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The Geometry of Involute Gears

J.R. Colbourne

The Geometry of Involute Gears

With 217 Illustrations

Springer-Verlag New York Berlin Heidelberg

London Paris Tokyo

J.R. Colbourne

Department of Mechanical Engineering The University of Alberta Edmonton, Alberta Canada T6G 2G8

Library of Congress Cataloging in publication Data Colbourne. J.R.

The geometry of involute gears. Bibliography: p. Includes index. I. Gearing. Spur. I. Title.

TJ 189.C65 1987 621.8'331 87-4843 © 1987 by Springer-Verlag New York Inc. Softcover reprint of the hardcover I st edition 1987

All rights reserved. This work may not be translated or copied ill whole or in part without the written permission of the publisher (Springer-Verlag, 175 Fifth Avenue. New York, New York 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication. even if the former are not especially identified, is not to be taken as a sign that such names. as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

987654321

[SBN-13: 978-1-4612-9146-6 e-[SBN-13: 978-1-4612-4764-7 00[: 10.1007/978-1-4612-4764-7

Table of Contents

Introduction

PART SPUR GEARS

1. The Law of Gearing 9

The Requirement for a Constant Angular Veloci ty Ratio 9

Rack and Pinion 13

Law of Gearing for Two Gears 19

Conjugate Profiles and the Basic Rack 23

2. Tooth Profile of an Involute Gear

Basic Involute Rack

Standard Pitch Circle

The Involute Tooth Profile

The Involute Function

Pressure Angle of a Gear

Tooth Thickness

Specifying a Spur Gear

3. Gears in Mesh

A Pinion Meshed wi th a Rack

A Pair of Gears in Mesh

Imaginary Rack

Fundamental Circles of a Gear

Advantages of the Involute Profile

4. Contact Ratio Interference and Backlash

Contact Ratio

Interference

Backlash

24

24

25

27

30

33

41

49

53

53

65

72

78

79

83

83

91

97

vi

5. Gear Cutting I Spur Gears

Form Cutting

Shaping with a Pinion Cutter

Advantages of Generating Cutting

Shaping with a Rack Cutter

Hobbing

Cutter Tooth Tip Geometry

Undercutting

6. Profile Shift

Definition of Profile Shift

Geometric Design of a Spur Gear Pair

Alternative Names for Profile Shift

7. Miscellaneous Circles

Contents

110

110

112

122

123

128

133

140

148

151

155

168

174

Highest and Lowest Points of Single-Tooth Contact 174

Form Diameter 177

Undercut Circle 179

8. Measurement of Tooth Thickness

Gear-Tooth Vernier Caliper

Span Measurement

Measurement Over Pins

9. Geometry of Non-Involute Gears

General Theory

Fillet Shape Cut by a Rack Cutter

Fillet Shape of an Undercut Gear

Profile Modification

Fillet Shape Cut by a Pinion Cutter

10. Curvature of Tooth Profiles

Involute Radius of Curvature

Euler-Savary Equation

Gear Tooth Fillet Radius of Curvature

11. Tooth Stresses in Spur Gears

Contact Force Intensi ty

Contact Stress

Fillet Stress

191

192 196

200

207

207

212

216

218

221

229

229

229

235

241

243

244

248

Contents

12. Internal Gears

Tooth Profile of an Internal Gear

Meshing Geometry of an Internal Gear Pair

Tip Interference

Axial and Radial Assembly

Cutting Internal Gears

Shape of the Fillet

Undercutting

Rubbing

Geometric Design of an Internal Gear Pair

Measurement of Tooth Thickness

PART 2 HELICAL GEARS

vii

259

259

266

272

275

279

283

288

292

294

297

13. Tooth Surface of a Helical Involute Gear 305

The Basic Helical Rack 309

Standard pi tch Cylinder of a Helical Gear 313

The Helix and the Involute Helicoid 318

The Generator Through Point A 331

Direction of the Normal to the Tooth Surface at A 337

Tooth Thickness 350

Span Measurement 354

Tooth Profile in the Normal Section 359

Specifying a Helical Gear 362

14. Helical Gears in Mesh 366

A Pinion Meshed wi th a Rack 366

A Pair of Helical Gears in Mesh 375

Contact Ratio 382

Backlash 387

position and Orientation of the Contact Line 393

15. Crossed Helical Gears 405

Rack and Pinion 406

A Crossed Helical Gear Pair 414

Path of Contact 423

Contact Ratio 432

Backlash 437

Tooth Contact Force and Bearing Reactions 445

viii Contents

16. Gear Cutting II Helical Gears 451 Shaping wi th a Pinion Cutter 451

Shaping wi th a Rack Cutter 454

Hobbing 457

Hobbing Machine Gear Train Layout 469

Use of a Differential in the Hobbing Machine 472

Theoretically Correct Shape for the Hob Thread 479

Geometric Design of a Helical Gear Pair 483

17. Tooth Stresses in Helical Gears 489

Tooth Contact Force 489

Contact Length 491

Contact Stress 502

Fillet Stress, and the Equivalent Spur Gear 508

Bibliography 525

Index 527

a

b

c

e f

h

k

m

p

r

s

t

u

v

w

x,y,z

Notation

Lower Case Symbols

Addendum Dedendum

Clearance Profile shift

Special function defined by Equation (13.89) Length on tooth of rack cutter

Dimensionless factor Ratio pi tch

Radius of circles not centered at the gear axis Coordinate, measured along path of contact

Tooth thickness position of rack

Velocity Space width; Tooth contact force intensity

Coordinates

Upper Case Symbols

A Point on tooth profile

B Point where the involute meets the base circle; Backlash

C Gear center; Center distance

D Diameter

E Interference point; Young's modulus

F Face width

G Point where the generator meets the base cylinder H End points of the contact path during cutting

I,J AGMA geometry factors

x

K

L

M

N

P

Q

Notation

Upper Case Symbols (continued)

Dimensionless factor

Lead

Measurement over pins; Torque

Number of teeth

Pitch point; Diametral pitch

End points of single-tooth contact

R Radius of circles centered at gear axis; Polar coordinate

S Span measurement

T End points of the contact path

W Tooth contact force

X,Y

m n p

v

a

f

p

o

T

w

Ital ic Symbols

Greek Symbols

Coordinates

Module

Uni t vector

Posi tion vector

Veloci ty vector

General angle

Angular position of gear

Angle between involute tangent and tooth center-line

Roll angle

Coordinates

Polar coordinate

Lead angle

Generator inclination angle; Poisson's ratio

Radius of curvature

Stress; Swivel angle

Time

Profile angle; Pressure angle

Helix angle

Angular velocity, measured in radians/sec

Shaft angle

Notation

b

c

d

e

f

g

h

i

n

p

r

s

t

u

w

x,y,z

F

L

P

R

T

L71, S >..

c

xi

Subscripts

At the base cylinder

Contact; Cutter

At the defining circle of the equivalent spur gear

Equivalent spur gear

Fillet

Gear

Hob; At the highest point of the involute

1 or 2

Normal

At the pitch cylinder

Rack; Rack cutter

Standard; At the standard pi tch cylinder

Transverse; Tensile

Undercut

At the load point

In the x,y,z direction

Face

Limit

Profile

At radius R

At the tip cylinder

In the L71,S directions

In the direction of the helix binormal

In the direction of the helix tangent

Superscript

Cutting

INTRODUCTION

Of all the many types of machine elements which exist

today, gears are among the most commonly used. The basic idea of a wheel with teeth is extremely simple, and dates back

several thousand years. It is obvious to any observer that one

gear drives another by means of the meshing teeth, and to the person who has never studied gears, it might seem that no

further explanation is required. It may therefore come as a surprise to discover the large quantity of geometric theory that exists on the subject of gears, and to find that there is

probably no branch of mechanical engineering where theory and practice are more closely linked. Enormous improvements have been made in the performance of gears during the last two hundred years or so, and this has been due principally to the

careful attention given to the shape of the teeth. The theoretical shape of the tooth profile used in most modern gears is an involute. When precision gears are cut by modern gear-cutting machines, the accuracy with which the actual teeth conform to their theoretical shape is quite remarkable, and far exceeds the accuracy which is attained in the manufacture of most other types of machine elements.

The first part of this book deals with spur gears, which

are gears with teeth that are parallel to the gear axis. The

second part describes helical gears, whose teeth form helices

about the gear axis. The book is primarily about involute

gears, since this type of gear is by far the most commonly

used. However, the first chapter introduces the Law of

Gearing, which must be satisfied by any pair of gears, and the

statements made apply not only to involute gears, but are also

true for non-involute types of gear. There is one other

chapter of the book which also deals with non-involute gears. Chapter 9 is on the general theory of gear tooth geometry, and

2 Introduction

is included in the book simply because the tooth profiles of involute gears contain sections which are not involute. In particular, the part of each tooth near its root, known as the fillet, is not an involute, but its shape can be found from the general theory of gears. And in some gears, small alterations from the involute shape, known as profile modifications, are made in the teeth, and again the final shape of the teeth can be found by means of the general theory.

In helical gears, the angle between the helix tangent and the gear axis is known as the helix angle. Spur gears can be regarded as helical gears, in which the helix angle is zero. Since a spur gear is therefore simply a special case of a helical gear, it might be asked why the two types should be dealt with separately. However, the geometry of spur gears is considerably simpler than that of helical gears, and it is therefore convenient to describe it first. The cross-section of a helical gear perpendicular to its axis, known as its transverse section, is the same as the cross-section of a spur gear, so a knowledge of spur gear geometry makes a good starting point for the study of helical gears.

The treatment of spur gear theory in this book is fairly conventional, except in one respect. No distinction is made between a gear pair meshed at the standard center distance, and one at extended centers. The terminology and the notation are the same for both cases. In conformity with this principle, the name "pitch circle" is always used for the circle of a gear which passes through the pitch point, and its radius is always represented by the symbol Rp' whatever its value. It is important to make a clear distinction between the pitch circle when a gear is in operation, and the pitch circle when it is meshed with its basic rack, which is used as a reference circle. For this reason, the pitch circle when the gear is meshed with its basic rack is called the standard pitch circle, and its radius is labelled Rs' where the subscript s is used to indicate the word "standard".

Apart from this change, the definitions and notation in this book have been chosen to conform as closely as possible with those used in current North American practice. However, a few additional alterations have been made, in cases where

Introduction 3

the existing terminology is confusing. For example, the phrase "pre'ssure angle" is used at present for several

different angles. Its original meaning is the angle between the line of action and the common tangent to the pitch circles, but it is used also for the angle between the tooth profile and the tooth center-line of a rack, and the angle between the radius and the profile tangent of a gear tooth at either the standard pitch circle or the pitch circle. In addi t ion, the angle between the radi us and the prof i Ie tangent at a typical radius R of the tooth profile is also commonly known as the pressure angle. In current usage, all these angles are called either the pressure angle or the operating pressure angle, and they are all represented by the symbol ~ when they are equal to the pressure angle of the basic rack, and ~' when they are not. Over the years, several attempts have been made to rename some of these angles, but the proposed alternative names have not been widely accepted, so in this book the name "pressure angle" has generally been retained, while the notation has been altered to help identify the particular angle that is referred to. The angle between the radius and the profile tangent at a typical point of the gear tooth profile is referred to as the profile angle at radius R, and it is represented by the symbol ~R' The profile angles of a gear at the standard pitch circle and the pitch circle are called the pressure angle and the operating pressure angle of the gear, with the symbols ~s and ~P' and the pressure angle of the basic rack is represented by the symbol ~r' Finally, the angle between the line of action and the common tangent to the pitch circles is called the operating pressure angle of the gear pair, and this is the only angle for which the customary symbol ~ is still used.

The symbols for the different pressure angles are distinguished by their subscripts, and the same convention is used for all quantities, such as the circular pitch and the

tooth thickness, whose values are functions of the radius. The subscripts R, s, p, or b are used whenever a quantity is measured on a gear tooth at a general radius R, at the

standard pitch circle, at the pitch circle, or at the base circle, and the subscript r applies to the corresponding quantity measured on a rack tooth.

4 Introduction

The second part of the book deals with the geometry of helical gears, and the treatment differs substantially from

the traditional approach. The sUbject is essentially

three-dimensional, and in the past the geometr ic theorems

have usually been proved with the help of projective geometry. In other fields of mechanics, projective geometry

has been largely superseded by vector methods, and in the

author's opinion, most of the theorems relating to helical

gears can be proved far more easily using vector algebra than using projective geometry. The entire description of helical gears in this book is therefore given with the help of vector

theory. I t is believed that most younger engineers, and today's engineering grounding in vector

students, receive a more thorough

theory than they do in projective geometry, and should therefore find thi s new approach to helical gear geometry easier to understand.

The word "pitch" is used in this book in a manner which differs slightly from the customary North American usage, and is in fact based on current European practice. In North America, the phrase "circular pitch" describes a particular

length on a spur gear, while "diametral pitch" is a quantity used to indicate the tooth size, defined as the number of teeth divided by the diameter of the standard pitch circle. The original meaning of the word "pitch" is the distance between similar objects that are repeated at regular intervals. In this book, the word is only used in a manner which conforms with the original meaning. The of a spur gear is defined in the usual

circular pi tch way, and the

corresponding lengths in a helical gear are called the

transverse pitch, the normal pitch, and the axial pitch. In

general, the diametral pitch is not referred to in this book,

since it is not a pitch in the sense described above. In order

to specify the tooth size of a gear we use the module, which

is the method used throughout Europe and Japan. However,

since the diametral pitch is still in common use in North

America, the relation between the module and the diametral pitch is described in the text, and the diametral pitch is

used in some of the examples at the end of each chapter. A list of references is provided, and this consists of a

number of books and articles which the author has found

Introduction 5

particularly helpful in his own study of gear geometry. In

addition, several articles are listed because they describe,

in considerable detail, certain topics which are only

outlined in this book. The list is not intended to include all

possible references, and no attempt has been made in the text to identify a source for each idea or theorem. In some cases,

it would probably be very di ff icult to di scover where a

particular idea originated. When any reference is quoted in the text, it is identified by a number in square brackets,

which refers to the number in the list of references at the

end of the book. In the diagrams throughout the book, the gear tooth

profiles were drawn by a computer-driven plotter. The remaining parts of the diagrams were drawn by Mr. Hiroshi Yokota, of the University of Alberta. The author would like to

thank him for his excellent work. The author also wishes to

express his appreciation to the University of Alberta, and to

the Natural Sciences and Engineering Research Council of Canada, both of whom provided support for the project.

This book is about the theoretical geometry of involute

gears, and is not intended to replace the many books and manuals that exist on the design of gears. However, the full potential of the involute as a tooth profile can only be used by a designer who has a good understanding of its fundamental

geometric properties. It is hoped that the book will contribute to that understanding.

PART 1

SPUR GEARS

Chapter 1 The Law of Gearing

External and Internal Gears

A spur gear is a gear cut from a cylindrical blank, with

teeth which are parallel to the gear axis, as shown in Figure 1.1. If the teeth face outwards, the gear is called an external gear, and if they face inwards, like those of the gear shown in Figure 1.2, the gear is known as an internal

gear. Much of the geometric theory of gears applies equally to

both external and internal gears. However, for the sake of

clarity, this book is restricted to the subject of external gears, except for a single chapter. The exception is Chapter 12, where we will show which parts of the theory of

external gears are valid for internal gears, and we will discuss the special features that apply only to internal gears.

The Requirement for a Constant Angular Veloci ty Ratio

When two gears rotate together, as shown in Figure 1.3,

the teeth of each gear pass in and out of mesh with those of

the other gear, and this occurs in an area that lies somewhere

between the gear centers C1 and C2 . The teeth from the two

gears pass through the meshing area alternately, first one

from gear 1, then one from gear 2, and so on. Hence, if the

gears have N1 and N2 teeth, and during a certain time interval

T the number of teeth from each gear passing through the meshing area is n, then the gears will make respectively

(n/N 1) and (n/N2) revolutions. By expressing the number of

'0 The Law of Gearing

Figure'. ,. An external gear.

revolutions in radians and dividing by the time taken, we

obtain average- values for the gear angular velocities w, and w2 ,

(w, ) average

(w2 )average

(...!l.)211' N, T

_ (...!l.) 211' N2 T

Figure '.2. An internal gear.

( 1.1)

(1. 2)

Constant Angular Veloci ty Ratio 11

where the minus sign indicates that the direction of rotation for gear 2 must be opposite to that of gear 1. From Equations (1.1 and 1.2), we can immediately obtain a relation between the average angular veloc i ties,

- N (w ) 2 2 average ( 1. 3)

Equation (1.3) is true for all gears, whatever the shape of the teeth. However, if the tooth shape is arbitrary, the gears will not run smoothly. Suppose gear 1 is driving, and turns at a constant angular velocity. In general, the angular veloci ty of gear 2 wi 11 not be constant, but wi 11 be a periodic function, repeating itself as each pair of teeth are meshed, with an average value given by Equation (1.3). The variation in angular veloci ty of gear 2 leads to vibrations in the gear train, and will generally cause fatigue cracks to form in the teeth, resulting in early failure of the gears.

Theoretical studies of gear tooth profiles date back to the 16th Century, but for many years the craftsmen who cut gears made no use of the knowledge that was available. Most machinery was quite slow-moving, and vibration was not considered important. Toward the end of the 18th Century,

Gear 2

Figure 1.3. A gear pair.

12 The Law of Gearing

machine designers began to make greater demands on the gears

in the machines they bui It. The gears turned faster than

before, and were more heavily loaded. Tooth breakage then

developed into a serious problem, and it became necessary to

choose tooth profiles which would allow the driven gear to maintain a constant angular velocity, whenever the driving

gear angular velocity was constant. To achieve this end, the

angular velocity ratio (w 1/w2 ) must remain constant at all

times, and not simply on average, as described by Equation (1.3). The new requirement for the angular

veloci ties can be expressed by the equation,

( 1 .4)

The purpose of this chapter is to determine the condition that must be satisfied by the meshing tooth prof i les, if the gears are to have the constant angular velocity ratio given in Equation (1.4). However, before looking at the case of two gears meshing together, we will consider that of a gear meshing with a rack.

J 1

1 r Figure 1.4. A rack and pinion.

Rack and Pinion

Profile angle ¢Ar /

Tooth thickness = p/2

Space width = p/2

Pitch P

Xr

Tangent at Ar

Figure 1.5. Rack pitch and profile angle.

Rack and Pinion

13

A rack is a segment of a gear whose radius is infinite.

I f the number of teeth N2 of gear 2 in Figure 1.3 were extremely large, the radius of the gear would also be large, relative to the tooth size, and the teeth near the meshing

area would lie almost on a straight line. In the limit, as N2 becomes infinite, the teeth would lie exactly on a straight line, as shown in Figure 1.4.

When two gears mesh, the smaller of the two is called the pinion, and the larger is usually referred to as the gear. Any gear meshed with a rack is considered smaller than the rack,

since the rack is part of a gear with an infinite number of

teeth. Hence, it is common to speak of a rack and pinion.

Whereas a gear pair is used to transmit rotary motion between

shafts, a rack and pinion are used to convert rotary motion

into linear, or vice-versa. One well-known application is the

rack and pinion steering of many automobiles.

Part of a rack is shown in Figure 1.5. The pitch p is the

distance between corresponding points of adjacent teeth. If we draw any line along the rack parallel to the line of teeth,


the intersection of this line with the tooth profiles will determine the tooth thickness and the space width, measured along that particular line. We define the rack reference line as the line along which the tooth thickness and the space width are equal, and since their sum is equal to the pitch p, the tooth thickness and the space width measured along the rack reference line must each be equal to (p/2). We now introduce coordinates xr and Yr fixed in the rack, with their origin on the rack reference line. The xr axis lies along a tooth center-line, and the Yr axis coincides with the rack reference line, as shown in Figure 1.5. A typical point of the rack tooth profile is labelled Ar' and the tangent to the tooth profile at this point makes an angle ~Ar with the x axis. The angle ~Ar is called the rack profile angle a~ point Ar •

In relating the rack velocity to the pinion angular velocity, the reasoning is identical to that used earlier for two gears. During any time interval T, the number n of rack teeth passing through the meshing area is equal to the number of pinion teeth which pass through. Thus, average values for the rack velocity vr and the pinion angular velocity ware given by the following expressions,

(W)average

!!E T

( 1. 5)

( 1. 6)

where vr is defined as positive in the upward direction, and W

is defined as positive when the angular velocity is counter-clockwise.

The relation we require is obtained by eliminating (niT)

from Equations (1.5 and 1.6).

N 21r(w)average ( 1. 7)

Equation (1.7) is exactly analogous to Equation (1.3). As with a pair of gears, the satisfactory operation of a rack and pinion requires that the relation between vr and w should remain constant. Hence, the tooth shapes must be such that vr and w satisfy the following equation,

Rack and Pinion

vr p

Nw 271'

15

( 1. 8)

Figure 1.6 shows the rack tooth and pinion tooth profiles, with points Ar of the rack and A of the pinion in contact. Since the teeth remain in contact but do not penetrate into each other, the velocity components of Ar and A along the common normal must be equal. In order to describe the velocities, we introduce a fixed set of unit vectors nt, n~, and nS' The directions of n t and n~ are perpendicular and parallel to the rack reference line, as shown in Figure 1.6, and nS is perpendicular to the plane of motion. The unit vector nnr' in the direction of the outward-pointing normal to the rack tooth profile at point Ar' can then be expressed in terms of n t and n~,

. Ar Ar - 51n IP n - cos IP n t ~ ( 1. 9)

Since Ar is the contact point, the unit vector nnr lies in the direction of the common normal. The velocity of Ar' and its component along the common normal, are given by the following two equations,

x

I..-Rack reference line

Common tangent

Figure 1.6. The common normal at the contact point.


A - cos ~ r v r

(1.10)

(1.11)

where the dot indicates the scalar product between two vectors.

If the vector from the pinion center C to point A is (xnE+Yn~), then the velocity of point A and its component along the common normal can be expressed as follows,

VA wnS x (xnE+Yn~) - wYne + wxn~ (1.12)

A nnr vA . Ar Ar (1.13) vn wY Sln ~ - wX cos ~

where the symbol x in Equation (1.12) indicates the vector product.

We now equate the normal velocity components of Ar and A, given by Equations (1.11 and 1.13), and use Equation (1.8) to express the relation we require between vr and w. We then obtain the following equation that must be satisfied by X and Y, the coordinates of the contact point.

Pinion pitch circle

c

I-Rack reference line

Figure 1.7. pitch point of a rack and pinion.

Rack and Pinion

y

(x - ~) 211"

A cot 1/1 r

17

(1.14)

Equation (1.14) can be interpreted in the following

manner. There is a fixed point P, at a distance (Np/211") from C

on the line through C perpendicular to the rack reference

line, such that the slope of line PA is equal to cot I/IAr • This means that line PA makes an angle (11"/2 _I/iAq with the n~ direction, and it is therefore the common normal at the

contact point A, since the common tangent makes an angle I/IA r

with the n~ direction. The position of point P is shown in

Figure 1.7. The result just proved is known as the Law of Gearing, as

it relates to a rack and pinion. It may be stated in the following way. The condition that must be satisfied by the

tooth profiles of a rack and pinion, in order that the relation between rack velocity and pinion angular velocity should remain constant, is that the common normal at the

contact point should at all times pass through a fixed point P. The position of P is at a distance (Np/211") from the pinion center C, on the perpendicular from C towards the rack reference line.

The point P is called the pitch point. The circle passing through P whose center is at C is called the pinion pitch circle, and its radius Rp is equal to the length CP,

(1.15)

In Equation (1.8), we gave the relation that we require between the rack velocity and the pinion angular velocity. We used that relation to prove the Law of Gearing, which lead us to define the pitch point and the pinion pitch circle. Having

now derived an expression in Equation (1.15) for the pitch

circle radius, we can combine Equations (1.8 and 1.15), and we

obtain a simpler form for the relation between the rack

velocity and the pinion angular velocity,

(1.16)

The line in the rack which touches the pinion pitch circle at P, as shown in Figure 1.7, is known as the rack


pitch line. When the pinion and rack are in motion, the velocity of any point on the pinion pitch circle is equal to RpW, and the velocity of any point on the rack pitch line is equal to vr ' Since these velocities are equal, as we can see from Equation (1.16), the motion of a rack and pinion is identical to the motion that would be obtained if the rack pitch line and the pinio~ pitch circle were to make rolling contact with no slipping.

Ci rcular pi tch

The circular pitch of the pinion teeth at any radius R is defined as the distance between corresponding points of adjacent teeth, measured around the circumference of the circle of radius R. Thus, the circular pitch PR at radius R, which is shown in Figure 1.8, is given by the following expression,

211'R N

(1.17)

In the case when the circular pitch is measured on the pitch

R

C

Figure 1.8.

Circular pitch PR at radius R

Circular pitch Pp

at the pitch circle

Circular pitch.

Law of Gearing for Two Gears

circle, we use the symbol Pp' and its value can substituting Rp in place of R in Equation (1.17).

211'Rp Pp N

19

be found by

(1.18)

When we replace the pitch circle radius Rp in this equation by the expression given in Equation (1.15), it is clear that the circular pitch of the gear at its pitch circle is equal to the pitch p of the rack,

p (1.19)

This result can be used to provide an alternative definition of the pitch circle of a pinion, when it is meshed with a rack. The pitch circle can be defined as the circle on which the pinion circular pitch is equal to the rack pitch p.

Law of Gearing for Two Gears

It was shown in the previous section that a rack and pinion behave in the same manner as if the rack pitch line and the pinion pitch circle were to make rolling contact with no

Pitch circle of gear 1-

17 r-Pitch circle of gear 2

/nn Common normal

TJ

C

Figure 1.9. pitch point of a gear pair.


slipping. We now investigate whether the same idea can be used

for two gears. First, we find two pitch circles which, if they

made rolling contact with each other, would provide the same

angular velocity ratio as the gears. And then we will

establish that the Law of Gearing also applies for a pair of

gears, or in other words, that the common normal at the tooth

contact point always passes through a fixed point.

Figure 1.9 shows two gears, with point Al of gear 1 in

contact with point A2 of gear 2. The distance C between the

gear centers is called the center distance. Parts of the pitch

circles have been drawn in, and their radii are shown as RPl

and Rp2 . The point where they touch is the pitch point P.

If the pitch circles are to make rolling contact with no

slipping, their radii must satisfy the following equations,

C (1.20)

(1.21)

The angular velocity ratio that we require was given in

Equation (1.4),

(1.22)

Equations (1.21 and 1.22) imply that the ratio of the pitch

circle radii is equal to the ratio of the tooth numbers,

( 1. 23)

We now solve Equations (1.20 and 1.23), to obtain the radii of

the pi tch ci rc les,

( 1 .24 )

(1.25)

The pitch point P, which is the point where the pitch

circles touch, therefore lies on the line of centers and

divides C1C2 in the ratio N1:N2 . We use this point as the

origin of a fixed system of coordinates E, ~ and S, with axes

Law of Gearing for Two Gears 21

in the directions shown in Figure 1.9. The position of the contact point relative to the pitch point is then given by the

coordinates ~ and 1/. As we did in the case of the rack and pinion, we write

down the velocities of points Al and A2 , and then equate their components along the common normal. The direction nn of the common normal, which is unknown at present, can be written in the following form,

where s~ and s1/ are the components of nn in directions. Then the velocities of Al and components in the normal direction, are following four equations.

Al v = w 1nS x [(Rpl+~)n~+1/n1/]

A2 v

(1.26)

the coordinate A2 , and their given by the

( 1. 29)

( 1 .30)

A A Equating the expressions for vn 1 and vn 2 , we obtain a relation which must be satisfied by the vector components s~ and s1/'

o (1.31)

The expression between the square brackets is zero, as we can see from Equation (1.21). The angular velocities w1 and w2 can never be equal, because for a pair of external gears the angular velocities must be of opposite sign, and for a pinion

meshed with an internal gear, the pinion angular velocity

must be greater than that of the internal gear. Hence, the term (w 1-w2 ) cannot be zero, and it follows that the remaining term is zero. The condition given by Equation (1.31) therefore reduces to the following form,


( 1. 32)

Equation (1.32) can be interpreted as showing that the unit vector nn along the common normal is parallel to line PA 1. In other words, the common normal at the contact point must always pass through the pitch point, which is the point that divides the line of centers C1C2 in the ratio N1 :N2 • This is the statement of the Law of Gearing, as it applies to a pair of gears.

We proved in Equation (1.19) that when a pinion is meshed with a rack, the circular pitch of the pinion at its pitch circle is equal to the pitch of the rack. A similar result is also true for a pair of gears. The circular pitch of each gear at its pitch circle is given by Equation (1.18),

Pp1 211'Rp1

N1 ( 1. 33)

Pp2 211'Rp2

N2 (1.34)

When we use Equations (1.24 and 1.25) to express the pitch circle radii RP1 and Rp2 ' it is immediately clear that the circular pitches of the two gears must be equal,

(1.35)

Path of Contact

The locus of successive contact points between a pair of teeth is called the path of contact. One consequence of the Law of Gearing is that the path of contact must pass through the pitch point. To prove this statement, we need only consider the situation if it were not true. If the path of contact were to cross the line of centers at any point P' which is not the pitch point, then when the contact point was at P', the common normal at the contact point would not pass through the pitch point, and the gear pair would not satisfy

the Law of Gearing.

The Basic Rack 23

Conjugate Profiles and the Basic Rack

Any pair of tooth profiles that satisfy the Law of

Gearing are said to be conjugate. If the tooth profile of one

gear is chosen arbitrarily, it is possible to find a tooth

profile for the other gear, such that the two profiles are conjugate. In particular, we can specify a rack tooth

profile, and then define a system of gears as having tooth

profiles which are conjugate to the chosen rack. This is the method generally used by various organisations, such as the

American Gear Manufacturers Association (AGMA) and the

International Organisation for Standardization (ISO), to

define the tooth profiles for a system of gears. The rack

tooth profile is then known as the basic rack for the system

of gears.

In the next chapter, we will consider a particular basic

rack, which is the one used to define the tooth profile of an

involute gear, and we will then describe the geometry of the gear teeth.

Chapter 2 Tooth Profile of an Involute Gear

Basic Involute Rack

In general, the tooth profile of a rack may be curved, and the profile angle ~Ar would then vary from one point of the tooth to another. We now consider a particular rack, in which the teeth are straight-sided. This is the basic rack which we use to define the tooth shape of an involute gear. The profile angle for this rack is constant, and the value of the constant will be represented by the symbol ~r' which is called the pressure angle of the basic rack. Thus, for the basic rack used to define involute tooth profiles,

constant (2.1)

In some cases the profile angle ~Ar may vary near the tips and the roots of the basic rack teeth. For example, the teeth may be rounded at the tips. The rack is still called an involute rack, provided a substantial part of its tooth profile is straight-sided. For the purpose of finding the shape of the gear tooth, we will start by assuming that the

basic rack has teeth which are entirely straight-sided, as

shown in Figure 2.1. The pressure angle is ~r' and we use the symbol Pr to represent the pitch of the basic rack.

Base Pi tch of the Basic Rack

The dimensions of the basic rack are determined by the

values of Pr and ~r' In addition, there is a third quantity shown in Figure 2.1 called the rack base pitch, which is

Standard pitch Circle

Pr 2"

Pitch Pr

25

Figure 2.1. Basic rack used to define the involute profile.

defined as the distance between adjacent teeth, measured along a common normal. The reason why this particular length on a rack is called the base pitch will be made clear later in this chapter. For the moment, we will simply use Figure 2.1 to express the base pitch Pbr of the basic rack in terms of its pi tch and pressure angle,

(2.2)

The three quantities Pr' Pbr and 4J r are the parameters used to describe the basic rack. Since they are related by Equation (2.2), it is clear that only two of the quantities are independent. We can choose any two, and then use Equation (2.2) to find the third.

Standard Pi~ch Circle

For any tooth profile, there are a number of quantities whose values are functions of the radius R. These include the circular pitch PR' which was introduced in Chapter 1, and the profile angle and the tooth thickness, which will be

26 Tooth Profile of an Involute Gear

discussed later in this chapter. As part of the description of a gear, it is necessary to provide the values of each of these quantities at some specified radius.

An obvious choice for this radius is the pitch circle radius of the gear when it is meshed with its basic rack. In Chapter 1 we defined the pitch circle of a gear as the circle through the pi tch point, and we used the symbol Rp to represent its radius. We showed there that the value of Rp depends on the pitch of the rack, when the gear is meshed with a rack, and on the center distance when the gear is meshed with another gear. In order, therefore, to identify the particular pitch circle of a gear when it is meshed with its basic rack, we will call it the standard pitch circle, and we will represent its radius by the symbol Rs'

In Equations (1.15 and 1.18), we gave expressions for the pitch circle radius of a gear when it is meshed with an arbitrary rack, and for the circular pitch measured at the pitch circle,

We also showed, in Equation (1.19), that the circular pitch at the pitch circle is equal to the pitch p of the rack,

When we replace the rack pitch p in these equations by Pr' the pitch of the basic rack, the first two equations give the standard pitch circle radius Rs of a gear, and its circular pi tch p at the standard pi tch ci rcle, while. the thi rd s equation shows that the circular pitch of the gear at its standard pitch circle is equal to the pitch of the basic rack,

RS NPr

(2.3) 21r

Ps 211'Rs

N (2.4)

Ps Pr (2.5)

Tooth Profile of an Involute Gear 27

The Involute Tooth Profile

We will now determine the shape of gear tooth profiles which are conjugate to the basic rack in Figure 2.1. The word

conjugate means, as we defined it in Chapter 1, that the gear

teeth are shaped in such a manner that the Law of Gearing is

satisfied, when the gear is meshed with the basic rack. In Figure 2.2, a pinion is shown meshing with the basic

rack. The plnlon pitch circle radius is Rs' given by Equation (2.3), and the pitch line is the line in the basic

rack which touches the pinion pitch circle at the pitch point P. The Law of Gearing states that the common normal at the contact point must pass through P. For any particular position of the rack, there is only one point Ar of the rack tooth profile whose normal passes through P, and this point

must be the contact point. The pinion tooth must therefore be shaped so that its profile touches the rack tooth at Ar •

c

Standard pitch circle~

Pinion base circle---. ""

Basic rack pressure angle cf>r~

---=:::::

Operating pressure angle cf>

Line of action

Figure 2.2. Meshing diagram of a pinion and the basic rack.


The point of the pinion tooth profile in Figure 2.2 which

coincides with Ar is labelled A. The line joining the contadt

point to the pitch point is called the line of action, since

it coincides with the common normal at the contact point, and

therefore in the absence of friction the contact force must act along this line. The angle between the line of action and

the tangent to the pinion pitch circle at P is called the operating pressure angle ~ of the gear pair. Since the line of

action is perpendicular to the tooth profile of the basic rack, it can be seen from the diagram that, when a gear is

meshed with its basic rack, the operating pressure angle of

the gear pair is equal to the pressure angle of the basic rack,

(2.6)

If we were to consider the basic rack in a new position, the description of the meshing geometry would be essentially

a repetition of the last paragraph. The new contact point would again lie on the line which passes through the pitch

point in a direction perpendicular to the tooth profile of the basic rack. This result is true for any position of the basic rack. Hence, the path of contact, which is the locus of all contact points, is a segment of the same straight line. And since the line of action is always the line joining the pitch point to the contact point, the direction of the line of action is fixed, and the line of action coincides with the

line containing the path of contact. We now construct the perpendicular from the pinion

center C to the line of action, and the foot of this

perpendicular is labelled E, as shown in Figure 2.2. The

pinion circle with center C and radius equal to CE is known as

the base circle, and its radius is represented by the

symbol Rb • Since the rack tooth profile and line CE are both

perpendicular to the line of action, they must be parallel, and the angle ECP is equal to ~r' We can then use triangle ECP to express the base circle radius in terms of the standard

pitch circle radius,

(2.7)

Alternative Definition of the Involute 29

The shape of the pinion tooth must be such that the

normal to the tooth profile at point A passes through P. This

is a direct statement of the Law of Gearing. using the base

circle just defined, we can restate the property of the tooth

shape a little differently. The shape of the tooth profile

must be such that the normal at the contact point touches the

base circle. As the pinion rotates, the contact point moves

along the pinion tooth, and therefore at each point of the

profile the normal to the profile must touch the base circle.

A curve with this property is known as an involute of the base

circle, and this is the origin of the name "involute gear".

Alternative Definition of the Involute

There is another manner in which the involute can be

defined. If the base circle is fixed, and a rigid bar AD rolls

without slipping on the base circle, as shown in Figure 2.3, then the path followed by point A is an involute. It is easy

to prove that the two definitions are equivalent. If point E

is the contact point between the base circle and bar, then E is also the instantaneous center of the bar as it rolls. The

r-Involute path followed I by point A of the bar

c

Rigid bar

o

Figure 2.3. A rigid bar rolling on a fixed cylinder.


velocity of point A is therefore perpendicular to EA. This

means that the tangent to the involute at A is perpendicular

to EA, and therefore the normal is along EA, which is the

property by which the involute was originally defined.

The Involute Function

The alternative definition is useful in helping to derive some of the fundamental geometric equations of the

involute. The point in Figure 2.3 where the involute curve meets the base circle is labelled B. This is the point where

the end A of the bar would meet the base circle, if the bar rolled to the position where A was the contact point. Due to the fact that the bar rolls without slipping, we can say that the length of arc EB on the base circle must be equal to the length EA on the bar. In symbolic form, this can be written,

arc EB EA (2.8)

We now need to define a number of new symbols, and to derive the relations between them. Figure 2.4 shows the base

Figure 2.4.

Tangent to involute at A

Normal to involute at A

Profile angle and roll angle.

The Involute Function 31

circle, and an involute curve starting at point B, with a typical point A at radius R. The normal to the involute at A

touches the base circle at E. We define an angle ~R' called the profile angle at radius R, as the angle between the radius

through A and the involute tangent at A. The radius CE is

perpendicular to EA, since EA touches the base circle, and CE

is therefore parallel to the involute tangent at A. Hence, the angle ECA is equal to the profile angle,

angle ECA (2.9)

Referring to triangle ECA, we obtain two immediate results,

(2.10)

EA (2.11)

Next, we define an angle ER, called the roll angle at radius R, as the angle between the radius through B and the involute tangent at A. Since CE is parallel to the involute tangent at A, the angle ECB is equal to the roll angle,

angle ECB (2.12)

and the length of arc EB is therefore given by the following equation,

arc EB (2.13)

where ER must of course be expressed in radians. We now combine Equations (2.8, 2.11 and 2.13), in order

to obtain a relation between ER and ~R'

(2.14 )

The angle between the radii CA and CB is clearly a

function of ~R' and the name inv ~R (short for involute function of ~R) has been chosen for this function. We express the angle ACB as the difference between angles ECB and ECA, and since angle ECB is given by Equation (2.14) in radians,

32 Tooth Prof i Ie of an I nvol ute Gear

the angles in the following equation are all expressed in radians.

inv I/>R angle ACB (2.15)

The function inv I/>R is used throughout the geometry of involute gears. Since, as we have shown, it represents an angle measured in radians, it is generally convenient to express other angles also in radians. For this reason, the following convention will be used in this book. Unless it is explicitly stated that an angle is given in degrees, it will be understood that the value is expressed in radians. For example, the polar coordinate 9R of a point on the tooth profile is given by Equation (2.35). The units are not given, so it is understood that the equation gives the value of 9R in radians.

When I/>R is known, the value of inv I/>R is given by Equation (2.15). It is also sometimes necessary to find I/>R'

when the value of inv I/>R is known, and this can be carried out by means of the following two steps,

q (inv If> ) 2/3 R

1.0 + 1.04004q + 0.32451q2 - 0.00321q3 - 0.00894q4 + 0.00319q5 - 0.00048q6

(2.16)

(2.17)

The maximum error given by the procedure is 0.0001°, for values of I/>R between 0° and 65°, and this range of I/>R values is sufficient for most practical purposes. The coefficients in Equation (2.17) are a simplified version of a set of ceofficients developed by Polder [9].

For the purpose of describing the geometry of a tooth, we generally use the radius R to specify any point A on the tooth profile. The profile angle at A is then given by

Equa t i on (2. 1 0 ) ,

Rb R

(2.18)

and the angle between line CA and the fixed line CB is expressed by the involute function,


----+---- Cable unwinding from cylinder

"-Fixed cylinder of radius Rb

Figure 2.5. A cable unwinding from a cylinder.

angle ACB inv /fiR

33

(2.19)

Another common description of the involute is based on the same idea as the alternative definition given earlier. We consider a cable wrapped round a fixed cylinder of radius Rb , with one end of the cable attached to the cylinder. If the

other end of the cable is partly unwound, the path followed by that end will be an involute. If Figure 2.5 were to represent the cable and cylinder, then EA would be the section of cable unwound from the cylinder, and it is obvious that the length of this part of the cable is equal to the arc EB, where the cable was originally wrapped round the cylinder.


The pressure angle /fI s of a gear is defined as the gear

profile angle /fiR at the standard pitch circle. The profile angle at radius R was given by Equation (2.10),

and the pressure angle can be found by setting R equal to Rs in this relation,

(2.20)


When Equation (2.20) is compared with Equation (2.7),

it is clear that the pressure angle of the gear is equal to that of the basic rack,

"'r (2.21)

There is a more direct proof that the two pressure angles are equal. Figure 2.6 shows the gear meshed with its basic rack, in positions such that the contact point coincides with the pitch point. If As is the point on the gear tooth profile at radius Rs' the pressure angle of the gear is defined as the angle between the radius CAs and the tooth profile tangent at As. Since As is also the contact point, when the gear and the basic rack are in the positions shown, the tangent to the gear tooth profile lies along the rack tooth profile, and the pressure angle "'s of the gear is therefore equal to the basic rack pressure angle "'r.

It is evident that the name "pressure angle" is used for several angles, each defined in a different manner. The pressure angle "'r of the basic rack is the angle between the

Basic rack pressure angle

cPs Pinion profile angle at radius Rs

Pinion tooth

Figure 2.6. Meshing diagram, with contact at the pitch point.

Base pi tch 35

tooth profile and the tooth center-line, while the pressure

angle ~ of a gear is the profile angle at its standard pitch s circle. Each of these two angles is a constant quantity associated with a particular gear, and in principle it can be measured on the gear. On the other hand, the operating

pressure angle ~ of a gear pair only exists when the two gears

are meshed. For a rack and pinion, it is defined as the angle between the line of action and the tangent to the pinion pitch circle at P, while for a pair of gears it is the angle between the line of action and the common tangent to the two pitch circles. We have shown, in Equations (2.6 and 2.21), that for

a pinion meshed with its basic rack, the three pressure angles are all equal in value. In Chapter 3 we will show that, in general, the operating pressure angle ~ of a gear pair may differ from the other two pressure angles. However, even when the values are equal, it is important to know which angle is referred to when the name "pressure angle" is used, and for

this reason the symbols ~r' ~s and ~ will be used throughout this book to distinguish the three angles.

Base pitch

The circular pitch at radius R was defined in Chapter 1 as the distance between corresponding points of adjacent teeth, measured around the circle of radius R. An expression for the circular pitch at radius R was given by Equation (1.17),

211"R N

The base pitch Pb of a gear is defined in a similar manner, as

the distance between corresponding points of adjacent teeth,

measured around the base circle. In other words, the base

pitch is the circular pitch at the base circle,

211"Rb N (2.22)

In Equation (2.10), we gave an expression for the base circle radius in terms of a typical radius R and the corresponding


profile angle /fiR'

We combine the last three equations to derive a relation between the base pitch and the circular pitch at radius R,

(2.23)

and as a special case of this equation, we set R equal to Rs' and we obtain the corresponding relation between the base pitch and the circular pitch at the standard pitch circle,

(2.24)

There is a property of involute curves which we will make use of in the chapters that follow. The normal to a tooth profile at any point A is also normal to any other involute of the base circle, and if it cuts the next tooth profile at

point A', the length AA' is equal to the base pitch Pb' Thus, the distance between adjacent tooth profiles, measured along a common normal, is equal to Pb' These results can be proved

c Figure 2.7. Base pitch.

Gear Parameters 37

with the help of Figure 2.7. The normal to the involute at A must touch the base circle at some point E, since this is the

defining property of the involute. If line EA cuts the next tooth profile at A', the normal to the second tooth profile at

A' must also touch the base circle, and therefore coincides with line EAA'. Hence, a line which is normal to one involute

is also normal to other involutes of the same base circle. To

prove that the length AA' is equal to the base pitch, we make use of Equation (2.8), which states that EA is equal to arc EB. Referring again to Figure 2.7, we have the following

relations,

AA' EA' - EA arc EB' - arc EB arc BB'

Since the involutes shown in Figure 2.7 are the profiles of adjacent teeth, arc BB' is by definition equal to the base

pitch, and the equation can be written,

AA' (2.25)

We have therefore proved the statement made earlier, that the distance between adjacent tooth profiles, measured along a common normal, is equal to the base pitch.

The definition of the base pitch given in Equation (2.22) would not apply in the case of a rack, because both the base circle radius and the number of teeth are infinite. However, earlier in the chapter we gave a definition for the base pitch of a rack, as the distance between adjacent tooth prof i les, measured along a common normal. In view of the result given by Equation (2.25), it is now possible to see that there is no essential difference

between the two definitions.

Relations Between the Gear Parameters and Those of the Basic Rack

We pointed out earlier that the parameters Pr' Pbr and 'r can be used to describe the basic rack, and for gears we introduced three corresponding quantities, the circular


pitch Ps at the standard pitch circle, the base pitch Pb' and the pressure angle 41 s • We have already shown in Equations (2.5 and 2.21) that the two pi tches and the two pressure angles are equal,

The two base pitches were expressed in terms of the remaining quantities by means of Equations (2.2 and 2.24),

When we compare these equations, bearing in mind that the pitches and pressure angles are equal, it is clear that the two base pi tches are also equal,

= (2.26)

We stated in Chapter 1 that we can define a system of gears, simply by specifying the shape of the teeth in the basic rack. The teeth of each gear in the system must be shaped so that they are conjugate to the basic rack. We have now shown that, for any gear in an involute system, the circular pitch at the standard pitch circle, the base pitch and the pressure angle must each be equal to the corresponding quantity in the basic rack.

When the geometry of a gear is described, it is necessary to refer repeatedly to the circular pitch at the standard pitch circle. Since this phrase is so cumbersome, it is common practice to describe Ps simply as the "circular pitch". There is no danger of confusion, provided the circular pitch at any other radius is clearly identified, and therefore from now on in the book we will adopt this convention. The same convention will be used for the names of several other gear tooth quantities, whose values depend on the radius, and these will be pointed out as they occur.

Tooth Profile of an Involute Gear 39

Module and Diametral pitch

When we introduced the standard pitch circle of a gear,

we stated that a number of the gear parameters are defined on the standard pitch circle. As part of the specification of a

gear, we must therefore give the radius of the standard pitch circle, or alternatively some quantity from which the radius

can be calculated. The standard pitch circle radius Rs was given originally

by Equation (2.3) in terms of the basic rack pitch Pr'

NPr """21r (2.27)

and since the pitch of the basic rack is equal to the circular pitch of the gear, the radius of the standard pitch circle can be expressed directly in terms of the circular pi tch,

(2.28)

For a system of gears conjugate to a particular basic rack, it would therefore be necessary to specify only the

value of the circular pitch Ps' which is the same for every gear in the system, and we would then use Equation (2.28) to calculate the standard pitch circle radius of each gear. This method of specification was in fact used in the past, and gears in which the circular pi tch is specified as a convenient length are known as "circular pitch gears". However, they are seldom made today, as they have one slight disadvantage. If the value of the circular pitch is chosen as a round number, the standard pitch circle radius is always an inconvenient size, due to the presence of the factor w in Equation (2.28).

It has been found more practical to design gears in which the standard pitch circle radius is a round number. With this

consideration in mind, we introduce a quantity called the

module m, defined in terms of the basic rack pitch,

m (2.29)

We now combine Equations (2.27 and 2.29), in order to express the standard pitch circle radius in terms of the module,


(2.30)

and, since once again the circular pitch of the gear is equal to the basic rack pitch, a relation between the circular pitch

and the module can be found immediately from Equation (2.29),

(2.31)

The module, which we have shown is proportional to the circular pitch, is used not only in the calculation of the standard pitch circle radius, but also as a measure of the

tooth size. When two gears are meshed together, they must clearly have teeth of approximately the same size, and in prac t ice they are des i gned with the same module m and the same

pressure angle ~s. In other words, the two gears are both conjugate to the same basic rack. We will show in Chapter 3 that these conditions ensure correct meshing of the gears.

The module can be measured in either mms or inches. In practice, it is most commonly measured in mms, since the module is generally used in countries which have adopted the

metric system. In North America, the quantity used at present to specify the tooth size of a gear is known as the diametral pitch Pd. This is defined as the number of teeth in the gear, divided by the diameter of the standard pitch circle,

N 2Rs

(2.32)

A relation between the diametral pitch and the circular pitch

can be found from Equations (2.28 and 2.32),

..JL Ps

(2.33)

and when we use Equation (2.31) to express the circular pitch

in terms of the module, it is clear that the diametral pitch

is equal to the rec iprocal of the module,

1 m (2.34)

Since the diametral pitch is expressed in teeth per inch,

Equation (2.34) requires that the module be given in inches.

Tooth Thickness 41

It seems probable that the use of the diametral pitch

will eventually be abandoned in favour of the module. The gear

geometry in this book is therefore described in terms of the module, and the module is also used in most of the examples at

the end of each chapter. However, since the diametral pi tch is

still widely used in North America, a number of examples are also included in which the tooth size is specified by means of the diametral pitch. For these problems, the module will first be found, using Equation (2.34), and the remaining

calculations will then be carried out in terms of the module.

Tooth Thickness

The tooth thickness at radius R is defined as the arc length between opposite faces of a tooth, measured around the circumference of the circle of radius R. We will show in this

section that when we know the tooth thickness at one radius, we can calculate it at any other. Thus, it is only necessary to specify the tooth thickness at one particular radius, and for this purpose we generally choose the standard pitch

circle. The symbol ts is used to designate the tooth thickness at the standard pitch circle, and tR is used for the tooth thickness at radius R. The tooth thickness ts at the standard

Tooth thickness tR at radius R

C

R

Space width wR at radius R

Circular pitch p at radius R R

Figure 2.8. Tooth thickness and space width at radius R.


C x

Figure 2.9. Tooth thicknesses at different radii.

pitch circle is described simply as "the tooth thickness", in the same way that the circular pitch at the standard pitch circle is called the circular pitch. The gap between the teeth, measured around the circle of radius R, is called the space width at radius R, and like the tooth thickness, the space width is generally measured on the standard pi tch circle. Since the tooth thickness, the space width and the circular pitch are all defined as arc lengths, as shown in Figure 2.8, it is clear that the sum of the tooth thickness and the space width at any radius R is equal to the circular pi tch at radius R.

A gear tooth is shown in Figure 2.9, with points B, As and A on the tooth profile at radii Rb , Rs and R. We will derive an expression for the tooth thickness tR at radius R, assuming the tooth thickness ts is known, and we start by finding the polar coordinate 9R of point A,

angle xCA angle xCAs + angle AsCB - angle ACB

Tooth Thickness 43

The angle ACB is given by the involute function inv 'R' as we showed in Equation (2.19), and since the profile angle at the standard pitch circle is equal to the pressure angle 's' the angle AsCB is equal to inv,s. Hence, the expression for 8R can be written,

= (2.35)

Having found the polar coordinate 8R of point A, we can immediately write down an expression for the tooth thickness at radius R,

In order to find a relation between the thicknesses at any two radii R1 and R2, we use Equation twice to write down the tooth thicknesses tR and tR '

h 1 " b h ,1 2 t en e lmlnate ts etween t e two expressIons,

tR tR = R2[r + 2(inv 'R - inv'R )]

2 1 1 2

(2.36)

tooth (2.36)

and we

(2.37)

where 'R and /fiR are the prof ile angles at the two radi i. 1 2

y

,\sase circle

c

Figure 2.10. Tooth profile tangent at radius R.

44 Tooth Profi Ie of an I nvol ute Gear

There is another quantity which will be useful in the

description of a gear tooth profile, in particular in

Chapter 11, where we discuss the tooth strength of a gear. We

define an angle YR, as shown in Figure 2.10, as the angle

between the profile tangent at point A and the tooth

center-line, which coincides with the x axis. Since line CE in

Figure 2.10 is parallel to the profile tangen~ at A, the angle

between CE and the x axis is equal to YR' and we can therefore

express YR as follows,

(2.38)

We replace 8R by the expression given in Equation (2.35), and

the equation for YR then takes the following form,

ts . tan ~ - --- - lnv ~

R 2Rs s (2.39)

Some additional tooth dimensions of a gear are shown in

Figure 2.11. The circles through the tips and the roots of the

teeth are known as the tip circle and the root circle, and

their radii are shown as RT and Rroot • The radial distances as

c

Standard pitch circle

Figure 2.11. Addendum and dedendum.

Standard Basic Rack Forms 45

and bs between these circles and the standard pi tch circle are called the addendum and the dedendum, and for this reason the tip and root circles are also called the addendum and dedendum circles. The sum of the addendum and the dedendum is known as the whole depth of the gear teeth. Finally, we use Figure 2.11

to express the addendum and the dedendum in terms of RT, Rs

and Rroot '

Standard Basic Rack Forms

R - R T s (2.40)

(2.41)

Although it is possible to choose arbitrary values for

the module m and the pressure angle ~r of a basic rack, there are a number of standard values which are most frequently used. As we have shown, the module and pressure angle ~s of a gear are equal to the module and pressure angle ~r of its

c

Figure 2.12. Tooth profile, N=36, ~s=14.5°.


basic rack, so by choosing standard values for the basic rack parameters, we are also choosing the same values for the parameters of the gear.

These standard values are recommended by organisations such as the ISO and the AGMA, first because they have been found in practice to give satisfactory results, and secondly for economic reasons. The tools used for cutting gears have dimensions which depend on the module and the pressure angle of the gear to be cut. A gear manufacturer will normally keep in stock the tools necessary for cutting gears with standard values of module and pressure angle, but when di fferent values are required, the cutting tool must be made specially, and the cost of the gear is therefore increased.

Preferred values for the module, measured in mms, are as follows,

1, 1.25, 1.5, 8, 10, 12,

2, 2.5, 3, 16, 20, 25,

4, 5, 6,

32, 40, 50,

and the preferred values for the diametral pitch, measured in

c

Figure 2.13. Tooth profile, N=36, ~s=20°.

Standard Basic Rack Forms 47

teeth per inch, are given below.

120, 96, 80, 72, 64, 48, 40, 32, 24, 20, 16, 12, 10, 8, 6, 4, 3, 2.5, 2.25, 2, 1.5, 1.

Both sets of numbers are listed in the order of increasing

tooth size. In the North American system, gears with a diametral pitch of 20 or greater are called fine-pitch gears, and those with a diametral pitch of less than 20 are called

course-pitch gears. The standard values for the pressure angle ~r of the

basic rack are 14.5°, 20° and 25°, with 20° being by far the most commonly used. The corresponding systems of gears therefore have the same three values for the pressure

angle ~s. To illustrate the effect of the value of ~s on the tooth shape, three gears are shown in Figures 2.12-2.14, each with 36 teeth, and the pressure angles are equal to the three standard values. The pressure angle of 14.5° is no longer recommended for new designs, because the teeth are relatively weak, and the gears are subject to a problem known as

c

Figure 2.14. Tooth profile, N=36, ~s=25°.

48 Tooth Prof i Ie of an I nvolute Gear

Pr 2

Pr 2

Pr = rrm

Figure 2.15. A typical basic rack.

undercutting, which will be discussed in Chapter 5. A typical basic rack is shown in Figure 2.15. In order

that the same basic rack can be used to define the tooth profiles for gears of any size, the dimensions of the basic rack are expressed in terms of the module. The rack pitch is then equal to ~m, and the reference line is the line along

which the tooth thickness and the space width are each equal

to O.5~m. The essential difference between this basic rack and the one shown in Figure 2.1 is that, in this basic rack,

the tooth profiles are rounded near the tips of the teeth. For

a gear which is conjugate to the basic rack, the shape of the

involute part of each gear tooth is defined by the straight

part of the basic rack tooth, while the shape of the gear

tooth near its root is defined by the curved section at the tip of the basic rack tooth. This section of the gear tooth is

known as the fillet, and it is shaped in a manner that blends

smoothly into the root circle, in order to strengthen the

tooth near its base. In Chapter 9, we will describe how the

shape of the fillet can be calculated.

Speci fying a Spur Gear 49

In order to specify the tooth shape of the basic rack, we need three pieces of information. First, of course, we

require the pressure angle ~r' Secondly, we need the distance between the tooth tips and the reference line, which is known as the rack addendum a r • And thirdly, we must know the radius rrT of the circular sections near the tooth tips. For basic racks with pressure angles of 20° or 25°, the following values for the rack addendum and the tooth tip radius have been widely used.

1.250 m (2.42)

0.300 m (2.43)

Of course, these are not the only values which give satisfactory results, but in most practical cases the values chosen do not differ from these by very much.

The radius of the rack tooth circular section is represented by the symbol rrT for the following reasons. We use the upper-case R throughout this book to refer to radii which are measured from the center of a gear, while the radii of other circles are represented by the lower-case r. The two subscripts rand T indicate that the symbol rrT refers to a radius on the rack, at the tip of the rack tooth.

Specifying a Spur Gear

The complete specification of a gear naturally includes items such as the material and the required hardness, but in this book we will deal only with those parts of the specification that determine the geometry of the gear.

The quantities required in the specification can be

divided into four groups. First, there are the parameters describing the basic rack, in particular the module m, the

pressure angle ¢r' the addendum a r , and the radius rrT of the circular sections at the tooth tips. In the second group are the dimensions of the gear blank, as shown in Figure 2.16. The size of the tip circle of the gear is of course determined by the overall size of the blank. In many of the calculations


required for the gear design, we will need to know the radius RT of the tip circle. However, rather than include the value of the radius as part of the specification, it is normal practice to give the diameter, since this can be more easily measured. Another gear blank dimension, which will be used in the design calculations, is the length of the gear teeth measured in the axial direction. This dimension is shown in Figure 2.16, and is known as the gear face-width F.

The third group of quanti ties in the specification relate to the cutting of the gear. They include the number of teeth N, the whole depth of the teeth, and some dimensions that can be conveniently measured, from which the tooth thickness can be calculated. It is difficult to measure the tooth thickness directly, but in discussing the geometry of a gear, we will assume that the tooth thickness ts has been specified, and in Chapter 8 we will describe some of the actual measurements that can be made to determine the tooth thickness. Finally, the fourth group of quantities concern the assembly of the gear pair, and in this group are the center distance, the part number of the meshing gear, and the backlash when the gear pair is assembled.

Hole diameter

Diameter DT of the tip circle

Overall length

~ Face-width

F

Hub diameter

Figure 2.16. Dimensions of the gear blank.

Examples 51

Numerical Examples

In each of the remaining chapters, a number of examples will be presented, and the following conventions will be

observed. In examples where the tooth size is expressed in terms of the module, all gear dimensions are calculated in

mms, even when the units are not explicitly stated. When the tooth size is specified by means of the diametral pitch, the module is calculated in inches, and all remaining lengths are

then expressed in inches. All values given in the examples are rounded to the third

decimal place for lengths expressed in mms, and to the fourth decimal place for lengths expressed in inches. This level of accuracy is higher than can normally be achieved in practice, but the results in this form are useful for checking computer programs. In most calculations, it is necessary to find a number· of intermediate values, before the final result is reached. For example, to find the tooth thickness at any specified radius, it is first necessary to calculate the base circle radius. In all the examples given in this book, the intermediate values have been stored electronically in the form they were first calculated, for use in the later calculations. For this reason, if the rounded intermediate

values which are shown in the examples are used in the later calculations, there will be small errors in the final results.

To help the reader follow the sequence of calculations, the equation number used at each step is written beside the corresponding calculated value. However, for quantities which are required in almost every example, such as the radii of the standard pitch circle and the base circle, the equation numbers are only given in the first few examples.

Example 2.1

A 24-tooth gear has a module of 8 mm and a pressure angle

of 20°. The diameter of the tip circle is 212 mm, and the

tooth thickness is 14.022 mm. Calculate the standard pitch

circle radius, the base circle radius, the circular pitch, and the tooth thickness tT at the tip circle.

52

Example 2.2

Tooth Profile of an Involute Gear

RS = 96.000mm

Rb 90.210

Ps 25.133

Pb 23.617 inv ~s = 0.014904

~T = 31.675° inv ~T = 0.064175

t T =5.037mm

(2.30)

(2.20)

(2.31)

(2.22)

(2.15)

(2.18) (2.15)

(2.36)

A 16-tooth gear has a D.P. (diametral pitch) of 4, a

pressure angle of 25°, and a tip circle diameter of 4.75

inches. Calculate the minimum tooth thickness at the standard

pitch circle, if the tooth thickness at the tip circle is not to be less than 0.0625 inches.

Example 2.3

m = 0.2500 inches

Rs = 2.0000

Rb = 1.8126 ~T = 40.252°

ts = 0.5091 inches

(2.34) (2.30)

(2.20) (2.18) (2.36)

For the gear specified in Example 2.1, calculate the

radius R of the tooth profile point A whose polar coordinate

9R is4°.

9R = 0.069813 radians

Rs = 96.000 mm

Rb = 90.210

inv ~R = 0.018122

q = 0.068994

cos ~R = 0.9317 R = 96.823 mm

(2.35)

(2.16)

(2.17)

(2.18)

Introduction

Chapter 3 Gears in Mesh

In the previous chapter we described the tooth shape of an involute gear. We now discuss the meshing geometry, first

of a rack and pinion, and secondly of a pair of gears. In each case, we will determine the conditions that are necessary for correct meshing, in the sense that the Law of Gearing is

satisfied. We also derive a relation between the positions of two gears in mesh, and we determine the position of the contact point and the velocity of sliding at the contact point. We will introduce a new concept known as the imaginary

rack, which can often be used to simplify the geometric analysis of a gear pair. And finally, we will describe some of the advantages that involute gears possess over gears with other types of tooth profile.

A Pinion Meshed Wi th a Rack

We consider a pinion with module m and pressure angle I/>s'

as described in the previous chapter, and a rack wi th pi tch P~

and pressure angle I/>~. The rack has teeth which are straight-sided, but in other respects they are not

necessarily the same shape as those of the basic rack. In

other words, the values of P~ and I/>~ may be different from Pr

and I/>r' The description of the meshing geometry of a pinion and

rack is very similar to that of a pinion and the basic rack, described in Chapter 2. There is, however, a major difference

in the underlying purpose. In Chapter 2, the tooth shape of

54 Gears in Mesh

the basic rack was known, but not that of the pinion, and our intention was to find the pinion tooth shape that would satisfy the Law of Gearing. In the present case, the pinion tooth shape is known, but the shape of the rack is not yet decided, apart from the fact that its teeth are straight-sided. We now determine what conditions must be met by the rack parameters, if the rack is to mesh correctly with the pinion.

In order to check whether the Law of Gearing is satisfied, we adopt the following procedure. We find a typical position of the contact point between the pinion and the rack, and we draw the common normal through this point. We also draw the perpendicular from the pinion center to the rack reference line, and the point where this line intersects the common normal is labelled P'. The Law of Gearing is satisfied if pI coincides with the pitch point P.

Figure 3.1 shows the pinion base circle, and a pair of teeth in contact. The base circle radius is given by Equation (2.20),

(3.1)

P~

c

Base circle

Figure 3.1. A pinion meshed with a rack.

A Pinion Meshed With a Rack 55

and by making use of Equation (2.30), which gives the standard pitch circle radius in terms of the module, we can express the

base circle radius in terms of the module and the pressure

angle,

~Nm cos 4lS (3.2)

Since the module m and the pressure angle 4ls are part of the specification of the pinion, the base circle radius Rb is known, and its value is constant.

Any normal to the rack tooth in Figure 3.1 must be perpendicular to the tooth profile, while any normal to the

pinion tooth must touch the base circle. Hence, as shown in the diagram, the common normal must be the base circle tangent

which is perpendicular to the rack tooth profile, and the contact point must lie on this line. If E is the point where the common normal touches the base circle, the radius CE is parallel to the rack tooth profile, and the length CP' can therefore be found from triangle ECP' ,

CP' cos 4l~

(3.3)

To find the position of the pitch point, we use Equation (1.15), which gives the pitch circle radius of a pinion meshed with an arbitrary rack with pitch p,

The rack in Figure 3.1 has a pitch p~, so in this case the pinion pitch circle radius is as follows,

Np~

271" (3.4)

Point P' in Figure 3.1 will coincide with the pitch point if

the length CP' is equal to the pitch circle radius Rp' We

equate the two expressions given in Equations (3.3 and 3.4),

and rearrange the terms to put the condition in the following

form.,

p~ cos 4l~ 271"Rb

N (3.5)

56 Gears in Mesh

Figure 3.2. Base pitch of a rack.

The base pitch Pb of the pinion is given by Equation (2.22),

211'Rb N (3.6)

and the base pitch Pbr of the rack, defined as the distance between adjacent teeth measured along a common normal, can be expressed in terms of the pitch and the pressure angle with the help of Figure 3.2,

P~ cos ~~ (3.7)

Hence, the condition given by Equation (3.5) implies that the base pi tch of the rack must be equal to that of the pinion,

(3.B)

When we replace CP' in Equation (3.3) by the pitch circle

radius Rp' the equation takes the following form,

(3.9)

We have shown that if the length CP' is equal to the pitch circle radius, the base pitches of the pinion and the rack must be equal. The converse is also true, as we can prove

A Pinion Meshed Wi th a Rack 57

Operating pressure angle ¢

Line of action

I+------,'--Rack reference line

Base circle c

Figure 3.3. Meshing diagram of a rack and pinion.

by considering Equations (3.3 - 3.8) in the opposite order. If

the base pitches are equal, then CP' is equal to Rp' and the Law of Gearing is satisfied. We have therefore proved that an involute pinion can mesh correctly with any rack whose teeth are straight-sided, provided the two base pitches are equal. The rack does not need to be the same shape as the basic rack. The meshing diagram is shown in Figure 3.3, with the common normal at the contact point passing through the pi tch point P. As always, the line in the rack which touches the pinion pitch

circle is the rack pitch line, the common normal at the

contact point is the line of action, and the angle between the line of action and the tangent to the pitch circle at P is the

operating pressure angle 41 of the gear pair. Since the line of

action is perpendicular to the rack tooth profile, it can be

seen from the diagram that the operating pressure angle 41 is

equal to the rack pressure angle,

41' r (3.10)

58 Gears in Mesh

Equations (3.9 and 3.10) can be combined to give a relation between the pitch circle radius, the base circle radius, and the opera t i ng pressure angle ~,

(3.11)

In Chapter 2, we proved that the circular pitch of the pinion is equal to the pitch of the basic rack, and we showed that when the pinion is meshed with its basic rack, the pressure angles ~, ~s and ~r are all equal. We now prove the corresponding results for the case when the pinion is meshed with an ordinary rack. The circular pitch Pp at the pitch circle is known as the operating circular pitch, and its value was given by Equation (1.18),

2l1'Rp N

(3.12 )

When we substitute the expression for Rp given by Equation (3.4), we prove that the operating circular pitch is equal to the pitch of the rack,

p' r (3.13 )

The profile angle of the pinion at its pitch circle is called the operating pressure angle of the pinion, and is represented by the symbol ~p. The profile angle at radius R was given by Equation (2.10),

and the operating pressure angle can therefore be found by

substituting Rp in place of R,

(3.14 )

If we compare Equations (3.9 and 3.14), it is evident that the operating pressure angle of the pinion is equal to the

pressure angle of the rack,

~' r (3.15)

Pinion and Rack Posi tions 59

and finally, Equations (3.10 and 3.15) are combined, to show that the operating pressure angle of the gear pair, the

operating pressure angle of the pinion, and the rack pressure

angle are all equal,

4l' r (3.16)

To conclude this section, we derive an expression for the tooth thickness tp of the pinion, measured at the pitch circle, and then we express the polar coordinate 8R of a typical point A in terms of the tooth thickness tp' The tooth thickness tR at radius R was given by Equation (2.36),

where ts is the tooth thickness at the standard pitch circle. Hence, the value of tp is found by substituting Rp in place of R,

t R [2 + 2(inv 4ls - inv 4lp )]

p Rs (3.17)

The polar coordinate 8R of point A on the tooth profile at radius R was given by Equation (2.35),

If this equation is combined with Equation (3.17), we obtain an expression for 8R in terms of the quantities defined at the pitch circle,

~ + inv 4lp - inv 4lR 2Rp

Relation Between the Pinion and Rack positions

(3.18)

In order to obtain a general relation between the posi tions of the pinion and the rack, we consider them

initially when the ~ontact point lies exactly at the pitch

point, and we then determine a relation between the

60

c

Gears in Mesh

x

I

~Rack reference line

Figure 3.4. Contact at the pitch point.

displacements from this position. We proved in Chapter 1 that, at some moment during the meshing cycle, the contact point must coincide with the pitch point, and in Figure 3.4 we

show the pinion and rack with the contact point in this position. As before, the x axis in the pinion and the xr axis in the rack each coincide with a tooth center-line, and the fixed ~ and ~ axes have their origin at the pitch point. The angular position of the pinion is specified by the angle fi, measured from line CP counterclockwise to the x axis, and the position of the rack is indicated by the distance ur of the xr axis above the ~ axis. The positions of the pinion and the rack in Figure 3.4 are as follows,

fi -~ (3.19) 2Rp

1 (3.20) ur 2"tpr

where tp and tpr are the tooth thicknesses, measured on the pinion pitch circle and the rack pitch line.

After the pinion has rotated an angle Ilfi from this

position, and the rack has displaced a distance Ilu r , the new

positions are given by the following two expressions,

fi (3.21)

position of the Contact Point 61

(3.22)

The plnlon rotation ~$ and the rack displacement ~ur are of

course related, and to find this relation we start from

Equation (1.16), which expresses the rack velocity vr in

terms of the pinion angular veloci ty w,

(3.23)

We integrate this equation, to obtain the relation between

the rack displacement and the pinion rotation,

(3.24)

We now use Equations (3.21 and 3.22) to express ~$ and ~ur in

Equation (3.24), and the resulting equation gives the

relation we require between the pinion angular position $ and

the rack position ur •

o (3.25)

Position of the Contact Point

When we discussed the condition for correct meshing of a pinion and rack, at the beginning of this chapter, we showed

that a typical contact point lies on the base circle tangent which is perpendicular to the rack tooth profile. Since the contact point always lies on this line, the path of contact is

a segment of the same line. We can therefore specify the position of a contact point by its distance along this line

from the pitch point. We introduce the length s, as shown in

Figure 3.5, and it is defined in the manner of a coordinate,

so that it is positive for points which lie above P, and

negative for those lying below P.

To derive an expression for s, we use a method which is

very similar to the procedure just used when we found a

relation between the pinion and rack positions. When the

contact point is at the pitch point, the angular position of the pinion is given by Equation (3.19),

62 Gears in Mesh

c

Figure 3.5. position of the contact point.

and the value of s is zero. After a pinion rotation ~~, the angular position of the pinion is given by Equation (3.21),

fJ (3.26)

The rack will have displaced a distance Rp~~' as we showed in Equation (3.24), and the corresponding value of s can then be read from Figure 3.5,

s (3.27)

We eliminate ~~ between Equations (3.26 and 3.27), and we use Equation (3.11) to replace (Rp cos~) by Rb• We then obtain an expression for s as a function of the pinion angular

position ~,

s Rb~ + lt cos ~ 2 P

(3.28)

If we use Equation (3.28) to find the values of s corresponding to any two positions of the subtract one result from the other, expression for the displacement ~s of

pinion, we can then and we obtain an the contact point

Sliding Velocity 63

corresponding to a rotation llfJ of the pinion.

lls (3.29)

It is interesting that the distance moved by the contact

point for a given rotation of the pinion is independent of the

rack pressure angle ~~. We will show later in the book that the same relation between lls and llfJ applies in the case of a pair of spur gears, and also in the case of a pair of helical gears wi th parallel axes.

Sliding Velocity

The sliding velocity at the contact point is defined as the difference between the velocities of the two points in contact. If point A of the pinion touches point Ar of the rack, then the sliding velocity is defined as follows,

Sliding veloc i ty (3.30)

In order to derive an expression for the sliding

Figure 3.6. Unit vectors associated with the rack tooth.

64 Gears in Mesh

velocity, we need to make use of the set of unit vectors nt, n~ and nS in the directions of the coordinate axes, and we also introduce the two unit vectors shown in Figure 3.6, which

are associated with a rack tooth profile. These vectors are

nnr' in the direction of the outward-pointing normal to the tooth profile, and nTr , in the tangential direction toward the tip of the tooth.

If the rack velocity is vr ' this is of course the velocity of any point in the rack, and therefore in vector

form the veloc i ty of Ar can be wri tten,

(3.31)

To obtain the velocity of point A, we form the vector product of the pinion angular velocity and the position vector from C to A. I f point A lies in posi tion s on the path of contact, the

. . CA . d th f th pos1t10n vector p 1S expresse as e sum 0 e vectors from C to P, and from P to A,

(3.32)

and the veloc i ty of A is found as follows,

(3.33)

Base circle of gear 2


Figure 3.7. Common tangent to the base circles.

A Pair of Gears in Mesh 65

When we subtract the velocity of Ar from that of A, the term (R w-v ) disappears, as we showed in Equation (3.23). Hence,

p r the sliding velocity is given by the following expression,

(3.34)

The sliding velocity is along the common tangent, as we

would expect, and its direction reverses when the quantity s changes sign, which occurs when the contact point passes

through the pi tch point.

A Pair of Gears in Mesh

We now consider the meshing of a pair of gears. If the

base pitches of the two gears are Pb1 and Pb2' the base circle radii are given by Equation (3.6),

(3.35)

(3.36)

We follow the same procedure outlined earlier to determine whether the Law of Gearing is satisfied. We find the position of a typical contact point, and then check to see if the common normal at the contact point cuts the line of centers at the pitch point. Any normal to the tooth profile of gear 1 must touch the base circle of gear 1, while any normal to the tooth profile of gear 2 must touch the base circle of gear 2. Hence, the common normal at the contact point must touch both base circles, which means it must coincide with the

common tangent to the base circles. If the common tangent

touches the base circles at E1 and E2 , as shown in Figure 3.7, then the posi tion of the tooth contact point must be somewhere

on line E1E2 , and the common normal at the contact point lies

along E1E2 . The point where this line intersects the line of

centers is labelled P', and we must now check to see whether P' coincides with the pitch point P.

The radii C1E1 and C2E2 in Figure 3.7 are each perpendicular to E1E2 , and they are therefore parallel.

66 Gears in Mesh

Hence, the triangles E,C,P' and E2C2P' are similar, and we obtain the following relation between the sides of the triangles,

The lengths E,C, and E2C2 are equal to the base circle radii, which we express by means of Equations (3.35 and 3.36), and the relation then takes the following form,

(3.37)

We proved in Chapter' that the pitch point P divides the line of centers C'C2 in the ratio N, :N2 • If point P' is to coincide with P, the right-hand side of Equation (3.37) must be equal to (N,/N2), and this is only true if the base pitches are equal,

(3.38)

To prove the converse, we consider a pair of gears in which the base pitches are equal. The base circle radii are then in the ratio N,:N2, and the common tangent E,E2 to the base circles divides C'C2 in the same ratio. Hence, the common normal at the contact point passes through the pitch point, and the Law of Gearing is satisfied. We have therefore proved that any pair of involute gears can mesh correctly, provided their base pitches are equal.

In practice, two gears intended to mesh together are almost always designed so that they have the same module m and the same pressure angle lP s • In other words, they are conjugate to the same basic rack. The base pitch of a gear is related to the module and the pressure angle by Equations (2.24 and

2.3'),

1rm cos I/I s (3.39)

Hence, if the two gears are designed with the same module and the same pressure angle, this ensures that they have the same base pi tch, and therefore that they wi 11 mesh correctly.


Operating pressure angle <p

Line of action



c

Figure 3.8. Meshing diagram of a gear pair.

The meshing diagram of a pair of gears is shown in Figure 3.8, with line E1E2 cutting the line of centers at the pitch point P. The pitch circles of the two gears are the circles which pass through P, and their radii are expressed in terms of the center di stance C by Equat ions ('.24 and 1.25),

(3.40)

(3.41)

We have proved that the common normal at the contact point lies along E,E2 , and this line is therefore the line of action. For a pair of gears, the operating pressure angle ~ is defined as the angle between the line of action and the common

tangent to the pitch circles at P. Since both lines E,C 1 and E2C2 are perpendicular to the line of action, they each make

the same angle ~ wi th the line of centers, and we can therefore use triangles E1C,P and E2C2P to derive the following relations,

68 Gears in Mesh

(3.42)

(3.43)

To find the value of the gear pair operating pressure angle ~,

we start by expressing the center distance as the sum of the pitch circle radii,

c

We then use Equations (3.42 and 3.43) to eliminate RP1 and RP2 in this relation, and we obtain an equation from which the value of ~ can be found,

cos ~ (3.44)

We complete this section by comparing the operating circular pitches and the operating pressure angles of the two gears. We have already proved, in Equation (1.35), that the

operating circular pitches are equal,

(3.45)

To compare the operating pressure angles, we make use of Equation (2.10), which gives the profile angle of a gear at radius R,

and we obtain the operating pressure angle ~P1 of gear 1 by

substi tuting RP1 and Rb1 in place of Rand Rb ,

(3.46)

When this relation is compared with Equation (3.42),

it is evident that the operating pressure angle of gear is

equal to the operating pressure angle ~ of the gear pair,


I/>pl (3.47)

A similar argument shows that the operating pressure

angle of gear 2 is also equal to 1/>,

I/>P2 (3.48)

and Equations (3.47 and 3.48) can of course be combined, to

show that the three angles are all equal,

I/>Pl I/>P2 (3.49)

Equation (3.49) could have been proved more directly,

simply by looking at the meshing diagram shown in Figure 3.9,

where the contact point coincides with the pitch point. The

common tangent to the tooth profiles at the contact point is perpendicular to the line of action, and it can be seen from

the diagram that the three angles I/>pl' I/>P2 and I/> are all

equal. We have shown that the operating circular pitches Ppl

and Pp2 are equal, and that the operating pressure angles I/>Pl and I/>P2 are also equal. It is often convenient, whenever we

Line of action

Icommon tangent

Figure 3.9. Meshing diagram, with contact at the pitch point.

70 Gears in Mesh

have proved that a particular quantity on one gear is always equal to the corresponding quantity on the other gear, to introduce a single symbol which can be used to stand for either quantity. We therefore use the symbol Pp to stand for

either Pp1 or Pp2' and 4>p to stand for 4>P1 or 4>p2. In a similar manner, the symbol 4>s is used for either ;S1 or ;S2. The same convention will be used throughout the remainder of the book, without further explanation.

Relation Between the Gear Positions

For gear 1, we have specified the angular posi tion by the angle f1 1, measured from line C1P counterclockwise to the x 1 axis. We now specify the angular position of gear 2 in the same manner, as the angle f12 measured from line C2P counterclockwise to the x2 axis. In this section we derive a relation between the angles f11 and f1 2•

The two gears are shown in Figure 3.9, with the contact point coinciding with the pitch point, and in these positions the angles f11 and f12 can be written down by inspection,

-~ 2Rp1 f11 (3.50)

-~ 2Rp2

(3.51)

After rotations ~f11 and ~f12' the angular positions of the two gears are given by the following expressions,

The angular Equation (1.21),

f11 -~+ 2Rp1

f12 -~+ 2Rp2

velocities

~f11 (3.52)

~f12 (3.53)

and are related by

(3.54)

and we integrate this equation to find a relation between the

gear rotations,

Path of Contact and Line of Action 7'

(3.55 )

Finally, we use Equations (3.52 and 3.53) to express ap, and

aP2 in Equation (3.55), and we obtain the relation we require

between the angular positions P, and P2 ,

o (3.56)

Path of Contact and Line of Action

We showed earlier that a typical contact point lies on

line E,E2 , the common tangent to the base circles. Since the contact point always lies on this line, the path of contact is a segment of the same line. We have also shown that, for any position of the contact point on this line, the common normal

at the contact point also lies along line E,E2 • Hence, the line of action, which is defined as the common normal at the

Line of action

C1

Imaginary rack c

Figure 3.'0. An imaginary rack.

72 Gears in Mesh

contact point, coincides with the path of contact. This is a

special property of involute gears, as we will prove in

Chapter 9. For non-involute gears, the path of contact is no longer straight, and the direction of the line of action

varies, depending on the position of the contact point.

Imaginary Rack

The gear pair of Figure 3.10 is shown with a rack profile

drawn between the teeth. Since the rack profile has no thickness, it is called an imaginary rack, or sometimes a

phantom rack. The concept of the imaginary rack is useful, because the geometric properties of a pinion and rack can generally be proved more easily than those of a pair of gears. Very often, a result proved for a pinion and rack can be applied directly to the case of a pair of gears, with the help of the imaginary rack.

If the imaginary rack is to fit between the gear teeth in the manner shown, the rack tooth profile must coincide with the common tangent to the gear teeth at their contact point, and the pressure angle 4>~ of the imaginary rack must therefore be equal to the operating pressure angle 4> of the gear pair,

4>' r (3.57)

In addition, for correct meshing between the imaginary rack and the two gears, the three base pitches must all be equal,

as we proved in Equation (3.8),

(3.58)

When the conditions given by Equations (3.57 and 3.58)

are satisfied, the imaginary rack can mesh simultaneously

with both gears. We have shown that the path of contact for

the two gears lies along the common tangent to the base circles, and when we consider the meshing of the imaginary

rack with either gear, the path of contact lies along the

tangent to the base circle which is perpendicular to the rack

tooth profile. Since the pressure angle 4>~ of the imaginary

position of the Contact Point 73

rack has been chosen so that its tooth profile is

perpendicular to the common tangent to the base circles, the

two paths of contact must coincide. This means that the

imaginary rack touches each gear at the same point that the

gears touch each other. In addition, whether we consider one

gear meshed with the imaginary rack, or both gears meshed

together, the position of the pitch point is always the same.

Hence, the pitch circle of either gear, when it is meshed with

the imaginary rack, coincides with its pitch circle when it is meshed wi th the other gear.

Position of the Contact Point of a Gear Pair

To find the position of the contact point between gear'

and gear 2, we can immediately make use of the imaginary rack.

We know that the contact point between the two gears coincides

with the contact point between gear' and the imaginary rack.

We can therefore use Equations (3.28 and 3.29) directly, to write down the position s of the contact point, and the

displacement ~s of the contact point corresponding to a

rotation ~~, of gear "

s (3.59)

~s (3.60)

Sliding Velocity

We calculate the sliding velocity of a gear pair in the

same manner that we did for the case of a rack and pinion. If

point A, of gear' is in contact with point A2 of gear 2, we

write down the velocities of A, and A2 , and subtract them to

find the sliding velocity. In Figure 3." the unit vectors

parallel with and perpendicular to the line of action are

labelled nnr and nTr • This notation reflects the fact that,

if we drew an imaginary rack between the gears, these would be

the directions of the normal and the tangent to the teeth of the imaginary rack.

74

C

Gears in Mesh

Common tangent

Figure 3.'1. Unit vectors associated with the contact point.

The position of the contact point is given by the coordinate s on the path of contact. The velocities of A, and A2 are then expressed in terms of s in the usual manner,

When we subtract the velocity of A2 from (Rp ,w,+Rp2w2) disappears, as we showed and the sliding velocity is given

expression,

A, A2 v - v

that of A" the term in Equation (3.54), by the following

(3.63)

Center Distance C, and Standard Center Distance Cs

When we proved that a pair of gears with equal base

pitches can mesh correctly at a center distance C, we made no restrictions on the value of C. An involute of a base circle is a curve which starts at the base circle, and then spirals

Standard Center Distance 75

Involute

Base circle

c

Figure 3.12. The involute curve.

around the base circle with ever-increasing radius, as shown in Figure 3.12. It is possible, in principle, to use any part of this curve as the tooth profile, and therefore a pair of

gears with base circle radii Rb1 and Rb2 could theoretically be designed to operate at any center distance which is larger than the sum of the base circle radii. In practice, of course, there are a number of considerations which limit the useful range of center distance values.

The most important of these practical considerations is the question of how the gears are to be cut. Let us assume that we have a cutting tool which can cut a pair of gears with N1 and N2 teeth, to mesh at a certain center distance. Suppose we now want to use the same cutter for a second pair of gears, also with N1 and N2 teeth, which is intended to operate at a different center distance. The second pair of gears will require tooth thicknesses which are different from those of the first gear pair, and this can only be achieved by the same cutter if the differences in the tooth thickness values are

not too great. Hence, the center distance of the second gear

pair may differ from that of the first, but not by too large

an amount.

A cutting tool is generally designed so that it can cut a

suitable tooth thickness in each gear for the case when the

pitch circles of the gear pair coincide with the standard

pitch circles. The center distance in this case is called the

standard center distance Cs ' and it is equal to the sum of the

76 Gears in Mesh

standard pi tch c i rc Ie radi i ,

(3.64)

The same cutter can then be used to cut a gear pair with the same tooth numbers, designed for any center distance C which is sufficiently close in value to Cs • As a rough guide, we can say that C should not be less than Cs ' and that when C is larger than Cs ' the difference should not normally exceed v(C B tan ~s). The quantity B in this expression is the gear pair backlash, which will be defined in Chapter 4. The two condi tions just given can be expressed by the following inequali ty,

C (3.65)

The reasons for this particular range of values will be discussed in Chapter 6.

In practice, the maximum difference between C and Cs generally varies between about 2% and 4%, depending on the values of Cs ' B and ~s. In other words, the value of C is always quite close to that of Cs • It then follows that the pitch circle radii do not differ substantially from the standard pitch circle radii, and that the values of ~p and ~ are within a few degrees of ~s.

Recommended Tooth proportions for a Gear

In Chapter 2, when we discussed the tooth geometry of a gear, we gave the addendum and the tooth tip radius values for some commonly used basic racks. We did not, however, give any recommended values for the tooth thickness or the addendum in a gear. The reason for this omission is that suitable values depend on the operating conditions of a gear pair, and in

particular, on the center distance. We defined the addendum as and the dedendum bs as the

radial distances from the tip circle and the root circle to the standard pitch circle. The addendum and dedendum can also be measured from the pitch circle, in the manner shown in

Recommended Tooth proportions

Standard pitch circle )...

Root circle~ ,

c

Tip circle

Pitch circle

Figure 3.13. Addendum and dedendum.

77

Figure 3.13, and in this case we use the symbols ap and bp to distinguish these quantities from as and bs • For a pair of gears, the working depth of the teeth is defined as the amount

by which the teeth overlap when the gears are in operation, and it is therefore equal to the sum of the addendum values, measured f rom the pi tch eire les,

Working depth (3.66)

The clearance at the root circle of each gear is the amount by which the dedendum of that gear exceeds the addendum of the meshing gear. Hence, the clearance values c 1 and c 2 for the

two gears of a pair are given by the following expressions,

(3.67)

(3.68)

Finally, the whole depth of the teeth, which was defined in

Chapter 2 as the sum (as+bs )' is of course also equal to the sum (ap +bp ) •

78 Gears in Mesh

For gears with pressure angles ~s of 20° or 25°, the values generally recommended for the working depth, the clearance and the whole depth are as follows,

Working depth 2.0m (3.69)

Clearance 0.25m (3.70)

Whole depth 2.2Sm (3.71)

In most designs, the actual values of these three quantities do not conform exactly to the recommended values, and it is not necessary that they should do so. The values given in Equations (3.69 - 3.71) are simply guides, which are found by experience to give satisfactory results, and it is sufficient if the actual quantities in a gear pair do not deviate too far from these values. In particular, as we will show in Chapter 4, the clearances in a gear pair should not be less than the value given by Equation (3.70).

It is not possible at this stage to give any recommendations for the addendum and dedendum values of each gear in a gear pair. They depend on the tooth thickness values, and these in turn depend on the center distance and the backlash required. We will return to this question in Chapter 6.

Fundamental Circles of a Gear

We have shown that the pitch circle radius of a gear depends on the center distance, and is therefore a variable quantity. On the other hand, the radii of the standard pitch circle and the base circle are constant, and these two circles

are therefore generally regarded as intrinsic to the geometry of the gear. However, there is a sense in which the base circle is more fundamental than the standard pitch circle.

Given any gear, it is possible in principle to find the base circle by a geometric construction. We draw the normal to the tooth profile at any point of the profile, and the base circle is the circle which touches this line. The base circle

Advantages of the Involute Profile 79

is therefore fixed in the gear, and its radius is always the

same. The situation of the standard pitch circle is different.

The standard pitch circle is defined as the pitch circle of the gear, when it is meshed with its basic rack. We have shown

in this chapter that the same gear can mesh with any rack,

provided their base pitches are equal. Hence, there is nothing unique about the particular basic rack used to define

a gear. The gear is also conjugate to any basic rack with the same base pitch, and if we had used a different basic rack to define the gear tooth geometry, we would have had a different radius for the standard pitch circle.

In order to describe a gear, a knowledge of the base circle radius is absolutely essential, since this determines the shape of the tooth profile. In addition, we choose some other circle as a reference circle, on which quantities such as the tooth thickness can be specified. The base circle is not suitable for this purpose, because in many gears the teeth lie entirely outside the base circle, and instead we use the standard pitch circle. The radius of the standard pitch circle is therefore primarily a matter of convenience. We have shown that the gear tooth geometry can be defined by any basic rack with the same base pitch, and by choosing a value for the standard pitch circle radius, we are in effect selecting one particular basic rack. It is because the standard pitch circle radius can be chosen, while the base circle radius is unique, that we stated at the beginning of this section that the base circle is the more fundamental of the two circles.

Advantages of the Involute Profile

We have shown that the center distance C of a gear pair

can vary, and this is the principal reason that has led to the

almost universal use of involute gears, in preference to other types of profile.

We will describe some of the properties of non-involute tooth profiles in Chapter 9, and we will show that a pair of

non-involute gears can only mesh correctly at one specified

80 Gears in Mesh

center distance. This means that the bearings must be positioned with extreme accuracy, and even then the gears may run badly if the shafts deflect under load, or if the center distance increases due to thermal expansion of the casing. Small changes of this type in the center distance, or small discrepancies from the design value due to errors during assembly, present no serious problem for a pair of involute gears.

A second major advantage occurs when the gears are cut. We stated that a pair of gears can be cut by a single cutter, to operate at any center distance within a certain range. It is therefore not necessary to use a special cutter, every time a gear pair is required to operate at an unusual center distance. In almost any design problem, it is possible to design a pair of involute gears that can be cut by a standard cutting tool, and this of course represents an important economic saving to the gear manufacturer.

Examples 81

Numerical Examples

Example 3.1 A 26-tooth pinion with a module of 12 mm and a pressure

angle of 25° is to be meshed with a rack whose pressure angle

is 22°. Calculate the pitch P~ of the rack, and the pitch

circle radius of the pinion.

Example 3.2

Ps = 37.699 mm

Pb = 34.167

Pbr = 34.167 P~ = 36.850

Rp = 152.488 mm

(2.31) (2.24)

(3.8) (3.7)

(3.4)

A gear pair has a module of 10 mm and a pressure angle

of 20°. The gears have 24 and 75 teeth, and the center distance is 500 mm. Calculate the pitch circle radii, and the operating pressure angle of the gear pair.

Example 3.3

RS1 = 120.000 mm Rs2 375.000 Rb1 112.763 Rb2 352.385 Rp 1 121 .212

RP2 = 378.788 mm I/J = 21.519°

(3.40) (3.41)

(3.44)

In traditional printing machines, the gearing was

designed so that the center distance was always equal to the

standard center distance. It was then necessary to use

circular pitch gears, in order that the circumferences of the standard pitch circles should be exact multiples of the paper

82 Gears in Mesh

sizes being printed. In one particular gear pair, the circular pitch Ps was 0.75 inches, the pressure angle 14.5°, and the tooth numbers 44 and 88. Redesign the gear pair for the same gear ratio and center distance, using a standard D.P. value and a pressure angle of 20°. Calculate the new operating pressure angle.

Old design, ps=0.75, N1=44, N2=88

Pd = 4.1888 m = 0.2387 inches

Rsl = 5.2521 Rs2 = 10.5042

C Cs = 15.7563

Try Pd = 4 m = 0.25 inches

TO obtain Cs slightly smaller than C, try N1=42, N2=84

Then we obtain the following values for Cs and tIl,

Rsl = 5.2500 Rs2 = 10.5000

Cs 15.7500 inches

Rbl = 4.9334 Rb2 = 9.8668 til = 20.063°

(2.33) (2.34)

(3.64)

(3.64)

(3.44)

Chapter 4-

Contact Ratio, Interference and Backlash

Contact Ratio

The meshing cycle of a tooth pair begins when the teeth first make contact, and ends when the contact is broken. If

one gear is to drive the other in a continuous manner, there must clearly be at least one tooth pair in contact at all

times. However, it is found that smooth operation is only possible when the contact at one tooth pair continues until

sometime after the contact has begun at a second pair. In other words, there must be parts of the meshing cycle during which two pairs of teeth are in contact simultaneously. In order to measure the amount of overlap, we introduce a quantity called the contact ratio, defined in the following

manner. If a~c is the angle through which a gear rotates during one meshing cycle, and aop is the angle subtended at the gear center by one tooth, the contact ratio mc is defined as follows,

( 4 • 1 )

The angle aop is known as the angular pitch of the gear,

and its value in radians is equal to 2~ divided by the number of teeth,

2~ N

(4.2)

The rotation a~c is called the angle of contact, and in

order to find its value, we need to describe the meshing

process in some detail. Figure 4.1 shows a pair of meshing gears, and we will assume that gear is driving and is

84 Contact Ratio, Interference and Backlash

Gear 1 driving Gear 2 driven

c

Figure 4.1. The ends of the path of contact.

turning counter-clockwise, so that gear 2 is being driven and is turning clockwise. When the gears rotate, the first contact between a pair of teeth occurs at T2 , the point where the tip circle of the driven gear cuts the common tangent to the base circles. This is because the contact point must lie on that line, as we showed in Chapter 3, and the teeth of gear 2 are not long enough to make contact at points on that

line below T2• At the initial contact, therefore, the tip of

the tooth on the driven gear is in contact with the tooth of the driving gear, at a point on its profile somewhere below

the pitch circle. After the initial contact, the gears continue to rotate,

and the contact point in Figure 4.1 moves upwards along the

path of contact. On the driving gear, the contact point moves outwards towards the tooth tip, while on the driven gear it

moves inwards. Contact ceases when the contact point reaches

the tooth tip of the driving gear. The position where this

occurs is at point T1, where the tip circle of the driving

gear crosses the common tangent to the base circles.

Contact Ratio 85

The displacement as of the contact point along the path

of contact was related to the gear rotation a~ by

Equation (3.60),

as (4.3)

Hence, if as is the contact point displacement between the c beginning and the end of the meshing cycle, the corresponding

gear rotation a~c can be expressed as follows,

asc Rb

(4.4)

We now combine Equations (4.1, 4.2 and 4.4), to obtain an

expression for the contact ratio,

asc 211'Rb

(-N-)

and since the denominator is equal to the base pitch Pb' as it was defined in Equation (2.22), the expression for the contact ratio can be simplified to the following form,

(4.5)

The initial and final points of contact are shown in Figure 4.1 as T2 and T1, and the length asc is equal to the distance between these points. The line segment from T2 to T1 is the path of contact, and asc is therefore equal to the length of the path of contact. From the manner in which the contact ratio was defined in Equation (4.1), it might appear that two meshing gears could have contact ratios of different values, but it is now clear from Equation (4.5) that the two

values must be the same, since the base pitches of the two

gears are equal.

Expressions for the positions sT1 and sT2 of the end points of the path of contact can be derived with the help of

Figure 4.2, which shows the essential features of Figure 4.1

in greater detail. It should be remembered that s is defined

like a coordinate, so that its value is positive for points

above P, and negative for those below. The positions of the end points of the path of contact are then as follows,


c

Figure 4.2. Essential details of Figure 4.1.

2 2 - Rb1 tan t/I + v'(RT1 -Rb1 ) (4.6)

In these equations,

circles, Rb1 and Rb2 operating pressure Equation (3.44),

2 2 Rb2 tan t/I - v'(RT2 -Rb2 ) (4. ?)

RT1 and RT2 are the radii of the are the base circle radii, and t/I is angle of the gear pair, given

tip the

by

cos t/I (4.8)

The final expression for the contact ratio, which is the one generally used to calculate its value, is found by expressing the contact length as the difference between the s

coordinates of points T1 and T2 , and then substituting into Equa t ion (4. 5) ,

(4.9)

Contact Ratio 87

Line of action

Length of path of contact dSc

c

Figure 4.3. A gear pair, with two pairs of teeth in contact.

A pair of meshing gears is shown in Figure 4.3, in

positions such that there is one pair of teeth in contact

near T l' and a second pair near T2• The contact points lie, as always, on the path of contact between Tl and T2• In order that there should be two simultaneous contact points, the distance between the contact points must of course be less than the length ~sc of the path of contact. The distance between the contact points is the same as the distance between a pair of adjacent tooth profiles of either gear, measured along the common normal, and we showed in Chapter 2 that this

distance is equal to the base pitch Pb' At the beginning of the present chapter, we pointed out that there must be parts

of each meshing cycle during which two pairs of teeth are in

contact. We have now shown that this requirement will be met,

provided the length ~sc of the path of contact is greater than

the base pitch Pb' This condition implies, as we can see from

Equation ~4.5), that the value of the contact ratio must be

greater than 1.0. It is possible to imagine a set of points, spaced at


intervals of Pb along the common tangent to the base circles, as shown in Figure 4.3, and moving upwards along that line as

the gears rotate. The points within the line segment from T2

to T 1 would represent the contact points between the gears. At any particular instant, there might be either one or two such points within this interval, but over a period of time the

average number of points in the interval would be equal.to the

length asc of the interval, divided by the distance Pb between the points. Since this quantity is exactly equal to the contact ratio, as it is represented by Equation (4.5), it is

clear that the contact ratio mc is equal to the average number of pairs of teeth in contact between the two gears. We have already shown that operation of the gears is impossible unless the value of mc is at least 1.0, and in general, the

higher the value of mc ' the more smoothly the gear pair will run. The minimum contact ratio required for satisfactory operation depends on the accuracy with which the teeth are cut, and on the speed at which the gears will turn. However, for spur gears a contact ratio of at least 1.4 is generally recommended.

Contact Ratio for a Rack and Pinion

The calculation of the contact ratio is very similar for the case when a pinion is meshed with a rack. As we proved in Chapter 3, the path of contact is a segment of the line through the pi tch point perpendicular to the rack tooth profile, so the operating pressure angle ~ of the gear pair is

equal to the rack pressure angle,

~' r (4.10)

The lower end of the path of contact is shown in

Figure 4.4 as Tr , the point where the tips of the rack teeth intersect the line containing the path of contact, and the

upper end T is the point where the tip circle of the pinion

intersects the same line. Hence, the end points of the path of

contact are in positions sTr and sT' given by the following two expressions,

Angles of Approach and Recess 89

I----~J--Line of action

c

Path followed by the tips of the rack teeth

Figure 4.4. Path of contact for a rack and pinion.

-~ sin tfJ

(4.11)

(4.12 )

where apr is the addendum of the rack, measured from its pitch line. These expressions are substituted into Equation (4.5),

and we obtain the contact ratio,

Angles of Approach and Recess

The tooth profile of the driving gear is shown in

Figure 4.5 in three positions, first in the position of

initial contact, secondly when the profile passes through the

pi tch point, and thirdly in the posi tion of final contact. The

profile in the three positions cuts the pitch circle at the

three points D, P and D'. When the tooth profile intersects

the pitch circle at any point between D and P, the contact

point is approaching the pitch point, and arc DP is therefore

known as the arc of approach. For a similar reason, arc PD' is called the arc of recess.


Pitch circle of gear 1 --Line of action

Tip circle of gear 2

\-Tip circle of gear 1

c

Gear 1 driviny

Figure 4.5. Angles of approach and recess.

The angles subtended at the gear center by the two arcs

are called the angles of approach and recess. During the approach phase of the meshing cycle, the gear rotation aft is equal to the angle of approach, and the contact point displacement As is equal to the length T2P. We can therefore use Equation (4.3) to find the angle of approach of gear 1, and the angle of recess can be found in a similar manner,

Angle of approach (4.14)

Angle of recess (4.15)

The values of sT1 and sT2 in these expressions can be found by means of Equations (4.6 and 4.7).

In the case of a pinion driving a rack, the quantities

sT2 and sTl are replaced by sTr and sT' given by Equations ( 4 • 11 and 4. 12) ,

Angle of approach (4.16)

Angle of recess (4.17)

Interference 91

Fillet circle

Involute

c

Figure 4.6. The fillet circle.

Interference

We pointed out in Chapter 2 that the part of a tooth profile near the root, known as the fillet, is shaped so that it blends smoothly into the root circle. The fillet profile does not coincide with the involute, and the fillet is therefore not intended to come into contact with the teeth of the meshing gear. If such contact does take place, the tips of the teeth in one gear will dig out material from the fillets of the other, and smooth running of the gear pair is impossible. This phenomenon is known as interference, and gear pairs must always be designed so that it will not occur. In the next few sections of this chapter, we describe a method by which the design of a gear pair can be checked to ensure that there will be no interference.

Fillet Circle

For the tooth shown in Figure 4.6, the point where the

fillet begins is labelled Af • Above this point, the tooth

profile coincides with the involute, and below this point the

fillet profile lies outside the involute, so that the tooth is

strengthened near its root. The circle through point Af will be referred to as the fillet circle. It is generally called


the true involute form circle, but there is a possibility of

confusion when this name is used, because there is another

circle, defined quite differently, which is called the form

circle. We will give the definition of the form circle in Chapter 7, and in order to avoid confusion later, we will use

the name "fillet circle" for the circle through the top point of the fillet.

The radius Rf of the fillet circle depends on the method by which the gear is cut, so we will not discuss how the value of Rf can be calculated until Chapter 5, when we describe the cutting of gears. We can show now, however, that in general the fillet circle is larger than the base circle, for the following reason. The involute was defined in Chapter 2 as a curve for which the normal at any point touches the base

circle. With this definition, it is impossible for the involute to extend inside the base circle. Since the fillet circle passes through Af , the point where the fillet joins the involute, the radius Rf is in general greater than the base circle radius Rb • In exceptional cases, Rf may be equal to Rb , but it is never smaller.

First Condi tion For No Interference

Figure 4.7 shows the meshing diagram for a typical gear pair. As usual, the path of contact lies along the common tangent to the base circles, which touches the two base

circles at El and E2• The ends of the path of contact are at Tl and T2 , the points where the tip circles intersect line E1E2 •

When the gears are in the positions shown, point Al of gear 1

is in contact with point A2 of gear 2. Hence, the contact on

the tooth of gear 1 takes place at a radius equal to C1Al . When the contact point is closer to El , the radius C1Al is

reduced, and if the contact point were to coincide with El the

radius C1Al would be equal to Rbl • We have stated that interference will occur if there is

contact at the tooth fillet. This means that the contact on

gear 1 must only take place on the involute section of the

tooth profile, or in other words, at radii greater than the

fillet circle radius Rfl • We have also pointed out that the

Condi tions For No Interference 93

Figure 4.7. Meshing diagram of a gear pair.

fillet circle radius is generally larger than the base circle radius. If, therefore, the contact point is allowed to move down the tooth profile of gear 1 as far as the base circle,

there will def inately be interference. Hence, in order to prevent the interference, it is first necessary that the path

of contact should end at a point on line E1E2 lying above E1• The length T2E2 in Figure 4.7 must therefore be less than E1E2 , and the necessary condition for no interference can be expressed in the following form,

< (4.18 )

The meshing diagram of a rack and pinion is shown in Figure 4.8. The line containing the path of contact touches

the pinion base circle at E, and the end of the path of

contact is shown as Tr • As before, interference would take

place at the tooth fillets of the pinion if the contact point

were to move down the tooth profile as far as the base circle,

so point Tr must lie above point E. The length TrP must therefore be less than EP, and we obtain the following

condition for no interference,

~ sin t/J

< (4.19 )


Pitch line Line of action

c

Path followed by the tips , I of the rack teeth ----+j

Figure 4.8. Meshing diagram of a rack and pinion.

where apr is the rack addendum, measured from its pitch line. If the condition given by Equation (4.18) in the case of

a gear pair, or by Equation (4.19) in the case of a rack and pinion, is not satisfied, then interference will take place. However, even when these conditions are satisfied, they are not always sufficient to prevent interference. We stated earlier that the tooth fillet of a gear extends to point Af , which generally lies a certain distance outside the base circle. We must now ensure that the top of the fillet, between the base circle and the fillet circle, does not come into contact with the teeth of the meshing gear. We therefore calculate the minimum radius at which contact takes place, and compare thi s wi th the radius of the fillet ci rcle.

Limi t Ci rcle

In Figure 4.7, the lower end of the path of contact is point T2 , and the radius C1T2 is therefore the minimum radius in gear 1 where contact occurs. The circle in gear 1 which

passes through T2 is called the limit circle or the contact circle of gear 1, and its radius is labelled RL1 • The length

Conditions For No Interference 95

E,T2 in Figure 4.7 is expressed as the difference between E,E 2 and T2E2, and we then use triangle C,E,T2 to derive an

expression for the radius RL"

(4.20)

In the case of the rack and pinion, the limit circle of the pinion is the circle passing through point Tr , the lower end of the path of contact. The radius RL of this circle can be read immediately from Figure 4.8,

2 ~2 Rb + [Rb tan 41 - sin 41] (4.21)

Second Condi tion For No Interference

Since the lowest contact point on a gear tooth is at the limit circle, and contact below the fillet circle must be prevented, it is clear that any gear pair must be designed so that the fillet circle of each gear is smaller than the corresponding limit circle,

< (4.22)

In order to allow for small errors in the center distance, it is advisable to design a gear pair so that the fillet circle of each gear is smaller than the limit circle by a definate margin. A suitable value for this margin is 0.025 modules, so the condition necessary to avoid interference can be stated as follows,

(4.23)

We are not yet in a position to use this equation, since we have not shown how the radius Rf of the fillet circle can be calculated. This will be done in Chapter 5, and we will give examples at the end of that chapter of the checks which are made During the design of a gear pair, to ensure that there will be no interference.

We have shown that two conditions must be satisfied, if


interference is to be avoided. For a gear pair, the conditions

are given by Equations (4.18 and 4.23), while for a rack and

pinion, they are given by Equations (4.19 and 4.23). These

conditions are sufficient to ensure that no interference will

take place at the top of the tooth fillet, close to point Af • However, there is still the possibility of interference at a point further down the fillet, midway between the fillet circle and the root circle. In order to avoid this possibility, a gear pair must be designed with adaquate

clearances at each root circle. The clearance c 1 at the root circle of gear 1 was defined

in Equation (3.67) as the difference between the dedendum of

gear 1 and the addendum of gear 2,

(4.24)

By expressing a p2 and bP1 in terms of the radi i of the corresponding tip circle, pitch circle and root circle, we can show that the clearance at the root circle of gear 1 is equal to the amount by which the tooth tips of gear 2 clear

Clearance c1

C

Figure 4.9. Clearance at the root circle of gear 1.

Backlash 97

the root circle of gear 1, as shown in Figure 4.9.

C-Rroot1-RT2 (4.25)

We stated in Equation (3.70) that a value of 0.25 modules

is the recommended minimum for the clearance at each root

circle. One of the reasons for this recommendation is to help

avoid the possibility of interference. The designer of a gear

pair should therefore ensure that the clearances are adaquate, as well as checking that the conditions for no

interference are satisfied. In practice, interference is more

likely to occur at the fillets of the pinion than at those of

the gear. For this reason, while the pinion must be checked

for both clearance and interference, the interference checks

are not usually necessary for the gear, provided there is

enough clearance at its root circle.

Interference Points

It was thought at one time that the condition given in

Equation (4.18) would be sufficient to prevent interference

in a gear pair. In other words, it was believed that there

would be no interference, provided the ends of the path of

contact both lay between El and E2 • For this reason, El and E2 were called the interference points. The single condition is

not sufficient, because it is quite possible for T2 to lie above E l' while at the same time the 1 imi t circle of gear 1 is smaller than the fillet circle. However, although it is now

recognized that both conditions are required, the points El

and E2 are still always referred to as the interference points.

Backlash

In a gear-box, as in all engineering products, there are

inevi table di fferences between the dimensions used in the

design, and the actual dimensions of the finished product. In

particular, the tooth profiles of the gears will not conform


exactly to the theoretical involute shapes, and the center distance will not be exactly the specified value. Moreover, any change in the temperature will cause changes in the dimensions of both the gear-box casing and of the gears. A gear pair must therefore be designed in a manner that makes allowance for thermal expansion, for center distance error, and for cutting errors in the gear teeth. To prevent the teeth of the two gears from jamming together, the tooth thickness of each gear is chosen so that contact will occur on one face only, leaving a small gap at the opposite face. This gap, which can be seen in the gear pair shown in Figure 4.10, is known as the backlash.

The size of the gap can be defined in a number of different ways, particularly in the case of a helical gear pair, and each method leads to a slightly different value for the backlash. It is common practice to refer simply to the backlash of a gear pair, without specifying which type of backlash is meant. This practice is quite acceptable, provided the word backlash is used in a general sense to refer to the gap between the teeth of the two gears. However, when a value is assigned to the backlash, it is preferable to specify which type of backlash is intended. In this chapter, we will describe the two methods commonly used to define the backlash of a pair of spur gears, and we will show how these two types of backlash are related.

Backlash

Figure 4.10. Backlash.

Backlash 99

Ci rcular Backlash

The circular backlash B of a gear pair is defined as the difference between the space width of one gear and the tooth

thickness of the other, both measured at the pitch circles. The tooth thickness and the space width of a gear at any

radius are together equal to the circular pitch at that

radius. Hence, if tP1 is the tooth thickness of gear 1 at its pitch circle, the space width wp1 is given by the following

expression,

(4.26)

where Pp is the operating circular pitch of either gear. An expression for Pp can be derived with the help of Equations (3.12 and 3.40),

(4.27)

If the circular backlash of a gear pair is defined as the

difference between the space width of gear 1 and the tooth thickness of gear 2, we use Equation (4.26) to express the space width of gear 1, and we obtain the following expression

for the circular backlash,

B (4.28)

Since the final expression for B is symmetric in tP1 and t p2 ' it obviously makes no di fference whether we subtract the tooth thickness of gear 2 from the space width of gear 1, or the other way round.

Physical Interpretation of the Circular Backlash

The space width is a curvilinear length measured round

one pitch circle, and from it we subtract a tooth thickness

measured round the other pitch circle. It is therefore not

entirely clear whether the circular backlash, defined in this manner, has any precise physical meaning. We will now discuss


that question.

When either gear of the pair is held fixed, the other can be rocked to and fro through a small angle, due to the small

gap which exists between the teeth. If the angle (measured in radians) through which the second gear can be turned is

multiplied by the radius of its pitch circle, we obtain the length of the path moved by a point on the pitch circle, during the rocking of the gear. We will now prove that the

length just described is equal to the circular backlash. For this reason, the circular backlash is also known as the backlash, measured at the pitch circle.

Before we discuss the general case of the backlash in a gear pair, we consider first a rack and pinion, and we will determine the tooth thickness of the rack, if its teeth are in contact with both faces of the pinion teeth. This is the situation known as close-mesh operation, when there is no backlash. Figure 4.11 shows the pinion and rack, in positions such that the contact point between one tooth of the pinion and the rack lies exactly at the pi tch point. The same pinion and rack are shown in Figure 4.12, and the pinion has been rotated until the opposite profile of the same tooth passes through the pitch point. Since both faces of the pinion tooth are in contact with the rack, the pitch point again coincides wi th a contact point.

The rotation ~p between the two positions of the pinion

c

Figure 4.11. Rack and pinion, in close-mesh operation.

Backlash 101

c

Figure 4.12. Displaced positions of the rack and pinion.

can be expressed in terms of its tooth thickness at the pitch

circle,

A{1 (4.29)

and the displacement AU r of the rack is equal to its space

width wpr ' measured at the pitch line,

(4.30)

The rack displacement and the pinion rotation are related by Equation (3.24),

(4.31)

By combining Equations (4.29 - 4.31), we prove that the space width of the rack is equal to the tooth thickness of the

pinion,

(4.32)

We now return to the discussion of the physical

interpretation of the circular backlash. A gear pair is shown

in Figure 4.13, with two imaginary racks drawn between the

teeth, and the tooth thicknesses of the imaginary racks are chosen so that each imaginary rack is in close mesh with one


Imaginary rack 1

Figure 4.13. Gear pair with two imaginary racks.

of the gears. Figure 4.14 shows the section through the imaginary racks corresponding to the pitch plane. A typical pair of teeth are in contact along line ArA~, and the gaps between the teeth are represented by the narrow un shaded bands.

We showed in Equation (4.32) that, when a rack is in close mesh with a gear, the space width of the rack is equal to the tooth thickness of the gear. For the imaginary racks in Figures 4.13 and 4.14, the positions of the teeth and the gaps are reversed from those of a real rack, so the tooth thickness of each imaginary rack is equal to that of the corresponding gear. In addition, as we proved in Equation (3.13), the pitch of each imaginary rack is equal to Pp' the circular pitch of either gear. These values for the pitch and tooth thicknesses of the imaginary racks are shown in Figure 4.14, and it is immediately clear that if ei ther rack is held fixed, the other

Backlash 103

Imaginary rack 1 maginary rack 2

P P'

Figure 4.14. Section through the imaginary racks.

can make a displacement ~ur given by the following

expression,

(4.33)

A comparison of Equations (4.33 and 4.28) shows that the circular backlash is equal to the displacement of the movable imaginary rack,

B (4.34)

We use Equation (3.24) to relate the displacement of the

imaginary rack to the angle ~p through which the movable gear can be rocked,

(4.35)

and Equations (4.34 and 4.35) can be combined to give the following &esult,

B (4.36)


This equation proves the statement made earlier, that the circular backlash can be interpreted as the length of the maximum arc through which a point on the pitch circle of one

gear can move, when the other gear is held fixed.

Backlash Along the Common Normal

The second manner in which the backlash is commonly defined is the backlash along the common normal. When we first introduced the topic of backlash, we pointed out that the tooth thicknesses of a gear pair are chosen so that, for each tooth in the meshing zone, there is contact on one face only, leaving a small gap at the opposite face. The backlash B'

along the common normal is defined as the shortest distance across this gap.

A gear pair is shown in Figure 4.15, and the contact point lies as usual on line E1E2 , which is one of the common

tangents to the base circles. The diagram also shows EiEi, the other interior common tangent to the base circles, and this

line cuts the profiles of two adjacent teeth at Ai and Ai. Since these profiles are both involutes, line A;E; is normal to the tooth profile of gear 1, and AiEi is normal to the tooth profile of gear 2. Hence, line EiEi is normal to both

Figure 4.15. Backlash along the common normal.

Backlash 105

profiles, and the length A;Ai is equal to the backlash B'.

When a gear rotates through an angle ~p, the distance ~s

moved by its tooth profile along the common tangent to the

base circles is given by Equation (3.60),

~s (4.37)

If ~p is chosen equal to the angle through which one gear can

be rocked while the other is held fixed, the corresponding

length ~s is then equal to the backlash along the common

normal. Hence, the backlash is given by the expression for ~s

in Equation (4.37),

B' (4.38)

When we combine Equations (4.36 and 4.38), we can

express the backlash along the common normal in terms of the

circular backlash,

B' (4.39)

The base circle and the pitch circle radii are related by

Equation (3.42),

where ~ is the operating pressure angle of the gear pair, given by Equation (3.44). Hence, the relation between the two

types of backlash can be expressed as follows,

B' B cos ~ (4.40)

Typical Design Backlash Values

The relation between the tooth thicknesses of a gear

pair and the circular backlash was given by Equation (4.28).

The designer specifies the tooth thicknesses so that they are

consistent with the backlash value required, which we will call the design backlash. This value determines the nominal


size of the gap at the non-driving face of the contact teeth. The actual size of this gap will differ from the nominal size, due to tolerances in the center distance and tooth thickness values, and to other errors in the tooth profiles. The design backlash must always be large enough to prevent contact occuring at the non-driving faces of the contact teeth.

The center distance tolerance for a gear pair is generally approximately proportional to the center distance, so the design backlash should depend in a linear manner on the center distance. The tooth thickness tolerance and the magnitude of allowable errors in the tooth profiles generally increase with the tooth size, but not exactly in a linear manner. Hence, if the design backlash is expressed as a power series in the module, the first term will be linear. When these considerations are taken into account, we obtain a function for the design backlash of the following form,

B (4.41)

The values of the constants k1' k2' k3 and k depend on the values used for the tolerances, and these in turn depend on the quality of the gears, and the application for which they are intended. The following equations give sui table values for the design backlash of commercial quality gear pairs used in power gearing,

B [0.0677m - 0.0137m4/ 3 + 0.0004C] mm (4.42)

B [0.0677m - 0.0403m4/ 3 + 0.0004C] inches (4.43)

In the first of these equations, the module and the center distance must be expressed in mms, while in the second equation they must be in inches. The constants have been chosen so that the values of B given by the equations agree

well with typical recommended values. For control gearing, the design backlash values would be considerably reduced. Moreover, if it is possible to adjust the center distance after the gear pa i r has been assembled, there is no longer any need to allow for errors in the center distance, apart from changes caused by thermal expansion, and smaller backlash

Backlash 107

-Dial gauge

Base CirCle~

c

Figure 4.16. Measuring the backlash.

values can be used. For additional information, the reader is referred to design references, such as the Gear Handbook [2].

The specification of a pair of spur gears generally includes maximum and minimum values for the circular backlash. When the gear pair is checked to see whether the actual backlash falls within the specified range, the simplest procedure is to measure the backlash B' along the common normal, and then to calculate the circular backlash, using the relation given by Equation (4.40). The backlash B' can be measured directly with a feeler-gauge, or by means of a dial gauge, as shown in Figure 4.16. The gauge is positioned so that its moveable arm lies along a base circle tangent of one of the gears. If this gear is rocked while the other gear

is held fixed, the displacement measured by the dial gauge is

then equal to the backlash along the common normal.


Numerical Examples

Example 4.1

A gear pair has a module of 8 mm, a pressure angle

of 20°, and a center distance of 453 mm. The tooth numbers are

16 and 95, the tooth thicknesses 14.90 mm and 16.76 mm, and

the diameters of the tip circles are 150.4 mm and 787.6 mm.

Calculate the contact ratio, the circular backlash, and the radii of the two limit circles.

m=8, </Is=20°, C=453, N 1=16, N2=95 t s1 =14.90, t s2=16.76, RT1 =75.2, RT2 =393.8

Example 4.2

RS 1 = 64.000 mm

Rs2 = 380.000

Rb 1 = 60.140

Rb2 = 357.083 </I = 22.924°

Ps = 25.133 Pb = 23. 6 1 7 mm

mc = 1.471

Pp = 25.642 </I p = 22.924°

RPl = 65.297 RP2 = 387.703 tpl = 14.170

tP2 = 10.969 B 0.503mm

RL 1 = 61.034

RL2 = 380.458 mm

(3.44)

(2.31) (2.24)

(4.9)

(4.27) (3.49)

(3.40) (3.41)

(2.36) (2.36)

(4.28)

(4.20)

A pinion, with D.P. 2 and pressure angle 25°, has 24

teeth and a tip circle diameter of 13.2 inches. A rack, with

the same D.P. and pressure angle, is mounted so that the line

through the tips of its teeth lies a distance 5.6 inches from

the pinion center. Calculate the contact ratio.

Examples

Example 4.3

m = 0.5000 inches

Rs 6.0000

Rb = 5.4378

t/> = 25°

apr = 6.0 - 5.6 = 0.4 Pb = 1.4236 inches

me = 1.511

109

(3.10)

(4.13 )

Use Equation (4.43) to choose a suitable design backlash

for a gear pair with D.P. 3 and a center distance of 32.0 inches.

Example 4.4

m = 0.3333 inches B = 0.026 inches (4.43)

Find the maximum sliding velocity, for the gear pair specified in Example 4.1, when the pinion is turning at 800 rpm.

W 1 = 800(~~) = 83.776 radians/sec

W2 = - 14.110 radians/sec ( 1 .4)

ST 1 = 1 9 • 7 1 2 mm (4.6)

At T 1 , A A

v Lv 2 = 1930nTr mm/sec (3.63)

ST2 = - 15.029 mm (4.7)

A A v Lv 2 = - 1471 nTr mm/sec (3.63)

Maximum sliding veloci ty = 1.930 m/sec

Chapter 5 Gear Cutting I, Spur Gears

Form Cutting

The simplest method by which a gear can be cut is to use a tool shaped exactly like the tooth space of the gear. This method of cutting gears is known as form cutting. In practice, the cutting tool is generally a milling cutter, such as the one shown in Figure 5.1. The gear blank is held stationary while the cutter is fed slowly in the direction of the gear axis, until one tooth space is cut. Then the gear blank is indexed through exactly one angular pitch, and a second tooth space is cut. The process is repeated for each tooth space, until the entire gear is completed.

Gears can be cut by this method, using special milling machines with automatic indexing mechanisms. These machines are designed for cutting gears, and are called gear-cutting machines. However, the main advantage of form cutting is that it can also be carried out on an ordinary milling machine, so the method can be used in small machine shops where there are no special-purpose machines for cutting gears. A second advantage is that the milling cutters are generally cheaper than the tools used to cut gears by other methods. There are, though, a number of disadvantages to form cutting, which we

will now discuss. The shape of the tooth space in a gear depends on the

tooth size, the pressure angle, the number of teeth in the gear, and the tooth thickness. In theory, therefore, a different cutter would be necessary for every combination of values of the module m, the pressure angle ~s' the number of teeth N, and the tooth thickness ts. There is no problem when a large number of identical gears are required, because it is

Form Cutting 111

Figure 5.1. A milling cutter.

then economic to order a particular cutter for the job. However, a difficulty arises when a range of different gears are to be cut, using cutters that are already in stock. In order to keep the minimum number of cutters within practical limits, it is usually assumed that the cutters will only be used to cut gears which will operate at the standard center distance, and the cutters are designed so that the tooth thickness of each gear is equal to one half of the circular pitch Ps' with a small reduction to allow for backlash. In addition, since the tooth space of a gear with N1 teeth is similar in shape to that of a gear with N2 teeth, provided the values of N1 and N1 are sufficiently close, it is customary for each cutter to be used for a specified range of N values. A set of eight cutters is generally regarded as adaquate for each combination of m and IPs.

The disadvantages of form cutting are obvious from the foregoing discussion. Since the cutter shape is only correct

for one particular value of N, there are unavoidable errors in

the gear tooth profile whenever the same cutter is used to cut

a gear with any other value of N. Secondly, the accuracy of

the gear depends on the accuracy of the indexing mechanism.

And thirdly, the tooth thickness of a gear cut by a particular

cutter cannot be changed. We will show later in this chapter

that, for other gear cutting methods, the same tool can be

used to cut gears with different tooth thicknesses. This

property is particularly important, because the tooth

112 Gear Cutting I, Spur Gears

thicknesses required in a gear pair depend on the intended center distance C. For all these reasons, form cutting is not

nearly as widely used for commercial gear production as the other methods described in this chapter.

Generating Cutting

Most modern gears are cut by a process known as generating cutting. When a gear is generated, the cutter has

the shape of another gear, and during the cutting process the cutter and the gear blank are each rotated, at the same angular velocities as if they were a meshing gear pair. The

two most common methods by which gears are generated are shaping and hobbing. In this chapter we will describe each of these methods, and we will discuss the advantages that gear generating possesses over other methods of cutting gears.


Figure 5.2 shows a pair of gears, in which the left-hand gear is made of a softer material than the other, and has

Figure 5.2. Gear blank and pinion cutter.

Shaping wi th a Pinion Cutter 113

teeth around only part of its circumference. If the two gears were forced to rotate in opposite directions at constant angular velocities, the material of the left-hand gear would be plastically distorted, and the remaining teeth would be

formed. These teeth would be exactly the shape to mesh correctly with the teeth of the right-hand gear, so if the

right-hand gear had involute teeth, the teeth formed on the left-hand gear would also have involute profiles.

It is possible to roll gears in the manner just

described, but it is more common to cut them. We now regard Figure 5.2 as showing a gear blank on the left, and a cutter on the right. The cutter is shaped exactly like a pinion, and is therefore called a pinion cutter. As before, the gear blank and the cutter are driven with constant angular velocities, as if they were a meshing gear pair. In addition, the cutter

Figure 5.3. positions of cutter, relative to gear blank.


must have some other motion to provide the cutting action, and it is therefore given a reciprocating motion in the direction of its axis. This method of cutting gears is known as shaping, in common with other cutting processes in which the cutting tool has a reciprocating motion.

Figure 5.3 shows the positions of the cutter, relative to the gear blank, as it makes a number of cutting strokes. The shape of the gear tooth, after the cutting has been completed, is the same as the envelope of the cutter positions. It can be seen that this shape is not exactly an involute, but consists of a series of arcs, whose sizes depend on the number of strokes that occur during the cutting of each tooth.

The amount of material that can be removed with each cutting stroke is of course limited, and it is therefore impossible to cut the first tooth of the gear blank immediately to its full depth. Before the cutting process begins, the cutter and the gear blank are given the correct angular velocities, and the reciprocating motion of the cutter is started. The cutter is then fed radially to.wards the gear blank, so that each tooth space of the gear blank is cut in turn to an increasing depth, until eventually the correct depth is reached. At this point the radial feed of the cutter is discontinued, and the cutting process continues while the gear blank makes one more complete revolution. In some cases a single cut is sufficient, while in others, it may be necessary to carry out one or more roughing cuts before the finishing cut, or else to finish the tooth surfaces by some other

method. A typical pinion cutter is shown in Figure 5.4, and the

quantities used in its specification will be identified by

the subscript c. If the number of teeth is Nc ' the radius of the standard pitch circle is given in terms of the module m by

Equation (2.30),

( 5. 1 )

The tooth thickness at the standard pitch circle is shown as t sc ' and the addendum measured from the standard pitch

circle is shown as a sc '


( •• \ J Front Standard pitch circle

clearance angle

Figure 5.4. A pinion cutter.

We now discuss how the shaper must be set up, if it is to cut a gear with Ng teeth, module m, pressure angle ~s' tooth thickness t sg ' and tip circle diameter DTg • The subscript g stands for the word "gear", and is included in order to distinguish quantities on the gear blank from those on the cutter. For the gear, the radius Rsg of the standard pitch circle, on which the pressure angle ~s and the tooth thickness tsg are defined, is given as usual by Equation (2.30),

(5.2)

The gear blank and the cutter are shown in Figure 5.5,

and all calculations can be carried out exactly as if they

were two gears meshed together. A number of symbols were

defined in Chapter 3, relating to the operation of two gears

in mesh. These included the center distance C, the pressure

angle ~, and the operating circular pitch Pp of each gear.

When we consider the situation of a gear blank and a pinion cutter, the corresponding quantities will be identified by


c/>C Cutting pressure angle

-Pinion cutter

J--+---->.....Cutting pitch circles

CC Cutting center distance

Figure 5.5. Meshing diagram of a gear and pinion cutter.

the superscript c, which represents the word "cutting". Hence, the quantities corresponding to C, ~ and Pp during the cutting process are called the cutting center distance, the cutting pressure angle, and the cutting circular pitch, and they will be represented by the symbols CC , ~c and P~.

The tip circle diameter of the finished gear has nothing to do wi th the tooth cutting process, but is simply determined by the diameter of the gear blank. Hence, the blank must be cut with a diameter of DTg , before the shaping begins. Next, in order to obtain the correct module and pressure angle on the gear, a cutter is used wi th the same values of m and ~s as those required for the gear. We will prove in the next section of this chapter that the module and pressure angle of the gear are always equal to those of the cutter.

The number of teeth Ng which are cut in the gear blank depends on the angular velocity ratio of the gear blank and the cutter. During the cutting process, the gear blank and the cutter must be rotated so that their angular velocities Wg and Wc satisfy Equation (1.4),


- N w c c

117

(5.3)

We define the gear ratio mg as the number of teeth required on the gear blank, divided by the number of teeth on the cutter,

(5.4)

The shaper must then be set up so that the angular velocities of the gear blank and the cutter are related as follows,

(5.5)

The required ratio is generally achieved by means of a

gear train connecting the work table and the cutter, with a

number of change gears in the gear train. However, in some of the most recent shapers, the work table and the cutter are each driven by stepping motors, and the angular velocities

are controlled electrically. In ei ther case, the machine must be set so that Equation (5.5) is satisfied exactly, or else

the teeth will be cut incorrectly. It is therefore preferable to specify the value of mg as a ratio of two integers, as in Equation (5.4), rather than in decimal form.

The remaining quantity for which the shaper must be

adjusted is the tooth thickness tsg required in the gear. We showed in Chapter 3 that the center distance C of an involute gear pair can be altered, and the gears will continue to rotate with a constant angular velocity ratio. The

corresponding conclusion, when we are considering a gear blank and a cutter, is that provided the constant angular velocity ratio is maintained, we can alter the cutting center

distance CC , and the cutter will continue to cut teeth in the

gear blank, with the correct involute profiles. However, the

tooth thickness of the gear is of course reduced, if the

cutter is fed more deeply into the gear blank. We can

therefore cut gears with any specified tooth thickness t sg '

simply by feeding the cutter radially into the blank, until

the correct tooth thickness is obtained.

Although the procedure just outlined is quite feasible,

it involves stopping the machine from time to time, in order

to measure the tooth thickness of the gear. Such stoppages


increase the total time required to cut the gear, and necessitate continued attendance by a machine operator. If the machine is to operate automatically, it is essential to know in advance the cutting center distance at which the specified tooth thickness will be reached. We now describe

how the tooth thickness tsg of the gear can be calculated, when the shaper is operated with a particular center distance Cc, and we will use this result to determine the necessary value of CC, if the tooth thickness cut in the gear is to be equal to the value specified.

The standard pitch circle radii of the gear and the cutter were given by Equations (5.2 and 5.1), and we use Equation (2.20) to obtain the corresponding base circle radii,

(5. 6)

(5. 7)

The meshing diagram of the gear blank and the cutter was shown in Figure 5.5. Since the diagram relates to the situation when the gear is being cut, the pitch circles are known as the cutting pitch circles. Their radii are given by Equations ( 3 • 40 and 3.41) ,

N CC RC 9 (5.8) pg (Ng+Nc )

RC N CC c (5.9) pc (Ng+Nc )

The angle ,C between the line of action and the common tangent to the cutting pitch circles is the cutting pressure angle, and its value is given by Equation (3.44),

(5.10)

We know from Equation (3.49) that the operating pressure

angles 'p of two meshing gears are equal to each other, and are also equal to the pressure angle, of the gear pair. When we consider the case of a gear blank and a pinion cutter, their operating pressure angles are defined as the profile


angles at the cutting pitch circles, and they are known as the cutting pressure angles of the gear blank and the cutter. Since they are equal to each other, they are both represented by the same symbol ~~, and their value is equal to that of the cutting pressure angle ~c,

~c (S.l1)

Because both faces of the cutter teeth are simultaneously cutt ing material f rom the gear blank, the

cutting process can be thought of as equivalent to the meshing of a gear pair with no backlash. In Equation (4.28) we gave an expression for the backlash of an ordinary gear pair, in terms of the circular pitch and the tooth thicknesses at the pitch circles,

B

If we set the backlash equal to zero, corresponding to the situation when we consider a gear blank and a cutter, this equation can be used to give the tooth thickness of the gear at its cutting pitch circle,

pc _ t P pc (S.12)

where p~ is the circular pitch of both the gear and the cutter at their cutting pitch circles, and is given by Equation (4.27),

(S.13)

Equation (S.12) can be interpreted as stating that the

tooth thickness of the gear is equal to the space width of the

cutter, both measured at the cutting pitch circles. In most

cases, however, we are interested in the value of the tooth

thickness at the standard pitch circle, rather than at the

cutting pitch circle. For both the gear and the cutter, the

tooth tliicknesses at the cutting pitch circle and at the

standard pitch circle are related by Equation (3.17),

120

t R [~+

pg Rsg t

R [~+ pc RSC

Gear Cutting I, Spur Gears

(5.14 )

(5.15)

We now define the standard cutting center distance C~ as the sum of the standard pitch circle radii of the gear and the cutter,

(5.16 )

The expressions for tpg and tpc are substituted into Equation (5.12). We then multiply by the ratio (C~/cc), and solve the resulting equation to find the gear tooth

thickness t sg '

(5.17)

In this expression, ps is the circular pitch of either the gear or the cutter at their standard pitch circles, and is equal to 7rm, as we showed in Equation (2.31).

Equation (5.17) gives the tooth thickness that will be cut, when the shaper is set with a center distance Cc • The result seems to be odd, because the quantity CC does not appear in the equation, so it might look as if the tooth thickness t does not depend on the cutting center

c sg distance C. However, the independence is only apparent, because the value of the cutting pressure angle ~c of the gear

c p blank and the cutter does depend on C , as we can see from

Equations (5.10 and 5.11).

In order to determine the value of CC necessary to obtain

a specified tooth thickness t sg ' we rearrange the equation in

the following manner,

1 inv ~s - 2Cc(PS - tsg - t sc )

s (5.18 )

The right-hand side of this equation contains only quantities

whose values are known, so we can use the equation to find the

value of the involute function (inv ~~). We calculate the

corresponding angle ~~ by means of Equations (2.16 and 2.17),

and the required cutting center distance CC is then found

Shaping with a Pinion Cutter 121

from Equations (5.11 and 5.10),

tPc P

(5.19)

Rbg+Rbc

cos tP c (5.20)

Later in this book, when we calculate the tooth fillet

shape of the gear, it will be necessary to know the relation between the angular positions of the gear blank and the

cutter. The corresponding relation for a pair of meshing gears was given by Equation (3.56),

In the case of a gear blank and a pinion cutter, the sum of the

tooth thicknesses is equal to the circular pitch p~ at the cutting pitch circles, as we showed in Equation (5.12), so the

relation can be written as follows,

o

We multiply this equation by the ratio (C~/cc), in order to obtain the relation in a more convenient form,

R /3 + R /3 + 1 sg g sc c 2Ps o (5.21)

Module and Pressure Angle of the Gear

When a pinion cutter is specified as having a module m and a pressure angle tP s ' we know that its circular pitch Ps at the standard pitch circle is equal to ~m, and its profile

angle on this circle is equal to tP s • It is generally taken for

granted that any gear cut by the pinion cutter will have the

same module, or in other words the same circular pitch Ps' and the same pressure angle tP s •

In the theory of meshing described in Chapter 3, we

proved that the operating circular pitches Pp and the

operating pressure angles tPp of two meshing gears are equal. It is therefore clear, when we consider a gear blank and a


pinion cutter, that their circular pitches and pressure angles are equal at the cutting pitch circles. However, since the cutting pitch circles do not generally coincide with the standard pitch circles, it is still necessary to prove that the circular pitches and the pressure angles are also equal at the standard pitch circles.

The pi~ion cutter is conjugate to the basic rack, so its circular pitch Ps and its pressure angle ~s are equal to the corresponding quantities Pr and ~r on the basic rack. We proved in Chapter 3 that the minimum condition for correct meshing between a pair of spur gears is that their base pi tches should be equal. The equivalent result, in the case of a gear blank and a pinion cutter, is that the base pitch of the gear will always be equal to that of the pinion cutter. It then follows that the base pitches of the gear and the basic rack are equal, so that the gear is capable of meshing correctly with the basic rack. Its standard pitch circle is, by definition, the pitch circle when it is meshed with the basic rack, and its circular pitch and pressure angle on this circle are therefore equal to Pr and ~r. Hence, they are also equal to Ps and ~s' the corresponding quantities on the pinion cutter.

Advantages of Generating Cutting

We are now in a position to discuss some of the advantages of generating cutting. We have shown that only a single cutter is required, for any specified combination of

module m and pressure angle ~s' and that this cutter can be used to cut gears wi th any number of teeth, and wi th any tooth thickness within certain limits. The practical range of tooth thickness values will be discussed in the next chapter. These

advantages can be summarized by the statement that non-standard gears can be cut by standard cutters.

If the pinion cutter shown in Figure 5.4 is compared with an ordinary pinion, it can be seen that they differ in certain respects. The cutter has teeth of greater depth than those of an average gear, because it must be able to cut the teeth of a gear blank to their full depth, while at the same time


maintaining some clearance at its own root circle. A second difference is that the teeth of the pinion cutter are not exactly parallel to the cutter axis, like those of a gear, but instead they are relieved, as shown in Figure 5.4, in order to

provide clearance behind the cutting edges. Due to this relief, the shape of the cutter teeth will change slightly

after each sharpening. Another of the advantages of involute

gears, and of the generating method of cutting, is that the correct tooth profiles can still be cut in the gear blank, in spite of these changes in the cutter tooth shape. The cutter teeth are designed so that the shape projected through the

cutting edges onto a plane perpendicular to the cutter axis is always an involute to the same base circle. Hence, the teeth on the gear blank will always be cut with the same base pitch. However, after each sharpening of the cutter, the tooth

thickness tsc will be slightly reduced. It should therefore be measured each time, so that the correct value can be used in Equation (5.18), when the cutting center distance is calculated.

Shaping wi th a Rack Cutter

The principle underlying the use of the pinion cutter, and indeed all methods of generating cutting, is that we use one gear to cut another. The same principle is valid, of course, when the cutter has the shape of a rack. A cutter of this type is called a rack cutter, and it has some advantages and some disadvantages, compared with the pinion cutter.

Figure 5.6 shows a partly finished gear blank, being cut by a rack cutter. During the cutting process the gear blank is

rotated, and the cutter is moved in the direction of its

reference line, exactly as if the two were meshed together. In

addition, the cutter is given a reciprocating motion parallel

to the gear axis, which provides the cutting action. We showed

in Chapter 3 that a pinion can mesh correctly with any rack

whose base pitch is equal to that of the gear. Hence, in

theory the cutter could be designed with any values of pitch

and pressure angle, provided its base pitch is equal to the required base pitch of the gear being cut. In practice,


Figure 5.6. Gear blank and rack cutter.

though, the cutter normally has exactly the same shape as the

basic rack, and in describing the operation of the rack

cutter, we will assume that this is the case. The pitch and the pressure angle of the cutter are then represented by the

symbols Pr and ~r' which are also used for the basic rack. In order to cut a particular gear, we use a rack cutter in which

the values of Pr and ~r are equal to of Ps and ~s' the circular pi tch and the pressure angle requi red in the gear.

The number of teeth cut in the gear blank depends on the

angular velocity Wg of the gear blank, and the velocity vr of

the rack cutter in the direction of its reference line. The

relation between Wg and vr is achieved in the shaper by means

of change gears or stepping motors, and in order to choose the

correct settings, we must determine the required relation, so

that the shaper will cut a gear with Ng teeth. The gear blank

and the cutter are each driven, as if they were a pinion and

rack in mesh, so Wg and vr must satisfy Equation (1.8),

vr ~ Pr 211'

(5.22)

The pitch Pr of the rack cutter is equal to 1I'm, like that of

the basic rack, and the relation we require between Wg and vr


can therefore be expressed in the following form,

(5.23)

This equation, like the corresponding equation for a pinion

cutter, must be satisfied exactly. As usual, the tooth thickness of the gear depends on the

depth to which the cutter is fed into the blank. Since the

cutter has the same shape as the basic rack, the cutting pitch circle of the gear coincides with its standard pitch circle,

and its radius is given by Equation (2.30),

(5.24)

The meshing diagram of the gear blank and the rack cutter is shown in Figure 5.7. The cutting pressure angle ~c is always

equal to the cutter pressure angle ~r' and does not depend on the position of the cutter, as it does in the case of a pinion cutter. The cutting pitch line in the rack cutter is tangent to the cutting pitch circle of the gear, and the tooth thickness of the gear is equal to the space width of the

Gear blank

Cutting pitch circle and

standard pitch circle

Cutting pitch line---+I

Rack cutter

Cutter reference line

Figure 5.7. Meshing diagram of a gear and rack cutter.


cutter, measured on the cutting pitch line,

(5.25)

where tpr is the tooth thickness of the rack cutter at its

cutting pitch line.

The standard cutting position is defined as the position

of the rack cutter when its reference line lies a distance Rsg

from the center of the gear blank, so that the cutter

reference line coincides with the cutting pitch line. We now

determine the tooth thickness cut in the gear, if the cutter

is offset a distance e from this standard position. In other

words, the cutter is fed into the gear blank until its

reference line lies a distance (RSg+e) from the center of the

gear blank. The offset e is the distance between the reference

line of the cutter and the cutting pitch line, and it is

positive when the pitch line lies between the gear center and

the cutter reference line. The rack cutter is shown in

Figure 5.8, with the cutting pitch line and the reference line

marked on the diagram. The reference line is the line at which

the tooth thickness is equal to the space width, so that each

Cutting pitch line i+--+--Cutter reference line

Figure 5.8. Cutter tooth thickness

at the cutting pitch line.


is equal to 1rm/2. The tooth thickness tpr at the cutting pitch line can be read from the diagram,

~1rm - 2e tan I/J r (5.26)

To obtain the tooth thickness of the gear, we substitute this expression into Equation (5.25), and replace the cutter

pressure angle I/J r by the gear pressure angle I/J s • The gear

tooth thickness tsg is then given by the following

expression,

(5.27)

As we pointed out during the discussion on the use of the pinion cutter, it is sometimes necessary to know the relation between the positions of a gear blank and the cutter. We now derive the corresponding relation for the case of a gear cut

by a rack cutter. For a pinion meshed wi th a rack, the relation between the

angular position fi of the pinion and the position ur of the rack was given by Equation (3.25),

When we consider a gear blank and a rack cutter, there is no backlash, so the sum of the tooth thicknesses is equal to the rack pitch. In addition, since the rack cutter has the same shape as the basic rack, the radius of the cutting pitch circle is equal to that of the standard pitch circle, and the

rack pitch Pr is equal to the gear circular pitch ps. The relation then takes the following form,

o (5.28)

Rack cutters have the advantage that, because their

teeth are straight-sided, they can be made and measured with

great accuracy. The teeth are relieved, like those of a pinion

cutter, to provide clearance behind the cutting edges. When

the teeth are sharpened, their shape remains unchanged, but the position of the reference line will be slightly altered.


There is, unfortunately, one serious disadvantage

possessed by rack cutters. To cut a gear in one continuous

motion, the rack cutter would require at least as many teeth

as the gear being cut. This is not practical when the gear has

a large number of teeth, since a cutter of the size required

would be too heavy to perform the reciprocating motion, and most rack cutters are therefore made with only a small number

of teeth. In order that the total number of teeth can be cut in the gear, the cutting process is stopped each time the gear

has turned through one or two angular pitches, and the cutter is moved back the same number of pitches. This procedure naturally adds to the time required to cut the gear, and also contributes to errors in the shape of the gear.

Bobbing

It is impossible to give a complete explanation of the hobbing process, except in the context of cutting helical gears. The description given in this chapter is therefore very brief, and many statements are made without proof. A full description, containing all the necessary proofs, will be given in Chapter 16.

A typical hob is shown in Figure 5.9. It has basically the same shape as a screw, with one or more threads, and each thread is cut by a number of gashes, either at right angles to the thread, or parallel with the hob axis, so that cutting

Lead angle

Figure 5.9. A hob.

Hobbing

Swivel angle a

129

Gear axis

Hob axis

~Gear blank

angle ASh

rWork table

Figure 5.10. Gear blank and hob.

faces are formed. Figure 5.10 shows the positions of a hob and

a gear blank during the cutting process. The cutting action of the hob is very similar to that of a rack cutter. The gear blank and the hob are each rotated about their axes, the gear blank slowly and the hob more quickly. Since the hob has the shape of a screw, its threads appear to move in the direction of its axis, so that they simulate the movement of a rack cutter. In addition, since the hob is rotating, the cutting faces move continuously in the tangential direction, and this movement provides the cutting action. The hobbing process therefore has two advantages over that of a rack cutter. The motions of the cutting faces are continuous in both the axial

and the tangent ial di rect ions, so there is no need to move the

hob back after one or two teeth are cut, as we must in the case

of a rack cutter, and the cutting action is obtained without

any reciprocating motion of the hob.

In order to cut each tooth across the entire face-width

of the gear, the hob is fed slowly in the direction parallel

to the gear axis. Each cutting face of the hob simulates a

tooth of the rack cutter, and in order that the cutting faces should meet the gear blank at the same angle as a rack cutter,


the hob is tipped over by a small angle, as shown in Figure 5.10. This angle is known as the swivel angle. On the hob, the angle between the thread tangent and a line

perpendicular to the hob axis is called the lead angle ~sh' and its value is part of the specification of the hob. When a hobbing machine is set up to cut a spur gear, the swivel angle is adjusted, so that it is equal to the lead angle of the hob.

The threads on a hob are equivalent to the teeth in a helical gear. The number of threads is also called the number of starts, because the number can be determined most easily by looking at the end face of the hob, and seeing how many threads start there. Before the hobbing machine is set up, we must calculate the angular velocity ratio required, if a hob with Nh threads is to cut a spur gear with Ng teeth.

At any particular instant, a thread on the hob will be making a cut on one tooth of the gear blank. After the hob has turned through exactly one angular pitch, the former position of the thread will now be occupied by the adjacent thread. The second thread will be making an identical cut on the next tooth of the gear blank, so that the angle through which the gear blank has turned is equal to the angle between adjacent teeth on the gear. If the gear is to have Ng teeth, this angle must be equal to (2W/Ng ). The hobbing machine must therefore be set so that, in the time the hob rotates an angle (2w/Nh ), the gear blank is turned an angle (2w/Ng ). Hence, if the gear ratio mg is defined as the number of teeth required in the gear, divided by the number of threads in the hob,

(5.29)

then the angular velocity ratio required in the hobbing

machine is equal to mg ,

(5.30)

This result is essentially the same as Equation (5.5), which gave the angular velocity ratio when a pinion cutter is used. The similarity is no coincidence, because in both cases we are using one gear to cut another. In the derivation of Equation (5.30), we discussed only the magnitudes of the

Hobbing 131

angular velocities, and the directions were not considered.

It is probably simplest to use physical considerations, to determine the rotation directions of the hob and the gear blank. A rotation of the hob in the direction shown in

Figure 5.10 would cause an apparent movement of the threads to the right, so that to remain in mesh the gear blank must be

rotated in the direction shown. For the purpose of calculating the shape of the teeth cut

in the gear blank, it is generally assumed that a hob can be

regarded in exactly the same manner as a rack cutter. This assumption has a number of consequences, which will be stated without proof in the next paragraph.

A section through the hob thread perpendicular to the direction of the thread is called a normal section, and many hobs are designed so that the threads are shaped in the normal section exactly like the teeth of a rack cutter. In other words, the thread profile in the normal section is straight-sided, with a pressure angle equal to the basic rack pressure angle ~r. If the gashes in the hob are cut perpendicular to the di rect ion of the threads, then the cutting faces are the same shape as the teeth of a rack cutter. During the cutting process, the cutting pitch circle in the gear blank coincides with its standard pitch circle,

exactly as if a rack cutter were used. The tooth thickness cut in the gear blank depends both on the tooth thickness of the hob, and on the depth to which the hob is fed into the gear blank. If the hob position is regarded as the standard position when the tooth thickness cut in the gear is equal to 'lfm/2, then an offset e of the hob from this standard position will produce a tooth thickness in the gear given by Equation (5.27),

1 "2'1fm + 2e tan ~s (5.31)

The assumption stated earlier, that a hob behaves in the

same manner as a rack cutter, is not completely accurate.

However, any errors introduced by the assumption are

extremely small, and the statements made in the last

paragraph are generally used for the purpose of gear design. In other words, when quantities such as the tooth thickness or


the tooth strength are calculated, the calculations are

generally carried out as if the gear is to be cut by a rack cutter, even though it will in fact be cut by a hob. A more

detailed analysis of the hobbing process will be given in

Chapter 16, and we will show there that some of the statements in the previous paragraph are only approximately true. In

particular, we will show that a hob whose threads are straight-sided in the normal section will not cut exact involute tooth profiles in the gear blank, and we will determine the correct shape of the hob thread, if it is to cut accurate involute gears.

Cutting Point

A part of a gear blank and a tooth of a pinion cutter are shown in Figure 5.11, with the cutter tooth profile drawn in several positions relative to the gear blank. The set of cutter tooth profiles shown in the diagram can be thought of as the positions of the cutter during a sequence of cutting

strokes. Since the motions of a generating cutter and a gear blank correspond exactly to the motions of two meshing gears,

the cutter tooth profile touches the gear tooth profile in

Gear blank

Figure 5.1" positions of cutter, relative to gear blank.

Cutter Tooth Tip Geometry 133

each position of the cutter. At every stroke of the cutter, material is removed from

part of the gear blank. However, in each cutter position there is only one point of the cutter tooth face which touches the

finished tooth profile of the gear, as we can see in

Figure 5.11. During the cutting stroke, this point of the cutter makes a final cut on the gear blank, and after the

stroke is completed, the cutter recedes from that particular

point of the gear blank, leaving a part of the finished tooth

surface. The points of the cutter and the gear blank where the

final cut is made will be referred to as the cutting points.

It is evident that they correspond exactly to the contact points in an ordinary gear pair. In Figure 5.11, we showed the positions of the cutter relative to the gear blank, as if the gear blank were fixed. During the actual cutting process, both the cutter and the gear blank rotate, in the same manner

as a pair of meshing gears. Hence, the position of the cutting point, when a generating cutter is used to cut a gear blank, is identical with the position of the contact point in the

corresponding gear pair.


For most gears, the tooth profile coincides with the involute, down to a point somewhere outside the base circle. The tooth fillet begins at this point, and in the fillet the tooth thickness is slightly greater than it would be, if the profile continued on down the involute. In the remaining sections of this chapter, we will determine the point at which

the fillet begins, and we will show that in certain circumstances the tooth thickness in the fillet may be

reduced, rather than increased. Since the tooth fillet of the

gear is cut by the tooth tip of the cutter, it is first

necessary to make a study of the cutter tooth tip geometry.

The tooth profile in some pinion cutters is rounded at

the tip, so that the profile makes a smooth transition with

the tip circle. In other pinion cutters, the profile follows

the involute right out to the tip circle, and there is a sharp


corner where the profile meets the tip circle. We will consider the geometry of a cutter wi th a ro'unded corner at the

tooth tip, of radius r cT ' and we can then regard the cutter with no rounding as a special case, in which the value of r cT is equal to zero.

A pinion cutter with rounded tooth tips is shown in

Figure 5.12. The circular section ha~ a radius r cT ' and it meets the tooth profile at Ahc ' and the tip circle at ATc • The symbol Ahc was chosen, because it is the highest point of the involute section of the cutter tooth profile. The center of the circular arc is shown as A~. In order to calculate the fillet shape in the gear, which is cut by the circular section of the cutter tooth, it is first necessary to find the coordinates of points Ahc and A~.

Since the circular section of the profile is tangent to the tip circle, the polar coordinate R~ of point A~ is equal to the radius of the tip circle, minus the radius of the rounded tip section,

Figure 5.12. Pinion cutter tooth tip geometry.


R' c

135

(5.32)

The normal to the prof ile at Ah passes through A', and c c touches the base circle at E. If the profile angle at Ahc is qJhc' the length EAhc is equa I to (Rbc tan qJhc)' as we proved in Equation (2.11). By expressing this length as the

sum of EA' and A'Ah ' we obtain a relation which can be used c c c . to calculate the value of qJhc'

Rbc tan t/lhc y'(R,2_R2 ) + rcT c bc (5.33)

The polar coordinates of point Ahc are then found by means of Equations (2.18 and 2.35),

Rhc Rbc

cos qJhc (5.34)

8hc tsc

inv t/l s - inv t/lhc -- + 2Rsc

(5.35)

The tangent to the tooth profile at Ahc makes an angle

Yhc with the tooth center-line, and the value of Yhc is given by Equation (2.38),

(5.36)

The Cartesian coordinates x~ and Y~ of point A~ can be read from the diagram, and the polar coordinate 8~ is then expressed in terms of x~ and y~,

x' c Rhc cos 8hc - rcT sin Yhc (5.37)

y' Rhc sin 8hc - rcT cos Yhc (5.38) c

8' arctan (y~) (5.39) c x' c

For a pinion cutter with no rounding at the tooth tips,

we set the value of r rT equal to zero, and the radi i Rhc and R~

are then equal to the tip circle radius RTc '

In.the cutter tooth profile shown in Figure 5.12, the

center A' of the circular tip section lies above the tooth c center-line. It is essential that A' should always be above c


the center-line, in order to allow the circular section of the

profile to merge smoothly with the tip circle. If A' were to c lie below the center-line, the cutter tooth would be slightly

pointed. Hence, the value of y~ given by Equation (5.38) must always be positive, and a negative value would indicate that

the radius r cT of the circular section is too large. When a

tutter is designed, there is sometimes an advantage in making the radius r cT as large as possible, particularly when the cutter is to be used for cutting internal gears, since this helps to minimize the fillet stress in the gears. The largest

usable value of r cT is the one that makes y~ equal to zero, but there is no simple method for calculating this value. The

most practical procedure is to choose a value for r cT ' and then to check that y~ is positive.

A diagram showing the tooth tip geometry of a rack cutter is shown in Figure 5.13. The circular section at the tip of

the tooth has a radius rrT' and meets the tooth profile at Ahr and the tip line at ATr • Point Ahr is the highest point in the straight section of the rack cutter tooth profile, and the distance between this point and the reference line is

1f4rr m

Cutter reference line

Figure 5.13. Rack cutter tooth tip geometry.

Cutter Tooth Tip Geometry 137

labelled h. The center of the circular section is A~, and the

line A~Ahr' which is perpendicular to the tooth profile, makes an angle ~r with the Yr axis. The value of h can be expressed in terms of the rack cutter addendum a r , which is

the distance from ATr to the reference line,

h (5.40)

The Cartesian coordinates of Ahr and A~ are then given by the

following expressions,

xhr - h (5.41)

Yhr 1

- "'41Tm + h tan ~r (5.42)

x' r - a r + rrT (5.43)

y' r - 11Tm 4 + h tan ~r + rrT cos ~r (5.44)

In Figure 5.13, the center A~ of the tooth profile circular section lies below the tooth center-line, as it must if the circular section is to merge smoothly with the tip line. In a properly designed cutter, therefore, the value of

y~ given by Equation (5.44) must always be negative, and a positive value would imply that the radius rrT is too large. The maximum usable value of rrT can be found by setting y~ equal to zero in Equation (5.44), using Equation (5.40) to express h in terms of rrT' and solving the resulting equation

for r rT'

rrT max 0.251Tm cos ~r - a r sin ~r

(1 - sin ~r) (5.45)

For most rack cutters and hobs, the values of a r and rrT are chosen so that the length h is slightly larger than 1.0m.

For example, in a 20° pressure angle rack cutter, with an

addendum of 1.25m and a circular tip radius of 0.3m, the

length h has the following value,

h 1.0526m (5.46)


Radius of the Fillet Circle

We defined the fillet circle of a gear in Chapter 4, as the circle through point Af on the tooth profile, where the

involute part of the profile ends and the fillet begins. To

find the position of Af on a gear cut by a pinion cutter, we consider the meshing diagram of the gear blank and the cutter, which is shown in Figure 5.14. If the cutter is turning

counter-clockwise, the cutting point moves outwards on the cutter tooth towards the tip, inwards on the gear tooth towards the root, and downwards along the path of contact. The involute part of the gear tooth profile is cut by the involute section of the cutter tooth. The lowest point Af on the gear tooth involute is therefore cut by Ahc ' the highest point on the cutter tooth involute, and this occurs when Ahc lies on the path of contact. The path followed by Ahc is a circle of radius Rhc ' and the point where this circle intersects the common tangent to the base circles is labelled Hc. Since point Af of the gear coincides with point Ahc of the cutter when the

cutting point is at Hc' the fillet circle of the gear must

Path followed by point Ahc of the cutter-~--....

Figure 5.14. Meshing diagram of a gear and pinion cutter.

Radius of the Fillet Circle 139

Path followed by (h - e )-t---t--

point Ahr of the cutter ---+\

Figure 5.15. Meshing diagram of a gear and rack cutter.

pass through point Hc' and its radius Rfg is given by the following expression,

(5.47)

The radius Rhc in this equation can of course be replaced by the tip circle radius RTc , for pinion cutters with no rounding at the tooth tips.

The meshing diagram is shown in Figure 5.15, for the case when a rack cutter is used to cut the gear. We consider the rack cutter when it is offset a distance e from its standard position, so that its reference line lies a distance (Rsg+e) from the center of the gear blank. The distance between point Ahr and the rack cutter pitch line is then equal to (h-e).

The involute section of the gear tooth profile is cut by the straight-sided part of the cutter tooth, and the end point Af of the gear tooth involute is therefore cut by point Ahr on the cutter. This cut is made when Ahr lies on the path of contact, which takes place at Hr , the point where the path followed by Ahr intersects the path of contact. Hence, the


fillet circle of the gear passes through point Hr , and its

radius is found as follows,

R2 [ h-e ] 2 bg + Rbg tan 4J s - sin 4J s (5.48)

Undercutting

In Figure 5.14, Hc is the end point of the straight-line path of contact, which corresponds to the involute sections of the gear and the cutter tooth profiles. If we were to find the position of the cutting point corresponding to a point on the gear tooth fillet, it would not lie on the same straight line. However, for the present we are only interested in the involute part of the tooth profiles, so we will regard point

Hc as the end of the path of contact. Hc is shown in Figure 5.14, lying between the pitch point P and the interference point Eg • As we stated in the previous section, the position of Hc is determined by the intersection of the cutter circle of radius Rhc ' and the common tangent to the base circles. If point Hc lies below Eg , the following situation will occur. While the cutting point moves down the path of contact as far as point Eg , the gear tooth is cut with the correct involute profile, right down to point B on the base circle. However, when the cutting point reaches Eg , there is still part of the cutter involute near point Ahc ' which has not yet made a final cut on the gear tooth. It is impossible to cut the gear tooth involute any further, since

the involute does not exist inside the base circle. In

Chapter 9, we will describe the exact shape cut in the gear by

the tooth tip of the cutter. At this stage, we will simply

state that the path followed by point Ahc ' relative to the gear blank, is a curve that intersects the involute near the

base circle, as shown in Figure 5.16. The result is that part

of the involute in the gear tooth profile is cut away. This

phenomenon is called undercutting, because a section of the correct involute profile is undercut by the tip of the cutter

tooth. Undercutting is essentially the same as interference

between the cutter and the gear blank. There is, however, one

important difference between interference and undercutting.

Undercutting 141

I--\--Base circle

Figure 5.16. Undercutting.

Path followed by point Ahc of

the cutter

Interference occurs in a gear pair, if there is non-conjugate

contact between the tooth tips of one gear and the tooth fillets of the other, and the gear pair is then unusable. When we consider a gear blank and a cutter in the situations where interference would occur, the cutter removes extra material from the gear blank, and part of the involute is lost. It is clearly undesirable for gear teeth to be severely undercut, since they are considerably weakened, and the missing part of the involute profile may cause a reduction in the contact

ratio. However, a small amount of undercut is not generally

harmful.

It is found that, near the top of the gear tooth fillet,

the shape cut by the rounded section of the cutter tooth is

almost identical to the path followed, relative to the gear

blank, by point Ahc of the cutter. For this reason, provided

there is no undercutting by point Ahc ' there is also no undercutting by the rounded section of the cutter tooth. When

we check to see whether undercutting will take place, we can


therefore treat the cutter tooth as if it ~nded at point Ahc ' We make the check, simply by determining the position of point

Hc on the path of contact. For no undercutting, Hc must lie between the pitch point and the interference point E • In g other words, the length HcEc in Figure 5.14 must be less than or equal to EgEc' and the condition for no undercutting can be expressed as follows,

(5.49)

The same considerations on undercutting apply when a gear is cut by a rack cutter or a hob. The path of contact in Figure 5.15 must end between the pitch point and the interference point, so the length HrP must be less than EgP, and we obtain the following condition for no undercutting,

Rbg tan tPs (5.50)

It can be seen that, for any particular value of the cutter offset e, there is a lower limit to the base circle radius in the gear being cut, if there is to be no undercutting. This means, in effect, that there is a lower limit to the number of teeth in the gear. For example, if a 20° rack cutter is used with a value of h given by Equation (5.46), and the cutter offset e is equal to zero, then there will be undercutting in any gear which has less than 18 teeth. On the other hand, we can always avoid undercutting by increasing the cutter offset. We will discuss this topic in more detail in the next chapter.

In deriving expressions for the fillet circle radii, given by Equations (5.47 and 5.48), we assumed in each case that the path of contact ended above the interference point. In other words, we assumed that there was no undercutting. Equations (5.47 and 5.48) are therefore only valid for gears which are not undercut. When undercutting occurs, the gear tooth fillet starts at the point where the path followed by point Ahc of the cutter intersects the gear tooth involute.

The circle through the top of the fillet is then called the undercut circle, and we will show in Chapter 7 how the radius of this circle can be found.

Examples 143

Numerical Examples

Example 5.1 A 24-tooth pinion cutter with a module of 6 mm and a

pressure angle of 20° is used to cut a 54-tooth gear. The

tooth thickness of the cutter is 0.5ps' and the cutting center distance is 233.7 mm. Calculate the radii of the

cutting pitch circles, the cutting pressure angle ~c, and the

tooth thickness cut in the gear.

RC pg 161.792 mm (5.8)

RC pc 71.908 mm (5.9)

Rsg 162.000 (5.2)

Rsc 72.000 ( 5. 1 )

Rbg = 152.230 (5.6)

Rbc = 67.658 (5.7)

~c 19.797° (5.10)

~c p 19.797° (5.11)

CC s 234.000 (5.16)

tsg 9.207mm (5.17)

Example 5.2

A 16-tooth pinion cutter with a module of 8 mm and a

pressure angle of 20° has a tip circle diameter of 151.2 mm

and a tooth thickness of 14.313 mm. Determine whether the

cutter can be designed with a tooth tip radius r cT of 2.4 mm.

RSC = 64.000 mm

Rbc = 60.140 R~ = 73.200 (5.32)


I/>hc = 36.271° (5.33)

Rhc = 74.594 (5.34)

9hc 0.025983 radians 1.489° (5.35)

Yhc 34.782° (5.36) y' = -c 0.033 mm (5.38)

Since y~ is negative, the radius r cT is too large. A similar

calculation shows that a value of 2.0 mm for r cT would be satisfactory.

Example 5.3

A rack cutter with D.P. 1 and pressure angle 20° is designed with an addendum of 1.335 inches. Determine the

largest usable value of r rT' and the corresponding value of h.

m = 1.0000 inches

rrT max = 0.4277 inches (5.45)

h = 1.0536 inches (5.40)

Example 5.4 The pinion cutter specification in Example 5.2 is

modified, so that the tooth tip radius is 2.0 mm. The cutter is to be used to cut a 12-tooth gear with a tooth thickness of

14.0 mm. Check to see whether there will be undercutting.

m=8, I/>s=20°, Nc =16, Ng=12 t sc =14.313, RTc =75.6, r cT=2.0, t sg=14.0

C~ = 112.000mm

. c lnv I/>p = 0.029102

24.767°

(5.16)

(5.18)

(2.16,2.17)

(5.19)

Examples

Rbg = 45.105 Rbc =60.140

R' = 73.6 c ¢hc = 36.454°

Rhc 74 . 77 1 mm

145

(5.32) (5.33)

(5.34)

(5.49)

Since this quantity is positive, there is no undercutting.

Example 5.5 A rack cutter with D.P. 4 and pressure angle 14.5° has an

addendum of 0.290 inches and a tooth tip radius of 0.040 inches. Determine the minimum cutter offset, and the corresponding tooth thickness of the gear, if the cutter is to be used to cut a 20-tooth gear, and there is to be no undercutting.

Example 5.6

m = 0.2500 inches h = 0.2600

emin = 0.1033 inches

tsg = 0.4461 inches

(5.40)

(5.50)

(5.27)

A gear pair has a module of 6 mm and a pressure angle

of 20°. The tooth numbers are 19 and 86, the tooth thicknesses

are 9.64 mm and 11.09 mm, the tip circle diameters are

127.0 mm and 533.0 mm, and the center distance is 318.0 mm.

Determine whether there will be interference at the tooth

fillets of the pinion, if the pinion is cut by a hob with

addendum 7.5 mm and tooth tip radius 1.8 mm.

Procedure: Start by checking that the first condition for no interference is satisfied. Next, calculate the limit


circle radius of gear 1. Then calculate the fillet circle radius. And finally, check whether the second condition for no interference is also satisfied.

m=6, ~S=20°, N1=19, N2=86, C=318.0 t s1 =9.64, t s2=11.09, RT1 =63.5, RT2 =266.5

Rbl = 53.562 mm Rb2 = 242.441 ~ 21.436 0

(4.18)

Since this quantity is positive, the first condition for no interference is satisfied.

RLl = 53.850 Dim (4.20)

The remaining equations deal with the cutter geometry, and the cutting of gear 1.

h = 6.316 e 1 0.296

Rfl = 53.596

RLl - Rfl - 0.025m = 0.104 mm

(5.40) (5.27) (5.48)

(4.23)

Since this quantity is positive, the second condition for no interference is also satisfied. In addition, the clearance is adaquate, so there will be no interference.

Example 5.7 Repeat Example 5.6, assuming that gear 1 is cut by the

pinion cutter specified in Example 5.1. The diameter of the cutter tip circle is 159.0 mm, and the radius of the rounding

at the tooth tips is 1.0 mm.

Examples

Rbc = 67.658 mm R' = 78.500 c

q,hc 31.096 Rhc = 79.012

Cc = 129.000 s

. c lnv q,p = 0.015739

q,c p 20.354°

q,c = 20.3540

Rfl = 53.724

RLl - Rfl - 0.025m = - 0.024 mm

147

(5.32)

(5.33)

(5.34)

(5.16)

(5.18)

(2.16,2.17)

(5.19)

(5.47)

(4.23)

Since this quantity is negative, there will be interference.

This example proves that the first condition is not sufficient to prevent interference. We will show in the next

chapter how the design can often be altered, so that the

interference is avoided.

Introduction

Chapter 6 Profile Shift

Since the use of involute gears is now so widespread, it is difficult to appreciate that a different type of gear, known as the cycloidal gear, was still commonly used within the last hundred years or so. In 1890, George B. Grant found it necessary to argue, in his "Treatise on Gears", for the merits of the involute tooth profile compared with the cycloidal.

One of the disadvantages of a cycloidal gear pair is that

it can only mesh at its standard center di stance. Hence, there is no distinction between the pitch circle and the standard pitch circle of each gear, since they must always coincide. Perhaps partly for this reason, when involute gears were first introduced, it was some time before full advantage was taken of the properties of the involute. Like cycloidal gears, the early involute gears were always designed to mesh at the standard center distance. The tooth thickness of each

gear was made equal to half the circular pitch ps' with a

small reduction to allow for backlash, and the addendum as was chosen equal to one module.

These values are still recommended for standard systems

of involute gears, where the main requirement is for

interchangeability. Systems of this sort are used for

catalogue gears, in which each gear of a set must be capable of meshing with every other gear of the same set. In spite of

the advantages that standardization may bring to gears

designed according to a standard system, they suffer from the

drawback that each gear pair can only mesh at the standard

center distance. In addition, many modern gears are cut from

Profile Shift 149

blanks which are integral with the shafts, and this is not practical in the case of catalogue gears. The use of catalogue

gears is therefore quite restricted, and the majority of gears are designed for the requirements of a particular

application. It is then evident that the designer of a gear

pair has a great deal of flexibility in the choice of values for the center distance, the tooth thicknesses, and the gear blank diameters. The only restriction which must normally be

accepted is that the gears should be cut by standard cutters. This is not generally a problem, since, as we showed in

Chapter 5, it is one of the principal advantages of the involute profile that non-standard gears can be cut by standard cutters.

A gear intended to mesh at the standard center distance will differ in several respects from one designed to mesh at a non-standard center distance. In particular, the tooth thicknesses and the tip circle radii of the two gears will usually be different. Figures 6.1 and 6.2 show two such gears,

c

Figure 6.1. A gear with zero profile shift.

150 Profile Shift

c

Figure 6.2. A gear with positive profile shift.

each with the same number of teeth, and cut by the same cutter. Since both gears have the same standard pitch circle radius Rs and the same pressure angle ~s' they must have base circles of the same size, and therefore the tooth profiles of each gear are formed from parts of the same involute. However, the tooth thickness and the tip circle radius of the second gear are larger than those of the first. In Figure 6.2, the involutes forming the opposite faces of each tooth are further apart than those in Figure 6.1, so that the tooth profiles of one gear are shifted, relative to those of the

other. Any gear whose tooth thickness ts is not equal to 0.5ps is said to be cut with profile shift. In this chapter, we will

discuss how the amount of profile shift can be defined, and we will show how it is related to the tooth thickness. We will then describe a method by which the tooth thickness and the

addendum values can be chosen, for a gear pair intended to mesh at any specified center distance C. And finally, we will present some of the most common reasons why gears are designed

with profile shift.

Profile Shift 151

Definition of Profile Shift

A gear is said to have a profile shift of e, if the

involute part of its tooth profile is conjugate to the basic

rack, when the basic rack is positioned so that its reference

line lies a distance (Rs+e) from the center of the gear.

When a rack cutter is shaped exactly like the basic rack,

the shape of any gear cut by the rack cutter is automatically

conjugate to the basic rack. We have already derived an

expression, in Equation (5.27), for the tooth thickness of a

gear cut by a rack cutter, when the cutter reference line is

situated a distance (Rs+e) from the gear center. We can

therefore use the same expression for the tooth thickness of a

gear, conjugate to the basic rack in the same position. Hence,

the relation between the tooth thickness of a gear and its

profile shift e is given by the same equation,

This relation is valid for any involute spur gear,

whether or not the gear is cut by a rack cutter. However, it

is clear from the manner in which the equation was derived

that, when a rack cutter is used to cut a gear, the profile

shift of the gear is equal to the distance between the cutter

reference line and the cutting pitch line. This distance is called the cutter offset, so for a gear cut by a rack cutter,

the profile shift is equal to the offset of the cutter.

As we pointed out in Chapter 5, it is common practice to

assume that a gear cut by a hob is identical to one cut by a

rack cutter. It would follow from this assumption that, when a

gear is cut by a hob, its profile shift is equal to the offset

of the hob. We are not yet in a position to prove this

statement, but in Chapter 16 we will discuss the hobbing

process in greater detail, and we will show that the statement

is indeed true, provided certain conditions are satisfied.

When a pinion cutter is used to cut a gear, the cutter

offset ac~ is defined as the difference between the cutting

center distance CC and the standard center distance C~ during

the cutting process,

152

CC - (R +R ) sg sc

Profile Shift

(6.2)

where Nand N are the numbers of teeth in the gear and the g c cutter. There is, however, no simple exact relation between the cutter offset ~c~, and the resulting profile shift in the

gear. In order to determine the profile shift in the gear,

corresponding to any specified value CC of the cutting center

distance, we must calculate the tooth thickness tsg of the gear at its standard pitch circle. Before we can carry out this calculation, we need to know the tooth thickness t of sc the cutter, and this value must be measured. The method by which we calculate the tooth thickness was outlined in Chapter 5. We use the following sequence of equations, which are taken from Equations (5.6 - 5.17),

Rbg Rsg cos I/I s (6.3)

Rbc Rsc cos I/I s (6.4)

cos I/Ic Rb9+Rbc

CC (6.5)

I/Ic P

tPc (6.6)

ps - tsc - 2C~ (inv I/I s - inv I/I~) (6.7)

Finally, the profile shift is given by Equation (6.1),

e 1 (t - 11l'm) 2 tan I/I s sg 2 (6.8)

In the case when the cutter tooth thickness tsc is equal

to half its circular pitch ps' the cutter will cut a gear with zero profile shift when the cutting center distance CC is set

equal to the standard value C~. I f the cutting center distance is now altered, the resulting profile shift e in the gear is approximately, though not exactly, equal to the cutter offset ~CC. This result can be verified numerically by means s of Equations (6.3 - 6.8). However, the tooth thickness of a

pinion cutter is not necessarily equal to O.5ps' since whatever value it has when the cutter is new, the value is

Profile Shift Coefficient 153

changed slightly every time the cutter is sharpened. Hence,

for a gear cut by a pinion cutter, the profile shift is not generally equal to the offset of the cutter.

Profile Shift Coefficient

In Equation (6.1), we gave the relation between the

profile shift of a gear, and its tooth thickness ts. This equation is always true, because it follows immediately from the definition of the profile shift. The addendum as of the gear is also usually related to the profile shift value, but

in a manner which is under the control of the designer. Since

the addendum as is the radial distance between the tip circle and the standard pitch circle,

R - R T s (6.9)

the value of as is clearly determined by the diameter DT of the gear blank. It is normal practice to choose the gear blank size so that, for a gear with zero profile shift the addendum

as is approximately equal to the module m, and for a gear with profile shift e, the addendum is increased by an amount which

is about equal to the profile shift. Hence, in all cases the addendum value is chosen in a manner which satisfies the following relation,

(6.10)

The tooth thickness in Equation (6.1) and the addendum in Equation (6.10) can each be expressed as multiples of the module,

(1 + ~)m m

(6.11)

(6.12)

The coefficients in both these equations depend on the ratio

(elm). It is therefore clear that, while the size of the tooth

is determined by the value of the module, its shape is largely

154 Profile Shift

influenced by the quantity (elm). This ratio is called the profile shift coefficient.

Limits to Profile Shift Values

The profile shift of a gear may be either negative or positive. In other words, starting from the position of zero profile shift, the cutter may be moved either towards the gear blank, or away from it.

If the cutter is moved too close to the center of the gear blank, the teeth will be undercut. Although gears with undercut teeth are sometimes used, it is preferable to design gears which are not undercut, so the onset of undercutting generally provides a lower limit to the range of useful profile shift values. At the other extreme, when the profile shift is increased, the corresponding increase in the addendum causes a reduction in the tooth thickness at the tip circle. If the profile shift is too great, the teeth may become pointed. It is normal practice to design gears with a minimum tooth thickness at the tip circle of O.25m, and this condition places an upper limit on the value of the profile shift.

In addition to the requirements just discussed, of no undercutting and of adaquate tooth thickness at the tip circle, there are several conditions to be met when the gear pair is assembled, and these also depend primarily on the tooth thickness and the addendum values. First, there must be no interference. Secondly, there should be a suitable amount of backlash. And finally, the gear pair must have an adaquate working depth and contact ratio, and there must be sufficient

clearance at the root circle of each gear. In the next two sections of this chapter, we will

describe a design procedure for a gear pair, in which the tooth thickness and the addendum of each gear are chosen to give suitable values for the backlash, the working depth and the clearances. It is then necessary to check that the conditions are satisfied for no undercutting, for sufficient tooth thickness at each tip circle, for no interference, and for an adaquate contact ratio. It will generally be found,

Geometric Design of a Spur Gear Pair 155

when all these requirements are met, that the values of the profile shift coefficients lie between the following limits,

-0.5 1.0 (6.13)

However, although the vast majority of gears are designed

with profile shift values within this range, there are always a few exceptions, and the number has been increasing in recent

years.

Geometric Design of a Spur Gear Pair

In this section, we describe a procedure for choosing the tooth thickness and the addendum values of a gear pair,

intended to mesh at an arbitrary center distance C. Initially, we design the gear pair so that the tooth

thicknesses are equal at the pitch circles. However, it is often possible to improve the design by increasing the tooth thickness of one gear, and reducing that of the other. We will show how a gear pair can be designed with unequal tooth thicknesses, and we will discuss some of the advantages of this type of design.

We first select the module m and the pressure angle ~s' according to the cutters that are available. The tooth numbers N 1 and N2 are then chosen so that we obtain the required angular velocity ratio, and a standard center Cs that is slightly smaller than the center distance C. It was suggested in Chapter 3 that Cs should lie within the range given by Equation (3.65),

C - v (C B tan ~ s) C (6.14)

In order for this condition to be met, the sum (N 1+N2 ) should

be equal to, or slightly less than, the ratio (2C/m).

Once the four parameters m, ~s' N1 and N2 are chosen, the radii of the standard pitch circles and the base circles are

established, and their values are given by Equations (2.30 and 2.20) ,

156 Profile Shift

RS1 1 '2N1m (6.15 )

Rs2 1 '2N2m (6.16)

Rb1 Rs1 cos tP s (6.17 )

Rb2 Rs2 cos tP s (6. 18)

The remaining quantities in the gear pair all depend on the value of the center distance C. In each gear, the pitch

circle radius Rp ' the operating pressure angle tPp and the operating circular pitch Pp can be found from Equations (3.40,3.41,3.44,3.49 and 4.27),

RP1 N1C

(6.19 ) (N 1+N 2)

RP2 N2C

(6.20) (N 1+N2 )

cos tP Rb1 +Rb2

(6.21) C

tPp tP (6.22)

Pp 211'C

(N 1+N2 ) (6.23)

An expression for the backlash of the gear pair was given in Equation (4.28),

B (6.24)

We first select a suitable value for the backlash, and then

choose the tooth thickness of each gear in conformity with

this equation. The simplest choice is to make the two values

equal,

1 -(p -B) 2 p (6.25)

The corresponding profile shift values are then found from

Equations (3.17 and 6.1),

(6.26)


e 1 41 (t s1 - 11l'm) 2 tan 2 s

(6.27)

t 2 ts2 R [~+ 2 (inv 4Ip - inv4ls)]

s2 Rp2 (6.28)

1 41 (t s2 - 11l'm) e 2 2 tan 2 s

(6.29)

The dedendum of each gear is effectively determined,

once the tooth thickness has been chosen. The dedendum bs is equal to the depth to which the cutter penetrates below the standard pitch circle. It is therefore equal to the addendum

of the cutter, less the amount of the cutter offset, which depends on the type of cutter and the tooth thickness required. For the gear pair being designed, we will first assume that the gears are to be cut by a rack cutter or a hob. The cutter offset for each gear is then equal to the required profile shift, and the dedendum values are as follows,

(6.30)

(6.31)

where a r is the cutter addendum. Later in this chapter, we will give the corresponding equations for the case when the gears are to be cut by a pinion cutter.

For the purpose of the gear pair design, we need the dedendum values bP1 and bp2 ' The dedendum bp in a gear is obviously related to bs ' since bp is the radial distance from the pitch circle to the root circle, while bs is the corresponding distance, measured from the standard pitch circle. The two lengths are shown in Figure 6.3, and we can use this diagram to express bP1 and bP2 in terms of bs 1

and bs2 '

(6.32)

(6.33)

We now choose the sizes of the two gear blanks, to give a

sui table working depth for the gear pair, and adaquate clearances at each root circle. The working depth and the

158 Prof ile Shi ft

c

Standard pitch circle---"

Figure 6.3. Addendum and dedendum, measured

from the pitch circle.

clearances were defined by Equations (3.66 - 3.68), in terms

of the addendum and dedendum values of each gear, measured

from the pitch circles. In addition, recommended minimum

values were given in Equations (3.69 and 3.70),

working depth a p1 + a p2 (6.34)

c 1 bP1 - a p2 (6.35)

c 2 bP2 - a p1 (6.36)

Minimum working depth = 2.0m (6.37)

Minimum clearance = O.25m (6.38)

It is not generally poss i ble to choose the addendum

values a p1 and a p2 of the two gears in such a manner that the

working depth and the two clearances are all equal to the

minimum values given in Equations (6.37 and 6.38). It is

therefore suggested that a p1 and a p2 should be chosen so that

the working depth is given by Equation (6.37), while the


clearances at the two root circles are equal to each other.

2.0m (6.39)

(6.40)

These two equations can be solved, giving the following

values for a p1 and ap2 '

1 m - 2(bp1 - bp2 ) (6.41)

1 m + 2(bp1 - bp2 ) (6.42)

and the diameters of the two gear blanks are then found as

follows,

(6.43)

(6.44)

Unequal Tooth Thickness Design

There are several reasons why the initial design just

described may need to be improved. For instance, there may be interference, or one of the gears may be undercut. For the moment we will simply describe how the design can be altered, and in the later sections of this chapter we will discuss some of the reasons for making the changes.

In the initial design, we chose to make the tooth

thicknesses tP1 and tP2 equal to each other. We now increase

one of the tooth thicknesses by an amount ~tp' and in order to retain the same backlash in the gear pair, we must reduce the

other tooth thickness by the same amount. Hence the tooth

thicknesses, which in the initial design were given by

Equation (6.25), are now as follows,

l(p -B) + ~t 2 P P

(6.45)

= l(p -B) - ~t 2 P p (6.46)

160 Profile Shift

The rest of the design follows exactly the procedure described earlier. In the original description, the equations

used to calculate the gear blank diameters were interspersed with other equations, which were necessary to explain the

reasons for each step in the procedure. For the sake of

clarity, we will now repeat the equations that are specifically required for the calculation of the gear blank

diameters. Whenever the equations for the two gears are identical, apart from the interchange of subscripts, the equation for gear 1 only will be given.

tsl ~ 2(inv q,p - inv q,s)] (6.47) Rsl [R +

pl

41 (t sl - 17Tm) (6.48) e l 2 tan 2 s

bsl a -r e l (6.49)

bPl bsl + RPl - Rsl (6.50)

apl m - 1 '2(bpl - bp2 ) (6.51)

ap2 m + 1 '2(bp1 - bp2 ) (6.52)

DT1 2RT1 2(Rp1 +ap1 ) (6.53)

Equations (6.45 - 6.53) form a design algorithm, which can of course be programmed for a micro-computer. The value of

~tp is chosen by the designer, and it may be either positive or negative. Whatever value is chosen, we obtain a gear pair

in which the working depth is equal to 2.0m, and the

clearances at the two root circles are equal to each other.

For the case when the center distance C is equal to the

standard center distance Cs ' the initial design procedure

gives a gear pair in which the tooth thicknesses tsl and ts2 are equal, and the addendum as in each gear is equal to 1.0m.

If a non-zero value is then chosen for ~tp' the addendum in one gear is increased, while that in the other gear is reduced by the same amount. This type of design is known as the "long

and short addendum system". The name is not appropriate for

gear pairs with non-standard center distances, because the


addendum values are unequal in both the initial and the final

designs.

Dedendum Values of Gears Cut by a Pinion Cutter

In the design procedure just described, the dedendum bsl of gear 1 was given by Equation (6.49), for the case when the

gears are cut by a rack cutter. If the gears are cut by a pinion cutter, the dedendum of each gear is still equal to the

cutter addendum minus the cutter offset, but for a pinion cutter the offset is no longer equal to the profile shift of

the gear. A method for finding the offset of a pinion cutter,

corresponding to a tooth thickness tsg in the gear, was described in Chapter 5. The relevant equations will be repeated here, with no further explanation. The pressure

angle ~~ of the gear and the cutter at their cutting pitch circles can be found from Equation (5.18),

. c lnv ~p (6.54)

Once the value of (inv ~~) is known, we use Equations (2.16

and 2.17) to calculate the corresponding value of ~~. The cutting pressure angle, the cutting center distance, and the cutter offset are then given by Equations (5.19, 5.20 and 6.2),

~c P

(6.55)

Rbg+Rbc

cos ~c (6.56)

CC _ CC s (6.57)

We use these equations to find the cutter offset ~C~l'

required to cut the gear tooth thickness tsl given by

Equation (6.47). The dedendum in gear 1 is then given by the following expression,

a - ~Cc sc sl (6.58)

162 Profile Shift

where a sc is the addendum of the pinion cutter. This equation replaces Equation (6.49) in the design procedure, and the dedendum of gear 2 is of course found in the same manner.

Avoidance of Undercutting, Interference and Pointed Teeth

If the initial design produces a gear pair in which one of the gears is undercut, the tooth thickness in that gear should be increased until the undercutting disappears, while the tooth thickness in the meshing gear is correspondingly reduced. If there is interference, it generally occurs at the tooth fillets of the pinion, and in this case the pinion tooth thickness should be increased until the interference ceases. Finally, if the teeth of either gear are too close to being pointed, or in other words, the tooth thickness at the tip circle is less than O.25m, then the tooth thickness of that gear should be reduced. The design procedure will automatically reduce the addendum, and the tooth thickness at the tip circle will be increased.

In every case, the best value for ~tp can be found most simply by trial and error. When a micro-computer is used, the program can carry out the checks to determine whether there is undercutting or interference, and whether the teeth are too pointed. The designer can increase or decrease ~tp until a satisfactory gear pair is found, or alternatively, until it becomes clear that the requirements cannot all be met.

Balanced Strength Design

The subject of tooth strength, and the calculation of

the tooth stresses, will be discussed in Chapter 11. At this stage it is sufficient to state that the tooth strength of a gear depends not only on the module, the pressure angle and the tooth thickness, but also on the number of teeth in the gear. If two meshing gears have equal tooth thicknesses ts1 and t s2 ' the teeth of the pinion are weaker than those of the gear.

When a gear pair is designed, the initial choice of zero

Recess Action Gears 163

for Atp often produces a gear pair which satisfies all the geometric requirements. However, if the tooth strengths of the two gears are significantly different, the design can be improved by increasing the tooth thickness of the pinion, and reducing that of the gear. If Atp is chosen so that the tooth strengths are equal, the design is known as a "balanced

strength design".

Recess Action Gears

I n Chapter 4, we defined the angles of approach and recess as the angles through which the driving gear turns, during the periods when the contact point is approaching towards, and receding from, the pitch point. It is found in practice that the performance of a gear pair is smoother during the recess part of the meshing cycle than during the approach. We proved in Chapter 3 that the sliding velocity

Line of action

Gear 1 driVing\ /Gear 2 driven

T 1 Last point of contact

M---T 2 First point of contact

Figure 6.4. A gear pair with partial recess action.

164

Gear 1 driVing\

Profile Shift

Line of action

T 1 Last poi nt of contact

r Gear 2 driven

T 2 First point of contact (Coincides with P)

Pitch circle of gear 2 (Coincides with tip circle)

Figure 6.5. A gear pair with complete recess action.

between the meshing teeth changes sign as the contact point passes through the pitch point. The change in the direction of the sliding velocity causes the friction force to change direction, and this is responsible for the smoother operation of the gear pair during the recess phase of the meshing cycle.

When one gear is always the driver, and the other is

always driven, it is possible to design a gear pair to take

advantage of the smoother recess action. The gear pair is

designed so that most, or even all, of the path of contact

lies on the recess side of the pitch point. This can be done

by increasing the tooth thickness of the driving gear, and

reducing that of the driven. If the recess side of the contact

path is significantly longer than the approach side, the gear pair is said to have partial recess action. If the entire

contact path lies on the recess side, the gear pair has recess

action only. An example of each type of gear pair is shown in

Figures 6.4 and 6.5.

Profile Shift 165

Contact Ratio and Root Circle Clearance

In the three previous sections, we showed how the design of a gear pair can often be improved by the use of unequal tooth thickness design. There are, however, two aspects of gear pair design which cannot be improved by this type of

modi fication. When gear pairs are designed according to the procedure

described earlier, it is very unusual for the contact ratio to

be less than the recommended minimum of 1.4. This is because the gear pairs always have a working depth of 2.0m, and this

generally ensures an adaquate contact ratio. In the rare cases where the contact ratio in the initial design is too low, it is possible to increase its value by reducing the tooth thickness of the pinion and increasing that of the gear. However, this solution to the problem is normally impractical, since it involves weakening the teeth of the pinion, which are already weaker than those of the gear. A better solution is to try a completely different design, in which the module is reduced and the number of teeth in each gear is increased.

In some designs, the clearances at the root circles may be inadaquate. It will be found that, for gears cut by a rack cutter or a hob, the value of atp has absolutely no effect on the size of the clearances, so if they are too small in the ini tial design, they will not be improved by any choice of ~tp. A rack cutter with an addendum of 1.25m would give clearances of O.25m in a gear pair meshing with zero backlash at the standard center distance Cs • When the gear pair is designed with a normal amount of backlash, the teeth of each gear are cut slightly deeper, and the clearances are of course

increased. If the same cutter is used to cut a gear pair for a

center distance C which is greater than the standard center

distance, and the tooth thicknesses are chosen so that the

backlash is unchanged, the clearances become smaller for

larger values of C. Hence, inadaquate clearances are an

indication that the the value of (C-Cs ) is too large. This is

the reason why there is a lower limit for the value of Cs ' and

in the next section we will show how the expression given in Equation (6.14) was derived.

166 Profile Shift

Recommended Range of Values for Cs

We stated earlier in this chapter that the designer of a gear pair chooses the module and the tooth numbers so that the standard center distance Cs is equal to, or slightly less than, the specified center distance C. We have now shown that if the difference between C and Cs is too great, the clearances in the gear pair will be inadaquate. It is therefore helpful to know the value of (C-Cs ), at which the clearances are equal to their minimum recommended value.

For simplicity, we consider a gear pair in which the tooth thicknesses are equal, and each gear has the same number of teeth N. This does not restrict the generali ty of the conclusions, even though it may appear to. We have already pointed out that the clearances do not depend on the value of ~tp' so it makes no difference to the clearances if we choose equal values for the tooth thicknesses. We will also find, when we follow the design procedure described by Equations (6.45 - 6.53), that the clearances depend on the sum (N 1+N2), but not on the individual values of N1 and N2 •

We will assume that the gears are cut by a rack cutter, and that the cutter addendum is equal to one module, plus the minimum clearance cmin required in the gear pair,

(6.59)

If, for example, we want a minimum clearance of O.25m, we would use a cutter with an addendum of 1.25m.

We now consider the design of a gear pair with backlash B, to operate at a center distance C. Our purpose is to find the value of C at which the clearances are equal to

the minimum value cmin. I f the tooth thicknesses are equal and each gear has the same number of teeth, the two gears will be identical. We follow the design procedure given by Equations (6.45 - 6.53), and calculate the dedendum bp and the addendum ap of each gear. The clearance c at each root circle is equal to (bp-ap ). We set this value equal to cmin ' substituting the expression in Equation (6.59) for the cutter addendum a r , and we obtain a relation that must be satisfied by the operating

pressure angle tl>p of each gear,

Recommended Range of Values for Cs

sin I/I s cos I/Ip

- ..1L + inv 1/1 + I/I s 2C P

We express I/Ip in the following form,

167

(6.60)

(6.61)

where the angle 61/1 is small compared with I/I s • We substitute this expression into Equation (6.60), expanding each term

containing I/Ip as a power series in 61/1, and solve to find the

approximate value of 61/1,

B 2C tan I/I s

The relation between Cs and C is then found as follows,

cos I/Ip (cos I/I)C e C - t/(C B tan I/I s )

s

(6.62)

(6.63)

This equation gives the maximum difference (C-Cs ) for which the gear pair will be cut with adaquate clearances. In other words, if w-e are designing a gear pair for a specified

center distance C, the equation gives the lowest acceptable value of Cs ' and this expression is therefore used for the lower limit in the range of Cs values given by Equa t i on (6. 1 4 ) •

We now give the reason for the upper limit in Equation (6.14), which states that Cs should not be greater than C. The maximum bending strength in a gear pair is achieved when the tooth thicknesses are chosen to give a balanced strength design. If we design a number of gear pairs for different values of (C-Cs )' each with the same backlash and wi th balanced tooth strength, then the tooth strength increases as the value of (C-Cs ) increases. It is unusual to

design gear pairs in which (C-Cs ) is negative, because this

causes an unnecessary weakening of the teeth. Hence, if we are

designing for a specified value of C, we generally choose the

values of m, N 1 and N2 so that Cs is less than or equal to C. It should be emphasised that the limits for Cs in

Equation. (6.14) are not inflexible. Clearly, if the tooth

strength is unimportant, gear pairs can be designed with Cs greater than C, provided this does not lead to any of the

168 Profile Shift

geometric problems, such as interference. At the other

extreme, it is also possible to design gear pairs in which the

clearances are adaquate, even though Cs is less than the lower limit in Equation (6.14). For example, when gears with

positive profile shift are cut by a pinion cutter, the dedendum values are slightly larger than when a rack cutter is

used, and the clearances in the gear pair are therefore increased. Alternatively, in cases where the gears are cut by a rack cutter, we can increase the clearances in the gear

pair, simply by choosing a cutter whose addendum is longer than the value given by Equation (6.59). It is therefore evident that Equation (6.14) merely provides a recommended

range of values for Cs ' but that in exceptional cases the value of Cs may lie outside this range.

Alternative Names for Profile Shift

The phrase "profile shift", which corresponds exactly to the German word Profilverschiebung, was introduced by Merritt [4] into the gear terminology of the English language. The concept of profile shift is also known by several other names, including hob offset, cutter offset, correction, and addendum modif icat ion.

In this book the name profile shift is used, simply because there are minor objections to each of the alternatives. For example, it would seem odd to refer to the hob offset of a gear, if it is not cut by a hob. The name cutter offset is perhaps an improvement, but it is not

suitable for a gear cut by a pinion cutter, because in this

case the profile shift is not equal to the cutter offset. The

name correction is a shortened version of an older name,

correction for undercutting, which implies that the only

reason for using profile shift is to avoid undercutting. This

name has now been largely replaced by the phrase "addendum

modification", but here again there is a problem, because the profile shift is a measure of the tooth thickness, rather than

of the addendum length. As we pointed out earlier in this

chapter, the addendum 6s of a gear is determined simply by the

size of the gear blank. It is therefore possible, though

Alternative Names for Profile Shift 169

unusual, to cut a profile-shifted gear with an addendum of one

module, and it is equally possible to cut a stub-toothed gear

with no profile shift, and an addendum of less than one

module. The value chosen for the addendum is therefore

independent of the profile shift, and the name "addendum

modification" is actually misleading, since it is used to

describe a quantity which is really related to the tooth

thickness.

170 Profile Shift

Numerical Examples

Example 6.1

A 24-tooth pinion cutter with module 6 mm and pressure

angle 20° has a tooth thickness of 0.5ps' Calculate the

cutter offset required to cut a 17-tooth gear with a profile

shift of 3.0 mm.

m=6 tP =20° , s ' Nc =24, Ng=17, t sc =9.425, e=3.0

tsg = 11.609mm (6.1)

CC = s 123.000 (5.16)

inv tP~ = 0.023782 (5.18)

tPc p 23.230° (2.16,2.17)

tPc 23.230° (5.19)

Rb9 47.924

Rbc 67.658

CC 125.780 (5.20) acc

s 2.780 mm (6.2)

Note that the cutter offset is similar in value to the profile

shift, but that the two quantities are not identical.

Example 6.2

A gear pair is to be cut by a hob with D.P. 4, pressure

angle 25°, and addendum 0.3125 inches. The tooth numbers are

24 and 61, the center distance is 10.8 inches, and the

backlash is to be chosen according to Equation (4.43). Use the

procedure outlined in Equations (6.45 - 6.53) to design the

gear pair, first for atp equal to zero, and then for atp equal

to 0.04 inches. For each design, calculate the tooth

thicknesses, the tip circle diameters, and the clearances at

the root circles.

Examples 171

Pd=4, 4>5=25°, a r =0.3125, N1=24, N2=61, C=10.8

m = 0.2500 inches

RS1 3.0000

RS2 7.6250

Rb1 2.7189

Rb2 6.9106

RP1 3.0494 (6.19)

RP2 7.7506 (6.20)

4> = 26.922° (6.21 )

4>p = 26.922° (6.22)

Pp = 0.7983 (6.23)

B = 0.0149 inches (4.43)

Initial design,

~tp 0.0000

tp1 0.3917 (6.45)

tP2 0.3917 (6.46)

tS1 0.4331 (6.47)

tS2 0.5068

e 1 = 0.0434 (6.48)

e 2 = 0.1223

bS1 0.2691 (6.49)

bs2 0.1902

bP1 0.3186 (6.50) bP2 0.3158

a p1 0.2486 (6.51)

a p2 0.2514 (6.52)

DT1 6.5961 (6.53)

DT2 = 16.0039 c 1 = 0.0672 (6.35 )

c 2 = 0.0672 inches (6.36)

Second design,

~tp 0.0400

tP1 0.4317 (6.45)

tP2 0.3517 (6.46)

tS1 0.4725 (6.47)

tS2 0.4674 e 1 = 0.0855 (6.48)

172 Profile Shift

e 2 = 0.0801

bs1 0.2270 (6.49) bs2 0.2324

bP1 0.2764 (6.50) bP2 0.3580

a p1 0.2908 (6.51) a p2 0.2092 (6.52)

DTl 6.6804 (6.53)

DT2 = 15.9196

c 1 = 0.0672 (6.35) c 2 = 0.0672 inches (6.36)

Note that the clearances c 1 and c 2 are not affected by the

value of L'ltp.

Example 6.3

The gear pair described in Example 5.6 was designed with L'ltp equal to zero. We showed in Example 5.7 that there is

interference at the tooth fillets of the pinion, when the gears are cut by the pinion cutter specified in Examples 5.1

and 5.7. Redesign the gear pair, assuming that the gears are to be cut by the same pinion cutter. Use L'ltp equal to 1.0 mm, and choose the backlash according to Equation (4.42).

m=6, ~s=20o, Nc =24, t sc =9.425, RTc =79.5, r cT=1.0 C=318.0, N1=19, N2=86

RS 1 = 57.000 mm

Rs2 = 258.000

Rbl = 53.562

Rb2 = 242.441

RPl = 57.543

RP2 = 260.457 ~ = 21.436°

~ = 21.436° P

P = 19.029 P

B = 0.384 mm (4.42)

We start the design by finding the required tooth

thicknesses, which do not depend on the type of cutter used.

Examples

atp = 1.000

tp1 = 10.323

tp2 = 8.323

ts1 10.634

ts2 10.095

173

(6.45)

(6.46)

(6.47)

In the next group of equations, we find the dedendum of

gear 1, when it is cut by the pinion cutter. For the sake of

brevity, the intermediate steps are left out for gear 2, and

only the result is given.

c Cs1 129.000

. c lnv ~p1 = 0.019592 (6.54)

~~1 = 21.836° (2.16, 2.17)

(6.55)

Rbc = 67.658

130.590 (6.56)

bS1 = 5.910 (6.57, 6.58)

b S2 = 6.589mm

Once the dedendum values have been found, the design is

completed in the usual manner.

bP1 6.453 (6.50)

bp2 9.046

a p1 7.297 (6.51)

a p2 4.703 (6.52)

DT1 = 129.679 (6.53)

DT2 = 530.321 mm

It can be verified that, in this new design, there is no

interference.

Chapter 7 Miscellaneous Circles

The specification of a gear includes the diameters of a number of circles, such as the standard pitch circle and the tip circle. In addition to these circles, there are several others whose sizes must be determined by the designer, in order to check that the design will perform satisfactorily. We have already discussed the limit circle and the fillet circle of a gear. In this chapter we will introduce some other circles which are frequently needed, and we will show how their radii can be calculated.

Highest and Lowest Points of Single-Tooth Contact

A typical meshing diagram is shown in Figure 7.1. The ends of the path of contact are labelled T1 and T2 , and these lie between the interference points E1 and E2 , as they must in any properly designed gear pair. We define two additional points Q and Q' on the path of contact, where Q lies a

distance Pb below T l' and Q' lies a distance Pb above T2• We stated in Chapter 4 that, when there are two pairs of

teeth simultaneously in contact, the distance between these

contact points is equal to the base pitch Pb' We can see from Figure 7.1 that, whenever there is a contact point between T2 and Q, there must simultaneously be a second contact point between Q' and T1. However, when a contact point lies between Q and Q' , this is the only contact point.

The total contact force must remain constant, if there is to be a uniform transmission of power. It is clear, therefore, that the contact force is roughly halved, when it is shared between two pairs of teeth in contact. Figure 7.2

Highest and Lowest Points of Single-Tooth Contact 175

Figure 7.1. The end points of single-tooth contact.

shows how the contact force acting on one tooth depends on the position of the contact point, as it moves along the path of contact. The part of the meshing cycle when the contact point lies between Q and Q' is called the period of single-tooth contact. During this period, only a certain part of the profile of each meshing tooth comes into contact with the tooth of the other gear. The ends of this section of the tooth profile are called the highest and lowest points of

single-tooth contact.

The maximum stresses in a spur gear tooth occur when the

tooth force is at its largest value, or in other words, during

the period of single-tooth contact. There are two types of

stress which are of primary interest to the gear designer, the

tensile stress in the fillet of each tooth, and the contact

stress at the point of contact. A typical loaded tooth is

shown in Figure 7.3, with the critical regions of fillet stress and contact stress marked on the diagram. For a

176

Q) u .... o --U ell -C o ()

Q Q'

Miscellaneous Circles

Position of contact point on contact path

Figure 7.2. Variation of the contact force.

constant tooth force, the fillet stress increases as the load moves towards the tooth tip. The maximum value is reached when the load is applied at the highest point of single-tooth

contact. Beyond this point the tooth force is approximately halved, as we showed earlier, and the fillet stress is therefore reduced. The contact stress also varies as the

contact point moves along the tooth profile. The subject of spur gear tooth stresses is discussed more fully in Chapter 11, and we will show there that the maximum contact stress in both gears of a pair occurs when there is contact at the lowest point of single-tooth contact in the pinion. In

Region of maximum tensile

fillet stress Tooth contact force

~--Region of contact stress

Figure 7.3. Regions of maximum stress.

Form Diameter 177

order to calculate the maximum stresses in a tooth, it is

therefore necessary to determine the highest and lowest

points of single-tooth contact. We will show how this is done,

by finding the radii of the circles through these points for

the two gears in Figure 7. ,.

As the position of the contact point moves up the path of

contact, the contact point on gear 1 moves up the tooth profile, towards the tip, and the contact point on gear 2

moves down the tooth profile, towards the root. On gear 1, the

highest point of single-tooth contact is therefore reached

when the contact point is at Q'. The radius RHSC , of the circle through this point is equal to the length C,Q' , and its

value can be found from triangle C,E 1Q' ,

2 RHSC1

When the contact point lies at Q, its position on the

tooth profile of gear' is at the lowest point of single-tooth contact. The radius RLSC , of the circle through this point is

equal to C,Q, and is given by the following expression,

( 7 • 2 )

On the tooth profile of gear 2, the highest and lowest points of single-tooth contact correspond to points Q and Q' on the path of contact. The radii RHSC2 and RLSC2 of the circles through these points can be found by interchanging the subscripts' and 2 in Equations (7.' and 7.2).

Form Diameter

The active part of a gear tooth profile is defined as the

part which comes into contact with the teeth of the meshing

gear. This is the part stretching from the tooth tip, down to

the limit circle. The radius of the limit circle was given by

Equation (4.20), for a gear meshed with another gear,

(7.3)

178 Miscellaneous Circles

and for a pinion meshed with a rack, the limit circle radius was given by Equation (4.21),

(7.4)

After a gear is cut, it should be checked to ensure that any errors in the tooth profile remain within the tolerance specified. Theoretically, the lowest point on the profile at which a measurement would be useful is at the limit circle, since this is the end of the active part of the profile. In practice, though, there is always the possibility of tooth contact taking place slightly below the calculated limit circle, due to tolerance in the center distance, and other possible errors. The profile is therefore checked, down to a point lying a short distance inside the limit circle, and the circle through this point is called the form circle. The radial distance between the two circles is generally chosen as O.025m, so the form circle radius Rform is defined as follows,

RL - O.025m (7.5)

The diameter of the form circle is known as the form diameter. There is a possibility of confusion between the form

circle and the fillet circle of a gear, since the reason for calculating the radii of both circles is to avoid the danger of interference. In order to eliminate any confusion, we will point out a number of differences between the two circles.

The fillet circle of a gear was defined in Chapter 4, as the circle through the point where the involute part of the profile joins the fillet. Hence, the fillet circle radius

depends only on the geometry of the gear itself, and not on that of the meshing gear. On the other hand, the form circle radius was defined in terms of the limit circle radius, and we can see from Equations (7.3 and 7.4) that this value depends on the geometry of the entire gear pair.

One of the conditions for no interference was given by Equation (4.23),

(7.6)

Undercut Circle 179

The fillet circle radius Rf was given by Equation (5.47) for a gear cut by a pinion cutter, and by Equation (5.48) for a gear

cut by a rack cutter or a hob. The designer of a gear pair should calculate the fillet circle radius of each gear as part

of the design procedure, in order to check that Equation (7.6) is satisfied. These checks ensure that there will be no interference, provided the gears are correctly cut. The

profile accuracy is measured after the gears have been cut, and it is at this stage that the form diameter is required.

The form circle radius of a gear is defined by Equation (7.5), and the fillet circle radius must satisfy

Equation (7.6). A comparison of these two equations shows that the fillet circle of a gear is either smaller than the form circle, or sometimes equal in size, but never larger.

Undercut Ci rcle

In Chapter 5, we described the conditions for no undercutting in a gear, and we pointed out that it is

Undercut circle

Path followed by point Ahc of the cutter

-Involute

Figure 7.4. An undercut gear.


preferable, though not essential, to design gear pairs in such a manner that neither gear is undercut. There may be times, however, when some undercutting cannot be av·oided, and in such cases it is important to know what effect this will

have on the contact ratio and the tooth strength. An undercut tooth profile is shown in Figure 7.4. The

point where the fillet starts is labelled Au' and the circle through this point is called the undercut circle. The diagram also shows the path followed, relative to the gear, by point Ahc on the cutter, the highest point on the involute section

of the cutter tooth profile. This path lies extremely close to the fillet, as we can see in the diagram, particularly near

the top of the fi llet. The only methods known to the author of this book, for

finding the radius Ru of the undercut circle, all involve some form of trial and error. The problem is simplified slightly,

if we find the point where the locus of Ahc intersects the involute, rather than the point where the fillet intersects

the involute. The two points are not identical, so the value we obtain for Ru is only approximate, but the error is negligible.

The value of Ru can then be found by the following procedure. An undercut tooth profile is shown in Figure 7.5,

Base circle --Locus of

point Ahc

-Undercut circle

Figure 7.5. Calculation of the undercut circle radius.

Undercut Circle 181

with the involute extended to the point where it meets the

base circle at B, and the diagram also shows the locus of

point Ahc ' A typical circle of radius R cuts the involute and

the locus at points A and A', and the polar coordinates of

these points are labelled 9R and 9R. If 9R is larger than 9R, the radius R is smaller than Ru' as we can see in Figure 7.5.

The radius Ru of the undercut circle can be found by

calculating 9R and 9R at a number of different radii, and eventually finding the value of R at which 9R and 9R are

equal.

The polar coordinate 9R of the point on the involute at radius R was given by Equation (2.35),

(7. 7)

The corresponding value of 9R depends on the type of cutter

used, and we will deal first with the case when the gear is cut by a pinion cutter.

Figure 7.6 represents the gear and the pinion cutter, in

their positions when point Ahc of the cutter lies on the gear circle of radius R. In order to keep the diagram as simple as

possible, the positions of the gear and cutter teeth are shown

only by the tooth center-lines, which lie at angles P and P g c with the line of centers. The polar coordinates (Rhc ,9hc ) of point Ahc ' where 8hc is measured from the tooth center-line, were given by Equations (5.34 and 5.35).

Our purpose is to find the polar coordinate 9R of

point Ahc ' relative to the coordinate system fixed in the gear. We start by using the cosine law in triangle C C Ah ' to g c c obtain the angular posi tion of the cutter,

arccos

{Cc )2+R2 _R2 hc

{CC )2+R2 _R2 [ hc]

2Cc R hc - 9 hc (7 .8)

The corresponding angular position of the gear is given

by Equation (5.21),

(7.9)

182

Gear tooth center line

Miscellaneous Circles

Cutter tooth center line

Figure 7.6. position of point Ahc •

We then use the sine law in triangle CgCcAhc' to find an expression for 8R,

_1_ sin (-fJ -8') Rhc g R

8' R

(7.10)

A similar procedure is used to find 8R, when the gear is cut by a rack cutter or a hob. As before, we look for the intersection of the involute with the locus of Ahr , the end

Cutter pitch line

C p

Gear tooth centerline

x

I--cutter reference line I

e

Yr

r-~~--------+Xr

Cutter tooth centerline

h 1f4rrm- h tan cJ>s

Figure 7.7. Position of point Ahr •

Undercut Circle 183

point of the straight section of the cutter tooth profile. Figure 7.7 represents the gear and the cutter, and again the positions of the teeth are indicated only by their center-lines. The coordinates of point Ahr , relative to the

(xr'Yr) coordinate system in the cutter, were given by Equations (5.41 and 5.42). The cutter is in the position where point Ahr lies on the gear circle of radius R. The cutter has an offset e, and ur is the distance it has moved in the direction of its reference line. Since the cutter tooth

center-line lies below the line CP, the distance between these two lines is (-ur ). The angular position of the gear is

shown by the angle ~g' In order to avoid confusing the diagram, the

right-angled triangle with hypotenuse CAhr is drawn again in Figure 7.8. The third corner of the triangle is labelled H, and the lengths of the sides are taken from Figure 7.7. We start by expressing the length of side HAhr in terms of the other two sides,

- ur + t'lTm - h tan t/I s (7.11)

This equation gives the cutter position ur at which point Ahr lies on the gear circle of radius R. The

corresponding angular position ~g of the gear is found from Equation (5.28),

(7.12 )

C ~-------r--------------~

-u r +1;4rrm

-h tan <Ps

Figure 7.8. Detail of Figure 7.7.


We now write down a second relation from the triangle in

Figure 7.8, and we obtain the required expression for the angle 8R,

8' R

R +e-h sg R cos (-fJ -8') g R

(7.13)

As we pointed out earlier, we find the radius of the

undercut circle by calculating 8R and 8R at a number of radii,

starting with the base circle radius Rbg , and gradually increasing the value of R until 8R is larger than 8R•

Contact Ratio When One Gear Is Undercut

We consider the case when gear 1 is undercut, and we

first calculate the radii Ru1 and RL1 of the undercut and limit circles of gear 1, in order to determine which circle is

larger. The meshing diagram is shown in Figure 7.9, for the

case when the undercut ci rc Ie is smaller than the limi t

Figure 7.9. A pinion with moderate undercut.

Contact Ratio When One Gear Is Undercut 185

circle. The two circles are nearly equal in size, and to avoid

complicating the diagram, the limit circle is not shown. However, it is possible to verify that the limit circle is larger than the undercut circle, by the following argument. The point where the undercut circle intersects line E1E2 , the

common tangent to the base circles, is labelled U1• The lower

end T2 of the path of contact lies between U1 and the pitch point, so T2 lies outside the undercut circle. Hence, the limit circle of gear 1, which is the circle through T2 , is

larger than the undercut circle. Since T2 lies outside the undercut circle, the contact on the tooth face of gear 1 ceases at a point on the tooth profile above the undercut circle. Although the profile is cut away inside the undercut circle, the meshing is not affected, because the missing part of the involute would not in any case be touched by the meshing gear. The contact ratio mc is therefore given by Equation (4.9), exactly as if there was no undercutting.

The meshing diagram is shown in Figure 7.10 for the second case, when the undercut circle of gear 1 is larger than the limit circle. The point where the undercut circle intersects the path of contact is again labelled U1, and in this gear pair U1 lies between point T2 and the pitch point. In a gear pair with no undercutting, the path of contact would be the line T1T2• However, if the path of contact in Figure 7.10 were to continue below point U1, there would be contact inside the undercut circle of gear 1, which is impossible because the involute tooth profile has been cut away. In this case, therefore, the path of contact extends

from Tl only to point U1. The length of this line can be read

from the diagram, and we divide by the base pitch to obtain the corresponding contact ratio,

(7. 15)

It is interesting that the value of the contact ratio given by Equation (7.15) depends only on the geometry of

gear 1. It does not depend on the center distance, nor on the


Figure 7.10. A pinion with severe undercut.

addendum of gear 2. The contact ratio is only affected by a change in either of these quantities, if the change is so great that the limit circle of gear 1 becomes larger than the undercut circle. In this case, Equation (7.15) no longer applies, and we must use Equation (7.14) to calculate the

contact ratio. The point of the tooth profile where the undercut begins

was labelled Au in Figures 7.4 and 7.5. At this point of the profile the tangent suddenly changes direction, as we can see

in the diagrams. If point Au comes into contact with the

meshing gear, the contact stress is extremely high, due to the discontinuity in the profile tangent. It is therefore

important to design the gear pair so that the active part of the tooth profile in gear 1 ends above point Au. In other words, the undercut circle of gear 1 must be smaller than the

limit circle, and to allow for some tolerance in the center

distance, the undercut circle should be no larger than the

form circle. If this condition is not satisfied in the initial

Contact Ratio When One Gear Is Undercut 187

design, the tooth thickness of gear 1 should be increased and that of gear 2 reduced, either until the form circle of gear 1 is as large as the undercut circle, or until gear 1 ceases to be undercut. The gear pair shown in Figure 5.10 is one in which the limit circle of gear 1 is smaller than the undercut circle, so the contact point will move right down the tooth face of gear 1 to the point where the undercut begins. It is obvious from the diagram that this is a thoroughly bad design.

At the beginning of the previous section, we mentioned the effect of undercutting on the tooth strength of a gear. A method will be described in Chapter 9 for calculating the fillet shape of a gear tooth, and in Chapter 11 we will show how the tooth stresses can be calculated. Neither method is affected by any undercutting in the tooth. The strength analysis of an undercut gear is therefore carried out in the same manner as that of an ordinary gear, but the effect of the undercut is of course to increase the calculated stresses.


Numerical Examples

Example 7.1 A gear pair is designed with D.P. 2.5 and pressure

angle 25°. The tooth numbers are 50 and 99, the diameters of the tip circles are 20.98 inches and 40.62 inches, and the

center distance is 30.0 inches. Calculate the radii of the highest and lowest points of single-tooth contact on each gear.

Example 7.2

Pd=2.5, ~S=25°, N1=50, N2=99 RT1 =10.49, RT2 =20.31, C=30.0

m = 0.4 inches

Rb1 = 9.0631 Rb2 = 17.9449 ~ = 25.807°

Pb = 1. 1389

10.2034

9.9652

RHSC2 = 20.0383 RLSC2 = 19.8022 inches

(7.1)

(7.2)

(7.1)

(7.2)

A 12-tooth gear with module 10 mm and pressure angle 20 0

is cut by a hob with an addendum of 12.5 mm and a tooth tip

radius of 3.0 mm. Show that the gear is undercut, if there is

no profile shift, and calculate the radius of the undercut

circle.

m=10, ~ =20°, N =12, e=O, a =12.5, rrT=3.0 s g r

R = 60.000 mm s9 Rb9 = 56.382

h = 10.526

h-e Rbg tan ~s - . Sln ~s

- 10.255 mm

(5.40)

(5.50)

Examples 189

This quantity is negative, so the gear will be undercut. The

radius of the undercut circle is found by trial and error,

using Equations (7.11-7.13). In order to save space, only the

final set of calculations is given below.

Choose R = 56.555 mm

ur - 23.378

- 0.651431 radians = - 37.324°

9' 8.345° R

tsg = 15.708

4>R = 4.488° 9R = 0.145643 radians = 8.345°

(7.11)

(7.12)

(7.13)

(5.31)

(2.18) (7.7)

Since 9R is equal to 9R, we have chosen the correct value

of R, and the undercut circle radius Ru is equal to 56.555 mm.

This particular gear is shown in Figure 5.16.

Example 7.3

The gear pair shown in Figure 7.9 has the following

specification. Gear 1 is the gear which was described in

Example 7.2, with a tip circle diameter of 140 mm, and the

radius of its undercut circle was calculated in that example.

Gear 2 has 25 teeth, and the diameter of its tip circle is

285 mm. The center distance is 195 mm. Show that the form circle of gear 1 is larger than its undercut circle, and then calculate the contact ratio.

m=10, 4>5=20°, N1=12, N2=25 RT1 =70.0, RT2 =142.5, C=195.0

Rbl = 56.382 mm

Rb2 = 117.462 4> = 26.937°

RL 1 = 56.899

Rf 1 = 56.649mm orm,

(7.3)

(7.5)

We showed in Example 7.2 that the radius of the undercut

circle in gear 1 is 56.555 mm. Since the form circle is larger

than the undercut circle, the gear pair is quite usable, even


though gear is undercut. Of course, the tooth strengths are very unequal, and the design would therefore be improved by increasing the tooth thickness of gear 1, which would then eliminate the undercut. We have also shown that the limit circle is larger than the undercut circle, so to find the contact ratio we use Equation (7.14).

Pb = 29.521 mm mc = 1.146 (7.14)

Introduction

Chapter 8 Measurement of Tooth Thickness

The tooth thickness ts of a gear is defined as the arc length between opposite faces of a tooth, measured around the standard pitch circle. This is a length which cannot be measured directly, so in practice a different dimension of the gear is measured, which is then used to calculate the tooth thickness. In this chapter we will describe three of the most commonly used methods, by which the tooth thickness of a

gear can be found. After a gear has been cut, the tooth thickness is

generally measured before the gear is moved from the worktable. In this way, if the measured tooth thickness is larger than the specified value, it is possible to make another cut without repositioning the gear on the worktable. It is also possible to calculate the addi tional depth to which the cutter must be fed in, in order to obtain the required tooth thickness. The relation between the offset e of a rack cutter and the resulting tooth thickness was given by Equation (5.27),

1 "27rm + 2e tan q, s (8.1)

In order to obtain a change ~ts in the tooth thickness, the cutter offset must therefore be altered by an amount ~e, given by the following expression,

2 tan q,s (8.2)

Since we are considering a reduction in the tooth thickness,

192 Measurement of Tooth Thickness

the values of ~ts and ~e are obviously negative. The equation is also valid when the cutter is a hob, and although it is not exactly correct for the case of a pinion cutter, it can be used with negligible error, provided the required change in the cutter setting is small compared with the module.

Gear-Tooth vernier Caliper

The most direct method for measuring the tooth thickness of a gear makes use of an instrument called a gear-tooth caliper. This instrument, which is shown in Figure 8.1, is a vernier caliper, with an adjustable stop that determines the radius on the tooth at which the measurement is made. In order to make a measurement, the instrument is placed over one of the gear teeth, as shown in Figure 8.2. The two vernier scales on the caliper are used to measure the distance between the caliper jaws, and the depth of the jaw tips below the stop.

Figure 8.1. A gear-tooth vernier caliper.

Measurement by Gear-Tooth Vernier Caliper 193

Gear-tooth caliper

-Tooth centerline

Figure 8.2. Measurement at the standard pitch circle.

On the gear, these lengths are known as the chordal tooth thickness and the chordal addendum.

The tooth thickness measurement should be made with the jaw tips of the caliper touching the tooth faces near the middle of the tooth profile. For gears with zero profile shift, this would mean that the contact is at the standard pitch circle. However, in a gear with profile shift e, the addendum as is extended by approximately e, so the middle of the profile lies at a radius approximately equal to (Rs+e). In practice, the measurement is normally made at the standard pitch circle for gears with small amounts of profile shift,

and it is only for gears with large values of e that the

alternative radius is used. The chordal tooth thickness and

the chordal addendum are represented by the symbols t hand sc a sch ' when the measurement is made at the standard pitch

circle, and we use t Rch and aRch when the measurement is made at radius R.

The gear-tooth caliper is shown in Figure 8.2, adjusted

so that the jaw tips touch the tooth faces at the standard pitch circle. The relations between the tooth thickness, the


chordal tooth thickness and the chordal addendum can be read from the diagram,

8s ts

(B.3) 2Rs

tsch 2Rs sin 8s (B.4)

asch RT - Rs cos 8s (B.5)

These values of tsch and a sch are generally included in the specification of a gear.

Figure B.3 shows the tooth thickness of a gear being measured, when the caliper jaws touch the tooth faces at radius R. The tooth thickness tR at this radius is related to ts by Equation (2.36), and the corresponding values of the chordal tooth thickness and the chordal addendum are again read from the diagram,

(B.6)

Gear-tooth caliper

R

-Tooth centerline

c Figure B.3. Measurement at radius R.

Measurement by Gear-Tooth Vernier Caliper 195

(8.7)

2R sin OR (8.8)

(8.9)

The expressions in Equations (8.4, 8.5, 8.8 and 8.9) for the chordal tooth thickness and the chordal addendum represent the correct values, when the tooth thickness has

the specified value ts' In other words, these values are sui table for checking whether the tooth thickness of a gear is equal to its required value. It may sometimes be necessary to measure the tooth thickness of a gear when it is not equal to its specified value, or when the specified value is unknown. In such cases, the gear-tooth caliper is adjusted so that ~he tips of the jaws meet the faces of the tooth near the middle

of each tooth profile. The values of t Rch and aRch are then read, and the tooth thickness is calculated from the following set of equations, which are simply a rearrangement

of Equations (8.6 - 8.9),

R2 1 2 + (R - 2 (8.10) <'2t Rch) T aRch)

tan OR t Rch

(8.11) 2 (RT - aRch)

tR 2ROR (8.12)

t ts R [-B + 2(inv fR - inv fs)] (8.13) s R

There are a number of sources of error when measurements are made with a gear-tooth caliper. The tip circle of the gear

is used as a datum, so any error in the value used for RT

causes an error in the calculated value of ts' The tips of the caliper jaws are subject to wear, which of course alters the

measured values. And lastly, for correct measurement, the

caliper must be held so that it is symmetrical with respect to

the tooth center-line, and the jaws touch each face of the

tooth at the same height. The other two methods described in

this chapter for measuring the tooth thickness of a gear are not subject to these objections.


Span Measurement

By measuring over several teeth, as shown in Figure 8.4, it is possible to make the measurement using the parallel faces of the caliper jaws, instead of the tips. This procedure is known as span measurement. An ordinary caliper can be used, without the adjustable stop of the gear-tooth caliper. We measure the length AA' over N' teeth, and from this measured

length S, it is possible to calculate the tooth thickness ts of each tooth.

Since line AA'is normal to the tooth profiles at A and at A', the line must touch the base circle at some point E. The involutes through A and A' meet the base circle at B and B', and we know that AE and A'E are equal to arc BE and arc B'E. Hence, the length S is equal to arc BB', which can be expressed as (N'-1) base pitches, together with the tooth thickness of one tooth at the base circle,

S AA' arc BB' (8.14)

In an ordinary tooth, the profile does not coincide with

c Figure 8.4. Span measurement over 3 teeth.


the involute all the way down to the base circle. Inside the fillet circle the tooth profile lies outside the involute, so that the tooth is strengthened near its root. However, the tooth thickness t b , which is used in Equation (S.14), is the tooth thickness that the tooth would have, if the profile at the base circle did coincide with the involute. Its value is therefore found from Equation (2.36), when we substitute the

value zero for the profile angle at the base circle,

(S.15)

We use Equations (2.20 and 2.30) to express the base circle radius in terms of the module,

and the tooth thickness tb is then given by the following expression,

(S.16)

The base pitch Pb is also expressed in terms of the module, by means of Equations (2.24 and 2.31),

(S.17)

We now combine Equations (S.14, S.16 and S.17), and we obtain an expression for the span measurement S,

S (S.lS)

In order to find the tooth thickness ts of a gear, we measure the length S, and ts is then found from Equation (S.lS),

- (N'-ll1rm - Nm inv ~ cos ~s s S (S. 19)

We have pointed already that, in the span measurement,

the contact between the gear teeth and the caliper jaws takes

place on the flat faces of the jaws. These faces are less subject to wear than the tips of the jaws, so the span


measurement is generally more accurate than the measurement made with a gear-tooth caliper. In addition, there is no need for exact symmetry in the span measurement. If the caliper in Figure 8.4 is held so that the lengths AE and A'E are not qui te equal, there is no change in the measured length S.

Number of Teeth in the Span

We have not yet considered the number N' of teeth over which the span measurement should be made. Ideally, the caliper jaws should touch the tooth faces near the middle of their profiles. This means, as we pointed out earlier, that the contact should take place at a radius of approximately (Rs+e). The value of N' must therefore be chosen with this consideration in mind.

The points of contact between the gear and the caliper are shown in Figure 8.5. To simplify the analysis, we will now assume that the caliper is positioned symmetrically, so that the length AE is equal to half the span measurement S. The

R

-Tooth centerline

c Figure 8.5. Radius of the span measurement contact points.

5pan Measurement 199

radius R of the circle through point A is approximately equal

to (Rs+e), and we therefore obtain the following equation

for 5,

(8.20)

We express Rb in terms of Rs' and expand the expression in the

square brackets as a power series in (e/Rs )' neglecting terms of second degree and higher. The equation for 5 then takes the

following form,

5 5!! (8.21)

We now have two expressions for 5, given by Equations (8.18

and 8.21). We equate these expressions, using Equation (6.1)

to relate the tooth thickness ts to the profile shift, and we solve the resulting equation for N' ,

N' (8.22)

The angle ~s in the second term of this expression has

been derived from the function inv ~s' so it must be expressed in radians. It is generally more convenient to express the

angle in degrees, and we then obtain the following expression

for N' ,

N' (8.23)

The purpose of Equation (8.23) is to provide a value for

N' such that, when the span measurement is made, the contact between the tooth faces and the caliper jaws takes place at a

radius close to (Rs+e). It is not always possible to achieve

this ideal, but in all cases the contact must occur between

the tip circle and the fillet circle, and preferably a certain

distance from both. In the derivation of Equation (8.23), a

number of approximations were used, and it is therefore

important to know whether the resulting value of N' is always

acceptab~e. The equation can be checked by the following

method. For any particular gear, the expression on the right-hand side of Equation (8.23) is evaluated, and N' is


chosen as the integer closest to this value. Equation (8.18)

is used to calculate the span measurement S, and the radius R at which contact takes place can then be read from Figure 8.5,

R (8.24)

If this procedure is carried out for a large number of

gears, we obtain the following results. The value of N' is always satisfactory when e is small. However, for gears with large values of e, particularly those with a small number of teeth, the value of N' given by Equation (8.23) is sometimes too large. I n these cases the contact would take place theoretically outside the tip circle of the gear. The error occurs because, when e is large and N small, the power series expansion in (e/Rs )' which was used to derive Equation (8.21), is not sufficiently accurate. To correct this problem, the last term in Equation (8.23) is multiplied by a factor [0.75-(2/N)]. This factor has the effect of reducing the value of N' for gears where e is large, and the reduction is greatest when N is small. It has been shown [10] that, for gears with profile shift values between -0.5m and 1.0m, the contact between the caliper and the tooth face will now take place in the required position, near the middle of the tooth profile. with this modification, the value of N' is given by the following expression,

N' 1 N~~ 2e[0.75-(2/N)] 2" + 180 + 7rm tan ~s (8.25)

The right-hand side of this equation is evaluated, and

N' is chosen as the integer closest to this value. The

corresponding span S is then given by Equation (8.18). When

the tooth thickness inspection of a gear is to be carried out

by means of a span measurement, the values of N' and S are

included in the specification of the gear.

Measurement Over Pins

A third method by which the tooth thickness can be measured is called the measurement over pins. Two cylindrical

Measurement Over Pins 201

112M

M

Figure 8.6. Measurement over pins.

pins of known diameter are inserted into opposite tooth spaces of the gear, as shown in Figure 8.6, and we measure the distance M between the outer points of the pins. We will now show how the measured value of M can be used to calculate the tooth thickness ts of the gear.

Figure 8.7 shows one tooth of a gear, with the pin positioned so that it is touching the tooth profile at A, and its center A' is lying on the center-line of the adjacent tooth space. The radius of the pin is r, and the distance between A' and the gear center C is shown as R'. We first need

to find a relation between R' and the tooth thickness ts' The tooth profile involute meets the base circle at B. We

construct a second involute, passing through point A', and

this involute meets the base circle at B'. The line AA', which

is normal to the tooth profile, touches the base circle at E.

Due to the fundamental property of the involutes that arcs BE and B'E are equal to AE and A'E, the arc BB' is equal to AA' , and the angle BCB' is therefore equal to (r/Rb ).


R' r

c

Figure 8.7. Radi us of the pin center.

The polar coordinate 9R at a typical point of the gear tooth involute was given by Equation (2.35),

ts 2R + inv tPs - inv tPR

s

The corresponding angle 9b of point B, the end point of the involute, is found by setting the profile angle tPR equal to zero,

(8.26)

If the polar angle of the line through B' is 9b, its

value can be found by adding the angle BCB' to 9b ,

9' b

t 2: + inv tPs + ..L

s Rb (8.27)

The angle between CB' and CA' is equal to (inv tPR')' and the angle between the center-lines of the tooth and the tooth

space is (~/N). These two angles are together equal to 9b, so

we obtain the following relation between the various angles,

. ~

lnv tPR' + N (8.28)

Measurement Over Pins 203

Figure 8.6 shows the measurement over pins for a gear

with an even number of teeth. The relation between the length

M and the radius R' can be read from the diagram,

M 2R' + 2r (8.29)

To find the tooth thickness of a gear, we measure the

length M, and R' is calculated from Equation (8.29). The

corresponding profile angle 9>R' is given by Equation (2.18),

cos 9>R' (8.30)

and Equation (8.28) is then used to calculate the tooth

thickness ts.

The order of the calculations is reversed when the tooth

thickness is specified, and we want to find the corresponding

value of M for inspection purposes. Equation (8.28) is used to

M

Figure 8.8. Measurement over pins, for a gear

wi th an odd number of teeth.


give the value of (inv ~R')' and the angle ~R' is found by means of Equations (2.16 and 2.17). The values of R' and Mare then found from Equations (8.30 and 8.29).

When a gear has an odd number of teeth, the pins are

placed in tooth spaces which are as closely as possible opposite to each other, in the manner shown in Figure 8.8. The radii through the pin centers no longer form a straight line, and the angle between them is equal to [180° - (180 0jN»). The relation between R' and M is then given by the following equation,

M 2R' cos (9~0) + 2r (8.31)

Apart from this change, the equations for finding the tooth thickness are exactly the same as those for a gear with an even number of teeth.

Examples 205

Numerical Examples

Example 8.1 The gear in Figure 8.2 has 36 teeth, a module of 8 mm,

and a pressure angle of 20°. Calculate the correct settings

for a gear-tooth caliper, if the tooth thickness is half the

circular pi tch, and the addendum is one module.

Example 8.2

RS = 144.000 mm

ts = 12.566 0.043633 radians = 2.500°

tsch = 12.562 RT = 152.000

a sch = 8.137 mm

(2.31)

(8.3) (8.4)

(2.40) (8.5)

The gear shown in Figure 8.3 has 15 teeth, D.P. 2, and

pressure angle 20°. It is cut from a blank with a diameter of

9.0 inches, and the specified profile shift is 0.25 inches. After a roughing cut, the tooth thickness is measured by means

of a gear-tooth caliper. with the chordal addendum set at 0.5 inches, the chordal tooth thickness is found to be 0.810 inches. Determine how much further the cutter should be fed in for the final cut, assuming the gear is cut by a hob.

N=15, Pd=2, ~s=20°, RT=4.5, e=0.25 aRch=0.5, t Rch=0.810

m = 0.5000 inches

Rs = 3.7500

Rb = 3.5238 R 4.0205

5.781° = 0.100906 radians

tR = 0.8114

~R = 28.779°

ts = 0.9974

Required value, ts = 0.9674

(8.10)

(8.11)

(8.12)

(2.18)

(8.13)

(6.1)


0.0301 toe - 0.0413 inches (8.2)

The cutter must be fed a distance 0.0413 inches towards the gear axis.

Example 8.3 For the gear specified in Example 8.2, the final tooth

thickness is to be checked by a span measurement. Calculate the number of teeth over which the span should be measured, the corresponding value of the span, and the radius at which the caliper jaws will touch the faces of the teeth.

N' = Integer closest to 2.7060 = 3 S 3.9662 inches R = 4.0435 inches

(8.25) (8.18 ) (8.24)

This radius is satisfactory, since the contact points are well below the tooth tips, and well above the fillet circle.

Example 8.4 Calculate the measurement over pins for the 11-tooth

gear shown in Figure 8.8, which has module 10 mm, pressure angle 20°, and profile shift 5.0 mm. The diameter of the measuring pins is 26 mm.

N=ll, m=10, ~s=200, e=5.0, r=13.0

RS = 55.000 mm

Rb 51. 683

ts = 19.348

inv~R' = 0.156726

~R' = 41.227° R' = 68.718

M = 162.038 mm

(8.28)

(2.16, 2.17) (8.30)

(8.31)

Chapter 9 Geometry of Non-Involute Gears

Introduction

In this chapter, we will first show how to calculate the tooth shape of a gear, when it is conjugate to a basic rack of arbitrary shape. This material is not required when the basic rack has straight-sided teeth, since we have already shown how to calculate the tooth profile shape of an involute gear. However, the method described will useful for finding the shapes of gear tooth fillets, which are conjugate to the

circular tips of the cutter teeth.

General Theory

We consider a basic rack with a curved tooth profile, as shown in Figure 9.1. The pitch is Pr' and the reference line is defined in the usual way, as the line along which the tooth thickness is equal to the space width. The rack coordinate

system (xr'Yr) is chosen so that the Yr axis lies along the reference line, and the xr axis coincides with a tooth center-line. A typical point Ar of the tooth profile has

coordinates (xr,y r ), and the profile angle at this point is ~Ar. Since the shape of the tooth profile is known, Y and A r

~ r are known functions of xr '

We now consider a gear meshed with the basic rack, as

shown in Figure 9.2. The pitch circle radius of a plnlon

meshed with a rack was given by Equation (1.15), and in

Equation (1.19) we showed that the circular pitch of the

pinion at its pitch circle is equal to the pitch of the rack. Since the gear in Figure 9.2 is meshed with its basic rack, we

208 Geometry of Non-Involute Gears

Yr Reference line-

I~ -r-.~----+

I~ Pr

Figure 9.1. A non-involute basic rack.

use the symbol Rsg for the radius of the pitch circle, and the two equations take the following form,

( 9. 1 )

(9.2)

The pitch point P is the point lying a distance Rsg from the center of the gear, on the line perpendicular to the reference line of the basic rack. We consider the basic rack when it is positioned with an offset e, so that its reference line lies a distance (Rsg+e) from the center of the gear. We will make use of the fixed (E,~) coordinate system, whose origin is at the pitch point, and the (x,y) coordinate system

in the gear, where the x axis coincides with a tooth center-line.

In order to find the shape of the tooth profile in the gear, we start from the Law of Gearing, which was proved in Chapter 1. This law states that the common normal at the contact point passes through the pitch point. We first

General Theory 209

I---Reference line of basic rack

Pitch circleof gear I-+-----''--Pitch line of basic rack

Figure 9.2. Meshing diagram of a gear and basic rack.

determine the position u r of the basic rack, when point Ar is

the contact point. Since the tangent at Ar makes an angle ~Ar with the ~ axis, the normal makes the same angle with the

1/ axis, and we can express the position of Ar in the following

way,

(9.3)

tan ~Ar (9.4)

The position ur of the basic rack, defined as the distance

between the e axis and the xr axis, is determined by the

position of point Ar'

(9.5)

For a gear being cut by a rack cutter, the relation

between the position of the rack cutter and the angular


position of the gear was given by Equation (5.28),

The same equation applies when we consider a gear meshed with

its basic rack, and we can therefore find the angular position ~g of the gear, corresponding to the position ur of the basic rack,

1 (u _ lp ) R r 2 s sg (9.6)

Since point Ar of the basic rack is the contact point, there is a point A of the gear which coincides with Ar , and its polar coordinates can be read from Figure 9.2,

R (9.7)

1/ arctan (~) - ~g sg

(9.8)

Finally, the tangent to the gear tooth profile at A coincides with the tangent to the basic rack tooth profile at Ar , and therefore makes an angle ,Ar with the ~ axis. The angle YR, which is defined as the angle between the gear tooth tangent at A and the tooth center-line, is then given by the following expression,

(9.9)

Equations (9.3-9.9) can be used to find the polar coordinates (R,8R) of a point A on the gear tooth profile, and the angle YR, corresponding to any specified point Ar on the basic rack tooth profile. By taking a number of points on the basic rack, we can construct the entire profile of the gear

tooth. The general theory just described is helpful for

clarifying some of the definitions used in gearing. If we use Equations (9.3 and 9.4) to calculate the position of the contact points, corresponding to a number of points on the basic rack tooth profile, we obtain a series of points on the path of contact. When the basic rack tooth profile is curved,

General Theory 211

Line of action

/,,4---Path of contact

Contact point

Figure 9.3. Path of contact, and line of action.

the path of contact is also curved, as we can see in Figure 9.3. The line of action, which is the line along which

the contact force acts in the absence of friction, coincides with the common normal at the contact point. The Law of Gearing states that this line passes through the pitch point, so the line of action is always the line from P to the contact point. The operating pressure angle ~ of the gear pair is the angle between the line of action and the tangent at P to the pitch circle. It is evident that, when the path of contact is not straight, the value of ~ varies as the contact point moves

along the path of contact.

Two basic racks that fit together exactly, in the manner

shown in Figure 9.4, are said to be complementary. If each of

these basic racks is used to define the tooth profile of a

gear, then these two gears can mesh together correctly, and

the Law of Gearing will be satisfied. However, the gears can

only operate at the standard center distance Cs ' equal to the sum of the standard pitch circle radii. If the center distance is altered, the common normal at the contact point will no


Figure 9.4. Complementary basic racks.

longer intersect the line of centers at a fixed point, and the Law of Gearing is not satisfied. Involute gears are the only type which will continue to mesh correctly, when the center

distance is changed. Hence, in all other types of gear, there

is no distinction btween the pitch circle radius Rp and the

standard pitch circle radius Rs' since they must always have the same value.

Fillet Shape Cut by a Rack Cutter

The general theory just described, which gives the shape

of tooth profiles conjugate to an arbitrary basic rack, can

also be used to find the shape of gear tooth profiles, when

they are cut by a rack cutter with curved teeth. In

particular, when we consider involute gears, we will use the

Fillet Shape Cut by a Rac k Cutter 213

c

x

Figure 9.5. Gear tooth fillet and rack cutter.

method to find the shape of the tooth fillet, which is cut by

the circular section at the tooth tip of the rack cutter. The tooth profile of the rack cutter is shown in

Figure 9.5. The circular section of the profile extends from

point Ahr to point ATr • The radius of this section is rrT' and its center is at point A~, whose coordinates (x~,y~) are given by Equations (5.43 and 5.44). In the general theory described in the previous section, we chose a point Ar of the cutter tooth profile, and determined the position ur of the cutter,

at which Ar would be the cutting point. For the special case when the cut ter tooth profi Ie is circular, it is sl ightly more

convenient to reverse this order. We choose the position of

the cutter, and we then determine which point Ar of the

circular profile is the cutting point.

We therefore consider the cutter in the position shown

in Figure 9.5, with an offset e and with the x axis lying a r distance (-ur ) below line CPo In this position of the cutter,

the center A~ of the tooth profile circular section has

coordinates (~' ,~'), which are given by the following

equations,

~ I e + x' r (9.10)


1/' (9.11)

If the line from the pitch point to A~ intersects the

tooth profile at Ar' and Ar is a point on the circular part of the profile, then the normal to the profile at Ar passes through P, and Ar must be the cutting point. The line ArP is the line of action, when the cutter is in the position shown in Figure 9.5. We now reintroduce the coordinate s, measured from the pitch point along the line of action, with points above P being positive. The positions s' and s of points A' r and Ar on the line of action can be expressed as follows,

s'

s s' - r rT

(9.12)

(9.13)

and we then write down the position of the cutting point Ar'

(9.14)

( ...§...) ., • s' 'f

(9.15)

It is evident that the negative signs of s· and s cancel out, and it would have been simpler to have used positive values. However, we will make use of these equations again in Chapter 10, when we discuss the fillet curvature, and for that purpose it is important to retain the correct sign fors.

To find the position of point A on the gear tooth fillet, corresponding to point Ar on the cutter, we now follow exactly the remaining steps of the general method. The angular

position fJg of the gear is given by Equation (9.6),

fJg _l_(u _ lp ) Rsg r 2 s

(9.16)

and the polar coordinates of the point on the gear tooth fillet are found from Equations (9.7 and 9.8),

R (9.17)

(9.18)

Fillet Shape Cut by a Rack Cutter 215

To complete the analysis of the gear tooth fillet shape, it is necessary to determine the initial and the final

positions of the cutter, at which points on the fillet are being cut. The highest point on the fillet is cut by point Ahr on the cutter, the point where the circular section of the tooth profile begins. When Ahr is the cutting point, the line

of action must make an angle tP r with the 1/ axis, since this is the direction of the normal to the cutter tooth profile

at Ahr • The coordinates (~' ,1/') of point A~ must then satisfy the following equation,

~ , 1/' tan tP r (9.19)

We substitute this expression in Equations (9.10 and 9.11) for ~' and 1/', and we obtain the value of ur at which the highest point of the gear tooth fillet is cut,

Yr

Reference line

c

Path followed by point Ahr-I

I Path followed by point AEr

Figure 9.6. Meshing diagram of an undercut gear and a rack cutter.


1 (e+x ') _ y I

tan ~r r r (9.20)

The other end of the gear tooth fillet, the point where

it meets the root circle, is cut by point ATr on the cutter.

The normal to the cutter tooth profile at ATr is parallel to the ~ axis, and ATr becomes the cutting point when the line of action is in the same direction. This will occur when point A'

r lies on the ~ axis, and its coordinate ~' is equal to zero. The corresponding position of the cutter can then be found from Equation (9.11),

- Y~ (9.21)

To construct the entire fillet of the gear tooth, we consider a number of positions of the cutter, starting with the ur value given by Equation (9.20) and ending with the value given by Equation (9.21). We then use Equations (9.10 - 9.18) to calculate the positions of the corresponding points on the fi llet.

Fillet Shape of an Undercut Gear

The method described in the previous section for finding the shape of a gear tooth fillet can still be used when the gear is undercut. The meshing diagram for the gear and the cutter is shown in Figure 9.6. Since the gear is undercut, we

are considering a situation in which the end point Hr of the path of contact lies below the interference point Eg • The

point of the cutter tooth profile which passes through Eg is

labelled AEr • The tooth profile of the undercut gear is shown in

Figure 9.7. Although the shape of the involute part of the

profile is already known, and it is therefore unnecessary to

use the general theory described at the beginning of this chapter, it is nevertheless helpful to consider the results

we would obtain if we were to do so. A typical point of the

cutter tooth profile is shown as Ar in Figure 9.6. In order to

use the general method, we need a relation between the

coordinates xr and Yr of point Ar' and this relation can be

Fillet Shape of an Undercut Gear 217

Base circle

Figure 9.7. Tooth profile of an undercut gear.

read from the diagram,

- 11Tm - x t .. 4 r an "'r (9.22)

Equations (9.3-9.8) can then be used to find the position of the corresponding point A on the gear tooth profile, which is

shown in Figure 9.7. We consider a sequence of points on the rack cutter

tooth, starting near the root and moving towards the tip. As we move along the profile to AEr , the cutting point moves down the path of contact to Eg , and on the gear tooth the corresponding points lie on the involute, down to point B where the involute meets the base circle.

We next consider a number of points on the cutter tooth

between AEr and Ahr • The positions of the cutting point, given

by E and 77 in Equations (9.3 and 9.4), lie between Eg and Hr on the path of contact. The corresponding points on the gear,

found by means of Equations (9.5 - 9.8), lie on the curve shown

in Figure 9.7, joining point B to point Af • It can be seen

that the straight section of the cutter tooth generates a

curve on the gear which has a cusp at the base circle. This is

a characteristic sign of a gear which is undercut. Finally, we consider points on the curved section of the


cutter tooth, lying between Ahr and ATr • We use Equations (9.10 - 9.18) to find the corresponding points on the gear,

and we obtain the curve in' Figure 9.7 which starts at point Af , and ends at the root circle.

From a physical point of view, it is clear that the final shape of the gear tooth is given by the shaded profile in

Figure 9~ 7, and that parts of the curve followed by the

cutting point have no effect on the final shape of the tooth. Since we do not need to know the path of the cutting point between Band Af , it is not necessary to use Equations (9.3 - 9.8) to find this path. We know the shape of the

involute part of the tooth prof ile, and the shape of the fillet curve is given by Equations (9.10 - 9.18). To complete the description of the tooth shape, it is only necessary to

find the position of point Au' where the fillet curve intersects the involute.

We have, in fact, already described a method for finding the approximate position of Au' when we discussed the undercut circle in Chapter 7. The method was based on the fact that the path followed by point Ahr , relative to the gear blank, is almost identical to the fillet shape near the top of the fillet. We are now in a position to verify that statement, since we can calculate the shape of the fillet, and compare it with the path of Ahr • We will find that the path of Ahr is tangent to the fillet curve at point Af , and is extremely close to the fillet curve at the point where it intersects the involute. Hence, the method described in Chapter 7 can be used

to find the position of Au' with sufficient accuracy for all practical purposes.

Profile Modification

The meshing cycle of a tooth pair has already been

described in the earlier chapters of this book, and it can be summarized in the following manner. We consider one

particular tooth pair, as it passes through the meshing

cycle. The tooth pairs that make contact immediately before

and immediately after will be referred to as the leading and

the trailing tooth pairs.

Profile Modification 219

The initial contact takes place at the limit circle of the driving gear, and at the tooth tip of the driven gear. The contact point then moves up the tooth face of the driving gear, and during the first phase of the meshing cycle there is another pair of teeth already in contact, so that the total

tooth force is shared between the two contacting tooth pairs. The first phase ends wh.en the contact point reaches the

lowest point of single-tooth contact on the driving gear. At this instant, in the leading tooth pair, the contact point has reached the tooth tip of the driving gear, and contact in this

tooth pair ceases. The second phase of the meshing cycle is the period of single-tooth contact, which lasts until the

contact point on the driving gear reaches the highest point of single-tooth contact. This is the moment when contact begins in the trailing tooth pair, so that during the third phase of the mashing cycle, there are again two pairs of teeth in

contact. The third phase continues while the contact point on the driving gear moves up the remaining part of the tooth face, until it reaches the tooth tip, and the contact ends.

In ideal conditions, there is a smooth transition from the second phase to the third, at the instant when the trailing tooth pair makes initial contact. In other words, the relative angular posi tions of the two gears are such, that they allow the tooth pair to come smoothly into contact. However, if the tooth spacing on each gear is not exactly constant, there may be an impact at the initial contact of some tooth pairs. Also, in cases where the gear pair is heavily loaded, the tooth flexibility allows the driven gear to lag slightly behind its correct angular position, particularly during the period of single-tooth contact. As a

result of this small error in the relative angular positions of the gears, there is an impact at the initial contact of

each tooth pair. In addition, with heavily loaded gears,

there may be damage to the tooth faces at the end of the

meshing cycle, due to the very high contact stress when the

contact point reaches the tooth tip of the driving gear.

In order to avoid these problems, it is possible to

remove some material from the tooth face of each gear, either

near the tip, or near the tip and near the limit circle. The

profile is not altered between the highest and the


lowest points of single-tooth contact. This procedure is

called tip and root relief, or profile modification. The extra material is removed from the tooth face of each

gear by means of a rack cutter or a hob whose teeth are not straight-sided along their entire profile. The purpose of

this section of the chapter is to show how the tooth profile of the gear is calculated, when the cutter tooth profile is

modified in this way. The tooth profile of the rack cutter is shown in

Figure 9.B. Near the root of the tooth, the tooth thickness is increased, compared to that of a conventional cutter, so that the gear will be cut with tip relief. The straight section of

the tooth profile ends at point Aqr' a distance q from the Yr axis, and beyond this point the profile is formed by a circular arc of radius rq • The values of q and rq are chosen by the tool designer, but there are some recommended limits on the values that should be used. For example, in the basic rack recommended by the ISO for general and heavy engineering, the minimum value of q is 0.4m, and the value of rq should be such that, at xr =1.0m, the profile does not deviate from the

straight line by more than 0.02m.

Yr

.A:~

Figure 9.B. A rack cutter with profile modification.

Fillet Shape Cut by a Pinion Cutter 221

A typical point A on the circular part of the profile

has coordinates (xr'Yr)' and profile angle ~Ar In order to calculate the shape of the gear tooth profile cut by this section of the cutter, we must express ~Ar and Yr in terms of x • The circular arc blends smoothly into the straight r section of the profile, and the center A; of the circular arc lies a distance (q - r sin ~ ) from the y axis. Hence, the

A q r r profile angle ~ r and the coordinate Yr can be found from the

following two equations,

r sin ",Ar q 'I' (9.23)

1 A - "41rm - q tan ~r - rq cos ~r + rq cos ~ r (9.24)

These equations are only valid for values of xr greater

than q. When xr is less than q, we are considering a point on the straight section of the profile, and the value of Yr is given by Equation (9.22).

Since we have now expressed Yr as a function of xr for all points on the cutter tooth profile below Ahr , we can use Equations (9.3 - 9.8) to calculate the corresponding section of the gear tooth profile. We could, of course, also express

Yr as a function of xr for the tip section of the cutter tooth, but the method described earlier is generally more

convenient.

Fillet Shape Cut by a Pinion Cutter

To find the fillet shape of a gear cut by a pinion cutter, we use a procedure which is essentially the same as

the method described earlier, for the case when the gear was

cut by a rack cutter. Figure 9.9 shows the pinion cutter, in

position to cut a point on the tooth fillet of the gear. The

circular section of the cutter tooth profile starts at point

Ahc ' and ends at ATc • The radius of the section is r cT ' and

its center A~ has polar coordinates (R~,8~) given by Equations (5.32 and 5.39).

We consider the posi tion of the cutter when the line C A' c c makes an angle a with the line of centers. The coordinates


Figure 9.9. Gear tooth fillet and pinion cutter.

(E' ,1/') of point A~ are then given by the following equations,

~ , RC - R' cos a pc C (9.25)

- R~ sin a (9.26)

The cutting point Ac is found by extending line PA~ to the point where it meets the cutter tooth profile. The positions 5' and 5 of points A~ and Ac on the line of action can be expressed as follows,

(9.27)

5 S' - r cT (9.28)

The coordinates (~,1/) of the cutting point Ac are then given

in terms of ~' and 1/',

1/

(..E....) t , S' ~ (9.29)

(9.30)

Fillet Shape Cut by a Pinion Cutter 223

The angle a between the line of centers and line CcA~ is

made up of the polar coordinate e~, together with the angle ~c through which the tooth center-line has turned. The angular

position ~c can therefore be expressed in terms of a, and the corresponding angular position of the gear can then be found

from Equation (5.21),

a - e' c

1 1 - ~(R ~ + -2Ps) sg sc c

( 9.31)

(9.32)

Since Ac is the cutting point, it coincides with a point

A on the gear tooth fillet. In order to determine the position

of A on the gear, we need to find its polar coordinates (R,eR). An expression for R can be written down immediately,

since we know the position of Ac' and therefore that of A. We

can also find eR, since the angle between line CgA and the line of centers is made up of the coordinate eR, together with

the angle ~g through which the tooth center-line has rotated. The coordinates of A are therefore given by the following

expressions,

R (9.33)

arctan (~) - fJ Rpg+~ g

(9.34)

The last step required before we can calculate the shape of the gear tooth fillet is to determine the initial and final values of a, corresponding to the upper and lower ends of the fillet. The upper end is cut by point Ahc ' the end of the involute section of the cutter tooth profile. When Ahc is the

cutting point, the cutting pressure angle is equal to ~c, the

same value as the cutting pressure angle when the involute

part of the gear tooth profile is being cut. The corresponding

position of A' is shown in Figure 9.10, and the angle E C A' c c c c c

is then equal to (~ +a). We can therefore express a in terms

of R~ and the base circle radius Rbc '

a R

arccos (R~c) _ ~c c

(9.35)


Figure 9.10. Cutting the end point of the fillet.

The other end of the fillet, where it meets the root circle, is cut by point ATc on the cutter. ATc will be the cutting point when the line from P to A~ passes through ATc • Since ATc ' A~ and Cc are collinear, ATc will be the cutting point when the line through the three points coincides with the line of centers. The value of a corresponding to the lowest point of the fillet is therefore equal to zero,

a o (9.36)

To construct the entire fillet, we consider a number of

values of a between the values given in Equations (9.35

and 9.36), and we use Equations (9.25-9.34) to find the coordinates of the corresponding points on the fillet.

Examples 225

Numerical Examples

Example 9.1 A hob has a module of 6 mm and a pressure angle of 20°.

Its addendum is 7.5 mm, and the tooth tip radius is 1.8 mm.

The hob is used to cut a 24-tooth gear, with a profile shift

of 1.5 mm. Calculate the polar coordinates in the gear of the

point where the fillet meets the involute, and of the point

where the fillet meets the root circle.

Rsg = 72.000 mm

Ps = 18.850

We start by finding the coordinates (x~,y~) of the center of

the circular section at the tip of the hob tooth.

h = 6.316 (5.40)

x~ - 5.700 (5.43)

y~ = - 0.722 (5.44)

Note that y~ is negative, which is essential in a correctly

des i gned hob.

A t the top of the fi llet , ur - 10.817 ~'= -4.200

1'/' - 11.539 s' = - 12.280 s = - 14.080 ~=-4.816

7j = - 13.231

13g - 0.281138 radians = - 16.108°

R = 68.475 mm, 8R = 4.967°

At the root circle,

Ur = 0.722

~' = - 4.200

7j' = 0 s' = - 4.200

(9.20) (9.10) (9.11)

(9.12) (9.13) (9.14)

(9.15)

(9.16)

(9.17,9.18)

(9.21)

(9.10)

(9.11)

(9.12)


s = -6.000 - 6.000

11 = 0

- 0.120869 radians = - 6.925°

R = 66.000, 9R = 6.925°

(9.13)

(9.14) (9.15)

(9.16) (9.17,9.18)

It should be noted that this point is not on the center-line of the tooth space, where the value of 9R is 7.5°. The tooth profile contains a very small circular section, coinciding with the root circle, which is generated by the flat section

at the tip of the hob tooth.

Example 9.2

A hob with D.P. 2.5 and pressure angle 20° is designed to cut gears with tip relief. The tooth profile of the hob is shaped like Figure 9.8, with the values of q and r being 0.16 . q and 4.8 inches. Determine the reduction in the tip tooth thickness of a 40-tooth gear, compared with a gear cut by a conventional hob, if the gear has zero profile shift and an

addendum of 0.4 inches.

m = 0.4000 inches

Rsg 8.0000 RTg = 8.4000

The value of xr on the hob which generates the tip of the tooth on the gear must be found by trial and error. We will

use the correct value immediately, and proceed with the

remaining calculations.

xr 0.3560

tJ>A r = 22.5110 (9.23)

Yr = - 0.4486 (9.24)

~ 0.3560 (9.3)

11 = 0.8590 (9.4)

u = r 1.3076 (9.5)

f3g 0.084916 radians 4.865° (9.6)

R = 8.4000 (9.7)

Examples

8R = 1.004° = 0.017526 radians tR = 2R8R = 0.2944 inches

For a gear with no tip relief,

Rb = 7.5175 41T = 26.499°

tT = 0.3043 inches Reduction in tooth thickness = 0.0098 inches

Example 9.3

227

(9.8)

(2.36)

Repeat the calculations of Example 9.1, assuming that the gear is cut by a 16-tooth pinion cutter with a tip circle diameter of 113.4 mm, a profile shift of 1.8 mm, and rounding at the tooth tips with a radius of 1.5 mm. This is the cutter shown in Figure 5.12.

m=6, 41 s =200, Nc =16, RTc =56.7, ec =1.8, r CT=1.5 Ng=24, eg=1.5

We must first find the polar coordinates (R~,8~) of the center of the circular section at the tip of the cutter tooth.

Rsc = 48.000 mm Rbc = 45.105 R~ 55.200

41hc = 36.454° Rhc = 56.078

tsc = 10.735 0.024246 radians

35.065° x' c 55.200

y~ = 0.132 8' = 0 137° c •

(5.32) (5.33) (5.34) (6.1)

(5.35) (5.36) (5.37) (5.38) (5.39)

Next, we determine the center distance at which the cutter wi 11 cut the requi red tooth thickness in the gear.

Rsg = 72.000 Rbg = 67.658

Ps = 18.850 t sg = 1 0 • 5 1 7 ( 6. 1 )


CC = s 120.000

inv ~~ = 0.024914 (5.18)

~c p 23.577° (2.16,2.17)

~c 23.577° (5.19)

CC 123.033 (5.20)

RC pg 73.820 (5.8)

RC pc = 49.213 (5.9)

Lastly, we find the coordinates in the gear of the end points

of the fillet.

At the top of the fillet,

a = 11.626°

~'=-4.854

71' - 11 • 124

s' - 12.137 s = -13.637

e - 5.454 71 = - 12.498

~c = 11.489° = 0.200516 radians ~g = - 0.264577 radians = - 15.159°

R = 69.499 mm, 9R = 4.799°

At the root circle,

a = 0

~' = - 5.987

71' = 0

s' = - 5.987

s = - 7.487

~ = - 7.487

71 = 0

~c - 0.137° = - 0.002388 radians

~g - 0.129308 radians = - 7.409°

R = 66.333 mm, 9R = 7.409°

(9.35)

(9.25) (9.26)

(9.27) (9.28) (9.29) (9.30) (9.31)

(9.32) (9.33, 9.34)

(9.36)

(9.25)

(9.26)

(9.27)

(9.28)

(9.29 )

(9.30)

(9.31)

(9.32)

(9.33, 9.34)

Chapter 10 Curvature of Tooth Profiles

Involute Radius of Curvature

The radius of curvature at any point of an involute is found most easily, by making use of one of the special propert ies of the curve.

We pointed out in Chapter 2 that the involute can be represented as the path followed by point A of a rigid bar, while the bar rolls without slipping on a circle of radius Rb • The bar and the circle are shown in Figure 10.1, and the

contact point is labelled E. Since E is also the instantaneous center of the bar, the path of A coincides momentarily with the circle whose center is E, and the radius of curvature p of the involute at point A is therefore equal to the length EA,

p EA ( 10. 1)

If A is the point of the involute at radius R, as shown in Figure 10.2, the angle ECA is equal to the profile angle ~R' as we proved in Equation (2.9), and the radius of curvature can therefore be expressed as follows,

p (10.2)

Euler-Savary Equation

The equation just derived gives a very simple method for

calculating the radius of curvature, at any point on the

involute section of a gear tooth profile. However, the equation is of course not suitable for finding the radius of

230 Curvature of Tooth Profiles

Involute path followed by point A of the bar

c

D

Figure 10.1. A rigid bar rolling on the fixed base circle.

curvature in the fillet. For this purpose, we make use of the Euler-Savary equation, which gives a relation between the radii of curvature of two conjugate profiles. The fillet of a gear tooth is conjugate to the circular section at the tip of the cutter tooth, whose radius is known. We can therefore use the Euler-Savary equation to find the radius of curvature at any point on the fillet.

Before we make use of the Euler-Savary equation, we will show how the equation is derived. We consider a pair of meshing gears, as shown in Figure 10.3, and we first discuss the motion of gear 2 relative to gear 1. We regard gear 1 as fixed, so that the motion of gear 2 can be represented by the motion of its pitch circle, rolling without slipping on the pitch circle of gear 1. If the angular velocity of gear 2

relative to gear 1 is w radians/second counter-clockwise, then the velocity of the gear center C2 is (Rp2w), and the angular velocity of the line of centers is equal to the velocity of C2, divided by the center distance. The velocity of P, the point at which the pitch circles touch each other, is then given by the following expression,

Euler-Savary Equation 23'

R

c

Figure '0.2. Radius of curvature of the involute.

('0.3)

We now consider the motion of a 'particular curve in gear 2, which we will call curve 02. As gear 2 rolls on the

pitch circle of gear " the various positions occupied by curve 02 form an envelope, and this envelope is called

curve 0,. Figure '0.4 shows the positions of gear 2 at two distinct times l' and 1" • At time 1', the center of gear 2 is C2 ' and the point where the pitch circles touch is P. The corresponding points at time 1" are shown as C2 and P'. The

two positions of the curve in gear 2 are shown as 02 and 02' and these curves touch the envelope 0, at points A and A' •

The instantaneous center of gear 2 at time l' is P. Hence, the point of gear 2 which coincides with A has a velocity that

is perpendicular to PA. The tangent to the envelope 0, must

lie in the same direction, and the normal therefore coincides

with PA. The two lines PA and P'A' are each normal to curve 0,

at adjacent points, so the point 0, where these lines meet is

the center of curvature of curve 0, at A, and the length O,A

is the radius of curvature Pl.

The velocity of A can be related to the velocity of P. If

PA makes an angle ~ with the tangent to the pitch circles at P, the velocity of P has a component perpendicular to PA of

232

Pitch circle of gear 1, fixed

Curvature of Tooth Profiles

Pitch circle of gear 2, rolling on that of gear 1

Figure 10.3. Gear 2 rolling on the pitch circle of gear 1.

(vp sin 1/1), and the velocity vAil can be expressed as follows,

(10.4)

In this expression, s is the distance from P to A. The symbol vAil is used to represent the velocity of A, to indicate that

the velocity is measured relative to gear 1.

We next determine the velocity of A, measured relative

to gear 2. Figure 10.5 shows the system with gear 2 at rest, and gear 1 rolling on the pitch circle of gear 2 with a

clockwise angular velocity w. As before, the positions of the

moving gear at the two times T and T' are indicated by the

unprimed and the primed symbols. Curve 02 is now the envelope

of curve 01' and the two positions shown of curve 01 touch curve 02 at A and A'. The lines PA and P'A' are normal to

curve 02' so the point 02 where they meet is the center of curvature, and the length 02A is the radius of curvature P2.

If we calculate the velocity of P, we will find that it

Euler-Savary Equation 233

Figure 10.4. Envelope 01 formed by a curve 02 in gear 2.

is the same as before. In other words, it is still given by

the expression in Equation (10.3). Its component

perpendicular to PA is also unchanged, and we can now write

down the veloc i ty of A,

(10.5)

The velocity of any point A, measured relative to gear 1,

is equal to the velocity of A measured relative to gear 2,

plus the velocity relative to gear 1 of the point in gear 2

which coincides with A. This is a standard theorem of

dynamics, and for the case we are considering, it can be

written as follows,

vA/ 2 + ws (10.6)

We now combine Equations (10.3 - 10.6), and we obtain a

relation between the radii of curvature Pl and P2 of curves 01

and °2 ,

Pl P2 (Pl- s ) - (P2+ s )

_, s_ (_1_ + _1_) Sln ~ Rpl RP2

(10.7)


Figure 10.5. Envelope formed by curve 01.

Finally, the left-hand side of this equation is simplified, and the relation takes the following form, which is known as the Euler-Savary equation for envelopes,

_1_ (_1_ + _1_) sin q, Rpl Rp2

(10.8)

In the derivation of this equation, we have used exactly

the same notation used throughout the rest of this book. The

two curves 01 and 02 represent the tooth profiles, A is the

contact point between the teeth, and P is the pitch point. The

angle q, is the operating pressure angle of the gear pair, which, as we have shown, is not constant when we are dealing

with non-involute profiles. Finally, s is the coordinate

giving the position of A on the line of action, and it is

positive when A lies above P. In Figures 10.4 and 10.5, the

two curves 01 and 02 are shown as convex, and we assumed in

the derivation that Pl and P2 are positive. The equation remains valid if either curve is concave, but the

corresponding radius of curvature is then negative.

Curvature of Tooth Profiles 235

Gear Tooth Fillet Radius of Curvature

We can use Equation (10.8) to obtain the radius of curvature at any point on a gear tooth profile, when the gear is cut by a non-involute cutter. If gear 1 represents the gear

being cut, and gear 2 the cutter, it is convenient to replace

the subscripts 1 and 2 by g and c. In Equation (10.8), the

radii RP1 and Rp2 of the pitch circles are then replaced by the cutting pitch circle radii R~g and R~c. We simplify the equation by introducing a length RO' defined as follows,

_1_ + 1 RC RC

pg pc (10.9)

so that the quantity (1/RO) represents the relative curvature of the two cutting pitch circles. We then solve

Equation (10.8) for Pg ' obtaining the following expression,

(p +s)2 c (10.10)

In order to calculate the radius of curvature Pg at a point on the gear tooth profile, we use the methods described in Chapter 9 to find the coordinates (~,~) of the cutting point, corresponding to any specified point on the cutter

tooth. The radius of curvature of the cutter tooth profile at

this point gives the value of Pc to be used in Equation (10.10), and expressions for If> and s can be read from Figure 9.9, which shows the meshing diagram during the cutting process,

tan If>

s

£. ~

-~sin If>

(10.11)

(10.12)

These values for P, If> and s are substituted into c Equation (10.10), and we obtain the radius of curvature Pg at

the corresponding point on the gear tooth profile.

We first use this method to find the radius of curvature

at points on a gear tooth fillet, for the case when the gear

is cut by a rack cutter. The cutting pitch circle radius of the rack cutter is infinite, and the cutting pitch circle of


the gear coincides with its standard pitch circle, so the

length RO defined by Equation (10.9) is equal to the standard

pitch circle radius of the gear,

( 10.13)

The radius of curvature of the circular tip of the rack cutter

tooth is rrT' and the values of ~ and ~ are given by Equations

(9.14 and 9.15). The profile radius of curvature at a point on

the gear tooth fillet can then be found from Equation (10.10).

However, since the fillet is concave, the radius of curvature

given by this equation would always be negative. It is rather

more convenient to change the sign in the equation, so that we

obtain an expression for Pf' the magnitude of the fillet

radius of curvature,

(10.14)

A point of particular interest is the point where the

fillet meets the root circle, since the radius of curvature at

this point is required for the calculation of the maximum

fillet stress in the tooth. The position of the cutting point

is given by Equations (9.10 - 9.15). To find the position

corresponding to the point where the fillet meets the root

circle, we choose the cutter position ur equal to (-y~), as we

proved in Equation (9.21). We then obtain the following

values,

'11 0

These values are substituted into Equation (10.14), glvlng an

expression for r f , the magnitude of the fillet radius of

curvature, at the point where the fillet meets the root

circle,

Gear Tooth Fillet Radius of Curvature 237

(a r -e-r rT }2 ( 10.15)

The fillet radius of curvature is found by the same

procedure, for a gear which is cut by a pinion cutter. The radii of the cutting pitch circles were given by Equations

(5.8 and 5.9),

When these values are substituted into Equation (10.9), we

obtain the following expression for the length RO'

NgNCCC

(N +N }2 g c (10.16)

The gear tooth fillet is cut by the circular section at

the tip of the cutter tooth, whose radius is r cT ' If there is no rounding at the cutter tooth tip, so that there is a sharp corner where the tooth profile meets the tip circle, then the value of r cT is zero. We use Equations (9.25 - 9.30) to find

the coordinates (~,~) of the cutting point, corresponding to any specified point on the cutter tooth profile. The values of ~ and s are found, as before, from Equations (10.11

and 10.12). The radius of curvature at the corresponding point of the gear tooth fillet is given by Equation (10.10),

and again we change the sign so that we obtain the magnitude P f of the radius of curvature,

(rCT+s)2 P r + f cT RO sin ~ - (rCT+s) (10.17)

To find the magnitude r f of the fillet radius of

curvature, at the point where the fillet meets the root

circle, we must determine the corresponding values of ~ and s.

When we described in Chapter 9 how the shape of the gear tooth

fillet is calculated, we specified the position of the cutter

by the angle a between the line of centers, and the line C A' c c in the cutter. We showed in Equation (9.36) that, when a is

zero, the cutter is in the correct position to cut the point

on the gear tooth where the fillet meets the root circle. With


this value of a, the coordinates (~,~) of the cutting point

are found from Equations (9.25 - 9.30), and the values of , and

s are then given by Equations (10.11 and 10.12),

t RC pc - R Tc

~ 0

, 90 0

s RC pc - R Tc

We substitute these values into Equation (10.17), and we

obtain an expression for r f ,

(RTC-Rpc-rCT)2 r f r cT + c ( 10. 18)

RO + (RTC-Rpc-rCT)

Examples 239

Numerical Examples

Example 10. 1

For each of the fillet points whose positions were found

in Example 9.1, calculate the fillet radius of curvature.


At the root circle,

s = - 14.080 mm

£=-4.816

17 = - 13.231

~ 20.000°

Pf = 5.886 mm

s=-6.000

- 6.000

17 = 0

~ = 90°

2.031 mm

(Example 9.1)

(Example 9. 1 )

(Example 9.1)

(10.11)

(10.14)

(Example 9.1)

(Example 9.1)

(Example 9.1)

(10.11)

(10.14)

The second of these results can be confirmed by

Equation (10.15), which gives directly the fillet radius of

curvature at the point where the fillet meets the root circle.

It might be expected that the first result could be confirmed

by Equation (10.2), which gives the radius of curvature at any

point on the involute. However, no such confirmation is

possible, because the tooth profile radius of curvature is

discontinuous at the point where the fillet meets the

involute. It should also be remembered that the fillet radius

of curvature given by Equation (10.14) is concave, while the

involute radius of curvature given by Equation (10.2) is

convex.

Example 10.2

Calculate the fillet radius of curvature at each end of

the tooth fillet, for the gear described in Example 9.3.


CC = 123.033 mm

RO = 29.528


At the root circle,

s = - 13.637

E = - 5.454

7'/ = - 12.498

~ 23.577 0

Pf = 7.651 mm

s=-7.487

E - 7.487

7'/ = 0

t/J = 90 0

Pf = 2.509 mm

(Example 9.3)

(10.16)

(Example 9.3)

(Example 9.3)

(Example 9.3)

(10.11)

(10.17)

(Example 9.3)

(Example 9.3)

(Example 9.3)

(10.11)

(10.17)

It is worth drawing attention to the results of Examples

10.1 and 10.2. The minimum radius of curvature in the gear tooth fillet, when the gear is cut by the pinion cutter, is

larger than when the gear is cut by the hob. This is true, even though the tooth tip radius r cT in the pinion is smaller than the corresponding radius rrT in the hob. It is therefore advisable, when a hob is designed, to choose the largest possible value for rrT' which is given by Equation (5.45), in order to keep the minimum radius of curvature cut in the gear

tooth fillet as large as possible.

Chapter 11 Tooth Stresses in Spur Gears

Introduction

When a gear pair is designed, it is important to calculate the maximum tooth stresses, to ensure that the teeth will not be damaged during the operation of the gear pair. However, the shape of a gear tooth makes it impossible to calculate the stresses exactly, using the theory of elasticity. There are a number of numerical methods, such as the finite element method, by which the stresses can be found, but these methods require large computers and take considerable computing time, so they are not suitable for general design purposes. There is a need for a simple theory, even if the results are only approximate, and it is essential that the calculations can be carried out on a programmable calculator or a micro-computer.

Approximate methods for calculating tooth stresses have, of course, existed for a long time, and descriptions of some of these methods have been made available in a number of the AGMA reports. Over the years, the methods have been refined, and the most recent AGMA report on this subject, published in 1982, is entitled "AGMA Standard for Rating the

Pitting Resistance and Bending Strength of Spur and Helical

Involute Gear Teeth" [6]. All statements in this chapter and in Chapter 17 about recommendations made by the AGMA refer to

this report. The material in these two chapters differs in

certain respects from that in the AGMA report, so when a gear

pair is to be rated according to the AGMA Standard, the report

should be followed exactly.

In this book, the present chapter contains a description

of the theory by which the tooth stresses are calculated, for

242 Tooth Stresses in Spur Gears

a pair of spur gears. The AGMA does not provide the background theory, but the formulae presented in the AGMA Standard, when applied to spur gears, are based on the same theory that is described in this chapter.

The stresses in a gear tooth depend primarily on the load and the tooth shape, but there are several other phenomena which must be taken into account. The stresses are first calculated, considering only the load and the tooth shape, and these are called the static stresses. Then the static stresses are multiplied by a number of factors, to compensate for each of the other influences. These factors include the application factor, which allows for momentary overloads, depending on the type of application: the size factor, which allows for non-uniformity of the material properties in the gears: the load distribution factor, which allows for non-uniformity in the tooth loading: and the dynamic factor, which compensates for the increase in tooth load caused by dynamic effects, as the teeth enter and leave the meshing zone. The AGMA report contains a description of how each of these factors should be chosen or calculated. In this book, we will describe only the calculation of the static stresses, since the subject of the book is the tooth geometry, and it is the static stresses, rather than the actual stresses, which are determined by the geometry.

There are two distinct types of stress which are calculated in the design of a gear pair, because each of these types can cause damage to the teeth, and eventual failure. First, there is the contact stress which occurs at the points where the meshing teeth are in contact. If the contact stress is too high, the tooth surface becomes pitted with small holes. This pitting may not be harmful, so long as the pits remain small, but if they become larger, the tooth surface is eventually destroyed. The second type of stress which is often responsible for tooth damage is the tensile stress in

the fillet, caused by a tooth load on the face of the tooth. If the tensile stress is too large, fatigue cracks will be formed in the fillet, and the tooth will eventually fracture. It is clear that both the contact stress and the fillet stress must always be calculated, and compared with values which the gear material can sustain without damage.

Contact Force Intensity 243

Figure 11.1 •. Tooth force applied to gear 1.

Contact Force Intensity

In the abSence of friction, the tooth force W is directed

along the normal to the tooth prof i Ie, which is tangent to the

base circle, as shown in Figure 11.1. If the torque applied to

gear 1 is M1, the corresponding tooth force W is found by

taking moments about the gear axis,

(11.1)

A similar relation exists between the contact force and the

torque applied to gear 2,

(11.2)

and, by eliminating W from these equations, we obtain a

relation between M1 and M2 ,

(11.3)

When a gear pair is designed, the value of either M1 or

M2 is known. We use Equation (11.3) to find the other torque,


and the contact force is then found from Equation (11.1 or 11. 2) •

Point Aw in Figure 11.1 is usually referred to as the contact point, but of course the gear is a solid object, and

the contact really takes place along the entire axial line

through Aw' whose length is equal to the gear face-width F. The contact length Lc is the total length of all the contact lines in the gear pair, and is therefore equal to either F or 2F, depending on whether the gear pair has one or two pairs of teeth in contact.

The load intensity w is defined as the tooth force per unit length of the contact line. The maximum load intensity occurs when there is only one tooth pair in contact, and is then equal to the tooth force divided by the gear face-width,

w (11.4)

When there are two pairs of teeth in contact, the total tooth force W is shared between both the contacting tooth pairs, so the load intensity is halved. Since the maximum stresses in spur gears always occur when the load intensit~ is a maximum, we need consider only the period of single-tooth contact, and the load intensity that is given by Equation (11.4).

Contact Stress

The contact stress at the points where the teeth touch each other is found by means of the Hertz contact stress

theory, described in all books on elasticity. Since the contact takes place along a line, the teeth are represented in

the vicinity of the contact line by two circular cylinders.

The radi i of these cylinders are equal to the radi i of

curvature P1 and P2 of the tooth profiles at the contact

point. The maximum contact stress Gc is then given by the following expression,

P +p Cp v[w( p\P22 )] (11. 5)

The quantity w in this expression is the load intensity, and

Contact Stress 245

Cp is an elastic coefficient, defined as follows,

'/1"( 1-v 2 ) '/1"( 1-v22) .J.... v[ 1 + ] Cp E1 E2

(11.6)

where E1,v 1 and E2 ,v 2 are Young's modulus and Poisson's ratio

for the material of each gear. The meshing diagram for a gear pair is shown in

Figure 11.2, with the contact point in position s on the path of contact. We proved in Equation (10.1) that the radius of

curvature P1 of the tooth profile of gear 1 at point A1 is equal to the length E1A1, while the corresponding radius of curvature P2 in gear 2 is equal to A2E2• Hence, the sum (P 1+P 2 ) is equal to E1E2 , which can be expressed in terms of the center distance C and the gear pair operating pressure

angle t/>,

C sin t/> (11.7)

This relation is used to simplify the expression for the contact stress,

c Figure 11.2. Tooth prof ile radi i of curvature.


(11.8)

The lengths E1A1 and A2E2 depend on the position of the contact point, so the radii of curvature of the two profiles are expressed in terms of the coordinate s,

Rb 1 tan q, + s (11.9)

Rb2 tan q, - s (11.10)

It is evident that the contact stress depends on the value of s, and therefore varies during the meshing cycle. For the purpose of the gear pair design, we are interested in the maximum value reached by uc • As we pointed out earlier, we consider only the period of single-tooth contact, so the load intensity w is constant, and the variation in Uc depends only on the denominator (P1 P2). A change in s increases one radius of curvature, and decreases the other by the same amount. It can be verified that we obtain the maximum value for Uc when the smaller radius of curvature is as small as possible. If

the gears are numbered so that gear 1 is the pinion, then Rb1 is smaller than Rb2 , and we obtain the smallest value for P1

when s reaches its largest negative value. The contact point is then as far as possible below the pi tch point, and since we are considering only the period of single-tooth contact, the contact point must lie at the lowest point of single-tooth contact on the pinion.

Figure 11.3 shows point Q on the path of contact, corresponding to the lowest point of single-tooth contact on the pinion, and the highest point of single-tooth contact on the gear. The value of P 1 can be read directly from the diagram, for the situation when the contact point coincides with Q, and the corresponding value of P2 is found from Equations (11.9 and 11.10),

(11.11)

(11.12)

To calculate the maximum contact stress during the meshing

Contact Stress 247

Figure 11.3. Contact at the lowest point of single-tooth contact on gear 1.

cycle, we substitute these values of Pl and P2 into Equation (11.8).

We can see from the expression for Uc that the maximum value depends on the load intensity, the material properties and the geometry of the gear pair. It is convenient to describe the influence of the gear pair geometry on the contact stress by means of a single dimensionless factor kc ' and we therefore express the contact stress in the following manner,

(11.13)

A comparison of Equations (11.8 and 11.13) shows that the

geometry factor kc is then given by the following expression,

(11.14)

where the radii of curvature are found from Equations (11.11

and 11. 12) •

The quantity kc is a geometry factor for the gear pair, rather than for either the pinion or the gear individually. To


find the maximum contact stress in the gear pair, the factor

kc is first calculated, and the contact stress is then given by Equation (11.13). This contact stress acts on the teeth of both gears, at the lowest point of single-tooth contact in the

pinion, and at the highest point of single-tooth contact in the gear.

In the case of a rack and pinion, there is no center distance C, and therefore Equation (11.14) cannot be used. To calculate the contact stress, we first put Equation (11.5) into the following form,

a c (11.15)

If we number the gears so that gear 1 is the pinion and gear 2 is the rack, then the curvature (1/P2) is zero, since the teeth of the rack are straight-sided. We now combine Equations (11.15 and 11.13), to obtain an expression for the geometry factor,

(11.16)

where the radius of curvature P1 is given, as before, by Equation (11.11).

Fillet Stress

A loaded gear tooth is shown in Figure 11.4, with the

contact force applied at a typical point Aw. The actual tooth

force is W, but for the stress analysis it is convenient to

consider a tooth of unit face-width, so we apply a force equal

to the load intensity w, given by Equation (11.4).

The stress is calculated as if the tooth were a beam,

fixed at its root. This method was first proposed by Wilfred

Lewis, but has been modified many times since his original suggestion. The force at Aw is considered to act at D, the

point where the normal to the profile at Aw cuts the tooth center-line, and it is then resolved into two components, one

perpendicular to the tooth center-line, and the other along

the center-line. The first component causes stresses in the

Fillet Stress 249

y

c 'Yw x

Figure 11.4. Geometric parameters at the contact point.

tooth like the bending stresses in a cantilevered beam. In particular, there are tensile and compressive stresses at points A and A', shown on the fillets in Figure 11.4. The second component causes a radial compressive stress throughout the tooth, like the axial stress in a beam, and this is smaller in magnitude than the bending stresses at A and A' •

When the bending and the radial stresses are combined, we obtain a tensile stress at A, and a compressive stress at A'. Although the compressive stress is larger in

magnitude, it is the tensile stress at A which is found to cause fatigue cracks, so it is this stress which we calculate. Since the stresses are found using elementary beam theory, and the tooth shape is totally unlike the shape of a beam, it is to be expected that the results are very inaccurate. We therefore multiply the calculated stress at A by a stress concentration factor, based on a series of photelastic experiments performed by Dolan and Broghamer [7]. The

resulting stress is called the fillet stress at A, and is

represented by the symbol a~. The subscript t is used to indicate that the symbol refers to the tensile fillet stress.

The value of a~ obviously depends on the fillet point A at which the stress is calculated. We are generally

interested in the maximum stress, which we call simply the

fillet stress 0t' and we will discuss later how this can be found.

We consider initially the load point Aw at an arbitrary


radius Rw' The profile angle at A , the coordinates of A , and w w the angle between the tangent at Aw and the tooth center-line are given by Equations (2.18, 2.35 and 2.38),

cos ~w Rb

(11.17) Rw

9 ts

inv ~s - inv ~w (11.18) 2R + w s

Xw Rw cos 9w (11.19)

Yw R w sin 9w (11.20)

Yw ~ - 9 (11.21) w w

We then find the x coordinate of D, the point where the normal at Aw intersects the tooth center-line,

(11.22)

The shape of the fillet depends on the type of cutter, and on its dimensions. Once the details of the cutter are known, the coordinates (x,y) of a typical point A on the fillet are found by the methods described in Chapter 9. The cross-section at AA' has a unit thickness and a depth 2y. The bending moment at this section is equal to [w cos Yw (xo-x)], and the radial force is (w sin Yw)' The bending stress and the radial stress are calculated, using elementary beam theory,

°bending

°radial w sin Yw (1 )(2y)

The stress concentration factor Kf for the fillet stress

is given by the following expression,

Kf k1 + (~)k2(..ll....)k3 (11.23) rf xo-x

where the term r f is the fillet radius of curvature, at the

point where it meets the root circle. The value of r f was

Fillet St ress 251

given by Equation (10.15), for a gear cut by a rack cutter, and by Equation (10.18) for a gear cut by a pinion cutter. When a hob is used to cut the gear, the value of r f is assumed to be the same as if the gear was cut by a rack cutter.

Values for the three constants k1, k2 and k3 were given by Dolan and Broghamer for gears with pressure angles ~s of

14.5° and 20°, and additional values are now given by the AGMA for 25° pressure angle gears. For other values of ~s between

14.5° and 25°, it is possible to find k1, k2 and k3 by interpolation. The following three equations represent k1, k2

and k3 as functions of ~s' giving the Dolan-Broghamer and the AGMA values when ~s is equal to 14.5°, 20° and 25°.

k1 0.3054 - 0.00489~; - 0.000069(~;)2 ( 11.24)

k2 0.3620 - 0.01268~; + 0.000104(~;)2 (11.25)

k3 0.2934 + 0.00609~; + 0.000087(~;)2 (11.26)

We combine the bending and the radial stresses, and multiply by the stress concentration factor, to obtain the fillet stress at point A,

(11.27)

The expression inside the square brackets has been multiplied by the module m, to make the expression dimensionless, and this accounts for the factor (11m) outside the brackets.

We now have to find the fillet point A where the tensile stress is a maximum. There are a number of methods used to find the critical section. Fortunately, if the stress is

calculated as a function of position along the fillet, the

function is very smooth, and is close to its maximum value

over a considerable length of the fillet. For this reason, it

is not very important to find the critical section exactly,

since a substantial error in the position of the critical

section gives only a very small error in the maximum fillet

stress. The method recommended by the AGMA is based on the original graphical method of Lewis, but is inconvenient for

use on the computer. The simplest method for finding the


critical section is to calculate o~ at a number of points along the fillet, and choose the largest value. The ~et of points should include the top point of the fillet, since in

some gears this is the point where the highest value of o~ occurs. Once this value has been found, it can be expressed in the following manner,

( 11.28)

The fillet stress 0t depends on the load intensity, and on a number of other quantities that are all geometric in nature. As we pointed out in connection with the contact stress, it is convenient to combine the geometric effects into a single dimensionless factor. We therefore express the fillet stress as follows, in terms of a geometry factor kt ,

(11.29)

We compare the last two equations, to obtain an expression for

the geometry factor,

1.5m(xo-x) O.5m tan Yw cosyw[Kf( 2 - )]max

y y (11.30)

So far, we have shown how to calculate the fillet stress when the load is applied at an arbitrary radius Rw' The tooth load moves along the tooth face during the meshing cycle, and the fillet stress depends on the position of the load. We now consider the load position which causes the fillet stress to

reach its max imum value. I n general, the fi llet stress

increases as the load moves towards the tip of the tooth. However, when the contact point is near the tooth tip, the

total tooth force is shared between two pairs of teeth,

provided the gears are accurately cut. The maximum fillet

stress is therefore reached when the load is applied at the

highest point of single-tooth contact. Figure 11.5 shows the meshing diagram for a typical gear pair. The length of the

path of contact T1T2 can be expressed as mcPb' where mc is the contact ratio, given by Equation (4.9). On gear 1, the load is

at the highest point of single-tooth contact when the contact

point is at Q', a distance Pb above T2 • Hence, the maximum

Fillet Stress

Figure 11.5. Contact at the highest point of single-tooth contact on gear 1.

253

fillet stress occurs when the load is applied at a radius Rw' given by the following expression,

(11.31)

When the gears are not cut with sufficient accuracy to ensure load-sharing, we have to allow for the possibility of the total tooth force continuing to act on one tooth, right up to the tip. In this case, the load intensity remains constant at (W/F) , and the load radius Rw must be chosen equal to RT, the radius of the tip circle. The AGMA report gives maximum

allowable values of variation in the base pitch, which can be

used to determine whether load sharing will take place.

In the design of a gear pair, the kt values are

calculated for both gears, and the fillet stress in each gear

is then given by Equation (11.29). If the kt values for the

two gears are very different, it is possible to reduce the

difference by altering the tooth thicknesses, in the manner

described in Chapter 6. A balanced-strength design is one in which the two kt values are equal.


AGMA Geometry Factors I and J

In the AGMA report, the methods for calculating the

contact stress and the fillet stresses in a spur gear pair look rather different from the methods presented in this

chapter. However, the underlying theory is the same, apart from the manner in which the critical section in the fillet is found.

The stresses are expressed in terms of a fictional

force Wt , known as the transmitted force, which has the value that the contact force would have, if it acted along the common tangent to the pitch circles. The transmitted force is related to the applied torque on gear 1 by the following

equation,

(11.32)

The corresponding relation between M1 and the contact force W was given by Equation (11.1), and a comparison of these equations gives the relation between Wt and W,

W cos '" (11.33)

The static contact stress and fillet stress are expressed in terms of the AGMA geometry factors I and J by the following two equations,

(11.34)

(11.35)

where dp is the diameter of the pinion pitch circle. When

these equations for ac and at are compared with Equations (11.13 and 11.29), we obtain the following relations between

the AGMA geometry factors I and J, and the factors kc and kt used in this chapter,

I (11.36)

J (11.37)

Examples 255

Numerical Examples

Example 11.1

A gear pair has the following specification: module

10 mm, pressure angle 20°, tooth numbers 28 and 75, center

distance 525.0 mm, face-width 35.0 mm, tooth thicknesses

18.51 mm and 20.10 mm, blank diameters 307.8 mm and 782.2 mm.

The gears are cut by a hob with addendum 12.5 mm and tooth tip

radius 3.8 mm. Calculate the static contact stress, and the

static fillet stress in gear 1, if the torque applied to

gear 1 is 2500 N-m, and the material constant Cp is

190.3 (MPa)0.5.

m=10, 's=200, Nl =28, N2=75, C=525.0, F=35.0

t sl =18.51, t s2=20.10, RT1 =153.9, RT2 =391.1

a r =12.5, rrT=3.8, Cp=190.3, Ml =2500

RSl = 140.000 mm

Rs2 375.000

Rb 1 = 1 3 1 • 557

Rb2 = 352.385 , = 22.810°

Pb = 29.521 mm

mc = 1.558

We fi rst calculate the contact stress.

W = 19003 N W

Fm = 54.295 MPa

Pl 50.341 P2 = 153.191

kc = 0.5137

o c = 720.4 MPa

(4.9)

(11.1)

(11.11)

(11.12)

(11.14)

(11.13)

To calculate the fillet stress, we start by finding the load

position, which is at the highest point of single-tooth

contact.

Rwl = RHSCl = 146.035

'w = 25.728°

9w = 0.048178 radians = 2.760°

(11.31)

(11.17)

(11.18)


Xw 145.865

Yw = 7.033

Yw = 22.968°

xn = 142.884 mm

(11.19)

(11.20)

(11.21)

(11.22)

Next, we calculate the minimum fillet radius of curvature r f , and some dimensions of the hob tooth.

e 1 = 3.849

a r - e 1 - rrT = 4.851

r f = 3.962

h 10.000

x~ = - 8.700

y~ = - 0.644 mm

(6.1)

(10.15)

(5.40)

(5.43)

(5.44)

Lastly, we find the positions of a number of points on the fillet, calculate the stress at each of these points, and choose the largest stress. To save space, we will only give the final calculation. The value of ur ' the hob tooth position when the critical fillet point is cut, has therefore been found by trial and error.

ur = - 6.017 mm ~' - 4.851

'1/' = - 6.661

S' = - 8.240

s = - 12.040

~ = - 7.088

'1/ = -9.732

- 0.155178 radians

R = 133.268

fiR = 4.703°

x = 132.819

Y = 10.927 mm

kl 0.180

k2 0.150

k3 0.450

Kf 2.011

kt 1.982

o = t 107.6 MPa

- 8.891°

(9.10)

(9.11)

(9.12)

(9.13)

(9.14)

(9.15)

(9.16)

(9.17)

(9.18)

(11.24)

(11.25)

( 11.26)

(11.23)

(11.30)

(11.29)

Examples 257

The gear pair specified in this example was designed by

the procedure given in Chapter 6, and the tooth thicknesses were chosen to give balanced strength. The critical point on

the tooth fillet of gear 2 is cut when the hob position ur is

equal to -4.182 mm. Using this value, the reader can verify

that gear 2 has the same kt value as gear 1, and hence the same

fillet stress.

Example 11.2

A 36-tooth plnlon with D.P. 2.5, pressure angle 20°, tooth thickness 0.620 inches, face-width 2.0 inches, and tip

circle diameter 15.2 inches, is cut by a hob whose addendum

and tooth tip radius are 0.534 inches and 0.171 inches. The

pinion is meshed with a rack, which is mounted so that its

addendum a is 0.4 inches. The material constant C has the pr 0 5 P

value 2290 (psi) .• Calculate the static contact stress, and the static fillet stress in the pinion, when the torque

applied to the pinion is 60000 lb-inches.

Pd=2.5, ~s=20°, N1=36, F=2.0, t s1 =0.620, RT1 =7.6 a r =0.534, rrT=0.171, apr =0.4, Cp=2290, M1=60000

m = 0.4000 inches

RSl = 7.2000

Rbl = 6.7658

t/I = 20.000°

Pb 1.1809 m = 1.8366 c

w 8868 lbs Ji. 11085psi Fm

P1 = 2.2810

k c = 0.4188

a 101000 psi c

(4.10 )

(4.13 )

(11.1)

(11.11)

(11.16)

(11.13)

The fillet stress calculation is essentially the same as

that in Example 11.1, so only the most important steps will be

wri t ten out.


Rwl = RHSC1 = 7.2039

'Yw = 17.629°

xo = 7.0992

e 1 = - 0.0114

a r - e 1 - rrT = 0.3744

r f = 0.1895

u r = - 0.4937

x = 6.7386

y = 0.4209 inches

Kf = 2.012

kt = 2.052

o = t 22740 psi

(11.31)

(11.21)

(11.22)

(6.1)

(10.15)

(11.23)

(11.30)

(11.29)

Introduction

Chapter 12 Internal Gears

As we stated at the beginning of Chapter 1, an internal gear is a gear whose teeth face inwards towards the center of

the gear. The geometry of an internal gear is very similar to that of an external gear, so in this chapter we will cover the geometric theory qui te rapidly, and emphasize only those aspects where there are differences between internal and external gears. In general, the treatment follows the same order of development that was used in the geometry of external gears. After first defining the shape of the internal gear tooth profile, we then discuss the meshing geometry of a pinion meshed with an internal gear. A gear pair of this type is called an internal gear pair. In the next part of the chapter we describe how internal gears are cut, and finally we outline a procedure for the geometric design of an internal gear pair.

Tooth Profile of an Internal Gear

When we discussed the tooth shape of an external gear, we

defined its profile as the shape which is conjugate with the

basic rack. In the case of an internal gear, it is clear that

the gear cannot mesh with any rack, so a different method is

required for defining its tooth shape. We first determine

what shape the tooth profile of the internal gear must have,

if the gear is to mesh correctly with an involute pinion. And

we will then show that the tooth profile of the internal gear can still be regarded as conjugate with the basic rack,

260 Internal Gears

Figure 12.1. An internal gear pair.

provided the basic rack is an imaginary rack of the sort discussed earlier.

Figure 12.1 shows an involute pinion meshed with an internal gear. The pinion is numbered as gear 1, and the internal gear as gear 2. We proved in Chapter 1 that, for the Law of Gearing to be satisfied, the common normal at the contact point must always pass through the pitch point P, which is a fixed point on the line of centers. The position of P is such that, when we draw the pitch circles touching each other at P, the angular velocity ratio of the two gears is the same as it would be if the two pitch circles were to make roll ing contact wi th no slipping.

The ratio between the angular velocities w1 and w2 is determined by the values of N1 and N2• We know that the teeth from each gear must pass alternately through the meshing zone, and this condition leads to the following relation between w1 and w2 ,

Tooth Profile of an Internal Gear 261

(12.1)

The proof of this equation is exactly the same as that of Equation (1.4), the corresponding relation for an external

gear pair. The direction of rotation of the pinion is always the

same as that of the internal gear, so w1 has the same sign

as w2 , as indicated by Equation (12.1). If the radii of the

pitch circles are RP1 and Rp2 ' and the circles roll together without slipping, their angular velocities must be related as follows,

(12.2)

It is evident from Equations (12.1 and 12.2) that the pitch circle radii are proportional to the tooth numbers,

(12.3)

In addition, we can see from Figure 12.1 that the difference between the pitch circle radi i is equal to the center distance C,

C (12.4)

We solve Equations (12.3 and 12.4), to obtain the values of

RP1 and Rp2 '

( 12.5)

(12.6)

We have now found the position of the pitch point P, since it is the point where the pitch circles touch each

other. We next determine the shape of the internal gear tooth

profile, if the common normal at the contact point is to pass

through P. Since the tooth profile of the pinion is an

involute, the normal to the profile at any point touches the

pinion base circle. To find the position of the contact point, when the pinion is in the position shown in Figure 12.1, we

262 Internal Gears

first draw the tangent from P to the pinion base circle,

touching the base circle at E1. The point A1 where this line cuts the pinion tooth profile must be the contact point, since it is the only point of the profi Ie whose normal passes through P.

A point on the tooth profile of the internal gear must therefore coincide with A1, since A1 is the contact point. This point on the internal gear is labelled A2 , and the normal to the internal gear tooth profile at A2 lies along A1PE 1, because the normals to both profiles must coincide at the contact point. These conditions are sufficient to enable us to construct the tooth profile. We draw the perpendicular from the gear center C2 of the internal gear to line PE 1, and the foot of this perpendicular is labelled E2 • The circle with center C2 and radius equal to C2E2 is called the base circle of the internal gear, and we have shown that the normal to the internal gear tooth profile at point A2 touches this circle.

The same argument can be used for each point of the profile, when it becomes the contact point. We have therefore proved that the normal to the tooth profile at every point of the profile must touch the base circle. This is exactly the manner in which the involute was defined in Chapter 2, so the tooth profile of an internal gear is identical to that of an external gear with the same number of teeth. The difference between the two types of gear, however, is that the teeth of the internal gear lie outside the profile, while those of the external gear lie inside it. In other words, the teeth of the internal gear have exactly the same shape as the tooth spaces in an external gear.

We now make use again of Figure 12.1. Since lines C2E2 and C1E1 are both perpendicular to line E2E1, the two triangles C2E2P and C1E1P are similar. The ratio of the base circle radii is therefore equal to that of the pitch circle radii, which we showed in Equation (12.3) is also equal to the ratio of the tooth numbers,

(12.7)

The base pitches of both gears are defined by Equation (2.22),

Tooth Profile of an Internal Gear 263

(12.8)

(12.9)

Since, as we have just proved, the base circle radii are

proportional to the tooth numbers, it follows from Equations

(12.8 and 12.9) that the base pitches of the two gears are the

same,

(12.10)

This result means that the base pitch of the internal gear is also equal to that of the basic rack used to define the tooth shape of the pinion. Although the internal gear cannot mesh with a real rack, it is quite possible to picture it meshing with an imaginary rack. Figure 12.2 shows a number of tooth profiles, which are conjugate to the basic rack. The same tooth profiles can be regarded either as the teeth of an

c

Figure 12.2. Internal gear conjugate to an imaginary rack.

264 Internal Gears

external gear meshing with a real rack, or as those of an

internal gear meshing with an imaginary rack. The two cases

are shown in the upper and lower halves of the diagram. We can therefore define the tooth shape of an internal

gear in the same manner as we define that of an external gear,

as conjugate to the basic rack with module m and pressure

angle ~r' Since the tooth profiles of the internal gear are identical to those of an external gear with the same number of teeth, we can immediately use many of the results derived in the earlier chapters of this book. Some of the more important results will be stated here, without further proof.

The profile angle ~R at a typical radius R of the tooth profile is given by Equation (2.18),

cos tPR Rb R

(12.11)

The standard pitch circle is defined as the pitch circle when the gear is meshed with the imaginary basic rack, and its

radius Rs is given by Equation (2.30),

(12.12)

The pressure angle tPs is defined as the profile angle at the standard pitch circle, and it is equal to the pressure angle

of the basic rack,

~s (12.13)

The circular pitch ps on the standard pitch circle is equal to the pitch of the basic rack, which can also be expressed in

terms of the module,

7rm (12.14)

By substituting R in place of R in Equation (12.11), we s obtain the usual relation between the radii of the base circle

and the standard pitch circle,

(12.15)

Internal Gears 265

Profile Shift, Tooth Thickness, and Other Geometric Relations

The profile shift of an internal gear is defined in exactly the same manner as that of an external gear. The internal gear has a profile shift e if it is conjugate to the basic rack, when the reference line of the basic rack lies a distance (Rs+e) from the center of the gear. In other words, the basic rack is offset a distance e from its standard position, and a positive value of e corresponds to an offset of the basic rack away from the gear center.

The relation between the tooth thickness of an external gear and its profile shift was given by Equation (6.1),

ts ~7I'm + 2e tan "'s

For an internal gear, the tooth shape is the same as the tooth space of an external gear, so the tooth thickness of the internal gear is equal to the space width of the external gear. Hence, the tooth thickness of an internal gear is related to its profile shift in the following manner,

~7I'm - 2e tan "'s (12.16)

To locate the position of point A at a typical radius R on the tooth profile, we again use a coordinate system in which the x axis coincides with a tooth center-line, as shown in Figure 12.3. The points on the involute at radii Rb , Rs and R are labelled B, As and A. The angle ACB is equal to inv "'R' as we proved in Equation (2.19), and angle AsCB is equal to inv "'5' The polar coordinate 6R of point A is therefore found as the angle xCAs ' minus angle AsCB, plus angle ACB,

ts 2Rs - inv "'s + inv "'R ( 12.17)

We can use this result, to write down the tooth thickness tR at radius R,

( 12.18)

It can be seen in Figure 12.3 that the tip circle of an

266 Internal Gears

Standard pitch circle

y

c x

Figure 12.3. Tooth thickness at radius R.

internal gear lies inside the standard pitch circle, while the root circle lies outside it. The addendum as and the dedendum bs are still defined in the usual way, as the radial distances from the standard pitch circle to the tip circle and the root circle. Hence, as and bs are related to the various radi i in the following manner,

( 12.19)

(12.20)

Meshing Geometry of an Internal Gear Pair

The meshing diagram of an internal gear pair is shown in Figure 12.4, with the pinion as gear 1 and the internal gear

as gear 2. The position of the pitch point, and the radii of

Meshing Geometry of an Internal Gear Pair 267

Line of action

Figure 12.4. Meshing diagram of an internal gear pair.

the pitch circles, were given by Equations (12.5 and 12.6). The common tangent E,E2 to the base circles is the line of act ion, and the angle t/I between thi s line and the common tangent to the pitch circles is the operating pressure angle

of the gear pair. The lines C1E1 and C2E2 are both perpendicular to the

line of action, so they each make an angle t/I with the line of

centers. By expressing the center distance C in terms of the

base circle radii, we obtain an equation for the operating

pressure angle t/I,

cos t/I (12.21)

We can also use Figure 12.4 to write down a relation between

Rb2 and Rp2 '

=

268 Internal Gears

The operating pressure angle ~P2 of the internal gear is defined as the profile angle at the pitch circle, and its value is therefore found from Equation (12.11), if we

substitute Rp2 in place of R,

cos ~P2

A comparison of the last two equations shows that ~P2 is equal to ~, and we can show in the same way that the operating pressure angle ~Pl of the pinion is also equal to ~. We therefore use the symbol ~p for the operating pressure angle of either gear, and its value is equal to the operating pressure angle of the gear pair,

(12.22)

The operating circular pitches of the pinion and the internal gear are defined as the circular pitches at the pitch circles, and their values are given by Equation (1.18),

When we substitute the expressions in Equations (12.5 and 12.6) for RPl and Rp2 ' it is clear that the operating circular pitches are equal. The symbol Pp is therefore used to represent the operating circular pitch of either gear, and its value is given by the following expression,

(12.23)

Relation Between the Gear positions

In an internal gear pair, the angular position of the internal gear is indicated in the same manner as that of the pinion, by the angle ~2 measured from the line of centers counter-clockwise to the x2 axis. An internal gear pair is shown in Figure 12.5, with the contact point coinciding with

Relation Between the Gear positions

Figure 12.5. Gear positions, with contact

at the pi tch point.

269

the pitch point. When the gears are in these positions, their

angular positions can be read from the diagram,

(12.24)

(12.25)

where tP1 and tP2 are the tooth thicknesses at the pi tch

circles. After rotations AP1 and AP 2 , the new angular

positions are as follows,

P 1 -~ + AP 1 (12.26) 2Rp1

P2 ~+ 2Rp2

AP 2 (12.27)

The angular velocities of the two gears are related by

Equation (12.2),

270 Internal Gears

Rp1 w1 Rp2w2

This equation is integrated, giving a relation between the gear rotations,

RP1~/J1 RP2~/J2 (12.28)

We now eliminate ~/J1 and (12.26 - 12.28), and we obtain the

the angular positions /J 1 and /J2'

~/J2 between Equations relation we require between

o (12.29)

Contact Ratio

We defined the contact ratio of an external gear pair in Chapter 4, as the rotation of either gear during one meshing cycle, divided by the angular pitch of the same gear. We then showed that this definition is equivalent to the length of the

contact path, divided by the base pitch. For an internal gear pair, the contact ratio is defined in the same manner as for an external gear pair. Once again, we can replace the definition by the alternative description, and this time we will not prove their equivalence, since the proof is identical to the proof given in Chapter 4.

In the meshing diagram shown in Figure 12.4, the line of action is the common tangent to the base circles, and the ends

T1 and T2 of the path of contact are the points where the two

tip circles intersect the line of action. The length T1T2 is related to the other lengths on the line of action,

We express each of these lengths in terms of quantities defined on the gears, and divide by the base pitch, to obtain

the contact ratio mc '

Internal Gears 271

Interference

As always, there is non-conjugate contact if the path of contact extends beyond the interference points. The first

condition, therefore, for no interference at the fillets of

the pinion in Figure 12.4 is that T2 should lie above E1, or

in other words, E2 T 2 must be larger than E2E 1 '

> (12.31)

There is no corresponding condition relating to the position of point T1, since in principle the involutes of the internal gear can extend out to any radius, and conjugate contact is theoretically possible, however large the tip circle radius

of the pinion. In practice, of course, the involute section of the

tooth profile in each gear ends at the fillet circle, and we must therefore ensure that contact ceases a suitable distance away from the fillet circle. The active section of the tooth profile in either gear is the part which comes into contact with the other gear. The end point of the active section of the profile nearest the root is called the limit point, and the circle through this point is the limit circle. In the pinion, the limit circle is the circle through point T2 , while in the internal gear, it is the circle through T1. The radii RL1 and RL2 of the limit circles can be read from Figure 12.4,

(12.32)

(12.33)

To ensure that there is no contact at the fillets, the

limit circle of the pinion must be larger than its fillet circle, and because the teeth of the internal gear face

inwards, its limit circle must be smaller than its fillet

circle. It is customary to leave a margin of at least O.025m

between the circles, to allow for possible errors in the

center distance, and we therefore obtain the following

conditions, which must be satisfied by the radii Rf1 and Rf2 of the fi llet ci rcles,

272 Internal Gears

(12.34)

(12.35)

The pinion fillet circle radius Rf1 was given by Equation (5.47), for a gear which is cut by a pinion cutter,

and by Equation (5.48) when the gear is cut by a rack cutter or a hob. Later in this chapter, when we discuss the cutting of internal gears, we will show how to calculate the internal gear fillet circle radius Rf2 •

In order to prevent the possibility of interference in an internal gear pair, it is essential that the three conditions given by Equations (12.31, 12.34 and 12.35) are all satisfied. In addition, the gear pair should also be designed with a minimum clearance of O.25m at each root circle.

In most internal gear pairs, if the interference conditions are satisfied at the pinion fillets, the clearance will be more than adaquate, while if there is sufficient clearance at the root circle of the internal gear, the

interference condition is satisfied automatically. If these general rules were invariably true, it would be sufficient to check for interference at the pinion fillets, and for clearance at the internal gear root circle. Since there are some exceptions to the general rules, it is still necessary to check that all three interference conditions are satisfied, and that both clearances are adaquate. However, the general

rules do form the basis of a design procedure, which will be

described later in this chapter.

Tip Interference

There is a second type of interference which can occur in

an internal gear pair. Figure 12.6 shows the path followed,

relative to the internal gear, by point AT1 on the tooth tip of the pinion. This curve is called a hypotrochoid, and it

touches the tooth profile of the internal gear at its limit

circle. In a well-designed gear pair, the path of AT1 lies within the tooth space of the internal gear, as shown in

Tip Interference 273

Path followed by point AT1 of the pinion~

Figure 12.6. Path followed by the pinion tooth tip.

Figure 12.6. However, in certain circumstances, the path of

AT1 passes through the corner of the internal gear tooth, and this phenomenon is known as tip interference. It is obvious

that when tip interference takes place, the gear pair is unusable.

The shape of the path followed by point AT1 is a convex curve. It is because the teeth of the internal gear are concave that tip interference can occur. I f the teeth of gear 2 were convex, as in the case of an external gear pair, then there would be no possibility of tip interference.

In order to determine whether tip interference will take place, we calculate the position, relative to the internal

gear, of the point where the tooth tip AT1 of the pinion

crosses the tip circle of the internal gear. We compare this

position with that of AT2 , the tooth tip of the internal gear.

To prevent the possibility of tip interference, the path of

AT1 must clear point AT2 by an adaquate margin, whose value

depends on the size and accuracy of the gears.

The polar coordinates 9T1 and 9T2 of points AT1 and AT2 are given by Equations (2.35 and 12.17),

ts1 -- + (inv tP - inv tPT1 ) 2Rs1 s

(12.36)

274 Internal Gears

ts2 - [-- - (inv ~ - inv ~T2)]

2Rs2 s (12.37)

where ~T1 and ~T2 are the profile angles at the tip circles. The value of 8T2 is negative, because point AT2 lies on the lower face of the tooth, and the polar angle is defined as positive when it is counter-clockwise.

Figure 12.7 shows the gear pair, at the instant when point AT1 is crossing the tip circle of the internal gear. The angles ~1 and ~2 are the angular positions of the two gears, and 82 is the polar angle, relative to the internal gear, of the position of AT1 • By considering triangle C1C2AT1 , we can write down two relations between the various angles,

222 (C +RT1 -RT2 ) 2CRT1

(12.38)

(12.39)

The angle 82 is then calculated by the following steps. The angular position of the pinion is given by Equation (12.38),

Tip circle of the gear

c Figure 12.7. Checking for tip interference.

Axial and Radial Assembly 275

(12.40)

The angular position of the internal gear is found from

Equation (12.29),

Finally, the value of 82 is given by Equation (12.39),

R arcsin [RT1 sin (fi 1+8T1 ) 1 - fi2

T2

(12.41)

(12.42)

The distance by which the path of AT1 clears point AT2 is

approximately equal to RT2(8T2-82)' Hence, if a clearance of 0.05 modules is adaquate for a particular gear pair, the condition for no tip interference takes the following form,

(12.43)

It will be found that this condition is generally satisfied for gear pairs where the difference (N2-N 1) is 8 or more, and in certain circumstances it may be satisfied when (N2-N 1) is 7 or even 6. However, the amount of clearance depends on several quantities, such as the center distance and the tip circle radii, so it is important to check for the possibility of tip interference whenever the value of (N 2-N 1) is small.


An internal gear pair can be assembled in either an axial

or a radial manner. In other words, if the internal gear is

held fixed, the pinion can be brought into its meshing

position, either by a movement in the direction of its axis,

or by a movement along a radius of the internal gear. The tooth shapes of the pinion and the internal gear will

always allow axial assembly, provided the gear pair has been

designed so that there is no interference or tip

interference. In some gear boxes, however, axial movement of the pinion may be obstructed, and axial assembly is then

276 Internal Gears

impossible. In this case, the gear pair must be assembled in the radial manner, and we must check that such assembly is

feasible. The check can be made by the following simple, though

long, procedure. Figure 12.8 shows an internal gear pair in position, so that a tooth of the pinion is lined up with a tooth space of the internal gear. We draw the ~ and ~ axes on

the diagram, so that the ~ axis coincides wi th the center-line of the pinion tooth. The points AT2 , AT2 , AT2 , etc, shown on the internal gear, are the corner points of the teeth closest to the ~ axis. On the pinion, points A1, A;, Ai, etc, are the points of each tooth which are furthest from the ~ axis. For the teeth near to the ~ axis, these points lie at the intersection of the tooth profiles with the vertical tangent to the base circle, and their distance from the E axis is equal to half the span measurement 5, which was given by

Tangent to the base circle of the pinion----''-++j

Figure 12.8. positions for radial assembly.


Tangent to the base circle of the pinion

277

Figure 12.9. Alternative positions for radial assembly.

Equation (8.14). For the remaining teeth, the point furthest

from the ~ axis is the corner point of the tooth. In order to determine whether the gear pair can be assembled radially, we calculate the ~ coordinates of the labelled points on the pinion, and check that in each case they are less than the ~

coordinates of the corresponding points on the internal gear.

If this condition is satisfied for each pair of points, then radial assembly can be carried out.

In cases where radial assembly is not possible in the

position of Figure 12.8, we can consider an alternative

position, as shown in Figure 12.9. We place the gear pair so

that a tooth space of the pinion is lined up with a tooth of

the internal gear, and again we calculate the ~ coordinates of

the various tooth points. As before, radial assembly is only

possible if each point on the pinion is closer to the ~ axis

than the corresponding point on the internal gear.

The minimum value of (N2-N 1) for which radial assembly is possible depends primarily on the pressure angle. For

278 Internal Gears

example, radial assembly of 20° pressure angle gear pairs can generally be carried out when (N2-N 1) is 17 or larger. In some cases it may also be possible, for gear pairs with lower values of (N2-N 1). However, since the positions of the tooth tip points in each gear depend on the tooth thickness and the radius of the tip circle, as well as on the pressure angle, the checks should be made whenever there is a danger that radial assembly may be impossible.

Highest and Lowest Points of Single-Tooth Contact

The meshing diagram for an internal gear pair is shown in Figure 12.10. The ends of the path of contact are labelled T1 and T2 , and the two points on the path of contact, a distance Pb inside each end, are shown as Q and Q'. There is single-point contact whenever the contact point lies between Q and Q' •

Figure 12.10. The end points of single-tooth contact.

Cutting Internal Gears 279

On the pinion, the highest and lowest points of single-tooth contact correspond to Q' and Q on the path of contact. The radii of the circles through these points can be read directly from the diagram,

(12.45)

Simi larly, on the internal gear, the highest and lowest

points of single-tooth contact correspond to Q and Q', and again the radii of the circles through these points can be read from the diagram,

(12.47)


The three methods of generating cutting which were described in Chapter 5 were shaping, with either a pinion cutter or a rack cutter, and hobbing. Of these three methods, the one which is almost always used to cut an internal gear is the method of shaping, using a pinion cutter. It is preferable to use a pinion cutter whose teeth are rounded at the tips, as shown in Figure 12.11, since otherwise the tooth fillets of the internal gear will have a very small radius of curvature, and this will cause high stresses in the fillets. On the

pinion cutter, the end point Ahc of the involute section of the tooth profile has polar coordinates (Rhc ,9hc )' given by

Equations (5.34 and 5.35), and the center A~ of the circular

section has coordinates (R~,9~), given by Equations (5.32 and 5.39) .

The meshing diagram for the cutting process is shown in

Figure 12.12, where the subscripts g and c refer, as usual, to

the gear and the cutter. The cutting pressure angle and the radii of the cutting pitch circles can all be expressed in terms of the cutting center distance Cc ,

280 Internal Gears

Figure 12.11. A pinion cutter.

cos q,c (12.48)

(12.49)

(12.50)

The cutting pressure angle q,~ of the gear or the cutter is equal to q,c, and the cutting circular pitch can be found from either of the cutting pitch circle radii,

( 12.51 )

(12.52)

Since the cutting process is equivalent to meshing with

no backlash, the tooth thickness of the gear is equal to the space width of the cutter, both measured at the cutting pitch

circles,

(12.53)

The tooth thicknesses at the cutting pitch circles are


Path followed by point Ahc of the cutter

281

Figure 12.12. Meshing diagram of a gear and cutter.

related to the tooth thicknesses at the standard pitch

circles by Equations (12.18 and 3.17),

t tpg R [~- 2 ( i nv tP s - inv tP~)]

pg Rsg

t inv IP~)] tpc R [2f. + 2 (inv IPs -

pc Rsc

In order to calculate the tooth thickness of the gear at

its standard pitch circle, we introduce the standard cutting

center distance c~, defined as the difference between the

standard pi tch circle radi i ,

(12.54)

We substitute the expressions for tpg and tpc into

Equation (12.53), multiply by the ratio (CC/CC), and we s

282 Internal Gears

obtain the required expression for t sg '

p - t + 2Cc (invt/l - invt/lCp ) s sc s s (12.55)

If we want to calculate the cutting center distance to be

used, in order to obtain a specified tooth thickness tsg in the gear, we start by using Equation (12.55) to find the value

of inv t/l~,

inv t/l~ inv t/l + _l_(p t - t ) s 2Cc s - sg sc

s (12.56)

We use Equations (2.16 and 2.17) to calculate the

corresponding value of t/l~, and the cutting center distance is then given by Equations (12.51 and 12.48),

t/lc P

Rbg-Rbc cos t/lc

(12.57)

(12.58)

5

Figure 12.13. Cutting a point on the gear tooth fillet.

Shape of the Fillet 283

The meshing diagram shown in Figure 12.12 can be used to derive an expression for the fillet circle radius of the internal gear. The involute section of the tooth profile on the internal gear is cut by the involute part of the cutter tooth, which ends at point Ahc • The path of contact is a segment of the common tangent to the base circles. The upper

end of the path of contact is therefore at Hc' the point where the path followed by Ahc intersects the common tangent to the base circles. When the cutting point reaches Hc on the path of contact, point Ahc of the cutter touches point Af on the gear tooth profile, the point where the involute section of the profile meets the fillet. The fillet circle of the internal gear is the circle passing through Af , and its radius Rfg is

equal to the length CgHc' which can be read from the diagram,

(12.59)

Shape of the Fillet

We use the geometry of non-involute generation, described in Chapter 9, to find the coordinates of points on the tooth fillet of the internal gear. Part of the meshing diagram during the cutting process is shown in Figure 12.13. On the cutter tooth, the center of the circular section at the tip is labelled A', and its polar coordinates (R' ,0') are c c c given by Equations (5.32 and 5.39). The line PA' meets the c cutter tooth profile at Ac' and this must be the point at which the cutter touches the tooth fillet of the gear, since the normal at Ac passes through the pitch point. We use the (~,~) coordinate system whose origin is at the pitch point,

and we consider the geometry when the line from the pinion

center to point A~ makes an angle a with the ~ axis. The coordinates (~' , ~') of point A' can then be wri tten, c

~ ,

~'

R' cos a - RC c pc

R' sin a c

(12.60)

(12.61)

The distances s' and s, from P to A~ and Ac' are found from

284

the following equations,

s'

s s' + r cT

Internal Gears

(12.62)

(12.63)

and we then wri te down the coordinates of point Ac'

(...§...) t , s' ~ (12.64)

1/ (12.65)

The angle a between line CcA~ and the ~ axis is made up of two parts, the coordinate 8~ of point A~, and the angle Pc through which the cutter has rotated. Hence, the angular position of the cutter can be expressed in terms of a,

a - 8' c (12.66)

The relation between the angular positions of a pinion and an internal gear was given by Equation (12.29),

When the equation is used for a plnlon cutter and an internal gear, the sum of the tooth thicknesses is equal to the

circular pitch P~ on the cutting pitch circles. The equation is simplified if it is multiplied throughout by the ratio

(C~/CC), and we then obtain an expression for the angular

position of the gear,

(12.67)

Since point Ac on the cutter is the contact point, it

coincides with a point A on the tooth fillet of the gear. The coordinates of A can be read immediately from Figure 12.13,

R (12.68)

arctan (_1/_) - Q

RC +~ 1'g pg

(12.69)

Shape of the Fi llet 285

Equations (12.60 - 12.69) are used to calculate the

coordinates of points on the internal gear tooth fillet,

corresponding to any chosen values of a. The first point which

needs to be considered is Af , where the fillet meets the

involute. In this case, the cutting point must lie at He on

the path of contact, and line PAc therefore makes an angle ~c

with the ~ axis, as shown in Figure 12.14. The value of a at

which this occurs can be read from the diagram,

a R

arccos (R~c) - ~c c

(12.70)

The last point to be considered is cut when line C A' c c lies along the E axis, and the value of a is then zero,

a o (12.71)

In this case, the cutter has reached its maximum penetration,

and A is the point at which the fillet meets the root circle.

The entire fillet shape can be constructed by choosing a

sequence of values for a, between the values given by

Equations (12.70 and 12.71), and then calculating the

corresponding coordinates Rand lI R•

Figure 12.14. Cutting the end point of the fillet.

286 Internal Gears

Fillet Radius of Curvature

In Chapter 10, we derived the Euler-Savary equation relating the radii of curvature of two conjugate profiles. For a pinion meshing with an internal gear, this equation takes the following form,

1 1 1 --(- --) sin q, Rpl Rp2

(12.72)

As before, a positive value of P1 corresponds to a convex tooth profile in the pinion. However, the entire tooth profile of the internal gear is concave, so the equation has been written with a different sign convention from the one used in Chapter 10, and a positive value of P2 now corresponds to a concave tooth prof i Ie in the internal gear.

We use this equation to calculate the radius of curvature at points in the internal gear tooth fillet, which is conjugate to the circular tip of the cutter tooth. We therefore write the equation as follows,

(12.73)

The expression in brackets is used to define a quantity (liRa)' which is the relative curvature of the cutting pitch circles,

1 1 -c- - -c-Rpc Rpg

(12.74)

We substitute for R~g and R~c' using Equations (12.49 and 12.50), and we obtain the following expression for RO'

NgNCCC

(N -N )2 g c (12.75)

Since the radius of curvature of the cutter tooth profile is known, and we want to calculate the corresponding radius of curvature in the gear tooth profile, we solve Equation (12.73) for Pg ,

(12.76)

Fillet Radius of Curvature 287

To calculate the radius of curvature Pf at a point A on

the tooth fillet of the internal gear, we first use Equations

(12.60 - 12.65) to find the position (L11) of A when it is cut. The corresponding values of ~ and s can be read from

Figure 12.13,

tan q,

s _E_ sin q,

(12.77)

(12.78)

The gear tooth fillet is cut by the circular tip section of

the cutter tooth, whose radius is r cT • We therefore find the

radius of curvature at A, by substituting r cT in place of Pg in Equation (12.76),

(12.79)

The minimum radius of curvature r f occurs at the point where the fillet meets the root circle. This point is cut when

line AcA~ in the cutter coincides with the E axis, and the angle a which we have used to describe the cutter position is equal to zero. We substitute this value of a into Equations (12.60 - 12.65) to find the corresponding position of the

cutting point, and we use Equations (5.32, 12.77 and 12.78) to

obtain the following results,

E RTc _ RC

pc

11 0

q, 90 0

s R - RC Tc pc

The value of r f is then given by Equation (12.79),

(RTC-Rpc-rCT)2 r f r cT + c

RO + (RTC-Rpc-rCT) (12.80)

Since the value of RO is generally quite large, the fillet radius of curvature r f is only slightly greater than

288 Internal Gears

the cutter tooth tip radius r cT • If the cutter tooth tip has no rounded section, the value of r CT is zero, and we obtain a

very small fillet radius of curvature in the gear. It is for this reason that a pinion cutter with rounded tooth tips is recommended for cutting internal gears.

Undercutting

To consider the possibility of undercutting, we refer again to the meshing diagram of the cutting process, shown in Figure 12.12. The interference point of the internal gear is Eg , the point where the common tangent to the base circles touches the internal gear base circle. For an external gear, if the cutting point lies near the interference point, the corresponding point on the gear tooth profile is close to the fillet. However, when we consider an internal gear, this relation is reversed. A cutting point near the interference point corresponds to a point near the tip of the gear tooth. ~t the other extreme, point Af of the gear tooth profile, where the involute meets the fillet, is cut when the cutting point lies at Hc' which is the end of the path of contact furthest from the interference point. Hence, no matter how large the cutter addendum is, there is no danger of conventional undercutting at the tooth fillets of an internal gear.

There is, though, a possibility of undercutting at the

tooth tips of the gear. The correct involute profile in a tooth of the gear is cut by the involute part of the cutter

tooth. The profile of the cutter tooth can only coincide with

the involute down to the base circle, and the tooth may be

designed so that the fillet starts slightly outside the base

circle, since the involute radius of curvature becomes zero

at the base circle. The involute part of the tooth, therefore,

ends at the fillet circle, which is either larger than the base circle, or the same size. On any particular cutter, the

radius Rfc of the fillet circle can be measured. In Figure 12.15, the fillet circle of the cutter is

shown, intersecting the line of action at Fc. The involute

section of the cutter tooth profile lies outside the fillet

Undercutting

Fillet circle of the cutter

Figure 12.15. Checking for undercutting by the cut ter tooth fillet.

289

circle, so the cutting of the involute part of the gear tooth profile can only take place outside this circle. This means that the path of contact must end at a point somewhere above Fc ' and we obtain the following condition that must be satisfied if there is to be no undercutting,

(12.81)

If the tip circle radius RTg required for the gear is smaller

than the minimum value given by this condition, then it is necessary to use a cutter wi th more teeth.

There is a second manner in which internal gears may be

undercut, and this is caused by tip interference between the

pinion cutter and the gear. This phenomenon can occur either

when the cutting center distance is equal to CC , or during the

initial cutting period while the cutter is being fed into the

gear, when the center distance is less than Cc • We therefore consider the possibility of tip interference at a center

290 Internal Gears

distance Cf , which is the value at some time during the feed-in period.

We have pointed out that internal gears are generally cut by a cutter whose teeth are rounded at the tips. However, in order to ensure that there is no tip interference, we will

assume for the purpose of the calculations that the involute profile of the cutter tooth extends right out to the tip circle, meeting it at point ATc • To check that there is no tip interference, we calculate the distance between the tooth tip corner ATg of the gear, and the point where the path of ATc intersects the tip circle of the gear. This distance should exceed a specified value, such as 0.02 modules.

The radius of point ATc on the cutter is of course equal

to RTc , the radius of the tip circle. The profile angle ~Tc at this point, and the polar coordinate 9Tc ' are found from Equations (2.18 and 2.35),

cos ~Tc Rbc

(12.82) RTc

9' tsc

inv ~s - inv ~Tc (12.83) -- + Tc 2Rsc

Figure 12.16. Checking for tip interference during cutting.

Undercutting 291

The remaining equations are similar in form to Equations (12.38 - 12.43), where we discussed tip interference in an

internal gear pair. Figure 12.16 shows the position of point ATc on the cutter, as it passes through the tip circle of the

gear. The angle 9g is the polar coordinate of ATc ' relative to

the axes fixed in the gear. By considering triangle CgCcATc' we can write down the following two equations,

( 2 2 2) Cf+RTc-RTg 2C f RTc

(12.84)

if- sin (fJ +9 ) Tc g g

(12.85)

The angular position of the cutter is found from Equation (12.84),

( 2 _ 2_ 2 ) RTg Cf RTc arccos 2C R

f Tc - 9' Tc (12.86)

The corresponding angular position of the gear is then given by Equation (12.67),

( 12.87)

Finally, we use Equation (12.85) to obtain the coordinate 9g of the point where the path of ATc crosses the tip circle of the internal gear,

R arcsin [ Tc sin (fJ +9' )] - fJ RTg c Tc g (12.88)

For a minimum clearance at ATg of 0.02 modules, the following condition must be satisfied,

(12.89)

where 9Tg is the polar coordinate of point ATg on the tooth tip of the gear, given by Equation (12.37).

During the cutting feed-in period, point ATc of the

cutter first penetrates the tip circle of the gear when the

cutting center distance is equal to (RTg-RTC ). To ensure that

there will be no tip interference, we should carry out the check for several values of Cf , starting with (RTg-RTc ) and

292 Internal Gears

ending with Cc • If tip interference is shown to occur at any

value of Cf ' the gear is probably unusable, since the cutter may remove a substantial part of the involute profile.

Rubbing

There is one other requirement that must be met, if the gear is to be cut in a satisfactory manner. During each return stroke of the shaping process, either the gear blank or the cutter is automatically displaced a short distance away from the cutting zone, to prevent rubbing between the cutter and the gear blank. If such rubbing occurs, the cutter wear is excessive, and burrs are left on the teeth of the gear. The designer should check, for any particular gear and cutter, that there is a displacement direction which will eliminate the rubbing. The situation is complicated by the fact that, during the shaping process, the tooth spaces of the gear have of course not yet been cut to their final shape. The cutter may therefore rub against parts of the gear blank which will later be cut away. Moreover, during the cutting strokes, there may be many points of each cutter tooth profile making contact with the gear blank, instead of the single point that would be in contact once the gear reaches its final shape.

In the discussion which follows, we will assume that the gear blank remains fixed, and the cutter is displaced away from the cutting zone. We will then calculate the direction required for the cutter displacement. If in fact the cutter

remains fixed and the gear blank is displaced, the direction

is obviously reversed. We consider first the trailing profile of one of the

cutter teeth. Figure 12.17 shows the positions of the gear and

the cutter, when the involute part of the cutter tooth profile

is just starting to cut into the gear blank. In other words,

point Ahc of the cutter tooth, the point on the trailing profile at the radius Rhc given by Equation (5.34), lies on the tip circle of the gear. The tangent to the cutter tooth at

Ahc makes an angle a' with the line of centers, and the value

of a' can be read f rom the diagram,

Rubbing 293

Figure 12.17. Displacement direction to avoid rubbing.

a' (12.90)

The cutter rotates as the tooth penetrates further into the gear blank, and the angles between the contact point tangents and the line of centers become smaller than a'. A displacement to prevent rubbing by the trailing profile of the cutter tooth

can therefore be made, provided the direction of the displacement makes an angle with the line of centers that is

greater than a'. We consider next the leading profile of the cutter

tooth, which is also in contact with a tooth of the gear. The

angles between the contact point tangents and the line of

centers reach their maximum values when the contact is with

the final tooth profile, and the angle is then equal to the

cutting pressure angle ~c. The angle of the cutter

displacement must therefore be less than ~c. To ensure that

there is a displacement satisfying both requirements, the

294 Internal Gears

designer should check that the angle a' given by Equation (12.90) is less than ~c by at least a few degrees, and then specify the displacement angle at a value between a' and ~c. It can be seen from the diagram that, if the displacement angle is chosen equal to ~c, the rubbing will be more severe than if the angle is chosen equal to a' • The angle should therefore be closer to a' than to ~c. If the value of a' given by Equation (12.90) is not less than ~c by a sufficient margin, it is then necessary to use a cutter with fewer teeth.

Geometric Design of an Internal Gear Pair

The design procedure is very similar to the procedure described in Chapter 6, for the de~ign of an external gear pair. We consider the situation when the center distance is specified, and we first choose the tooth numbers so that the standard center distance is approximately equal to the center distance,

(12.91)

The base circle radii, the gear pair operating pressure angle ~, the operating pressure angle ~p of each gear, the pitch circle radii, and the operating circular pitch Pp are then all determined, and can be calculated by means of the equations at the beginning of this chapter.

We now choose tooth thickness values which give the

required backlash B,

l(p -B) + at 2 p p (12.92)

1(Pp-B) - atp (12.93)

As before, atp is a free parameter that is chosen by the designer, and can be used to achieve balanced strength, or for

a number of other purposes. Once the tooth thickness values are chosen, the dedendum

of each gear is fixed. On the internal gear, its value can be

Geometric Design of an Internal Gear Pair 295

found from the following sequence of equations, which are

explained below,

t 2 ts2 R [~- 2(inv4>p- inv 4>s)] (12.94)

s2 Rp2

. 4>c inv 4>s 1 (12.95) lnv p2 + ---c-(ps-t s2-t sc )

2C s2

4>c c (12.96) 2 4>P2

CC Rb2-Rbc

( 12.97) 2 cos 4>~

t:J.Cc c c (12.98) 2 C2 - Cs2

bs2 a sc + t:J.Cc 2 (12.99)

bP2 bs2 - RP2 + Rs2 (12.100)

The relation between ts2 and tP2 was given by Equation (12.18). We then use Equations (12.56 and 12.57) to

find 4>~, the cutting pressure angle required to give the tooth thickness t s2 ' Equations (12.97 and 12.98) give the cutting center distance and the corresponding cutter offset, and the last two equations give the dedendum of the internal gear.

The pinion dedendum can be found using the equations in

Chapter 6. It is not of course necessary to use the same cutter to cut the pinion, and in fact it is preferable to use a hob, because there is then less likelihood of interference at the tooth fillets of the pinion. In any case, whether the plnlon is cut by a hob or by a pinion cutter, we can use the equations in Chapter 6 to calculate the pinion dedendum bp1 '

We now have to choose the addendum values, or in other

words, the radii of the tip circles RT1 and RT2 • In the case of an external gear pair, the values of ap1 and a p2 were

chosen so that we obtained a working depth of 2.0m, and equal

clearances at each root circle. If this procedure is followed

in the design of an internal gear pair, we find that there is

generally interference at the tooth fillets of the pinion.

It is not necessary, however, to maintain a working

depth of '2.0m in an internal gear pair. The path of contact for an internal gear pair is generally longer than that of an

296 Internal Gears

external gear pair with the same working depth. It is this

property which causes the interference, but it also means that the contact ratio of the internal gear pair is larger. We

can therefore reduce the working depth of the internal gear

pair, and still maintain an adaquate contact ratio. With this consideration in mind, we design the internal

gear wi th the largest possible addendum which causes no interference, and we choose the addendum of the pinion to give a clearance of 0.25m at the root circle of the internal gear.

The pinion fillet circle radius Rf1 is known, and in order to avoid interference, the limit circle radius RL1 must exceed Rf1 by at least 0.025m. If the internal gear addendum is to have the maximum possible value, the difference between RL1

and Rf1 will be exactly 0.025m,

Rf1 + 0.025m (12.101)

The tip circle radius RT2 of the internal gear is then found from Equation (12.32),

(12.102)

Occasionally, when RT2 is calculated by the procedure just described, the clearance at the root circle of the pinion is less than the recommended minimum of 0.25m. In such cases, it is best to increase the value of RT2 , until the clearance is equal to 0.25m. The radius of the tip circle is then given by

the following equation,

RP2 - bP1 + 0.25m (12.103)

The addendum of the pinion is chosen to give a clearance

of 0.25m at the root circle of the internal gear,

bP2 - 0.25 m (12.104)

The pinion is therefore designed with a tip circle radius of

the following value,

(12.105)

Measurement of Tooth Thickness 297

When a gear pair is designed according to the procedure

just described, the working depth is usually less than 2.0m,

but the contact ratio is still generally quite adaquate, with

typical values being greater than 1.5. The clearance at the

pinion root circle is often very large, but this is necessary

to prevent interference, and causes no problems. In some

cases there may be interference at the tooth fillets of the

internal gear, but this can almost always be avoided by

increasing the tooth thickness of the pinion, and reducing

that of the internal gear.

Measurement of Tooth Thickness

Of the three methods described in Chapter 8 for the tooth

thickness measurement of external gears, the most practical

method for use with internal gears is the measurement between

pins. Since the proof is almost identical to that in

Chapter 8, the equations for finding the inspection measurement M are presented here without further proof.

Two pins of radius r are inserted into opposite tooth

spaces of the internal gear, and the distance M between the

pins is measured. The correct value of M, when ts is the tooth

thickness specified for the gear, is given by the next four

equations. If the pin centers lie at radius R', and 4>R' is the corresponding profile angle, the values of ¢R' and R' are found as follows,

R'

1L_L_~+ N Rb 2Rs

Rb

inv ¢s ( 12.106)

(12.107 )

The relation between M and R' is given by one or other of the

following equations, depending on whether the internal gear

has an even or an odd number of teeth.

M

M

2R' - 2r

2R' cos (90°) - 2r N

(12.108 )

(12.109)

298 Internal Gears

Numerical Examples

Example 12. 1 An internal gear pair with D.P. 4 and pressure angle 20°

is to be designed for a center distance of 0.765 inches. Use the procedure described in Chapter 12 to spec ify the gear

pair, using the following information. The pinion has 35 teeth, and is to be cut by a hob with addendum 0.333 inches and tooth tip radius 0.107 inches. The internal gear has 41 teeth, and is to be cut by a pinion cutter with 20 teeth, tooth thickness 0.395 inches, and tip circle diameter 5.632 inches. In choosing the tooth thicknesses, assume a backlash of 0.013 inches, and use a value of 0.12 inches for the

quantity Atp' After the design is completed, calculate the working depth and the contact ratio, and check that there is

no tip interference.

Pd=4, ~s=200, N1=35, N2=41, C=0.765 a r =0.333, rrT=0.107, Nc=20, t sc =0.395, RTc =2.816

B=0.013, Atp=0.12

m = 0.2500 inches

Rsl = 4.3750 Rs2 = 5.1250 Ps = 0.7854

Rbl = 4.1112 Rb2 = 4.8159 Pb = 0.7380 ~ = 22.888°

Rpl = 4.4625

Rp2 = 5.2275

Pp = 0.8011 ~ = 22.888°

P

0.1200

0.5141 0.2741

(12.21)

(12.5)

(12.6) (12.23)

(12.22)

(12.92) (12.93)

First, we find the tooth thickness and the fillet circle

radius of the pinion.

Examples

tsl = 0.5722 e l = 0.2466

h = 0.2626

Rfl = 4.3592

299

(6.47)

(6.48)

(5.40) (5.48)

Next, we calculate the tooth thickness of the internal gear,

and hence the radii of the tip circles in both gears.

ts2 = 0.1888 inches

Rsc 2.5000

Rbc 2.3492

2.6250

inv 4>~2 = 0.053306

c C2 = 2.8461

JJ.cc = 0.2211 2

a sc = RTc - Rsc = 0.3160 bs2 0.5371

bp2 = 0.4346

RLl = 4.3654 RT2 = 5.1294 inches

apl = 0.3721

RTl = 4.8346 inches

(12.94)

(12.54)

(12.95)

(2.16,2.17)

(12.96)

(12.97)

(12.98)

(12.99) (12.100)

(12.101) (12.102) (12.104)

(12.105)

Finally, we calculate the working depth and the contact

ratio, and check to see whether there is tip interference.

Working depth = a pl + a p2 = 0.4702 inches

mc = 1.458 (12.30)

300 Internal Gears

'T 1 = 31.749 ° ( 2. 18)

IITl = 0.015626 radians 0.895° (12.36)

9>T2 = 20. 134 ° ( 12. 11 )

IIT2 = - 0.018731 radians = - 1.073° (12.37)

fJ 1 = 70.563° = 1.231564 radians (12.40)

fJ 2 = 1.126716 radians = 64.556° (12.41)

112 = - 1.226° = - 0.021401 radians (12.42)

(12.43)

Since the clearance is greater than 0.05m, there is no tip

interference. It can be verified that, if the gear pair had

been designed with atp=o, there would have been tip

interference.

The gear pair designed in this example is shown in

Figure 12.18, except that in the diagram the amount of

backlash is doubled, to make it more easily visible.

Figure 12.18. The gear pair designed in Example 12.1.

Examples 301

Example 12.2 The cutter specified in Example 9.3 is to be used to cut

an internal gear with 35 teeth, a tooth thickness of 6.168 mm, and a tip circle whose diameter is 205.968 mm. Calculate the

cutting center distance, and check that there is no tip

interference during the cutting feed-in process. Determine

also the angle at which the cutter should be displaced during

the return strokes.

m=6, ~s=20°, Ng=35, t sg=6.168, RTg=102.984

Nc =16, t sc=10.735, RTc =56.7, r CT=1.5

Rsg 105.000 mm

Rsc 48.000 ps = 18.850

Rbg 98.668

Rbc = 45.105

CC = 57.000 s

. c lnv ~p = 0.031979

~c p 25.516°

"'c = 25.5160

CC = 59.351 mm

(12.54)

(12.56)

(2.16,2.17)

(12.57)

(12.58)

The cutting feed-in process starts when the cutting

center distance is equal to (RTg-RTC )' which is 46.284 mm. The check for tip interference should be made at several values of

the feed-in center distance Cf ' starting with 46.284 mm, and

ending with Cc . As an example, we will carry out the

calculations for the midpoint value of 52.818 mm.

Cf = 52.818 mm

~' = 37.297° Tc eTc = 0.015971 radians = 0.915°

~c 38.894° = 0.678828 radians ~g = 0.400081 radians = 22.923°

(12.82)

(12.83)

(12.86)

(12.87)

302 Internal Gears

6g - 2.283° = - 0.039848 radians (12.88)

'Tg = 16.647° (12.11) 6Tg = - 0.022928 radians (12.37)

Clearance = RTg(9Tg-6g) = 1.742 mm (12.89)

The clearance is greater than 0.02m, so we have shown that

there is no tip interference at this particular value of Cf • Similar calculations, using different values for Cf , would show that there is no tip interference throughout the cutting process.

The final calculations are made to check that there will be no rubbing during the return strokes of the cutter.

R~ = 55.200 mm

'hc = 36.454° Rhc = 56.078 a' 17.271°

,c = 25.5160

(5.32) (5.33) (5.34)

(12.90) (12.48)

The angle a' is several degrees smaller than the cutting pressure angle ,C, so the cutter can be displaced in a manner that prevents rubbing. The angle of the displacement should be between a' and ,C, and rather closer to a', so a value of 20° would be suitable.

Example 12.3 Determine the inspection length M for the measurement

between pins, when a 49-tooth internal gear with D.P. 8,

pressure angle 25° and tooth thickness 0.150 inches, is measured using pins of diameter 0.240 inches.

m 0.1250 inches

Rs 3.0625 Rb 2.7756

inv'R' = 0.026365

'R' = 24.004° R' = 3.0383

M = 5.8336 inches

(12.106) (2.16,2.17)

( 12.107)

(12.109)

PART 2

HELICAL GEARS

Chapter 13

Tooth Surface of a Helical Involute Gear

Introduction

The spur gears discussed in Part 1 of this book have one principal disadvantage. During part of each meshing cycle there is a single pair of teeth in contact, while during the remainder of the cycle there are two. For each tooth pair in contact, the length of the contact line is equal to the gear face-width F. Hence, the total length of the contact line is either F or 2F, depending on the number of pairs of teeth in contact. And each time a tooth pair comes into contact or loses contact, the total length of the contact line either increases or decreases by F. These large and abrupt changes in the length of the contact line result in noisy operation of the gears.

The gear shown in Figure 13.1 is called a stepped gear. It is formed by two spur gears side by side, each with a face-width of (F/2), so that the total face-width is F. When a tooth pair comes into contact or loses contact, the change in the total length of the contact line is now only (F/2), compared with F for a conventional pair of spur gears, and the resulting operation is smoother. We could improve the

performance further by increasing the number of gears in the

stepped gear, keeping the total face-width constant, and if

the number were increased to infinity, we would obtain a helical gear.

In this chapter, we will describe the geometry of a

helical gear, in the same manner as we did for a spur gear in

Chapter 2. The description of the tooth surface in a helical

gear involves several parameters, such as the profile angles

and the pitches, and we will first define these parameters and

306 Tooth Surface of a Helical Involute Gear

Figure 13.1. A stepped gear.

then develop relations between them. The purpos~ of these relations will become clear in the remaining chapters, where we di scuss the meshing, the cutt ing, and the tooth strength of helical gears.

A Note on the Use of Vectors

There were a number of sections in the first part of this book where vectors were employed to help in the explanation of some aspects of spur gear geometry. However, the use of vectors was not very widespread, because spur gear geometry is essentially two-dimensional, and vector theory is not often required. On the other hand, the geometry of a helical gear is three-dimensional, and vectors can be very helpful in clarifying the various proofs and explanations.

Figure 13.2 shows a typical spur gear and rack, with two sets of unit vectors included in the drawing. One set, n x ' ny and n z ' are fixed in the spur gear, with nz in the direction of the gear axis. When the gear rotates about its axis, the direction of n z will not be affected, but the directions of nx

A Note on the Use of Vectors 307

1/ Y

nz X

nn n y n~ 0 ~n, Lnt

{

Figure 13.2. Directions of the unit vectors.

and ny will obviously change. The other set, n~, n~ and n S' are said to be fixed in space, which simply means that their

directions do not change. The unit vector n~ is chosen perpendicular to the plane formed by the tips of the rack teeth, in the direction from the gear towards the rack. The

remaining two unit vectors are perpendicular to n~, with nS parallel to the lines of the rack teeth, and n~ at right angles to them. When a spur gear is meshed with a rack, the axis of the gear is parallel with the teeth of the rack, so the two unit vectors nz and nS are parallel. It should be noted that a unit vector is used only to specify a direction. Since the rack does not rotate, it makes no difference whether we regard a unit vector as fixed in space, or as fixed in the

rack. Hence, the three vectors n~, n~ and n S' which we originally described as fixed in space, can equally well be

thought of as fixed in the rack.

The two vectors nz and nS are shown in Figure 13.2 by counter-clockwise circular arrows, and these are used in

conformi ty with the right-hand rule from the theory of

vectors. According to this rule, a vector can correspond to a

sense of rotation, using the following convention. The right hand is held so that the fingers point in the required sense


of rotation, and the thumb then points in the direction of the

vector. The circular arrows in Figure 13.2 therefore indicate that the vectors point in the direction upwards out of the

drawing. The spur gear and the rack are being viewed in the negative nS direction. A clockwise circular arrow would mean that the vector points downwards into the drawing. It will be the general pract ice, throughout Part 2 of thi s book, to

include in each diagram a unit vector represented by a circular arrow, in order to specify the direction of the view being taken.

The spur gear and rack of Figure 13.2 are shown again in

Figure 13.3, viewed this time in the direction perpendicular to the plane of the rack, as indicated by the circular arrow ,--, representing the unit vector n~. In Figure 13.4 we show a helical gear meshed with a rack, and we now have to choose the directions of the set of fixed unit vectors. They are shown in Figure 13.4 in the same directions, relative to the rack teeth, as the three fixed vectors in Figure 13.3. In other words, n~ is again perpendicular to the plane of the rack, and nS remains parallel with the rack teeth. This turns out to be the most convenient choice for the geometric analysis, even though the shapes of the two racks in Figures 13.3 and 13.4 are quite different. Since the axis of a helical gear meshed with a rack is not parallel with the teeth of the rack, the unit vectors n z and nS in Figure 13.4 are no longer parallel.

I I nTJ

0 ) ()

I I Figure 13.3. A spur gear and rack.

The Basic Helical Rack 309

Figure 13.4. A helical gear and rack.

The Basic Helical Rack

Figure 13.5 shows the basic helical rack, used to define the tooth surface of a helical gear. Just as the basic rack of a spur gear has teeth which are straight-sided, the basic rack

in Figure 13.5 has teeth whose faces are flat planes. The angle between the gear axis and the direction of the rack

teeth is shown as ~r' and it is called the basic rack helix angle. A plane cut through the rack perpendicular to the gear axis is known as a transverse section of the rack, and a plane

cut perpendicular to the rack teeth, in other words

perpendicular to n S' is called a normal section. The diagram also shows a transverse section and a normal

section through the basic rack. The distances in the two

sections between corresponding points of adjacent teeth are

called the transverse rack pitch Ptr and the normal rack

pitch Pnr. It can be seen from triangle A1A2A3 that there is a relation between the two pitches,

(13.1)

We define a transverse module mt and a normal module mn in terms of the pi tches of the basic rack,


\-Reference line , I ,

'LDirection. of , gear aXIs

----

--.~ f\ nz n /: Transverse ~ section

Figure 13.5. The helical basic rack.

(13.2)

(13.3)

and the modules are then related in the same manner as the pitches,

(13.4)

The transverse diametral pitch Ptd and the normal

diametral pitch Pnd are defined as the reciprocals of the two

modules,

(13.5)

(13.6)

The pressure angles shown in the transverse and normal

Independent Parameters of the Basic Rack 311

sections in Figure 13.5 are called the transverse rack

pressure angle ~tr and the normal rack pressure angle ~nr.

They can be expressed in terms of the tooth dimensions as follows,

tan ~tr ht H

tan ~nr hn H

where H is the tooth depth, and ht and hn are the lengths shown in Figure 13.5 in the transverse and normal tooth sections. These two lengths are related as follows,

and from the last three equations, we obtain a relation between the two pressure angles and the helix angle,

tan ~tr cos "'r (13.7)

The rack base pitches in the two sections are defined as the distances between adjacent tooth profiles, measured in each case along the common normal. The transverse base pitch

Ptbr and the normal base pitch Pnbr are shown in Figure 13.5, and are related to the rack pitches Ptr and Pnr in the following manner,

(13.8)

(13.9)

Independent Parameters of the Basic Rack

We have specified the basic rack by means of the

following seven quantities: Ptr' Pnr' Ptbr' Pnbr' ~tr' ~nr and "'r. However, we have shown that there are four relations

between the quantities, given by Equations (13.1, 13.7, 13.8

and 13.9). It is clear that only three of the quantities used

to specify the basic rack are independent. We will choose to


regard Pnr' ~nr and Wr as the three independent parameters, and we now repeat the relations in a form suitable for calculating the remaining four quantities.

tan ~tr

Basic Rack Reference Plane

Pnr cos Wr

tan ~nr cos Wr

(13.10)

(13.11)

(13.12)

(13.13)

In Chapter 2, when we discussed the basic rack profile, we defined the rack reference line as the line along which the tooth thickness and the space width are equal. In the context of a helical rack, the reference line would be a line in the transverse section, as shown in Figure 13.5. If we construct the reference lines in a number of different transverse sections, the lines will all lie in a plane perpendicular to n E, and this plane is known as the rack reference plane, or sometimes the datum plane. Because the tooth profile of an involute rack is straight-sided in the normal as well as the transverse section, a normal section will also intersect the reference plane in a line along which the tooth thickness and the space width are equal. Hence, the rack reference plane can

be described as the plane at which the tooth thickness is equal to the space width in both the transverse and the normal

sections.

pitch Cylinders of a Helical Gear Pair

A plane cut through a helical gear perpendicular to its

axis is called a transverse section of the gear. Suppose we

have a pair of helical gears with parallel axes, and we consider any transverse section through the two gears. The

tooth profiles in this section must be shaped so that the Law

Standard pi tch Cylinder of a Helical Gear 313

of Gearing is satisfied, and we can therefore apply the results proved in Chapter 1 for a pair of spur gears. There must be a pitch point dividing the line of centres in the ratio N1:N2 , and in each gear the pitch circle is the circle

which passes through the pi tch point. We can construct the pitch circles at any number of

transverse sections through the pair of gears, and in every case the ratio of the two pitch circle radii is the same. Hence, in each gear the pitch circles at the different transverse sections are all the same size, so that they form a cylinder, and this is called the pitch cylinder of the gear. We can therefore describe the pitch cylinders of a helical

gear pair as the cylinders with radii RP1 and Rp2 ' where the values of these radii are found by applying the theory of spur gear geometry to any transverse section through the gear

pair. A similar description can be given for the case of a

helical gear and a rack. In this chapter we will consider only the geometry of a gear meshed with its basic rack, and in the following chapter we will discuss the geometry of a gear meshed with an ordinary rack, and that of a pair of gears.

Standard pi tch Cylinder of a Heli cal Gear

We now study the geometry of a gear with N teeth, whose tooth shape is defined as being conjugate to the basic rack in Figure 13.5. If we consider a single transverse plane through both the gear and the basic rack, as shown in Figure 13.6, the tooth profile of the gear must be conjugate to that of the rack. Hence, the gear tooth profile in the transverse plane

can be found by means of the spur gear geometry described in

Chapter 2. The profile is therefore an involute defined by a

basic rack with pitch Ptr and pressure angle ~tr' The radii of the standard pitch circle and the base circle are then given

by Equations (2.27 and 2.7),

(13.14)

( 13.15)


rjJtr

I Rack pitch plane

R r-Rack reference plane s

Standard pitch cylinder of gear

Figure 13.6. Transverse section through gear and basic rack.

It is clear that at every transverse section of the gear we obtain a standard pitch circle, each with the same radius. The cylinder containing all these circles, in other words the cylinder of radius Rs' is the pitch cylinder of the gear when it is meshed with its basic rack, and it is called the standard pitch cylinder. It is used as a reference cylinder, in exactly the same manner as the standard pitch circle of a

spur gear. In particular, many of the quantities which define

the shape of the teeth, such as the pressure angles and the

tooth thicknesses, are specified by their values on the

standard pi tch cylinder.

The radius of the standard pitch cylinder was given by

Equation (13.14) in terms of the transverse pitch Ptr of the basic rack, and in Equation (13.2) the transverse module mt was defined as Ptr divided by ~. We combine these equations, in order to express the standard pitch cylinder radius

directly in terms of the transverse module,

(13.16)

Basic Rack pi tch Plane 315

Since the profile of a helical gear in the transverse section is identical with that of a spur gear, the quantities

defined in the transverse section, such as Ptr' mt , and ~tr' are exactly equivalent to the corresponding quantities Pr' m and ~ defined in connection with spur gears. For this reason, r all the geometric relations which were derived in Part 1 for a spur gear can be applied immediately to the transverse section of a helical gear.

Basic Rack pitch Plane

In Figure 13.6, which shows a transverse section through the gear and the basic rack, the rack pitch line lies parallel to the rack reference line, and touches the standard pitch circle of the gear. I f we were to consider a number of transverse sections, the rack pitch lines from the different sections would all lie in one plane, and this plane is called the pitch plane of the basic rack. The pitch plane can therefore be defined as the plane of the basic rack which is parallel to its reference plane, and which touches the standard pitch cylinder of the gear. In other words, it is the plane of the basic rack which lies at a distance Rs from the gear axis.

y

Figure 13.7. Coordinate systems fixed in the gear.


The Tooth Surface of a Helical Gear

In order to describe the tooth surface of a helical gear,

we use a set of Cartesian coordinates (x,y,z) fixed in the gear, with the z axis coinciding with the axis of the gear. This is the same system of coordinates that was used for spur gears in Part 1 of this book. In addition, we will use the cylindrical coordinates (R,e,z) shown in Figure 13.7, which are simply the polar coordinates used in Part 1, together with the axial coordinate z.

We now consider two transverse sections through the gear, one at plane z=O and the other at plane z. As we showed earlier, the tooth profile in any transverse section must always be conjugate to the corresponding transverse section through the basic rack, so the tooth profiles in the two sections through the gear must each have a standard pitch circle and a base circle with radii given by Equations (13.14 and 13.15). The only difference between the two transverse sections through the basic rack, as we can see from Figure 13.8, is that the rack tooth profile at plane z is

displaced upwards a distance (z tan "'r)' compared with that at plane z=O. The gear tooth profile in the transverse section at plane z is therefore identical with the profile at plane z=O,

~ -

~I ~~~~~I ~z ~n~ Section

z=o Section z

I. z .1

Figure 13.8. Transverse sections through the basic rack.

The Tooth Surface of a Helical Gear 317

except that it is rotated through a certain angle 1),,9, in order to mesh correctly with the rack tooth profile in its displaced

position at plane z. For a spur gear meshed with a rack, the rotation I),,~ of

the gear corresponding to a displacement I)"u r of the rack was given by Equation (3.24),

When the gear is meshed with its basic rack, the pitch circle coincides with the standard pitch circle, so the pitch circle radius Rp in the expression for I),,~ can be replaced by the

standard pitch circle radius Rs'

I)"u r Rs

( 13.17)

In our study of the tooth shape of a helical gear, we are not considering a ro~ation of the gear, but a rotation of the tooth profile as we move axially along the gear. We therefore replace I),,~ by the prof i Ie rota t i on 1),,9, and for I)"u we

r substitute the expression (z tan ~r)' which is the relative displacement between the rack tooth profiles in the two transverse sections. We then obtain the following expression for the angle 1),,9,

z tan 1),,9

Rs "'r

r ---n x

Figure 13.9. Helix through point AO'

(13.18 )


The Helix and the Involute Helicoid

A helix is a spatial curve which can be defined in the

following manner. If a rigid bar CA, as shown in Figure 13.9, moves so that one end C travels along a fixed line, while the bar remains perpendicular to the line and rotates through an angle proportional to the distance travelled by C, the path followed by the other end A is a helix. The motion can be described by the following equation,

kz (13.19)

where OA is the angle the bar makes with a fixed direction, OAO is the initial angle, z is the distance travelled by point C, and k is a constant.

The value of z at which the bar has made a complete revolution is known as the helix lead L. The length of the lead can be determined from Equation (13.19),

kL (13.20)

If we now choose any curve in the plane z=O, and through every point of the curve we construct a helix, each with the same lead L, we obtain a surface called a helicoid. A section through the helicoid surface at plane z is identical with the section at plane z=O, except that the entire shape is rotated through an angle tJ.O equal to kz.

When we described the gear tooth surface earlier, we showed that the tooth profiles in any two transverse sections

are the same, with one profile rotated relative to the otHer

by an angle proportional to the distance between the

sections. With this description, it is evident that the tooth surface is a helicoid, and since the profile in the transverse

section is an involute, the surface is called an involute

helicoid. We can regard the surface as formed by a family of helices, each helix lying in a cylinder coaxial with the gear,

and each with the same lead L. Hence, the intersection between any coaxial cylinder and the tooth surface is a helix with

lead L. In Equation (13.19), we introduced a constant k in the


relation describing a helix. The value of k for the helical gear can be obtained from a comparison of Equations (13.18

and 13.19),

tan I/J r Rs

k (13.21)

Expressions ,for the angular difference (eA_eAO ) and the lead

L of a helix were given in Equations (13.19 and 13.20). We now substitute for k from Equation (13.21), and we obtain the corresponding expressions for the helical gear,

eA _ AO tan I/J r e Rs

z (13.22)

L 211'Rs

tan I/J r (13.23)

A helical gear is called right-handed or left-handed, depending on the direction in which the helix rotates. The definition of a right-handed gear is essentially the same as the definition given earlier for the right-hand rule in the theory of vectors. The right hand is held with the thumb pointing along the gear axis, and we consider a movement along the gear in the direction of the thumb. If the helices formed by the gear teeth rotate in the direction of the fingers of the right hand, the gear is said to be right-handed, while if they rotate in the opposite direction, the gear is left-handed. For example, the gear shown in Figure 13.4 is a right-handed gear.

The definition just given would imply that, for a right-handed gear, the coordinate angle e would increase when we move along a helix in the direction of increasing z, while for a left-handed gear the value of e would decrease. In order

that Equation (13.22) should remain valid for both right and

left-handed gears, we will adopt the convention that the

helix angle I/J r of the basic rack is positive when the gears

conjugate to the basic rack are right-handed, and I/J is r negative when the gears are left-handed. For consistency with

this sign convention, the value of the lead L given by

Equation (13:23) will als~ be positive or negative, depending

on whether the gear is right or left-handed. If a cylinder is unrolled so that it lies in a plane, it


Transverse section Transverse

z=Q section z

-----.----._11--. ----'Z=---------.j·1

v~ Ao l-;n, 27TR

A-

Developed helix

Figure 13.10. Developed cylinder of radius R and length z.

is said to be developed. We will now prove that any helix lying in the cylinder becomes a straight line when the cylinder is developed. Figure 13.10 shows a cylinder of radius R, developed into a rectangle. AO is any point on the cylinder at plane z=O, and A is the point at plane z on a helix through AO. We introduce a symbol V to represent the depth of any point below the top edge of the rectangle, and the vertical difference ~V between points AO and A is then equal to R times the angular coordinate difference,

~V (13.24)

We now use Equation (13.19) to express the vertical

difference in the following form,

~V Rkz (13.25)

Since the ratio (~V/z) is constant, all points on the

helix through AO must lie on a straight line. The angle WR between this line and the z axis is called the helix angle of the helix through AO and A, and it can be expressed as

follows,

tan WR Rk (13.26)


When we are considering helices associated with a helical gear, we are interested primarily in those which have the same lead L as the gear, given by Equation (13.23). For example, as we pointed out earlier, the intersection of a tooth surface with any cylinder coaxial with the gear is a

helix with lead L. When it is important to distinguish between helices with lead L, and those with leads of different values, we will refer to the former as gear helices. Thus, although there are any number of helices passing through a typical point A of the gear, there is only one gear helix through A.

For any gear helix, the helix angle can be found from Equation (13.26), if we substitute the value of k given by Equation (13.21),

(13.27)

Since all gear helices lying in the cylinder of radius R have the same helix angle, the angle ~R given by Equation (13.27) is called the helix angle of the gear at radius R. The helix angle at the standard pitch cylinder is called simply the helix angle of the gear, and it is represented by the

symbol ~s' Its value can be found by setting R equal to Rs in Equation (13.27), and it is clear when we do so that the gear helix angle 'is equal to the helix angle of the basic rack,

(13.28)

The relation between the normal module mn and the transverse module mt was given by Equation (13.4) in terms of the basic rack helix angle ~r'

Since we have now proved that the gear helix angle ~ is equal s to the basic rack helix angle, the relation between the

modules can be expressed in terms of the gear helix angle,

(13.29)

The radius of the standard pitch cylinder was given by


Equation (13.16) in terms of the transverse module,

For many gears, the specification contains the value of the

normal module, but not that of the transverse module, so it is

convenient to combine Equations (13.16 and 13.29), in order

to express the standard pitch cylinder radius in terms of the normal module,

(13.30)

An alternative expression for tan "'R can be found by combining Equations (13.23 and 13.27),

tan "'R (13.31)

If we develop a cylinder of radius R and length equal to the gear lead L, we obtain a rectangle with sides 2~R and L, as

shown in Figure 13.11. It can be seen from Equation (13.31)

that the diagonal of this rectangle makes an angle "'R wi th the z axis. We have also shown that gear helices become straight

lines in the developed cylinder, making the same angle "'R with the z axis. Hence, all gear helices, such as intersections

between the cylinder and the tooth surfaces, will appear in the developed cylinder of length L as lines parallel with the diagonal of the rectangle.

...:::-_--.-:--_________ --,' 1_ n z

L

27TR

Developed helix

Figure 13.11. Developed cylinder of radius R and length L.

Tooth Surface of a Helical Involute Gear 323

Base Cylinder

We showed in Figure 13.6 that, at any transverse section of the gear, we can construct a base circle with radius Rb given by Equation (13.15). The cylinder of radius Rb , coaxial

with the gear, is therefore called the base cylinder. The

helix angle at radius Rb is known as the base helix angle ~b' and is found by setting R equal to Rb in Equation (13.27),

tan ~b Rb tan ~r

(13.32)

Equation (13.27) can be rearranged into the following form,

(13.33)

and since the right-hand side of this equation is constant, it

proves that the quantity (tan ~R/R) is independent of the radius R. By substituting Rb and Rs in turn for R, we obtain relations between the helix angles at the various cylinders of the gear,

(13.34)

If A is any point on a tooth surface at radius R, we construct the gear helix through A, and the point where this helix cuts plane z=O is labelled AO' We now use Equations (13.22 and 13.32) to obtain a new relation, which will be used later in this chapter, between the angular difference (eA_eAO ) and the z coordinate of point A,

(13.35)

Tooth pi tches

The circular pitch of a spur gear at radius R was defined

in Chapter 1 as the distance between corresponding points of

adjacent teeth, measured around the circumference of the circle of radius R. When we described the tooth shape of a


spur gear in Chapter 2, we introduced the circular pitch at the standard pitch circle and at the base circle, and the

operating circular pitch was defined in Chapter 3, when we

discussed the meshing of a gear pair. We follow the same

pattern in the description of a helical gear, except that in this case there is more than one type of tooth pitch. We first

define the tooth pitches at a cylinder of arbitrary radius R, and then we describe the corresponding quanti ties at the standard pitch cylinder and at the base cylinder. The operating pitches will not be introduced until Chapter '4, where we discuss the meshing of helical gears.

The tooth pitches at any radius R are defined as lengths on the developed cylinder of radius R, which is shown in Figure 13.12. The cylinder is drawn wi th a length equal to the gear lead L, so that the teeth appear as lines parallel to the diagonal, making an angle "'R with the z direction, and in this length each tooth makes exactly one revolution round the gear. A typical tooth is shown as a broken line such as

T,T2T3T4 • A transverse section through the cylinder appears on the developed surface as a vertical line with a constant value of z. Hence, if the gear has N teeth, there must be N teeth cutting any vertical line through the rectangle, and since the lines representing the teeth are parallel with the diagonal, there will also be N teeth cutting any axial line.

The axial pitch Pz is defined as the distance between adjacent teeth, measured in the axial direction. Since any axial line in Figure 13.12 is crossed by N teeth, the axial

L

2rrR l--~Z Direction of

gear axis

Figure 13.12. Relation between the pitches.

Tooth Pi tches 325

pi tch is equal to the lead di vided by the number of teeth,

~ N

(13.36)

The axial pitch is independent of the radius R, as we can see from Equation (13.36). It is also clear, from the same

equation, that the axial pitch Pz must obey the same sign

convention as the lead L. In other words, Pz is positive for right-handed gears, and negative for left-handed.

It should perhaps be pointed out that in North America

the standard symbol for the axial pitch is Px' The use of Pz was chosen here, because with the coordinate system used in this book, the axial pitch is measured in the z direction.

The transverse pitch PtR at radius R is defined as the distance between adjacent teeth, measured in the transverse direction. Its value can be read from Figure 13.12,

211'R N

(13.37)

From this definition, it is clear that the transverse pitch of

a helical gear at any radius is identical with the corresponding circular pi tch of a spur gear.

Next, the normal pitch PnR at radius R is defined as the distance between adjacent teeth, measured along a line perpendicular to the teeth in the developed cylinder, as shown in Figure 13.12. We can use the diagram to derive relations between the various pitches,

PnR PtR cos "'R (13.38)

Pz PnR (13.39)

sin "'R

The transverse pitch and the normal pitch at the standard pitch cylinder are generally called the transverse

pitch and the normal pitch of the gear. They are represented

by the symbols Pts and Pns' and their values can be obtained if we replace R by Rs in Equations (13.37 and 13.38),

211'Rs N (13.40)


Pts cos "'s (13.41)

We can also write down the relation between the axial pitch and the normal pitch at the standard pitch cylinder, corresponding to Equation (13.39),

(13.42)

The axial pitch in this equation is still represented by the same symbol Pz' since it is independent of the radius.

Lastly, we introduce the transverse base pitch Ptb and the normal base pitch Pnb' defined on the developed base cylinder, with similar relations existing between them and the axial pitch.

Ptb 211'Rb

(13.43) N

Pnb Ptb cos "'b (13.44)

Pz Pnb (13.45)

sin "'b

Unit Vectors Associated with the Gear Helix at A

The point at radius R on the tooth profile in the transverse section at plane z=O is labelled AO' as shown in Figure 13.13, and a typical point on the gear helix through AO is labelled A. We will now derive expressions for a set of unit vectors associated with the gear helix at A. These vectors are in the directions of the tangent, the principal normal, and the binormal to the helix. The first step is to calculate the position vector from the coordinate origin to point A, whose position is determined by the position of AO' and the value of the angular coordinate eA. The x and y coordinates of A can be written down immediately in terms of R and eA, and since A lies on the gear helix through AO' the z coordinate is given by Equation (13.35). Hence, if Co is the coordinate origin, the position vector from Co to A is given by the following expression,

Unit Vectors Associated with the Gear Helix at A 327

ny

n~ t ! n~ Cylinder of radius R I I I I \ \

z

Figure 13.13. Unit vectors at point A.

A A Rb A AO R cos e nx + R sin e ny + tan ..pb (e -e )nz (13.46)

To obtain a vector parallel to the helix tangent at A, we differentiate the position vector with respect to the

variable eA,

eAn eAn + Rb - R sin x + R cos y tan..pb n z (13.47)

We then divide this vector by its length, to obtain a unit vector nA in the direction of the helix tangent at A,

/.I

A eAn + Rb ----'--::::--{ - R sin e n + R cos n ) R2 x y tan..pb z

v[R2+ b ] tan 2..pb (13.48)

An alternative expression for the helix angle of the

gear at radius R can be found by rearranging Equation (13.34),

(13.49)

and from the triangle shown in Figure 13.14, we can write down

expressions for the sine and cosine of ..pR' which can be used

to simpiify Equation (13.48). We then obtain the final expression for n~,


R

Figure 13.14. Triangle to determine cos tPR and sin tPR•

We now differentiate n~ to obtain a vector in the direction of the principal normal to the helix at A.

dnA --E. d9A

(13.51)

This vector is in the direction from A towards point C, the centre of the transverse section through A. For the purpose of describing the tooth surface geometry, it is rather more convenient to define a uni t vector n~ in the opposi te direction, in other words from C towards A. Hence, to obtain n~, we change the sign of the vector in Equation (13.51), and divide by its length.

(13.52)

The third unit vector n~ of the of the orthogonal set is

found simply by forming the vector product (n~ x n~) ,

Equations (13.50, 13.52 and 13.53) give the directions

of the three unit vectors n~, n~, and n~, defined on the gear helix at a typical point A of the tooth surface. The lines

through A in these three directions are known as the tangent,

the principal normal and the binormal to the helix at A.


Transverse Profile Angle at Radius R

The transverse profile angle of a helical gear is defined in exactly the same manner as the profile angle of a

spur gear. Figure 13.15 shows the tooth profile in a

transverse section, and the point where it intersects the

cylinder of radius R is labelled A. The transverse profile angle at radius R is defined as the angle between CA and the profile tangent at A, as shown in Figure 13.15, and it is

represented by the symbol ~tR' Since line CE is parallel to the profile tangent at A, the angle ECA is also equal to ~tR' Two relations can now be read from Figure 13.15, each of them equivalent to the corresponding relations for spur gears.

cos ~tR Rb R

EA Rb tan ~tR

(13.54)

(13.55)

Since the tooth profile in the transverse section is an involute, we can make use of the involute property given by Equa t ion (2. 8) ,

arc EB EA

Base cylinder

(13.56)

Transverse profile

angle <PtA

Figure 13.15. Transverse section through point A.


The angle ECB can be found by combining Equations (13.55 and 13.56),

angle ECB arc EB Rb

tan I/l tR

and angle ACB can then be read from Figure 13.15,

angle ACB

(13.57)

(13.58)

Lastly, Equation (13.54) can be combined with Equation (13.34) to give one of the fundamental equations relating the angles in a helical gear,

Transverse Pressure Angle

tan "'b tan "'R

(13.59)

The transverse profile angle at the standard pitch cylinder is called the transverse pressure angle I/lts of the gear. We can find its value from Equation (13.54), if we substitute Rs in place of R,

(13.60)

ieCYlinder

Figure 13.16. Transverse section at plane z.

The Generator Through Point A 331

In Equation (13.15) we gave the base cylinder radius of

the gear in terms of the standard pitch cylinder radius and

the transverse pressure angle "'tr of the basic rack,

( 13.61 )

When we compare Equations (13.60 and 13.61), it is evident

that the transverse pressure angle of the gear is equal to the

basic rack transverse pressure angle,

(13.62)

The Generator Through Point A

The transverse section at plane z of a helical gear is shown in Figure 13.16, and as always in a transverse section, the tooth profile is an involute. The normal to the involute

at a typical point A touches the base circle at E, and the involute meets the base circle at B.

Figure 13.17 shows the transverse section through the same tooth at plane z=O, and in this section the involute

meets the base circle at BO. The two points Band BO both lie

on the curve forming the intersection of the base cylinder

Base cylinder

Figure 13.17. Transverse section at plane z=O.


with the tooth surface. Since this curve is a gear helix, BO

can be identified as the point where the gear helix through B cuts the plane z=O. The profiles in the two sections are identical, except that the profile at plane z is rotated

relative to the other by the angle tJ.8 given by Equation (13.18),

(13.63)

In Figure 13.16, B' is the point where the axial line

through BO cuts plane z, so that line CB' is parallel to CaBo in Figure 13.17. B'CB is the angle through which the tooth profile has rotated between the two transverse sections, and is therefore equal to tJ.8. In Figure 13.17, E' is the point where the axial line through E cuts plane z=O, and A' is the point where the base circle tangent at E' cuts the tooth profile. We now use Equation (13.56) to derive an expression for the di fference between the lengths EA and E' A' •

EA - E'A' arcEB - arcE'BO arc EB - arc EB'

EA - E'A' arc B' B (13.64)

The expression for tJ.8 in Equation (13.63) can be put into a different form by means of Equation (13.32),

(13.65)

and we substitute this expression into Equation (13.64).

EA - E'A' (13.66)

A three-dimensional view of the base cylinder is shown

in Figure 13.18, and the lines EA and E'A' are drawn in on the

diagram. In view of Equation (13.66), it is clear that the

line AA' makes an angle ~b with the gear axis. In addition, every point on this line is a point on the gear tooth surface,

since Equation (13.66) is satisfied at every point along the

line. When a curved surface contains a family of straight lines, these lines are called generators of the surface, so

The Generator Through Point A

G

Transverse plane z=o

Transverse plane z

Figure 13.18. The generator through point A.

333

the line AA' is known as the tooth surface generator through point A.

Since lines EA and E'A' are both tangent to the base circles, the four points A, E, A', and E' lie in a plane which touches the base cylinder along line EE'. Hence, the generator through A must touch the base cylinder at a point on line EE', as shown in Figure 13.18, and this point is labelled G.

It was stated in Chapter 2 that an involute can be thought of as the curve traced out by the end of a cable, when the cable is unwrapped from the base circle. In Figure 13.18, lines E'A' and EA would represent the cables corresponding to

the transverse sections at plane z=Q and plane z. Since every

point of line A'A lies on the tooth surface, it is possible to

think of the plane E'A'AE as forming part of a flexible sheet,

unwrapping from the base cylinder. We can therefore represent

an involute helicoid in the following manner. We consider a

flexible sheet, wrapped round the base cylinder, with its end

cut off so that it makes an angle ~b with the gear axis. The involute helicoid is the surface swept out by the end of the sheet, when it is unwrapped from the base cylinder.


This analogy can be used to prove one of the important properties of the tooth surface. Any plane which is tangent to the base cylinder cuts the tooth surface along a straight

line, which is one of the generators of the surface. To prove this statement, we consider the flexible sheet unwrapped to

exactly the position where it coincides with the tangent plane, and we know that in this position, or any other, the

end of the flexible sheet coincides with a generator of the surface. The tangent plane must therefore intersect the tooth surface along this generator.

Coordinates of Point G

For any point A of the tooth surface, there is a generator passing through the point, and this generator touches the base cylinder at some point G. Very often, we will want to find the position of G corresponding to a particular point A, and we can do this most conveniently by deriving expressions for the cylindrical coordinates (RG,eG,zG), in

terms of the coordinates (R,eA,Z) of point A. Since G lies on the base cyl inder, the radi us RG is equal

to the base cylinder radius Rb • And since G lies on the axial line through E, the angular coordinates of G and A differ by ~tR' as we can see from Figure 13.15. Finflly, length EA in Figure 13.18 is equal to (Rb tan ~tR)' as we stated in Equation (13.55), and the length GE is equal to EA divided by

(tan ~b). Hence, the coordinates of point G are related to those of A in the following manner,

eG eA - ~tR

Properties of Point G

Rb tan ~tR z -

tan ~b

( 13.67)

(13.68)

(13.69)

There are two important properties associated with

point G, shown in Figure 13.18. First, point G lies on the

Properties of Point G 335

gear helix through BO and B. And secondly, the generator GA is also the helix tangent at G.

Before we prove the two properties, we will explain why they are significant. In the first place, if the generator through A coincides with the helix tangent at G, it is a very

simple matter to find its direction. In addition, we know that the intersection of the tooth surface with any coaxial cylinder is a gear helix, and we know that the tooth profile in the transverse section at plane z meets the base circle at point B. The intersection of the tooth surface with the base cylinder must therefore be the gear helix through B, and this curve is called the base helix of the tooth surface. If, as we have stated above, the generator through any point A of the tooth surface meets the base cylinder at a point G on the gear helix through B, then the entire tooth surface can be thought of as the surface swept out by the generator through G, while

G moves along the base helix. To prove the first property, we start from

Equation (13.35), which gave the angular coordinate difference (eA_eAO ) between two points AO and A, lying on the same gear helix at plane z=O and plane z,

(13.70)

This equation can be adapted to give the angular difference ~e between any pair of points on the same gear helix, when nei ther point lies in the plane z=O,

(13.71)

where ~z is the difference between the z coordinates of the

two points. We now determine whether points G and B satisfy this equation.

Since point E lies on the axial line through G, as shown

in Figure 13.18, the difference between the e coordinates of B

and G is equal to the angle between the radii through Band E.

This is the angle ECB, shown in Figure 13.15, and its value is given by Equation (13.57).

angle ECB tan qJtR (13.72)


Point B lies in the transverse plane z, and the axial

distance between Band G can therefore be found from Equation (13.69),

Rb tan 9>tR tan 1/Ib (13.73)

It is evident that the coordinate differences given by Equations (13.72 and 13.73) satisfy Equation (13.71). We have therefore proved that the same helix passes through Band G, or in other words, that G lies on the gear helix through B.

To prove the second property, we start by writing down the posi tion vector from the coordinate origin Co to point A,

(13.74)

We use Equations (13.54, 13.68 and 13.69) to express R, eA and z in terms of Rb , eG and zG,

and the expression is then simplified as follows,

(RbCoseGnx + RbSineGny + zG nz )

+ Rb tan 9>tR(-sin eGnx + cos eGn + 1 n) y tan 1/Ib z (13.75)

The first bracket in this equation is the position

vector from Co to G. The second bracket is closely related to

the unit vector nG, which gives the direction of the helix /J

tangent at G. The expression for n: is obtained from Equation (13.50), if we replace A and R by G and Rb •

The position vector from Co to A, given by Equation (13.75), can now be put in the following form,

(13.77)

Direction of the Normal to the Tooth Surface at A 337

This equation can be interpreted as stating that to get from

Co to A, we can first go to G, and then move along the helix tangent at G. Since we know that line GA is the generator through A, we have proved that the helix tangent at G coincides with the generator through G and A. We can therefore use Equation (13.76) to give the direction of the generator

through A. In moving from Co to A by the route just described, the distance that we would travel along the generator is equal to the coefficient of nG in Equation (13.77). This qu&ntity

/.I is of course equal to the length GA, as we can see in Figure 13.18.

Direction of the Normal to the Tooth Surface at A

The simplest method for finding the normal to the tooth surface at A is to find the directions of any two tangents through A. The two tangents define the tangent plane, and the direction of the normal must be perpendicular to both of them.

One line which touches the tooth surface at A is the generator through A, since the entire generator lies within the surface, and is therefore tangent to it. Hence, the normal to the tooth surface at A is perpendicular to the generator direction nG A second tangent direction can be seen in

/.I'

c

E (and G)

Lse cylinder

/\~n~ t )nz

Figure 13.19. Transverse section through point A.


Figure 13.19, which shows the transverse section through A. The profile normal at A touches the base circle at E, and the

profile tangent at A is therefore parallel to CEo Point G lies on the axial line through E, so in Figure 13.19, G would

appear exactly behind E. The vector n~ is therefore parallel to CE, and hence is also parallel to the profile tangent at A.

We introduce a unit vector n~ to indicate the direction normal to the tooth surface at A. We have shown that nA must n be perpendicular to both n: and n~, and it can therefore be expressed as follows,

n: x n~ (13.78)

The unit vector n~ is found from Equation (13.53), if we substitute G and Rb in place of A and R. We then obtain an expression for the unit vector normal to the tooth surface at A,

Normal Section at A

We define the normal section at any point A as the section through the gear perpendicular to the helix tangent at A, or in other words, perpendicular to n~. Generally, we will consider the normal section through a point A which lies on the tooth surface, but the definition remains valid whether A lies on the tooth surface or not.

Normal Profile Angle at Radius R

Figure 13.20 shows the normal section through point A on

the tooth surface at radius R. The shape of the tooth profile in a normal section is unknown at present, but we will

describe later in this chapter how the shape can be

calculated. If C is the centre of the transverse section

through A, as shown in Figure 13.19, the line CA also lies in

the normal section, since the unit vector n~ along CA is

Normal Profile Angle at Radius R 339

Normal at A

nA

_~ __ -=~~====~~~R~ __ --------~It~::~-;.;::-~ Normal C profile

angle

Tangent at A Line perpendicular to CA

Figure 13.20. Normal section through point A.

perpendicular to n~. Earlier, we def ined the transverse

profile angle ~tR as the angle between CA and the profile

tangent in the transverse section. We now define the normal

profile angle ~nR in a similar manner, as the angle between CA

and the profile tangent in the normal section.

The unit vector n~ in Figure 13.20 points in the

direction of the normal to the tooth surface at A, and is

therefore perpendicular to any line touching the surface

at A. One such line is the helix tangent at A, so the vector n~ must be perpendicular to nA, which means that it lies in the

JJ plane of the normal section. A second line touching the

surface at A is the profile tangent in the normal section, so

n~ is also perpendicular to this direction. The angle in

Figure 13.20 between the unit vectors n~ and n~ is equal

to ~nR' since one is perpendicular to the profile tangent, and

the other is perpendicular to line CA. We can use this result

to derive an expression for cos ~nR.

cos ~nR


cos 'nR cos "'b cos "'R cos(BA-BG) + sin "'b sin "'R (13.79, 13.53)

cos 'nR cos "'b cos "'R (13.68)

cos 'nR cos "'b cos "'R (13.59)

cos 'nR

cos 'tR

tan "'b tan "'R

sin "'b sin "'R

+ sin "'b sin "'R

+ sin "'b sin "'R

(13.80)

Since there are quite a number of steps in the development just presented, no explanation has been given for each step, but at each line the number of the equation used to justify the step has been written in brackets under the equals sign. The same procedure will be used again later, wherever it is warranted by the number of steps in a proof.

The angle between the vectors n~ and n~ in Figure 13.20 is ('1'/2 - 'nR)' so we can derive an expression for sin 'nR in a similar manner.

sin 'nR

sin 'nR cos "'b sin(BA-BG) (13.79, 13.52)

sin 'nR cos "'b sin 'tR (13.81) (13.68)

Finally, we combine Equations (13.80 and 13.81) to

obtain an expression for tan 'nR'

tan 'nR

tan 'nR

(13.82)


Normal Pressure Angle

The normal profile angle of a gear at its standard pitch cylinder is called the normal pressure angle of the gear, and

it is represented by the symbol ~ns. We can find its value

from Equation (13.82), if the angles ~tR and ~R are replaced

by If>ts and ~s' their values at the standard pitch cylinder.

tan <P ts cos ~s (13.83)

In Equation (13.7) we gave an expression for <P nr , the normal pressure angle of the basic rack,

(13.84)

We have shown in Equations (13.62 and 13.28) that the transverse pressure angle and the helix angle of the gear are equal to the corresponding angles in the basic rack. A comparison of Equations (13.83 and 13.84) therefore shows

that the two normal pressure angles are also equal.

(13.85)

Generator Inclination Angle at Radius R

We introduce the symbol "R to represent the angle between the generator through A and the helix tangent at A, where A is a point on the tooth surface at radius R. There is no name in common use for the angle "R' but we will refer to it as the generator inclination angle. An expression for cos "R can be found in the usual manner.

cos "R sin ~b sin ~R cos(eA-eG) + cos ~b cos ~R (13.76, 13.50)


cos "R cos -JiR cos -Jib (13.86)

We now use Equation (13.86) to derive an expression for

sin "R.

sin "R

sin "R

sin "R

2 sin "R sin -JiR y'( 1 - cos 'tR) (13.59)

sin "R (13.87)

We will show in Chapter 14 that the orientation of the contact line between two meshing gears is related to the value of "R at the pitch cylinder, and this value is therefore called the contact line inclination angle.

Independent Angles

Early in the description of helical gear geometry, we

introduced the transverse profile angle 'tR and the helix angle -JiR, each defined at radius R. We then used these angles

to define a number of new angles, the base helix angle -Jib' the

normal profile angle 'nR' and the generator inclination angle "R. All five angles are useful in describing different

aspects of helical gear geometry. However, only two can be

regarded as independent, and it is already clear that there are a considerable number of relations between the angles. In

working through the trigonometric steps necessary for some of

the proofs that we will encounter, it is often convenient to

express all the angles in terms of the two which are regarded

as independent.

Independent Angles 343

We must now decide which of the angles to choose as the

independent pair, and it is perhaps a matter of opinion which

are the most important. We will choose the base helix angle "'b

and the normal profile angle ~nR' since the base cylinder is

certainly fundamental to an involute gear, and the normal

profile angle has a special significance, as we will show when

we describe the hobbing of gears in Chapter 16. We now express

the remaining three angles in terms of ~nR and "'b.

We obtain sin "'R directly from Equation (13.80),

sin "'R sin "'b

cos ~nR

and we then derive an expression for cos "'R'

v(cos2~nR - sin2"'b)

cos 4>nR

(13.88)

It is convenient to introduce a function f of ~nR and "'b'

defined by

(13.89)

and the expression for cos "'R can then be written,

f(~nR''''b) cos 4>nR

(13.90)

An expression for cos ~tR was given by Equation (13.59),

tan "'b tan "'R cos 4>tR

and we now use Equations (13.88 and 13.90) to express cos ~tR

in terms of ~nR and "'b'

cos 4>tR (13.91)

We obtain sin ~tR directly from Equation (13.81),

sin 4>nR

cos "'b sin 4>tR (13.92)

Lastly, cos PR and sin PR are given by Equations (13.86

and 13.87), if we express 4>tR and "'R in terms of 4>nR and "'b.


cos 4>nR cos IPb (13.93)

(13.94)

Several relations between the various angles have already been proved. These are given in Equations (13.59, 13.80-13.82, 13.86 and 13.87). There are some additional relations which are sometimes useful, and since they can all be proved simply by expressing every angle in terms of 4>nR and IPb' the relations will be stated here wi thout proof.

cos IPb cos 4>tR cos IPR cos 4>nR (13.95)

cos 4>tR cos 4>nR cos vR (13.96)

tan vR tan IPR sin 4>nR (13.97)

sin 4>nR cos vR sin 4>tR cos IPR (13.98)

All the relations discussed in this section are valid at any radius R. As a special case, they are of course true at the standard pitch cylinder. The relations can therefore all be rewritten with the pressure angles in place of the profile angles. Equation (13.88), for example, would then take the following form,

(13.99)

It is not necessary to repeat the entire set of relations, wi th the pressure angles in place of the prof ile angles. However, in the remaining chapters we will often need to make use of relations between the pressure angles, and for proof the reader will simply be referred to the corresponding

relations between the profile angles.

Relations Between the Gear and the Basic Rack Parameters

We have already shown, in Equations (13.28, 13.62 and 13.85), that the helix angle of a gear at its standard pitch

Parameters of the Gear and Basic Rack 345

cylinder is equal to the basic rack helix angle, and that the transverse and normal pressure angles of the gear are equal to those of the basic rack. We will now prove that a similar set of relations exist between the pitches of the gear and those

of the basic rack. I n Chapter 2, we showed f or a spur gear that the circular

pitch at the standard pitch circle is equal to the pitch of

the basic rack, and that the base pitches of the gear an? the basic rack are also equal. The same statements can therefore be made in relation to the transverse pitches and the transverse base pi tches of a helical gear and its basic rack,

( 13.100)

(13.101)

Expressions for the normal pitch of the gear at its

standard pitch cylinder, and for that of the basic rack, were given by Equations (13.41 and 13.1),

Since the transverse pitches and the helix angles are equal, as we showed in Equations (13.100 and 13.28), it is evident that the normal pi tches are also equal.

( 13.102)

TO complete this section, we will prove that the normal

base pitch of the gear is equal to that of the basic rack. In

Equations (13.39 and 13.45) we gave expressions for the axial

pitch of the gear in terms of the normal pitch at radius R, and of the normal base pitch,

PnR sin "'R

Pnb sin "'b


By equating the two expressions for Pz' we obt,in a relation between the normal base pitch and the normal pitch at radius R,

PnR sin "'b sin "'R

(13.103)

The helix angles at the base cylinder and at radius Rare related by Equation (13.80),

and we therefore obtain the following expression for the normal base pi tch of the gear,

(13.104)

The right-hand side of this equation is an expression that can be evaluated at any radius R. For our present purpose, we want to express the normal base pitch in terms of the normal pitch at the standard pi tch cylinder, and we therefore set R equal to Rs '

(13.105)

The normal base pitch of the basic rack was given by Equation (13.9),

We have proved in Equations (13.102 and 13.85) that the normal pi tches and the normal pressure angles are equal, so a comparison of the last two equations shows that the normal base pitch of the gear is equal to that of the basic rack,

( 13.106)

Normal Base pitch

In Chapter 2, we defined the base pitch of a spur gear as the distance between corresponding points of adjacent teeth,

Normal Base pitch

Base cylinder

Figure 13.21. Normals to the tooth surfaces at points A and A I.

347

measured around the base circle. We then showed that the base pitch is equal to the distance between adjacent tooth profiles, measured along a common normal.

The normal base pitch of a helical gear was defined earlier in this chapter, as the distance between corresponding points of adjacent teeth, measured on the developed base cylinder in a direction perpendicular to the lines of the teeth. We will now show that the normal base pitch, defined in this manner, is also equal to the distance between adjacent tooth surfaces, measured along a common normal.

Figure 13.21 shows the base cylinder of a gear, with the base helices of two adjacent teeth. We consider two points, A on one tooth and A' on the other, with generators starting at points G and G', and we will determine what conditions must be satisfied if there is to be a common normal at A and A'.

A first condition, necessary for the existence of a common normal, is that the normals to the tooth surfaces at A

and A' should be parallel. This condition can be written as

follows,

(13.107)

and when we use Equation (13.79) to express the vectors n~ and n~', the condition becomes:


- cos "'b sin eG n + cos "'b cos eG n G' x G' Y

sin "'b n z - cos "'b sin e nx + cos "'b cos e ny sin "'b n z (13.108)

The solution to this equation can be seen by inspection,

G' e (13.109)

G G' For the polar coordinates e and e to be equal, the two points G and G' must lie on the same axial line. The two generators and the axial line GG' then form a plane, as shown in Figure 13.22, and this plane is perpendicular to nGR• The

G G' unit vectors n# and n# must be equal, in view of Equation (13.109), and the two generators are therefore parallel. The plane containing these generators is called a base tangent plane, since it touches the base cylinder along line GG' •

The unit vector n~ normal to the tooth surface at A is parallel to n~, as we showed in Equation (13.78), and is

therefore perpendicular to n~ and to n~. It must then lie in the base tangent plane, in the direction perpendicular to the generators. The same is true of the uni t vector n~' , normal to the second tooth surface at A'. Hence, if we choose A and A' so that the line joining them is perpendicular to the

Figure 13.22. Coincident normals to the tooth surfaces

at points A and A' •

Normal Base pi tch 349

generators, the normals at A and A' will coincide, and we have

found a common normal. The length AA' can then be read from

Figure 13.22,

AA' Pz sin "'b (13.110)

We showed in Equation (13.45) that the axial pitch is related

to the normal base pi tch as follows,

and when the last two equations are combined, it is clear that

the length AA' in Figure 13.22 is equal to the normal base pitch,

AA' (13.111)

We have therefore proved the statement made at the beginning of this section, that the normal base pitch is equal to the distance between adjacent tooth surfaces, measured along a common normal.

We can also use Figure 13.22 to prove another important result. The generators AG and A'G' are the lines where the base tangent plane intersects the tooth surfaces containing A and A'. The same plane would intersect the other teeth of the gear in a set of parallel lines, each a distance Pnb apart, and each making an angle "'b with the z axis. Hence, the section formed by a base tangent plane looks the same as the developed base cylinder, in which the teeth also appear as

straight lines, a distance Pnb apart, making an angle "'b with the z axis. This result could also have been proved directly,

by considering the tooth surface in the manner described earlier, as the surface swept out by the edge of a flexible

sheet unwrapping from the base cylinder.

Strictly speaking, the base tangent plane is not

absolutely identical to the developed base cylinder. In the

base tangent plane, the side of each tooth furthest from line

GG' is the generator, and is therefore straight, while the

other side is slightly curved. In the developed base cylinder, as in every developed cylinder, both sides of the


teeth are straight. However, when we consider the base tangent plane, we are almost always interested in the sides of

the teeth which coincide with the generators, so it is common

practice to regard the base tangent plane and the developed

base cylinder as identica1.

Tooth Thickness

For helical gears, we define the tooth thickness in both

the transverse and the normal directions, in exactly the same

manner as we defined the transverse and normal pitches. Figure 13.23 shows the developed cylinder of radius R, and

each pair of diagonal lines represents the intersection

between the cylinder and the two faces of a tooth. The

transverse and the normal tooth thickness at radius Rare

defined as the distances between the tooth lines on the developed cylinder, measured in the transverse and normal

directions. The relation between the transverse thickness ttR and the normal thickness tnR can be read from Figure 13.23,

(13.112)

At the standard pitch cylinder, the tooth thicknesses

are represented by the symbols tts and t ns • When we refer to the transverse and normal tooth thickness, without specifying any particular radius, it is generally understood that we

21TR

Figure 13.23. Transverse and normal tooth thickness.

Profile Shift 351

mean tt and t , the tooth thicknesses at s ns the standard pitch

cylinder. These values must of course satisfy

Equation (13.112),

( 13. 113)

The transverse tooth thickness is defined in essentially the same manner as the tooth thickness of a spur gear, so the results derived in Part 1 for the spur gear tooth thickness apply equally to the transverse tooth thickness of a helical gear. In particular, we can use Equation (2.36), which gave the relation between the tooth thickness at radius R and that at the standard pitch circle. When applied to a helical gear, this equation can be written,

( 13.114)

If the transverse tooth thickness is known at one radius and we need to find its value at another, Equation (13.114) can of course be used twice. There is no corresponding equation relating the normal tooth thicknesses at different radi i. However, when we know the normal thickness at one radius, we can find its value at another by the following procedure. We calculate the transverse thickness at the first radius from Equation (13.112), then find the transverse thickness at the second radius by means of Equation (13.114), and lastly we use Equation (13.112) once more to obtain the normal thickness at the second radius.

Profile Shift

The relation between the tooth thickness ts of a spur

gear and its profile shift e was given by Equation (6.1),

111'm + 2e tan ¢s ( 13.115)

The profile shift of a helical gear is defined in the

same manner as that of a spur gear. A helical gear has a profile shift e if its teeth are conjugate to the basic rack,


when the distance between the gear axis and the reference

plane of the basic rack is equal to (Rs+e). Due to the equivalence between the tooth thickness of a spur gear and the transverse tooth thickness of a helical gear, we can

immediately write down an equation corresponding to Equation (13.115), relating the profile shift of a helical gear, and its transverse tooth thickness.

( 13.116)

The three relations between tts' mt , IP ts ' and the corresponding quantities in the normal section, were given by Equations (13.113,13.29 and 13.83),

When these equations are substituted into Equation (13.116), we obtain the relation between the normal tooth thickness and the profile shift,

(13.117)

Chordal Tooth Thickness in the Normal Section

The simplest method by which we can measure the tooth

thickness of a helical gear is to measure the normal tooth

thickness at the standard pitch cylinder, using a gear-tooth

caliper. However, the normal thickness is defined along a

line in the developed cylinder, as shown in Figure 13.24, so

that on the actual gear it is a measurement along a helix. As we described in Chapter 8, the gear-tooth caliper measures the tooth thickness along a straight line, or in other words,

it measures the chordal thickness. In order to derive a relation between the normal tooth thickness and the

corresponding chordal thickness, we must first find the

Chordal Tooth Thickness in the Normal Section 353

Normal helix

21TR

Figure 13.24. Normal helix in the developed cylinder of radius R.

radius of curvature of the helix along which the normal tooth

thickness is defined. We will use the symbol PR to represent the radius of

curvature of the helix at radius R, with helix angle ~R. When we consider a small movement along the helix, the helix tangent unit vector changes by an amount which can be found from Equations (13.51 and 13.52),

dn~ - sin ~R deA n~

Since the length of the unit vector is unchanged, the magnitude of the angle through which it has turned is (sin ~R deA). The corresponding distance moved along the helix is (R deA/sin ~R). Hence, the radius of curvature, which is equal to the distance moved divided by the angle through which the tangent turns, can be expressed as follows,

R (13.118) sin2~R

The normal tooth thickness at any radius R is measured along a

helix known as the normal helix, which is shown in

Figure 13.24 as a line perpendicular to the teeth. Its helix

angle is (?r/2-~R)' and its radius of curvature PnR can

therefore be found from Equation (13.118),

R (13.119)


The relation between the normal tooth thickness t ns at the standard pitch cylinder, and the corresponding chordal thickness t nsch ' can be read from Figure 13.25.

t 2 P n s sin ( 2 pnnss )

and we use Equation (13.119) to express Pns in terms of the radius and the helix angle at the standard pi tch cylinder,

2 Rs . t ns cos ~s --:2;-- sIn ( 2R ) cos ~s s

2 (13.120)

For gears with large amounts of profile shift, it may be more convenient to measure the normal tooth thickness at a radius R which is different from Rs. The chordal tooth thickness is then given by the following equation, proved in exactly the same manner as Equat ion (13.120),

R t cos2~R 2 sin( nR 2R ) (13.121) cos2~R

Span Measurement

In Chapter 8, we described the span measurement of the tooth thickness, in the case of a spur gear. The same method

Normal helix at the standard pitch cylinder

tnsch -::~~~_----l~ Chordal tooth

thickness

Figure 13.25. Tooth thickness and chordal tooth thickness in the normal section.


Figure 13.26. Span measurement.

can be used for a helical gear, and the measurement is made in a base tangent plane, since in this plane the normals to the surfaces of different teeth all lie in the same direction.

A base tangent plane is shown in Figure 13.26, with the shaded areas representing the sections through the teeth. The span measurement is made in the direction perpendicular to the teeth, and is shown in the diagram as the length S. Since the base tangent section is essentially the same as the developed base cylinder, as we pointed out earlier in this chapter, the pitch and the tooth thickness in the direction of the span measurement are equal to the normal base pitch Pnb and the normal tooth thickness t nb at the base cylinder. Hence, if the span is measured over N' teeth, the span length is equal to (N'-1) normal base pitches, together with the tooth thic kness,

S (13.122)

By expressing Pnb and t nb in terms of the corresponding quantities in the transverse plane, and then relating these

to the quantities in the standard pitch cylinder, we can

express the span length S in the following manner,

(13.123)


Once the length S is measured, the normal tooth thickness is found by rearranging Equation (13.123),

S - (N'-1)7I'm - Nm inv'ts cos 'ns n n

(13.124)

The radius R at which the measurement is made can be read from Figure 13.26,

R (13.125)

We now have to choose the value of N' so that the span measurement is made near the middle of the tooth face, or in other words, at a radi us of approximately (Rs +e). The value of S which would give the ideal measurement radius is found by setting R equal to (Rs+e) in Equation (13.125),

(13.126)

We follow the same procedure that was used in the case of spur gears. We express Rb in terms of Rs ' and expand the expression for S as a power series in (e/Rs )' retaining only the first two terms,

( 13.127)

The span length S was given earlier by Equation (13.123). We substitute this expression into Equation (13.127), and solve for N', the number of teeth over which the span should be measured,

N' 1 N'ts '2+-71'-+

We find, as we found with spur gears, that the value of N' given by this equation is too high in cases where e is large and N is low. This is because the expansion for S given by Equation (13.127) is inaccurate when (e/Rs ) is large. We therefore multiply the coefficient of e in Equation (13.128)

by the factor [O.75-(2/N)], exactly as we did for spur gears,

and once again the angle 'ts in the second term is expressed more conveniently in degrees, so that the final expression

Position of a Typical Point A on the Tooth Surface 357

for N' takes the following form,

N' ~ 1 + N1~8tos + N tan ~ns ::n2~b + 2e[0.75-(2/N)] (13.129) - 2 'II' cos 'II'mn tan cfl ns

This equation generally gives a non-integer value for N' , and

the number of teeth to be spanned is equal to the integer

closest to this value. There is one condition that must be met, for the span

measurement of a helical gear to be possible. The measurement

must obviously be made inside the end faces of the gear. The projection of the span length in the axial direction, which is shown in Figure 13.26, must therefore be less than the face

width,

S sin ~b < F (13.130)

Position of a Typical Point A on the Tooth Surface

We stated earlier that the tooth surface of a helical gear is formed by a family of helices, each passing through a specified profile in the transverse section z=O. So far in this chapter, whenever we have discussed the position of a typical point A, we have essentially given its position

relative to AO' the point where the gear helix through A cuts the transverse plane z=O. However, we have not yet

established the position of AO. Having now defined tts and t ns ' the two measures of tooth thickness in a helical gear, we are able to specify the involute profile in the plane z=O, and we can then determine the position of AO. We have used the coordinates (R,eA,Z) to represent the position of

point A, and we will now derive an expression for eA as a

function of Rand z. In other words, we will find the position

of the point on the tooth surface that lies at radius R in the

transverse section at plane z.

It is convenient to choose the (x,y,z) coordinate system

in the position shown in Figure 13.27, so that the x axis

coincides with a tooth centre-line in the transverse section

z=O. As always, since the profile in a transverse section is an involute, the results derived for spur gears can be used


(seCYlinder n 80

Ao ~

nx

x

/ Figure 13.27. Transverse section at plane z=O.

directly, apart from the changes in notation. Hence, if AO is the point at radius R on the tooth profile in the transverse section z=O, the angular coordinate BAO can be found from Equations (2.18 and 2.35).

(13.131)

(13.132)

If point A lies on the surface of the same tooth, at the same radius R as point AO' the gear helix through AO must also pass through A, and the difference between the angular coordinates of the two points is given by Equation (13.35),

tan l/Ib R z

b (13.133)

We now combine Equations (13.132 and 13.133) to obtain an expression for BA,

( 13. 134)

The Cartesian coordinates of point A can of course be found from the cylindrical coordinates by the usual transformation.


Tooth Profile in the Normal Section

A gear tooth surface is specified by its shape in the transverse section, but there are times when we may need to

know the corresponding shape in the normal section. For example, by shining parallel rays of light at a gear, a shadowgraph apparatus can project onto a screen a curve which is almost exactly the shape of a normal section through the

tooth. The apparatus can therefore be used to check the accuracy of the tooth surface, provided the theoretical shape of the normal section is known.

Before discussing the tooth profile in a normal section,

it is important to clarify which normal plane is referred to. The normal plane through a point D is defined as the plane through D, perpendicular to the helix tangent direction nD.

jJ.

The normal section profiles that we are most likely to require are the sections either through a point on the tooth profile,

or through a point on the standard pitch cylinder. The method is the same in both cases, so we will deal with the most general situation, where D is an arbitrary point at any radius R', not necessarily on the tooth surface. We introduce local coordinates xn and Yn in the normal plane, as shown in Figure 13.28, and in order to determine the shape of the

---.~nD R

Figure 13.28. Normal section through point D.


normal section profile, we will show how to calculate the

values of xn and Yn for points on the profile. We choose point D by specifying its cylindrical

coordinates R', 9D and zD. If the gear helix through D cuts

the transverse plane z=O at DO' the angular coordinate of DO is given by Equation (13.35),

( 13.135)

The position vector to D and the direction of the helix tangent at D can be found from Equations (13.46 and 13.50),

where the helix angle at radius R' Equation (13.49),

is given by

(13.138)

We now choose any radi us R, and we determine the posi t ion of A, the point where the cylinder of radius R is intersected by the tooth profile in the normal section through D. We use Equation (13.46) once again, this time to obtain the position vector to A,

COA A A Rb A AO P =Rcos9nx+Rsin9n + (9-9)n (13.139) y tan ~b z

where the angle eAO is given by Equations (13.131 and 13.132), and 9A is at present still unknown.

If A lies in the normal section through D, the vector from D to A must be perpendicular to the helix tangent at D, and this condition can be put in the following form,

o (13.140)

The expressions given above for the position vectors and the unit vector are substituted into Equation (13.140), and we obtain the following equation for (9A_eD),

Calculation of R 361

AO DO e - e (13.141)

The angle (eA_eD) is small in all cases of practical interest, and we obtain an approximate solution with negligible error when we replace the sine of the angle by the angle itself. Hence, the solution to Equation (13.141) can be

written,

( 1.3. 142)

The local coordinate system (xn'Yn) will be positioned with its origin at point D, and the coordinate axes in the

directions of the unit vectors n~ and n~. The position of A relative to D is then given by the following two equations,

x~ C A C D

(p 0 _ pO) R sin (eA-eD) cos 1/IR, (13.144)

For any chosen value of R, we find the value of (eA_eD) by means of Equation (13.142), and then the position of point A on the normal section profile is given by Equations (13.143 and 13.144). By carrying out the calculation for a number of different R values, we can construct the entire tooth profile in the normal section, including the fillet if required.

Calculation of R When the Transverse Profile Angle, the Helix Angle, or the Normal Prof i Ie Angle I s Known

It sometimes happens that we want to find the radius at

which one of the angles 'tR' 1/IR' or 'nR' takes a specified value. For example, knowing the value of one of the angles at the pitch cylinder, we might want to use this information to

calculate the radius Rp' First, we consider the case when the value of ~tR is

specified. The corresponding radius R is then given by Equation (13.54),

R (13.145)


If the helix angle ~R is specified, the radius can be found from Equation (13.34),

R ( 13.146)

Lastly, if ~nR is specified, we need to make use of Equations (13.54, 13.91 and 13.89) to obtain an equation relating R

and ~nR'

Rb R cos ~tR

V(COS2~nR - sin2~b) cos ~b

The value of R is then given by the following expression,

R (13.147)

Specifying a Helical Gear

The specification of a helical gear consists of the quantities given in Chapter 2 for the specification of a spur gear, together with a few additional values required particularly for helical gears.

In place of the module and the pressure angle, which are given for a spur gear, the specification of a helical gear contains either the transverse module mt and the transverse pressure angle ~ts' or the normal module mn and the normal pressure angle ~ns' For reasons which will be explained in Chapter 16, mt and ~ts are sometimes given when the gear is

cut by a pinion cutter, while mn and ~ns are generally given for gears cut by a rack cutter or a hob. It does not matter

which pair of quantities is provided, since they are related

by Equations (13.29 and 13.83),

(13.148)

(13.149)

The remaining quantities in the specification of a

helical gear, which have no counterpart in that of a spur

gear, are the helix angle ~s and the axial pitch Pz' The sign

Speci fying a Helical Gear 363

convention that we have used for these two quantities, where they are positive or negative depending on whether the gear is right or left-handed, is not used in the specification. The general practice is to give the magnitudes of the helix angle and the axial pitch, and to state whether the gear is right or

left-handed. For convenient reference, a number of the equations

derived in this chapter are repeated below, and they can be used to calculate the remaining important gear parameters, in terms of those that are specified.

RS Nmn

2 cos IPs ( 13.150)

tan tP ts tan tPns cos IPs

( 13.151 )

Rb Rs cos tP ts (13.152)

L 21rRs

tan IPs ( 13.153)

Rb tan IPs tan IPb Rs

( 13. 154)

Finally, we use the following equations to calculate the profile angles and the helix angle at a typical radius R,

cos tPtR Rb R ( 13.155)

R tan IPs tan IPR Rs

( 13.156)

tan tPtR cos IPR (13.157)


Numerical Examples

Example 13.1 A helical gear with 24 teeth has a transverse diametral

pitch of 2, a transverse pressure angle of 20°, and a helix angle of 23°. Calculate the following quantities: normal module, normal pressure angle, standard pitch cylinder

radius, base cylinder radius, lead, axial pitch, base helix angle, and normal base pitch. Also, calculate the generator inclination angle at the tip cylinder, if the addendum is equal to the normal module.

Example 13.2

mt = 0.5000 inches

mn 0.4603

IPns 18.523° Rs 6.0000 Rb 5.6382 L = 88.8135

Pz = 3.7006 !Jib = 21.746°

Pnb = 1.3710 inches

as = 0.4603 inches

RT = Rs + as = 6.4603

IPtT = 29.221° ~T 24.562° vT = 11.708°

(13.5) (13.148) (13.149)

(13.16) (13.152) (13.153)

(13.36) ( 13.154)

(13.45)

(13.155) (13.156)

(13.87)

A 78-tooth gear has a normal module of 8 mm, a normal

pressure angle of 20D, a helix angle of 30.176°, and a normal

tooth thickness of 12.35 mm. First, determine the axial pi tch, and then calculate the span measurement, and the

normal tooth thickness at the tip cylinder, if the addendum is

equal to the normal module.

Examples

RS = 360.908 mm

~ts = 22.832° Rb = 332.629

L = 3900.0

Pz = 50.000 mm

365

(13.150) ( 13.151) ( 13. 152 ) ( 13.153)

(13.36)

The value of the helix angle was chosen specifically to give a round number for the axial pitch. There is no particular advantage in choosing a round number, such as 30°, for the helix angle. However, we will show in Chapter 16 that it is very much easier to select the change gears for a hobbing machine, when the axial pitch of the gear is a round number.

"'b = 28.187° e = - 0.297 mm

N' = Integer closest to 13.349 = 13 5 = 308.218 mm

Measurement radi us R = 359.295 mm Ideal measurement radius = Rs + e = 360.611 mm

tts = 14.286 RT = Rs + as = 368.908

~tT = 25.623° ttT = 7.311

"'T = 30.725° tnT = 6.285 mm

(13.154) (13.117) (13.129) ( 13.123) ( 13.125)

(13.113)

( 13.155) (13.114) (13.156) (13.112)

Introduction

Chapter 14 Helical Gears in Mesh

In this chapter we will describe the geometry, first of a helical gear meshed with a rack, and then of a pair of helical

gears, and in each case we will determine what conditions must be satisfied for correct meshing. We will also discuss how the contact ratio of a pair of helical gears differs from that of a pair of spur gears, and we will show that the calculations for interference, undercutting and backlash in helical gears are essentially the same as for spur gears. Finally, we will

make a detailed study of the position and orientation of the contact line between any pair of meshing teeth.


We showed in Chapter 3 that a spur gear can mesh wi th any

rack whose teeth are straight-sided, provided the base pitch

of the rack is equal to that of the gear. We now investigate whether a helical gear can mesh with a rack whose tooth faces

are planar, but with parameters which are not necessarily

equal to those of the bas ic rac k. We consider a gear with normal module mn , normal

pressure angle ~ns and helix angle ~s' and we will determine the conditions that must be satisfied if the gear is to mesh

with a rack whose independent parameters are P~r' ~~r and ~~. The remaining parameters of the rack can be expressed in terms

of the first three by means of Equations (13.10 - 13.13),

P~r cos ~~

(14.1)


tan ¢~r

cos "'~

367

(14.2)

(14.3)

(14.4)

We consider first the meshing geometry in the transverse plane z=O, and, as always when we consider a transverse plane, we make use of the results derived earlier for spur gears to write down the corresponding results for the transverse plane of the helical gear. The condition for correct meshing is that the transverse base pitch of the rack must be equal to that of

the gear,

(14.5)

The pitch cylinder radius of the gear is then given by Equa t i on (3.4),

(14.6)

The transverse profile angle, the normal profile angle and the helix angle of the gear at its pitch cylinder are

called the operating transverse pressure angle ¢tp' the operating normal pressure angle ¢np' and the operating helix angle "'p. In addition, the transverse pitch and the normal pitch at the pitch cylinder are known as the operating

pi tches, and are represented by the symbols Ptp and Pnp Expressions for the helix angle and the profile angles

of the gear at an arbitrary radius R were given in Equations (13.34, 13.54 and 13.82). When we replace R in these equations

by the pitch cylinder radius Rp ' we obtain the corresponding expressions for the operating helix angle and the operating

pressure angles,

(14.7)

(14.8)

(14.9)

368 Helical Gears in Mesh

The operating tooth pitches are found in exactly the same manner, by substituting Rp for R in Equations (13.37 - 13.39),

271'Rp N

Ptp cos .pp

Pnp sin .pp

(14.10)

(14.11)

(14.12)

We proved in Equations (3.13 and 3.15) that when a spur gear is meshed with a rack, the operating circular pitch and the operating pressure angle of the gear are equal to the pitch and the pressure angle of the rack. We therefore know that for a helical gear, the operating transverse pitch and the operating transverse pressure angle are equal to the corresponding quantities in the rack,

(14.13)

(14.14)

We have now obtained all the information available from a consideration of the single transverse section at plane z=O. Next, we determine the value of the rack helix angle .p~

for correct meshing at the transverse plane z. The procedure is the same as that used in Chapter 13, except in one respect. We were trying then to find the tooth shape of a gear that would mesh with the basic rack, while now the tooth shape of the gear is known, and we are looking for the shape of a rack which wi 11 mesh correctly wi th the gear.

The helix rotation ~e of the gear tooth surfaces between plane z=O and plane z was given by Equation (13.65),

(14.15)

We use Equation (14.7) to relate the base helix angle .pb of the gear with its operating helix angle .pp, and the expression for ~e then takes the following form,

tan .pp R z

P (14.16)

A Pinion Meshed wi th a Rack 369

Due to the helical rotation of the gear teeth between the two transverse sections, the rack tooth profile at plane z

must be displaced relative to that at plane z=O. The rack displacement ~ur corresponding to a gear rotation ~f3 was given by Equation (3.24),

(14.17)

If we choose the value of ~f3 in this relation equal to the helical rotation ~e given by Equation (14.16), we obtain the required relative displacement between the transverse rack sections at plane z=O and plane z,

z tan I/Ip (14.18)

For a rack with helix angle I/I~, the value of this relative displacement is equal to (z tan I/I~), as we can see in Figure 14.1. We therefore substitute this expression for ~ur in Equation (14.18), and it is immediately clear that the rack helix angle must be equal to the operating helix angle of the gear,

Plane z=Q

1/1' r

Plane z

Figure 14.1. A helical rack.

(14.19)

370 Hel ical Gears in Mesh

A relation between the operating pressure angles and the operating helix angle of the gear was given by Equa t i on (14. 9) ,

tan I/Itp cos "'p

and in Equation (14.2) we gave the corresponding relation between the angles in the rack,

tan I/I~r tan I/Itr cos "'~

Since we have just proved that I/Itp and "'p are equal to I/I tr and ",~, it is clear that the two normal pressure angles must also be equal,

(14.20)

The operating normal pitch of the gear and the normal pitch of the rack were given by Equations (14.11 and 14.1),

Again, we have shown that the transverse pi tches and the helix angles are equal, so the normal pi tches must also be equal,

(14.21)

Lastly, we compare the normal base pitches of the gear and the rack. In Equation (13.104) we gave a relation between the normal base pitch of the gear, and its normal pitch at any radius R,

By setting R equal to Rp in this relation, we can express the normal base pitch in terms of the operating normal pitch,

= (14.22)

Minimum Condi tions for Correct Meshing 371

The normal base pitch of the rack was given by

Equation (14.3),

P~r cos q,~r

and since we have already proved that the normal pitches and

the normal pressure angles are equal, a compar i son of the last two equations shows that the normal base pitches of the gear

and the rack must be equal,

(14.23)

Minimum Condi tions for Correct Meshing

We have shown that if a rack can mesh correctly with a helical gear, the transverse and normal base pitches of the rack and the gear must be equal, and the remaining five parameters of the rack must be equal to the corresponding quantities of· the gear, measured at the pitch cylinder. We proved in Chapter 3, for a spur gear and rack, that the base pitches must be equal, and that the pitch and the pressure angle of the rack must be equal to the operating circular pitch and the operating pressure angle of the gear. However, we then showed that the condition of equal base pitches is sufficient to ensure correct meshing of the spur gear and rack, and that the other two results follow automatically. We will now prove the corresponding statements for a helical gear. We will show first, that a helical gear can mesh with any rack, provided the transverse base pitch and normal base pi tch of the rack are equal to those of the gear, and

secondly, that when these two conditions are met, the other

five requirements are automatically satisfied.

Since the steps in the proof are mostly the same, apart

from the order, as those we used in Equations (14.20 - 14.23),

we will make the the proof as brief as possible. We consider a

rack whose transverse and normal base pitches Ptbr and P~br

are equal to the base pitches Ptb and Pnb of the gear. First, we take the results proved in Chapter 3 for a spur gear, and

we apply them to the transverse section of the helical gear.


We know, therefore, that since the transverse base pitches of the gear and the rack are equal, the pitch cylinder radius is given by Equation (3.4), and the operating transverse pitch

Ptp and the operating transverse pressure angle ~tP of the gear are equal to the corresponding parameters Ptr and ~tr of the rack.

Next, we use the condition that the normal base pitch of the gear is equal to that of the rack. We use Equations (14.22, 14.11 and 14.9) to express the normal base pitch of the gear in terms of the operating transverse pitch and the operating pressure angles,

Ptp cos 1/Ip cos ~np sin 4>np

Ptp tan ~tP

and we obtain a similar expression for the normal base pitch of the rack by means of Equations (14.3, 14-.1 and 14.2),

P~r cos ~~r sin ~~r

Ptr tan ~i:r

We equate the two expressions for the normal base pitches, and since we have already shown that the transverse pitches and the transverse pressure angles are equal, we have proved that the normal pressure angles are also equal. Finally, by comparing Equations (14.22 and 14.3), we show that the operating normal pitch of the gear is equal to the normal pitch of the rack, and we use Equations (14.9 and 14.2) to prove that the operating helix angle of the gear is equal to the helix angle of the rack.

In summary, we have shown that when the transverse and normal base pitches of the gear and the rack are equal, the

other five rack parameters Ptr' P~r' ~tr' ~~r and 1/1~ are equal to the corresponding quantities of the gear, measured at the pitch cylinder. Since the transverse base pitches are equal, we know that the gear and the rack will mesh correctly in the transverse section at plane z=O. We also proved, in Equation (14.19), displacement of

that we the rack

obtain the correct relative teeth between the transverse

sections at plane z=O and at plane z, if the rack helix angle is equal to the operating helix angle of the gear. We have now shown that this condition is satisfied, provided that the

Plane of Act ion 373

transverse base pitch and the normal base pitch of the rack are equal to those of the gear. We have therefore proved that the gear and the rack will mesh correctly in the transverse section at any plane z, and hence throughout the axial length

of the gear.

Plane of Action

A typical transverse section through the gear and the rack is shown in Figure 14.2, and the line from the pitch point to the contact point is the line of action. The angle 't between this line and the rack pitch line is called the operating transverse pressure angle of the gear pair, and it is equal to the transverse pressure angle of the rack, as we proved in Equation (3.10),

c

(14.24)

Operating transverse pressure angle <Pt

Plane of action

t+----+-Rack reference plane

Figure 14.2. Transverse section through the gear and rack.

374

s

-1- p

I I z=O Transverse L;Jez

Helical Gears in Mesh

Axial line through the pitch points

Figure 14.3. The plane of action.

The lines of action at each transverse section form a

plane, which is called the plane of action. This is shown in Figure 14.3, with a coordinate system consisting of the axial

coordinate z, and a second coordinate s. The coordinate s is the same as the quant i ty introduced in Chapter 3, which indicates the position of the contact point on the path of contact, relative to the pitch point. The coordinate origin in Figure 14.3 is therefore PO' the pitch point in the transverse section at plane z=O.

The displacement ~s of the contact point between a spur gear and rack, corresponding to a rotation ~f3 of the gear, was

given by Equation (3.29),

~s (14.25)

Hence, for a helical gear, we can find the difference in the s

values of the contact points at plane z=O and at plane z, if

we replace the gear rotation ~f3 in Equation (14.25) by the helical rotation ~(} of the gear teeth, given by

Equation (14.15). The expression for ~s then takes the

following form,

~s z tan "'b (14.26)


The quantity As is therefore linear in z, with a

coefficient equal to tan ~b. Hence, we have shown that the contact points in the different transverse sections lie on a straight line, which makes an angle ~b with the z direction, as shown in Figure 14.3. The only straight lines in the gear tooth surface are the generators, and we proved in Chapter 13

that these also make an angle ~b with the gear axis. The contact line between the gear and the rack is therefore a straight line coinciding with one of the generators, and later in this chapter we will discuss in more detail the position of the particular generator which is in contact with

the rack.

A Pair of Helical Gears in Mesh

We now discuss the meshing geometry of a pair of helical

gears, mounted so that their axes are parallel. Once again, we start by considering the geometry in the transverse section at plane z=O, and we apply the results derived in Chapter 3 for a pair of spur gears. The condition for correct meshing in the transverse plane is that the transverse base pitches of the two gears should be equal,

(14.27)

The pitch cylinder radii and the operating transverse pressure angle of the gear pair are then given by Equations (3.40, 3.41 and 3.44),

RP1 N1C

(N 1+N 2) (14.28)

RP2 N2C

(N 1+N2 ) (14.29)

cos ¢t Rb1 +Rb2

C (14.30)

The operating transverse pressure angles of the two gears,

¢tp1 and ¢tp2' can be found from Equation (3.46),

cos ¢tp1 Rb1 Rp1

(14.31)


cos I/Itp2 (14.32)

We also know, from Equations (3.45 and 3.49), that the operating transverse pitches are equal, with a value given by Equation (4.27), and that the operating transverse pressure angle of each gear is equal to the operating transverse pressure angle of the gear pair,

(14.33)

(14.34)

Next, we determine the relation between the helix angles of the two gears, if there is also to be correct meshing in the transverse section at plane z. The helical rotation ~8 of the teeth of a gear, between the transverse sections at plane z=Q and at plane z, was given by Equation (14.16). Hence, for the two gears we are considering, the helical rotations are as follows,

~81 tan IP121

(14.35) = Rp1

z

~82 tan IP122

(14.36) RP2

z

In Equation (3.55), we gave the relation between the rotations ~1J1 and ~1J2 of a pair of spur gears,

RP1~1J1 - RP2~1J2 (14.37)

If we substitute the helical rotations ~81 and ~82 in place of ~1J1 and ~fi2' we obtain a condition that must be satisfied if the teeth of the two gears are to mesh correctly at plane z, and this gives the relation we require,

IPp1 - IPP2 (14.38)

The operating helix angles, therefore, must be equal in magni tude, and the gears must be of opposi te hand.

We are now in a position to derive the relations between the other parameters of the two gears. The base helix angle of


a gear was given by Equation (13.59), in terms of the helix angle and the transverse profile angle at radius R. We use

this relation to express the base helix angle of each gear in terms of the operating helix angle and the operating transverse pressure angle,

tan Wp1 cos 4>tp1

tan Wp2 cos 4>tp2

We have proved that the operating helix angles are equal and opposite, and that the operating transverse pressure angles are equal, so a comparison of these equations shows that the base helix angles are also equal and opposite,

(14.39)

The normal base pitch of a gear was given by Equation (13.44),

Again, we write down this relation for each gear, making use

of the results already proved, and we show that the normal base pi tches of the two gears are equal,

(14.40)

We follow the same procedure for the remaining parameters. The operating normal pressure angle, the operating normal pitch and the axial pitch of a gear are given by Equations (14.9, 14.11 and 14.12),

tan 4>tp cos Wp

Ptp cos Wp

Pnp sin Wp

We apply these equations to each of the gears, using the


earlier results, and we prove the following new relations,

tP np1 ( 14.41 )

(14.42)

- Pz2 (14.43)

We have therefore proved that, for a pair of helical gears with parallel axes to mesh correctly, the transverse and the normal base pitches of the two gears must be equal, and at the two pi tch cylinders the normal and transverse pitches are equal, the normal and transverse pressure angles are equal, and the helix angles are equal and opposite. In addition, we have shown that the base helix angles and the axial pitches are also equal and opposite.

Imaginary Rack

At the beginning of this chapter, we showed that a helical gear can mesh with any rack, provided their transverse and normal base pitches are equal. We now consider an imaginary rack inserted between the teeth of the two helical gears. The tooth surfaces of the imaginary rack are formed by flat planes of zero thickness. The transverse base

pitch Ptbr and the normal base pitch P~br are chosen equal to the corresponding quantities in the two gears.

( 14.44)

(14.45)

From the results proved earlier, we know that the other rack

parameters Ptr' P~r' tP tr ' tP~r' and "'~ are equal to the corresponding parameters of gear 1, measured at its pitch

cylinder. The same is true of gear 2, except that the helix

angles "'~ and "'p2 are equal and opposite, and thi.s is because the two gears lie on opposite sides of the imaginary rack

tooth surface.

Imaginary Rack 379

For the pair of helical gears, the plane which is tangent to both pitch cylinders is called the pitch plane, and of course thi s plane coincides wi th the pi tch plane of the imaginary rack. The line of action in any transverse section is the common tangent to the base circles. Hence, the plane of

action, which is the plane formed by all the lines of action, is the plane which is tangent to both base cylinders. The angle shown in Figure 14.4 between the pitch plane and the plane of action is the operating transverse pressure angle 4J t , whose value was given by Equation (14.30).

We can now use the imaginary rack to find the nature of the contact between the teeth of the two helical gears. We proved earlier that, when a gear is meshed with a rack, the contact points between the teeth form a straight line lying in the plane of action, and making an angle ~b with the gear axis. Hence, the lines of contact between the imaginary rack and the two helical gears lie in the plane of action, and make

angles ~b1 and ~b2 with the gear axes. These angles are of opposite sign, but once again, the reason is because the two

Pitch cylinder of gear 1

Base cylinder of gear 1

Operating transverse pressure angle cPt

Plane of action



c

Figure 14.4. Transverse section through a gear pair.


gears lie on opposite sides of the imaginary rack tooth

surface, and the lines of contact in fact make the same angle

with the gear axes. When we introduced the imaginary rack for spur gears in Chapter 3, we showed that the two gears and the imaginary rack all touch at the same point. Hence, for the

helical gears and the imaginary rack, there is at least one transverse plane where the contact points coincide, and since the contact lines are parallel, they must coincide throughout the axial length of the gears. If each gear touches the imaginary rack along the same line, they must also touch each other along that line, and we have therefore shown that the contact between two helical gear teeth takes place along a straight line.

We can also use the imaginary rack to establish the minimum condi tions required to ensure correct meshing between the gears. I f the two gears have equal transverse base pitches, equal normal base pitches, and helices of opposite hand, then it is possible to find an imaginary rack that can mesh simultaneously with each gear. Since the Law of Gearing is satisfied between each gear and the imaginary rack, it must also be satisfied between the two gears. These are therefore the minimum conditions for correct meshing between a pair of helical gears.

In practice, a helical gear pair is designed so that the two gears have the same normal modules mn, the same normal

pressure angles ~ns' and helix angles ~s which are equal and opposite. The normal and transverse base pitches can then be found from the following set of relations, which are derived

from Equations (13.102, 13.3, 13.105, 13.42, 13.45

and 13.44),

Pns 1rmn (14.46)

Pnb Pns cos ~ns (14.47)

Pz Pns (14.48)

sin ~s

sin ~b Pnb (14.49) Pz

Ptb Pz tan I/Ib (14.50)

Standard Center Distance 381

It can be seen from these equations that, if the normal modules and the normal pressure angles are equal, and the helix angles are equal and opposite, then the normal base pitches and the transverse base pitches of the two gears are also equal, and the gears will therefore mesh correctly.

Center Distance C, and Standard Center Distance Cs

In Chapter 3, we defined the standard center distance Cs of a pair of spur gears as the sum of the standard pitch circle radii, and we showed that in a gear pair for which the center distance C is equal to Cs ' the operating pressure angle ;p of each gear is equal to the pressure angle ;s. We also pointed out that, in cases where C is not equal to Cs ' the gear pair is always designed so that C and Cs are approximately equal, and in this case the value of ;p does not differ substantially from that of ;s.

Exactly the same considerations apply in the case of a helical gear pair. The radii of the standard pitch cylinders are given by Equation (13.30),

RS1 N1mn

(14.51) 2 cos "'s

RS2 N2mn

(14.52) 2 cos 1/1~

and the standard center distance is defined as the sum of these radi i ,

Cs RS1 + RS2 (14.53)

Helical gear pairs, like spur gear pairs, are also designed so that the values of C and Cs are either equal or approximately equal, and this is done for the same reason. A standard cutter can only be used to cut gears with a limited amount of profile shift, which means that the tooth thickness is only suitable when the radius of the pitch cylinder is approximately equal to that of the standard pi tch cylinder.

The angles ;nR' ;tR and "'R are all functions of the radius R. Hence, the values of ;np' ;tp and "'p are equal to


'ns' 'ts and 1/I s whenever the pitch cylinder radius Rp is equal to the standard pitch cylinder radius Rs' or in other words, whenever C is equal to Cs • And in other cases, the values of

'np' 'tp and 1/Ip are approximately equal to 'ns' 'ts and 1/Is • In addition, the operating transverse pressure angle tfl t of the gear pair, which we showed in Equation (14.34) is equal to the operating transverse pressure angle 'tp of each gear, is either equal or approximately equal to 'ts' depending on whether or not C is equal to Cs •

There is one small difference between spur gears and helical gears, in regard to the standard center distance. The values of Cs for a spur gear pair and a helical gear pair are given by the following two equations,

1 '2(N 1+N2)m

(N 1+N2 )mn 2 cos 1/Is

(14.54)

(14.55)

Whenever possible, standard values are used for the module of the spur gears, and for the normal module of the helical gears. If a spur gear pair is to be designed for a specified center distance C, it is clear that there may be no standard value of the module which makes Cs equal to C, and it is therefore often necessary to design spur gear pairs in which Cs differs from C. On the other hand, the helix angle 1/I s in a helical gear pair can generally be chosen so that Cs is equal to C. However, it should be emphasized that, for helical gears as well as for spur gears, there is no particular advantage to be gained by designing so that the center distance and the standard center distance are exactly equal.

Contact Ratio

The contact ratio mc of a spur gear pair was defined by Equation (4.1),

(14.56)

In this expression, apc is the angle of contact for either one

Contact Ratio 383

of the gears in the pair, and ~9p is the angular pitch of the same gear. We proved in Chapter 4 that we obtain the same

value for the contact ratio, whichever gear we consider. For a helical gear pair, the contact ratio is defined in

exactly the same manner, but due to the helical nature of the

teeth, the angle of contact ~f3c is found by a different procedure from that used for spur gears. The ends of the helical gear are formed by the transverse sections at plane

z=O and plane z=F, where F is the face-width of the gear. We

consider first the contact in the transverse section at plane

z=F. The symbol ~f3p is introduced to represent the rotation of the gear during one meshing cycle in this transverse section, and the calculation of ~f3p is therefore identical with the

calculation of ~f3c for a spur gear. However, when the contact

ends for the tooth section at plane z=F, the meshing cycle at

any other transverse section through the same tooth is not yet

finished, due to the helical rotation of the teeth. The tooth remains in contact until the meshing cycle is completed in the

transverse section at the other end of the gear, in the plane

z=O. I f ~f3F is the angle through which the gear rotates between the end of the meshing cycle at plane z=F, and the end

of the meshing cycle at plane z=O, the angle ~f3F is equal to the helical rotation ~9 between the two sections. The value of

~9 between the sections at plane z=O and at plane z was given by Equation (13.65),

and the value of ~f3F is therefore found by substituting the face-width F in place of z,

(14.57)

The contact period for one tooth lasts from the initial

contact at plane z=F until the final contact at plane z=O, and

the angle of contact ~f3c is therefore equal to the sum of the

two rotations ~f3p and ~f3F'

( 14.58)


We substitute this expression for apc into the definition of the contact ratio, given by Equation (14.56),

(14.59)

and the two terms in this expression are used to define two

new quantities. These are called the profile contact ratio mp and the face contact ratio mF,

(14.60)

(14.61)

When Equations (14.59 - 14.61) are combined, it is clear that

the contact ratio mc is equal to the sum of the profile contact ratio and the face contact ratio,

(14.62)

and for this reason, when we are considering a helical gear

pair, mc is generally known as the total contact ratio. The quantity mp is called the profile contact ratio, or

alternatively the transverse contact ratio, because its value depends only on the tooth profiles of the meshing gears in a transverse section. Its definition is equivalent to that of the contact ratio mc for a pair of spur gears, and we can therefore use Equation (4.5) to express mp in terms of the length asc of the path of contact in a transverse section, and

the transverse base pitch Ptb'

asc

Ptb (14.63)

We then use Equations (4.9 and 4.13) to write down expressions

for the profile contact ratio, first for the case of a pair of

helical gears, and secondly for a helical pinion and rack,

(14.65)

Contact Ratio 385

In order to obtain an expression for the face contact

ratio mF, we combine Equations (14.57 and 14.61). The angular pitch ~ep was given by Equation (4.2),

211 N

(14.66)

and the base circle radius is expressed in terms of the transverse base pitch by means of Equation (13.43),

211Rb N

(14.67)

The expression for mF then takes the following form,

F tan "'b (14.68)

Ptb

In Chapter 4, when we discussed the contact ratio of a pair of spur gears, we suggested that the positions of the contact points could be pictured as a series of points moving up the line of action, with those points within the path of contact representing the contact points. A similar analogy is possible for the case of a helical gear pair. We imagine a series of lines moving upwards in the plane of action. Each

line makes an angle "'b with the z axis, as shown in Figure 14.5, and the vertical spacing between the lines is

equal to the transverse base pitch Ptb. The region of contact is a rectangle of width F and height ~sc. The lines within this region represent the contact lines of the gear pair.

The diagram in Figure 14.5 shows the rectangular contact region, and in addition we have constructed a triangle T10T1FT', whose sloping side is parallel to the contact lines, so that the length of the side T1FT'

(F tan "'b). This diagram can be used to significance of the various contact ratios.

is equal to

explain the

The profile contact ratio mp is equal to the length ~sc

divided by the transverse base pitch Ptb' as we showed in Equation (14.63). It can therefore be interpreted as the

average number of contact lines passing between T2F and T1F ,

or in other words, the average number of contact points in any

transverse section. The face contact ratio mF , which we showed in Equation (14.68) is equal to (F tan "'b) divided

386

Region of contact


T' -------,-

=Xlb

F

Figure 14.5. The region of contact.

by Ptb' can be interpreted as the average number of contact lines passing between T1F and T'. Since each of these lines intersects the line T10T1F , the face contact ratio is equal to the average number of contact points in any axial line through the region of contact, and for this reason mF is sometimes called the axial contact ratio. Finally, the total contact ratio mc is equal to the sum of mp and mF• It can therefore be interpreted as the average number of contact lines passing between T2F and T', and this is equal to the average of the total number of teeth in contact.

It was pointed out in Chapter 13 that helical gears run

more smoothly and more quietly than spur gears cut with the

same accuracy. This is partly because the total contact ratio of a helical gear pair is invariably larger than the contact

ratio of a spur gear pair, and partly because there are no

sudden changes in the length of any of the contact lines. For

each tooth pair, the initial contact occurs at the point

labelled T2F in Figure 14.5. The initial length of the contact line is zero, and it then increases smoothly as the contact line moves up the plane of action. After the contact line

passes through point T1F , its length remains constant until the line reaches point T20 , after which the length diminishes

smoothly to zero. The intensity of the contact force is

Interference and Undercutting 387

determined by the total length of all the contact lines, and a method for calculating that length will be described in Chapter 17, where we discuss the stresses in the teeth.

A spur gear pair with tooth profiles identical to the transverse section through a helical gear pair would have a contact ratio equal to the profile contact ratio of the helical gear pair. In Chapter 4, we suggested that the contact ratio for a spur gear pair should be at least 1.4. However, for the helical gear pair, the total contact ratio is equal to the sum of the profile contact ratio and the face contact ratio, so a value for mp of less than 1.4 is acceptable, and a minimum of 1.0 is generally considered sufficient. Values in

use for the face contact ratio mF are occasionally less than 1.0, but in order to obtain the full benefit of the helical action,"a minimum value of 2.0 is recommended.

Interference and Undercutting

The checks for interference in a helical gear pair are identical with those made for a pair of spur gears. In other words, we check to see that there is no interference in a transverse section. When this condition is satisfied, then there is no contact between the tooth tips of one gear and the tooth fillets of the other, and hence there is no interference whatsoever.

The phenomenon of undercutting in a helical gear can be discussed in exactly the same manner as that of interference in a helical gear pair. The check for undercutting in a helical gear is made in the transverse plane, exactly as if we were checking a spur gear, and we are therefore ensuring that

there is no undercutting in a transverse section. Obviously,

if the tooth profile is complete in a transverse section, then

it is complete in any section, and there is no undercutting of any sort.

Backlash

In this section, we will describe three methods by which the backlash of a helical gear pair is most commonly defined.


These are called the circular backlash, the backlash along the common normal, and the normal backlash. The relations between the three types of backlash will be discussed in the next section.

The circular backlash B of a pair of spur gears was

defined in Chapter 4 as the difference between the space width of one gear and the tooth thickness of the other, both measured at the pitch circles. An expression for B was given by Equation (4.28),

B

We also showed that the circular backlash can be interpreted as the distance moved by a point on the pitch circle of one gear, if that gear is rocked to and fro while the other gear is held fixed.

We define the circular backlash of a helical gear pair in the same way, as the difference between the transverse space width of one gear and the transverse tooth thickness of the other, both measured at the pi tch cylinders,

B (14.69)

Since the geometry of a helical gear pair in any transverse section is identical with that of a spur gear pair, it is clear that the circular backlash of a helical gear pair has exactly the same physical interpretation as that of a spur

gear pair. The backlash along the common normal of a spur gear pair

was defined as the shortest distance between the trailing

profile of a meshing tooth in the driving gear, and the

adjacent tooth profile in the driven gear. We define the

backlash B' along the common normal of a helical gear pair in

exactly the same manner, as the shortest distance across the

gap between the adjacent tooth surfaces. When we discussed the circular backlash of a spur gear

pair in Chapter 4, we drew a diagram in Figure 4.13 showing

the tooth profiles, and two imaginary racks between the

teeth. The tooth thicknesses of the imaginary racks were chosen exactly equal to those of the two gears, measured at

Backlash 389

their pitch circles. In Figure 4.14 we showed the pitch plane section through the imaginary racks, and we then proved that the circular backlash of the gear pair is equal to the distance that either imaginary rack can move, when the other is held fixed.

We follow the same procedure for the case of a helical

gear pair. Figure 14.6 shows a transve~se section through the gear pair, with the imaginary racks drawn in, and Figure 14.7 shows the pitch plane section through the imaginary racks. A typical pair of teeth are in contact along line ArA~, and the gaps between the teeth are represented by the narrow un shaded bands. As we proved earlier in this chapter, the transverse pi tch, normal pi tch and helix angle of the imaginary racks are equal to the corresponding quantities in the gears, measured at their pitch cylinders. The transverse tooth thicknesses of the imaginary racks are chosen equal to those of the gears,

Imaginary rack 1

n~ n

Figure 14.6. Transverse section with imaginary racks.

390

Ir-------.I Ir+-------+-,


Imaginary rack 2

Imaginary rack 1

Figure 14.7. Pitch plane section through the imag i nary rac ks.

and it then follows that the corresponding normal tooth thicknesses are also equal. These values are therefore shown on Figure 14.7, and it is clear that the circular backlash of the gear pair, defined by Equation (14.69), is equal to the width of the gap between the teeth of the imaginary racks, measured in the transverse direction.

We now introduce a third method for defining the

backlash in a helical gear pair. We define the normal backlash

Bn of the gear pair in a manner similar to Equation (14.69),

as the difference between the normal space width of one gear and the normal tooth thickness of the other, both measured at

the pi tch cylinders,

p - t - t np np1 np2 (14.70)

The tooth thicknesses in the normal and transverse directions

are related by Equation (13.112),

(14.71)

Backlash 391

Hence, with the definition given in Equation (14.70), it can be seen in Figure 14.7 that the normal backlash is equal to

the width of the gap between the teeth of the imaginary racks, measured normal to the tooth profiles in the pitch plane.

Relations Between the Different Types of Backlash

The relation between the normal backlash Bn and the circular backlash B can be seen immediately from Figure 14.7,

B cos I/tp (14.72)

In order to find a relation between the backlash B' along the common normal and the circular backlash, we first consider a typical transverse section through the gear pair, as shown in Figure 14.8. The two interior common tangents to

the base cylinders are labelled E1E2 and E;Ei. The contact point in this section lies on line E1E2 , while line E;Ei cuts the non-contacting tooth profiles at points A; and Ai.

Figure 14.9 shows the plane through E; and Ei, which is tangent to both base cylinders. We proved in Chapter 13 that any plane which is tangent to the base cylinder of a helical gear intersects the tooth surface along a generator, and we

Figure 14.8. Transverse section showing the backlash.

392

E' 1

Plane z


Figure 14.9. Common tangent plane to the base cylinders.

showed that all the generators of a gear make the same angle

"'b with the .gear axis. The plane shown in Figure 14.9 is tangent to both base cylinders, and it therefore intersects the tooth surfaces along two parallel straight lines, which pass through A; and Ai. We will show, later in this chapter, that the normals to the tooth surfaces at A; and Ai lie in this plane, and that they are perpendicular to the generators. Hence, the backlash B' along the common normal, which is defined as the shortest distance between the tooth surfaces, is equal to the perpendicular distance between the two generators in Figure 14.9.

If as is the distance between points Ai and Ai in the transverse section shown in Figure 14.8, then in Figure 14.9,

as is the vertical distance between the two generators. We can

therefore express the backlash along the common normal in

terms of as,

B' as cos "'b (14.73)

We now consider holding one gear fixed, and rotating the other

to close the gap between Ai and Ai. The relation between the

displacement as and the corresponding rotation ap was given

by Equation (3.60),

Position and Orientation of the Contact Line 393

(14.74)

We use Equation (14.31) to express the base cylinder radius of

the moveable gear in terms of its pitch cylinder radius,

(14.75)

and we then combine Equations (14.73 - 14.75) to obtain an

expression for B' ,

B' (14.76)

The product Rp~~ is equal to the circular backlash B, as

we proved in Equation (4.36), and we can express the remaining

terms by means of Equat ion (13.95),

When these substitutions are made in Equation (14.76), we

obtain the relation we require between B' and B,

B' (14.77)

Finally, we combine Equations (14.72 and 14.77), to

express the backlash along the common normal in terms of the normal backlash,

B' (14.78)

Position and Orientation of the Contact Line

We now consider once again the contact line between a

pinion and rack. We stated earlier that the contact line

coincides with a generator of the pinion, and we will now

determine, for any specified angular position of the pinion,

which generator is in contact with the rack tooth, and the

orientation of the line of contact on the rack tooth. These

results were not required when we described the meshing geometry of a pinion and rack, and this is the reason why they


have been left to the end of the chapter. However, they will be used in the next chapter, when we discuss the meshing of a

crossed helical gear pair.

We start by specifying the exact directions of the various sets of unit vectors that will be used in the

analysis. The vectors n x ' ny and n z are fixed in the pinion, with n z along the gear axis, and nx in the direction of the x coordinate axis, which coincides with a tooth center-line in the transverse section z=O, as shown in Figure 14.10. The

vectors n~, nT/ and nS are fixed in the rack, with n~

perpendicular to the rack pitch plane, in the direction from

the pinion towards the rack. The vectors nT/ and nS form a plane parallel to the pitch plane, and their directions are perpendicular to and parallel wi th the rack teeth, as shown in Figure 14.11. We define a new set of fixed unit vectors,

nx(O), ny(O) and nz(O), as the directions of n x ' ny and n z when the pinion is in the reference position, from which the angle ~ is measured. For a spur gear, the angular position ~ was defined as the angle between the line CP and the x axis. In order to remain consistent with that definition, we now

y

x

Pitch cylinder

Base cylinder

Figure 14.10. Transverse section z=O.



choose the direction of nx(O) perpendicular to the rack pitch plane, so that it coincides with n~. The set of vectors nx(O), n (0) and n (0) is shown in Figure 14.11, and we can now write y z down the relations between these vectors and the set n~, n~ and nS' We have proved in Equation (14.19) that the rack helix angle ~~ is equal to the operating helix angle ~p of the pinion, and in relating the various sets of unit vectors, it will be convenient to describe all the angles in terms of those defined on the pinion. We therefore obtain the following relations from Figure 14.11,

(14.79)

cos ~p n~ + sin ~p nS (14.80)

(14.81)

Figure 14.12 shows the transverse section through the

pinion at plane z=O, when the pinion has rotated through an

angle p. The vectors n x ' ny and n z are expressed in terms of their reference directions by the following set of relations,

396

Tooth centerline


Figure 14.12. Transverse section z=O, when the pinion is in the angular position fl.

(14.82)

(14.83)

(14.84)

We now introduce another pair of unit vectors fixed in the rack. A normal section through the rack tooth is shown in Figure 14.13, and the vectors nnr and nTr are defined in the direction normal to the tooth surface, and in the direction of the tangent pointing towards the tooth tip. The normal pressure angle I/l~r of the rack is equal to the operating normal pressure angle I/l np of the pinion, as we proved in Equation (14.20). Once again, we use the angle defined on the pinion when we express the relations between the different

un i t vectors,

- sin I/l np n~ - cos I/l np n'T/ (14.85)

(14.86)

In order to determine which of the pinion generators is in contact with the rack, we make use of the following condition. If A is a contact point on the pinion tooth, then


Figure 14.13. Normal section through the rack tooth.

the unit vector n~, which is normal to the plnlon tooth surface at A, must point in exactly the opposite direction

to nnr' the unit vector normal to the rack tooth surface. In other words, the two unit vectors must be equal and opposite,

(14.87)

The vector nnr is given by Equation (14.85), and n~ was expressed by Equation (13.79) in terms of the unit vectors

n x ' ny and n z '

In this equation, G is the point where the generator through A

touches the base cylinder. By using the condition given in Equation (14.87), we can determine the value of eG, and hence

we can find the pos i t ion of G.

Before we substitute for n~ and nnr in Equation (14.87), we use Equations (14.79 - 14.84) to express both vectors in

terms of nx(O), ny(O) and nz(O),

n~ cosl/lb [- sin(eG+tnnx(O) + cos(eG+p)ny(O)] - sinl/lbnz(O) (14.89)


nnr - cos .pb [sin <l>tp nx(O) + cos <l>tp ny(O)] + sin .pb nz(O) (14.90)

The derivation of Equation (14.90) involves some of the trigonometric relations which were proved in Chapter 13. However, it is not practical, at this stage of the book, to include each step of every proof. The results can always be verified by the method described earlier, in which all the

angles are expressed in terms of <l>np and .pb. The expressions for n~ and nnr given by Equations (14.89

and 14.90) are substituted into Equation (14.87), and the value of 8G which satisfies the equation can then be seen by inspection,

- fJ - <l>tp (14.91)

This expression for 8G can be given a physical interpretation. Figure 14.14 shows a typical transverse section through the pinion, with the line of action in that section touching the base cylinder at E. If point A of the tooth profile is in contact with the rack tooth, it must lie

c

Figure 14.14. Transverse section through the pinion and rack.


in the plane of action, and the generator through A must touch the base cylinder at a point on the line where the plane of action touches the base cylinder. This means that G lies on the axial line through E, and its angular coordinate eG is given by the expression in Equation (14.91).

The direction n: of the generator through G and A was given by Equation (13.76),

G . .1. • eG • .1. eG .1. (14 92) n", - SIn 'l'b sIn nx + sIn 'l'b cos ny + cos 'l'b n z •

If we substitute the value for eG given by Equation (14.91), and then express the unit vectors in terms of nx(O), ny(O) and nz(O), we obtain the following expression for n:,

n: = sin VJb[sin "tp nx(O) + cos "tp ny(O)] + cosVJb nz(O) (14.93)

The value of eG is also substituted into Equation (14.89), and A we obtain the vector nn' also expressed in terms of nx(O),

ny(O) and nz(O),

n~ = cosVJb[sin"tpnx(O) + cos "tpny(O)] - sinVJbnz(O) (14.94)

In Equations (14.93 and 14.94), the terms in the square brackets represent a unit vector in the plane of action, which is shown in Figures 14.14 and 14.15. The equations therefore prove that the generator through A and the normal to the tooth surface at A each lie in the plane of action. The generator makes an angle VJ b with the z axis, as we proved in Chapter 13, and the normal to the tooth surface is perpendicular to the generator.

These conclusions apply to any point A of the tooth surface which lies in the plane of action. Hence, if we consider a point A' on the opposite face of the tooth, and A' lies in the other interior common tangent plane to the base

cylinders, then the same proof shows that the generator through"A' and the normal to the tooth surface at A' must lie in that plane. This is the result that was used earlier in this chapter, when we discussed the backlash along the common


G z=o E z=F

Plane z Plane of action

Figure 14.15. Position of the contact line.

normal in a helical gear pair. In order to describe the position of G, we find its

position relative to PO' the pitch point in the transverse plane z=O. The vector from Po to G can be expressed as follows,

(14.95)

where Co is the center of the gear in the transverse section at plane z=O. The vector from Po to Co can be written down by inspection, and we obtain the vector from Co to G, if we replace A and R in Equation (13.46) by G and Rb ,

POCO P

In the second of these equations, GO is the point

helix through G cuts the transverse section z=O.

helix through G and GO is the base helix, it

transverse section z=O at point BO' and GO

(14.96)

where the Since the meets the therefore


coincides with BO' The position of BO is shown in Figure 14.10, and we can use that diagram to express the angular coordinate eGO in terms of the transverse tooth

thickness at the pi tch cylinder,

~. 2R + lnv cP tp p

(14.98)

The angles eG and eGO in Equation (14.97) are replaced by trre

expressions in Equations (14.91 and 14.98). We now combine Equations (14.95 - 14.97), and express all the unit vectors in terms of n (0), n (0) and n (0). The resulting equation can be

x y z simplified by means of Equation (14.93), and we obtain the following expression for the position of G,

(14.99)

This equation can be Figure 14.15, which shows

interpreted wi th the help of

the plane of action. The s coordinate is defined, as always, as the vertical distance of

any contact point above the axial line through PO' If sb is the coordinate of the contact point in the transverse section z=O, its value is given by Equation (3.28),

(14.100 )

The first term in Equation (14.99) can therefore be interpreted as the horizontal distance between Po and Gp ' the point where the generator through A cuts the axial line through PO' The second term represents the distance along the generator from Gp to G, the point where the generator touches the base cylinder.

If we use the coordinate s to specify the position of point A on the generator, we can describe the position of A

with the help of Figure 14.15. The vector from Po to A is

written as the sum of the vector from Po to Gp ' and the vector from Gp to A,

Rb (ttn G -=..I;;.+/J)n + s n tan "'b 2Rp z sin "'b jJ

( 14.101 )

By using Equations (14.7 and 14.71) to express Rb and ttp in


terms of Rp and t np ' we can give the position of point A in a form that will be used in the next chapter,

The last question to be considered in this chapter is the orientation of the contact line on the rack tooth face. The direction of the generator in contact with the rack tooth was given by Equation (14.93). We now use Equations (14.79 - 14.81) to express nx(O), ny(O) and nz(O) in terms of nt, n~ and nS. The result we obtain can be expressed in the following form,

(14.103)

where nTr is the tangent vector in the rack tooth face, given by Equation (14.86). The angle ~p is the value at the pitch cylinder of the generator inclination angle ~R' which was defined in Equation (13.86). The expression for n: in Equation (14.103) shows that the contact line makes an angle ~p with the tip of the rack tooth, and ~p is therefore called the contact line inclination angle.

We have found the position and orientation of the contact line, for the case of a pinion meshed with a rack. There is no need to repeat the analysis for the case of two meshing gears. We simply consider an imaginary rack between the teeth, and we know that the contact line between the gears coincides with the contact line between either gear and the imaginary rack. The orientation of the contact line is therefore given by Equation (14.93), where the unit vector nx(O) is now defined in the direction of a common perpendicular joining the gear axes, and Equations (14.99 and 14.101) can then be used to give the positions of points G and A on the contact line.

Helical Gears in Mesh 403

Numerical Examples

Example 14. 1 A helical gear pair has a normal module of 8 mm, a normal

pressure angle of 20°, and a helix angle of 30°. The tooth

numbers are 36 and 75, with the pinion having the right-handed

helix. The normal tooth thicknesses are 12.847 mm and

12.750 mm; the tip cylinder diameters are 350.0 mm and

710.0 mm; the face-width is 60.0 mm, and the gears are to

operate at a center distance of 514.0 mm. Calculate the operating pressure angles, the operating helix angle, the

various types of contact ratio, and the three types of backlash.

mn=8, ~ns=20°, ~Sl=300, ~s2=-30°, N1=36, N2=75 t ns1 =12.847, t ns2 =12.750, RT1 =175.0, RT2 =355.0

F=60.0, C=514.0

RS1 = 166.277 mm

Rs2 346.410

~ts 22.796° Rb1 153.289

Rb2 319.352

~b = 28.024°

Ptb = 26.754

tts1 14.834 tts2 = 14.722

Rp 1 = 1 66. 703 mm

RP2 = 347.297

qlt = 23.142°

'" = 23 142° "'tp . ~p = 30.063°

~np = 20.299°

1.400

1.194

2.594

(13.150)

(13.150)

(13.151)

(13.152) (13.152)

(13.154) (13.43)

(13.113)

(13.113)

(14.28) (14.29)

(14.30)

(14.34)

(13.156)

(13.157)

(14.64)

(14.68)

(14.62)

404

Example 14.2

Ptp = 29.095 mm

ttp1 = 14.511 ttp2 = 14.007

B = 0.577

Bn = 0.499 B' = 0.468 mm


(14.33) (13.114) (13.114)

(14.69) (14.72)

(14.78)

Find the pressure angles, the helix angle, and the pitches for the imaginary rack which could mesh with the gear pair specified in the previous example. Also determine the angle that the contact lines would make with the tooth tips of the imaginary rack.

As we proved earlier in the chapter, the parameters of the imaginary rack are all equal to the corresponding parameters of the gears, measured at the pitch cylinders. And the angle between each contact line and the tooth tip of the

rack is equal to Pp ' the contact line inclination angle. We can therefore write down immediately the following results.

~tr ~tP = 23.142° w' W = 30.063° r p

~' ~ = 20 299° nr np • Ptr = Ptp = 29.095 mm P~r = Pnp = 25.181 mm

Pp= 11.355°

(14.33) (14.11) (13.87)

Introduction

Chapter 15 Crossed Helical Gears

In this chapter we will describe the meshing geometry of a pair of helical gears, when their axes are not parallel. Each gear of the pair is an ordinary helical gear, of the type described in Chapter 13. We proved in Chapter 14 that, when two helical gears are mounted on parallel shafts, they must have helix angles ~s that are equal and opposite. We will now show that two helical gears whose helix angles are not equal and opposite can be mounted on non-parallel shafts, and that, provided certain conditions are met, the angular velocity ratio of the two gears will remain constant.

Crossed helical gears can be used whenever it is necessary to transmit motion between two non-intersecting and non-parallel shafts. However, as we will show in this chapter, the contact between each pair of meshing teeth takes place theoretically at only one point, instead of along a line of contact, as it does when the shafts are parallel. In practice, the contact extends over a small area, due to the deformation of the teeth, but the contact stress is nevertheless very high. The amount of power that can be transmitted is therefore severely limited, and for this

reason, crossed helical gears are used mostly in instrument

gear trains rather than in power gearing. However, even

though their use is far less frequent than that of

parallel-axis gears, it is important to understand the

meshing geometry of a crossed helical gear pair, since it

forms the theoretical basis of the hobbing process. The

technique of hobbing is the most common method by which gears are cut. It is essentially a process in which the gear blank

406 Crossed Helical Gears

and the hob operate as a pair of crossed helical gears. A brief description of hobbing was given in in Chapter 5, but

for a more detailed explanation, it is first necessary to

describe the meshing geometry of crossed helical gears.

Rack and Pinion

A helical rack and pinion, of the type discussed in Chapter 14, is shown in Figure 15.1. The rack is supported in guides which allow movement only in the direction perpendicular to the pinion axis. We showed in Chapter 14 that a rack and pinion will mesh correctly, provided the normal base pi tch and the transverse base pi tch of the rack are equal to those of the pinion. The pi tch cylinder radi us of the pinion is given by Equation (14.6),

(15.1)

where Ptr is the transverse pitch of the rack. We also proved that on the pinion the normal and transverse pitches, the

Figure 15.1. A helical rack and pinion.

Rack and Pinion 407

helix angle, and the normal and transverse pressure angles, all measured at the pitch cylinder, are equal to the corresponding quantities on the rack.

The operating helix angle ~p and the pressure angle ~np of the pinion Equation (13.88),

sin ~b cos ~np

operating normal must satisfy

(15.2)

where ~b is the base helix angle of the pinion. Since ~p and

~np are equal to ~~ and ~~r' the corresponding angles on the rack must also satisfy the same relation,

sin ~~ (15.3)

Figure 15.2 shows a pinion which is identical to the pinion in Figure 15.1, but it is meshed this time with a rack whose guides allow movement in a different direction from the ra€k in Figure 15.1 The teeth of both racks have the same

(1T/2)-"'~

Figure 15.2. A crossed helical rack and pinion.

408 Crossed Hel ical Gears

normal section, and the helix angle "'~ between the pinion axis and the rack teeth is the same in each case, so for each rack and pinion, the tooth shapes in the regions of the contact I ines are ident ical.

In order to determine the velocity of the rack in Figure 15.2, we compare the motions of the two racks when the pinions have the same angular velocity. The velocity of the rack in Figure 15.2 can be resolved into two non-perpendicular components v~ and v~, the first in the direction perpendicular to the pinion axis, and the second parallel to the rack teeth, as shown in Figure 15.2. A motion of the rack in the direction of its teeth would cause no rotation of the pinion, so the only component of the rack velocity related directly to the pinion angular velocity is v~. The value of v~ for the rack in Figure 15.2 is therefore equal to the velocity of the rack in Figure 15.1, and the other component v~ must be such that the resultant velocity is in the direction allowed by the guides. Since the angle between the component directions is fixed, the components· v~ and v~ remain in a constant ratio, and a constant angular velocity of the pinion will therefore produce a constant velocity of the rack.

To distinguish between the two types of rack shown in Figures 15.1 and 15.2, we regard each rack as the limiting case, when the number of teeth becomes infinite, of a gear whose axis is perpendicular to the direction of motion of the rack. In the case of Figure 15.1, the axis of this gear is parallel to the pinion axis, so the rack and pinion can be thought of as a special case of a parallel-axis gear pair. On the other hand, the rack in Figure 15.2 is the limiting case of a gear whose axis is not parallel to the pinion axis, and the angle between these two axes is shown in the diagram as t. Since the axes are not parallel, the rack and pinion can be regarded as a crossed helical gear pair. The two types of rack and pinion will be referred to as either a parallel-axis, or a crossed helical, rack and pinion.

The conclusion reached in the earlier discussion, relating to the velocity of the crossed helical rack, can now be stated in a more general manner. In a crossed helical rack and pinion, the relation between the rack velocity and the

Transverse Direction in the Rack 409

pinion angular velocity will remain constant, provided the Law of Gearing is satisfied for a parallel-axis rack and pinion with the following characteristics. The parallel-axis rack has the same normal section and the same helix angle as

the crossed helical rack, and the two pinions are identical. A

rack and pinion defined in this way will be called the

e~uivalent parallel-axis rack and pinion.

Transverse Direction in the Rack

A transverse section of a gear was defined in Chapter 13 as any section perpendicular to the gear axis. This definition did not seem appropiate for a rack, since it does not turn about an axis. We therefore defined a transverse section of the rack, when we discussed the meshing geometry of a parallel-axis rack and pinion in Chapter 14, as any section

perpendicular to the pinion axis. If we now regard the rack and pinion in the manner suggested earlier, as a special case of a parallel-axis gear pair, it can be seen that the two definitions for the transverse sections in the rack are equivalent.

In a crossed helical gear pair, a transverse section of

ei ther gear is again defined as a section perpendicular to the corresponding gear axis. In this case, however, a transverse section in one gear is not parallel to a transverse section in the other, since the two gear axes are not parallel. In each gear, a transverse plane coincides with a plane of motion. For a crossed helical rack and pinion, we define a transverse section of the rack in a manner which is consistent with the definition of the transverse sections in a crossed helical

gear pair. We regard the rack, once again, as the limiting

case of a gear whose axis is perpendicular to the direction of motion of the rack, and a transverse section in the rack is

therefore defined as any section perpendicular to that axis.

In other words, the transverse direction in a helical rack is

parallel with the direction of motion allowed by the guides.

This definition remains valid, whether the rack is meshed in a parallel-axis or in a crossed helical manner.

The two racks in Figures 15.1 and 15.2 are identical in


their normal sections, so all quantities defined in the

normal section, such as P~r' P~br and '~r' have the same values in each rack. We also stated that the two racks are

oriented so that they have the same helix angle ~~. However, we have shown that the transverse directions in the two racks are different, so quantities defined in the transverse

section, such as P~r' P~br and '~r' do not have the same values in each rack.

Pinion Pitch Cylinder

We proved in Chapter 1 that the motion of a spur gear and rack can be regarded as if the pitch circle of the gear made rolling contact with the pitch line of the rack. The same analogy can be used for a parallel-axis helical rack and pinion, but not for a crossed helical rack and pinion, since the direction of motion of the rack is not perpendicular to the pinion axis. We therefore have to decide, for the case of the crossed helical rack and pinion, whether- there is any need for a pitch cylinder, and if so how it should be defined.

The concept of the pitch cylinder does turn out to be useful in the geometric analysis of a crossed helical rack and pinion. We start by defining it, and then we will explain the reasons for the way in which it is defined. The pitch cylinder of a pinion, meshed with a rack in a crossed helical manner, is defined as the cylinder with the same radius as the pinion pitch cylinder in the equivalent parallel-axis rack and pinion. With this definition, the radius of the pinion pitch cylinder in Figure 15.2 is equal to that in Figure 15.1.

The pitch cylinder radius for the pinion in Figure 15.1 was given by Equation (15.1) in terms of the transverse pitch

of the rack,

We cannot use the same equation for the pinion in Figure 15.2, because the transverse pitches of the two racks are different. We therefore use Equation (14.1) to express the transverse pitch of the rack in Figure 15.1, in terms of its

Pinion pi tch Cylinder 411

normal pitch and its helix angle. We then obtain the following expression for the pitch cylinder radius,

271' cos I/I~ (15.4)

This equation can now be used to give the pitch cylinder radius for the pinion in Figure 15.2, since the values of the rack normal pitch and the rack helix angle are the same in the

two diagrams. The most important advantage of defining the pi tch

cylinder in this manner is that we know, from the meshing geometry of the parallel-axis rack and pinion described in Chapter 14, that the normal pitch, the normal pressure angle, and the helix angle of the rack are equal to the corresponding quantities on the gear, measured at this particular radius,

1/1' r

4>np

(15.5)

(15.6)

(15.7)

There is, of course, no longer any relation between the

transverse quantities of the rack and those of the pinion, since the transverse quantities on the rack are all different from those on the parallel-axis rack in Figure 15.1.

The only disadvantage we find, when the pitch cylinder is defined in this way, is that the rack velocity is no longer equal to RpW, as it is in the case of a parallel-axis rack and pinion. However, it is not difficult to express the rack velocity in terms of the pinion angular velocity. We proved

earlier that if the pinions in Figures 15.1 and 15.2 have the

same angular velocity, the velocity of the rack in Figure 15.1

is equal to v~, the velocity component of the rack in Figure 15.2 perpendicular to the pinion axis. We know that the

rack in Figure 15.1 has a velocity equal to R w. For the rack p

in Figure 15.2, it is therefore the velocity component v~,

rather than the resultant veloc i ty, which is equal to Rpw,

v' r (15.8)


We now use the velocity vector triangle in Figure 15.2,

to express the resultant rack velocity vr in terms of the pinion angular velocity,

cos "'~ v~ cos("'~-~)

cos "'~ cos("'~-~) Rpw (15.9)

If we combine Equations (15.4 and 15.9), we obtain an alternative expression for the rack velocity,

211' cos("'~-~) W (15.10)

This equation could have been derived directly, using the method described in Chapter 1, from the consideration that in any period of time the same number of pinion teeth and rack teeth must pass through the meshing zone.

We proved in Equation (15.6) that the normal pressure angle of the pinion at, its pitch cylinder is equal to the normal pressure angle of the rack. This property is true, not only for a crossed helical rack and pinion, but also for a parallel-axis rack and pinion, and even for a spur gear and rack. For this reason, the pitch cylinder of a pinion meshed with a rack has sometimes been defined as the cylinder on which the normal pressure angle is equal to that of the rack. However, this particular definition was not used in this book, because it would have been difficult to explain the reason for the definition in the early part of the book. By defining the pitch point in terms of the Law of Gearing, and then defining the pitch circle as the circle through the pitch

point, the book has followed the order in which the geometric

theory of gears was originally developed.

Minimum Condi tions for Correct Meshing

We proved earlier in this chapter that the relation

between the rack velocity and the pinion angular velocity is constant for a crossed helical rack and pinion, provided the

Law of Gearing is satisfied for the equivalent parallel-axis rack and pinion. ~he minimum conditions for correct meshing

of a parallel-axis rack and pinion were discussed in

Minimum Conditions for Correct Meshing 413

Chapter 14, and we showed that the normal base pitch and the transverse base pitch of the rack must be equal to those of the pinion. The equivalent parallel-axis rack was defined as a rack with the same normal section as the crossed helical rack, so their normal base pitches must be equal, and the normal base pitch of the crossed helical rack must therefore

be equal to that of the gear,

(15.11)

The transverse sections of the two racks are not the same, so their transverse base pitches are different, and there is no relation between the transverse base pitch of the crossed helical rack, and that of the pinion. However, the equivalent parallel-axis rack was also defined as having the same helix angle as the crossed helical rack, and we showed that the helix angle of a parallel-axis rack must satisfy Equation (15.3). Hence, the helix angle of the crossed helical rack must satisfy the same equation,

sin 1/1~ sin 1/Ib

cos I/I~r (15.12)

We have proved that if the equivalent parallel-axis rack and pinion mesh correctly, then the crossed helical rack must satisfy Equations (15.11 and 15.12). To show that the reverse is true, we need to prove that when the two equations are satisfied, the normal and transverse base pitches of the equivalent parallel-axis rack are equal to those of the pinion.

We know immediately, from Equation (15.11), that the normal base pitches are equal. The proof that the transverse base pi tches are also equal is given by the following sequence of equations.

P~r cos 1/1~ Ptr cos I/Itr cos .1.' 2 2

'l'r v'(cos 1/1' + tan 1/1' ) (14.1,14.2) r nr

P~r cos I/I~r cos 1/Ib

(15.12) (14.3) (15.11) (13.44)


We have shown, in effect, that Equations (15.11 and 15.12) are equivalent to the two minimum conditions for correct meshing of the parallel-axis rack and pinion. The

crossed helical rack has the same normal section as the

parallel-axis rack, and its teeth make the same angle w~ with the gear axis. We now apply these results, to obtain the two minimum conditions which must be satisfied for correct

meshing of the crossed helical rack and pinion. The normal base pitch of the rack must be equal to that of the pinion,

and the rack must be oriented so that the angle w~ between its teeth and the pinion axis has the value given by Equation (15.12).

A Crossed Helical Gear Pair

We now consider the meshing of two helical gears, mounted on non-parallel, non-intersecting shafts. There is one line joining the two gear axes which is perpendicular to both axes, and this is called the common perpendicular to the

axes. It is also the shortest distance between the two axes. The points where the common perpendicular intersects the axes

are labelled COl and CO2 ' as shown in Figure 15.3, and the distance between these two points is known as the center distance C of the gear pair. The angle between the gear axes is called the shaft angle E.

In order to determine whether the gear pair will mesh

correctly, we consider whether it is possible to insert an imaginary rack between the teeth. I f a constant angular

velocity of gear 1 would cause a constant velocity of the

imaginary rack, and this in turn would cause a constant angular velocity of gear 2, then the angular velocity ratio is

constant for the two gears. When a rack meshes with a pinion, the pitch plane of the

rack is parallel to the plane of the rack tooth tips, and it

touches the pinion pitch cylinder. Hence, if an imaginary rack meshes simultaneously with both gears in Figure 15.3, the two pitch planes in the imaginary rack must be parallel to

each other, and in order that they can touch the pitch

cylinders of the two gears, they must be perpendicular to the

A Crossed Helical Gear Pair

Axis of gear 2

415

Line through C01 parallel to gear 2 axis

Pitch plane 1 of imaginary rack



Figure 15.3. The pitch cylinders in a crossed helical gear pair.

line of centers C01 C02 ' For correct meshing between each gear and the imaginary

rack, Equations (15.11 and 15.12) must be satisfied for each

gear,

P~br Pnb1 (15.13)

sin 1/Ib1 sin 1/1~ 1 cos qJ~r

(15.14)

P~br Pnb2 (15.15)

sin 1/Ib2 sin 1/1~2 cos qJ~r

(15.16)

In addition, since each rack helix angle is defined as the

angle between the direction of the rack teeth and the

corresponding gear axis, the sum of the rack helix angles must

be equal to the shaft angle,

(15.17)


If we can find values of ~'1' ~'2 and~' which satisfy r r nr Equations (15.13 - 15.17), then we can construct the imaginary rack, and we have shown that the crossed helical gear pair will mesh correctly. One condition that must be satisfied, which can be seen immediately from Equations (15.13 and 15.15), is that the normal base pitches of the two gears must be equal,

( 15.18)

To solve the remaining equations, we first eliminate ~~r from Equations (15.14 and 15.16),

sin ~~1 sin ~b1

(15.19)

Equations (15.17 and 15.19) can now be solved to give the rack helix angles,

sin ~b1 sin 2: tan ~~ 1 (sin ~b1 cos 2: + sin ~b2)

(15.20)

tan ~~2 sin "'b2 sin 2:

(sin "'b2 cos 2: + sin ~b1 ) (15.21)

and the normal pressure angle ~~r is then found from Equations (15.14 or 15.16),

cos ~~r (15.22)

We have therefore shown that, provided the normal base pitches of the two gears are equal, it is possible to find an imaginary rack which can mesh simultaneously with each of the gears. Hence, the requirement that the normal base pitches are equal represents the only condition that must be satisfied, to ensure that the crossed helical gear pair will

mesh correctly. Each gear makes contact with the imaginary rack along a

line in the rack tooth face. However, these lines are not parallel, as they are in the case of a parallel-axis gear pair. Later in this chapter, we will determine the angle between the two lines, but for the moment it is sufficient to

Pi tch Cylinders 417

note that they are not parallel. This means that the gears are in contact at one point only, which is the point where the two

lines intersect.

pi tch Cylinders

We showed in Equations (15.5 - 15.7) that in a crossed

helical rack and pinion, the operating normal pitch Pnp' the operating normal pressure angle ¢np' and the operating helix angle Wp of the pinion are equal to the corresponding quantities on the rack. The same results are true when a gear is meshed with an imaginary rack. Hence, for the crossed

helical gear pair, the operating helix angles WP1 and Wp2 are given by Equations (15.20 and 15.21), and the operating normal pressure angles of the two gears are equal, and are given by Equation (15.22),

sin wb1 sin l:

(sin Wb1 cos l: + sin wb2) sin wb2 sin l:

(sin Wb2 cos l: + sin wb1 )

sin wb1 sin wb2 sin WP1 sin Wp2

(15.23)

(15.24)

(15.25)

Since the helix angles at the pitch cylinders are now known, we can use Equation (13.146) to find the corresponding pi tch cylinder radi i ,

RP1 Rb1 tan WE1

(15.26) tan Wb1

RP2 Rb2 tan WE2

(15.27) tan Wb2

It is interesting to note that, for a crossed helical

gear pair, there is no relation between the pitch cylinder

radii and the center distance C. In particular, the center

di stance is not generally equal to the sum of the pi tch

cylinder radii. The pitch cylinders, therefore, do not touch

each other, as they do in the case of a parallel-axis gear

pair, and the analogy of rolling contact is no longer applicable.


Special Solution

In certain cases, the shafts of a crossed helical gear pair are oriented so that the shaft angle is equal to the sum of the gear helix angles,

(15.28)

We know that the helix angles are related to the normal pressure angle by Equation (13.99),

sin "'b1 sin"'sl cos cf>ns

(15.29)

sin "'b2 sin "'s2 cos cf>ns

(15.30)

We have assumed, in these equations, that the normal pressure angles of the two gears are equal. This assumption is correct in all cases of practical interest, since gear pairs are invariably designed so that each gear has the same normal module and the same normal pressure angle. By combining Equations (15.29 and 15.30), we obtain a relation between the hel ix angles of the two gears,

sin "'s1 sin "'b1

(15.31)

For the case we are considering, when the shaft angle is equal to the sum of the helix angles, "'s1 and "'s2 satisfy Equations (15.28 and 15.31), which are identical to Equations

(15.17 and 15.19) satisfied by "'~1 and "'~2. Hence, for this particular case, the imaginary rack helix angles "'~1 and "'~2' and therefore the gear operating helix angles "'p1 and "'p2' are equal to the gear helix angles'" s 1 and", s2.

(15.32)

(15.33)

Since the operating helix angles are equal to the helix angles, it means that the pitch cylinders coincide with the standard pitch cylinders. Once again, there is no relation

Standard Center Distance and Standard Shaft Angle 419

between the center distance C and the sum of the pi tch

cyl inder radi i.

Standard Center Distance and Standard Shaft Angle

We define the standard center distance Cs as the sum of the standard pitch cylinder radii, and the standard shaft

angle I:s as the sum of the gear helix angles,

(15.34)

(15.35)

There is no need to design a gear pair so that the actual

center distance C and shaft angle I: are equal to Cs and I: s ' However, we pointed out earlier that gears can only be cut with a limited amount of profile shift, if standard cutters are to be used. For this reason, crossed helical gear pairs are generally designed so that the values of C and I: are ei ther equal, or approximately equal, to Cs and I:s '

It is one of the major advantages of all involute gears that small errors or changes in the center distance do not affect the operation of a gear pair, since the value of C does

not have to be equal to the standard value Cs ' Crossed helical gears have the further advantage, that small errors or changes can also be made in the shaft angle, and in spite of this, the angular velocity ratio remains constant.

Relations Between the Uni t Vectors

Before we discuss the meshing geometry in detail, it is

necessary to define the directions of the various sets of unit

vectors, and this is done in a manner which is consistent with

the definitions given at the end of Chapter 14.

The n~ direction is chosen perpendicular to the pitch

planes of the imaginary rack, in the direction from COl

towards CO2 ' The two pitch cylinders and the imaginary rack pitch planes are shown in Figure 15.4, viewed in the n~

420

Tooth direction of imaginary rack


Crossed Helical Gears

Imaginary rack Pitch cylinder

of gear 1


direction. The unit vectors n~ and nS are defined, as usual, in the directions perpendicular to and parallel with the teeth of the imaginary rack.

We use points C01 and CO2 as the origins of the x,y,z coordinate system in each gear. This means, of course, that the origin of each coordinate system no longer lies in the end face of each gear, as it did when we discussed the geometry of parallel-axis gears in Chapter 14. There is a set of unit vectors nx1 ' ny1 and nz1 fixed in gear 1, and a second set nx2 ' ny2 and nz2 fixed in gear 2. The reference directions, from which the gear rotations are measured, are chosen so that nz1 (0) and nz2 (0) are parallel with the gear axes. nx1 (0) and nx2 (0) are parallel to line C01 C02 ' so that they are perpendicular to the imaginary rack pitch planes, and in each case they point in the direction from the gear towards the imaginary rack. Finally, ny1 (0) and ny2 (0) complete the

right-handed sets. The relations between the reference directions of the

gears and the vectors fixed in the imaginary rack can be read from Figure 15.4. Once again, it is more convenient to express these relations in terms of the operating helix angles of the

gears, rather than the rack helix angles.

Relations Between the Uni t Vectors 421

(15.36)

cos "'p1 n1/ + sin "'p1 nS (15.37)

(15.38)

(15.39)

(15.40)

(15.41)

We will make use again of nnr and nTr , the unit vectors perpendicular and tangent to the tooth profile in a normal

section through the imaginary rack. The vectors n E and n1/ can be expressed in terms of nnr and nTr by means of Equations ( 1 4 • 85 and 1 4 • 86) ,

- sin 9>np nnr - cos 9>np nTr (15.42)

- cos 9>np nnr + sin 9>np nTr (15.43)

After the gears have rotated through angles P1 and P2 from their reference positions, the vectors fixed in each gear are given in terms of their reference directions by Equations (14.82-14.84). Since the relations are the same for each gear, they will only be written out once, and it is understood that the subscript i stands for either 1 or 2, depending which gear is referred to.

nxi

n. yl

(15.44)

(15.45)

(15.46)

The two pitch cylinders and the imaginary rack pitch

planes are shown again in Figure 15.5, viewed this time in the

negative n1/ direction. Since the sum of the pitch cylinder radii is not generally equal to the center distance, we define

422


CO2 --.-----+-+ ---- --+--


r Axis of gear 2 -+---,-

Rp2

C /,"Pitch plane 2 of

---"'----r-'---t--:.;::.......,-----"-....<'---~I i magi nary rack

/ Axis of gear 1

\ Pitch cylinder of gear 1


Figure 15.5. View of the pitch cylinders in the direction perpendicular to the teeth of the imaginary rack.

a length ~Cp as the difference,

( 15.47)

When we discussed the geometry of a helical rack and pinion at the end of Chapter 14, we introduced a point PO' defined as the pitch point in the transverse section z=O. If Co is the center of the pinion in the same transverse section, then Po lies at the foot of the perpendicular from Co to the pitch plane of the rack. If the two gears in Figure 15.5 are regarded as meshing with the imaginary rack, they each have a

pitch point in their transverse sections zl=O and z2=O, and these points are shown in the diagram as POl and P02 • They lie on line C01 C02 ' and the distance between them is ~Cp.

The points COl and CO2 are satisfactory as reference points for each gear separately, but we need a single point to serve as a reference for the gear pair. We introduce a point P

on line C01 C02 ' as shown in Figure 15.5, at a distance ~Cl

from POl and ~C2 from P02 • The values of ~Cl and ~C2 will be chosen later, but of course their sum must be equal to ~Cp'

Path of Contact 423

(15.48)

The point P is not a pitch point, since it does not lie on either pitch cylinder. It will simply be used as a reference

point, from which we can locate the position of the contact

point.

Path of Contact

For

typical

Equation

an ordinary

point A

(14.102),

rack and pinion, the vector from Po to a on the contact line was given by

sin1 1/Ip (~tnp + Rp cos I/Ip tnnz(O) + sins I/Ib n: (15.49)

where n: is the direction of the generator through A, and the coordinate s is the perpendicular distance between point A

and the axial line through PO. For the crossed helical gear pair shown in Figure 15.5,

we can use Equation (15.49) to write down expressions for the vector from P to a typical point A1 on the contact line between gear 1 and the imaginary rack, and for the vector from P to a corresponding point A2 on gear 2.

s 1 G 1 + . ,t, n Sln "'b1 /.11

(15.50)

aC2n~ - Sin1I/1p2(~tnp2 + Rp2cOSl/lp2P2)nz2(O)

( 15.51 )

If A1 and A2 are to be the points where the two gears are

in contact, the vectors from P to A1 and A2 must be equal,

(15.52)

We substitute the expressions for these vectors given by Equations (15.50 and 15.51), and we obtain a vector equation


that must be satisfied by Pl' P2 , sl and s2' if Al and A2 are to be the contact points.

In order to replace the vector equation by three scalar equations, we express every unit vector in terms of a single set of unit vectors, and the most convenient set turns out to

be nnr' nTr and nS. The vector n~ was given already by Equation (15.42), and we can use Equations (15.38, 15.41 and

15.43) to express n zl (O) and n z2 (0) in terms of nnr' nTr and n S'

sin "'p1 cos 4>np nnr - sin "'pl sin 4>np nTr + cos"'pl nS (15.53)

- sin "'p2 cos 4>np nnr + sin "'p2 sin 4>np nTr + cos "'p2 nS (15.54)

The vectors n: ~ and n:~ were given by Equat ion (14.93),

G, njlf sin "'bi [sin 4>tpi nxi(O) + cos 4>tpi nyi(O)]

+cos"'b' n .(0) (15.55) 1 Zl

We have already made the set of vector transformations for gear 1, with the result given in Equation (14.103),

- sin vpl nTr + cos vpl nS (15.56)

The corresponding transformations for gear 2 are the same, apart from some differences in sign, and we obtain the

following expression for n:~,

(15.57)

The unit vectors n:~ and n:~ represent the directions of the lines where each gear is in contact with the imaginary

rack. The angle between these unit vectors is therefore the angle between the two contact lines, and it can be seen from Equations (15.56 and 15.57) that this angle is equal to

(VPl +V P2 )· b · h 'f PA 1 d PA2' We now su stltute t e expressIons or p an pInto

Equation (15.52), expressing all the unit vectors in terms of

Path of Contact 425

nnr' nTr and n S' and then we equate the coefficients of the three unit vectors on each side of the equation. We make use of the relations between the various angles, derived in Chapter 13, and we obtain the following three scalar equations,

flC p tan tP np (15.58)

+ s2 flC p cos ~b2 sin tP np

(15.59)

s1 s2 cos ~b1 tan ~p1 cos ~b2 tan 1/IP2

cos tP np 1 tan 1/1 ('2t np1 + RP1 cos 1/IP1 ~ 1 )

p1 cos tPnp 1 tan 1/1 ('2t np2 + RP2 cos 1/IP2 ~2) (15.60)

p2

The first of these three equations gives the relation

between the angular positions ~1 and ~2 of the two gears. The equation is exactly comparable with Equation (3.56), which gave the corresponding relation for a pair of spur gears. If we differentiate the equation with respect to time,

o (15.61)

we confirm that the angular velocity ratio is constant. The coefficients of w1 and w2 can be expressed in the following manner,

R . cos 1/1 • pl pl Ni Pnpi

2 'IT

The operating normal pitches Pnp1 and Pnp2 are each equal to the normal pitch of the imaginary rack, and are therefore

equal to each other, so that Equation (15.61) can be

simplified to the following form,

o (15.62)

In other words, the angular velocity ratio is inversely


proportional to the ratio of the numbers of teeth, as we would expect.

The remaining two scalar equations that were derived earlier, Equations (15.59 and 15.60), can be solved to give expressions for s1 and s2'

s1 _ ~Cp sin "'p1 cos "'p2 + 1 cos "'b1 - cos I/lnp ( tan I/lnp sin I: 2t np1 + Rp1 cos "'p1 fj1)

(15.63)

cos "'b2

It is now a good moment to choose values for ~C1 and ~C2' bearing in mind that their sum must be equal to ~Cp. We define ~C1 and ~C2 as follows,

~CE sin "'E1 cos "'E2 ~C1 sin I: ( 15.65)

~C2 ~CE sin "'E2 cos "'E1 (15.66) sin I:

and the expressions for s1 and s2 then simplify to the following form,

cos "'b2

+ ~tnp1 + RP1 cos "'p1 fj1) (15.67)

1 + "2t np2 + Rp2 cos "'p2 fj2) (15.68)

The expressions in brackets in these two equations occur frequently in the geometric theory of crossed helical gears,

and it is therefore more convenient to use s1 and s2 to identify the angular positions of the gears, rather than fj1

and fj2. When either s1 or s2 is known, Equation (15.59) can be used to find the other s value, and then of course the corresponding fj values can be found from Equations (15.67 and 15.68).

The position of the contact point is now given by Equation (15.50), when we use Equation (15.67) to express fj1 in terms of s1. As before, we express all the unit vectors in

terms of nnr' nTr and n S' and we obtain the following expression for the vector from P to the contact point,

Path of Contact 427

AC 1 (. ~

sl cos'" ) nnr + b1 tan ~np tan "'p1 nS (15.69)

Different values of sl give the positions of different contact points, so this equation can regarded as the equation

of the path of contact. Since the coefficient of nS is constant, the path of contact lies in a plane perpendicular to

the nS direction, as shown in Figure 15.6. A rotation AP1 of gear 1 causes a change AS 1 in the value

of sl' which can be found from Equation (15.67),

AS 1 (15.70)

The corresponding displacement of the contact point is then given by Equation (15.69),

(15.71)

It can be seen that the displacement is always in the

direction of nnr' or in other words, perpendicular to the plane of the imaginary rack tooth face. Since the direction is always the same, it means that the path of contact is a

Path of contact

dCpCOSl/lp1 COSI/IP2

tan <pnpsin!

Figure 15.6. View in the direction of the line of centers, showing the path of contact.

428




Figure 15.7. View of the pitch cylinders in the direction along the ,teeth of the imaginary rack.

straight line, and it makes an angle ~np with the n~

direction, as shown in Figure 15.7. The common normal at the contact point is also perpendicular to the tooth face of the imaginary rack, so the path of contact coincides with the line of action, exactly as it does in the case of a spur gear pair.

In order to find the relation between the gear rotation

and the distance moved by the contact point along the path of contact, we combine Equations (15.70 and 15.71), and then use

Equation (13.95) to simplify the resulting expression,

flp (15.72)

It is not easy to visualize the path of contact, when it is given by Equation (15.69) in terms of the unit vectors

associated with the imaginary rack. However, we can determine

its position and direction, in a form that can be more easily understood, if we construct the vector from C01 to the contact

point,

Path of Contact 429

(15.73)

We substitute the expression in Equation (15.50) for pPA1,

and we use Equation (15.55) to express n~l in terms of nx1 (0),

n y1 (0) and n z1 (0). We then obtain the following equation for

the path of contact,

P C01 A1 (Rp1 + sl sin I/Itp1) n x1 (0) + sl cos I/Itp1 ny 1 (0)

+ (tan ~C1

- sl tan 1/Ib1)nz1 (0) (15.74) I/I np sin 1/Ip1

This equation demonstrates a result which may be considered somewhat remarkable. If the gear is viewed in the axial direction, as shown in Figure 15.8, the path of contact appears exactly the same as the path of contact in a spur gear pair, and we can see that the straight line containing the path of contact touches the base cylinder of the gear. We can also use the diagram to obtain a relation between the radius

R1 of the contact point, and the corresponding value of sl'

<Ptp1

Path of contact

Figure 15.8. pitch cylinder of gear 1,

viewed along the gear axis.


2 2 v'(R 1-Rb1 ) - Rbl tan 4>tpl (15.75)

Point POl was defined earlier as the point where line C01 C02 ' the common perpendicular to the gear axes, intersects the pitch cylinder of gear 1. When the gear is viewed in the

axial direction, as in Figure 15.8, POl appears as the point where the radius in the nxl (O) direction meets the pitch

cylinder. By setting sl equal to zero in Equation (15.74), we obtain the position vector to the point where the path of contact intersects the pitch cylinder. This point is labelled P l , and its position is given by the following expression,

(15.76)

It is clear that P l lies on the axial line through POl' so that in Figure 15.8 the two points appear to coincide, and the distance between the two points is equal to the coefficient of nzl (O) in Equation (15.76).

An alternative form of Equation (15.74) can be found,

simply by bringing together the terms containing the

variable sl'

(15.77)

The terms in the square brackets represent a unit vector,

giving the direction of the path of contact. Since the

coefficient of nzl (O) has a magnitude of sin ~bl' the path of

contact must make an angle C7r/2 - ~bl) with the gear axis. The coefficients of nx1 (0) and nyl (O) are in the ratio of

tan 4>tpl : 1, which confirms that when the path of contact is viewed in the axial direction, it appears to make an angle

4>tpl with the nyl (O) direction, as we showed in Figure 15.8. By using the relations developed in Chapter 13 between

the various angles, we can express the position vector from

Co 1 to the contact point in yet another form,

Path of Contact 431

(15.78)

Once again, the terms in the square brackets represent a unit

vector. Since the coefficient of n x1 {O) is sin ~np' the path of contact must make an angle (7r/2-~np) with the n x1 {O)

direction. The coefficients of nz1 {O) and ny1 {O) are in the

ratio of (- tan lPp1 ) : 1, so when the path of contact is viewed

in the nx1 {O) direction, as shown in Figure 15.9, it appears to make an angle lPP1 with the ny1 {O) direction.

We obtain the corresponding set of results for gear 2 if

we express the direction of the path of contact in terms of

nx2{O), ny2{O) and nz2{O). Since the equations are exactly analogous, they will not be repeated here. The only equation which we will write out is the one corresponding to Equation (15. 75), giving the relation between s2 and the

radi us R2 of the contact point,

Path of contact

ny1 (O)

{in,,(o) ( n,,(O)

Line in pitch plane 1 parallel with gear 1 axis

(15.79)

Figure 15.9. Path of contact, viewed in the direction

of the line of centers.


By substituting these expressions for sl and s2 into Equation (15.59), we obtain a relation between the radii of the contact points in each gear,

2 2 2 2 i(R 1-Rb1 ) i(R2-Rb2 ) +

cos Wb1 cos Wb2

Rb1 tan ~tp1 +

cos Wb1

Contact Ratio

Rb2 tan ~tp2 + ~6_C~p~ cos wb2 sin ~np

(15.80)

The contact ratio mc of a crossed helical gear pair is defined, as always, as the rotation of either one of the gears during a single meshing cycle, divided by the angular pitch of the same gear,

(15.81)

In order to prove that we obtain the same value for the contact ratio, whether we use gear 1 or gear 2 in the definition, we express the angular pitch 68p1 by means of Equation (4.2), and we use Equation (15.72) to relate the gear rotation 6fJ 1 to the magnitude 16pl of the contact point displacement along the path of contact,

271') (if'" Rb1 cos wb1 1

The denominator in this equation is equal to the normal base pitch of gear 1, as we can see from Equations (13.43 and 13.44),

The contact ratio is therefore equal to the displacement of the contact point during a meshing cycle, divided by the normal base pitch of gear 1, and since the normal base pitches in the two gears are equal, we could clearly have defined the contact ratio in terms of gear 2, with no change in its value. Hence, the contact ratio can be expressed in the following form,

Contact Rat io 433

(15.82)

This equation proves that the contact ratio is equal to the length of the path of contact, divided by the normal base

pitch. In Chapter 13, we showed that the normal base pitch is

equal to the distance between adjacent tooth surfaces, measured along a common normal. The contact ratio therefore

represents the average number of tooth pairs in contact at any

particular time, in exactly the same way as the contact ratio

in a spur gear pair or in a parallel-axis helical gear pair.

The expression for mc given by Equation (15.82) is not particularly convenient when we want to calculate the value

of the contact ratio. We obtain a more useful expression, if

we use Equation (15.71) to substitute for IApl in terms of As 1,

cos lPb1 Pnb (15.83)

In this relation, s~l and s~2 are the values of sl at the ends of the path of contact, or in other words, at the points where

the path of contact intersects the two tip cylinders. The

contact ratio can be calculated most easily if it is expressed

in terms of RT1 and RT2 , the radii of the tip cylinders. In order to do so, we first use Equation (15.59) to relate s~2

T2 and 52 ' resulting in the following expression for mc ' T1 T2

1 sl s2 -( + Pnb cos lPb1 cos lPb2

(15.84)

We then use Equations (15.75 and 15.79) to express sT1 and sT2 1 2

in terms of the tip cylinder radii, and we obtain the final

expression for the contact ratio,

2 2 2 2 1 v(RT1 -Rb1 ) v(RT2 -Rb2 ) m -[ +

cos lPb2 c Pnb cos lPb 1

Rb1 tan 4>tE1 Rb2 tan 4>tE2 ACE ] (15.85)

cos lPb1 cos lPb2 sin 4>np

We pointed out earlier that the maximum power which can

be transmitted by a crossed helical gear pair is very limited,

since theoretically there is only point contact between the meshing teeth, and the contact stress is therefore very high.


For this reason, it is advantageous if there are at least two

pairs of teeth in contact at all times, so crossed helical gear pairs are designed whenever possible with a contact ratio greater than 2.

Interference

The conditions necessary to ensure that there will be no interference are very similar to the corresponding conditions

for a spur gear pair, described in Chapter 4. Once again, we will discuss only the conditions relating to interference at the tooth fillets of gear 1. The corresponding conditions for gear 2 can be found simply by interchanging the subscripts 1 and 2 in every equation.

We proved earlier in this chapter that the line containing the path of contact touches the base cylinders of the two gears. The points where the line meets the two

cylind~rs are again called the interference points E1 and E2 , exactly as they are in the case of a spur gear pair. The first condition for no interference is that the ends of the path of contact should lie between E1 and E2• The end point T2 of the path of contact is the point where line E1E2 intersects the tip cylinder of gear 2. The value of s2 at this point is given by Equation (15.79),

(15.86)

and the corresponding value of s1 at the same point can be found from Equation (15.59),

(15.87)

The interference point E1 is shown in Figure 15.8, and the

value of s1 at this point can be read from the diagram,

- Rb1 tan IP tp1 (15.88)

For T2 to lie between E1 and E2 , the value of s~2 must be greater than s~1, and we therefore obtain the first

Minimum Face Width 435

condition for no interference,

> 0 (15.89)

The limit circle of a gear was defined in Chapter 4 as

the circle with radius RL, where RL is the minimum radius at which contact takes place. For gear 1, the minimum value of Rl occurs at point T2 , provided Equation (15.89) is satisfied, and the limit circle radius is then given by Equation (15.75),

(15.90)

In order to eliminate the possibility of contact at the tooth fillets of gear 1, the radius Rfl of the fillet circle must be less than that of the limit circle, exactly as in the case of a spur gear pair. As before, it is customary to allow a small margin for possible errors in the center distance, and the second condition is therefore the same as Equa t i on (4. 23 ) ,

(15.91)

The two conditions given by Equations (15.89 and 15.91)

must both be satisfied, in order to ensure that there is no interference at the tooth fillets of gear 1.

Minimum Face Width

When we discussed the path of contact, the contact

points on each gear were labelled Al and A2• The position C01 Al f vector p 0 Al relative to COl was given by

Equation (15.77), in terms of the fixed unit vectors nxl (O),

ny1 (0) and nz1 (0). It would be possible to express the

position vector in terms of nx1 ' nyl and nz1 ' using Equations (15.44 - 15.46). In this case, we would obtain the coordinates

x~l y~l and z~l of the contact point in gear 1 as the

coefficients of the three unit vectors. If we calculated the

values of x~l, y~l and z~l for different values of sl' we would obtain the locus of the contact point on the tooth face


of gear 1.

It is not generally necessary to carry out this procedure, since the exact shape of the locus is not important. The only coordinate which we will need is z~l, and

this can be obtained immediately as the coefficient of nz1 (O) in Equation (15.77),

tan ~np sin ~P1 - sl tan ~b1 (15.92)

The corresponding coordinate of the contact point on gear 2 can be wri t ten down wi thout further proof,

tan ~np sin ~P2 - s2 tan ~b2 (15.93)

The locus of the contact point is a curve on the tooth face, and this curve reaches the edge of the tooth face either at the tooth tip, or at the end face of the gear. We assumed, when we discussed the contact ratio and the possibility of interference, that the path of contact ends when the contact point reaches the tooth tip of ei ther gear. I n other words, we

assumed that the contact locus on the tooth face of each gear stops at the tooth tip, rather than at the end face of the gear. A crossed helical gear pair should be designed so that this condition is satisfied, since otherwise part of the tooth face on each gear is unused, and the contact ratio is reduced. We will now determine the minimum face-width necessary for each gear, and the correct axial positioning,

in order that this condition should be satisfied.

In Equations (15.87 and 15.86), we gave the values of sl

and s2 for the contact point when it reaches the end T2 of the path of contact. If these values are substituted into

Equations (15.92 and 15.93), we obtain the axial coordinates

z~2 and z~2 at one end of the contact locus on each tooth

face. I n a simi lar way, we can calculate the axial coordinates

z~l and z~l at the other end of each contact locus. On gear 1, the contact locus lies between the two transverse sections

Tl T2 zl=zl and zl=zl • The gear must therefore be designed so that the two end faces lie outside this interval, and a similar

consideration applies to gear 2. It will be found that if the

face-width of each gear is to be kept to an absolute minimum,

Backlash 437

the gears must be mounted unsymmetrically relative to points

COl and CO2 • However, we have so far considered rotation of each gear in one direction only. If rotation occurs in the

opposite direction, the new contact locus on the tooth face of

gear 1 will lie between the two transverse sections zl=-z~2 and Zl=-z~l. Hence, in order to allow for rotation in both

directions, the distance in gear 1 between COl and each end

face must be equal to the magnitude of either z~l or z~2, whichever is larger, and the gear is then positioned

symmetrically relative to COl. These positions of the two end faces determine the minimum theoretical value of the gear

face-width, but the actual value should be increased

slightly, to ensure that the end points of the contact locus

lie at a certain distance inside each end face.

Backlash

The three different types of backlash in a crossed

helical gear pair are defined in essentially the same manner

as they were in the case of parallel-axis helical gears. However, there are some differences in the expressions used

to calculate the backlash values, and in the relations

between the different types of backlash. When we discussed the normal backlash of a parallel-axis

gear pair, we showed in Figure 14.7 a section through the pitch plane of two imaginary racks, whose tooth thicknesses

are chosen so that each imaginary rack is in contact with both faces of the teeth in the corresponding gear. As a consequence, the normal and transverse tooth thicknesses of

each imaginary rack at its pitch plane are equal to the normal

and transverse tooth thicknesses of the corresponding gear at

its pitch cylinder. The normal backlash Bn of the gear pair

was then defined as the gap width, measured in the normal

direction, between the tooth profiles of the two imaginary

racks in the pitch plane section.

The corresponding diagram for a crossed helical gear

pair is shown in Figure 15.10. However, in this case the pitch

planes of the two imaginary racks do not coincide,. so it is

impossible to show a section through both pitch planes


Imaginary rack 1 Imaginary rack 2

Figure 15.10. Section through the imaginary racks.

simultaneously, and Figure 15.10 shows a section through the pitch plane of imaginary rack 1. The normal tooth thickness of imaginary rack 1 at its pitch plane is labelled t npr1 ' but in the case of imaginary rack 2 the symbol t~r2 is used, to indicate that this is not the tooth thickness at its own pitch plane. As before, each imaginary rack is in contact with both tooth faces of the corresponding gear, so the normal tooth thicknesses t npr1 and t npr2 of the imaginary racks at their

pitch planes are equal to t np1 and t np2 ' the normal tooth

thicknesses of the gears at their pitch cylinders. Figure 15.11 shows a normal section through a tooth of

imaginary rack 2, and we can use the diagram to obtain a

relation between t~r2 and t npr2 ' the normal tooth thicknesses of the rack at the two pitch planes.

(15.94)

The normal backlash Bn is defined as the gap width, measured in the normal direction, between the tooth profiles of the two imaginary racks in any section parallel to the

Backlash

Pitch plane of imaginary rack 1 Pitch plane of imaginary rack 2

dCp

d<t>~r=<t>np t nTJ

'np!"'np2 t 0 n:

n(

Imaginary rack 2

439

Figure 15.11. Normal section through imaginary rack 2.

pitch planes. Since the adjacent non-contacting tooth faces are parallel to each other, the gap has the same width in any

section parallel to the pitch planes, and we can therefore

read the value of Bn directly from Figure 15.10,

p' - t - t' nr npr1 nr2 (15.95)

We now combine Equations (15.94 and 15.95), replacing

all quantities defined on the imaginary racks by the corresponding quantities defined on the gears, and we obtain

the final expression for Bn'

(15.96)

The circular backlash B in a parallel-axis helical gear

pair was defined as the difference between the transverse

space width in one gear, and the transverse tooth thickness in

the other. This definition is unsuitable in the case of a

crossed helical gear pair, since the transverse sections in

the two gears are no longer parallel.


For a parallel-axis gear pair, we showed that, in addition to this definition, there are two other ways in which the circular backlash can be described. It can be regarded either as the product (Rpap) of one gear, when the other is held fixed, or as the displacement aUr of the imaginary rack associated with one gear, when the other imaginary rack is held fixed. Either one of these descriptions can be used to suggest a definition for the circular backlash in a crossed helical gear pai r.

We choose the first of these methods, and we define a quantity B1 as the pitch cylinder radius of gear 1, multiplied by the maximum rotation aP1 of gear 1 when gear 2 is held fixed,

(15.97)

The reason for the use of the symbol B1, rather than simply B, will be explained shortly. We know, from Equation (14.17), that the product (Rp1 a p 1) is equal to the displacement aUr1 of an imaginary rack, provided it is only free to move in the direction perpendicular to the gear axis. Hence, the quantity B1 is also equal to the imaginary rack displacement,

B1 (15.9B)

Figure 15.10 shows the two imaginary racks associated with the crossed helical gear pair, and each imaginary rack is guided so that it can only move in a direction perpendicular to the corresponding gear axis. Since the initial gap between the teeth of the two imaginary racks is Bn in the normal direction, the maximum displacement of rack 1, when rack 2 is held fixed, is given by the following expression,

cos IPP1

By combining the last two equations, we obtain a relation

between B1 and the normal backlash,

Bn (15.99)

Backlash 441

It is clear that if we defined another quantity B2 as the product (Rp2~P2) when gear 1 is held fixed, we would obtain a similar relation,

(15.100)

We can see from Equations (15.99 and 15.100) that in

general the values of B1 and B2 are different, so neither can be regarded as the circular backlash of the gear pair. It is more accurate to describe B1 as the circular backlash of gear 1 when gear 2 is held fixed, and B2 as the circular backlash of gear 2 when gear 1 is held fixed.

The third type of backlash is defined in exactly the same manner as it was before. The backlash B' along the common normal is the shortest distance between adjacent non-contacting tooth faces, when the opposite faces are in contact. Since the common normal to the tooth faces coincides with the path of contact, the shortest distance between the faces is equal to l~pl, the magnitude of the contact point displacement during the rotation of one gear, when the other is held fixed. The contact point displacement was given by Equation (15.72), and the backlash along the common normal is

therefore equal to the coefficient of nnr in that equation,

B' Rb1 cos "'b1 ~P1

We express the base cylinder radius in terms of the pitch cylinder radius, then use Equation (13.95) to relate the various angles; and finally we use Equations (15.97 and

15.99) to express the rotation ~P1 in terms of the normal backlash. We finally obtain the same relation between B' and

Bn that we found for a parallel-axis gear pair,

B' (15.101)

Sliding Velocity

If A1 is the point of gear 1 which is in contact with gear 2, the velocity of A1 is found in the usual manner,


(15.102)

We substitute the expression for pC01Al given by

Equation (15.74), and the velocity of Al is then expressed as follows,

The velocity of A2 , the contact point in gear 2, is found in exactly the same way,

Finally, in order to obtain the sliding velocity, we subtract the velocity of A2 from that of A1, and express the result in terms of a single set of unit vectors,

cos", (w 1s 1 cos "'tpl + w2s2 cos tl>tp2)nTr np

+ [W 1Rp1 sin "'pl - w2Rp2 sin "'p2 + tan "'np(w1s 1 tan "'bl - w 2 s 2 tan "'b2)]nS (15.105)

It can be seen that the sliding velocity lies in the plane of the imaginary rack tooth surface. This is to be expected, since the velocity components of Al and A2 along the common normal at the contact point must of course be equal. If the sliding velocity is evaluated for various positions of

the contact point along the path of contact, it will be found

that the coefficient of nTr changes sign, in the same manner

as the sliding velocity in a spur gear pair, but that the

coefficient of nS is generally larger, and remains essentially constant. The high value of the sliding velocity

in a crossed helical gear pair is somewhat of a disadvantage, but in the hobbing process, it is the relative velocity between the hob and the gear blank which provides the cutting

action, and a high value of the sliding velocity in a

relatively constant direction is therefore an essential

requirement.

Crossed Helical Gears 443

Directions of the Vector Systems

When the geometric analysis is required for a crossed helical gear pair, it is not always easy to decide which are

the correct directions for the various unit vectors. For

example, the magnitude of the shaft angle E is defined as the

angle between the unit vectors nz1 (O) and nz2 (O) along the gear axes, but it is not immediately clear which way along the

axes these vectors should point. The answer is important,

since it determines whether the shaft angle is the acute or the obtuse angle between the axes. Even when the magnitude of

E is known, it is still necessary to determine whether the

value is positive or negative. The following procedure is

helpful in deciding which are the correct directions for the

unit vectors. We consider first the case of an existing gear pair. We

. choose one of the gears as gear 1, and the vector n E is then

directed along the common perpendicular to the gear axes, pointing from gear towards gear 2. The approximate

direction of the teeth in the meshing zone can be seen, simply

by inspection of the gear pair, and nS is parallel with this direction. The exact direction will be determined later, once

the values of "'p1 and "'p2 have been calculated. At this stage, it is only necessary to know the general direction, and we are free to choose which way along this line the vector should point. The third fixed vector, n , must be directed so that

1'/ the three vectors form a right-handed set. The vector n z1 (O) is parallel with the axis of gear 1, in the direction which

has a positive component in the nS direction. This is because

the angle "'~1 between nz1 (O) and nS is equal to the operating helix angle on gear 1, and must therefore be acute. Finally,

nx1 (O) is parallel to nE, and ny1 (O) completes the

right-handed set. The directions of nz2 (O), nx2 (O) and ny2 (O)

are found in the same way, except that nx2 ( 0) is in the

direction opposite to n~.

The magnitude of E is equal to the angle between n Z1 (O)

and nz2 (O). Since we are considering the analysis of an

existing gear pair, we know the helix angles "'s1 and "'s2' and we therefore know the standard shaft angle E • The value of E s is always close to that of Es ' and this enables us to


determine the sign of ~, and also serves as a check that we have used the correct angle between the shafts.

The second situation where the analysis of a gear pair

may be required is at the design stage. The positions of the two axes are then known, and we must also know the required direction of rotation of each shaft. Again, we start by choosing one of the gears as gear 1. We proved in

Equation (15.62) that the two angular velocities must be opposite in sign. We therefore choose the directions of nz1 (O) and nz2(O) along the gear axes, so that one angular velocity is positive and the other negative. As before, the magnitude of ~ is equal to the angle between the two unit vectors.

We can now choose the gear helix angles ~s1 and ~S2' so that their sum is equal, or approximately equal, to ~.

Generally, if the shaft angle is sufficiently large, the two gears are designed with helix angles of the same sign, but if the shafts are almost parallel, we choose gears with helix angles of opposite sign, so that no gears are required with

very small helix angles. As soon as the values of ~s1 and ~s2 are chosen, the value of ~s is known, and the sign of ~ is therefore determined. The directions of the remaining unit vectors can be found in the manner described earlier.

In many of the diagrams in this chapter, the vector nT/ is shown in the upward vertical direction. When we found the position of the contact point, we considered the contact between two teeth where the contact tooth surface of gear 2 lies above the contact tooth surface of gear 1. For a gear

pair which is oriented differently, so that nT/ is no longer vertical, the contact point analysis applies to a tooth pair

such that the contact tooth surface of gear 2 lies in the

positive nT/ direction, relative to the contact tooth surface

of gear 1. If the driving gear is now rotated in the opposite

direction, the contact point will shift to the opposite face

of each tooth, and the equations given earlier in this chapter no longer apply. When the geometry of a gear pair is being

analysed, we should check to see that the contacting tooth surfaces of gears 1 and 2 are in the positions we have

assumed. If this condition is not satisfied, it would be

possible to repeat the derivation of the path of contact


equations, for the case when the contact occurs on the opposite tooth faces. However, there is an alternative solution, which is much quicker. We can simply reverse the directions of every unit vector except n~, nx1 (0) and nx2(0). The condition will then be satisfied, and all the equations derived earlier in this chapter can therefore be used

directly.

Tooth Contact Force and Bearing Reactions

Since the contact force acting on each gear has a component along the gear axis, it is obvious that each gear must be supported by at least one thrust bearing. The purpose of this section is to determine the value of the contact force, and hence the magnitude of the reaction to be carried by each thrust bearing.

We use the symbols M1 and M2 to represent the torques applied to the shafts, and thevalues are positive or negative depending whether the moments are right or left-handed about the corresponding unit vector directions. We showed in Equation (15.62) that the two angular velocities are opposite in sign, and a consideration of the energy balance for the gear pair shows that the applied torques must both be the same sign. In the following discussion, we will assume that the directions of n Z1 (0) and n z2 (0) are chosen so that M1 and M2 are both positive. This assumption corresponds exactly to the assumption made in the previous section, regarding the tooth faces which are in contact. Once again, if the condition is not satisfied, then the directions of the unit vectors should be reversed, so that the equations we derive can be used directly.

When we discussed the path of contact, we stated that the

line of action coincides with the path of contact, whose

direction is given by the unit vector contained in square

brackets in Equation (15.77). The direction of the

outward-facing normal to the tooth surface of gear 1 at the

contact point is therefore given by the following expression,

cos !Jib1 sin IP tp1 n x1 (0)

+ cos !Jib1 cos IP tp1 ny1 (0) - sin !Jib1 n z1 (0) (15.106)


The tooth force on gear 1 acts in the direction opposite to n~l. If its magnitude is W, we can see from Equation (15.106) that its component along the gear axis is

(W sin ~bl)' so the component perpendicular to the gear axis must be (W cos ~bl). Since the line of action touches the base cylinder, we can calculate the moment of this component about the gear axis, and equate it to the applied torque,

(W cos ~bl) Rbl (15.107)

We write down the corresponding equation for gear 2,

(15.108)

and then we eliminate W, to obtain a relation between the applied torques,

(15.109)

When either torque is known, we can use these equations to calculate the contact force and the second torque. The contact force acting on gear 1 has a component along the gear axis of (W sin ~bl) in the n zl (O) direction. Similarly, the contact force on gear 2 has an axial component of (W sin ~b2) in the n z2 (0) direction. The thrust bearings must therefore be designed to resist axial movements in the positive nz(O) direction for a gear with a right-handed helix angle, and in the negative nz(O) direction for a gear with a left-handed

helix angle. The remaining components of the contact force acting on

gear 1 can be seen from Equation (15.106) to be

(-wcos~bl sinq,tpl) in the n xl (O) direction, and

(- W cos ~bl cos q,tpl) in the nyl (O) direction. For the purpose of calculating the bearing reactions, these forces

may be regarded as acting at any point on the line of action, and the most convenient point is P l , whose position is given

by Equation (15.76). Once the positions of the bearings are

chosen, the equations of static equilibrium can then be used

to determine the reaction at each bearing. Finally, we calculate the bearing reactions for gear 2


by exactly the same method. The direction of the

outward-facing normal n~2 at the contact point on gear 2 is

given by Equation (15.106), if the subscript 1 is replaced by 2 throughout the equation. The contact force has a

magnitude W, and acts in the direction opposite to n~2. Once

again, for the purpose of finding the bearing reactions, it is

convenient to regard the contact force as if it acted at

point P2' and the position of P2 is given by Equation (15.76), when the subscripts 1 and 2 are interchanged.


Numerical Examples

Example 15. 1

A crossed helical gear pair has a shaft angle of 64° and

a center distance of 320.3 mm. The gears have the following spec i f icat ions: normal module 4 mm; normal pressure

angle 20°; tooth numbers 42 and 95; helix angles 35° and 28°;

tip cylinder diameters 216.3 and 440.3 mm; and normal tooth

thicknesses 7.342 and 6.874 mm. Calculate the contact ratio, the angle between the two generators which are in contact, and the backlash.

t=64°, C=320.3, mn=4, ~ns=20°

N1=42, N2=95, ~Sl=35°, ~s2=28° RT1 =108.15, ~2=220.15, t ns1 =7.342, t ns2=6.874

RS 1 = 102.545 mm

Rs2 = 215.188

~ts1 23.957° ~ts2 = 22.403°

Rb 1 = 93.711 Rb2 198.948

~b1 = 32.615° ~b2 = 26.178° Pnb = 11.809

tts1 8.963 tts2 = 7.785

~P1 35.569°

~P2 28.431°

~np 22.089°

~tP1 = 26.515°

~tp2 = 24.773°

RP1 = 104.727

RP2 = 219.111 tlC p = - 3.538 mm

mc 1.755

IIp1 15.051°

IIp2 11.508°

(13.105) (13.113) (13.113)

(15.23)

(15.24)

(15.25)

(13.82)

(13.82)

(15.26)

(15.27)

(15.47)

(15.85)

(13.87)

(13.87)

Examples 449

Angle between generators = vp1 +P p2 26.559°

Pnp = 12.744 (13.104)

ttp1 7.072 (13.114)

ttp2 4.466 (13.114)

t np1 5.753 (13.112)

t np2 3.927 (13.112)

Bn 0.192 mm (15.96) B' 0.178 mm (15.101)

Example 15.2 Suppose it is required to increase the contact ratio for

the gear pair specified in Example 15.1. Determine how much the tip cylinder radius of each gear can be enlarged, without causing interference, and calculate the new contact ratio. Assume the gears are cut by a hob with an addendum of 5.40 mm, and a tooth tip radius of 1.52 mm. Then calculate the minimum face-width of each gear, assuming the gears may rotate in either direction.

We start by calculating the radius at the top of the fillet in each gear, or in other words, the radius of the fillet cylinder.

h = 4.400mm (5.40)

e 1 1.455 (13.117)

e 2 = 0.812 (13.117)

Rf1 = 99.820 (5.48)

Rf2 211. 779 (5.48)

The maximum values for RT1 and RT2 , if there is to be no

interference, are obtained when the limit cylinder radii are

greater than the fillet cylinder radii by only 0.025mn •

RL1 = 99.920

RL2 = 211.879

The radius RT2 is found directly from the equations given


earlier in the chapter. To find RT1 , the subscripts 1 and 2 must be interchanged.

- 12.080

RT2 = 221.003

s~l = - 18.926

T1 sl = 9.838

RT1 = 109.474 mm

(15.90)

(15.87)

(15.86)

(15.85)

Finally, we calculate the axial coordinates zl and z2 of points T1 and T2 , the ends of the path of contact, and we use these values to find the minimum face-width required for each

gear.

- 2.014 (15.65)

~C2 = - 1.525 (15.66)

Z~l = - 14.826 (15.92)

- 0.800 (15.92)

2 [ I z 1 I max + O. 1 mn] 30.451 mm

1.413 (15.93)

Z~2 = - 10.066 (15.93)

Chapter 16

Gear Cutting I I, Helical Gears


We showed in Chapter 14 that two helical gears mounted on

parallel axes can mesh correctly together, provided their

normal base pitches and transverse base pitches are equal. We

can therefore use a helical pinion cutter to cut helical gears, and the normal base pi tch and the transverse base pi tch of the gear will then be equal to those of the cutter.

During the cutting process, the pinion cutter and the

gear- blank are each rotated, exactly as if they were a pair of

meshing gears. In order to obtain the cutting action, the cutter is given a reciprocating motion in the axial

direction. Since both the cutter and the gear have helical

teeth, the cutter must also be rotated during the reciprocating strokes, and this motion is achieved by means of a helical guide, whose lead is equal to that of the cutter. The end face of each tooth in the cutter is generally angled

so that it coincides with a normal section through the tooth, as shown in Figure 16.1.

The specification of a pinion cutter usually includes

the transverse module mt , the transverse pressure angle ~ts'

and the helix angle ~sc. The normal module and the normal

pressure angle are then found from Equations (13.148

and 13.149),

( 16. 1)

tan ~ts cos ~sc (16.2)

In this chapter, we will use the symbol ~Sg for the helix

452 Gear Cutting I I, Helical Gears

Cutting faces

Figure 16.1. A helical pinion cutter.

angle of the gear, when it is necessary to distinguish it from that of the cutter. In many cases, no such distinction is required, and then we will use the ordinary symbol 1/I s •

Before deciding how the shaper can be set up to cut a particular gear, we will prove that any gear cut by the pinion

cutter must have the same values of mt , mn , ~ts and ~ns as the cutter, and a helix angle 1/Isg which is equal in magnitude to 1/I s ' but opposite in sign. This result would be obvious if c . the cutting pitch cylinders were to coincide with the standard pitch cylinders. In general, however, they do not coincide, so although we know that the various gear quantities are equal to those of the pinion cutter on the cutting pitch cylinders, it is still necessary to prove that they are also equal on the standard pi tch cylinders.

When we say that the gear has the same values of mt and mn as the cutter, this is equivalent to stating that the

pitches Pts and Pns of the gear are equal to those of the

cutter. The proof, for the entire set of quantities Pts' Pns'

~ts' ~ns and 1/I s ' is essentially the same as the proof given in Chapter 5 for a spur gear. The pinion cutter is conjugate to the basic rack with parameters equal to those of the cutter.

Any gear cut by the pinion cutter has a transverse base pitch

and a normal base pitch equal to those of the cutter, and

therefore equal to those of the basic rack. Hence, the gear can mesh correctly with the basic rack. The standard pitch

cylinder of the gear is, by definition, the pitch cylinder

when it is meshed with the basic rack. The meshing theory of

Chapter 14 then shows that the transverse and normal pitches,

the transverse and normal pressure angles, and the helix

Shaping with a Pinion Cutter 453

angle of the gear, all measured on this cylinder, are equal to

the corresponding quantities on the basic rack. They are therefore also equal to the corresponding quantities on the pinion cutter, except that the helix angles of the gear and

the cutter must be opposite in sign, as in any parallel-axis

helical gear pair. When we determine how the shaper should be set up to cut

a particular helical gear, it is simplest to consider the gear and cutter geometry in the transverse plane, since this is essentially the same as the geometry of a spur gear and a pinion cutter. We now describe the settings necessary to cut a helical gear with Ng teeth, transverse module mt , transverse

pressure angle ~ts' helix angle ~sg' and transverse tooth thickness ttsg. The radius of the standard pitch cylinder of the gear is given by Equation (13.16),

(16.3)

and this of course is the cylinder on which the quantities

Pts' ~ts' ~sg and ttsg are defined. As we proved earlier in this section, the cutter must

have the same transverse module mt and transverse pressure

angle ~ts as those required in the gear, and the helix angle ~sc of the cutter must be equal and opposite to that of the gear,

- ~sg (16.4)

In order to obtain the required number of teeth in the gear blanks, the change gears or the stepping motor speeds in the shaper must be chosen so that the angular velocity ratio

of the gear blank and the pinion cutter is the inverse of the ratio of the tooth numbers,

(16.5)

where Nc is the number of teeth in the cut ter.

Finally, the cutting center distance CC required to give

the correct tooth thickness can be found by the same method used for spur gears, which was described in Chapter 5. The

454 Gear Cutting II, Helical Gears

transverse pressure angle '~p of the cutter and the gear at their cutting pitch cylinders is given by Equation (5.18),

. c lnv 'tp (16.6)

In this equation, C~ is the standard cutting center distance, equal to the sum of the standard pitch cylinder radii of the

gear and the cutter, Pts is the transverse pitch, equal

to ~mt' and ttsc is the transverse tooth thickness of the pinion cutter. Once the value of inv '~p has been calculated, the angle '~p is found by means of Equations (2.16 and 2.17). The transverse cutting pressure angle ,~ and the cutting center distance CC are then given by Equations (5.19 and5.20),

c 'tp

Rbg+Rbc

cos 'i where Rbg and Rbc are the two base cylinder radi i.

Shaping with a Rack Cutter

(16.7)

(16.8)

Since we know that a rack can mesh correctly with a helical gear, it is evident that we can also use a rack cutter to cut a helical gear. A rack cutter and a gear blank are shown in Figure 16.2. During the cutting process the gear

blank is rotated, while the cutter is moved perpendicular to

the gear axis, so that the movements of the cutter and the gear blank correspond to those of a meshing rack and pinion.

In addition, the cutter is given a reciprocating motion in the

direction of its teeth, and this motion provides the cutting

action. In Chapter 15 we discussed the minimum conditions for

correct meshing, in a crossed helical manner, of a rack and a helical gear. We showed that correct meshing is possible,

provided their normal base pi tches are equal, and the teeth of

the rack are set at the correct angle with the gear axis, given by Equation (15.12). It is therefore possible, in

Shaping with a Rack Cutter 455

I~-----,I

Reciprocating strokes of cutter

I~--I Figure 16.2. Shaping a helical gear with a rack cutter.

principle, for the rack cutter to have arbitrary values of normal pitch and normal pressure angle, so long as the normal base pitch of the cutter is equal to the normal base pitch required in the gear. In practice, however, rack cutters are almost always the same shape as the basic rack, so this is the only case we will consider. Equation (15.12) then takes the following form,

sin "'b cos I/lnr

(16.9)

and this relation can be used to give the base helix angle "'b that will be cut in the gear, when the cutter is set at an

angle "'r' Since the rack cutter has the same shape as the basic

rack, it is an immediate consequence that a gear cut by the

rack cutter is conjugate to the basic rack, and the cutting

pitch cylinder of the gear coincides with its standard pitch

cylinder. The gear parameters Pns and I/lns are therefore equal

to Pnr and I/lnr on the cutter, and the gear helix angle "'sg is


equal to the angle "'r at which the cutter is set. We now consider how the shaper is set up to cut a gear

with N teeth, normal module m , normal pressure angle ~ , g n ns and normal tooth thickness t nsg • First, as we have just shown, we choose a rack cutter with the same normal module and normal

pressure angle as those required for the gear. The angle "'r between the cutter teeth and the gear axis is adjustable, as shown in Figure 16.2, and this angle must be set equal to the

helix angle "'sg required in the gear. In order to obtain the correct number of teeth in the

gear, the cutter velocity vr perpendicular to the gear axis must be equal to the pitch line velocity of the gear. The pitch line velocity is defined as the velocity of any point on the pitch cylinder of the gear, and is therefore equal to

(R~gWg)' where Wg is the angular velocity of the gear blank, measured in radians per second, and R~g is the radius of the cutting pitch cylinder. As we pointed out earlier, the cutting pitch cylinder coincides with the standard pitch cylinder, and the radi i of both cylinders are given by Equation (13.30),

2 cos "'s (16.10)

The settings in the shaper must therefore be chosen so that the cutter velocity vr is related to the gear blank angular velocity Wg in the following manner,

Ngmn (16.11)

This equation must be satisfied exactly, if the gear is to be cut correctly. When the drives are controlled by stepping motors, the speeds are infinitely variable, and there is no difficulty in satisfying the equation. However, for machines using change gears, it may sometimes be difficult, or impossible, to find a combination of change gears giving the required relation between the rack cutter veloci ty and the gear blank angular veloc i ty. In thi s case, it

is necessary to alter the helix angle "'sg of the gear, and therefore also the cutter setting "'r' in order that the exact relation can be achieved.

Hobbing 457

The tooth thickness of the gear is determined by the depth to which the cutter is fed into the gear blank. Since the cutter has the shape of the basic rack, the cutter offset is equal to the profile shift e. The relation between the required normal tooth thickness t nsg and the profile shift

was given by Equation (13.117),

}rmn + 2e tan 9'lns ( 16.12)

The cutter should therefore be positioned so that its reference plane, which is the plane at which the normal tooth thickness and the normal space width are equal, lies a

distance (Rsg+e) from the gear axis, and the offset e has the value given by Equation (16.12).

Robbing

The most commonly used method for cutting helical gears is by hobbing. As always in generating cutting, one gear is used to cut another. A typical hob is shown in Figure 16.3, and it can be seen that, apart from the gashes forming the cutting faces, the hob is simply a helical gear, in which each tooth is referred to as a thread.

Lead angle Ash

Figure 16.3.

~angle !/Ish

A hob.


Since the hob is similar in shape to a screw, its helix

angle "'sh is always large, particularly when there is only one thread. It is cust~mary to specify the shape of a hob by means

of its lead angle, rather than its helix angle. For a

right-handed hob, the lead angle Ash is defined as the complement of the helix angle,

(16.13)

where Ash and "'sh are measured in degrees. For a left-handed hob, whose helix angle is negative, the lead angle can be def ined as follows,

- 90 0 - '" sh ( 16.14)

so that we obtain a negative lead angle for a left-handed hob. In practice, it is generally the magnitude of the lead angle which is given in the specification, together with a statement to indicate whether the hob is right or left-handed.

It is clear that the lead angle can be determined from the helix angle, and vice versa. In describing the geometry of the hobbing process, we will specify the shape of the hob by means of its helix angle, since the symbols will then agree with the notation used in Chapter 15, where we described the geometry of crossed helical gears.

Figure 16.4 shows a hob in position to cut a gear blank, and since their axes are not parallel, it is clear that they

form a crossed helical gear pair. During the cutting process, the hob and the gear blank are rotated about their axes with

angular veloci ties wh and wg ' I n order to cut the teeth of the gear across the entire face-width, the hob is moved slowly in

the direction of the gear axis, and the velocity of the hob

center is called the feed velocity vh .

The values required for the three variables wh ' Wg and vh are achieved by means of change gears or stepping motors in the hobbing machine. There are two additional settings which

must be made when the hobbing machine is being set up. These

are the shaft angle ~, which is the angle between the axes of

the hob and the gear blank, and the cutting center

Hobbing 459

Hob

Figure 16.4. A hob cutting a gear.

distance CC, which is the distance between the two axes. In

the remainder of this section, we will determine the values

required for the machine parameters wh ' wg ' vh ' E and CC, if the hob is to cut a gear with Ng teeth, normal module mn , normal pressure angle ~ns' helix angle ~sg' and normal tooth thickness t nsg

Before we discuss the details of the cutting process, we will first prove that, as usual, the gear will have the same normal module and normal pressure angle as those of the hob. We showed in Chapter 15 that the minimum condition for correct meshing of two crossed helical gears is that their normal base pitches should be equal. The corresponding result, when we

consider a gear being cut by a hob, is that the normal base

pitch of the gear will always be equal to that of the hob.

Since the normal base pitch of the hob is equal to that of the

basic rack, we can conclude that the normal base pitches of

the gear and the basic rack are equal, and the gear can

therefore mesh correctly with the basic rack. As always, the

standard pitch cylinder of the gear is defined as its pitch

cylinder when it is meshed with the basic rack. The normal

pitch Pns and the normal pressure angle ~ns of the gear must


then be equal to those of the basic rack, and hence equal to

those of the hob. This result remains true, whether or not the cutting pitch cylinders of the gear and the hob coincide with the i r standard pi tch cylinders.

In order to cut the gear described earlier, we must

therefore use a hob with the same normal module and normal pressure angle as those specified for the gear. We consider next how to cut the required number of teeth, and the correct

helix angle. When a rack cutter is used to cut a gear, the helix angle of the gear depends on the angle at which the cutter is set, so it might be expected that the helix angle of a gear being hobbed would be determined by the value of the shaft angle ~. This is not the case, however, and we will now show that the number of teeth cut in the gear blank, and the helix angle at which they are cut, depend only on the values

chosen for wh ' Wg and vh • In Chapter 5, we defined the cutting point as the point

where the cutter makes a cut on the final tooth surface, and we showed that this point corresponds to the contact point when the gear blank and the cutter are regarded as a pair of meshing gears. The situation is no different when the cutter is a hob. We described in Chapter 15 how to find the position of the contact point in a crossed helical gear pair, and this point becomes the cutting point when we consider a hob cutting a gear.

As in any metal-cutting process, the shape of each tooth cut in a gear blank is the envelope of positions through which

the hob moves, relative to the gear blank. For the purpose of determining this shape, it is helpful to neglect the gashes in

the hob thread, so that the threads are regarded as

continuous, and we can imagine that the teeth are formed in

the gear blank by grinding, rather than by cutting. If the hob and the gear blank were a pair of crossed

helical gears, there would always be at least one thread of

the hob making contact with the gear. Hence, for the hob and the gear blank, there is always at least one thread which is in contact with the final tooth surface of the gear. We label

the points in contact AOh on the hob, and AOg on the gear. After the hob turns through exactly one angular pitch, the

position of the thread containing point AOh is occupied by the

Hobbing 461

next thread, and the corresponding point Alh on this thread will now be the cutting point, touching a point A1g on the

gear tooth adjacent to the tooth containing AOg. As the hob rotates, we can identify a sequence of points

such as AOh and Alh on the hob threads, and AOg and A1g on the gear teeth. The points on the hob lie in the same transverse section and are evenly spaced, at angular intervals equal to the angular pitch. Of course, if the hob has only one thread,

the angular pitch is 360 0 , and the points all coincide. On the other hand, the gear points do not lie in one transverse section, due to the feed of the hob in the direction of the

gear axis, and each point is displaced axially a small amount relative to the next point.

We now consider the position on the gear of point ANg ,

the cutting point when the hob has turned through Ng angular pitches. Since the gear is to have Ng teeth, points AOg and ANg must be on the same tooth. Hence, if the gear is a spur

gear, ANg must lie on the axial line through AOg' while if the gear is a helical gear, ANg must lie on the gear helix

through AOg. The distance through which the hob is fed during one revolution of the gear blank is called the feed rate f. Since the magnitude of f is small compared with the tooth dimensions, point ANg always lies close to the axial line

through AOg. The gear blank must therefore turn through approximately one revolution while the hob turns through Ng angular pitches, which is a rotation equal to (Ng/Nh ) revolutions.

In order to meet this requirement, the angular velocity ratio (wh/wg ) must be approximately equal to (Ng/Nh ), or exactly equal, when a spur gear is being cut. In the case of a helical gear, the small difference between the two ratios is

one of the factors which determine the helix angle of the gear, as we will show later in this section.

Once the settings are chosen for the hobbing machine,

the value of (wh/wg ) is established, and the number of teeth

that will be cut in the gear is then given by the following expression,

NhWh Integer closest to (--)

Wg (16.15)


Having found how the value of Ng depends on the hobbing machine angular velocities wh and wg ' we now consider the

helix angle. If points AOg and A1g lie at radius R, the positions of these points at various times can be plotted on a developed cylinder of radius R, as shown in Figure 16.5. The times at which the hob touches points AOg and A1g are called T

and T', and the diagram shows the positions of AOg at time T,

and A1g at time T'. Since the feed of the hob is in the direction of the gear axis, the line in the diagram joining AOg and A1g is in the same direction. The diagram also shows the gear helices through these points, which appear as straight lines making an angle ~Rg with the gear axis, and these are labelled helix 0 and helix 1. The point on helix 0 in the transverse section through A1g is labelled Ag • The position of helix 0 at time T' is shown by the dotted line,

and the positions of Aog and Ag at this time are shown as AOg and Ag •

In Figure 16.5, the length AOgA1g represents the hob feed between the times T and T', and AgAg represents the arc

!-Direction of gear axis

, , ,

A' , g,'

, , ,

, , , ,

~-------.. ' , ,/ Aog /..,'

0,' .$,' -..;,

..... ' Cl)"

0/ .;r,' af'

.:t,/

, , ,

Figure 16.5. Cutting points, shown on the developed cylinder of radius R.

Hobbing 463

length moved by point Ag in the same time interval. Since

helix 0 and helix 1 are gear helices on adjacent teeth at the

same radius, their positions at any instant are exactly one

tooth pitch apart. A1g and Ag lie on the two helices in their positions at time T', so the distance between these points is

equal to the transverse pitch. We therefore obtain the

following expressions for the three lengths,

A A' 9 9

A A' 19 9

The time interval required for the hob to rotate through one

angular pitch can be expressed in terms of the hob angular velocity,

T' - T

We now use triangle AOgA1gAg to relate the three lengths,

A A' - A A' 19 9 9 9

and when their values are substituted, we obtain the following relation between wh ' Wg and vh '

211' vh -N-- tan l/IR hWh 9 ( 16.16)

The feed rate f of the hob was defined earlier as the distance moved by the hob during one revolution of the gear

blank. It is customary to express the feed velocity vh in terms of f,

_f_ (211') Wg

(16.17)

and with this substitution, Equation (16.16) takes the

following form,

tan l/IRg 211'R ( 16.18)


The helix angle of the gear at radius R is expressed in terms of the lead Lg by Equation (13.31),

tan IPRg

and Equation (16.18) then becomes a relation giving the lead that will be cut in the gear,

l(NhWh ) f - Ng Wg

( 16.19)

The quantity (Ng/Lg) is equal to the reciprocal of the axial pitch, as we showed in Equation (13.36), and this can be expressed in terms of the helix angle IPsg by means of Equation (13.42),

_1_

Pzg

sin IPSg

Pns (16.20)

Hence, Equation (16.19) can be put into two alternative forms, giving either the axial pitch or the helix angle of the gear,

_1_

Pzg l(NhWh ) f - Ng Wg

(16.21 )

(16.22)

It is an interesting result that, as we pointed out earlier, the helix angle cut in a gear is not affected by the shaft angle ~ of the hobbing machine. This angle is generally

set equal to the standard shaft angle ~s' or in other words, equal to the sum of the helix angles of the gear and the hob,

~s (16.23)

We showed in Chapter 15 that a pair of crossed helical gears can mesh correctly, even when the shaft angle is not equal to the standard shaft angle. It therefore follows that a hob can cut an accurate involute gear, even when ~ is not exactly equal to ~s. However, for the remainder of this section, we

will assume that the shaft angle is set equal to ~s' and in a

Hobbing 465

later section of the chapter we will discuss the consequences of a small change in this value.

The last setting of the hobbing machine to be considered is the cutting center distance CC , and its effect on the tooth thickness of the gear. As we discussed earlier, the cutting process can be considered as equivalent to meshing with zero

backlash. An expression for the normal backlash in a crossed helical gear pair was given in Equation (15.96),

The length ~Cp

Equation (15.47),

in this expression

(16.24)

was defined by

(16.25)

and all the other quanti ties are defined on the pitch cylinders, as indicated by the notation.

We are considering, at present, a hob cutting a gear blank when the shaft angle ~ is set equal to the standard value ~s. In this case the cutting pitch cylinders coincide with the standard pitch cylinders, as we proved in Chapter 15.

If we replace RP1 and Rp2 in Equation (16.25) by Rsg and Rsh ' and set the backlash in Equation (16.24) equal to zero, these two equations give an expression for the normal tooth thickness cut in the gear,

(16.26)

The expression in brackets in this relation represents the hob offset. When the normal tooth thickness t h of the hob is ns equal to O.5Pns' Equation (16.26) has exactly the same form as Equation (16.12), which gave the normal tooth thickness of

a gear cut by a rack cutter. If the normal tooth thickness of

the hob is greater than O.5Pns' the normal tooth thickness of the gear is reduced by the same amount. Whatever the value

of t nsh ' the effect of a change in the hob offset on the tooth thickness of the gear is identical to the corresponding

effect caused by a change in the offset of a rack cutter. In Chapter 5, we stated that the tooth thickness of a gear cut by


a hob is generally calculated as if the gear were cut by a rac k cutter. We have now shown that thi s procedure is essentially correct, prpvided the hobbing machine is set with

its shaft angle ~ equal to the standard value ~s' There is a second manner in which the cutting action of a

hob resembles that of a rack cutter. In the discussion following Equations (15.74 and 15.77), we showed that the path of contact in a crossed helical gear pair touches each

base cylinder, and makes an angle (~- 'tp1) with the line of centers, when viewed in the direction of the axis of gear 1. Hence, in the case of a hob cutting a gear, the path followed by the cutting point touches the base cylinders of the gear

and the hob, and makes an angle (~- 'tpg) with the line of centers, when viewed in the direction of the gear axis. If the shaft angle is set at the standard value, this angle becomes (1[2 - 't ), as shown in Figure 16.6, and the path of the sg . cutting point then appears identical with the corresponding path when the gear is cut by a rack cutter. It is for this reason that, when we check for undercutting in a gear, we can regard the hob as equivalent to a rack cutter. We check that

Hob base cylinder

Gear base cylinder

Path of the cutting point

Figure 16.6. Path of the cutting point, viewed in the direction of the gear axis.

Swivel Angle 467

there would be no undercutting if the gear was cut by the rack cutter, and this implies that there will also be no

undercutting when in fact the hob is used.

Swi vel Angle

Earlier in this chapter, we stated that it is common practice to specify the lead angle of a hob, instead of its helix angle. It is also customary to specify the angular setting of the hobbing machine by means of the swivel angle,

rather than by the shaft angle. Since a hob is shaped like a screw, its helix angle is

always large, particularly in the case of a single-thread hob, for which the helix angle is typically about 85°. The

helix angle of the gear being cut may of course have any value, but in the majority of gears, the magnitude of the helix angle is between 0° and 30°. In general, right-handed hobs are used to cut right-handed gears, and left-handed hobs are used for left-handed gears. In most cases, therefore, the shaft angle is approximately equal to a right angle, and the swivel angle is defined as the amount by which the shaft angle differs from a right angle. For example, if the axis of the gear is vertical during the cutting process, the swivel angle a is defined as the angle which the hob axis makes with the

Figure 16.7. Right-handed hob and spur gear.


Figure 16.8. Right-handed hob and right-handed gear.

horizontal. The standard shaft angle was defined by

Equation (16.23), as the sum of the gear and the hob helix angles. We express the helix angle of the hob in terms of its lead angle, by means of Equation (16.13 or 16.14), and we obtain the following expression for the standard shaft angle,

,f, + 90° - A "'sg - sh (16.27)

where the plus and minus signs refer to a right or

Figure 16.9. Left-handed hob and spur gear.


Figure 16.10. Left-handed hob and left-handed gear.

left-handed hob. We now define the standard swivel angle os' so that it differs by a right angle from the standard shaft angle,

(16.28)

As discussed earlier, the hobbing machine is generally set so that the shaft angle is equal to the standard shaft angle, and it then follows that the swivel angle is equal to the standard swivel angle. Figures 16.7 and 16.8 show the relations between the shaft angles and the swivel angles when a right-handed hob is used to cut a spur gear or a right-handed helical gear, while Figures 16.9 and 16.10 show the corresponding relations when a left-handed hob is used to cut a spur gear or a left-handed helical gear.

Hobbing Machine Gear Train Layout

We showed in Equations (16.15 and 16.22) that the number

of teeth and the helix angle cut in a gear depend on the feed

rate f and the angular velocity ratio (wh/wg ) in the hobbing machine,

NhWh Integer closest to (----)

Wg (16.29)


sin I/I sg l(NhWh ) f - Ng Wg

(16.30)

It is helpful to examine how the gear trains in some typical hobbing machines are arranged, in order to achieve the values of Ng and I/I sg required in the gear.

One type of hobbing machine is shown schematically in Figure 16.11. The rectangular boxes in the diagram represent gear pairs or gear trains, with the output-input ratio in each case given by the constant k. The symbol ki stands for the ratio of the index change gears, kf is the ratio of the feed change gears, and the other k values represent the gear trains built into the machine, whose ratios cannot be altered by the user. The values of wh and wg , and of the hob feed velocity vh ' can be read from the diagram in terms of the input angular velocity w 1 '

Index change gears

Wg Table

(16.31)

(16.32)

Hob

Figure 16.11. Hobbing machine gear train layout.


(16.33)

The feed rate f was given by Equation (16.17), in terms

of Wg and vh ' and when these are expressed by means of Equations (16.32 and 16.33), we obtain a relation between the

feed rate and some of the gear ratios in the hobbing machine,

f

The terms in brackets are combined into a single constant,

known as the machine feed constant Cf , whose value is provided

by the manufacturer of the hobbing machine. The feed rate is then expressed solely as a function of the ratio kf of the

feed change gears,

f (16.34)

To obtain the ratio (wh/wg ) in terms of the hobbing

machine gear ratios, we express wh and Wg by means of Equations (16.31 and 16.32),

As before, the terms in brackets are combined into another constant, the machine index constant Ci , whose value is also provided by the manufacturer, and the angular velocity ratio is then given by the following expression,

C· 1

k." 1

(16.35)

We substitute this expression into Equations (16.29

and 16.30), and we obtain the number of teeth that will be cut

in the gear, and its helix angle, in terms of the hobbing

machine gear ratios,

NhC. Integer closest to ( __ l)

k i (16.36)

{16.37l

We now determine how the machine ratios should be

472 Gear Cutting I I, Helical Gears

chosen, in order to cut a gear with the number of teeth and helix angle required. The feed rate f and the hob angular velocity wh are chosen to obtain good metal-cutting characteristics. The values depend on the size of the hob, the hardness of the material being cut, and the surface finish required. For more details, the reader should consult references such as the Gear Handbook [2]. Once a value for f is. chosen, the required ratio kf for the feed change gears is found from Equation (16.34),

f Cf

(16.38)

The value chosen for wh is obtained by setting the input speed change, shown in Figure 16.11, to a suitable value.

With the feed change gear ratio already selected, the index change gear ratio is used to determine both the number of teeth cut in the gear, and its helix angle. We choose the ratio ki so that it satisfies Equation (16.37), in order to obtain the required helix angle,

NhCi ki f sin ~Sg (16.39)

(1rm + N ) n g When this value for ki is substituted into Equation (16.36), we find that we also obtain the correct number of teeth, because the magnitude of the term (f sin ~Sg/1rmn) in the expression for ki is always very much less than 0.5.

It is sometimes difficult to find change gears which provide exactly the value of ki given by Equation (16.39). Once the change gears have been chosen, their actual ratio ki should be calculated, and this value is substituted into Equation (16.37), to give the helix angle that will in fact be

cut in the gear.

Use of a Differential in the Hobbing Machine

There is one major problem associated with hobbing machines, when they are designed in the manner shown in Figure 16.11. If a second cut is required, as is often the case, it is necessary to disconnect the feed drive, in order


to return the hob quickly to its starting position. It is then very difficult to reset the machine, with the work table and the hob in exactly the correct positions. This problem can be overcome if a differential is incorporated into the hobbing

machine. In order to determine the relation that must be

maintained between the hob feed, the work table rotation and the hob rotation, we once again consider Equation (16.30),

We use Equation (16.17) to express the hob feed rate f in

terms of the feed velocity vh ' and we obtain the relation which must be maintained throughout the cutting process between the hob feed velocity, the table angular velocity, and the hob angular veloci ty,

(16.40)

A gear train with one degree of freedom can always be represented by a linear equation relating the angular velocities of the input and the output shafts. A differential is a gear train with two degrees of freedom, and it has three shafts, either two input and one output, or one input and two output. The angular velocities of the three shafts are always related by a single linear equation. Hence, as we can see from Equation (16.40), if the hob feed, the work table drive and the hob drive were all connected to the three shafts of a suitable differential, they would then always maintain the correct relative positions.

The differential may be a simple planetary gear train,

or one which is constructed of bevel gears. In either case,

the output angular velocity w3 is a linear combination of the

input angular velocities w1 and w2 ' and can therefore be represented by the following equation,

(16.41)

The constants k7 and ka of the differential depend on the design of the gear train, and need not concern us here.


Table

Feed Hob

Figure 16.12. Hobbing machine with differentiaL

The complete layout of the hobbing machine is shown in Figure 16.12, where the differential is represented as a simple planetary gear train. The hob drive is connected to the sun gear of the differential, the table drive is connected to the planet carrier, and the feed is connected to the internal gear. As before, the index change gears and the feed change gears are represented by symbols ki and kf , and now there is a third set of change gears, the differential change gears,

represented by the symbol kd • The constants k1 to k6 are the fixed ratios of the gear trains in the hobbing machine. The

constant k6 represents the ratio of a worm and gear, connecting the differential change gears to the internal gear

of the differential. This ratio is shown with a minus sign, since the hand of the helix in the worm is chosen so that a positive angular velocity in the worm produces a negative angular veloci ty in the gear.

We pointed out earlier that the number of teeth and the helix angle cut in the gear depend on the feed rate f and the


angular velocity ratio (wh/wg ). We therefore need to express these two quantities in terms of the hobbing machine gear ratios. We start by writing down a number of relations between

the angular velocities,

Wh k1w1 (16.42)

Wg k2ki k3w3 (16.43)

vh k2kik4kfw3 (16.44)

w 2 k2kik4kfkd(-k6)w3 (16.45)

The feed rate f, which was given by Equation (16.17), can now be expressed in terms of the gear ratios,

f

As before, the terms in brackets are combined into a single quantity, the machine feed constant Cf , and the feed rate is then given simply in terms of the feed change gear ratio,

f (16.46)

When Equations (16.42 and 16.43) are used to express wh and wg ' the angular velocity ratio takes the following form,

wh k1w1 Wg k2k3kiw3

and the relation between w 1 and w3 is found from Equations (16.41 and 16.45),

The last two equations are combined to give the angular

velocity ratio in terms of the gear ratios, and we use

Equation (16.46) to express the ratio kf in terms of the feed rate f,

(16.47)


We now define the machine index constant Ci and the machine di fferent ial constant Cd as follows,

C. 1

k1 k2 k3k7

k3 k7Cf k1k4 k6kS

As usual, the values of the machine constants Cf ' Ci and Cd are all provided by the manufacturer of the hobbing machine. Ci is a ratio, but Cf and Cd are lengths, since they are defined in terms of the feed rate f, which is the distance moved by the hob during one revolution of the work table. When the constants are substituted into Equation (16.47), we obtain the final expression for the angular velocity ratio,

(16.4S)

This expression is substituted into Equations (16.29 and 16.30), and we obtain the number of teeth and the helix angle that will be cut in the gear, corresponding to the feed rate f and the change gear ratios ki and kd in the hobbing machine,

C· fkd Integer closest to [Nh(k: + c)]

1 d (16.49)

(16.50)

Once again, we must determine how the change gear ratios kf , k i and kd should be chosen, in order to cut a gear with Ng teeth and helix angle ~sg' As before, we choose the feed rate f from metal-cutting considerations, and the ratio kf is then

given by Equation (16.46),

(16.51)

If we are cutting a spur gear, or in other words a gear with zero helix angle, we can satisfy Equation (16.50) by setting the value of kd equal to zero, and choosing the value

of ki as follows,

Use of a Differential in the Hobbing Machine

k. 1

477

(16.52)

The conventional method for cutting a helical gear is to use the same value for ki , and to choose kd in a manner which then satisfies Equation (16.50),

Cd sin IPsg Nh '/I'mn

(16.53)

An alternative expression for the required differential change gear ratio is found by combining Equations (16.20 and 16.53),

(16.54)

When we compare the last two equations, it is clear that it is much easier to select suitable change gears, giving the correct value for kd , if we design the gear so that its axial

pitch Pzg is a round number, rather than its helix angle IPsg. There are times when it is difficult, or even

impossible, to find change gears which provide the exact values for ki and kd , given by Equations (16.52 and 16.53). For example, when the value required for Ng is a large prime number, we cannot obtain the exact value for ki , since most

sets of change gears do not contain gears with more than 120 teeth. Also, when the helix angle ~sg is very small, it may be difficult to obtain a sufficiently accurate value for kd •

When these situations occur, we can choose the index gears so that their ratio ki differs slightly from the value given by Equation (16.52), and the differential change gears are then used to ensure that Equation (16.50) is still satisfied with sufficient accuracy. Since the index change

gear ratio is close to the value given by Equation (16.52), it

can be represented by an expression with the following form,

k. 1

(16.55)

where the quantity ~ may be either positive or negative. This

expression for ki is substituted into Equation (16.50), and

we obtain the corresponding value of the differential change gear ratio required to cut the correct helix angle,


(16.56)

We have determined the values of ki and kd in a manner that satisfies Equation (16.50), so we know that the correct helix angle will be cut. It is now necessary to substitute the

values of ki and kd into Equation (16.49), in order to confirm that the gear will also be cut with the correct number of teeth. The expressions for ki and kd given by Equations (16.55 and 16.56) are substituted into the right-hand side of Equation (16.49), with the following result,

(16.57)

As we pointed earlier, the magni tude of the term

(f sin ~sg/wmn) is always less than 0.5, so with these values of ki and kd , the number of teeth cut in the gear will indeed be equal to the number required. The change gear ratios given be Equations (16.55 and 16.56) can be used for cutting either helical or spur gears, whenever it is difficult to obtain the values given by Equations (16.52 and 16.53).

It is interesting that the quantity a has cancelled out from the expression in Equation (16.57). This means that there is no theoretical limit to the value of a which can be used, and the ratio ki may therefore differ considerably from the value given by Equation (16.52). In practice, however, it is usually easier to select the differential change gears to obtain an accurate value for kd , if the index gears are chosen so that their ratio is close to the value given by

Equation (16.52), and the magnitude of a is therefore small compared wi th 1.

It is evident that a differential is useful in the design

of a hobbing machine, since it facilitates the selection of

the necessary change gears. However, the original purpose for

which the differential was introduced, as we discussed

earlier in the chapter, was to maintain the correct relation between the hob feed, the work table rotation and the hob

rotation, during a rapid return of the hob to its starting

position. Figure 16.13 shows how this purpose is achieved. The drive is disconnected, by means of a dog clutch, between

the feed change gears and the feed drive. An auxiliary motor,

Theoretically Correct Shape for the Hob Thread 479

Hob

Figure 16.13. Machine set for hob rapid traverse.

known as the hob rapid traverse motor, is then used to drive the hob feed. The drive passes through the di fferent ial, causing the work table to turn at exactly the correct speed, so that the helical teeth in the gear mesh continuously with the threads of the hob. During the entire return motion of the hob, only a very small rotation of the table is required, compared with the many revolutions that take place while the gear is being cut. Hence, the return of the hob can be carried out quite quickly, without damage to the gearing driving the

work table.

Theoretically Correct Shape for the Hob Thread

We stated in Chapter 5 that a hob whose thread profile is

straight-s;ded in the normal section will not cut exact

involute tooth profiles. We are now in a position to estimate the amount of error, and to determine the correct normal


profile in the hob thread.

In Chapter 15, we proved that two involute helical gears can mesh with crossed axes, and maintain a constant angular velocity ratio. The hobbing process is essentially the same as the meshing of a pair of crossed helical gears. It therefore follows that, in order to cut correct involute profiles in the gear, the thread of the hob must also have the shape of an involute helicoid. In other words, the thread has an involute profile in the transverse section. The corresponding profile in the normal section is a convex curve, and not a straight line. However, because the helix angle of a hob is so large, the profile in the normal section is extremely close to the straight line. Hence, when a straight-sided hob is used to cut gears, the resulting error in the gear tooth profiles is generally small.

We can estimate this error in the following manner. We described a method in Chapter 13 for calculating the profile of the normal section through a helicoid. We now use this method to find the profile of the normal section through the hob thread. We calculate the distances, at the thread tip and at the top of the fillet, between this profile and its tangent at the standard pitch cylinder, as shown in Figure 16.14. The profile of a straight-sided hob would coincide with this tangent, and a hob of that type would therefore cut too deeply into the teeth of the gear, in the regions near the fillet and near the tip. Figure 16.15 shows the normal section through an

Involute hob thread profile

Figure 16.14. Normal section through an involute hob thread.

Effect of a Non-Standard Shaft Angle

Approximately llT

Tooth profile cut 1- __ _ by a straight-sidedJ !

hob

Tooth profile of involute gear

Approximately llf

Figure 16.15. Normal section through a gear tooth.

481

exact involute helicoid tooth, and it also shows the profile we obtain when the gear is cut by a straight-sided hob. The maximum differences between the two profiles are approximately equal to the distances described earlier, by which the normal section profile of the involute hob deviates

from the straight line. As we can see in Figure 16.15, the tooth shape cut by a

straight-sided hob is similar to the shape of a tooth cut with tip and root relief. The errors caused by the use of a straight-sided hob are therefore sometimes beneficial, and this is one of the reasons for the continued use of straight-sided hobs, when true involute hobs are also readily obtainable. There are times, however, when the errors caused by straight-sided hobs may be excessive. This is often the case for gears cut by multi-thread hobs, or by single-thread hobs of large module, whose helix angles are usually less than 85°. Whenever there is a possibility that a straight-sided hob may cut too much tip and root relief in a gear, the procedure just described can be used to determine whether a true involute hob should be used.

Effect of a Non-Standard Shaft Angle

In an earlier section of this chapter, we described how

to calculate the tooth thickness cut in a gear, when the shaft

angle ~ of the hobbing machine is set equal to the standard value ~s. We stated at that time that we would still obtain a


correct involute profile in the gear tooth, even if the values

of E and Es wer~ not the same. The only effect of the altered shaft angle is a change in the tooth thickness, and we will

now discuss briefly how the new tooth thickness can be found.

Since it is not generally necessary to make this calculation, we will simply outline the steps, without presenting all the equations.

When the shaft angle is not equal to its standard value, the cutting pitch cylinders of the gear and the hob do not coincide with their standard pitch cylinders. The first step is therefore to calculate the cutting pitch cylinder radii

R~g and R~h. Knowing the normal thread thickness t nsh of the hob at its standard pitch cylinder, we then calculate its normal thread thickness t nph at the cutting pitch cylinder. To find the normal tooth thickness cut in the gear, we regard the hobbing process as the meshing of a crossed helical gear pair with zero backlash. An expression was given in Equation (15.96) for the normal backlash in a crossed helical gear pair,

The length ~cp in this equation was defined by Equation (15.47), as the difference between the center distance and the sum of the pitch cylinder radii. For the situation of a hob cutting a gear, ~cp would represent the difference between the cutting center distance and the sum of the cutting pitch cylinder radii,

We combine these equations, and set the backlash Bn equal to zero, to obtain the normal tooth thickness t npg cut in the

gear. The final step is to calculate the corresponding normal

tooth thickness t nsg of the gear at its standard pi tch

cylinder. If we carry out this calculation, we will find that the

normal tooth thickness t nsg cut in the gear is almost

independent of the shaft angle E. In other words, the tooth thickness is hardly affected by a small change in the shaft

Geometric Design of a Helical Gear Pair 483

angle, provided of course that the cutting center distance is left unchanged. However, the radii of the cutting pitch cylinders are very sensitive to the shaft angle value. In particular, a small change in the value of ~ can move the cutting pitch cylinder of the hob right off the surface of the

hob thread. In the absence of experimental evidence, it is not certain what effect this may have on the tooth surface

quality. Therefore, although the shaft angle need not theoretically be set equal to its standard value, it is nevertheless recommended that in practice this value should

continue to be used.

Geometric Design of a Helical Gear Pair

In the final section of this chapter, we outline a procedure by which we can choose the helix angle, the profile shift values and the gear blank diameters, for a pair of helical gears intended to mesh on parallel shafts at an arbi trary center di stance C.

Since the standard center distance depends on the helix angle,

(16.58)

it would appear that we can always choose the helix angle so that the standard center distance Cs is equal to the center distance C. In this case, the pitch cylinder of each gear would coincide with its standard pitch cylinder. However, as we will show, it is not always practical to choose the helix angle in this manner, and there is no particular advantage in doing so.

When the gears are cut by a pinion cutter, the helix

angle of each gear is equal to that of the cutter, so the

choice of ~s is limited by the cutters that are available. When a rack cutter is used to cut the gears, the cutter

veloci ty v r and the gear blank angular veloci ty must be related by Equation (16.11),


This equation must be satisfied exactly, because an incorrect value for vr would result in uneven spacing of the teeth on the gear. However, when ~s is chosen so that Cs is equal to C, it may be impossible to find change gears giving the exact relation between vr and wg • In general, it is probably easiest to obtain the required helix angle when the gear is cut by a hob, and the differential change gear ratio is given by either Equation (16.53) or Equation (16.56). Even in this case, it may be difficult to find a set of change gears giving a sufficiently small error. The effort required is seldom justified, because a gear pair can be designed quite satisfactorily, assuming only that Cs is approximately equal to C. The procedure is essentially the same as the one described in Chapter 6, for the design of a spur gear pair.

For the reasons just outlined, it is generally best to choose the helix angle ~s so that the gears can be cut wi thout difficulty, and at the same time the standard center distance Cs is slightly less than the center distance C. The value of Cs should lie within the range given by Equation (6.14),

C (16.59)

The design procedure now consists in the choice of suitable profile shift values, and the gear blank diameters, in order to obtain the backlash required, and adaquate values for the working depth and the clearances at each root cylinder. We consider the meshing geometry in a transverse plane, and the design steps are then identical to those used in the design of a spur gear pair. For a helical gear pair, it

is customary to specify the normal backlash Bn' rather than the circular backlash B. It is therefore necessary to calculate a number of the gear parameters in the transverse plane, before we can consider the transverse plane geometry.

The values of mt' Rs1 ' ~ts' Rb1 , Rp1 ' ~p' ~tP' Ptp' and Bare found from Equations (13.148, 13.150, 13.151, 13.152, 14.28, 14.7, 14.8, 14. 10 and 14.72).

(16.60)

(16.61)

Geometric Design of a Helical Gear Pair

B

tan tl>ns cos "'s

Rp1 tan "'s

Rs1

Rb1 Rp1

21TC

485

(16.62)

(16.63)

(16.64)

(16.65)

(16.66)

(16.67)

(16.68)

The design of a helical gear pair with parameters mn

and tl>ns' and normal backlash Bn' has now been effectively replaced by the design of a spur gear pair with parameters mt and tl>ts' and circular backlash B. We use the method described in Chapter 6, and in particular Equations (6.45 - 6.53), to carry out the necessary steps. Since the procedure was explained in Chapter 6, the equations will be presented here with very little explanation.

We start by writing down the transverse tooth thicknesses at the pi tch cylinders,

1 2"(Ptp-B) + tlttp (16.69)

1 2"(Ptp -B) - tlttp (16.70)

where tlttp is a quantity chosen by the designer, to increase the tooth thickness in one gear, and reduce it in the other.

The next four equations are given for gear 1 only, since the

corresponding equations for gear 2 are found by interchanging

the subscripts 1 and 2.

t ttsl R [.:..!E..!. + 2(inv tP tp - invtl>ts») (16.71)

s 1 Rpl

1 tI> (tts1-

1 (16.72) e 1 2 tan 2"1Tmt ) ts

bs1 a r - e 1 (16.73)


b + R - R sl p1 sl (16.74)

The addendum values ap1 and ap2 are chosen to give a working depth of 2.0mn , and equal clearances at each root cylinder,

mn - ~(bp1 - bp2 )

1 mn + 2(bp1 - bp2 )

(16.75)

(16.76)

And finally, we obtain the diameters of the two gear blanks,

(16.77)

(16.78)

Once the dimensions of the gear pair are all chosen, the designer should of course check, as in the design of a spur gear pair, that there is no interference or undercutting, and that the contact ratio, the root cylinder clearances, and the tip cylinder tooth thicknesses are all adaquate.

Gear Cutting II, Helical Gears 487

Numerical Examples

Example 16. 1 A 55-tooth helical gear with normal module 4 mm, normal

pressure angle 20 0 and helix angle 30 0, is to be cut with a

normal tooth thickness of 6.915 mm. Calculate the cutting

center distance, and the radius of the root cylinder in the

gear, if it is cut by a 32-tooth pinion cutter with a normal

tooth thickness of 6.40 mm, and a tip cylinder diameter of

158.12mm.

mn=4, ~ns=200, ~s=300, Ng=55, t nsg=6.915 Nc =32, t nsc =6.40, RTc =79.06

Example 16.2

Rsg = 127.017 mm

~ts 22.796 0

Rbg = 117.096

ttsg = 7.985

73.901 68. 129

7.390

inv ~~p = 0.024565

~~p = 23.471 0

201.932 mm

122.872 mm

(13.113)

(13.113)

(16.6)

(2.16,2.17)

(16.7)

(16.8)

A hobbing machine has an index constant C. of 24, and a 1

differential constant Cd of 25 mm. Calculate the change gear

ratios required to cut a 49-tooth gear with a normal module of

5 mm and a l)elix angle of 23 0, using a 2-thread hob.

C.=24, Cd=25, N =49, Nh=2, m =5, ~ =23 0 1 g n s


k. = (48/49) 1

kd = 0.3109340

40.201 mm

(16.52)

(16.53)

(16.20)

The index change ratio can obviously be provided by a

single gear pair. The differential ratio can be achieved with good accuracy by two gear pairs, having ratios of (24/66) and

(59/69). It is not always easy, however, to find change gears which give the required ratio. In the case described in this example, it would have been simpler if the gear pair had been designed with an axial pitch of 40 mm, in which case the required differential change gear ratio would have been exactly (25/80).

Example 16.3 When lead screws and other transfer mechanisms are

converted from inches to mms, it is sometimes necessary to introduce a factor of 25.4 into their drives. This factor requires a gear with 127 teeth, which is difficult to cut using conventional change gear ratios, because 127 is a prime number, and most sets of change gears do not contain gears with more than 120 teeth. Use Equations (16.55 and 16.56) to choose the ratios to cut a 127-tooth spur gear wi th a single-thread hob, when the hobbing machine has a feed rate of 0.020 inches, and the machine constants Ci and Cd are 24 and 0.5 inches.

Required ki = 0.1889764 (16.52)

Choose index change gears with ratios (24/41) and (31/96).

(24/41) x (31/96)

- (1/31)

0.8064516

(16.55) (16.56)

The differential ratio can be provided by a single gear pair

with a ratio of (25/31).

Chapter 17 Tooth Stresses in Helical Gears

Introduction

The calculation of the tooth stresses in a helical gear is considerably more complicated than the corresponding calculat ion for a spur gear. The contact stress and the fillet stress in each tooth depend on the intensity of the load, and on its position. Since the load intensity varies, as the position of the contact line moves up or down the tooth face, it is not easy to decide when the maximum stresses will occur.

As we pointed out in Chapter 11, we consider in this book only the static stresses that would occur if the gears were not rotating. The actual stresses that exist in normal operation are found by multiplying the static stresses by various factors, to account for dynamic effects, type of loading, and so on. Values for these factors are given in the AGMA Standard referred to in Chapter 11 [6]. The method described in this chapter for calculating the static stresses is based on the AGMA method, but differs from it in certain respects. A summary of the differences will be presented at the end of the chapter.

Tooth Contact Force

In a helical gear pair, there are generally several tooth pairs which are simultaneously in contact. The contact in each tooth pair takes place along a straight line, which coincides with one of the generators in each tooth. In order

to calculate the tooth stresses, we assume that the load intensity w is constant along all the contact lines. The value

490 Tooth Stresses in Helical Gears

of w at any instant is then equal to the total contact force W, divid.d by the total contact length Lc'

w (17.1)

In this section of the chapter, we determine the value of W, corresponding to any specified value of the applied torque. And in the following section, we will describe how to calculate the contact length Lc.

The direction n~ of the normal to the tooth surface at A, when A is a point on the contact line, was given by Equation (14.94),

n~ = cos "'b [sin t/ltp nx(O) + cos t/ltp ny(O)] - sin "'b nz(O) (17.2)

In the absence of friction, the contact force acts in the direction opposite to n~, and its component parallel to the gear axis is therefore (w sin "'b). Hence, the component perpendicular to the gear axis, which is the useful component, is equal to (W cos "'b) •

The base cylinder of gear 1 is shown in Figure 17.1, with

Figure 17.1. Tooth force component in the transverse plane.

Contact Length 491

the plane of action of the contact force touching the base cylinder. The diagram also shows the component of the contact force perpendicular to the gear axis. We take moments about the axis, to obtain a relation between the applied torque M1

and the contact force W,

(17.3)

and we use the same method to find the corresponding relation between the contact force and the torque M2 appl ied to gear 2,

(17.4)

The contact force is found from either of these equations. By combining the two equations, we obtain a relation between M1 and M2 , which is the same as Equation (11.3), the corresponding relation between the torques applied to a pair

of spur gears.

(17.5)

Contact Length

As we stated earlier, there generally several tooth pairs in contact at any instant, and the contact length Lc is the sum of the contact lengths on each of these tooth pairs. In this section, we will derive a general expression for Lc' It turns out that we do not often need to make use of the general expression, since the cases required for the stress analysis are always special, and therefore simpler. However,

it is a matter of interest to have the general result, and it

also helps to determine when the maximum and minimum values of

Lc occur. A transverse section through the gear pair is shown in

Figure 17.2, with the plane of action touching the two base

cylinders. As usual, the ends T1 and T2 of the path of contact

are the points where the tip cylinders intersect the plane of

action. Figure 17.3 shows the plane of action, with the axial lines through T1 and T2 meeting the transverse plane z=O at

492

Tip cylinder of gear 1


Tooth Stresses in Helical Gears

Plane of action


Tip cylinder of gear 2

Figure 17.2. Transverse section through the gear pair.

T10 and T20 , and meeting the transverse plane z=F at T1F

and T2F • The region of contact is the rectangle T10T20T2FT1F. We stated in Chapter 14 that the lines of contact on the

different contacting tooth pairs can be represented by a set

of diagonal lines in the region of contact, each making an

angle "'b with the gear axis, and with a vertical spacing equal

to the transverse base pi tch Ptb.

To find the length of the contact lines in the rectangle,

it is helpful to construct two additional triangles T'T 10T1F and T10T"T20 , as shown in Figure 17.3. The value of Lc is then

found as the length of the diagonal lines in triangle T' T"T 2F'

minus the lengths in triangles T'T 10T1F and T10T"T20 •

We proved in Chapter 14 that the lengths T' T 1 F and T 1 F T 2F

are equal to mFPtb and mpptb' where mF and mp are the face

contact ratio and the profile contact ratio, given by

Equations (14.68 and 14.64),

_1_ Ptb F tan "'b (17.6)

Contact Length 493

T20 I

z=O

I. Figure 17.3.

T2 I

Plane z

F

The plane of action.

In addition, the length T'T2F is equal to mcptb' where mc is the total contact ratio, equal to the sum of mF and mp ,

(17.8)

In order to find the value of Lc' we first consider a

general triangle of height mPtb' where m can represent any of the contact ratios me' mF or mp. This triangle is shown in Figure 17.4, and the upper contact line is shown in a typical

position, lying a vertical distance ePtb below the top corner of the triangle, where e is any number between 0 and 1.

The number of contact lines in the triangle is equal to

Ptb

Figure 17.4. A general triangle with contact lines.

494 Tooth Stresses in Hel ical Gears

(n +1), where n represents the integral part of the number e e (m-e). If e is greater than m, there are no contact lines in

the triangle, and the value required for ne is -1. We therefore define a function,

n int(f) (17.9)

where f is any number, and n is the largest integer which is less than or equal to f. If, for example, f has the values

2.2, 1.0 and -0.3, the corresponding values of n are 2, 1

and -1. The val ue of n e can then be expressed by the funct ion,

n e int(m-e) (17.10)

In the triangle shown in Figure 17.4, the upper contact

line has a length (m-w)ptb/sin ~b' The next contact line is

shorter than the first by Ptb/sin ~b' and so on. The total contact length Le can therefore be expressed as an arithmetic series, whose sum is given by the following expression,

(17.11)

We now apply this result to the three triangles in Figure 17.3. Once again, we consider the gear pair when the upper contact line lies a distance ePtb below point T'. The contact length Lc in the gear pair is then found from the following equations,

n ce int(mc-e)

Si~t~ [(n +l)(m -e-0.5n ) b Cf C Cf

(17.12)

(17.13)

(17.14)

(17.15)

If we use this method to calculate the contact length Lc for various values of e, we will find the following results.

Minimum and Maximum Values for Lc 495

The value of L is always a minimum when e is zero, and a c

contact line passes through the upper corner T10 of the region of contact. And the value of L is a maximum when e is equal to c [mF - int(mF)], and a contact line passes through the other

upper corner T1F of the contact region.

Minimum and Maximum Values of Lc

For the purpose of the stress analysis, we would expect to be most interested in the minimum value of Lc' since this corresponds to the maximum load intensity. Now that we know that the contact length is a minimum when a contact line

passes through T10 , it is possible to find simpler expressions for the value Lcmin • We can simplify the expressions further, if we consider only gear pairs in which every transverse section has either one or two contact

points. This condition means that the profile contact ratio lies between the following limits,

< 2 (17.16)

and this range includes all gear pairs of normal design. In the transverse section shown in Figure 17.2, there

are two points Q and Q' marked on the plane of action. Point Q

lies a distance Ptb below T l' and Q' lies a distance Ptb above T2• If the diagram represented a spur gear pair, Q and Q' would be the points on the path of contact corresponding to the ends of the period of single-tooth contact. In a helical gear pair, there is generally no period of single-tooth contact, because the total contact ratio mc is usually larger

than 2. However, Q and Q' would represent the ends of the period of single-tooth contact in any particular transverse

section, and it is therefore still customary to refer to these

points as the end points of single-tooth contact.

The region of contact is shown again in Figure 17.5, with

the axial lines through Q and Q' cutting the transverse plane

z=O a~ QO and QQ' and cutting the transverse plane z=F at QF and Qp.. We have stated that the value of Lc is a minimum when a contact line passes through point T 10. There must


T10 "'b T1F (mp -1)Ptb

Q' Q' 0 F (2 - mp)Ptb A~mFPtb (mp -1)Ptb

QO QF

T201 • .I T2F

F

Figure 17.5. Minimum contact length for gear pairs wi th mF S; 2 - mp

simultaneously be a second contact line through QO' since the distance between T10 and QO is equal to the transverse base

pitch Ptb' Due to the symmetry of the rectangle, we can also argue that Lc is again a minimum when there are contact lines

through T 2F and QF' We now consider a particular gear pair, with the contact

lines shown in Figure 17.5. One contact line passes through point T10 , while a second line starts at QO' and intersects the plane z=F at point AF , somewhere between QF and QF' For this situation to be possible, the length mFPtb must be less than (2-mp)ptb' as we can see from the diagram. Such a gear pair is therefore defined by the condition,

(17.17)

and will be referred to as a very low face contact ratio (VLFCR) gear pair. A spur gear pair, in which the face contact

ratio is zero, would fit into this category. The condition given by Equation (17.17) is equivalent to

the statement that the total contact ratio mc is less than 2. This means that there are periods of the meshing cycle when only one tooth pair is in contact, which is the situation

shown in Figure 17.5. The contact length Lcmin for this case can be read directly from the diagram,

L . cmln F

cos "'b (17.18)

Minimum and Maximum Values for Lc

Figure 17.6. The same contact region as Figure 17.5, with the contact line passing through Qp..

497

The same region of contact is shown in Figure 17.6, but the contact line has moved up, so that it now passes through point Qp., and a new contact line is about to enter the region at T2F • During the period when the contact line moves between the positions of Figures 17.5 and 17.6, there is only one tooth pair in contact, and the contact length Lc remains constant, with the value equal to Lcmin given by Equation (17.18).

The region of contact for a second gear pair is shown in Figure 17.7. The contact line which starts at QO now intersects the plane z=F at a point between QF and T1F • The face contact ratio must lie wi thin the following range,

T10 "'b (1 - mF)Ptb

T1F !

r -t- O' AF Ptb 0

F (2- mp)Ptb mFPtb

0 0 OF

T2°1. t2F (mF + mp - 2)Ptb

F

Figure 17.7. Minimum contact length for gear pairs with 2 - mp < mF ~ 1


< (17.19)

and this type of gear pair will be described as a low face contact ratio (LFCR) gear pair.

We know that the contact length is at its minimum value, since one contact line passes through T10 • To calculate the

value of Lcmin ' we take the length of the contact line through QO' and we add the length of the short contact line near T2F ,

L . cmln F

cos 1/Ib + Ptb (m +m -2)

sin 1/Ib F P

The expression is simplified if we use Equation (17.6) to

express Ptb in terms of F,

L . cmln (17.20)

Lastly, we consider gear pairs in which the face contact ratio is greater than 1,

> (17.21)

Gear pairs that fall within this category are known as normal helical gear pairs, since most helical gear pairs are designed with a face contact ratio larger than 1.

The region of contact for a gear pair of this type is shown in Figure 17.8, with the contact lines in the positions

Figure 17.8. Minimum contact length for gear pairs with mF > 1

Minimum and Maximum Values for Lc 499

corresponding to the minimum contact length. Starting from

the left, there is one contact line passing through QQ' then there are a number of complete contact lines stretching from the bottom edge to the top edge of the region, and finally there are either one or two lines which intersect the

right-hand edge. To find the value of Lcmin' we consider in turn each of the three groups of contact lines just described.

We start by defining two new quantities nc and nF , as the

integer parts of mc and mF ,

(17.22)

(17.23)

The number of contact lines crossing the upper edge of the region is nF , which means that the number of complete lines is (nF-1). The total number of contact lines is nc ' so the number crossing the right-hand edge is (nc-nF). Hence, the contact

length Lcmin is found as follows,

L . cmln Ptb •• 1. [1 + mp (nF-1) + (m -n ) + (m -n +l)(n -nF-1)]

sIn "'b c c c c c

The result is then simplified, and expressed in terms of the face-width F,

L . cmln

Figure 17.9. Maximum contact length for gear pairs

wi th mF S mp - 1


We pointed out earlier that the maximum load intensity

on a gear tooth corresponds to the minimum contact length, and

for this reason we derived expressions for Lcmin ' However, as we will show later in this chapter, the fillet stress is often a maximum when the contact line passes through the corner T1F of the contact region, and this occurs when the load intensity

is a minimum. We therefore also need expressions for Lcmax ' the maximum contact length.

A region of contact is shown in Figure 17.9, with one

contact line through T1F , and a second passing through QF' Both lines extend to the left-hand edge of the region. For

this to be possible, the lower line must intersect the left-hand edge above point T20 , which requires the following

condition,

(17.25)

The contact length is t~en equal to the combined lengths of the two contact lines,

2F (17.26) cos IItb

When the face contact ratio increases above the value given by Equation (17.25), we obtain the region of contact shown in Figure 17.10. We are still considering the case where there are only two contact lines, so the upper contact line must intersect the left-hand edge above point QO' and mF must

~T O...--------r----",T 1 F

mFPtb 1 0'0 O'F Ptb !

0 0 OF--hmp -1 )Ptb

T 20 ..... 1, -----..:::::....---'f 2F-f

F

Figure 17.10. Maximum contact length for gear pairs

wit h mp - 1 < mF :s; 1

Minimum and Maximum Values for Lc

lie in the following range,

<

The contact length is again the sum of the two lengths,

and, as usual, we express the result in terms of F,

F '" (m -1) mF cos b c

5{)1

(17.27)

(17.28)

Figure 17.11 shows the region of contact for a gear pair with mF greater than 1. Starting from the right, there is one

contact line through QF' then a number of complete contact lines, and lastly either one or two lines passing through the left-hand edge of the region.

If the contact line through T1F meets the lower edge of

the contact region. at T, the distance T20T is equal to (mF-mp)pz' We define a quantity nFP as follows,

(17.29)

where the function int is the same function introduced in Equation (17.9). The number of contact lines crossing the

(mF -nFP -l)pz

I' 'I T 10 ,-----r------r-----..,.,T, F

0 /0 O/F

QO QF=*mp -l)ptb

T201:-. --"-----.T=I---~~f 2F

(mF - mp)pz mppz

Figure 17.11. Maximum contact length for gear pai rs with mF > 1


lower edge of the region is then equal to (nFP+2), and the total number of contact lines in the region is (nF+2). Hence, the number of complete contact lines is (nFP+1), and the number of lines crossing the left-hand edge of the region is (nF-nFP)' By considering the three groups of contact lines described earlier, we-obtain the following expression for the contact length,

As before, we simplify this result, and express Lcmax in terms of F,

Contact Stress

In the last section of Chapter 14, we proved that when A is a contact point between a rack and a pinion, the line of contact through A and the common normal at A both, lie in the plane of action. The same is true when A is a contact point between a pair of helical gears. To prove this statement, we need only consider the imaginary rack between the gears, and make use of the result just stated, first for one gear and the imaginary rack, and then for the second gear and the imaginary rack.

The plane of action for a gear pair is shown in Figure 17.12, with the contact line GA making an angle..pb with the n direction, as we proved in Equation (14.93). Near z point A, the tooth surfaces can be represented by two circular cylinders in contact, with their axes lying in the plane of

action. Their radi i are shown as Pc 1 and Pc2' with the subscript c indicating that these are the radii of curvature when we make a section through the cylinders perpendicular to the line of contact. If we make a transverse section, as shown in the diagram, the cylinders appear as ellipses. For gear 1, the semi-minor axis of the ellipse is Pc1 ' while the

semi-major axis is equal to (pc 1/cOS ..pb)' The radius of

Contact Stress 503

G

Plane of action

----.~nz

Section normal to contact line

/ G (itA / np' /"(3lPcl

.Y-~"""'::"~ / ~ Pc2

~--".L:"""""'~~ Pt2W A cos r/lb

Ptl~e~ -+--- COSr/lb Pel

Transverse section

E

I Transverse plane z

Figure 17.12. The contact line through point A,

in the plane of action.

curvature Pt 1 at point A in the transverse section through the tooth profile is then equal to the radius of curvature at the corresponding point of the ellipse,

2 Pc l

(17.31)

The corresponding equation for gear 2 can be written down immediately,

(17.32)

The maximum contact stress 0c between two cylinders of radii Pc1 and Pc2 was given by Equation (11.5),

C v[w(PC 1+PC2)] p Pc l Pc2

(17.33)

where w is the load intensity, and Cp is the elastic coefficient given by Equation (11.6),

w(1-v 2 ) w(1-v 2 ) v[ 1 + 2 ] El E2

= (17.34)


Plane of action

cPt

c Figure 17.13. Radii of curvature in the transverse section.

The radii Pc1 and Pc2 are expressed by means of Equations (17.31 and 17.32) in terms of Pt1 and Pt2' the radii of curvature of the tooth profiles in the transverse section through A. We proved in Equation (10.1) that Pt1 and Pt2 are equal to the lengths E1A and AE2 in the transverse section, which is shown in Figure 17.13. We can see from the diagram

that the sum (P t1 +Pt2 ) is equal to (C sin ~t)' where C is the center distance, and ~t is the operating transverse pressure angle of the gear pair. The contact stress is then given by

the following expression,

(17.35)

We showed in Equation (14.34) that ~t is equal to ~tP' the operating transverse pressure angle of either gear. And, as

we proved in Equation (13.92), the product (sin ~tP cos l/Ib) is equal to sin, • We can therefore simplify the expression for np 0c to its final form,

C sin 'np Cp v' [ w ( P P ) ]

t 1 t2 (17.36)

Contact Stress 505

The values of Pt1 and Pt2 depend on the position s of point A on the path of contact in the transverse section. We

now have to determine at which point the contact stress should be calculated. We use Figure 17.13 to express Pt1 and Pt2 in

terms of s,

Rb1 tan 4>t + s (17.37)

(17.38)

We showed in Chapter 11 that the minimum value of the product (P t1 Pt2 ) is obtained when the smaller of the two radii of curvature is as small as possible. In other words, if gear 1 is the pinion, (P t1 Pt 2) is a minimum when s reaches its largest negative value, which occurs at the lowest possible

posi tion in the region of contact. However, the contact stress also depends on the load intensity w, and we therefore need consider only the contact lines for which the load intensity is a maximum.

In VLFCR gear pairs (mF ~ 2-mp)' the lowest point of a

contact line for which the load intensity is a maximum is QO' as shown in Figure 17.5, which corresponds to the lowest point of single-tooth contact in the pinion. The contact region for a LFCR gear pair (2-mp < mF ~ 1) is shown in Figure 17.7, with one contact line passing through point QO' The load intensity is a maximum when there are contact lines through T10 and QO' and due to the symmetry of the contact region, the load intensi ty is also a maximum when the contact lines pass through T2F and Qp' If the contact lines in Figure 17.7 were moved down the contact region until one of them passed through Qp, the other end of this line would intersect the

left-hand edge of the region below point QO' In other words,

there are positions of the contact line extending below the

lowest point of single-tooth contact on the pinion, for which

the load intensity is a maximum. For normal helical gear pairs

(mF > 1), there are contact lines of maximum load intensity

reaching right to the bottom edge of the contact region, which

corresponds to the limi t radi us on the pinion.

It would seem that the contact stress should be calculated at the points closest to the pinion limit radius of


the contact lines just described. However, it is found that

the tooth surface pitting is generally initiated close to the

pitch cylinder. It is therefore more realistic to calculate

the contact stress at the radius where damage is expected to occur. Until recently, the contact stress in a spur gear pair was calculated with the load applied at the lowest point of

single-tooth contact in the pinion, while in a helical gear pair it was calculated with the load applied at the pitch cylinder. It is now recognized that there should be a smooth transition between these two cases. This transition can be achieved if, for gear pairs with a face contact ratio of less than 1, the contact stress is calculated at the following position,

s (17.39)

The term in the square brackets corresponds to the lowest point of single-tooth contact in the pinion. The factor (1-mF) causes the point at which the calculation is made to move up the pinion tooth face as mF inc~eases, reaching the

pitch cylinder when mF is equal to 1. Having determined the value of s at the point where the contact stress is to be calculated, we substitute this value into Equation (17.37), and we obtain the radius of curvature Pt l'

(17.40)

For normal helical gear pairs (mF > 1), the contact

stress is calculated at the pitch cylinder, and the value of

Pt1 is found by setting s equal to zero in Equation (17.37),

(17.41)

The pinion radius of curvature Pt1 is given by Equation (17.40

or 17.41), depending on the value of mF• In either case, the corresponding value of Pt2 can be found from Equations (17.37

and 17.38),

(17.42)

Contact Stress 507

Once the radii of curvature Pt 1 and Pt2 have been calculated, the values are substituted into Equation (17.36), and we obtain the contact stress. In Chapter 11 we introduced the factors kc and kt , to represent the influence of the gear pair geometry on the values of the contact stress and the

fillet stress. As before, we now express the contact stress in

the following manner,

(17.43)

The expression for kc is then found by comparing Equations ( 17.36 and 17.43),

(17.44)

The expressions in Equations (17.36 and 17.44) for the contact stress 0c and the geometry factor kc must be modified for a rack and pinion, because the radius of curvature of the rack tooth face is infinite. We return to Equation (17.33), which we put in the following form,

C v[w(--1-- + __ 1 __ )] P Pc 1 Pc 2

We set the rack tooth curvature (1/Pc 2) equal to zero, and then complete the analysis in exactly the same manner as before.

(17.45)

(17.46)

We conclude this section of the chapter by summarizing

the steps necessary to calculate the contact stress. We start

by using Equation (17.3) to find the total contact force W.

The maximum contact stress always occurs when the load

intensity is a maximum, so for the contact length L we use c

the minimum value Lcmin ' given by Equation (17.18, 17.20

or 17.24), depending on the value of the face contact ratio mF • We then use Equations (17.40 - 17.42) to find the

radii of curvature Pt1 and Pt2 at the point where the contact


stress is to be calculated. And finally, the factor kc is

given by Equation (17.44 or 17.46), and we substitute into

Equation (17.43) to obtain the contact stress.

Fillet Stress, and the Equivalent Spur Gear

Before we calculate the maximum fillet stress in the

tooth of a helical gear, we first consider a more general case. Figure 17.14 shows a tooth profile, loaded by a force of

intensity w at a typical point Aw' which is at radius Rw' The tooth profile shown is the profile in the normal section

through Aw' so that the tooth force acts in the plane of the section, and along the normal to the profile. Since Aw does

not lie on the tooth center-line, the tooth profile is not

exactly symmetrical. Although it is possible to calculate the shape of the

tooth profile in the normal section, by the method described

in Chapter 13, the procedure is very long. It would be much

simpler if we could represent the profile by a spur gear tooth

of similar shape, because the tooth shape is then easy to calcula te, and we can use the method of Chapter 11 to find the fillet stress. The spur gear used for this purpose is known as

the equivalent spur gear, and the quantities used in its specification will all be distinguished by the subscript e.

c

Figure 17.14. Load of intensity w acting at radius Rw'


In order that the tooth profile of the equivalent spur

gear should resemble as closely as possible the normal tooth profile of the helical gear, we define the equivalent gear in the following manner. We construct a circle of radius Rde ,

equal to the radius of curvature of the normal helix in the pitch cylinder of the helical gear. This circle will be called

the defining circle of the equivalent spur· gear, and quantities measured on this circle will be indicated by the

subscript d. The teeth of the equivalent spur gear are shaped

so that on the defining circle their circular pitch Pde'

pressure angle ~de' dedendum bde and addendum ade , are equal

to the corresponding quantities Pnp' ~np' bp and a p on the pi tch cylinder of the helical gear.

The radius Rde of the defining circle of the equivalent spur gear is given by Equation (13.119),

cos21/1p (17.47)

and the radii of the root circle and the tip circle are then found as follows,

Rroot,e

Rd + a e p

(17.48)

(17.49)

By expressing the circumference of the defining circle as the number of teeth multiplied by the circular pitch, we obtain the number of teeth Ne in the equivalent spur gear,

cos31/1p

N (17.50)

The base pitch Pbe of the equivalent spur gear is equal

to (Pde cos ~de)' and since Pde and ~de are chosen equal to Pnp and ~np' the base pitch is equal to the normal base pitch Pnb of the helical gear,

(17.51)

The equivalent spur gear is therefore conjugate to a basic

rack with module m and pressure angle ~ • The radius Rse of n ns


the standard pitch circle, and the radius Rbe of the base circle, are then given by Equations (2.30 and 2.20), where the

module and the pressure angle now have the values mn and q,ns'

(17.52)

(17 .53)

As part of the specification of the equivalent spur gear, it

is also convenient to calculate its dedendum bse ' measured from the standard pitch circle,

(17.54)

To determine the tooth shape of the equivalent spur gear, including its tooth thickness and the shape of the fillet, it is necessary to decide how the gear might be cut. It seems likely that the fillet shape of the equivalent spur gear will be closest to that of the normal section through the fillet of the helical gear~ if the cutter used to calculate the equivalent spur gear shape is as similar as possible to the actual cutter used for the helical gear.

If the helical gear is cut by a rack cutter or a hob, then the choice is obvious. We calculate the tooth shape of the equivalent spur gear, assuming it is cut by the same cutter. To obtain the correct dedendum in the equivalent spur gear, given by Equation (17.54), the cutter offset ee must have the following value,

(17.55)

where a is the cutter addendum. The tooth profile shape can r now be calculated, including the fillet, by the method described in Chapter 9. We are not free to choose the tooth thickness of the equivalent spur gear, since this is effectively determined by the dedendum value. However, we will find that the tooth thickness tde on the defining circle is extremely close in value to the helical gear tooth thickness t np • Although the two quantities are not generally identical, the difference is absolutely negligible.


When the helical gear is cut by a pinion cutter, it is not possible to use the same cutter for the equivalent spur gear, since a helical pinion cutter can only cut gears with the same helix angle as itself. Instead, we use a spur pinion

cutter, and we must now choose its shape, which cannot be identical to the normal section through the helical cutter,

since this is not an involute. We can nevertheless choose an involute cutter with module, pressure angle, addendum,

dedendum and tooth thickness, all measured at the standard

pitch circle, equal to the corresponding quantities mn , ~ns' a sc ' bsc and t nsc in the normal section of the helical cutter. If the teeth of the helical cutter are rounded at their tips,

then the teeth of the equivalent spur cutter should be rounded in the same manner.

The only quantity remaining to be chosen in the equivalent cutter is the number of teeth Nce ' The best choice

is to relate the value of Nce to that of the helical cutter Nc in the same manner as Equation (17.50),

cos 31/1p (17.56)

The radius Rsce of the standard pitch circle in the equivalent cutter is given by Equation (2.30),

(17.57)

and the radius RTce of the tip circle is found by adding the addendum of the helical cutter,

(17.58)

The cutting center distance Cc is then chosen to give the correct dedendum in the equivalent spur gear,

R + R root,e Tce (17.59)

As before, we use the method of Chapter 9 to calculate the

shape of the tooth profile. We will find, once again, that the

tooth thic kness tde of the equivalent spur gear on its defining circle is almost identical to the normal tooth


thickness t np of the helical gear. This is a consequence of

the choice of Nce given by Equation (17.56). If we choose any

other value for Nce ' the error in the tooth thickness may sometimes be substantially larger.

Tooth Load on the Equivalent Spur Gear

On the helical gear, we consider the tooth force acting

at point Aw' which lies at a height (Rw-Rp) above the pitch cylinder. We therefore apply the load on the equivalent spur

gear at a radius Rwe' the same height above the defining circle,

(17.60)

The contact line on the helical gear coincides with a generator, and therefore makes an angle with the tooth tip. The angle between the generator and the helix tangent at

radius R is the generator inclination angle vR' which was defined in Chapter 13. In order to use the equivalent spur gear to calculate the fillet stress in a helical gear, we must represent the real load on the helical gear, which acts along an oblique line at an angle with the tooth tip, by an equivalent load on the spur gear acting along a line parallel to the tooth tip.

This equivalent load was studied by Wellauer and

Figure 17.15. Oblique line load on a cantilevered plate.

Tooth Load on the Equi valent Spur Gear 513

Seireg [8], who considered the cantilevered plate shown in Figure 17.15. The plate is loaded by a force of intensity w, acting along an oblique line at an angle v with the plate edge. They showed how to calculate the bending moment

intensity at point A, which can then be expressed in the form (WH/Ch ), where H is the height of point Aw above point A. The

quantity Ch is defined as the bending moment intensity if the plate were loaded along a line parallel with the plate edge,

divided by the maximum bending moment intensity when the plate is loaded with the same force intensity along the oblique line. In the original paper, the quantity (1/Ch ) is plotted in a diagram as a function of v. The values of Ch given by the diagram are believed by the author of this book to be inaccurate. They have therefore been recalculated [11], using essentially the original method, and they can be represented by the following expression,

(17.61)

The angle v must be expressed in degrees, as indicated by the notation, and the equation is valid for values of v between 0° and 25°.

In order to calculate the value of Ch required for a particular helical gear, we must choose the value of v to be used in Equation (17.61). On the helical gear, the generator inclination angle vR varies with the radius R. Hence, the load on the plate in Figure 17.15 should really lie along a curve, as shown in Figure 17.16, instead of a straight line. However, the bending-_moment intensity at A is determined primarily by

the load in the immediate vicinity of point Aw' so it is sufficiently accurate to represent the load curve by a straight line at an angle v , where v is the generator w w inclination angle at point A • The value of v is given by w w Equation (13.87), in terms of the helix angle and the

transverse profile angle of the helical gear at radius Rw'

sin v w sin "'w sin 9>tw (17.62)

When this value of Vw is substituted into Equation (17.61), the quantity Ch is called the helical factor of the gear.


Load curve on the equivalent spur gear A

Figure 17.16. The ideal load curve.

The fillet stress in a spur gear was given by Equation (11.28),

w 1.5m(xD-x) O.Sm tan Yw (17.63) °t iii cos Yw [Kf( 2 - )]max

y y

The first term in the round brackets represents the tensile stress caused by the bending moment in the fillet, while the second term represents the reduction in tensile stress caused by the radial component of the tooth load. To calculate the fillet stress in the tooth of a helical gear, we use Equation (17.63), applied to the equivalent spur gear. However, the first term in the round brackets is divided by the helical factor Ch , to compensate for the oblique loading. The second term is left unchanged, since the oblique loading does not affect the radial component of the load. We also

replace the module m by the normal module mn of the helical gear, since this is the value used for the module of the

equivalent spur gear. The fillet stress is then given by the

following expression,

O.Smn tan Yw --.:..:....-----"-) ]max (17.64)

y

The quantities x, y, xD' Yw and Kf in this equation are all defined on the equivalent spur gear, and the expression for 0t is evaluated in the manner described in Chapter 11.

Equation (17.64) gives the fillet stress in the tooth of

a helical gear, when the load point Aw lies at any radius Rw'

Critical Load position 515

The value of the fillet stress obviously depends on the position of the contact line, and in the next section of this chapter we will determine the value of Rw that gives the

maximum fillet stress.

Critical Load position

The fillet stress given by Equation (17.64) depends on

the load intensity w, as well as on the load position. If w were constant, the maximum fillet stress would occur when the contact line passed through the corner of the tooth, and the

radius Rw would then be equal to the radius RT of the tip circle. However, the load intensity w is not constant, and we already know that for a spur gear, the maximum fillet stress occurs when the load is applied at the highest point of

single-tooth contact. In the design of a gear pair, the fillet stress

calculation is carried out separately for each gear. However, in order to be specific in the description of the method, we will assume at present that we are calculating the fillet stress in gear 1. Hence, when a contact line passes through the corner of a tooth, this contact line is represented by the

line through point T1F in the region of contact, which is shown in Figure 17.11.

We stated at the beginning of this chapter that, when a contact line passes through T1F , the load intensity is at its minimum value. We would therefore expect this position of the contact line to be the critical position for gear pairs in which the load intensity is fairly constant. On the other hand, if there are large changes in the load intensity, we

would expect the fillet stress to be a maximum when the

contact line is in a position of maximum load intensity.

The ratio of the maximum load intensity to the minimum load intensi ty is equal to (L /L.). The values of L cmax cmln cmax and Lcmin depend on the face contact ratio mF , as we showed earlier in this chapter. For gear pairs with a profile contact

ratio between 1 and 2, the ratio (L /L.) starts at 2 cmax cmln when mF is zero, and remains at 2 for very small values of mF . Then the ratio drops, reaching the value 1 when mF is equal


to 1. As mF increases above 1, the ratio is 1 whenever mF is exactly equal to any integer, and is fairly close to 1 at other values of mF •

With these considerations in mind, we can determine the

position of the contact line corresponding to the maximum fillet stress. For gear pairs with mF less than 1, there is

considerable variation in the load intensity, and the maximum

fillet stress occurs when the load intensity is a maximum. However, when mF is greater than 1, the load intensity is relatively constant, and the maximum fillet stress occurs when the contact line passes through the tooth corner. These conclusions can be confirmed, for any particular gear pair, by ~&ing Equations (17.1 and 17.15) to calculate the load intensity at a number of different positions of the contact line, and then finding the corresponding fillet stress by means of Equation (17.64).

We now consider the three types of gear pair defined earlier, and for each type we will determine the contact length Lc and the radius Rw for the critical position of the contact line. For VLFCR gear pairs, the highest point reached

by a contact line with maximum load intensity is QF' shown in Figure 17.6, which is the highest point of single-tooth contact. The transverse section z=F is shown in Figure 17.17, with QF lying a distance (mp-1)ptb below the upper end of the path of contact. When the contact line is in this position,

the contact length is equal to Lcmin ' and the radius Rw of point Aw can be read from Figure 17.17.

When mF :!> 2-mp L L . F (17.65) c cmln cos .,pb

R2 R2 + [v'(R2_R2) 2 (17.66) - (mp-1)ptb1 w b T b

Equations (17.65 and 17.66) were derived from a

consideration of gear 1. If we considered gear 2 instead, the

cri tical position of the contact line would pass through point QO in the region of contact, and the contact length

would be unchanged, due to the symmetry of the contact region.

The equations are therefore valid for either gear, and this is

why there are no subscripts 1 or 2 in the equations. The same

Critical Load position 517

T1F

Q';)mp - 1lPtb

Figure 17.17. Transverse section at plane z=F.

comments apply to Equat ions (17.67 - 17.70), which give the critical contact lengths and load positions for gear pairs wi th larger values of mF•

The region of contact for an LFCR gear pair was shown in Figure 17.7. The highest position of a contact line with maximum load intensity is the position passing through QO'

and this line reaches within (l-mF)ptb of the upper edge of the contact region. The contact length is again Lcmin' and the radius Rw is found by the same method as before.

When 2-mp < mF S 1,

Lc Lcmin (17.67)

( 17 • 68)

Lastly, for normal helical gear pairs, the critical

contact line passes through the tooth corner, so the contact

length is equal to Lcmax' and the radius Rw is equal to the radius of the tip cylinder,

When mF > 1,

Lc Lcmax F

mF cos ~b [mp (nFP+2) + (mF-nFP-2)(nF-nFP)] (17.69)


(17.70)

Once again, it is convenient to express the maximum fillet stress in terms of the geometry factor kt , defined as follows,

(17.71)

A comparison of Equations (17.64 and 17.71) shows that kt is given by the following expression,

0.5mn tan Yw ----'''----.!!.) ] max (1 7 • 72 )

Y

Comparison with the AGMA I and J Factors

In Chapter 11, we presented relations between the geometry factors kc and kt defined in this book for spur gears, and the corresponding AGMA factors I and J. Relations of this sort are not possible in the case of helical gears, because the stresses calculated by the method of this chapter are not the same as those calculated by the AGMA method. There are a number of differences between the two procedures, of which the following are the most important. The procedure described in AGMA 218.01 [6] uses an approximate method for calculating the load intensity in gear pairs with mF less than 1, and this obviously affects the calculated values of both the contact stress and the fillet stresses. To find the

fillet stresses in gear pairs with mF greater than 1, the AGMA procedure uses the maximum load intensity, even though the

stress is calculated with the contact line passing through

the tooth corner, which is a position of minimum load

intensity. And finally, the AGMA procedure uses an expression

for calculating the helical factor Ch which gives

significantly higher values than those given by Equation (17.61). Each of these variations between the two

methods contributes to differences between the calculated

stress values, and for this reason it is not possible to

express I an J in terms of kc and kt •

Tooth Stresses in Helical Gears 519

Numerical Examples

Example 17.1 The pinion described in Example 16.1 is meshed with a

124-tooth gear at a center distance of 415.0 mm. Calculate the

normal tooth thickness of the pinion at its pitch cylinder.

Then determine the parameters of the equivalent spur gear for

the pinion, and calculate the tooth thickness tde of this spur

gear at its defining circle.

mn=4, ~ns=200, ~s=300, C=415.0, N1=55, N2=124 t ns1 =6.915, Nc =32, t nsc =6.40, RTc =79.06

RS1 = 127.017 mm

~ts 22.796° Rb 1 = 117.096

ttS1 = 7.985 Rp 1 = 1 27 • 5 1 4

qltp = 23.321° ~ = 30.097°

P ttp1 = 7.592

t np1 = 6.569 mm

(13.113)

(13.155)

(13.156)

(13.114)

(13.112)

The specification of the equivalent spur gear is then found as follows,

Rde = 170.352 mm Ne 84.927

169.854 159.611

(17.47) (17.50)

(17.52)

(17.53)

The dedendum bde of the equivalent spur gear, measured

from the defining circle, is chosen equal to the dedendum bP1 of the helical pinion, which must therefore be calculated.

Rroot ,1 = 122.872 (Example 16.1)

bP1 = RP1 - Rroot ,1 = 4.642

Rroot,e = 165.710 (17.48)


Finally, we introduce the equivalent cutter, and we

calculate the tooth thickness of the equivalent spur gear, first at its standard pitch circle, and then at radius Rde •

The superscript c indicates quantities during the cutting of the equivalent spur gear by the equivalent cutter.

Nce = 49.412

Rsce 98.824

Rbce 92.864 tsce = t nsc = 6.400

a sc = RTc - Rsc = 5.159 RTce = 103.983

269.693

,pc 20.584°

20.585°

268.678

tse = 6.916 ~de = 20.455° tde = 6.568mm

(17.56) (17.57)

(17.58)

(17.59)

(5.10)

(5.11)

(5.16)

(5.17) (2.18) (2.36)

In this example, the value of tde is almost identical to that of t np1 ' It is obviously essential, when the equivalent spur gear is used to represent the helical gear in the fillet

stress calculation, that both the dedendum and the tooth

thickness of the equivalent spur gear should be as close as

possible to those of the helical gear.

Example 17.2

Two helical gears are cut by a hob with normal diametral

pitch 4, normal pressure angle 20°, addendum 0.333 inches, and tooth tip radius 0.107 inches. The gears are cut with helix angles of 25°, and the face-width of each gear is 1.2

inches. The tooth numbers are 65 and 136, the profile shift

values are 0.1797 and 0.0872 inches, and the tip cylinder

diameters are 18.8 and 38.2 inches. The gears are meshed at a

Examples 521

center distance of 28.0 inches. Calculate the static contact

stress, and the stat ic fillet stress in each gear, if the material constant Cp is 2290 (psi)O.5, and the torque applied

to the pinion is 50000 lb-inches.

Pnd=4, ~ns=20°, a r =0.333, rrT=0.107

N 1 = 6 5, e 1 = 0 • 1 797, RT 1 = 9 • 4 N2=136, e 2=0.0872, RT2 =19.1

~s=25°, F=1.2, C=28.0, Cp=2290, M1=50000

mn = 0.2500 inches

Rs1 = 8.9649

Rs2 = 18.7574 .. = 21 880 0 "'ts . Rb1 = 8.3192

Rb2 = 17.4062

~b = 23.399 0

Ptb = 0.8042 ~t = 23.254 0

mF 0.6457

mp 1.4736 m 2. 1194 c

Lcmin 1.5492

Lcmax 2.2666

RP1 = 9.0547 Rp2 = 18.9453 ~p = 25.219 0

tP tp 23.254 0

.. = 21 244 0 "'np •

Pt1 = 3.5739 Pt2 = 7.4807

kc = 0.2711 W 6549 lbs

.Ji... = 21829 psi Fmn

91720 psi

( 13. 1 54 )

(13.43) (14.30)

(17.6)

(17.7)

(17.8)

(17.20) (17.28)

(17.40)

(17.42)

(17 .44)

(17.3)

(17.43)


When we calculate the fi llet stresses, the contact

length Lc is equal to Lcmin' because we are considering a gear pair whose face contact ratio mF is less than 1.

R wl = 9.2708 (17.68)

IP twl = 26.188° (13.155)

"'wl = 25.744° (13.156) /I wl = 11.051° (17.62) Chl = 1 • 173 (17.61)

Rdel = 11.0632 (17.47)

bPl = 0.2431

Rroot ,el = 10.8202 (17.48)

Nel = 87.7858 (17.50)

Rsel = 10.9732 (17.52)

Rbel = 10.3115 (17.53)

bSel = 0.1531 (17.54)

eel = 0.1799 inches (17.55)

We carry out the fillet stress analysis for the equivalent spur gear, using the method described in Chapter 11, and we obtain the following results.

ktl = 1.4134 0tl = 30850 psi (17.71)

We then repeat the procedure for gear 2. Since the method is identical, we will present only the final results.

Example 17.3

kt2 = 1.5515

0t2 = 33870 psi (17.71)

Repeat the calculation shown in Example 17.2, with the

following differences. The face-width is increased to 2.4

inches, and the torque applied to the pinion is increased to

100000 lb-inches.

F=2.4, Ml =100000

Many of the quantities to be calculated are the same as

Examples 523

in the previous example, so they will not be repeated here.

The quant i ties which are changed are those which depend on the

face contact ratio, and the position of the load point.

mF = 1.2914

mc = 2.7651 Lcmin = 3.5741 inches

Lcmax 4.1643 inches

Pt1 = 3.5749 Pt2 = 7.4797

kc = 0.2524 W 13098 lbs

-.!L = 21829 psi Fmn

85390 psi

(17.6)

( 17.8)

(17.24)

(17.30) (17.41)

(17.42)

(17.44)

( 17.3)

(17.43)

In this gear pair the face contact ratio is greater

than 1. We therefore use Lcmax for the contact len~th Lc' when we calculate the fillet stress, and the load is applied at the

tooth tip.

Rw1 = RT1 = 9.4 inches

4>tw1 = 27.746 0

"'w1 = 26.056 0

"w1 = 11. 800 0

Ch 1 = 1.185

kt1 = 1.3411 O't1 = 29280 psi

(17.70) ( 13.155)

(13.156) (17.62) (17.61)

(17.71)

Once again, we give only the final results for gear 2, without the intermediate steps.

kt2 = 1.4336

O't2 = 31290 psi (17.71)

The last two examples illustrate an interesting property

of helical gears. If we double the face-width, and double the

applied torque, the stresses do not remain exactly constant.

This is because the contact length is not exactly doubled, and the position of the load point is also changed.

Bibliography

The first five references listed are books on gearing.

Some of them deal not only with the geometry, but also with many other aspects of gearing. However, the books are

included in this bibliography because they all contain

excellent material on the geometry of gears.

The sixth reference is a Standard published by the American Gear Manufacturers Association. It is the only AGMA publication referred to directly in the text of this book, but

there are many other useful publications on gearing available from the AGMA, at 1500 King Street, Suite 201, Alexandria,

Virginia 22314, U.S.A. The remaining five references are papers reporting the

results of specific research projects. There are a few

results in this book which, because of space limitations,

have been quoted without proof. The full derivations of these

results are given in the papers listed in references [7 - 11].

[1] Buckingham, Earle: "Analytical Mechanics of Gears", McGraw-Hill, New York, 1949, and republished by Dover, New York, 1963.

[ 2 ] Dudley, Dar le W., (Edi tor) : "Gear Handbook",

McGraw-Hill, New York, 1962.

[3] Tuplin, W.A.: "Involute Gear Geometry", Chatto and

Windus, London, 1962, and also published by Ungar, New York.

[4] Merritt, H.E.: "Gear Engineering", Pitman, London,

1971.

[ 5 ] Dudley, Darle W.: "HandtJook of Pract ical Gear Design", McGraw-Hill, New York, 1984.

526 Bibliography

[6] "AGMA Standard For Rating the Pitting Resistance

and Bending Strength of Spur and Helical Involute Gear Teeth", AGMA 218.01, Dec. 1982.

[7] Dolan, T.J. and Broghamer, E.L.: "A Photoelastic

Study of Stresses in Gear tooth Fillets", Univ. Illinois Eng. Expt. Sta. Bull. 335, March 1942.

[8] Wellauer, E.J. and Seireg, A.: "Bending Strength of Gear Teeth by Cantilever-Plate Theory", Journal of Engineering for Industry, Trans. ASME, Vol 82, Series B, No.3, pp. 213-222, Aug. 1960.

[9] Polder, J .W.: "Overcut Interference in Internal Gears", Proc. International Symposium on Gearing and Power Transmissions", Tokyo, 1981.

[10] Colbourne, J.R.: "Optimum Number of Teeth for

Span Measurement", AGMA Paper No. 85 FTM 9, Oct. 1985.

[11] Colbourne, J .R.: "Effect of Oblique Loading on the Fillet Stress in Helical Gears", AGMA Paper No. 86 FTM 6, Oct. 1986.

Index

Addendum c i rc Ie

Addendum, measured from the standard pitch circle

measured f rom the pi tch c i rc Ie

Addendum modification

Angle of contact

Angles of approach and recess

Angular pi tch

Axial assembly of internal gear pairs

Axial contact ratio

Backlash, in a spur gear pair

in a helical gear pair

in a crossed helical gear pair

Balanced strength design

Base cylinder, of a spur gear

of an internal gear

of a hel ical gear

Base helix

Base helix angle

Base pitch, of a spur gear

transverse, normal

Basic rack, for gears in general

for involute gears

for helical gears

Bearing reactions, crossed helical gears

Center distance, in a spur gear pair


Chordal addendum

Chordal tooth thickness, of spur gears

of helical gears

Clearance

Close-mesh operation

Conjugate profiles

45

45

76

168

83

89

83

275

386

97

387

437

162

28,78

262

323

335

323

35

326,346

23

24

309

445

20

414

193

193

352

77

100

23

528

Contact force, in spur gear pairs

in helical gear pairs

in crossed helical gear pairs

Contact length, in helical gear pairs

Contact line inclination angle

Contact ratio, in a spur gear pair

when one gear is undercut

in a rack and pinion

in an internal gear pair



Contact stress, in spur gears

in hel ical gears

Correction

Cutting center distance, spur gear and pinion cutter

internal gear and pinion cutter

helical gear and pinion cutter

Cutting circular pi tch

Cutting point

Cutting pressure angle

Cutting, spur gears

internal gears

helical gears

Cycloidal gears

Dedendum circle

Dedendum, measured from the standard pitch circle

measured from the pi tch circle

Developed cylinder

Diametral pi tch

transverse, normal

Differential

End points of the path of contact

during cutting


Equivalent spur gear

Euler-Savary equation

External gears

Face contact ratio

Face-width

minimum value, for crossed helical gears

Index

243

489

445

491

342,402

83

184

88

270

382

432

244

502

168

116,121

282

453

116

132

116

110

279

451

148

45

45

76

320

39

310

472

85

140

433

508

229

9

384

50

435

Index

Fillet

Fillet circle

Fillet shape, cut by a rack cutter

of an undercut gear

cut by a pinion cutter

of internal gears

Fillet stress, in spur gears

in helical gears

Form cutting

Form diameter

Gear-tooth vernier caliper

Generating cutting

Generator

Generator inclination angle

Geometric design, of spur gear pairs

of internal gear pairs

of helical gear pairs

Geometry factor kc

Geometry factor kt Geometry factors I and J

Helical factor

Helical gears

Helix

Hel ix angle, of a gear

of a rack

of a crossed helical rack

Highest point of single-tooth contact

Hobbing machine gear train layout

Hobbing, of spur gears

of helical gears

Imaginary rack

Interference, in a spur gear pair

in an internal gear pair



Interference points

Internal gears

Inverse involute function

Involute

Involute function

529

48

91,138

212

216

221

283

248

508

110

177

192

112,122

331

341

155

294

483

247,507

252,518

254,518

513

303

318

321

309

408

174

469

128

457

72,378

91

271

387

434

97

9,259

32

27

30

530

Involute helicoid

Law of Gearing

Lead

Lead angle of a hob

Left-handed helical gear

Limit circle

Line of action

Long and short addendum system

Low face contact ratio gear pairs

Lowest point of single-tooth contact

Measurement between pins

Measurement over pins

Module

transverse, normal

Normal direction to involute helicoid surface

Normal section

Normal section tooth profile

Operating helix angle

Operating pressure angle, of a spur gear pair

of a rack and pinion

of a pinion meshed wi th a rack

of a spur gear

transverse, of a helical gear pair

transverse, normal, of a helical gear

Path of contact, of a spur gear pair

of a crossed helical gear pair

Pinion

Pinion cutter, spur

helical

Pitch, circular

operating circular

transverse, normal, axial

of a rack

pitch cylinders, of a spur gear pair

of a pinion meshed wi th a rack

of a helical gear pair


Pitch line of a rack

pi tch point

Plane of action

Index

318

9

318

130,458

319

94

28,57,67,71

160

498

174

297

200

39

309

337

338

359

367

67

28,57

58

68 373

367

22,28,71

423

13

112

451

18

58,99

323

13

20,67

17,55

312,375

417

17,57

17

373

Index

Pressure angle, of a rack

of a spur gear

transverse, normal

Profile angle, of a rack

of a spur gear

Transverse, normal

Profile contact ratio

Profile modification

Profile shift, of spur gears

Rack

of internal gears

of helical gears

Radial assembly of internal gear pairs

Radius of curvature, of the involute tooth profile

of spur gear tooth fillets

of internal gear tooth fillets

Recess action gears

Reference line of a rack

Reference plane of a helical rack

Region of contact

Right-hand rule of vectors

Right-handed helical gear

Roll angle

Root circle

Root relief Rubbing, during cutting of internal gears

Shaft angle

Shaping, spur gears with a pinion cutter

spur gears wi th a rack cutter

internal gears

helical gears wi th a pinion cutter

helical gears with a rack cutter

Sliding velocity, in a rack and pinion

in a spur gear pair


Space width

Span measurement, of spur gears

of helical gears

Specification, of a spur gear

of a hel ical gear

531

24

33

330,341

14

31

329,338

384

218

148

265

351

13

275

229

235

286

163

14

312

385

307

319

31

44

220

292 414

112

123

279

451

454

63

73

441

42,99

196

354

49

362

53-2

Spur gears

Standard center distance, of a spur gear pair

of a helical gear pair


Standard cutting center distance

Standard pi tch cyl inder, of a spur gear

of a helical gear

Standard shaft angle

Stress concentration factor

Swivel angle

Threads on a hob

Tip circle

Tip interference

Tip relief

Tooth thickness

transverse, normal

Tooth thickness measurement, of spur gears

of internal gears

of helical gears

Total contact ratio

Transverse section, through a gear

through a rack

through crossed helical gears

Undercut circle

Undercutting, in a spur gear

in an internal gear

in a helical gear

Vectors, use of

Very low face contact ratio gear pairs

Whole depth of teeth

Working depth

Index

7

74

381

419

120

25,78

313

419

249

130,467

128,479

44

272

220

41

350

191

297

352

384

312

309

409

142,179

140

288

387

306

496

45

77

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