THE ANALYSIS AND SYNTHESIS
OF ROLLING CURVES

Thai: for the Doom of M. S.
MICHIGAN STATE COLLEGE
Kenneth Ward SidonI

I954

 

   

'IW1IMWUHWI‘TM5?

01 692 3082

This is to certify that the
thesis entitled

The Analysis and Synthesis of Rolling Curves

presented by

Kenneth Ward Sidwell

 

has been accepted towards fulfillment
of the requirements for

M. 3. degree in Mechanical Engineering

a? We

Major professor

Date November 211, 19511

 

0-169

 

55.3.? a; . .13.... .Iifiwlllt
.T. . ; :

i... i

THE ANALYSIS AND SYNTHESIS

OF ROLLING CURVES

By

Kenneth ward Sidwell

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1954

THESIS

55.x... tars... fiaevilhi
.... H
s.

R J

ACKNOWLEDGMENT

The author wishes to express his sincere appreciation to Dr.
Rolland T. Hinkle of the Mechanical Engineering Department for his
guidance in the preparation of this thesis and for the use of his

comprehensive library.

I
Ho
" H-
l

n
‘1
3*
C4)

Kenneth Ward Sidvell
candidate for the degree of

Master of Science

Thesis: The Analysis and Synthesis of Rolling Curves

Outline of Studies
Major subject: Mechanical Engineering
Minor subject: Mathematics
Biographical Items
Born. March 29. 1932. Kenmore, New York
Undergraduate Studies. Michigan State College. 1949-53
Graduate Studies, Michigan State College. 1953-54

Experience: Graduate Assistant. Michigan State College,
1953-54

Membership: Tau.Beta Pi
Pi Tau Sigma
Pi Mu Epsilon
Phi Kappa Phi
Michigan Society of Professional Engineers (BIT)

- iii -

ABSTRACT

Rolling curves are used to produce a cyclic variation in the
angular velocity of a shaft. In any application. rolling curves are
used as the surfaces of non-circular cams or the pitch lines of non-
circular gears.

The purpose of this thesis is:

1. To develop the conditions and the equations for rolling

curves.

2. To explore some specific examples of the design and

analysis of rolling curves,

3. To deveIOp a general equation for the synthesis of rolling

curves, and

4. To outline methods for manufacturing non-circular cams

and gears.

The conditions and the equations for rolling curves have been

derived previously. The general relationship between the angular

displacements of the two rolling curves is expressed by Equ. (1).

94 ‘ {(9.1) m

where 6‘ a the angular displacement of the follower curve
in radians (clockwise is positive)
6d =the angular displacement of the driver curve in
radians (counterclockwise is positive)

-17-

The basic equations for rolling curves are derived from the con-

dition that the curves must have pure rolling at the point of contact.

(2a) and (2b)

 

.. L— ... g _,
ff-F‘(ed)*ld rd L fi

#794): Tea 9(edlz Lad (3)

where ‘8‘. athe radius of the follower curve
Ta 3 the radius of the driver curve
L 3 the constant distance between the axes of rotation
Ug=the angular velocity of the follower curve
wdathe angular velocity of the driver curve

Several applications are discussed by showing the derivation of
Equ. (l) in each case.

The most common example of rolling curves is a pair of identical
ellipses. The ellipses rotate around a focus point and the distance
between the axes of rotation equals the major axis. The relationships
between the angular displacements and velocities are found by using

the polar equation of the ellipse and Equ. (2a).

 

(4)4.- =w ' "4" (4a)
J I+L‘+Z.eco¢94

= -1 (I ‘4‘) 44:90.) ad b
er fine 21’ +(ez+l)m9d (a)

where at. the eccentricity of the ellipses

-v-

It is proved that the equivalent linkage of rolling ellipses
(a nonsparallel equal crank linkage) also produces Equs. (#a) and (4b).

Using the equations for rolling curves, the problem of design
reduces to the problem of finding the desired motion pattern.

The specified data can be in either of two forms. First. it
could be specified in the form of Equ. (1). Here the rolling curve
equations are found directly by using Equs. (2a) and (2b). Second.
the motion pattern could be specified only at several points of the
driver curve rotation. Here it is necessary to deve10p a complete

motion pattern which satisfies the given data.

Red): 3 (Gail/((30;
((91)., (adJ/C (9;):

94

Figure 1. Displacement diagram showing the specified data

 

 

The specified data is shown in Fig. (l) in the most complete

form. Only one interval of the cycle is shown as each interval is

handled separately.

-vi-

A general equation is develoPed that satisfies all the given
data and the second derivative of which equals zero at the end points
of the interval. With these preperties the resulting composite curve
is continuous and its first and second derivatives are continuous.
Thus. the resulting rolling curves are continuous and have no cusps.

In some applications the values of the angular displacement of
the follower curve are not specified. Further equations are deve10ped
from the general equation which enable these values to be determined
such that the angular accelerations are a minimum.

The last chapter contains a brief description of a method for
manufacturing nonrcircular cams and gear blanks. Also. a method of
forming non-circular gears by using standard formed-tooth milling

cutters is described.

-v11-

I.
II.
III.
IV.
V.
VI.
VII.

VIII.

Introduction

TABLE OF CONTENTS

Conditions for Rolling Curves . . . . . .

Representation of the Motion Pattern
Examples of Design

Rolling Ellipses, an Example of Analysis

General Equation for Synthesis . .

Preperties of the General Equation

Manufacture..........oo

Appendix

- viii -

O O I O O O O O O O 0

page 1

. . 7
. . 13
. . 18
. . 25
. . 31
. . 38
. . 50
. . 57

I. INTRODUCTION

One of the problems in the kinematics of machinery is producing
cyclic variations in the angular velocity of a shaft. Rolling curves
offer an interesting solution of this type of problem.

The use and study of rolling curves is not new. The first known
design is credited to Leonardo da Vinci.1 Some of the mathematical
preperties of rolling curves were stated and proved by luler.2'3

However. it has been only recently that the problems of design
and manufacture have been solved to an extent that enable non-circular
cams and gears to be produced accurately and fairly economically.l+
This is especially true of non-circular gears.

This thesis presents a study of some of the aspects of rolling
curves. More specifically. the purpose of this thesis is:

1. To develOp the conditions and the equations for rolling

curves.

2. To explore some specific examples of the design and

analysis of rolling curves.

1Uno Olsson, o -circular C drical ars Acta Polytechnica.

Mechanical Engineering Series, vol. 2. no. 10 (Stockholm: lsselte
Aktiebolag, 1953). p. 1.

2mm... p. 7.

3Robert Willis, Principles of MechanismI (2nd ed.: London:
LOngmans. Green, and 00.. 1870), p. 62.

l“Now-circular" is the common spelling but ”noncircular" is also
need.

3. To develop a general equation for the synthesis of rolling

curves , and

4. To outline methods for manufacturing non-circular cams

and gears.

In any application. rolling curves are used as either nonpcircular
gears or non-circular cams or, in some cases, a combination of non-
circular gear and cam segments.

Rolling cylinders and circular gears are also rolling curves.
However, it is generally more convenient to handle these separately
rather than as a special case of rolling curves.

Bolling curves are used to produce a cyclic variation in the
angular velocity of one of the shafts. Since they produce a variation
in angular velocity. they also produce a variation in the mechanical
advantage between the two shafts. This fact has been used in a number
of cases, particularly in the earlier applications.

Probably the first design of rolling curves was a set of non-
circular cogwheel segments which appear in the works of Leonardo da
'Vinci. These segments were apparently designed for use in the ten-
sioning of crossbows. thus utilizing the variation in mechanical
aurvantage.5

Another example of the use of the variation in mechanical advan-
tauge is Harfield's steering gear shown in Fig. (l). The gears are
inserted between the steering wheel and the boat's rudder. As the

rTuider angle is increased the mechanical advantage is also increased.

k

solsson, op. git.. p. l.

The change in mechanical advantage offsets the increased water forces

on the rudder at the larger angles.6

   

 

Steering wheel axis

Driving link of the
rudder mechanism

 

Figure 1. Harfield's steering gear

However. the main use of rolling curves is to produce a cyclic
variation in the angular velocity of one shaft.

A variety of rolling curves has been deveIOped for use in yarn
and silk winding machinery. If thread is wound on a cone frustum or
on any other solid of revolution with varying radius. the velocity of
winding varies as the thread travels up and down the axis of revolution.
Non-circular gears have been designed for controlling the angular
velocity of the solid of revolution so that the thread is wound at a
constant velocity.7

In recent years rolling curves have been widely used in multi-
plying mechanisms. These mechanisms are used in range finders as

8

they are compact. quite accurate. and completely automatic.

 

6S. Dunkerley. MechanismII ed. by Arthur Morley (3rd ed.: London:
Longmans. Green. and Co.. 1912). p. 339.

7"Gearing for Yarn Winding Machinery.” Engineering, XI (Jan. 6.
1871). p. 20.

8”The Works and Products of Messrs. Barr and Stroud. Limited.”
Eggineerigg. CVIII (Dec. 12. 1919). pp. 778-79-

Rolling curves are used to produce a cyclic variation in angular
velocity. Therefore. in the design of rolling curves there are basi-
cally two problems. First. the desired cyclic variation must be
expressed mathematically. Second. the rolling curves must be designed
to produce the specified variation in angular velocity.

Many authors. particularly in the field of mechanics of machin-
ery, have preferred to design the rolling curves by graphical means.
The general method is to assume one curve and to graphically plot the
mating curve. Thismethod is well adapted to drafting work but
sacrifices both accuracy and the control of the cyclic variation
pattern.

The mathematical conditions for rolling curves have been known
for a long time. The primary condition was stated and proved by
Euler.9 Extending the mathematical conditions. it is possible to
reduce the problems of rolling curve design to the problem of deriving
the desired pattern of variation of the angular velocity from the
physical conditions.

Once the equations for the required rolling curves have been
found. the curves are used as either the surface of nonpcircular cams
or the pitch lines of non-circular gears. The choice between the
cams and the gears depends upon the particular rolling curves. If
the radius of the driving curve is increasing. there is positive
action and cam surfaces are used. If the radius is decreasing or

constant. gear teeth must be used. In most applications it is

 

9Olsson. op, cit.. p. 7.

possible to use a combination of gear and cam segments although
usually gear teeth are used for the entire curve. Occasionally gear
teeth are not used and the cams are held together by a continuous
flexible connector which is wound around them.10

It is imperative that this discussion be limited to the most

common uses of rolling curves. Therefore. the ensuing discussion
of rolling curves has the following limitations:

1. The discussion is limited to plane rolling curves. These
are by far the most common. but non-circular bevel gears
also have been designed and manufacturedyt’12

2. The discussion is limited to the case where one of the
rolling curves rotates at a constant angular velocity.
This limitation does not affect the usual application
of rolling curves. but it eliminates such cases as gear
trains. However. some of the discussion may easily be
extended to the case where both curves have a varying
angular velocity.13

3. The discussion is limited to the case where the ratio
of the average angular velocities of a.pair of mating

rolling curves is unity. In other words one complete

 

10"Centre-turning Mobile Cranes.” Eggineerigg. 160 (Sept. 28.
19AI5) s P0 2h?’

11Stillman W. Robinson. Principles of Mechanism. (New York: John
Wiley and Sons. 1896). pp. 69-74.

12Olsson. Op. citn pp. 16h—65.

138ee Appendix B. pp. 63-64.

5.

revolution of one curve produces one complete revolution
of the other curve. Several rolling curve pairs have been
designed where a complete revolution of one curve pro—
duces two or three revolutions of the mating curve:

e. g.. Harfield‘s steering gear. Also. non-circular rack
and pinion mechanisms have been designed. However, the
ratio of the average angular velocities equals unity in
the usual application.

The discussion is limited to external rolling curves.

The discussion is limited to the case where the axes of
the rolling curves are fixed. This limitation is gener-
ally included in the definition of rolling curves. How-
ever. the theory has been extended to non-circular

planetary gears.1“

 

1“Olsson. Op, cit,. pp. 159-60.

II. CONDITIONS FOR ROLLING CURVES

The primary condition for rolling curves is that they satisfy
the requirements for pure rolling. Or. in other words. they roll on
each other and do not slide. Pure rolling imposes two mathematical
requirements.

First. the point of contact of the rolling curves falls on the
line joining the axes of rotation.1 This requirement is expressed
mathematically by considering that the point of contact is on both
curves. This point is on the line joining the axes of rotation if
the sum of the two radii to this point equals the fixed distance

between the axes of rotation.

‘YZI I“ lip := L.. (2'1)

where ‘0‘ the radius of the driver curve
ff. ' the radius of the follower curve
L 8 the constant center distance (distance between
the axes of rotation)
Second. the rolling curves must satisfy the angular velocity
ratio theorem for rolling contact. The angular velocity ratio of

. the driver and follower is inversely proportional to the contact

radii.2

 

101sson. Non-cirgglar Cylindrical Gears. p. 7.

2Rolland T. Einkle.‘§inematic§ of MechanismL (New York: Prentice-
HBll. Ines, 1953). Po 26s

This gives the second equation for rolling curves.

__Lt)+‘ =__rd (2.2)
”J lof-
where “’4 =the angular velocity of the driver curve
U; =the angular velocity of the follower curve

The reasoning behind these two equations is based upon the veloc-
ities of the points of contact. 'Consider the point of contact as a.
pair of points. one on each curve. If the velocities of each of the
two points are equal. there is no relative velocity between the two
points and, therefore. no sliding between them. Fbr the two veloc-
ities to be equal. they must be parallel (actually colinear) and
numerically equal.

If the point of contact is on the line of centers. the two rolling
curve radii are colinear. Since. in circular motion the velocity of
a point is perpendicular to its radius. the two velocities are perpen-
dicular to the same line and therefore parallel. The angular velocity
ratio theorem assures that the velocities are numerically equal.

The pair of equations can also be derived by using instant centers
of velocity. If there is no relative velocity between the two curves
at the point of contact. the point of contact is the instant center
for the two rolling curves. Since the two curves rotate about fixed
points. Kennedy's theorem dictates that the point of contact lies on
the line of centers. The angular velocity theorem for instant centers

gives Equ. (2.2).3

3Hinkle. Op. cit.. pp. 33. 37.

Physical considerations demand that the rolling curves must
satisfy two further conditions. First. the arc lengths between any
two points of contact on continuous arc segments must be equal.
Second. at any point of contact the values of the angles between the
radius and the tangent to the two curves must be supplements. It can
be shown that both of these conditions are met if the rolling curves
satisfy the conditions for pure rolling.“

Therefore. the only requirements for rolling curves are lqus.

(2.1) and (2.2).

 
  
 

Follower
curve

Driver
curve

 

Polar axes

Figure 2. Sign convention for rolling curves

Using the condition that both rolling curves have external
contact. a definite statement may be made about the directions of
rotation. This condition dictates that the point of contact lies
between the axes of rotation. When the point of contact is on the
line of centers and lies between the centers. the driver and follower

rotate in apposite directions.5 This results in the sign convention

 

“See Appendix A. pp. 58-62.

5mm... op, cit“ p. 24..

10

shown in Fig. (2). The driver curve is assigned the usual angular
sign convention used in polar coordinates: namely. a counterclockwise
angular displacement is considered positive. The follower curve has
the apposite sign convention in order to give positive angular values.
A sign convention may be deveIOped for angular velocities. However.
it is more convenient to discuss the ratio of the angular velocities.
This ratio is considered positive. Also. the condition of external
contact for the rolling curves dictates that the radii values are
always positive.

Using the equations for pure rolling. polar equations for both
curves are derived which depend solely upon the desired motion pattern
of the follower curve. The equations for the rolling curves may be
expressed either as functions of time or of the angular displacement
of the driver curve. The latter is the better choice for the special
case of constant angular velocity of the driver curve. The main
reason for this choice is that angular values are dimensionless.

Thus. the angular displacement of the follower curve is a function

of the angular displacement of the driver curve.

9+. = .F (9d) (2.3)

where 6.: 3the angular displacement of the follower curve
62‘ 3 the angular displacement of the driver curve
The distinction between the driver and the follower curve depends
'upon.Equ. (2.3). The terms “driver“ and "follower" are artificial.
though they do correspond to the usual application. Accordingly. in
the following discussion the driver curve is the one which.has a

constant angular velocity.

11

The angular velocity of the follower curve is found by differ-

entiating Equ. (2.3).

d9 __A___'F(9d)._. ' 2.1+
do: " ale. “9" < )

d6 -d9d _ .
jiTF- FCGJ)

60; = F '(94) we ‘2‘“)

Equs. (2.2) and (2.4a) are combined.

 

 

BL: rd 3 ‘ (2.5)
L‘dd ng: In (kid)

Equs. (2.1) and (2.5) are combined.

Lg}? ‘ F794)

 

(2.6)

n_L._

+‘(e.)+l

r = L__,_L-__=_L_f_'(_9i (2.7)
J W94)“ 19794)”

These equations have been derived previously in a similar

manner.6'7 The equations may also be derived by using Equ. (2.1)

 

51. E. Lockenvitz. J. B. Oliphint, w. c. Wilde. and James M. Young.
"Noncircular Cams and Gears." gachine Design, 24 (May. 1952). p. 142.

7H. E. Golber. Rollcurve Gears. Preprint of a speech presented on
Dec. 6. 1938 to the Graphic Arts Section at the annual meeting of the
A. So N. Es. p. 5.

12

and the condition that the arc lengths between any two points of
contact must be equal.8 Again. the equations may be derived by using
Equ. (2.1) and the condition that the angles between the radii and
the tangents to the rolling curves must be supplements.9

Using Equs. (2.6) and (2.7). the problem of designing rolling
curves reduces to the problem of finding the desired motion pattern.
The specified data of an application may be in either of two forms.
First. it might be specified in the form of Equ. (2.3) or (2.4).
In this case the rolling curve equations are found directly by using
Equs. (2.6) and (2.7). Second. the motion pattern might only be
specified at several points of the driver curve rotation. Here it
is necessary to develop a satisfactory form of Equ. (2.3) or (2.4)

which satisfies the prescribed data points.

 

8Olsson. 0p. cit.. pp. 8-11.

9Julius Weisbach and Gustav Herrmann. The Mechanic; of the
Machinery of TransmissionI Translated by J. F. Klein. Mechanics of
Engineering and of Machinery. vol. III. part 1. sec. 1 (2nd ed.:
New York: John Wiley and Sons. 1902). pp. 190-92.

13

III. REPRESENTATION OF THE MOTION PATTERN

The main problem in designing rolling curves is that of finding
the motion pattern which satisfies the given application. This is
especially true when the data is specified only at a few points of
the driver curve rotation.

In constructing an artificial function to satisfy the given data
points. it is convenient to study the motion pattern with the aid of
graphs. There are two graphs which can be used. Both have the
angular displacement of the driver curve as the abscissa. The two
ordinates are the angular displacement of the follower curve. #(ed).
and the ratio of the angular velocities. (0."?de or £794).

In using either of these graphs. five factors must be considered
from the viewpoints of how easily each may be studied on the graph
and of how severely each restricts the formation of motion patterns.
These five factors are:

l. The value of the angular displacement of the follower

curve,

2. The value of the angular velocity of the follower curve.

3. The value of the angular acceleration of the follower

curve.

4. The continuity of both curves. and

A?
JG

 

5. The smoothness of both curves or the continuity of

for both curves.

14

In discussing factors four and five it is assumed that the given
data itself satisfies the continuity conditions. For discussing

factor five. two equations are useful.

dfi: ..
de

(In (If Jed ”L‘FYQJ, (3.11:)
«19—7 Jed 86—: [Hedi-IF We.)

where (‘76:! )_ 5—9— d...‘F(9d) -géf‘F(9d)

Fig. (3) represents a motion pattern plotted on the graph using

(3.1a)

 

the angular displacement of the driver curve as the abscissa and the
ratio of the angular velocities of the follower and the driver curves
as the ordinate. This graph is generally called a velocity diagram

or a speedgraph (or ”speedgraf').1

9.:

 

 

Figure 3. Velocity diagram or speedgraph

 

1Golber. gollcurve figars. p. 2.

15

Obviously. a speedgraph is very well suited for studying the
value of the angular velocity of the follower curve. It is also well
suited for studying the value of the angular acceleration of the
follower curve. which is directly preportional to the slepe of the
speedgraph curve.

For the facility of studying the angular velocity and acceler-
ation. the speedgraph sacrifices the ability to study angular dis-
placements. The angular displacement of the follower curve is found

by combining Equs. (2.3) and (2.5).

9.4M.)
Nedlhgf‘

9+. aferdded =f—Ejj— Jed (3.2)

The right side of Equ. (3.2) is precisely the area under a speedgraph
curve. Therefore. to control the values of the follower curve angular
displacement it is necessary to control the area under the speedgraph
curve. In practice this is fairly difficult.

It is specified that the follower curve must make exactly one

revolution for each revolution of the driver curve.

21:
£04 ,_. (3.3)
f; u) Jed 2 1T

 

J

16

This condition makes it difficult to choose a satisfactory speed—
graph curve. In one application the area was computed to ten signif-
icant figures to insure accurate gears.2

From Equs. (2.6) and (2.7) the resulting rolling curves are
continuous if the speedgraph curve is continuous. From Equs. (3.1a)
and (3.1b) the rolling curves are smooth-m df/de is continuousm-if
both the speedgraph curve and its first derivative are continuous.
If dze is not continuous the rolling curves have sharp points or
cusps. Such rolling curves have been used in.practice but they are

avoided if possible as the cusps cause a sudden change in the follower

curve acceleration and thus prevent smooth Operation.

(Zr, 2r)

 

Eid'

Figure 4. Displacement diagram

 

Fig. (4) represents a motion pattern plotted on the graph using
the angular displacement of the driver curve as the abscissa and the
angular displacement of the follower curve as the ordinate. This

graph is generally called a displacement diagram.

 

ZGolber, op. cit.. p. 4.

17

The displacement diagram is very well suited for studying the
value of the angular displacement of the follower curve. Also. it is
well suited for studying the value of the angular velocity. which is
directly preportional to the slepe of the displacement diagram curve.
However. the displacement diagram is not satisfactory for studying the
angular acceleration of the follower curve because the acceleration
is preportional to the second derivative of the displacement diagram
curve.

From Equs. (2.6) and (2.7) the resulting rolling curves are
continuous if the first derivative of the displacement diagram curve
is continuous. Similarly. from Equs. (3.1a) and (3.1b) the rolling
curves are smooth-«- dv/de is continuous-~if both the first and the

second derivatives of the displacement diagram curve is continuous.

18

IV. EXAMPLES OF DESIGN

Non-circular cams and gears have applications in various mechan-
ical devices. This chapter develops the design of rolling curves as
applied to several specific applications. No attempt is made to
extensively cover any design features other than the rolling curves.

It is noted in these applications that the only problem is to
express the desired motion pattern by a mathematical equation. The
problem of finding the rolling curves from the motion pattern is
easily met by using Equs. (2.6) and (2.7).

Sometimes Equ. (2.3) is expressed in parametric form. There is

no need to eliminate the parameter as this form is satisfactory for

computing purposes.

§cotqh Yoke with Constant Eelocity1
A Scotch yoke is generally used for obtaining a sinusoidal trans-
lation from a uniformly rotating shaft. It is possible to obtain a
constant velocity translation by using rolling curves. In this case
the Scotch yoke is driven by the follower curve.
The velocity of the Scotch yoke equals the parallel component of

the velocity of a point on the follower curve.

 

lMillis. Principles of Mechanism. pp. 231-33. These rolling
curves were originally designed to be used with a slider crank mechan—
ism to give a constant velocity to the slider. Based upon Equ. (4.1)
the application to the slider crank mechanism is theoretically incor-
rect although it is a good approximation for a high connecting rod
to crank ratio (about 8.5:1 in this text). However. if the example
is applied to a Scotch yoke. Equ. (4.1) is exact.

l9

' 4.1
V ' jb (A); Jaw 91c ( )
where v ‘ the constant velocity of the Scotch yoke
.41: the length of the driving crank on the follower

curve

441.142; V I _ K

 

F(94)- a). - d9..- wMG¢°W-M9¢

. V
where K [2,: ”J

Gd =f"é—"¢Csveg°d9;3 ‘Zé'cébeg‘l'c

 

Boundary conditions: 6d . 0 when 9‘ = 0

ads" when 6‘ :‘fl'

Therefore, 63—211;- and K =_%_

9d ag-ZE-w 6; “Er-W 9;
r = L. = LMG;
F {"(94)+I 2/11 +44lsv94:

ZL
{fir/0.6.223;

In one complete revolution of the follower curve the Scotch yoke

 

 

 

T.“ L-fpgz

travels in two directions. For the velocity to be linear for both

directions, the yoke must have infinite accelerations at the dead

20

center positions. In any application this is impossible. Therefore.
the actual rolling curves only approximate the theoretical curves
near the dead center positions.

The rolling curves must be used as non—circular gears. In this
application the gears are designed as a pin and cog arrangement.

The rolling curve equations are based upon only one direction of
yoke travel. Therefore. they are valid for only one half of a revolu-
tion. The second half of the rolling curves are easily obtained
because in this case the rolling curves are symmetrical to the polar

axis.

Non-cirgula; Game for Obtaining Linear Measurementgz

 

In any continuous measurement it is advantageous to use an
instrument which gives a linear record of the desired variable. Some-
times this is not possible. Either a linear instrument can not be
used or it is easier to use a non-linear instrument. A couple of
possibilities are:

1. Measuring flow by means of the differential pressure. and

2. Measuring temperature by means of the saturated vapor

pressure by a specific liquid.

The non-linear variation of the instrument is changed to a linear
variation by using rolling curves. This is actually a reverse appli-
cation. Bolling curves are generally used to obtain a varying motion

pattern from a linear rotation.

 

2!. V. Hannula, “Designing Noncircular Surfaces for Pure Rolling
Contact,“ Machine DesignI 23 (July. 1952). pp. 111-14.

21

In using the equations deveIOped in Chapter II. it makes no
difference which curve drives and which curve follows. The terms
“driver” curve and “follower” curve are purely artificial. 'Since the
driver curve generally rotates at a constant angular velocity. this
distinction is used here.

For a specific example. consider a case where it is necessary to
measure the temperature of a fluid by indirect means. One method of
obtaining a linear measurement of temperature is:

1. To measure the saturated vapor pressure exerted by another
fluid in a closed system subject to the fluid in question,
and

2. To convert the pressure variation to a linear temperature
variation.

The variation in pressure is converted to rotation of the follower

curve (actually the driving shaft in this case) by means of'a Bourdon
tube or a similar mechanism. The shaft rotation must be linear with

respect to the pressure.

6¢=KP

where P3 the pressure in psi.
K 3 a constant
It is necessary that the relationship between the pressure and

temperature in step one must be known. In this example
mt
P311, ET“

where t ’- the temperature in degrees Fahrenheit

22

Let I: 10. meé, and n: 470.

The driver shaft is to rotate linearly with respect to temperature.

94:“.t
62F 3 l(§,F> 3 l<qg'(()'_‘L'£V*ELVZE

L.. K,=o.:5 and K.=o.cu.

The constants K. and K1 are used to control the maximum

variation in the angular rotations.

t

l a d9f;_d_91_ r" [7*J't'I-9'70
“a" on «It ‘ amora—

 

The rolling curves are determined by using Equs. (2.6) and (2.7).
In this case they are used as non-circular cams. The cams are
held together by gravity. The constants limit the maximum possible
angular variation to forty-five and thirty degrees for the driver
and follower respectively. Thus. it is not necessary to use gear

teeth to keep the cams in contact during rotation in either direction.

iral are 1 a Multi 1 In trument3
Spiral gears are used in a multiplying instrument which has
wide application in range finders. The instrument was used as a

part of British naval range finders in World War I.“ A similar

 

3The non-circular spiral gears should not be confused with
spiral bevel gears.

1“'The Works and Products of Messrs. Barr and Stroud. Limited.“
Eggineerigg. pp. 778-79.

23

instrument is used in the United States Army T41 range finder.5

The motion pattern is a logarithmic function.

9: = Ned) " 0. 94 1' b ("'2’
. f?
Heavy;-
Leda

 

(“Sam rim-6

where Q. and b :3 constants

Three pairs of these rolling curves are used in the multiplying
mechanism shown in Fig. (5). The values of the variables X and 43/
linearly control the driver curve rotation. The two sets of spiral
gears convert the linear rotation into a rotation which is a logarith-
mic function of the variable by Equ. (4.2). These logarithmic func-
tions are added by a differential mechanism like rolling cones or
bevel gears. This produces a shaft rotation which is a logarithmic
function of the product of 7Com? The logarithmic function is
eliminated by using a reversed set of spiral gears: i. e.. the
”follower” gear drives the “driver.“ This produces a rotation which
is the product of x and #.6

As an example of the use of spiral gears in range finders. con-

sider the problem of finding the altitude of an approaching airplane.

 

5"Non-circular Gears Generated Automatically.“ Iron Age, 171
(Feb. 19. 1953). p- 123.

6Francis J. Murray. The Theory of Matheqqatical Mpchines. (rev.
ed.: New York: King's Crown Press. 1948). Part II. p. 18.

21.

The range finder gives the distance and the angle of inclination of

the airplane. From trigonometry

‘1 “AIM“ AND

.341, h " 1o? Jb + Jo? M at .
where ‘1 =~the unknown altitude
JL= the distance between the range finder and the
airplane
o( = the angle betweenJL and the horizontal
The range finder produces two shaft rotations. one preportional
tofir and one proportional to the sine of 0‘. It is generally neces-
sary to use gear reductions to eliminate preportionality constants.7
This type of multiplying mechanism has several advantages. It

is small. It is completely automatic. By using spiral gears. it has

sufficient accuracy for fire control purposes.

 

 

(Reversed

Spiral _%
Gear

Uni t

 

 

 

   

Di ffe rent ial
Mechanism

Figure 5. Spiral gears in a multiplying mechanism

 

7"The works and Products of Messrs. Barr and Stroud. Limited."
02. Citgn pp. 778-790

25

V. ROLLING ELLIPSES, AN EXAMPLE OF ANALYSIS

The majority of the literature on rolling curves and non-circular
gears concerns elliptical gears. Elliptical gears have wide applican
tion as quick-return mechanisms for shapers, planers, and machine
tools. Elliptical gears have the advantage of being efficient. posi-
tive, and comparatively cheap.1

A pair of identical ellipses which rotate about one of the foci
satisfy the conditions for rolling curves if the center distance is
equal to the major axis. Several other combinations of simple geo-
metric curves satisfy the conditions for rolling curves. However. a

pair of identical ellipses is the only combination where both curves

are continuous and closed.

 

 

'3;
,r””:;’ \;:“-.\‘
1’ - ‘
‘CB ELL”" ’q:f{' F, §_ “:1: (2‘
Driver Fbllower

Figure 6. Rolling ellipses

 

1Reginald Trautschold, ”Machine-cut Elliptical Gears. Laying
Out and Machining Elliptic and Oval Gears," Machinery (New York},
x1111 (Aug.. 1917). p. 101+9.

26

A pair of identical ellipses is shown in Fig. (6). The distance
between the axes of rotation is AJAF' This distance is equal to the
major axis of each ellipse because AdPequals Aggfor identical

ellipses. From the properties of an ellipse

Ad (31+ 83:1:ch “"0
AfPf +PFBF= Cfip.

Therefore,

'4de + 334‘ Ara: 1" Pfgfg’qd’qf- (5'1)

The two points a and B: are chosen such that

ARC PB‘ARCPH.

Therefore, from the symmetry of the ellipses shown in Fig. (6)

Pd Bd = A; Pf AND (5.2a)
Ad Pd = 8+“ PF , (5.21:)

Combine Equs. (5.1) and either (5.2a) or (5.2b).

Ade+PfAf=AdAf (5.3)

Equ. (5.3) is a special case of Equ. (2.1). Thus, the point of
contact is on the line of centers and the curves satisfy the condi-
tions of pure rolling.2

The question naturally arises, what motion pattern does a pair

of rolling ellipses produce?

 

2Hinkle, ginematics of MechanismL p. 129-30.

27

The equation of an ellipse in polar coordinates with the origin
at a focus point is used.3

For the driver ellipse

Z
10: Ova—4) (5.4)
I +1, M 6
where a: half of the major axis
_12,= the eccentricity

This equation corresponds to the angular values shown in.Fig. (7).

     

Follower

 

Figure 7. Sign convention for rolling ellipses

Combine Equs. (5.4) and (2.7) and solve for the ratio of angular

velocities.

 

 

3Olsson, Nonpcircular Cylindrical GearsI p. 27.

28

2a.
70" = 2“— fi'(ed) + I

where 2Q =L=the major axis (the distance between the

 

axes of rotation)

CHI-1‘) = _ Za. ;
I+Lm9d 0' ¢'(ed)+l

 

 

' a); l-‘Qf . '- a-
-F(94) ”J; " (5 5) L

' l+.2.'+21.cw94

One of the main considerations in an application of elliptical
gears is the maximum and minimum values of the ratio of angular veloc—
ities. From geometrical considerations the maximum and minimum values

are reciprocals. These values are found from Equ. (5.5).

ELM/Ix: [-1.1 _ (I+,¢)(l-—,¢,)
(dd M'N- (heft-Zn' (1:42,)z

 

 

 

n=—'—+—""— -'_._._L:'_-9_
(“—2 VI 1+1-
ah‘l (5.6)
‘0' n+l

 

 

—--= the minimum value of

f
M

29

Solve for the angular displacement of the follower ellipse by

integration.4

_. 2 - 9
e = e : r’ (I J)“ J (5.7)
4: H d) Zn+(.e‘+l)w94
Again consider the symmetry of the ellipses in Fig. (6). Using

this symmetry

AAdeBJ=4AfEFBP (5'8)

From Equ. (5.8) the line 8‘3; in Fig. (7) is a straight line
containing the point of contact. Extend the discussion used to prove
that the point of contact lies on the line of centers for rolling

ellipses. Combine Equs. (5.1) and (5.2b).

3; P; ‘" Pd 34 "' Ad As (5'9)

Therefore, it is possible to connect the moving fbci with a
rigid link without changing the motion pattern. Further, it is pos-
sible to replace the rolling ellipses with the linkage shown in Fig.
(8). The lengths of the links correspond to the distances between

the foci of the rolling ellipses.

LINK A, A1,: LINK 3; 3.:

The linkage is called the equivalent linkage for rolling ellipses.5

 

“Richard S. Burington, comp., Handbook of Mathematical Tablgg
and Fbrmulasl (2nd ed.: Sandusky, Ohio: Handbook Publishers, Inc..
1940), p. 714-.

5Hinkle, op, cit.. p. 130.

30

 

 

Figure 8. Equivalent linkage for rolling ellipses

A relationship between the angular displacements of the driver

and follower ellipses is obtained from the equivalent linkage. The
6

derivation is based entirely upon trigonometric considerations.

n(l+med-M6;-megemeg)
- me; + w 64 = 0

where .£L"' llld’f3;"€é lq‘,"q.F

The definition of eccentricity corresponds to that for the

(5.10)

ellipse.
It can be proved that Equ. (5.10) is identical to Equ. (5.7).7

Continuing the analysis of the equivalent linkage, a relationship

is obtained for the ratio of angular velocities. This derivation uses

Equ. (2.2) (the angular velocity ratio theorem) and trigonometric

considerations. The resulting expression is identical to Equ. (5.5).8

 

6See Appendix C, pp. 65-67.
7See Appendix C. pp. 67-70.

8See Appendix C, pp. 70-72.

{

“an” M 9 ”an ‘n ”.38.!

‘44
.I

V I
_:_"I3‘

V.‘
'1'; '
It. _‘

”t. s
I.
I

31

VI. GENERAL EQUATION FUR.SYNTHESIS

In an application where data is given for several positions of
the rolling curves, it is necessary to connect these positions with
a proper curve which can be used to design the complete rolling curves.
This has been done by plotting the data upon a speedgraph and con-
necting these points by a series of algebraic curve segments.l Another
possibility is to derive a general analytical expression for connecting
given data points. The interval between each pair of points can be
handled by a separate expression which. except for its end points,
is independent of the expressions for all the other intervals.

Fbr deve10ping an analytical expression, it is easiest to work
with the displacement diagram. The main reason for this choice is
that it eliminates any conditions for the integral of the general
equation. Also, it is possible to directly control the angular dis-
placement of the follower curve.

The given data consists of three values at each data point:
the driver curve displacement. the follower curve displacement. and
the ratio of the angular velocities of the fellower and the driver
curves (which equals the slope of the displacement diagram curve).
Referring to the requirements for curves on the displacement diagram,
the general equation for an interval must meet the following condi-

tions:

 

lGolber, Rollcurve GearsI p. h.

32

1. It must connect two given end points,
2. It must have specific sIOpes at each of the end points.
3. It must have the same second derivatives as the adjacent
curves at its end points, and
4. The follower curve accelerations should be as small as
possible.
The fourth condition is general and is the main criterion in
choosing between possibilities which meet the other three conditions.
It is very desirable to have the curve for each interval com-
pletely independent of the curves for adjacent intervals, except for
the given data at the end points. This can be done by replacing the
third condition by specifying a particular value for the second deriv-
ative at the end points. The best choice is to have the second deriv-
ative equal zero at the end points. Any other choice would eliminate
the special case where two points might be connected by a straight
line, the case which gives circular gear segments.
Thus, the general formula must meet the following conditions:
1. It must connect two given points,
2. It must have specific s10pes at each of the end points.
3. Its second derivative must equal zero at the end points, and
4. The follower curve accelerations should be as small as
possible.
The given data for the general case is shown in Fig. (9).
Condition one is easily met by a simple linear expression. Also,

the second derivative of a linear eXpression is zero.

33

(Infid/sgqé

(1%" %%fi’l

9.:

 

 

Figure 9. Displacement diagram with the given data

The general formula must be the sum of several expressions.
Thus, the remaining expressions have the following prOperties at both
end points: the value and the second derivative both equal zero and
the first derivative does not equal zero. These conditions are met
by any sine curve whose period or the integer multiple of that period
is equal to twice the given interval. Since there are two slape
constants that the curve must satisfy, the general expression contains

two sine terms.

 

9d -X. . rm.
+ 'Wm‘l'
7.2- x. K’- xa—z.

+4r4' (94" 7")"“9’1
+74."

63%: (h(' .ALDOL’VY\IT
(6.1)

 

where K, and K2: constants which depend upon the given data.

m and n = unequal integers

3“

1’1, and Xag‘ the value of the driver curve displace-
ment at points one and two respectively
«I and g; ' the value of the follower curve displace-
ment at points one and two respectively
It is convenient to use a different set of coordinate axes for
each interval as the curve for each interval is independent of all
other intervals. By using a set whose axes are parallel to the orig-
inal set but whose origin is at the given point one, Equ. (6.1) is

simplified.

6p =,K m— m:_6_4+ K24..." :56” 4-1;:64 (W)

 

where x xt—x'
44 era:

The first derivative of Equ. (6.1a) is used to determine the

data constants K. and K2“
I
efimm me. +mmm+i (6.2)
76 26 7c ' 3”
It is apparent that m and 7' cannot be either both even or both
odd. Therefore, let?“ be odd and n be even. The values of the data
constants are found by substituting the given data at the end points

into Equ. (6.2) .

l_ Kym" Kin"; (6.2a)
V" x + X T:

 

 

35

 

. -K.vmr szr 14:.
= +____ (6.2b)
if" 7L 1 +74

where g; and q; 8 the value of the ratio of the angular
velocities of the follower and driver
curves (also the slope on the displace-
ment diagram) at points one and two

K. and K2 are found by solving Equs. (6.2a) and (6.21)) simul-

taneously.

-Zmfl —(#" ”#20 (6.3.)
(M +442 -4) (6.31:)

The angular acceleration of the follower curve should be as small

 

as possible. Condition four is used to find the best values for the
integers Mand n . The angular acceleration of the follower curve

is the second derivative of Equ. (6.1a) with respect to time.

2.
“4: «_(£%)1)= (Kym’dw-mi— ”64 +Kz' n 1M— £79; (6 1+)

_-TF 67—;Jd[m "((414 -434“)? 64
+(14:+4¢;——2—;¢)nm3£-64

(6.4a)

36

The integers appear both within the sine terms and as linear
factors in Equ. (6.4a). Thus. for minimum follower curve accelerations
the values on“ and n should be as small as possible. Let m equal
one and Y] equal two. The general equation can now be written in

definite form as the terms depend only upon the given data.

9; . K. M-g—Gd + K2 44312-27119); +£6¢ (6.5)

K. 7%(41-42') “'5‘”

K1“ ‘fn 1(49 “v: ii) “'5‘”

The general equations for the angular velocity and angular accel-

eration of the follower curve are written in their final form.

W]; 71 Zn .
w: =K,— 24W —ed +2 Kz-i—m—i-ed #21 ‘6 6’

z
oq: = w—ég-WJ) (K, M £94, +1“sz dag-£94,) (6'7)

Referring to Equs. (2.6) and (2.7). the general equations for

the rolling curves are written.

L
121%“ W-%9J +2 fining-’94) +-§-+ I

(6083)

37

t= L— L
1(K. WI ““9442sz J94) +¥+I

 

(6. 8b)

It should be remembered that all of the angular values used in
this chapter are in radian units. In applications it is generally

convenient to convert to degree values.

38

VII. PROPERTIES OF THE GENERAL EQUATION

§pecial Cases of the General Eqpation
The conditions for either of the constants to equal zero are

found from Equs. (6.5a) and (6.5b).
I<i 3):) 'F: “€f7':= “flit;
Kz=0 u: 4411-4421 =2;—

If both of these conditions are met, the displacement curve is

linear. This case gives circular rolling curve segments.

lgguations of the Genergligquation and Its Derivativeg

In the design of rolling curves it is important to know the arc
length, the angle between the radius and the tangent. and several
other properties. These prOperties may be found for the general
equation by substitution into the calculus formulas. However. for
the purposes of computation the resulting equations are burdensome.
It is generally easier to compute the value of the general equation
and its derivatives as intermediate values. These values are then
substituted into the necessary equations. This method is especially
advantageous because the intermediate values appear in several equa-

tions.

. 71' . 1a
1°(ed)= (flanged + mangled +1§ed 0* >

39

{176” "if-K, W-—;{- 94 +¥szgred +% ‘7'“)

{:"(9‘0 = " %2 MW WEGJ +4-sz 3%794 (7'1”

10"‘(ed)=-— {’— K,w"ed+3chosZ—"ed <7. m

It is noted that the above equations have mainly sine and cosine
terms. With a preper choice of values of the driver curve displace-

ment the computing work is cut in half.

Arc Length of the General£§guations for Rolling Curves
In the design of non-circular cams and gears it is desirable to
know the arc length of the rolling curves. This is especially true
in the design and manufacture of non—circular gears.
Since rolling curves satisfy the conditions for pure rolling.
the length of arc segments is the same for both curves. The equation
for the arc length is obtained by using either Equ. (2.6) or (2.7) and

the expression for the arc length of a curve in polar coordinates.1

flea)]‘[r"'(ea>+flz+meafl_ "‘
S Lf ‘[ Masha? 6" ( )

where S = the arc length of the curve between 9{ and 6

Substitute into Equ. (7.2) to find the arc length of the rolling

curves of the general equation.

 

13cc Appendix A. pp. 58-60.

x —(K: 1x; (44,164 +2 K2171; Wines! +%)z
o L. «$046116. 1'2Kzlirm'fz'n9d'ri112:

(K. g5. ado-,3 94+ 4:91;. mated)" :19
6f.%%%9¢l +2K2%W%EGJ+% I"): d

Since the rolling curves of the general equation correspond to

 

S=L

 

 

 

particular intervals only. the arc length of each interval must be
computed separately.
Equ. (7.3) must be computed by using the trapezoidal rule or

Simpson's rule.

Apgles getween the Radius and Both the Taggent and Earmal to the

Rolligg Curves of the General Equation
The expressions for the angles between the radius and both the

tangent and normal to the rolling curves are used in the manufacture
of both non-circular cams and gears. The angle between the radius and
the normal is especially important in producing non-circular gears.
This angle is used in adding the addendum to the rolling curve to
find the size of the gear blank. Also. the center lines of the gear
teeth are normals to the pitch line.

These angles are found by using the apprOpriate calculus formulas
in polar coordinates.

The angle between the radius and the tangent may be computed for
either the driver or the fellower rolling curve. Since these values

are supplements. it is only necessary to compute one of them.

1+1

 

 

Figure 10. Angles between the radius and both the tangent and normal

3*” Yd Alf? B‘IGJHL'I W

where Wd 3 the angle between the radius and the tangent to

the driver curve as shown in Fig. (10)
-(K,I£M%64 +2Kzgwé§ed+f)
X(K.%w%9d +2Kzgm§jflg and“)
1.}; (K3715- Aim/g'ed +4K21£Mé71§ 64)
All; a 130’... (p; (7.6)

where ‘8; 2the angle between the radius and the tangent to

Intr-

 

the follower curve as shown in Fig. (10) in
degree units
The value of the angle between the radius and the normal to the

rolling curve is found from Fig. (10).

 

2See Appendix A. pp. 61-62.

42

4"! ._.. q0°_ ('1; (7.7a)
4’.- = 902' Y; = L$2470" (7'71”

where ¢d =the angle between the radius and the normal to
the driver curve as shown in Fig. (10) in degree
units
¢+Pthe angle between the radius and the normal to
the follower curve as shown in Fig. (10) in

degree units

Aggglar Acceleration of the Follower Curve

A previous method for obtaining rolling curves from given.data
consists of plotting the given data upon the speedgraph and connecting
the given points by a combination of simple algebraic curves.3 A main
advantage of this method is that it provides a method for controlling
the angular accelerations of the follower curve.

In working with a general analytical equation. the control of
angular accelerations is lost. In the case where the complete data
is specified for all the given points. the value of the acceleration
follows directly from the general equation and there is no possibility
of changing it. However. in the case where the specified data is
incomplete. the value of the angular acceleration of the follower
curve is undetermined. Thus. the maximum angular acceleration of
the follower shaft furnishes a criterion for determining the incom-

plete values.

 

3Golber. Rollcurve Gears; p. 2.

43

For example. consider the case where the given data is complete
except for the follower curve position at both end points of an inter-
val. In order to find the general equations for the interval. it is
first necessary to specify the follower curve displacement increment.
The best choices are the positions which give the smallest possible
maximum angular acceleration of the follower curve. Since the general
equation has two sinusoidal terms. this condition occurs when the
maximum angular accelerations for all intervals are numerically equal.

In any problem there are four values which determine the general
equation for an interval. These are: the lepe constants (ratios of
angular velocities). 4: and 4;; the change in driver curve displace-
ment. % : and the change in follower curve displacementq. Thus.
four types of problems are possible depending upon which factor is
not specified. Generally. the follower curve increment..11y. is the
adjustable factor. This type of problem corresponds to the speedgraph
method for synthesis. The acceleration expression is develOped for
this type of problem. but the equations may be used in solving the
other types of problems.

The angular acceleration of the follower curve is given by

Equ. (6.7) e

Ida-IT - .
a "—72" K,M%6d+‘szm%F-9J <7~8>

W4 27» —[(f: “4a)4w_ed‘ “(MW ' awn/€16)

(7083)

where m emi—

X.

Assuming the type of problem where the follower increment is not

specified. % is given. Therefore. Equ. (7.8a) may be written

K; .2-‘I'Xv— (A): F. (7.9)

Fat—(4,244; mired—4(M—m):: 2,194 (7 9b)

The maximum angular acceleration of the follower curve occurs

where the first derivative of the factor F equals zero.

(dig-"0 g'ZITz *[k'Wfid +8Kz(2M§—ed '0]

léKzW z ”64+Kfl-m12E64-W8Kzs0 (7.10)

Solve for the location of the maximum follower curve acceleration

by using the formula for the roots of a quadratic equation.

Wfl= “KI __+JK +5IZKL (7.11a)
32K; 32 K2

-(‘¥ 4‘) +/ (#1 ‘42.) 2'"F572[£68,i'q;,)'?1::r-(7. 11b)
3213(4.‘ +443— -m]

 

W?

I+5

where p = the value of 1%" 94 which gives the maximum value
of the follower curve acceleration
Equ. (7.11b) gives a pair of values for the location of the
maximum follower curve acceleration. The value which gives the largest
numerical value of the factor F: is used. Table I shows the location
of these values and the sign of the maximum value of the factor F: for

various values of the data constants.

TABLE I. LOCATION AND SIGN OF THE MAXIMUM ACCELERATION

K, AND K2 AND 5 F mm
xvi-vi 2’44”va “*
+ + 1:61-90" --
+ - Cid-B5" -—
- + sot-135° +
—- ' — 452 90° +

 

 

 

 

 

 

 

 

 

Figs. (11) and (12) show the location of the maximum follower
curve acceleration and the maximum value of the factor F: for partic-
ular values of 4%: and 42' and for various values of the ratio m.
The location curve shows the predominance of the accelerations of the
double angle term over those of the single angle term. This is also
apparent in Equ. (7.1lb).

The process of computing the maximum value of the factor F: for
a satisfactory range of data constants becomes very involved. It is

therefore necessary to use an approximation. Several are possible.

 

4:40
«2' =05

 

 

 

 

—————— ———~‘_ *————--—*——

 

 

 

 

 

 

 

 

LO (.5 2.0 2.5 6‘?

0.5

0

0° 46’ 90' a 135° 180‘

Figure 11. Location of the maximum follower curve acceleration
versus the ratio m for particular values of 4" and “9?,-

1+7

 

4.51.0
/ I"; '05

 

 

 

 

7—
\
31F! u-

 

 

/
2.

\V/ _
o
O 0.5 IO m 1.5 2.0 2.5 3.0

Figure 12. Maximum value of the factor F: versus the ratio m for
particular values of 4, and V;

 

 

 

 

 

 

 

 

 

 

14-8

It is observed that the curve of Fig. (12) has a pair of asymp-

tOteso

BF, - J2; (13¢: ‘44] = i ‘fEn-ZL(«4:+14.§I<7.12>

One approximation is an hyperbola whose axis is parallel to the
F:-axis, whose asymptotes are Equ. (7.12), and which passes through

flFl-%(4:’-4é31‘_Lm-2L(14I+4£‘)] : m
0-?)‘W-45)‘ ‘50 *5) (44 142') '

The analytical expression describes a pair of hyperbolas. One

 

 

of these is extraneous. Solving for the factor F:, the particular

hyperbole used in the approximation is found.

 

 

“=1 =I?(e:—4.')I+/(Tj§W-4z'>‘+I613" -2'-(4i+#z‘flz

Another expression is possible which enables a graphical deter-
mination of the factor F:. As an approximation, consider the maximum
value of I: to be equal to the sum of the maximum values of both of
the terms. The resulting vee shaped curve is easily represented by
an alignment chart. The resulting maximum error is the difference

between the approximation and the asymptotes.

E a '(l -%z)[2L(1¢o'+Qi)-MJI (7.110

where E3 the maximum possible error in the alignment chart

shown in Fig. (13)

49

 

 

 

 

 

 

 

 

D
<90“ 4/ + F 4; m
>qo° #5 "F 44} m

 

 

 

 

 

 

 

 

30" -80-- 3.0-- Io-L
20" 0-— 2m 20--
I 0" zo-— I.O*~ 3&-
0 J- Ibo-«- o J- ltd-

1. Use Table I to find the location of 5.

2. Plot data on lines A. B, and D.

3. Use lines A.and B to find the point on line C.
1+. Use lines o and D to find the point on line E.

Figure 13. Alignment chart for an approximation of the angular accel-
eration of the follower curve

50

VIII. MANUFACTURE

Using the conditions for pure rolling, rolling curves are de-
signed with theoretically perfect accuracy. Therefore. the accuracy
of non-circular cams or gears depends entirely upon their manufacture

from the specified equations.

In general there are two methods of manufacture. namely continuous

cutting and increment cutting.1 Increment cutting is best adapted
for either small quantity production or production of masters for
large quantity production. Only increment cutting is discussed here.
It should be remembered that the follower curve equations are
not based upon the usual sign convention for polar coordinates. Any
confusion on this point may be easily avoided by manufacturing the
follower curve in the same manner as the driver curve and simply

inverting the follower curve before use.

Manufacture of flgn-circulgr Gang
The manufacture of non-circular cams is developed for increment
cutting with a milling cutter. The theory may easily be adapted to
other cutting tools.
Fig. (14) shows a method of milling a non-circular cam. The
values used in the theory apply to either the driver or the follower

cam and, therefore, the subscripts have been drapped.

1Lockenvitz, Oliphint, Wilde, and Young, "Noncircular Cams and
Gears." p. 143.

 

‘:
‘~_~v
v. 1 ~.‘

 

 

51

Killing
cutter
axis

 

 

 

 

 

 

/ man
I Y tangent
Y V
A O
9 Polar

1 axis

Axis of

rotation

Figure 14. Machining non-circular cams with a milling cutter

The cams are easily out if the location of the center line of
the milling cutter is known as a function of the angular displacement
of the driver curve. It is easiest to specify the milling cutter

position by using Cartesian coordinates.

t1 (b 3% .72? (8.1)

where ¢= the angle between the normal and the radius as

shown in Fig. (lb)

Vae—¢ (8.2)

 

52

where U= the angle between the vertical and the polar axis

as shown in Fig. (11+)

X = - 1’41... 4’ ‘83“2’3
Y= 1" wcf + C (8‘3“

where X and Y: the Cartesian coordinates of the center
line of the milling cutter

C=the radius of the milling cutter

 

'fianufacture of Blangggfor Non-circular Gears !
Blanks for non-circular gears are produced by the same method
used for non-circular cams.
The specified rolling curve forms the pitch line of the non-
circular gear. To find the gear blank it is necessary to add the
addendum to the rolling curve. Since the center lines of the gear
teeth are the normal lines, the addendum is added normal to the pitch
line. The equations for the position of the milling cutter are
similar to those for non-circular cams since the cutter radius is

also measured along the normals.“

X: _‘0M¢ (8.49.)
Y: ‘0 w¢ + C + A (8.1%)

where A 3the addendum

 

zLockenvitz, Oliphint, Wilde, and Young, op, c1t,. p. 143.

301sson, gen-circular Cylindrical Gears, p. 132.

41b1d., p. 128.

53

As with ordinary gears the addendum depends upon the diametral

pitch and the tooth form.

Manufacture of Non—circular Gearg

There are several methods of forming teeth in the non-circular
gear blanks. The method described here uses fermed-tooth milling
cutters and is well adapted to small quantity production.

Involute teeth are used on non-circular gears. Theoretically
the involutes are drawn from a non-circular base curve. However,
the true involutes closely approximate circular involutes and ordinary
formed-tooth milling cutters are used.

Since the radius of curvature is not constant, it is generally
impossible to use the same formed cutter for the entire gear. This
requires computing the radius of curvature at the center line of each
tooth space and then specifying the preper formed cutter.

In Fig. (15) the formed cutter center lines are located by the
same method used for non-circular cams. The settings of the blank
must be found so that the teeth are evenly spaced. This requires
that the distance between every tooth space center line be equal to

the circular pitch.

-_§_ 8.
P-“ (5)

where F): the circular pitch
8: the total circumference of the pitch curve

n= the number of teeth

 

51+

 

Formed-tooth
cutter axes
final positions

T Tooth space
center line

 

 

Addendum line

r \\“‘ Pitch line

B
L.—
Y K (")0 Dedendum line

 

 

/ Polar axis

Axis of rotation

 

Figure 15. Cutting non-circular gears with a formed-tooth cutter

TABLE II. SELECTION OF STANDARD romaine-room MILLING CUTTERSs

       

     

     

    

              
     

    

21
5

23
a

19
5%

       

1?
6

11+ 15

at

 

13
7%

12
8

Value of Z

             

    

    
   

      

      

     

Cutter No. 7

 

        
   

        

 

 

so
1%

#2 135

2%

    
  

    

26
I4.

35
3

 

30
3%

Value of Z

     
   

  

  

   

  

Cutter Ho.

 

 

SOlsson, Op, cit“ p. 131+.

 

55

The first tooth space center line is located arbitrarily. This
also locates a gear tooth and consequently all of the tooth space
center lines on the mating gear.

Using Equ. (8.5), it is possible to find all the proper values
of the angular displacement and accordingly the settings for the
gear blanks.

Again, the cutter positions is located by Cartesian coordinates.

X=_f.M¢ (8.6a)
Y: rw¢+B—D (8.6b)
V=e_4, (8.6a)

where B = the outside radius of the formed cutter
D-‘the dedendum

The specific formed cutter depends upon the radius of curvature

and the diametral pitch.

Z‘Za‘R (8.7)

where z: “The number of teeth for which the cutter is
designed when milling cylindrical gears.”6‘
Pd=the diametral pitch
R 1' the radius of curvature
To use Table II, round down the algebraic value of z. The
value of Z is negative for internal gears.

The radius of curvature is computed directly from the calculus.7

 

6Olsson, op. cit“ p. 128.

7See Appendix D, pp. 73-75.

 

 

56

in L _fl:‘l’6.)]‘[~"(ea)+fl’+ Phil?" (8...)

gym: [1"(9‘IWE‘VBJHWZ BYesz-f'leanmm.)
R‘, L ,{[£'(e.)]‘£°'<e»+:]‘+[479.37%(8...,
I379.) + I] 3 [.0 ‘(eafl ‘+ L? 19.)} EC “(MT-r {794) 4 “(6.)

where Ratthe radius of curvature of the driver curve

 

 

 

 

Rg‘the radius of curvature of the follower curve

For the purposes of selecting a formed cutter, compute the radius

 

of curvature at the midpoint of the tooth space.

Generally it is far simpler to use an approximation for computing
the radius of curvature. The portion of the pitch circle between two
tooth spaces is replaced by a circle with radius equal to the radius

of curvature . 8

 

Z gill. - (8.9)

where AV 3 the angular difference between two adjacent

blank positions for cutting tooth spaces

 

8Olsson, 0p, cit“ p. 134.

 

A.

B.

C.

D.

APPENDIX

Proof That the Arc Lengths Are Equal and the Angles
Between the Radius and the Tangent Are Supplements
for Any Pair Of BOlling Curvefi e e a e e e e e e e

Extension of the Rolling Curve Equations to the
Case of a Driver Curve with a Varying Angular
velOCity e a e e a e e e e e e e e e e e e e e e e

Derivation of the Equations of Motion for Rolling
Ellipses by the Use of the Equivalent Linkage and
Proof that These Equations Are Identical to the
Equations Obtained from the Theory of Rolling

Curves e e e e e e a e e e e e e e e e e e e e e e

Derivation of the Equations for the Radius
Of Curvature or 3011138 curves 0 e e e e e e e e e

57

page 58
e a 63
. . 65
. . 73

 

 

58

A. PROOF THAT THE ARC LENGTHS ARE EQUAL AND THE ANGLES BETWEEN
THE RADIUS AND THE TANGENT ARE SUPPLEMENTS FOR
ANY PAIR OF ROLLING CURVES
Physical considerations demand that a pair of rolling curves

satisfy two conditions. First, the arc lengths of both curves between
any two particular points of contact on continuous arc segments must
be equal. Second, at any point of contact the values of the angle
between the radius and the tangent for both curves must be supplements.
It was stated in Chapter II that both of these conditions are met if
the rolling curves satisfy the conditions for pure rolling.

.Using only the conditions for pure rolling, the equations for

the rolling curves are found.

- L 2.6
70“ ’ $790+! ( )

vie—4L (2.?)
a? (9d) + I

The arc length expression for a curve in polar coordinates is

found by calculus.1

P z 2’2
Sn]; [f+‘j“5‘] d9 ‘ (M)

where 53 the arc length of the curve between 0‘ andP

 

1William A. Granville, Percey F. Smith, and William B. Langley,
Elements of Cagulus, (Boston: Ginn and Co.. 1941), p. 292.

59

The proof that the arc lengths of a pair of rolling curves are
equal consists of separately evaluating Equ. (A.1) for both Equs. (2.6)
and (2.7) and then showing that the expressions for the two are lengths
are identical.

Let two points of contact on continuous arcs of a pair of rolling

curves be :

Point 1: ed a“ (NO

Point 2: ed :fl.

.fl. -.- L I". 9“ (2.7.
d 94 [419.) + Ii )
- " HYe.) 2 We.) J ’2
5" L {[(‘YGJHJ + [men + 13‘ “'9"
a ' z .. 2 a
5; = L f { EF (6‘)] +% ([94 (4.2.1)

(3"(64) +02 [30'(e.)+l "'

where Sd=the arc length of the driver curve between 0‘

and p

P 2 J 1’ 2. V?-
+ .‘ Y; d9; 4:
where Sg=the arc length of the follower curve between 0‘

and?

fli=____c_l{-L 'Fued ._L__.. 2.6a
J94: [$76,044] (“(94) ( )

 

 

 

60

d6; = ”94% Jed ‘2'“)

We.) kerzefl .
5“ f... £ij He .)+ I] Fume,

[#"(eaflM’z
5* liIL-fledhflfi R Med

B (9.01- [1"
5r L f [1‘ '(94) [H6122
”OWE" E” (Gd) “3

Therefore, 5d = 54:-

It is noted that the arc length is directly prOportional to the

 

 

 

aid 9d (A.2b)

distance between the axes of rotation. This fact is used in appli-
cations of non-circular gears. The gears are designed for an arbi-
trary distance between the axes of revolution. This value is then
adjusted to give a convenient value of the arc length, the circular

pitch and, accordingly, the diametral pitch.

 

\ed Id fl {9‘
VG.
/ ‘1’. fl \

Figure 16. Sign convention for the angle between the radius and the
tangent

61

The proof that the value+f the angles between the radius and

the tangent to the curve are supplements follows in a similar manner.

JI§i~Le (fl: ‘f’__,_ :+;;. (5.3)2

where Y3 the angle between the radius and the tangent to
the curve as shown in Fig. (16)
The two angles must be supplements of each other. In mathemat-

ical terms
134,31. +fiwfl=0. (1.4)

where v‘ = the angle between the radius and the tangent of
the follower curve
‘l’d: the angle between the radius and the tangent of
the driver curve
From here the proof consists of using Equ. (A.3) for both curves

and showing that the resulting equations satisfy Equ. (A.4).

.Ji:‘OLI ‘r;?= ‘f;:+

.. L- .B’gfljfi. .
”VI-$764)“?! -L 4276.!) 446")

 

 

 

e. a...

 

zGranville, Smith, and Langley, op. cit,, p. 207.

62

fiw'fl“ mg;

L +794) [fiwdhflz
My” F'(e)+l L-'+"(6d)

 

m ‘Ifi" - :2523; [-F (9d ) +0 (A.5b)

 

Th °f N" lbw Y4: +13.» 9’40 ‘0

63

B. EXTENSION OF THE ROLLING CURVE EQUATIONS
TO THE CASE OF A DRIVER CURVE WITH
A VARYING ANGULAR VELOCITY

In the usual application the driver curve rotates at a constant

- "I.
F
.5
.‘Yb"

angular velocity. However, this is not a necessary condition. For

"r‘
u
l

eprle when a very high maximum value of the ratio of angular veloc- I
ities is required, non-circular gear trains are used.

The theory is easily extended to cover the case where both curves 5

 

rotate with a varying angular velocity.
Consider the case where the complete motion pattern of both

curves is known.

ed: 3 (t) (3.1.)
6.; = -F (t) (3.1b)

where *3 time

It is possible to use the equations deve10ped in Chapter II if
the variable t can be eliminated so that 6.; is expressed as a func-
tion of Gd . A different set of equations must be used if the variable

t can not be eliminated.

Fred) gfl - 'th) (13.2)

 

 

- LEM 3.3.
If 'e-(e-I-re) ‘ ’

1" = LIN“) (3.31:)
3'(t)+f'(t)

The general equation can be used for a synthesis problem. The
only requirement is that the required data can be specified on a
displacement diagram.

All of the preperties of the general equation remain the same

except the acceleration expression.

=~ %LJJ UK: W94+4K2MJ9J
(3.1+)

+°Q K— EWEGJ+ZK2IM§~ 64+$

The discussion of angular accelerations deve10ped in Chapter VII
is invalid for the case of varying angular velocity of the driver

curves

65

C. DERIVATION OF'THE EQUATIONS OF MOTION FOR.ROLLING ELLIPSES
BY THE USE OF THE EQUIVALENT LINKAGE AND PROOF THAT THESE
EQUATIONS ARE IDENTICAL TO THE EQUATIONS OBTAINED
FROM THE THEORY OF ROLLING CURVES

In Chapter V it was proved that rolling ellipses may be replaced
by an equivalent linkage. This linkage is used to derive a relation-
ship between the angular displacements of the driver and follower
curves. The derivation depends only upon trigonometry.

Fig. (17) shows the equivalent linkage.

Figure 1?. Equivalent linkage for rolling ellipses

L“ ¢=4340Ad'=4B;DA-p
AdBd=A¢BflR
BdD=7b and B;D=Za.-7o

66

It is convenient to use the eccentricity from the rolling ellipses.

Zn.
The following relationships are obtained from trigonometric consid-

BdS=R Med .
BE?”= R we;
M 4): R ”6;!
X
- _ R we;
Mt" Za-x
Raves: Rash/91F
7K Zea-70
Zaeu'wed
m94+4ee94
Received
X
3T=Za—Rw9++ Rwed

g 3T
34 BE

 

 

it: =

m4>=

=.e(m ed ham/94:) “'1’

WA

67

00-04): Za-Rweri'RmeL
2a.

W¢= I+.s(w94-w9;) (0.2)

Equs. (0.1) and (0.2) are combined by using an elementary trigo-

nometric identity.

M2 C) + WZT = I
.e‘eee'wzed 1" 2.25m, 9; Med £9,544.36;
+1 +21, med ‘ZLMG; +1} 44% 6,:
“ZLZM-tef med +n‘m‘ed = 1

£2 (M294 + (LN/‘64 + safe, + M‘s.)
+Z1, (med -W6r)
+Z,&Z(-u'ov9;, med—we. meg-=0

.L(I +M94: sum/ed -' W94: W94)
(c.3)
+Wed-x wag-‘0

Equ. (5.7) gives the relationship between the angular displace-

ments for two rolling ellipses when considered as rolling curves.

2: .. (I "£Z)Med (an)
e; 21+(LZ+I)W94

68

Equ. (0.3) gives the relationship between the angular displace-
ments for two rolling ellipses as derived from their equivalent link-
age.

Equs. (C.3) and (0.1+) must be proved to be identical. The method
is to find the expressions for the sine and cosine of the angular

displacement of the follower curve from both Equ. (0.3) and (0.1+).

_.«J~z
Jib-76 04—623

 

1",. , 9+. .. 0‘93 M94 (0.5)

Z1. + (43+ I) We; (0.6)
H

The factor H is found by two methods. First, it is found by the

CNT1ZL162F

use of a trigonometric identity. Second, it is found by substituting
Equs. (0.6) and (0.5) into Equ. (0.3). The proof consists of showing

that these two methods give the same value for H and consequently

the same values for the sine and cosine of 9;.

The first method uses Equs. (0.5) and (0.6) and a trigonometric
identity.

mew MW =I
U’ ‘12)Meflz‘f‘ [Zn +(.of+ I) wear: H“
H'=.e."+21.‘+l 4- ‘ME cafe.)

. (0.7)
+ ’N.’ med + Hanson/ed

69

The second method uses Equs. (0.5) and (0.6) which are substi-

tuted into Equ. (0.3).

we. (Mimi __ weal2e+éf+0wg§

 

eel-I—

 

H H
.. ZL+(—;:I+I)wed + wed=0

(1.4- med) H Under/294 (£3- !) +8 Law 94 + £1.
+1.65“) (‘44—‘94 +(.e‘+ I) m 64
£2.61."- I)(I-Mo29d) +2130“ 9:! +31,
+1.65%!) I‘d-6' 64 + (car- I) w 6,;
=1? u. --_o’ m‘GJ +1 m‘QI-anwey
121.30%294 to 644.29,; +24te‘w 6,; + m 6.;
= (its) + we.) (33+ I) +21. w‘ed
= 21, mica + Bias 6,; +med (11+!)
+1? + .1.
= (w 94 +.c)(2.e m. 64 +.¢’-+ I)

H=leed +.ez+l

7O

Hz= [razed-Jed te’i- I+‘I'.£’W94
(0.8)
+41. wed +8.0"

Therefore, the value of H2 in Equ. (0.?) equals the value in
Equ. (0.8). This proves only that the two methods give the same
relation between the numerical values of the angular displacements
of the two ellipses.

The proof is incomplete because two values of the factor *4 may
be obtained from Equ. (0.7). These two values are numerically equal
and differ only in sign. One of these values is eliminated by physi-
cal considerations. Fbr a small rotation of the driver ellipse in
the positive direction from the polar axis both 62d and £2; assume
first quadrant values. This also applies to the equivalent linkage
for a small positive rotation of the driver link. From this consid-
eration one of the values of the factor II obtained from Equ. (0.7)
is eliminated.

Therefore, the two methods give the identical relationship
between the angular displacements of rolling ellipses.

Also, Equs. (0.5) and (0.6) may now be completed.

(I-J’z )‘M’ 6L. (0.5a)
{21.04494 +1, z-I-I

Z£+ («9’ +) (wed. (0.6a)
2.1, we... +1} +I

The equivalent linkage is used to obtain a relationship between

map:

we:

the angular velocities of the driver and follower ellipses. The

71

derivation uses the angular velocity ratio theorem and trigonometric
considerations. The angular velocity ratio theorem states that the
angular velocities of the driver and follower vary inversely as the
segments on the line of centers cut by the line of transmission.1
For the equivalent linkage the line of transmission is link 8d B;

in Fig. (17).

(A); = 7’4 ._. AJD (0.9)
00d f; A90
The following relationships are obtained from trigonometric

considerations.

4,: Zamed

4431/64 tun/64,- ”(’4’

 

= __ 3 game; ‘
TD (20. Z)W¢ med +M94: W¢

A40: SD ’SAJ

Am 2:41:36; MI " Rmed

A D= 24mg (“Muted-mgfl-Rmea (seemed)
d We; +4.44.» 64:

game. «maimed —Rmedme,c
My 9J+4iovef

 

 

AaD"

 

1Hinkle, Kinematics of MechanismI p. 22.

 

72

AfD = ApT +TD

- Basal/ye; .
Aio’RW6*+med+me. WI

R we (megawhzaabafidmerwsl
44;... e.. + 4141/ 6.0

 

Ang

Aim/ed +M6§

Ad D = 2a. (415661-24 meIerQi-wweawacl (0.10)
A45 Za(we¢+nwe.med+emedwee.)

Substitute Equs. (0.5a) and (0.6a) into Equ. (0.10) and simplify.

A! D we; (21. meg te‘I- 0:44.08, [21+(e‘+l)coe64]
AF D 4' 6M ed ( I 2.901» 94}

M63 (P u‘) tgfahu 64 [2.2. HEB-01344.93
+ med (I-n‘)u~9d

 

A4 D s 21. we; +4.2” “2&2—ZLW'9J)MQ¢_
A; D I-_¢,Z+Z.¢z +4.4. 64 (flu. Isa—Lil «Um/94

Equ. (0.11) is identical to Equ. (5.5).

(4)4: _ Ad D ,___ (“1} (0.11)
00d AIcD l+1f+2lm9d

 

D. DERIVATION OF THE EQUATIONS FOR THE RADIUS

OF CURVATURE OF ROLLING CURVES

73

The equations for the radius of curvature of rolling curves are

derived by using the general equation for the radius of curvature

in polar coordinates.

z . 3!:
R: [r +(r)‘]

f‘+Z(I”)‘— f-I’"

d__T;_
r To

an“
“—7" de

The driver curve equation is used.

 

I"'(+" D-ZH')‘
TJ‘L' ’ (1:44.03

 

(11.1)1

(D.2a)

(D.2b)

(D.2c)

1Granville, Smith, and Langley, Element; of Calculus, p. 222.

7’4-

 

 

.19. da
432-: 429* ... 0'z “94)
def d9:
.. J36¢= d’ N94)
’c 461, def

Equs. (D. l) and (D. 2a-c) are combined.

R _ L gr“)‘(4=+l)‘+(¢ )‘J”
" (M) Kat')’+(+')‘+2(¢')‘- 1““th

vhe ere Rd: the radius of curvature of the dr ive er curve

 

 

The follower curve equation is used.

 

 

f4: :41}, (mun)
nPLG'F ) (p.413)
1""6”"le
1..th (LI? Jaye. 13694) +7, (__d 491)...)2
‘° (194 49.! «161:: d9+ c194 d9; ’ Jade;

”ac—+157??— EG'J‘Wv -(¢9‘r"-N'J‘”'“°’

2Ivan S. Sokolnikoff. Advanced Calculus, (New York and London:
McGraw-Hill Book Company. Inc.. 1939). p. I+8.

 

Equs. (13.1) and (D. IJ's-c) are combined.

R L [(4%- flf’"))‘+(~c )J;é
“ (To [(4’) +(+)’- (#0ng

where {=3 the radius of curvature of the follower curve

 

75

76

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"Non-circular Gears Generated Automatically,” Iron AgeI 171 (Feb. 19.
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Engineerigg. CVIII (Aug. 1. Aug. 15. Aug. 29. Sept. 19. Oct. 24,
Nov. 21. Dec. 12. and Dec. 26. 1919). 133-3“. 198-200. 263-66.
363‘650 536-“0. 669‘720 776.79. 846-49.

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