pitches

Pa = Pc cot ψ = Pcn sin ψ

Normal pressure angle

Transverse pressure angle

tan φn = tan φ cos ψ

Pitch diameter

Number of teeth and pitch

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 pitch diameters

a = 0.5 ( Do - D ) = 1 Pd

Dedendum

Pitch diameter and root diameter (DR)

b = 0.5 ( D - DR )

Clearance

Addendum and dedendum

c = b-a

Working depth

Addendum

hk = 2a

Transverse pressure angle

Base circle diameter and pitch circle diameter

cos φt = Db / D

Pitch helix angle

Number of teeth, normal diametral pitch and pitch diameter

cos ψ = N Pn D

Lead

Pitch diameter and pitch helix angle

L = π D cos ψ

To Obtain

Symbols

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

Length of action

ZA

ZA = (C² - (Rb+rb)²)½ (maximum) ZA = (Ro²-Rb²)½ (ro²-rb²-C sin φr)½

Start of active profile

SAP

SAPp = -(Ro²-Rb²)½ SAPg = Zmax-(ro²-rb²)½

Contact ratio

Rc

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

Pdn =

D =

Pd cos ψ

N = Pd

N Pdn cos ψ

INVOLUTE GEAR PAIRS

T50

To Obtain

TABLE 1.7 CONT. - SUMMARY OF FUNDAMENTALS WORM MESHES From Known

Pitch diameter of worm

Number of teeth and pitch

Pitch diameter of worm gear

Number of teeth and pitch

Lead angle

Pitch, diameter, teeth

Lead of worm

Number of teeth and pitch

Normal circular pitch

Transverse pitch and lead angle

Center distance

Pitch diameters

Center distance

Pitch, lead angle, teeth

Velocity ratio

Number of teeth

To Obtain

BEVEL GEARING From Known

Symbol and Formula dw = nw Pcn p sin λ Dg = Ng Pcn π cos λ -1 λ = tan nw = sin-1 nw Pcn Pddw pdw L = nwpc = nw pcn cos λ Pcn = Pc cos λ C = dw + Dg 2 C = Pcn [ Ng + nw ] 2π cos λ sin λ Z = Ng nw Symbol and Formula Z = N1 N2

Velocity ratio

Number of teeth

Velocity ratio

Pitch diameters

Z = D1 D2

Velocity ratio

Pitch angles

Z = sin γ1 sin γ2

Shaft angle

Pitch angles

Σ = γ1 + γ2

T51

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 the spur 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 around tooth base support. 2. contact ratio is increased due to the axial tooth overlap. Helical gears thus tend 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 tight angles.

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 three dimensions 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 taut string of the spur gear in Figure 12. On the plane there is a straight line AB, which when wrapped on the base cylinder has a helical trace AoBo. As the taut plane is unwrapped any point on the line AB can be visualized as tracing an involute from the base cylinder. Thus, there is an infinite series of involutes generated by line AB, all alike, but displaced in phase along a helix on the base cylinder. Again a concept analogous to the spur-gear tooth development is to imagine the taut plane being wound from one base cylinder on to another as the base cylinders rotate in opposite directions. The result is the generation of a pair of conjugate helical involutes. If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, a complete involute helicoid tooth is formed.

T52

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 the outside radnjs, the helix angle, w~ is detned as the angle between the tangent to the helicoidal tooth at the intersection of the pitch 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. In addition there is an axial pitch. These are defined and related as follows: Referring to Figure 1.25, the two circular pitches are related 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 being equal 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 cos ψ

(26)

The axial pitch of a helical gear is the distance between corresponding points of adjacent teeth measured parallel to the gears axis—see Figure 1.26. Axial pitch, p1. is related to circular pitch by the expressions: Pa = Pc cot ψ = Pcn = axial pitch sin ψ

(27)

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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, looking at one tooth, there is a resemblance to an involute tooth of a pitch corresponding to the normal pitch. However, the shape of the tooth 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 tooth form spur gear is given in Figure 1.27. The result of the transposed geometry is an equivalent number of teeth given as: NV = N (28) cos³ψ This equivalent number is also called a virtual number because this spur gear is imaginary. The value of this number is its use in determining helical tooth 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 individual gear. For the helical gear there is a normal pressure angle as well as the usual pressure angle in the plane of rotation. Figure 1.28 shows 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 as spur gears. However, this means the normal pitch is the common denominator, and usually is taken as a standard value. Since the true involute features are in the transverse plane, they will differ from the standard normal values. Hence, there is a real need for relating parameters in the two reference planes. T54

f

5.7 Helical Tooth Proportions These follow the same standards as those for spur gears. Addendum, dedendum, whole depth and clearance are the same regardless of whothor measured in tho piano of rotation er the normal piano. Pressure angle and pitch are usually specified as standard 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, the helix 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 normal pitch 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 to standard spur gears. Further, by manipulating the helix angle (ψ) the center distance can be adjusted over a wide range of values. Conversely, it is possible a. to compensate for significant center distance changes (or erçors) without changing the speed ratio between parallel geared shafts; and b. to alter the speed ratio between parallel geared shafts without changing center distance by manipulating helix angle along with tooth 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 is the sum of the transverse contact ratio, calculated in the same manner as for spur gears (equation 18), and a term involving the axial pitch. (mp)total = (mp)trans + (mp)axial (32) where T55

New Page 4 (mp)trans = value per equation 18 (mp)axial = F = F tan ψ = F sin ψ Pa Pc Pcn and F = face width of gear. 5.8.5 Involute interference — Helical gears cut with standard normal pressure angles can have considerably higher pressure angles in the plane of rotation (see equation 29), depending on the helix angle. Therefore, referring to equation 19, the minimum number of teeth without undercutting can be significantly reduced and helical gears having very low tooth numbers without undercutting 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 can be 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 shaft angle: ψ1 + ψ2 = θ (33) where: ψ1, ψ2 = the respective helix angles of the two gears θ = 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 distance becomes: 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 be altered 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 from the ratio of pitch diameters, since these can be altered by juggling of helix angles. The speed ratio can be determined only from the number of teeth as follows: velocity ratio Z = N1 (35) N2 or 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 loads gear teeth and bearings and must accordingly be considered in the housing and bearing design. In some special instrument designs this thrust load can be utilized to actuate face clutches, provide a friction drag, or other special purpose. The magnitude of 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 herringbone gears (Figure 1.30a) is worth considering.

T57

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 the tooth is of simple V form. Racks can be both spur and helical. A rack will mesh with all gears of the same pitch. Backlash is computed by the same formula as for gear pairs, equation 22. However, the pressure angle and the gears pitch radius remain constant regardless of changes in the relative position of the gear and rack. Only the pitch line shifts accordingly as the gear center 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 mates with an ordinary external gear. This offers considerable versatility in the design of planetary gear trains and miscellaneous instrument 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 contact If instead, one of the pitch circles rolls on the inside of the ether, it forms the basis of internal gearing. In addition, the larger gear must have the material forming the teeth on the convex side of the involute profile, such that the internal gear is an inverse of 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 spur gears there is a taut generating string that winds and unwinds between the base circles. However, in this case the string does not cross 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. T58

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 the addendum of the gear is reduced to avoid trimming of the teeth in the fabrication process.

T59

Tooth thickness of the internal gear can be calculated with equations 9 and 20, but one must remember that the tooth and space thicknesses are reversed, (see Figure 1.35). Also, in using equation 10 to calculate tooth thickness at various radii, (see Figure 1.36), it is the tooth space that is calculated and the internal gear tooth thickness is obtained by a subtraction from the circular pitch 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 this time 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 T60

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 are characterized by high velocity ratios. Also, they offer the advantage of the higher load capacity 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. A central 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 are only 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 exact duplicate of that used in generating the wormgear. To increase the length of action the wormgear is made of a throated shape to wrap around the Worm. T61

8.2 Worm Tooth Proportions Worm tooth dimensions, such as addendum, dedendum, pressure angle, etc., follow the same standards as those for spur and helical gears. The standard values apply to the central section of the mesh, (see Figure 1.40a). A high pressure angle is favored and 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 one continuous tooth or thread. For slightly smaller angles them can be two, three, or even more threads. Thus, a worm is characterized 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 following defines the geometry of worm mesh components. T62

8.4.1 Pitch Diameters, Lead snd Lead Angle Pitch diameter of worm = dw = nw Pcn π sin λ

(40)

Pitch diameter of wormgear = Dg = Ng Pcn π cos λ

(41)

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 2

2π

cos λ

+

nw ] sin λ

(42)

T63

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 the ratio of tooth numbers: velocity ratio = Z = no. teeth in worm gear = no. threads in worm

Ng

(43)

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

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 a sphere. Pitch diameters of mating bevel gears belong to frusta of cones, as shown in Figure 1.42. In the full development on the surface 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 rack of spur gearing, and is the basic tool for generating bevel gears. However, for practical reasons the tooth form is not that of a spherical involute, and instead, the crown gear profile assumes a slightly simplified form. Although the deviation from a true spherical involute is minor, it results in a line of action having a figure-S trace in its extreme extension, see Figure 1.44. This shape gives rise to the name "octoid" for the tooth form of modem bevel gears.

T64

T65

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, the pressure angle of all standard design bevel gears is limited to 200. Pinions with a small number of teeth are enlarged automatically 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). Since the narrow end of the teeth (toe) vanishes at the pitch apex (center of reference generating sphere) there is a practical limit to the 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 = N2

sin γ1 D2

(44) sin γ2

where:

γ = pitch angle (Figure 1.45)

T66

In the simplest design the tooth elements are straight radial, converging at the cone apex. However, it is possible to have the teeth curve along a spiral as they converge on the cone apex, resulting in greater tooth overlap, analogous to the overlapping action 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 bevel generators are of the Ceniflex form which gives an almost imperceptible convexity to the tooth surfaces. Older machines produce true straight elements.. See Figure 1 .46a. Straight bevel gears are the simplest and most widely used type of bevel gear for the transmission 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 in their lengthwise direction. The teeth, therefore, tolerate small amounts of misalignment in the assembly of the gears and some displacement of the gears under load without concentrating the tooth contact at the ends of the teeth. As a result, these gears are 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 the other. Imagine cutting a straight bevel into an infinite number of short face-width sections, angularly displace one relative to the other, and one has a spiral bevel gear. Well-designed spiral bevels have two or more teeth in contact at all times. The overlapping 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 of the 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 same grinding 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 be used in the place of of straight bevels without mounting changes. Zerol bevels are widely employed in the aircraft industry, where ground-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. T67

10.0 GEAR TYPE EVALUATION The choice of gear type is dependent upon a number of considerations involving physical space and shaft arrangement, load, gear ratio, and desired precision or quality level. A general guide is to choose the simplest gear type that can accomplish the objectives. Spur gears are the first choice if they can do the job, as they are the easiest to make. That means they are the least expensive 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 of undercutting in small tooth number pinions; and helicals can be designed to neatly span non-standard center distances. Crossed helicals are an acceptable skew shaft drive only if the loads are small. Worm gearing and bevels offer right angle drives for skew and 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 the desired 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 special features, and the fundamental and essential gear data. This gear data can be efficiently and consistently specified on the gear drawing in a standardized block format. The format varies in accordance with gear type. A typical data block for standard fine-pitch spur gears is given in Figure 1.47. Formats for coarse pitch gears, helical gears and other gear types are given in detail in the appendix of Ref. 5. T68

Type Spur

Helical

Crossed - helical

Internal spur

Bevel

Worm mesh

Specials (face, Spiroid, Helicon, Beveloid)

TABLE 1.8 SUMMARY AND EVALUATION OF GEAR TYPES Precision Comments Features Applications Rating Regarding Precision excellent Parallel shafting Applicable to all Simplest tooth elements offering max High speeds types of trains and imum precision. First choice, recom and loads a wide range of mended for all gear meshes, except Highest efficiency velocity ratios, where very high speeds and loads or special features of other types, such as right - angle drive, cannot be avoided. good Parallel shafting Most applicable to Equivalent quality to spurs except for Very high speeds high speeds and complication of helix angle. and loads loads; also used Recommended for all high- speed Efficiency slightly wherever spurs and high-load meshes. Axial thrust less than spur mesh are used. component must be accommodated. poor Skewed shafting Relatively low To be avoided for precision meshes. Point contact velocity ratio; Point contact limits capacity and precision. High sliding low speeds and Suitable for right - angle drives if low speeds light loads only. A less expensive substitute for bevel light load. Any angle gears. Good lubrication essential because skew shafts. of point contact and high sliding action. fair Parallel shafts Internal drives Not recommended for precision meshes High speeds requiring high because of design, fabricabon, and High loads speeds and high inspection limitations. Should only be loads; offers low used when internal feature is necessary. sliding and high stress loading; good for high capacity, long tie. Used In planetary gears to produce large reduction ratios. fair to Intersecting Suitable for 1:1 and Good choice for right-angle drive, good shafts higher velocity particularly low ratios. However, High speeds ratios and for rightcomptcaled tooth form and fabrication High loads angle mashes limits achievement of precision. (and other angles) Should be located at one of the less critical meshes of the train. fair to Right - angle High velocity ratio Worm can be made to high precision, good skew shafts Angular meshes but worm gear has inherent limitations. High velocity High loads To be considered for average precision ratio meshes, but can be of high precision with High speeds care. Best choice for combination high and loads velocity ratio and right- angle drive. High Low efficiency sliding requires excellent lubrication. Most designs nonreversible poor to Intersecting and Special cases To be avoided as precision meshes. Sig fair skew shafts nificant nonconjugate action with depart Modest speeds ure from nominal center distance and and loads shaft angles. Fabrication requires special equipment and inspection is limited. T69

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 classes have been established by AGMA in Gear Classification and Inspection Handbook (ANSI/AGMA 2000 - A88 ). This information is reproduced in Table 1.9. The previous issue of this specification, (390.02), offered a more detailed table of backlash allowance and tolerance which is still a useful design guide. See Table 1.10. Although no longer part of current AGMA standards, it is consistent 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 gear pair due to tooth thickness tolerance will be the sum of two values from Table 1.9, Most often the same tolerance is applied to each 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 is known by several names: transmission linearity, angular transmission accuracy, and index accuracy. Theoretically, involute gears will 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 is transmitted to the instrument or machine involved. T70

The single most important criterion of the above position errors is the total composite error of the gear (TCE). This is defined simply as the maximum variation in center distance as the gear is rolled, intimately meshed with a master gear, on a variable-center-distance fixture. The device has one floating center, and as the gears are rolled any eccentricity, tooth-to-tooth variation, and profile deviation results in center distance variation. This variation can be measured and plotted, as shown in Figure 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 specify both 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 2

θ , where θ = angular position of the gear

(45)

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 Tolerance Codes Quality Diametral Number Pitch A B C

3 and 4

5

6

7 thru 15

0.5

0.074

1.2

0.031

2.0

0.019

3.2

0.012

5.0

0.0075

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

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

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 12.0 20.0 32.0 50.0 80.0 120.0

0.005 0.003 0.0024 0.0016 0.0012 0.0008 0.00067

0.0025 0.0018 0.0012 0.0008 0.0006 0.00045 0.00034

0.00125 0.0009 0.0006 0.00043 0.0003 0.00022 0.00017

D

0.00063 0.00044 0.0003 0.0002 0.00014 0.00011 0.00009

*Extracted from AGMA Standard 2000-ABB, Gear Classification and Inspection Handbook Tolerances and Measuring Methods for Unassembled Spur and Helical Gears, with permmision of the publisher, American Gear Manufacturers Association, 1500 King Street, Alexenderia , Virginia 22314 T71

Equation 45 yields a linear position error measured in inches along the pitch circle. If an angular transmission error, aET, is desired it is necessary to divide by the pitch radius of the gear. Thus: (46) aET = Etc sin q (radians) = 3440 Etc sin θ (arc minutes) 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 both gears. 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) R1,2

(47)

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.

Center Distance ( Inches ) Up to 5 Over 5 to 10 . Over 10 to 20 Over 20 to 30 Over 30 to 40 Over 40 to 50 Over 50 to 80 Over 80 to 100 Over 100 to 120

Backlash Designation

A

B C D E

**TABLE 1.10 AGMA BACKLASH ALLOWANCE AND TOLERANCE COARSE- PITCH GEARS Normal Dlametral Pitches 0.5 - 1.99 2 - 3.49 3.5 - 5.99 6 - 9.99

0.040 - .060 0.050-.070 0.060 - .080 0.070 - .095 0.080 - .110

Normal Diametral Pitch Range 20 thru 45 46 thru 70 71 thru 90 91 thru 200 20 thru 60 61 thru 120 121 thru 200 20 thru 60 61 thru 120 121 thru 200 20 thru 60 61 thru 120 121 thru 200 20 thru 60 61 thru 120 121 thru 200

0.030-.040 0.035 - .045 0.040 - .055 0.045 - .065 0.050 - .080

0.020 -030 0.025 -.030 0.030 - .040 0.035-.O50 0.040 - .060

FINE - PITCH GEARS Tooth Thinning to Obtain Backlash Allowance Tolerance ( per gear) ( per gear ) .002 0 to .002 .0015 0 to .002 .001 0 to .00175 . 00075 0 to .00075 .001 0 to .001 .00075 0 to .00075 .0005 0 to .0005 .0005 0 to .0005 .00035 0 to .0004 . 0002 0 to .0003 .00025 0 to .00025 .0002 0 to .0002 . 0001 0 to .0001 0 to .00025 0 to .0002 Zero 0 to.0001

0.010 - .020 0.015 - .025 0.020 - .030 0.025- .040 0.030 - .040

Resulting Approximate Backlash (par mesh) Normal Plane .004 to .008 .003 to .007 .002 to .0055 . 0015 to .003 .002 to .004 .0015 to .003 001 to .002 .001 to .002 .0007 to .00013 .0004 to .001 .0005 to .001 .0004 to .0008 . 0002 to .0004 0 to.0005 0 to .0004 0 to.0002

*See Reference 5 for the case of considering phase angles. **Extracted from AGMA Gear Classification Manual AGMA 390.02, September 1964 T72

10 - 19.99 0.005 - .015 0.010 - 020 0.020 - .030

11.4 AGMA Quality Classes Using criteria that are indicators and measures of gear quality, the AGMA has established a convenient standardization that forms a continuous spectrum of quality classes ranging from the crudest to the most precise gears. For all gears, coarse and fine pitches, them am 13 classes numbered 3 through 15. AGMA Gear Classification and Inspection Handbook (ANSI/AGMA 2000-A88) specifically defines various gear quality parameters for these 13 classes. This includes tolerance ranges for runout, pitch, profile, lead, total composite error, and tooth-to-tooth composite error. These values are for spur and helical gearing. In addition, them are separate table values for rack and pinions, bevel and hypoid gears, and fine pitch worm gearing. Also presented are class tolerances of key parameters for spur and 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 to all those who wish to obtain additional information and tables related to this subject. Many designers, however, may not have access to the tables published in previous AGMA 390.02 and AGMA 236.04 standards. For this reason, Tables 1.10A and 1 .10B are presented. Furthermore, as a result of increased international trade and the influx of metric gears, it is useful to compare different national gear standard values. Such a comparison giving approximate equivalence of values is given in Table 1.10 C. T73

AGMA Quality No. 5

6

7

8

9

10

11

12

13

14

15

16

TABLE 1.10A FINE-PITCH GEAR TOLERANCES FOR AGMA QUALITY CLASSES No. of Teeth Diametral Tooth-to-Teeth And Pitch Composite Pitch Diameter Range Tolerance Up to 20 teeth inclusive 20 to 80 0.0037 Over 20 teeth, up to 1.999” 20 to 32 0.0027 Over 20 teeth. 2” to 3.999” 20 to 24 0.0027 Over 20 teeth, 4” & over 20 to 24 0.0027 Up to 20 teeth inclusive 20 to 200 0.0027 Over 20 teeth,up to 1.999” 20 to48 0.0019 Over 20 teeth, 2” to 3.999” 20 to 32 0.0019 Over 20 teeth, 4" & over 20 to 24 0.0019 Up to 20 teeth Inclusive 20 to 200 0.0019 Over 20 teeth, up to l.999" 20 to200 0.0014 Over 20 teeth, 2" to 3.999” 20 to 48 0.0014 Over 20 teeth, 4” & over 20 to 40 0.0014 Up to 20 teeth inclusive 20 to 200 0.0014 Over 20 teeth, up to l.999” 20 to200 0.0010 Over 20 teeth, 2” to 3.999” 20 to 100 0.0010 Over 20 teeth,4” & over 20 to64 0.0010 Up to 20 teeth inclusive 20 to 200 0.0010 Over 20 teeth, up to 1.999” 20 to 200 0.0007 Over 20 teeth, 2” to 3.999” 20 to 200 0.0007 Over 20 teeth, 4” & over 20 to 120 0.0007 Up to 20 teeth inclusive 20 to 200 0.0007 Over 20 teeth, up to l.999” 20 to200 0.0005 Over 20 teeth, 2” to 3.999” 20 to 200 0.0005 Over 20 teeth,4” &over 20 to200 0.0005 Up to 20 teeth inclusive 20 to 200 0.0005 Over 20 teeth, up to 1.999” 20 to 200 0.0004 Over 20 teeth, 2” to 3.999” 20 to 200 0.0004 Over 20 teeth,4” & over 20 to200 0.0004 Up to 20 teeth inclusive 20 to 200 0.0004 Over 20 teeth, up to 1.999” 20 to 200 0.0003 Over 20 teeth, 2” to 3.999” 20 to 200 0.0003 Over 20 teeth, 4” &over 20 to 200 0.0003 Up to 20 teeth inclusive 20 to 200 0.0003 Over 20 teeth,upto 1.999” 20 to200 0.0002 Over 20 teeth, 2” to 3.999” 20 to 200 0.0002 Over 20 teeth, 4” & over 20 to 200 0.0002 Up to 20 teeth inclusive 20 to 200 0.00019 Over 20 teeth, up to 1.999” 20 to 200 0.00014 Over 20 teeth, 2” to 3.999” 20 to 200 0.00014 Over20 teeth,4” & over 20 to200 0.00014 Up to 20 teeth inclusive 20 to 200 0.00014 Over 20 teeth, up to1.999” 20 to 200 0.00010 Over 20 teeth, 2’ to 3.999” 20 to 200 0.00010 Over 20 teeth, 4” & over 20 to 200 0.00010 Up to 20 teeth inclusive 20 to 200 0.00010 Over 20 teeth,upto 1.999” 20 to200 0.00007 Over 20 teeth, 2” to 3.999” 20 to 200 0.00007 Over 20 teeth, 4” & over 20 to 200 0.00007

*From AGMA “Gear Classification Manual for Spur, Helical and Heningbone Gears.” AGMA 390.02, Sept. 1964. T74

Total Composite Tolerance 0.0052 0.0052 0.0061 0.0072 0.0037 0.0037 0.0044 0.0052 0.0027 0.0027 0.0032 0.0037 0.0019 0.0019 0.0023 0.0027 0.0014 0.0014 0.0016 0.0019 0.0010 0.0010 0.0012 0.0014 0.0007 0.0007 0.0009 0.0010 0.0005 0.0005 0.0006 0.0007 0.0004 0.0004 0.0004 0.0005 0.00027 0.00027 0.00032 0.00037 0.00019 0.00019 0.00023 0.00027 0.00014 0.00014 0.00016 0.00019

TABLE 1.10B COMPARISON OF NEW AND PREVIOUS FINE-PITCH AGMA QUALITY CLASSES* PREVIOUS FINE-PITCH SYSTEM , AGMA 236.04 FINE-PITCH SYSTEM, AGMA 390.02 AGMA Tooth-to-Tooth Total AGMA Tooth-to-Tooth Total Quality Composite Composite Quality Composite Composite No. Error Error No. (Error) Tolerance (Error) Tolerance 0.0027 or 0.0052 or Commercial 1 0.0020 0.0060 5 or 6 0.0019 0.0037 0.0019 or 0.0037 or Commercial 2 0.0015 0.0040 6 or 7 0.0014 0.0027 Commercial 3 0.0010 0.0020 8 0.0010 0.0019 Commercial 4 0.0007 0.0015 9 0.0007 0.0014 0.0005 or 0.0010 or Precision 1 0.0004 0.0010 10 or 11 0.0004 0.0007 Precision 2 0.0003 0.0005 12 0.0003 0.0005 0.0002 or 0.0004 or Precision 3 0.0002 0.00025 13 or 14 0.00014 0.00027 * Extracted from AGMA Gear Classification Manual AGMA 390.02, Sept. 1964. For more current standard, consult ANSI/AGMA 2000-A88, March 1988.

International ISO 4 5 6 7 8 9

TABLE 1.10C QUAUTY NUMBER COMPARISON OF DIFFERENT NATIONAL GEAR STANDARDS W. Germany Japan DIN JIS 4 0 5 1 6 2 7 3 8 4 9 5 T75

U.S.A. AGMA 13 12 11 10 9 8

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 gear parameters, 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 be grouped according to the following categories: I.

II.

III.

IV.

V.

Design backlash allowance 1. Gear size allowance — any reduction of tooth thickness (or testing radius) below nominal value 2. Center distance — any increase in center distance above nominal value Major tolerances 1. Gear size tolerance (tooth thickness or testing radius) 2. Center distance tolerance Gear center shift due to secondary sources 1. Fixed bearing eccentricities a. Outer-race eccentricity of ball bearing b. Inside-diameter and outside-diameter runout of sleeve bearing 2. 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 rotation 1. Total composite error 2. Clearance between gear bore and shaft 3. Runout at point of gear mounting 4. Eccentricity of rotating race of ball beating Miscellaneous sources 1. Dimensional changes due to thermal expansion or contraction 2. 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 backlash sources, 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 shift due to eccentricities, must be converted to backlash measured along the pitch circle in accordance with equation 22a prior to addition of sources such as tooth- thickness tolerances, etc. Also, note that sources of backlash can be divided into two categories: those of constant magnitude; and those the magnitude of which varies with gear rotation. The latter sources are associated with runout. Thus, backlash can be expressed as follows: T76

B = Bc + Bv

(49)

where: Bc = constant backlash Bv = 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.

Position error in the individual gears 1. Total composite error a. Single-cycle errors (pitch-line runout) b. High-frequency tooth-to-tooth composite errors (TTCE)

II.

Installation errors 1. 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 assembly 2. 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 total transmission error for each mesh is proportional to the sum of all eccentricity error sources: a(ET)mesh = ± 3440 ΣEi (arc minutes) R

(50)

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 gear performance. For example, in a servomotor gear train, backlash may be very important, whereas position error may be immaterial. Alternatively, in a unidirectional position sensor gear-train, backlash may be of little concern, while transmission error might 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 T77

errors. However, this combination is not necessarily simple since many of the transmission-error sources are identical to those associated with variable backlash. In addition, transmission error varies between maximum lead and lag values. Details of the integration 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 ) 2

(51)

where: Bc = backlash constant with rotation ET = 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 of narrowing all allowances and tolerances on sources is effective. Accordingly, precision gear qualities are specified, particularly with regard to testing radius (tooth thickness) and TCE. However, there are practical limitations since cost increases exponentially with precision. Some method of circumventing extremes of precision must be used. An alternate means of controlling backlash is to use adjustable centers or to spring-load the gears by one of several different designs. In this regard, the spring-loaded scissor gear has particular merit since all backlash is continually eliminated. However, it is limited to low torque applications. Consult Reference 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. The most 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 pitchline runouts to provide cancellation of error. However, besides being costly and not foolproof, this method is very limited since it requires 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 tooth strength and the latter as durability. Strength is determined in terms of tooth-beam stresses for both static and dynamic conditions, following well established formulas and procedures. Durability ratings are evaluated in terms of surface stresses including the influence not only of dynamics, but also of material combinations, lubrication and a considerable number of empirically derived factors. 13.1 Bending Tooth Strength Tooth loading produces stresses that can ultimately result in tooth breakage. This is not a prevalent T78

type of failure because mechanical properties of gear materials are well known, and the design equations are sufficiently accurate. 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 the tip to the base, as shown in Figure 1.49. The load can be resolved into a tangential force, W1, causing bending, and a normal force, 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 now reknowned 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 take into account the dynamics of meshing teeth. In that regard, later investigators have modified and improved the original Lewis equation. 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 and is satisfactory for commercial gears at pitch-line velocities up to 1,500 fpm: Wt

= SFY Pd

(

600 ) 600+V

where: Wt = transmitted load (52) S = maximum bending tooth stress, at the root outer fibers. F = face width of gear Y = Lewis factor Pd= diametral pitch V = 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 standard teeth 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 be evaluated from an S-N curve, modified Goodman diagram, Soderberg line, or equivalent data. Reference 6 contains helpful information on fatigue stress analysis. Table 1.12 gives safe stresses for a number of engineering materials. An estimate for the maximum allowable bending stress, S in equation 52, can then be obtained by multiplying the stress given in Table 1.12 by two factors: a service factor given in Table 1.13 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. For acceptable designs, Wb>= Wt The tangentially transmitted load is calculated from the transmitted horsepower as follows: Wt = 126,000 Pt DNr

where:

Pt = transmitted horsepower Nr= gear speed in revolutions per minute D = gear pitch diameter T79

(53)

T80

D190 Catalog

TABLE 1.11 LEWIS Y FACTORS No. of Teeth 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 32 34

Full Depth Involute 14½o

20o

0.176 0.192 0.210 0.223 0.236 0.245 0.255 0.264 0.270 0.277 0.283 0.292 0.302 0.308 0.314 0.318 0.322 0.325

0.201 0.226 0.245 0.264 0.276 0.289 0.295 0.302 0.308 0.314 0.320 0.330 0.337 0.344 0.352 0.358 0.364 0.370

No. of Teeth

Full Depth Involute 14½o

20o

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

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

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

Rack

0390

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) Safe Stress Ultimate Strength sw sw 8,000 24.000 10,000 30,000 12,000 36,000 20,000 65,000

Material* Cast iron, ordinary Cast iron, good grade Semisteel Cast steel Forged carbon steel : SAE 1020 casehardened SAE 1030 not treated 1035 not treated 1040 nat treated 1045 not treated 1045 hardened 1050 hardened Alloy steels: Ni, SAE 2320, casehardened Cr-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 Bakelite Micarta

Yield Stress sw

18,000

55,000

30.000

20,000 23,000 25,000 30.000 30,000 35.000

60,000 70,000 80,000 90.000 95,000 100,000

33,000 38,000 45,000 50.000 60,000 60,000

50,000 65,000 67,500 20.000 10,000 12,000 15,000 6,000 6,000 6,000 6,000

100,000 120,000 130,000 60.000 30,000 36,000 65.000

80,000 100,000 110,000 30,000 15,000 20,000 25,000

18,000 bending 18,000 bending

* 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.

T81

36,000

D190 Catalog

TABLE 11.3 SERVICE FACTORS Type of Load Steady Light shock medium shock Heavy shock

8-10 hr per day 1.00 0.80 0.65 0.55

Type of Service 24 hr Intermittent, 3 hr per day per day 0.80 1.25 0.65 1.00 0.55 0.80 0.50 0.65

TABLE 1.14 LUBRICATION FACTORS Type of Lubrication Submerged in oil Oil drip Grease Intermittent Lubrication

Lubrication Factor 1.00 0.80 0.65 0.50

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 factor is 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 is shared by a second pair of teeth. The following tables are useful in determining gear load ratings: Table 1.15 Tables 1.16 & 1.17 Table 1.18 Tables 1.19 & 1.20

: Ratings for steel spur gears : Ratings for small-pitch spur gears : Ratings for hardened steel helical gears : Ratings of worms and worm gears.

13.2 Dynamic Strength Equations 52 and 54 give adequate results for gear meshes that are in a static situation. When gears are in action, however, tooth loading is greater than the static value due to dynamic effects. In a gear system, dynamic forces arise from a combination of the masses involved, their elasticity and the forcing function representing the prescribed motion. Inaccuracies in gear-tooth profiles 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 of pitch-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 that safety represents the dynamic condition. A T82

NOTE: All charts are based on 30,000 p.s.i. yield stress. For other yield stress values multiply gear by thickness matetial stress ratio. 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 *70,000-150,000 *50,000-115,000 40,000 30,000 20,000 20,000 20,000 8,000 6,000

3140 Steel Stainless Steel 416 Aluminum Alloy 24 S-T4 Stainless Steel 303 Phosphor Bronze SAE 1020 Brass Phenolic Nylon

Stress Ratio *0.43-0.20 *0.60-0.26 0.75 1.00 1.50 1.50 1.50 3.75 5.00

* depends upon heat treatment. **By permission, Product Engineering, October 1955 TABLE 1.18 RA11NGS FOR HARDENED STEEL HELICAL GEARS**

Number of Teeth 100 8 .03 10 .04 12 .04 15 .05 16 -

Horsepower at Various R.P.M.* 20 D.P. - 3/8" Face 24 D.P. - 1/4" Face 200 300 600 900 1200 1800 100 200 300 600 900 1200 1800 .05 .07 .14 .20 .26 .37 .05 .11 .16 .30 .43 .55 .76 .07 .10 .19 .27 .34 .47 .07 .14 .21 .40 .56 .71 .47 .08 .12 .23 .32 .41 .55 .09 .18 .26 .48 .68 .85 .55 .11 .16 .29 .41 .51 .68 .12 .23 .33 .61 .85 1.05 .68 -

18 20 24 25 30

.04 .05 .06 .08

.09 .10 .12 .14

32 36 40 48 50 60 72

.13 .14 .17 .21

.23 .26 .30 .36

.32 .35 .41 .48

.40 .44 .50 .58

.53 .57 .64 .73

.11 .21 .14 .26 .16 .31

.30 .54 .37 .65 .43 .75

.09 .17 .12 .22 -

.25 .42 .31 .52 -

.55 .66 -

.66 .77 -

.81 .92 -

.22 .40 .27 .49

.56 .93 1.19 1.38 .67 1.08 1.35 1.55

.15 .27 .17 .32

.37 .60 .43 .67

.75 .82

.86 .93

1.01 1.07

.32 .57 -

.77 1.20 1.48 -

*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. T85

.73 .86 .98

.89 1.04 1.16

1.67 -

1.14 1.30 1.43 1.65 1.81 1.91 -

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 transmitted force as a function of: profile errors; acceleration forces; elasticity properties; forces required to deform the teeth an amount equivalent 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 minute F = 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

Tooth Form

Materials, Pinion and Gear

Cast Iron & Cast Iron Steel & Cast Iron Steel & Steel

14½o 20o

Cast Iron & Cast Iron Steel & Cast Iron Steel & Steel

full depth

Cast Iron & Cast Iron Steel & Cast Iron Steel & Steel

20o stub tooth T87

Error in Action, Inches 0.0005 0.001 0.002 0.003 0.004 400 800 1,600 2,400 3,200 550 1,100 2,200 3,300 4,400 800 1,600 3,200 4,800 6,400 415 830 1,660 2,490 3,320 570 1,140 2,280 3,420 4,560 830 1,660 3,320 4,980 6,640 430 860 1,720 2,580 3,440 590 1,180 2,360 3,540 4,720 860 1,720 3,440 5,160 6,880

0.005 4,000 5,500 8,000 4,150 5,700 8,300 4,300 5,900 8,600

13.3 Surface Durability The Lewis formula and its modification to Incorporate dynamic conditions is limited to beam-stress analysis. In addition, there are stresses generated in the surface layers of the teeth by the direct crushing action of the forces. These stresses can exceed the material 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 is caused by high tooth loads leading to excessive surface stress, a high local temperature due to high rubbing speeds, or inadequate lubrication. Minute cracking of the surface develops, spreads and ultimately results in small bits breaking out of the tooth surface. Scoring — This is a heavy scratch pattern extending from tooth root to tip. It appears as if a heavily-loaded tooth pair has dragged 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 by loose 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 considerable intermingling. There have been many attempts to derive expressions for calculating safe surface stress. The Buckingham durability equations based on Hertzian contact stresses and the work of others can be found in the references. All of the various design equations and procedures are closely related to specific empirical data and experience. The AGMA equations are in wide use in the United States. 13.4 AGMA Strength and Durability Ratings The AGMA rating formulas again represent a combinations of analysis, approximations, and empirical data. A complete treatment of AGMA practices is too extensive for this discussion and only an introductory survey is offered. More details are available from AGMA 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 Kv

.

Pd . F

KsKm J

(58)

Surface Durability:

Sc = Cp WtCo . Cs . Cv dF

CtCm l

(59)

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 the application. The allowable stresses are as follows: Allowable surface durability stress:

St =< Sat KL KrKr

(60)

T88

Allowable surface durability stress:

Sc =< Sac CLCH CRCT

(61)

Definition 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 departure and tie improved accommodation to actual performance is dependent upon the coefficients associated with each term The 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 FORMULAS Term Strength

Durability

LOAD: Transmitted Load Dynamic Factor Overload Factor

Wt Kv Ko

Wt Cv Co

SIZE:

-F Pd Ko

d F -Co

Km

Cm I Cf

St Sat

Sc Sac CP CH CL CT CR

Pinion Pitch Diameter Net Face Width Transverse Diameteral pitch Size Factor STRESS DISTRIBUTION: Load Distribution Factor Geometry factor Surface Condition factor

J --

STRESS: Caiculated Stress Allowable Stress Elastic Coetficient Hardness-Ratio Factor Life Factor Temperature Factor Factor of Safety

---

KL Kt KR 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 increment increases linearly. However, the dynamic effects of tooth errors is much more complex. Tooth-to-tooth errors, which arise in a variety 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 can be 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 clear fashion. Temperature factors —

KT & CT

These factors modify the design in accordance with adverse temperature effects on lubricant performance. Usually this factor does 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 the gear-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. The latter 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 are somewhat akin the Lewis Y factors. For standard tooth proportions, 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 is unimportant or insufficient information exists for its adequate evaluation it is usually safe to equate the factor to unity. In most cases, this results in a conservative or mid-value rating. Evaluations of equations — The above information constitutes an outline of the procedures offered by AGMA for determining strength and durability ratings. As an outline it cannot include detail;and to apply the procedures the reader should refer to the references. Additional design equations — The AGMA beam strength and durability equations have been custom modified and refined by a number of gear designers and manufacturers, creating a variety of design techniques and equations. Often this may be proprietary information, but will be available for specific use with customers’ needs. In addition, there are a host of varied design equations used by T90

lou near designers foreign gear designers. This multiplicity of equations underlines that gear strength and durability is not an exact engineering science, but rather is empirical and experience dependent. Also, the user should be aware that most gear equations and empirical results pertain to coarse pitch gears. The literature offers much less about tine pitch instrument gearing. Computer programs — The AGMA design equations involving various parameters are defined with specific 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 suitable gear material is very important. Often not all design requirements are compatible. High load capacity requires a tough, hard material which is difficult to machine; whereas high precision favors materials that are easy to machine and, therefore, have lower strength and hardness ratings. Light weight and small size favors light non-ferrous materials, while high capacity requires the opposite. Thus, tradeoffs and compromise are required to achieve an optimum design. Gear materials vary widely, ranging from ferrous metals, through the many non-ferrous and light-weight metals, to the various plastics. The gear designer and user faces a myriad of choices. The final 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 most widely used far gears, because they offer high strength, response to 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 costs by molding the gear blank. Cast steels also offer this advantage together with higher tensile and yield strengths, but cast iron is superior under dynamic conditions, providing excellent internal damping properties. 14.1.2 Steels are divided into two main divisions: plain carbon and alloy. The carbon steels offer low cost reasonably easy machining and ability to be hardened. A major disadvantage is the lack of resistance to corrosion. When elements other than carbon are added to the iron, the steel is termed "alloy steel". These cover 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 heat treatment 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, which resist nearly all corrosive conditions; and the 400 series, which although not truly stainless, offer less corrosion resistance only in certain environments (such as certain acids and salt water) and are otherwise considered stainless. The further significant distinction between the two series is that the 300 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 T91

file:///C|/A3/D190/HTML/D190T91.htm [9/27/2000 6:49:31 PM]

heat treatable and have a much better index of machinability corresponding to some of the carbon steels. Table 1.23 lists mechanical 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-ferrous metals 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 It is also corrosion resistant and easy to machine. A major disadvantage is the large coefficient of thermal expansion compared to steels. Many aluminum alloys differ in ease of forming, machining and casting. Aluminum alloys respond to cold working and heat treatment 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 with steel gears. They are particularly advantageous in worm meshes and crossed-helical meshes because of the large amount of sliding. Bronzes are extremely stable and offer excellent machinability. The material can be cast, but bar stock and forgings are superior. 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. This table 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 a few 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 is a 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 sintered powder 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 low cost. On the debit side, they are difficult to machine to high precision and are subject to large temperature-induced dimensional changes and instability. Gears can be directly finish molded with teeth, entirely machined from bar and plate stock, or cut from molded blanks. Phenolic laminates have bases of either paper, linen, or cotton cloth with relative strengths in that order. They offer relatively good strength and in cotton-canvas base are suitable for large gears and high loads. Properties for gear phenolics are given in Table 1.28. T92

TABLE 1.25 COMPARATIVE PROPERTIES OF MATERIALS Tensile Yield 108 Cycles Elongation Strength Strength Material (ASTM No.) Endurance (%in 2") (psi) (psi) Limit(psi) Aluminum Alloys: Wrought: 2011-T8 59000 45000 18,000 12 68.000 47,000 19,000 18 2024-T4 83,000 73,000 23,000 11 7075-T6 38,000 31,000 18,000 13 5025-H34 45,000 40,000 14,000 12 6061-T6 Cast: 195-T6 356-T6

Bronzes: Aluminum Bronze-B150-2 (annealed) Phosphor Bronze- B139C Silicon Bronze - B98B (Hard) Manganese Bronze - B138-A Brasses: Free Cutting -B16 Yellow B-36-8 Naval - B124-3 (¼H) Cartridge - B134-6

36,000 40,000

24,000 27,000

8,000 13,000

5 5

100,000 80,000 65,000 80,000

60,000 45,000 35,000 65,000

28,000 31,000 25,000 17,000

25 33 10 25

55,000 61,000 70,000 72,000

44,000 50,000 48,000 52,000

20,000 16,0001 22,0002

32 23 25 30

Hardness Brirtell 100 Brinell 120 Brinell 150 Brinell 68 Brinell 95 Brinell 75 Brinell 90

Rockwell B90 Rockwell B80 Rockwell B80 Brinell 80

Rockwell Rockwell Rockwell Rockwell

B75 B70 B80 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

Material

TABLE 1.26 PROPERTIES OF DIE-CASTING ALLOYS SUITABLE FOR GEARS Comp108 Cycles Tensile Yield Shear Elongaressive Nominal Endurance Strength Strength Strength tion (% Strength Composition (%) Limit (psi) (psi) (psi) in 2") (psi) (psi)

Aluminum Alloys: 13 85 380

12 Si 5 Si , 4 Cu 8.5 Si, 3.5 Cu

Magnesium Alloys ASTM-AZ91

9 AI,0.2 Zn 0.13 Mn

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

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

Hardness (Brinell)

37,000 40,000 31,000

18,000 24,000 31,000

28,000 23,000 31,000

21,000

19,000

2

25,000

21,000

4

80 75 80

33,000

22,000

20,000

22,000

10,000 14,000

3

60

41,000

31,000

60,000

6,900

10

82

47,000

38,000

87,000

8,200

7

91

Taken from: Michalec, G.W., i’redsion Gearing, Wiley 1968 T95

TABLE 1.27 TYPICAL SINTERED POWDER GEAR ALLOYS Name

Specification Designation

Composition(%)

Copper 7-11 Iron-Copper Alloy Iron-remainder

SAE Type 3 ASTM B222-58

Ultimate Apparent Tensile Hardness Strength (Rockwell) (psi) 40,000

Iron 94,0 min. ASTM Copper-Steel Alloy Copper 1.0 - 4.0 60,000 B310-58T Other 2.0 max. Class A Type II Iron 95.5 min. SAE Type 6 Silicon 0.3 max. Class C Carbon-Steel Alloy 50,000 Aluminum 0.2 min ASTM B310-58T Other 3.0 Carbon 0.30 Manganese 0.50 Alloy Steel Silicon 0.25 Z2* 160,000 AISI 4630 Nitrogen 1.7 Molybdenum 2.5 Iron balance Iron 97.90 Copper 0.15 Iron High Density Silicon 0.20 Aluminum 0.15

ASTM B309-58T Class A

52,000

H-95

B-56

* Designation of Keystone Carbon Co. t Designation of The Brush Beryllium Co. Taken from: Michalec. G.W., "Precisjon Gearing". Wiley 1966 T96

Offers a controlled amount of porosity suitable for lubricant Impregnation Good for Gear applications subject to high impact

A-40

Excellent wear resistance

C-35

The highest strength sintered powder material

A-60

Good gear material for impact, strength and hardness. High density allows it to be case hardened by carburizing or nitriding

H-75

One of the strongest sintered bronzes

B-85

A maximum strength beryllium alloy

Other 1.60

SAE Type 1 Class A Copper 87.0 Min. ASTM Tin 9.5 - 10.5 Phosphor Bronze 30,000 B202-58T Phosphor 0.3 - 0.5 Type 1 Class A Other 1.5 max. Mil B 5687A Type 1 Comp.A Beryllium 1.5 75,000 to Beryllium Copper Cobalt 0.25 150Pt 100,000 Copper balance

Comments

Catalog D190 TABLE 1.28 PROPERTIES OF PLASTIC MATERIALS Property Tensile Strength Yield Strength Compressive Strength. psi Water Absorption % (24 hrs.) Saturation % Density- lbs/in3 Modules of Elasticity. psi (Flexural) Coefficient of Linear Thermal Expansion oF

NYLON ASTM Type 66 ASTM Type 6 0.2% 2.5 % 0.2% 2.8% Moisture Moisture Moisture Moisture 11.800 13.800 11.200 9,000 7.400 8.700 13.000 8.000 5.900 8.500 1.5 1.6 to 2.0 7,200 8 9.5 .041 .041 5 4.1 x 105 2 .5 x 10 -5 1.1 x 105 4.6 to 5 x 10 1.75 x 105 -5 5.4x10

Data at 70o F DELRIN*

Property

100 10.000 9.500 5,200 .25 .9 0.514 4.1 x 105 5.5 x 10-5

Yield Strength - psi Shear Strength - psi Compressive Stress - 1% deformation - psi Water Absorption %(24 hrs) Saturation % Density- lbs/in3 Modules of Elasticity. psi (Flexural) Coefficient of Linear Thermal Expansion oF Data at 70o - 75o F Registered trade name of E.I. DuPont de Nemours & Co. Properties Base Tensile Strength - psi Lengthwise Crosswise Flexural Strength - psi Lengthwise Crosswise Compressive Strength - psi Flatwise Modules of Elasticity. psi (Flexural) Lengthwise Crosswise Water Absorption %(24 hrs) Coefficient of Linear Thermal Expansion oF Lengthwise Crosswise

X

PHENOLIC LAMINATES NEMA Grade XXX C

Kraft Paper

Paper

Cotton Canvas Fabric

21,000 17,000

16,000 13,000

11,500 9,500

26,000 24,000

14,000 12,000

22,000 18,000

36,000

32,000

37,000

1.8 x 108 1.3 x 108 0.9

1.3 x 108 1.0 x 108 0.3

1.0 x 108 0.9 x 108

1.1 x 10-5 1.4 x 10-5

0.94 10-5 1.4 x 10-5

1.04 x 10-5 1.22 x 10-5

T97

500 10.000 9.500 5,200 .25 .9 0.514 4.1 x 105 4.5 x 10-5

L Fine Weave Cotton Linen Fabric 14,500 11,000 23,000 18,000 35,000 1.1 x 108 0.8 x 108 0.77 x 105 1.04 x 10-5

Catalog D190

Material Ferrous: Cast Irons Cast Steels Plain-Carbon Steels Alloy Steels Stainless Steels: 300 Series

400 Series Nonferrous: Aluminum Alloys Brass Alloys Bronze Alloys

Magnesium Alloys Nickel Alloys

Titanium Alloys Die-Cast Alloys Sintered Powder Alloys Nonmetalic: Delrin Phenolic Laminates Nylons Teflon (Fluorocarbon)

TABLE 1.29 SUMMARY OF MATERIAL FEATURES AND APPLICATIONS Obtainable Precision Outstanding Features Applications Rating Low cost, good machining,high Internal damping Low cast, high strength Good machining, heat - treatable Heat treatable, highest strength durability High corrosion resistance, non magnetic. nonhardenable Hardenable, magnetic. moderate stainless steel properties

Large-size, moderate power rating, commercial gears Power gears, medium ratings Power gears, medium ratings Severest power requirements Extreme corrosion, low power ratings

Commercial quality Commercial to medium precision Precision and high and precision Precision

Low to medium power ratings, moderate corrosion High precision Extremely light-duty instrument gears Low-cost commercial equipment

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

Commercial quality

High precision Medium precision

Mates for steel power gears High prectsion

Excellent machinability, low friction, and good compatibility with steel mates

Special lightweight, low-load uses

Extreme light weight,poor corrosion resistance

Special thermal cases

Low coefficient of thermal expansion, poor machinability

Special lightweight strength

High strength for moderate weight, corrosion resistant

High production, low quality,commercial

Medium precision

Commercial grade Medium precision applications

Low-grade commercial High production, low quality commercial Commercial

Low cost, no precision, low strength Low cost, low quality, moderate strength

Wear resistant, long life, low water Long life, low noise, low absorption loads

Commercial

Quiet operation, highest strength plastic

Commercial

Medium loads, low noise Long life, low noise, low loads

Low friction, no lubricant, high water absorption

Special low friction

Low friction, no lubricant

Taken from: Michalec, GW., "Precision Gearing", Wiley 1966 T98

Commercial Commercial

Nylon has good wear resistance, even when operating without lubricant. A major disadvantage is instability in the presence of moisture 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 gears are 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, the more 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 instrument fields because of their corrosion resistance. For fine-pitch precision applications, stainless steels are excellent. Although the 400 series is easier to machine and can have superior properties as a result of heat treatment, the 303 type of stainless steel has reasonable 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 in spur 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 machining characteristics and greater instability make them undesirable for precision applications. Their quiet operation and minimal lubrication 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 type of 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, or surface finish. This limits coatings to thin coverings of oxides or a substance that permanently adheres to the base, and not all are 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 aluminum gears is usually limited to the gear blank prior to tooth cutting. * Registered trade name of E.I. duPont de Nemours and Co. T99

Catalog D190 Because the oxide film is somewhat porous. it can be impregnated with dyes of various colors. Anodized gears possess not only improved 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 film of chromate which is extremely thin and does not alter dimensions. However, the thin film has little wear resistance and offers corrosion protection only against non-abrasive environments. Coating color varies with the particular metal and alloy. Most often there is an iridescent color, which generated the common trade name Iridite, Dyes can be added to produce a wide assortment of colors. Because there is no dimensional change, chromating can be applied to all gears, including precision, after tooth cutting. 15.3 Passlvatlon This is not a coating, in the strict sense, but a conditioning of the surface. It is particularly applicable to stainless steels. The process 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 changes and does not discolor or otherwise alter the natural surface. If anything, it prevents random staining due to "free iron" particles left from machining. All quality stainless steel gears can be passivated after complete machining since dimensions and stability are unaffected. 15.4 Plating. The common electroplating materials, such as cadmium, chromium, nickel and copper, are not suitable for gear surfaces since they alter dimensions. Also, susceptibility to localized buildup precludes their use on any precision part. Use of these platings should be limited to the application of coatingt prior to cutting of the teeth and of any other gear dimensions requiring close tolerances. 15.5 Special Coatings In 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 successful applications 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 military equipment 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. T100

Catalog D190 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 galvanic interaction with other parts. Bronze gears could be chromate coated after cutting or cadium plated in the blank state, followed by chromating after tooth generation. Table 1.30 summarizes features of the various coatings. 16.0 LUBRICATION Lubrication serves several purposes, but its basic and most important function is to protect the sliding and rolling tooth surfaces from seizing, wear, and other phenomena associated with surface failure by film separation. This is particularly pertinent to power gearing. 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 and speed, it may be necessary to use a circulating system with lubricant cooling. Lubricant can be supplied as a liquid bath or fine spray. Lubrication of small, low-power gear trains can be accomplished with a grease pack in some cases. Many consumer home products 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, instrument gear 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 as power gears. The difference is that, in these extremely low powers, the heat dissipation is not a problem, therefore the unit can be 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 assembly and reapplied only as maintenance and usage dictate. In such applications, it is important that gear speeds are not so great that the 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, the outer enclosures of which must often be removed far maintenance of other items, should be.worked on in clean and controlled environments. Where prolonged or uncontrollable exposure occurs, temporary or permanent inner dust covers for the gear train are recommended. This is particularly advisable in hybrid electronic instrument boxes in which the danger of solder splatter and other debris is high. 16.3 Oil Lubricants Oils are the most common lubricants and come in various-types. The compounding of oils provides combinations and generation of various properties. The most basic lubricant is petroleum to which animal, vegitable and synthetic oils and additives are combined to yield specific properties. T101

Catalog D190

Catalog D190 Oils offer a wider range of operating speeds than greases. Also, they are easier to handle and are more effective because of their liquid nature. 16.4 Grease Grease is a combination of liquid and solids, in which the latter serve as a reservoir for the liquid lubricant as well as imparting certain of their own properties. Grease has the advantage of remaining in place and not spreading as oils, and has a much lower evaporation rate. Also, it can provide a lubricant film at heavy loads and at low speeds. 16.5 Solid Lubricants In 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 to use in open gearing since they do not contaminate as rapidly as oils and grease. However, most solid films alter dimensions significantly - and some drastically. The latter cannot be tolerated in quality gearing. Dry-film lubricants represent a one-shot application of lubricant that must 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 Lubricants The choice of lubricants is very wide. Military specifications govern most types and classes of lubricants, to which many manufacturers’ products qualify. Table 1.31 is a list of typical gear oils and grease lubricants and their applications. 17.0 GEAR FABRICATION The fabrication of a complete gear normally includes most or all of the following operations: 1. 2. 3. 4. 5. 6.

Blank fabrication Tooth generation Refining of tooth shape (shaving, grinding, honing) Heat treatment Deburring and cleaning 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 of a gear. Blank fabrication involves all the general and special features of the gear body. Tooth generation involves only machine-cut or around 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 to gears requiring surface hardness and/or strength. Deburring and cleaning is essential for all gears irrespective of method of manufacture or quality. Finish coats are limited to certain materials and environments requiring corrosion protection or improved appearance. Modern methods of producing gear teeth cover a wide variety: T103

Catalog D190 Table 1.31 Typical Gear Lubricants Lubricant Type

Military Specification

Useful Temp. Range (o F)

Oils: Petroleum oil

MIL-L-644B

-10 to 250

Diester oil

MIL-L-6085A

-67 to 350

Diester oil

MIL-L-7808C

-67 to 400

Remarks

Applications

Good general purpose lubricant. All quality gears having a narrow range of operating temperature. General purpose, low starting Precision instrument gears and torque, stable over a wide small machinery gears. temperature range. Suitable for oil spray or mist system, at high temperature.

High speed gears.

Best load carrier of silicone oils, Power gears requiring wide widest temperature range. temperature ranges. Silicone oil Greases Diester Oil- lithium soap

MIL-L-7808C

-100 to 600

MIL-G-7421A

-100 to 200

Particularly suited for low starting torques, low temperatures.

Moderately loaded gears.

Diester Oil- lithium soap

MIL-G-3278A

-67 to 250

General purpose light grease.

Precision instrument gears, and generally lightly loaded gears.

High temperature only.

High speed and high loads.

Petroleum oil- sodium MIL-L-3545 soap Silicone oil- non-soap MIL-G-2501 3B Silicone oil- lithium soap Solid Lubricants Molybdenum disulfide(MoS2) powder Graphite in resin binder

MIL-G-1 5719 -A

-20 to 300 -65 to 400 0 to 350 -350 to 750;> 20O0 in vacuum

MIL-M7866A

-100 to 450

-100 to 450 MoS2 in resin binder

T104

Good high temperature features. High temperature, moderately loaded gear trains. High temperature use only. Light to moderately loaded gears, low speeds. Highly stable, radiation resistant, useable in vacuum over wide temp. range. Application by spray and baking up to 3500 F. Film thickness .0003 to .001 in. Application by spray and baking up to 350oF. Film thickness .0003 to .001 in. Stable in vacuum.

Light duty precision gears.

Low precision and commercial quality gears. Light loads. Space gear trains and vacuum.

Catalog D190 1. 2. 3. 4. 5. 6.

Machine cut Grinding Casting Molding Forming (drawing, extruding, rolling) Stamping

Each method offers special characteristics relating to quality, production quantity, cost, material and application. 17.1 Generation of Gear Teeth Machining constitutes the most important method of generating gear teeth. It is suitable for high precision gears in both small and large quantities. 17.1.1 Rack generation — This is the basic method of producing involute teeth. The rack cutter forms conjugate tooth profiles on the blank as the rack and blank are given proper relative motion by the drive mechanism of the generating machine. As the rack traverses the gear blank, it is reciprocated across the blank face. Cutting edges on the rack teeth generate mating conjugate teeth on the blank. The chief disadvantage of this method is that the rack has a limited length which necessitates periodic indexing. 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 that the 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 and gear. The differences are that the thread of the hob is axially gashed or fluted in several places so as to form cutting edges, while the 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 slowly traverses 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 all angles with the same tooling. With regard to accuracy, hobbing is superior to the other cutting processes. Gears can be directly hobbed to ultra-precision tolerances 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 the equivalent. Uke a rack cutter, a given gear-shaped cutter is conjugate to all tooth numbers of that pitch. Thus, a gear made as a cutting tool can generate the teeth of a blank when the two are rotated at proper speeds. The cutting tool can be imagined as a gear that axially traverses the blank with a reciprocating axial motion as it rotates. The teeth on the gear cutter are appropriately relieved 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 as hobbing, 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 are designed 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 the gear. For internal gears, the shaping process is the only basic method of 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 T105

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 teeth simultaneously 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. The main 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 Grinding Although grinding is often associated with quantity fabrication of high quality gears as a secondary refining operation, it is also a basic process for producing hardened gears. In addition, many high-precision fine-pitch gears have their teeth entirely ground from the blank state. Gear-tooth grinding can involve either form grinding or the generating process. The latter is basically more accurate because the dressing 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 endurance stress levels by very significant amounts. See Table 1.23. Often it is the difference between a reliable and unreliable gear. In particular, case hardened, carburized or nitrided gears offer outstanding strengths and performance. They are typically used for the 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 also because of the finer surface finish. A fine finish enables maintenance of a good continuous oil film versus boundary lubrication and 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 root 5. 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 precision and strength. T106

Catalog D190 17.3 Plastic Gears These can be produced by the normal hobbing and shaping processes. In addition, they can be produced by various molding techniques. 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 due to temperature instability, material flow, and generally poorer cutting qualities. Attainable quality is less than for metals and varies with the particular plastic. See Section 20.10, which deals with plastic gears in greater detail. 18.0 GEAR INSPECTION The performance of a gear can be assured only by confirmation of its critical dimensions and parameters. With increasing gear precision, 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 of the 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 Testers Both TCE and tooth thickness can be measured by means of roll testing with a variable-center-distance fixture. There are many varieties, but essentially all consist of a fixture having two parallel shafts (or precision centers), one fixed arid the other floating on smooth, low-friction ways. The test gear is mounted on one shaft while an accurate known quality master gear is mounted on the other shaft. The pair is held in intimate contact by spring loading or the equivalent. As the test gear is rotated, tooth-to-tooth errors and runout are revealed as a variation in the center distance of the pair. This variation can be sensed, amplified and displayed as a dial reading or recorded on a chart. See Figure 1.48. Sensitivity of the measurement is on the order of 50 to 100 millionths 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 to Figure 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 corrective action. 18.1.2 Gear Size — If the center distance setting of the roll tester is carefully established, the absolute readings are an indication of tooth thickness. Thus, in Figure 1.48, the mean line of the trace is a measure of tooth thickness. The high and low readings indicate the extreme variation of tooth thickness at the nominal pitch radius. Changes in center-distance are an indirect measure 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 it reveals characteristics that occur in the real application. Also, the method is rapid and, therefore, suitable for production gear inspection. 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 .0001 inch. On the other hand, size measurement is not as reliable as an absolute measure. This is due to the nature of the fixture and the integration of several error sources in the calibration process. A repeatability of .0002 inch is considered good, and often it is even better. T107

Catalog D190 18.2 Over-Pin. Gaging The equations relating tooth thickness and a measurement over cylindrical pins or rolls inserted between the teeth were given in Paragraph 4.6. This is a widely used method for gaging gears during fabrication (while they are still in the gear generating machine) 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. This is because over-pins gaging is insensitive to pitch-line runout. On the other hand, rolling a gear necessarily involves the TCE and its runout component. The best correlation is obtained by equating the over-pins measurement to the average value of center distance found in the roll test. Apart from the correlation problem, over-pins measurements by themselves are inadequate because the undetected runout can be out-of-control causing interference with its mate. It is necessary, therefore, to control and to inspect runout. 18.3 Other Inspection Equipment In addition to the basic inspection methods and equipments described in Paragraphs 18.1 and 18.2, other special-purpose equipment is available. This includes involute-profile form checkers, tooth-spacing gages and runout checkers. Also, for high precision gears, equipment is available for inspecting the position error of individual gears and the transmission error of a gear train. 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 measurements suitable for large, coarse-pitch gears. Hence, fine-pitch gears are almost exclusively inspected by functional testing on a variable-center-distance fixture. Over-pins measurements are also used, but generally are restricted to a reference measurement, This is primarily used in the fabrication 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 Implementation The inspection operation is essential to obtaining a quality product. In effect, it is a policing operation that ensures conformance to 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 effort than that for low quality gears. Equipment must be of the best grade, calibrated periodically and restricted to use by qualified personnel. Control of temperature environment is essential for measurements on the order of .0001 inches. The cleanliness of equipment, gears and working area are also very important. T1O8

Catalog D190

GEAR DESIGN - METRIC 19.0 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 major obstacle 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 D

(62)

where: N = number of teeth D = pitch diameter Pd = diametral pitch From this relationship there are particular integer - values of diametral pitch that yield integer values for center distance in inches. Thus 8, 16, 32 and 64 diametral pitches, to mention only some, can be associated with tooth numbers which can result in 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 pitch diameter per tooth); whereas diametral pitch is the number of teeth to a unit length of pitch diameter. Again convenient center distances 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 are non - interchangeable. It should be noted that the term diametral pitch is associated with the inch system. In the metric system the nearest analogue to pitch is termed "module", and the word pitch is reserved for tooth spacing along the pitch circle. In the inch system, the tooth spacing 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 the conversion 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 conversion results in decimal values, often awkward numbers, for one or the other measure. It follows that convenient values in one system will not be convenient values in the other. For this reason each system (inch diametral pitch and metric module) has adopted preferred 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 valued pitches and modules are usually not tooled for. It becomes obvious, therefore, that direct replacement of conventional inch gearing with metric gearing is impossible. The best that 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 and module is best measured by the circular pitch and the circular tooth thickness. These values are given in inches and millimeters in Table 1.32. T109

Catalog D190 As a consequence, metrification of gearing requires a completely new design with regard to gear dimensions and center distances. 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 (see Table 1.33). Judging by their acceptance by the industrialized metric countries, the following modules are expected to be preferred 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 to designers 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-wire measurements. Similarly for module-type gears, computerized Table 1.34 was produced. This lists both pitch diameters as well as over-wire measurements in both millimeters and inches. TABLE 1.33 Modules and Diametral Pitches of Cylindrical Gears for General and Heavy Engineering* (ISO Recommendation R54 1977) Modules m

I 1

II 1.125

1.25 1.375 1.5 1.75 2 2.25 2.5 2.75 3 3.5 4 4.5 5 5.5 6 7 8 9 10 11 12

Diametral Pitches P III I II 20 18 16 14 12 11 10 9 8 7 6 (3.25) 5.5 (3.75) 5 4.5 4 3.5 3 2.75 (6.5) 2.5 2.25 2 1.75 1.5

14 16

1.25 18

20

1 22

25

0.875 0.75

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Catalog D190 19.2 Metric Design Equations Some of the gear design equations are dimensionless and are derived from geometric proportions and relationships. These equations 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 approached in two steps. The first step is to express the present inch-base units in metrics and to modify the constants and coefficients accordingly. This procedure will yield results expressed in the form presently used in engineering practice in industrialized metric countries. The second step is to express these results in Si units which differ slightly from the conventional metric units. Thls is true for stress 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 metric countties, but its use has not spread to the practical design engineering profession. For. these countries, transition to the SI system represents a change which is accompanied by a degree of reluctance. The standardization related to transition to metrics in 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 no problem with metrification. Most of the kinematic design equations that appear American gear texts. and are associated with inch-system gears, are suitable for use with metric gear dimensions, provided that a proper substitution of module (in) is made for-pitch. For equations involving diametral pitch:

Pd

is replaced by 25.4

(65)

m Recalling that:

Pd * Pc

=π 25.4

m

we find that for equations involving circular pitch:

Pc

is replaced by

π

(66)

m

25.4 Note: When converting between metric module and the inch diametral pitch, the conversion factor and relationship can be remembered 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 design equations given in inch units. Basic kinematic and geometric design equations for spur gears in both metric module and inch diametral-pitch forms are given in table 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. 2. 3. 4.

Over-pins measurements. Relationship between tooth thicknesses at different radii from gear center. Long and short addendum equations. Profile-shifted gear-design equations: i.e., enlarged gear teeth, non-standard center distance T122

Catalog D190

Catalog D190 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 DIN standard, is widely used throughout Europe. However, the Japanese have their own version, defined in JIS standards. The deviations among these and other national metric standards are fortunately minor: the various metric standards, differ only with regard to dedendum size and root radii. Even these minor deviations are resolved by a new unified module standard sponsored and promoted by the International Standards 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 ISO standard, 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 American gears: 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 the basic rack, as shown in the illustration. Dimensions, in millimeters, are normalized for module m = 1. Corresponding values for other modules are obtained by multiplying each dimension by the value of the specific module, m. Major tooth parameters are described 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 specify rounding. However, as a maximum or limit value, American gear makers are not prevented from specifying a tip radius as near zero as possible. Note that the basic racks for metric gears and for American inch gears are essentially identical. For metric gears, specific size dimensions are obtained from multiplying by m (the module). Gears conforming to diametral pitch American standards are sized by 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 of Machine 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. T124

Catalog D190 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 and ready 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 been decreed 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 long time to metric measurement countries have developed their own standards for metric gearing. In general they have similar standards, 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 appropriate foreign standards have become important. To serve that need Table 1.37 offered as a listing of key gear standards in use in several major countries and geographic areas. ISO 53:1974 ISO 54:1977 ISO 677:1976 ISO 678:1978 ISO 701:1979 ISO 1122-1:1983 ISO 1328:1975 ISO 1340:1976 ISO 1341:1976 ISO 2490:1976 ISO/TR 4407:1902 ISO 4468:1982

TABLE 1.36 ISO METRIC GEARING STANDARDS Cylindrical gears for general and heavy engineering — Basic rack Cylindrical gears for general and heavy engineering — Modules and diametral pitches Straight bevel gears for general and heavy engineering — Basic rack Straight bevel gears for general and heavy engineering-. Modules and diarmetral pitches International gear notation — Symbols for geometrical data Glossary of gear terms — Part 1: Geometrical definitions Parallel involute gears -. ISO system of accuracy Cylindrical gears.- Information to be given to the manufacturer by the purchaser In order to obtain the gear required Straight bevel gears - Information to be given to the manufacturer by the purchaser in order to obtain the gear required Single-start solid (monobloc) gear hobs with axial keyway, 1 to 20 module and 1 to 20 diametral pitch - Nominal dimensions Addendum modification of the teeth of cylindrical gears for speed-reducing and speed-increasing gear pairs Gear hobs - Single start- Accuracy requirements T-126

Catalog D190

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

NF NF NF NF NF

E E E E E

23-001 23-002 23-005 23-006 23-011

1972 1972 1965 1967 1972

NF E 23-012 1972 NF L 32-611 1955

DIN 37 12.61 DIN 780 Pt 1 05.77 DIN 780 P12 05.77 DIN 867 02.88 DIN 868 12.76 DIN 3961 08.78 DIN 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 08.86 DIN 3965 Pt 2 08.86 DIN 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

TABLE 1.37 FOREIGN METRIC GEAR STANDARDS AUSTRALIA 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)

FRANCE 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 producer Calculating spur gears to NFL 32-610

GERMANY - DIN (Deutsches Institut für Normung) 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] T127

Catalog D190 TABLE 1.37 CONT. FOREIGN METRIC GEAR STANDARDS DIN 3971 07.80 DIN 3972 02.52 DIN 3975 10.76 DIN 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 09.76 Suppl 1 DIN 3998 Ptl 09.76 DIN 3998 Pt2 09.76 DIN 3998 P13 09.76 DIN 3998 P14 09.76 DIN 58405 Pt1 05.72

GERMANY CONT. - DIN (Deutsches Institut für Normung) 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 of worm gear drives [6] Measuring element diameters for the radial or diametral dimension for testing tooth thickness 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 geometrical limits of internal gear-pinion matings [15] Geometrical design of cylindrical internal involute gear pairs —Diagrams for the determination of addendum modification coefficients [15] Geometrical design of cylindrical internal involute gear pairs -. Diagrams for limits of internal gear-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

DIN 58405 P12 05.72 DIN 68405 P13 05.72 DIN 68405 Pt 4 05.72 DIN ISO 2203 06.76

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.

UNI UNI UNI UNI UNI

3521 3522 4430 4760 6586

1954 1954 1960 1961 1969

UNI 6587 1969 UNI 6588 1969

ITALY 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)

UNI 6773 1970 T128

Catalog D190

B B B 8 B B B B B B B B B B B B B B B B B

0003 0102 1701 1702 1703 1704 1705 1721 1722 1723 1741 1751 1752 1753 4350 4351 4354 4355 4356 4357 4358

1989 1988 1973 1976 1976 1978 1973 1973 1974 1977 1977 1976 1989 1976 1987 1986 1988 1988 1985 1988 1976

TABLE 1.37 CONT. - FOREIGN METRIC GEAR STANDARDS JAPAN - JIS (Japanese Industrial Standards) Drawing office practice for gears. Glossary of gear terms Involute gear tooth profile and dimensions Accuracy for spur and helical gears Backlash for spur and helical gears Accuracy for bevel gears Backlash for bevel gears Shapes arid dimensions of spur gears for general engineering Shapes and dimensions of helical gears for general use Dimensions of cylindrical worm gears Tooth contact marking of gears Master cylindrical gears Methods of measurement of spur and helical gears Measuring method of noise of gears Gear cutter tooth profile and dimensions Straight bevel gear generating cutters Single thread hobs Single thread fine pitch hobs Pinion type cutters Rotary gear shaving cutters Rack 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 1972 Specification of gears for electric traction BS 438 Pt1 1987 Spur and helical gears — Basic rack form, pitches and accuracy (diametral pitch series) BS 436 Pt 2 1984 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) BS436 Pt3 1986 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) BS721 Pt 1 1984 Specification for worm gearing — Imperial units BS721 Pt2 1983 Specification for worm gearing — Metric units BS978 Pt1 1984 Specification for fine pitch gears — Involute spur and helical gears BS978 Pt2 1984 Specification for fine pitch gears — Cydoidal type gears BS978 Pt3 1984 Specification for fine pitch gears - Bevel gears BS978 Pt4 1965 Specification for fine pitch gears - Hobs and cutters BS1807 1981 SpecifIcation for marine propulsion gears and similar drives: metric module BS2007 1983 Specification for circular gear shaving cutters, 1 to 8 metric module,accuracy requirements BS2062 Pt 1 1985 Specification for gear hobs — Hobs for general purpose: 1 to 2O dp., inclusive BS2082 Pt2 1986 Specification for gear hobs — Hobs for gears for turbone reduction and similar drives BS2518 Pt 1 1983 Specification for rotary form relieved gear cutters - Diametral pitch BS2518 Pt2 1983 Specification for rotary relieved gear cutters - Metric modules T129

Catalog D190

BS 2519 Pt 1 1976 BS 2519 Pt 2 1976 BS 2697 1976 BS 3027 1968 BS 3696 Pt 1 1984 BS BS BS BS BS

4517 4582 4582 5221 5246

1984 Pt 1 1984 Pt 2 1986 1987 1984

BS 6168 1987

TABLE 1.37 CONTD. FOREIGN METRIC GEAR STANDARDS UNITED KINGDOM CONT. — BSI (British Standards Institute) 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, Milton Keynes MK146LE, United Kingdom.

ADDITIONAL GEAR DESIGN LITERATURE AND SOFTWARE From noted authorities In 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 description and ordering information on pages T159 and T160 T130

Catalog D190

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 also being enhanced with new developments, both in materials and processing. In this regard plastics contrast somewhat dramatically from metals, in that the latter materials and processes are essentially fully developed and, therefore, are in a relatively static state of development. Among the various methods of producing plastic gears, molding is unique in many respects. For that reason, it is singled out for in-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 most significant 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

Catalog D190 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 vary with regard to strength, rigidly, dimensional resistance, fabrication requirements, moisture absorption etc. Standardized tabular data is available from various manufacturers catalogs. In general, the information and data is less simplified and fixed than for the metals. This is because plastics are subject to wider formulation variations and are often regarded as proprietary compounds and mixtures. Tables 1.38 through 1.43A are representative listings of physical and mechanical properties of gear plastics taken from a variety of sources. It is common practice to use plastics in combination with different metals and materials other than plastics. Such is the case when gears have metal hubs, inserts, rims, spokes, etc. In these cases one must be cognizant of the fact that plastics have an order of magnitude different Coefficients of Thermal Expansion as well as Density and Modulus of Elasticity. For this reason TABLE 1 .43A is presented. Other properties and features that enter into considerations for gearing are given in Table 1.44 (Wear) and Table 1.45 (Poisson’s Ratio). 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 Table 1.46. In this table, rate refers to expansion from dry to full moist condition. Thus, a 0.20% rating means a dimensional increase of 0.002 inch per inch. Note that this is only a rough guide as exact values depend upon factors of composition and processing, both the 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

Acetal ABS Nylon 6/6 Nylon6/1O Polycarbonate High Impact Polystyrene Polyurethane Polyvinyl Chloride Polysulflon MoS2 Filled Nylon

Tensile Strength Flexural Compressive Strength Modulus (psi x (psi x 103) (psi x 103) 103) 8.8-1.0 13-14 410 4.5-8.5 5-13.5 11.2-13.1 7-8.5 8-9.5

14.6 10.5 11-13

1.9-4

5.5-12.5

4.5-8 6-9

7.1 8-15

10.2 10.2

16.4 10

120-200 400 400 350

300-580 85 300-400 370 330

Heat Water Mold Distortion Rockwell Absorption Shrinkage Temperature Hardness (% 24 hr) (in./in.) (oF @ 264 psi) 230-255 180-245 200 145 285-290

0.25 0.2-0.5 1.3 .4 0.15

M94 R120 R80-120 R115-123 R111 M70 R112

0.022/0.003 0.007/ 0.007 0.015 0.015/0.005 0.007/ 0.003

160-205

.05-.10

M25-69

0.005

180-205 140-175

.60-.80 07-.40

M29 R90 R100-120

0.009/0.002 0.004

345 140

0.22 0.4

M69-R120 D785

0.0076 0.012

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, Celanese Corporation, 26 Msin St., Chaitham, N.J. 07928 T132

Catalog D190 TABLE 1.39 PROPERTY CHART FOR BASIC POLYMERS FOR GEARING Water Mold Tensile Flexural Izod deflect. Absorption Shrinkage Strength Modulus Impact Temp. 24 hr *Yield Strength @264 psi #Break Notched

Units ASTM 1.Nylon 6/6 2.Nylon6 3.Acetal 4.Polycarbonate 30% G.F 15% PTFE 5.Polyester (thermoplastic) 6.Polyphenylene Sulfide 30% Sulfide 15% PTFE 7.Polyester elastomer 8.Phenolic (molded)

% D570 1.5 1.6 0.2 0.06

In./In. D955 .015/.030 .013/.025 .016/.030 .0035

psi D638 *11,200 *11,800 *10,000 *17,500

psi D790 175,000 395,000 410,000 1,200,000

ft-lb/in. D256 2.1 1.1 1.4/2.3 2

oF

D648 220 150 255 290

Coeff. Linear Thermal Expan.

Specific Gravity

10-5 oF D696 4.5 varies 4.6 5.8 1.50

1.13/1.15 1.13 1.42 1.55

D792

1.3 1.69

0.08

.020

0.03

.002

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

340,000

1.2

130

1,300,000

1.10

500

5.3

1.25 1.42

1.50

0.3

.012

0.45

.007

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

--

--

122

340,000

.29

270

10.00 3.75

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

"DELRIN" Properties - Units

ASTM

Yield Strength, psi Shear Strength, psi Impact Strength, (Izod) Elongation at Yield,% Modulus of Elasticity, psi Hardness, Rockwell Coefficient of Linear Thermal Expansion, in/in. oF Water Absorption 24hrs.% Saturation, %

500

100

D638 10,000 D732 9,510 D256 1.4 D638 15 410,000 D790 M94 R120 4.5 x 10-5 D785 D696

Specific Gravity

D570

2.3 75

0.25 0.9 1.425

"ZYTEL" 100 .2% Moisture

2.5% Moisture

11,800 9,600 0.9 5 410,000 M79 R118

8,500 2.0 25 175,000 M59 R108

4.5 x 10-5

1.5 8.0 1.14 1.14

D792 Test conducted at 73o F Reprinted with the permission of E.l. DuPont de nemours and Co.; see Ref. 8 T133

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Catalog D190 TABLE 1.46 MATERIAL RANKING BY WATER ABSORPTION RATE Material

Rate of change %

Polytetrafluoroethylene Poly ethylene: medium density high density high molecular weight low density

0.0