NASA

Contractor Report 185200

T-6"_

AVSCOM

Technical Memorandum 90-C-003

Computer Simulation of Gear Tooth

Manufacturing Processes

_?.A!:, rUc'I_! _A_drACIUi_l ¢_ D_c_C_sqL ¢. Fin,:_l

_,'uort (Cincinn_ti Univ.) :>3 _ CqCt !Jl

NoO-e9171

Dimitri Mavriplis and Ronald L. Huston

University of Cincinnati

Cincinnati, Ohio

February 1990

Prepared for

Lewis Research Center

Under Grant NSG-3188

N/kSA

National Aeronautics and

Space Administration

AVIATION RIS,T ACTIVITY

CONTENTS

ABSTRACT ................................. 1

1. INTRODUCTION ............................. 2

2. REVIEW OF SPUR AND BEVEL GEARS AND THEIR MANUFACTURE ......... 3

2.1 External Spur Gears ....................... 3

2.2 Straight Bevel Gears ................. _ ..... 5

2.2.1 Milling ......................... 7

2.2.2 Hobbing ......................... 8

2.2.3 Shaping ......................... 9

3. DIFFERENTIAL GEOMETRY ENVELOPES .................. 10

3.1 Theoretical Considerations ................... 10

3.2 Envelope of a One Parameter Family of Curves .......... 12

3.3 Envelope of a Straight Sided Rack Cutter on the Gear Blank 14

4. ENVELOPE OF INVOLUTE PROFILE CUTTER ................. 16

4.1 Introduction .......................... 16

4.2 Involute Analysis ........................ 17

4.3 Rolling Analysis ........................ 20

4.4 Envelope Analysis ........................ 22

4.5 Envelope of Involute ...................... 23

5. COMPUTER SIMULATION OF MANUFACTURING PROCESSES ............ 28

5.1 Introduction .......................... 28

5.2 Gear and Straight Sided Rack .................. 29

5.3 Involute Profile Cutter on a Wheel ............... 35

5.4 Straight Bevel Gears ...................... 41

DISCUSSION ................................ 46

CONCLUSIONS ............................... 47

REFERENCES ................................ 48

ABSTRACT

This report discusses the use of computer graphics to simulate gear tooth

manufacturing procedures. An analytical basis for the simulation is established

for spur gears. The simulation itself, however, is developed not only for spur

gears, but for straight bevel gears as well.

The applications of the developed procedure extend from the development

of finite element models of heretofore intractable geometrical forms, to

exploring the Fabrication of nonstandard tooth forms.
I. INTRODUCTION

This report discusses the geometry of gear tooth manufacturing procedures.

It presents results of research supported by the National Aeronautics and Space

Administration, Lewis Research Center under Grant NSF 3188 to the University of

Cincinnati. It is based upon a portion of the first author's doctoral

dissertation [1].

The gear tooth surface geometry is viewed as the envelope of a cutting tool

as it sweeps through a gear blank. The analysis is based upon earlier research

also supported by the Lewis Research Center and reported in [2,3].

Recently there have been dramatic advances in procedures for computeraided design of mechanical systems. In particular, there have been significant

advances in finite element methods and in associated computer graphic procedures.

Indeed, in the past decade the finite element method and computer graphic

analyses have become standard procedures in mechanical design.

For power transmission and gear design, however, the geometry has often

precluded accurate applications of the finite element method. That is, the gear

tooth profile is difficult to define -- especially for helical and bevel gears.

The contact path of mating gears is even more difficult to obtain.

In this report we present a computer graphics technique for developing gear

tooth geometry. It is based upon a geometric modeling of the gear fabrication

procedure. That is, the gear tooth surface is viewed as the "envelope" of a gear
cutter passing through the gear blank. In the report we develop analytical

procedures For determining the envelopes. Theseprocedures in turn are used to

validate computer graphic Inumerical) methods for obtaining the envelopes.

The balance of the report is divided into five parts with the first part

providing a brief review of spur and bevel gear generating processes. The next

two parts discuss the envelopes of spur gear cutters. The subsequent part

discusses a computer graphic technique for simulating the cutting process. The

final part contains a brief discussion and concluding remarks.

2. REVIEWOFSPURANDBEVELGEARSANDTHEIRMANUFACTURE

In the following paragraphs we review the nomenclature of spur and bevel

gears. Their cutting and Fabrication methods are also discussed.

2.1 External Spur Gears

The involute spur gear is the most predominant tooth profile used in power

transmission devices. A spur gear and its rack are depicted in Figure 2.1.1.
Circular pllch

Tooth thickness

Spur rock

Circular pitch

Figure 2.1.1 Spur Gear Nomenclature

The "involute" spur gear is so-named since the tooth profile has the shape of

an involute curve.

teeth. However,

watches.

The cycloid is the only other curve affiliated with spur gear

the cycloid tooth applications are limited to clocks and

The involute profile has the advantage of easy manufacturability, and the

kinematic feature that the velocity ratio of mating gears does not change as the

center distance varies. Figure 2.2 depicts mating spur gears.
_. :_' . _,_,_

Figure 2.1.2 Mating Spur Gears

2.2 Straight Bevel Gears

Bevel gears are used in connecting shafts with intersecting axes. The

pitch surface is conical. Rolling without slippage takes place between the pitch

cones of the two mating bevel gears. The teeth are tapered in both thickness

and height. The tooth profile in a section normal to the tooth, closely

corresponds to that of a spur gear tooth with a pitch radius equal to the back

cone distance, and a diametrical pitch equal to that of the bevel gear. Such

a spur gear is called an "equivalent spur gear".

Bevel gear terminology and nomenclature are given in Figure 2.2.1.
Back

angle

Face

angle

Oedendum

angle

t_ Pitch .._J

r_ apex to back

_-Pitch apex to crown

Crown

Pitch

apex

Shaft angle

Pitch

angle clearance /

/

Figure 2.2.1 Bevel Gear Nomenclature
The equivalent spur gear nomenclature is presented in Figure 2.2.2.

Whote Depth

Working Depth

Fit[et

Top

Land

Chordal

thickness

TRANSVERSE

SECTION A-A

C[I oraRce

Backlash

DedendU_um

Figure 2.2.2 Equivalent Spur Gear Nomenclature

2.2.1 Milling

Gear milling produces spur gear teeth and straight bevel gear teeth using

a form cutter. One tooth space is milled at a time. The gear blank is then

indexed to the next cutting position. A disadvantage of this method is

inaccurate tooth spacing.

Gear milling is a non-generating process.
Figures 2.2.1.Ia and 2.2.1.Ib showthe milling cut of a spur and straight

bevel gear tooth.

(a)

Figure 2.2.1.1

(b)

Milling of Spur and Straight Bevel Gear Teeth

2.2.2 Hobbinq

Hobbing is a generating process of cutting spur gears. The gear teeth are

gradually formed in a series of cuts. The generation of the gear tooth is a

continuous indexing process in which both the cutter and the gear blank rotate

in a constant ratio while the hob is fed into the gear blank. The most common

shape of the hob has straight sides, thus producing involute spur gear teeth.

Figures 2.2.2.1 depict the generation of a spur gear with a hob.
--I,OHob

\ -I

Figure 2.2.2.1 Spur Gear Hobbing

The constant indexing makes hobbing rapid and accurate. It is also

economical since one hob will cut a number of teeth of a given pitch and pressure

angle.

2.2.3 Shapinq

Gear shaping is also a generating process. Here the cutter is a gear

itself. It generates the proper tooth curve to mesh with itself. The cutter

is rotated in a constant ratio with the gear blank. Using a reciprocating

motion, material is then removed from the gear blank. A diagram depicting this

process is shown in Figure 2.2.3.1, for spur gear teeth.
[xlemal

spur ll_r

! .. CutlJnll

critter _ -

I Pitchcircle ,_-_ -

k I

_2.. "1-'- .i _,,_&

1f,.,,,,,,.,, i:i ikii !! •,..,,.-',,.--.- ,, <,} ..,,' ..,,-.,,.J.. - "

[- I

Figure 2.2.3.1 Shaping of Spur Gear

3. DIFFERENTIAL GEOMETRY - ENVELOPES

3.1 Theoretical Considerations

Consider a family of surfaces defined by the equation

f(x,y,z,t) = 0 (3.1.1)

where t is the parameter. As t changes, a new surface member of the family is

defined. Consider a typical surface, S, which intersects with a neighbor

surface, S', in a curve, C. If S and S' correspond to the values t and t+At,

10
then the intersection C is described by the two equations

f(x,y,z,t) = 0 and f(x,y,z,t+At) = 0 (3.1.2)

The surface

f(x,y,z,t+At) - f(x,y,z,t) = 0 (3.1.3)

also passes through the curve C. Consequently, the curve C is described by the

equations

f(x,y,z,t) = 0 and f(x,y,z,t) f(x,t,z,t+At) = 0 (3.1.4)

C# .

In the limit, as Ate0 and S approaches S', the curve C approaches a curve

C' is defined by the equations:

f(x y,z,t) = 0 and a[f(x,v,z_t)l = 0 (3 1 5)

, at " "

As t varies, the curve C' also varies, generating a surface called the envelope

of the family [17]. The envelope is a surface tangent to each surface S of the

family at all points of the curve C' on S. The elimination of t, the parameter

af

between f and _-_, produces an equation G(x,y,z) = O, which precisely describes

the envelope of the family.

As an example, consider the family of planes equidistant from a fixed

point. It is readily seen that the envelope of this family is a spherical

11
surface with a radius equal to the distance of the plane from the fixed point.

The family is a two-parameter family, since two angles are needed to determine

the orientation of the plane relative to the point.

It is readily seen that deriving an equation that describes the envelope

analytically is a complex process. In the following section, the development

of an envelope for a one parameter family of curves is presented. Several

examples are presented to demonstrate the use of this theory, the most advanced

being the determination of the envelope of an involute profile cutter.

3.2 Envelope of a One Parameter Family of Curves

The best way to illustrate the application of the concepts presented in

the previous section is to consider a simple case: a one parameter family of

lines equidistant from a fixed point in a plane.

In Figure 3.2.1 a family of lines L, each a distance r from point O, are

shown. The position of each line L can be defined by the single parameter @.

12
Y

0

Figure 3.2.1 Family of lines equidistant from a point

The equation of L is a straight line

Y " YD : m (x - xp) (3.2.1)

where m is its slope and (xp,y_) are the coordinates of P, the point of

intersection of L and its normal line through O. Since m is tan@ and y is

rsinO(= -rcos@) and xp is rcos#(= rsin@), we can write 3.2.1 as

or

y + rcosO : tan@ + (x

ycos@ : xsin@ + r : 0

rsin@) (3.2.2)

(3.2.3)

This is the parametric representation of the family with ¢ as its parameter:

f(x,y,@) : 0 (3.2.4)

13
Differentiating with respect to @, we have:

af

a@ = -ysin¢ + xcos¢ = 0

(3.2.5)

Solving for x and y simultaneously in equations (3.2.3) and (3.2.5), we obtain

the expressions

x = -rsin@ (3.2.6.a)

y = -rcos¢ (3.2.6.b)

Substituting equations (3.2.6) into (3.2.3) and eliminating @ we have

or

y(-Y) + x(-x) + r- 0 (3.2.7)

r r

x2 + y2 = r2 (3.2.7)

which is, as expected, the equation of a circle. This expression represents the

equation of the envelope of the lines L.

3.3 Envelope of a Straight Sided Rack Cutter on the Gear Blank

S. Chang, R. L. Huston, and J. J. Coy [18] have developed an analytical

expression for the involute tooth profile of a spur gear, generated by using a

straight sided rack cutter as described in section 2.3.2. The results of this

analysis are included in this section as an example of the theory discussed in

14
section 3.1. As an aid to visualizing this geometry, consider a perfectly

plastic wheel rolling over a straight sided step. The impression of the step

on the plastic wheel is the envelope of the Family of line segments representing

the side of the step. The only difference from the previous example is the fact

that the wheel is rolling. Hence, it is advantageous to describe the geometry

relative to a coordinate system fixed on the wheel (Figure 3.3.1).

W/ ^

Y

Y

0 -x A

X

Figure 3.3.1 Wheel and step

The equations of the line and its derivative with respect to the parameter are

found to be:

y(cos0 - tan¢ sin0) - x(sin0 + tan¢ cos#)

+ r + (xo - r0) tan¢ = 0 (3.3.1)

and

y(sin0 + tan@ cos0) + x(cos0 tan¢ sin0) + rtan@ = 0 (3.3.2)

Solving these for x,y we obtain the parametric representation of the envelope

of L, relative to the wheel W. The envelope equation represented by these two

15
parametric equations is indeed the involute of a circle, since the locus of the

center of the curvature of E is a circle having the equation

xc2 + y2 . r2sin2@ (3.3.3)

This circle is the base circle of the involute. The pressure angle _ is the

complement of the angle @ of the step's inclination.

4. ENVELOPE OF INVOLUTE PROFILE CUTTER

4.1 Introduction

As a final example of determining an analytical representation of the

envelope of a family of curves, consider the case of spur gear tooth generation

by shaping. This process can be modeled by two rolling disks, one having

penetrating involute profile teeth and the other a gear blank to be cut. The

task is to determine an expression that describes the "footprint" of the cutter

onto the gear blank, and then to show that the footprint is of involute shape.

Note that because of the constant profile shape along the width of the disks,

the problem is reduced to two dimensions.

16
Figure 4.1 Spur Gear Shaping.

4.2 Involute Analysis

The parametric equations of an involute curve in rectangular coordinates

may be described as follows:

Consider an arbitrary point P on the involute curve located by a position

vector p as in Figure 4.2.1. Then p may be expressed as:

p = rnr + r@n 8

(4.2.1)

17
Cadl

Y

o

X

Figure 4.2.1 Involute Curve

where n r and n8 are unit vectors as shown, r is the generating base circle radius

and @ is the involute angle. The coordinate axes are X and Y and n xand n yare

unit vectors parallel to these axes. The unit vectors are then related to one

another by the expressions:

nr = cosSn x + sinOny

n0 = - sinOn x + coSOny

where 0 is the inclination angle of nr as shown.

(4.2.2,a)

(4.2.2.b)

Equation 4.2.1 may then be written as

p = (rcosO rCsinO)n x + (rsinO + r_cosO)ny (4.2.3)

18
However

Hence,

and

# + @ = _ (4.2.4.a)

cosO = cos(_ - ¢) = sin@

sin8 = sin(_ - @) = cos@

Therefore from Equation (4.2.3) p becomes

(4.2.4.b)

(4.2.4.c)

p = (rsin¢ - r¢cos¢)n x + (rcos@ + r@sin@)ny (4.2.5)

The parametric equations of the involute line in rectangular coordinates are then

and

or

with slope

x : rsin@ - r@cos¢

y = rcos@ + r@sin@

y : f(x)

dy

dv = d@ : cot@

dx d_xx

(4.2.6)

(4.2.7)

(4.2.7.a)

19
4.3 Rolling Analysis

4.3.1

Consider two rolling wheels w, and w2 of unequal radii as shown in Figure

Figure 4.3.1 Rolling of Gear Blank over an Involute Profile Cutter

Let wheel w, be the cutter, and with a tooth of involute profile attached to it.

Let wheel w2 be the gear blank which rolls on the cutter and eventually over the

involute curve. Consider a point P on the involute with coordinates x,y. The

^ A

coordinate system (X-Y) is. fixed on wheel w2. Consider wheel w2 at position O*

^

making an angle 02 with the Y axis. Angle 0, is measured between the Y axis and

the line 020* making an angle 02 with the Y axis. Angle 0, is measured between

2O
the Y axis and the line 020* of the centers. Nx, Nyare unit vectors along the

X-Y axes and NR,N _ are unit vectors along the X-Y axes. The position vector p

P is then

p = xNx + yNy = 0,0" + p* (4.3.1)

where p* is the position vector of p relative to X,Y at the O_ position.

Using the geometry from Figure 4.3.1 we can write:

p = (rI + r2)N + _Nx + _Ny

A

where N is a unit vector along the line of centers of the wheels as shown.

Expressing p in terms of the unit vectors Nx,Ny in X-Y we have:

p = [(r, + r2)sinO2 + _cose]N_+ [(r, + r2)cosO2 - Rsin_] N) (4.3.2)

Thus the coordinates of P with respect to the X-Y coordinate system may be

written as:

x = (r, + r2)sinO2 + Rcose + _sin:

y = (r, + r2)cos82 - _sine +?cos:

(4.3.4.a)

(4.3.4.b)

solving for R,_ we have:

= xcos_ ysine + (rI + r2)sin(e - #2)

) = xsine + ycos_ (rl + r2)cos(e - 02)

(4.3.4.a)

(4.3.4.b)

21
To simplify these equations, suppose r, = r 2 = r , then 0, = _and

x = Rcose + sine + 2rsin_

y - Rsine + _cose + 2rcos_

(4.3.6.a)

(4.3.6.b)

and

- xcose - ysine + 2rsin_

9 = xsine + ycose - 2rcos_

(4.3.7.a)

(4.3.7.b)

4.4 Envelope Analysis

Suppose y=f(x) describes a curve (that is, an involute tooth profile) in

wheel w,. Then this curve will have in the coordinate system of wheel w2 a form

expressed as

y(_,),e) : f[x(_,_,e)] (4.4.1)

wherey(R,_,e) and x(R,_,e) are given by equations (4.3.4) and (4.3.6). Equation

(4.4.1) can be rewritten in the form

F(R,_,e) = 0 (4.4.2)

Then the envelope of the curves in w2 is given by the intersection of the surface

of Equation (4.4.2) and the surface defined by the equation:

aF(2,_,e}= o (4.4.3)

ae

22
4.S Envelope of Involute

Suppose the involute curve of Equations (4.2.6) and (4.2.7) is written as

y : f(x) (4.5.1)

From Figure 4.3.1 it is observed that

0, = (r2) 82 (4.5.2)

r,

and

e = 0, + #2 = (r__2+ I) 02 = r,r,+ r2 #2 (4.5.2.a)

Hence

r,

02 = r, + r2 0, (4.5.3)

Using Equations (4.3.4) and the above expression, Equation (4.5.1) takes

the form:

(r, + r2)cos r, + r2

f I(r, + r2)sin [(r)] r, + r2 e

- Rsine + _cose =

(4.5.4)

23
Hence, by differentiating with respect to e, we have:

- r,sin r, + r,

_-_ ,cos r, + r2

(4.5.5)

However, from Equations (4.3.4), (4.5.2), (4.5.3), and (4.2.7a) we have

Rcose + _sina = x

(r' + r2)sin[( r'r,+ r2 ) or]

(4.5.6)

( r, )_]

- _sine +)cos(z --y - (r, + r2)cos[ r, + r2

(4.5.7)

and

df cot@

dx =

Hence, Equation (4.5.5) becomes:

- xsin@ + r2sin (r,r,+ r2

or

xsin@ + ycos@ = r2cos

>]e sin@ -ycos@ - r2cos Ec >] r, + r2 e cos@

(4.5.8)

24
Observe though that from Equations (4.2.6) and (4.2.7) of the involute we have

or

xsin@ + ycos¢ = r2sin2@ - r2@cos@sin@ + r2cos2@ + r2@sin@cos@

xsin¢ + ycos@ = r2 (4.5.9)

Comparing Equations (4.5.8) and (4.5.9) we note that in order for the right sides

to be equal, we must have:

or

COS [¢ - r, + r2 I](_ = I

@ : (r,r,+ r2 ) e

(4.5.10)

Next, examine the involute curve in w2 in terms of _ and _:

From Figure 6.3.1 we note that

Ot - 0 2 = ( r, )(2

r, + r2

(4.5.11)

From Equations (4.3.5) we have:

R = xcos_ ysine + (r, + r2)sin [(rr, + r2

(4.5.12.a)

and

25
.9 - xsine + ycos_ - (r, + r_)cos [cr,r,+ r, (4.5.12.b)

Then from Equations (4.2.6) and (4.2.7) we can substitute x and y for functions

of ¢. This leads to:

R - r2sin@cose - r2@cos¢cose - r,cos@sine - r2¢sin@sine + (r,+r2)sin (r,rl+ r2 )e

and

= r2sinCsine - r2@cosCsina + r,cos@cosa + r2sin@cose - (r'+r2)c°s (r,rl+ r2 )e

or

- r2sin(@ - e)

and

) - r2cos(@ - e)

- r2@cos(@ - e) + (r, + r2)sin (4.5.13.a)

+ r2@sin(@ - e) (r, + r2)cos r,rl+ r2 e (4.5.13.b)

Observe from Equation (4.5.10) that

@ - e m (r) r, + r2 e - e- -(r.) r, + r2

and

(4.5.14)

26
Hence, Equations (4.5.13) become:

R : rlsin (r2_ @ ) -r2@cos (r_ ¢)

and

r 2

Using the substitution

@ = /_ (4.5.15)

we obtain:

= r,sin_ - r_cos_ (4.5.16.a)

and

) = - r,cos_ - r,Bsin_ (4.5.16.b)

Observe by comparison with Equations (4.2.6) and (4.2.7), that Equations

(6.5.16) describe an involute on wheel w2 going downwards. Thus the footprint

of the involute profile of the cutter tooth on the gear blank is seen to be of

involute shape.

27
5. COMPUTERSIMULATIONOFMANUFACTURING PROCESSES

5.1 Introduction

Although the analytical development of the involute profile of a spur gear

tooth is relatively simple, the analogous development for bevel gears is much

more cumbersome. An alternative to the analytical development is to use computer

graphics technology. A feature of this technology is "solid modeling" which is

used for developing geometry for finite element analyses. The settings and

parameters of the gear manufacturing processes may be specified to simulate the

motion of a cutter and gear blank. Thus analytical expressions of the geometry

are not needed.

For an accurate analysis, the computer model cut gear tooth must be an

accurate model of the actual manufactured gear tooth. To obtain such a model

it is necessary to have exact specifications and dimensions of the cutting tool

and gear blank, the relative positions of the two gears and their motion relative

to each other.

This part of the report discusses computer simulation of gear manufacturing

procedures. It is shown that this method accurately defines the geometry of the

gear tooth surfaces. The method will be demonstrated through a series of

examples.

The software used is the solid modeling family in the SDRC I-DEAS package,

called GEOMOD [13]. GEOMOD provides the user with a set of modular software

28
tools to aid in the generation of geometric design of a mechanical system. In

particular, object modeling is a modulewhere the user can create 3-dimensional

geometric shapes and manipulate the geometry by deforming, orienting, joining,

and cutting objects with one another. Overall, the package offers a powerful

tool, for defining and evaluating the geometry of a structural component or

system.

5.2 Gear and Straight Sided Rack

Figure 5.2.1 Rack and Spur Gear Generating Process

The generating process of forming involute gear teeth is shown in Figure

5.2.1, where a gear blank and a rack cutter are shown. The cutting process

occurs as the cutter moves horizontally and the gear blank rolls upon it. Gear

29
The teeth formed are the "footprint" of the rack onto the gear blank [18].

Figure 5.2.2 illustrates the concept of the "footprint".

Figure 5.2.2 Footprint of a Step on the Wheel Rolling Over It

Applying this procedure with solid modeling we create two objects: the

gear blank (a cylinder of radius equal to the outside radius of the gear) and

the cutter. Figure 5.2.3 shows the straight sided rack cutter. Figure 5.2.4

shows the position of the cutter before and after the rolling motion. It also

shows the gear blank with the "footprint". Figure 5.2.5 shows two spur gear

teeth generated by the process.

30
Figure 5.2.3 Solid Model of Straight Sided Rack Cutter

31
Figure 5.2.4 Solid Model of Gear Blank and Rack Cutter

32
Figure 5.2.5 Solid Model of Resulting Spur Gear Teeth

Whatremains to be shownis that the profile of the generated tooth surface

is of true involute shape. To see this the coordinates of points on the

generated curve profile must be calculated and comparedwith those of an involute

curve.

At each time step of the cutting process, a facet on the tooth surface is

created. The coordinates of points on the facets may be extracted during the

generation process. At the sametime, the angle betweena line from the origin

to the point of intersection between the tangent to the base circle and the Y

33
involute curve to obtain an analytical description of the matching involute that

is:

and

x - r sin@ - r ¢ cos@

y = r cos@ + r @ sin@

(5.2.1)

(5.2.2)

Figure 5.2.6 Coordinates of Points of an Involute Curve

Figure 5.2.7 shows a comparison of the two curves. It is seen that the two

curves are the same. (In the figure the curves are offset for the purpose of

comparison.)

34
II_I l_l'lll

I

, : : i

I , ,

I , ._, I

: ..-'7: I

I ' , ,..'Jr , , t

I : : ! ' '

! : : ..-"Y. i : /

.I : : ..'?"! ' : I

-'r.............. .-'/---;- .... ; ...... 1-.....

i I .," o | o

! , ,...'/ : , , I

i • a/ . , o /

i : "2 : ' ' '

a , "/: : : ' I

.a J -/; ; , : i

i _ i I ! I I I ! I ! b I I ill Ii_ i I! I i IIi I ! i I i ill il

, , _'" ..... T...... : ' I

i :'- ' " ' : I

I

_'.._..'/.' t : - : I

i .." -.---- i

..... : ..... ............ 4

| 2-- I

I , , ,

! !

i.?&. ., 4

I

I

I '

I

1, I

lil|ll_I

ill

Figure 5.2.7 Generated and Calculated Involute Profiles

5.3 Involute Profile Cutter on a Wheel

The foregoing process may be generalized with the cutter tooth being an

involute on a circular disk, as in Figure 5.3.1. Figures 5.3.2 to 5.3.5 show

the resulting computer graphic simulation.

35

C_R_4NAL PAGE IS

OF POOR QUALITY
Figure 5.3.1 Footprint of an Involute Cutter on a Gear Blank

36
Figure 5.3.2 Solid Model of Cutter of Involute Pr,)file

37
Figure 5.3.3 Generated Gears (Equal Radii)

38
Figure 5.3.4 Generated Gears (Unequal Radii)

39
Figure 5.3.5 Close-up View of Generated Teeth

4O
5.4 Straight Bevel Gears

Consider the extension of these concepts to the development of straight

bevel gears. The involute profile of a spur gear tooth is generated from a "base

circle". Extended in the axial direction, the set of base circles form a

cylinder -- the "base cylinder". In an analogous manner, the basic rack for a

spur gear becomes a "crown gear rack" for a bevel gear. For a bevel gear the

base surface is a cone. See Figure 5.4.1.

Figure 5.4.1 Bevel and Crown Gears

The pitch surface of the crown gear is planar. The teeth of the crown gear

are straight sided with an inclination corresponding to the pressure angle. When

generating methods are employed in manufacturing bevel gears, the relative motion

specified For the cutting tool and the gear blank is the same as would be

obtained if the bevel gear to be cut is rotating in mesh with the crown gear.

This concept provides the procedure for computer solid modeling simulation of

the straight bevel gear tooth.

41
The steps in this simulation are the sameas those of the spur gear.

Figure 5.4.2 shows a bevel gear blank and a crown gear cutter. Figures 5.4.3

to 5.4.5 show a generated tooth on a pinion.

Figure 5.4.2 Solid Model of Pinion Blank and CrownGear Cutter

42
Figure 5,4.3 Cutter Before and After Cut and Pinion with Generated Tooth

43
Figure 5.4.4 Close-up View of the Cut Tooth on the Pinion Blank

44
Figure 5.4.5 Pinion with Generated Teeth

45
DISCUSSION

The foregoing concepts establish the principles of a computer simulation

of gear tooth manufacturing procedures. The analysis shows that gear tooth

surfaces may be viewed as envelopes of gear cutters sweeping through the gear

blank. The examples demonstrate the feasibility of computer based analysis and

design of the cutting process.

The analysis is developed for spur gears. As such, it is a planar

analysis. Even with the simplification of planar modeling, however, it is seen

that the analysis is cumbersome. Indeed, a strictly analytical development of

bevel gear surfaces -- particularly spiral bevel gear surfaces -- is virtually

intractable. The computer simulation, however, is not restricted to a planar

analysis. With the computer simulation, three-dimensional surfaces, such as

occur with spiral bevel gear teeth, are developed in the same manner as twodimensional spur gear tooth surfaces.

The value of the analysis is that it establishes the validity of the

computer simulation. Hence, the surfaces of standard tooth forms can be

developed. These surfaces in turn, can be used to develop finite element

modeling of the gear teeth.

The capabilities of the computer analysis extend beyond this to the

analysis of non-standard tooth forms and their associated fabrication procedures.

That is, the procedure can be used to obtain a representative of non-standard

46
tooth surfaces• The fabrication procedure to develop these surfaces is a byproduct of the simulation.

Finally, by extending the computer graphic analysis it is seen that it can

be applied with other machine surfaces, such as cams and bearings.

CONCLUSIONS

• The analysis demonstrates the utility of computer graphics in simulating

gear design and manufacture• The use of these procedures eliminates the

need to rely upon unwieldy analytical representations of gear tooth

geometry. (Analytical representations have heretofore been intractable

except for the simplest of gear tooth surfaces•)

• The computer graphics representation is ideally suited for developing

finite element models of gear teeth. (In the past, geometrical complexity

has precluded an easy development of such models.)

. The developed procedure may be used for conceptual exploration of new

designs (that is, the cutter shape and movement) is an immediate byproduct

of the graphics process.

47
REFERENCES

•

t

o

.

5.

o

o

J

o

10.

11.

12.

13.

14.

D. Mavriplis, "Computer Simulation and Analysis of Gear Teeth Manufacturing

Processes," Ph.D. Dissertation, University of Cincinnati, Cincinnati, Ohio

45221, 1988.

S. H. Chang, R. L. Huston, and J. J. Coy, "Computer Aided Design of Bevel

Gear Tooth Surfaces," NASA Technical Memorandum 101449 and AVSCOM Technical

Memorandum 88-C-005, December, 1989.

S. H. Chang and R. L. Huston, "A Computer Aided Design Procedure for

Generating Gear Teeth," Fourth International Power Transmission and Gearing

Conference, ASME Paper No. 84-DET-184, Boston, MA, 1984.

W. C. Graustein, Differential Geometry, MacMillan, New York, 1935.

F. Litvin, W. Tsung, H. Lee, Generation of Spiral Bevel Gears with

Con_uqate Tooth Surfaces and Tooth Contact Analysis, NASA Technical Report

87-C-22, August 1987.

R. Drago, B. Uppaluri, G. Fries, Larqe Rotorcraft Transmission Technoloqy

Development Proqram, Volume I, Technical Report to NASA, 1982.

J. J. Coy, D. P. Townsend, and E. V. Zavetsky, Gearinq, NASA Reference

Publication 1152, AVSCOM Technical Report 84-C-15, December 1985.

American Gear Manufacturers Association, AGMA Desiqn Manual for Bevel

Gears, AGMA 330.01, Washington DC, 1972.

Y. Terauchi, K. Nagamura, W.-S. Peng, and H. Sentoku, Study of Bevel Gears

- ]st Report, Theoretical Calculation of Tooth Bearinq and Experiment,

Bulletin of the JSME, Vol. 29, No. 250, 1986, pp. 1294-1302.

Y. Terauchi, K. Nagamura, W.-S. Peng, and H. Sentoku, Study of Tooth

Bearinq of Bevel Gears 2nd Report, Relation Between Tooth Bearing and

Assembly Error, Shaft Anqle Error, Bulletin of the JSME, Vol. 29, No. 250,

1986, pp. 1303-1312.

J. E. Shigley and L. D. Mitchell, Mechanical Enqineerinq Oe$ign, McGraw

Hill, 1983, pp. 665-667.

R. H. MacNeal (Ed.), MSC/NASTRAN - Handbook for Linear Static Analysis,

The MacNeal-Schwendler Corp., Los Angeles, 1985.

L. Peterson, Ed., MSC/NASTRAN - MSGMESH Analyst's Guide, The MacNealSchwendler Corp., Los Angeles, 1983.

Darle W. Dadley, Gear Handbook, McGraw Hill, 1962, First Edition, pp. 5.54-

5.59, Table 5.22.

48
15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

Gleason, Straiqht Bevel Gear Desiqn, Gleason Works, Publication SD3160,

1972.

SDRC I-DEAS Geomod * Solid Modelinq and Desiqn, User's Guide (level 3),

SDRC, Milford, Ohio.

S. H. Chang, R. L. Huston, and J. J. Coy, A Finite Element Stress Analysis

of Spur Gears Includinq Fillet Radii and Rim Thickness Effects, Journal

of Mechanics, Transmissions, and Automation in Design, Vol. 105, pp. 327-

330, September 1983.

SDRC I-DEAS, Supertab, Model Solutions, User's Guide, level 3, Structural

Dynamics Research Corp., Milford, 1987.

R. L. Huston and C. E. Passerello, Finite Element Methods:

Introduction, Marcel Dekker, 1984.

An

C. A. Brebbia, J.C. Telles, L. C. Wrobel, Boundary Element Techniques,

Springer-Verlag, Berlin, 1984.

S. Oda, K. Nagamura, K. Aoki, Stress Analysis of Thin Rim Spur Gears by

Finite Element Method, Bulletin of JSME, Vol. 24, No. 193, July 1981, pp.

1273-1280.

William C. Graustein, Differential Geometry, The MacMillan Company, New

York, 1935.

S. Chang, R. L. Huston, and J. J. Coy, A Computer Aided Desiqn Procedure

for Generatinq Gear Teeth, ASME 24-DET-184, 1985.

Werner F. Vogel, Involutometry and Triqonometry, Michigan Tool Company,

Detroit, 1945.

T. W. Khiralla, On the Involute Geometry of External Involute Spur Gears,

C/I Leaming, California, 1976.

R. L. Huston and J. J. Coy, Ideal Spiral Bevel Gear - A New Approach to

Surface Geometry, NASA Technical Memorandum 81446, 1980.

F. Litvin, K. Krylov, and M. Erichov, Generation of Tooth Surfaces by Two

Parameter Envelopinq, Mechanism and Design, Vol. 10, 1975, Pergamon Press,

pp. 365-373.

R. L. Huston, Y. Lin, and J. J. Coy, Tooth ProFile Analysis of Circular

Cut, Spiral Bevel Gears, ASME 82-DET-79.

T. R. Kane, Analytical Elements of Mechanics, Academic Press, NY, 1961.

P. Altidis, M. Savage, Flexibility Effects on Tooth Contact Location in

Spiral Bevel Gear Transmissions, NASA 4055, 1987.

49
31.

32.

33.

34.

37.

38.

39.

40.

41.

W. Coleman, Desiqn and Manufacturinq of Spiral Bevel and Hvooid Gears for

Heavy-DutY Drive Axles, SAE, January 1962.

A Dyson, A General Theory of the Kinematics and Geometry of Gears in Three

Dimensions, Clapendon Press, Oxford, 1969.

W. Bultmann, Die Arbeitszeitermittlunq der Vezzahnunqsarbeiten, SpringerVerlag, Berlin, ]960.

Earle Buckingham, Analytical Mechanics of Gears, Dover Publications Inc.,

NY, 1949.

Formulas in Gearinq, Brown & Sharpe Mfg. Co., 1936.

Luther P. Eisenhart, A Treatise on the Differential Geometry of Curves and

Surfaces, Ginn and Company, 1909.

C. E. Weatherburn, Differential Geometry of Three Dimensions, University

Press, London, 1927.

R. H. Fowler, The Elementary Differential Geometry of Curves, University

Press, Cambridge, 1920.

Irving J. Levinson, Machine Desiqn, Reston Publishing Company Inc.,

Virginia, 1978.

Dirk J. Struik, Classical Differential Geometry, Addison-Wesley Press,

Cambridge, 1950.

Chuan-Chih Hsiung, A First Coarse in Differential Geometry, John Wiley &

Sons, NY, 1981.

5O
Report Documentation Page NatdonalAeronauticsand

Space Adminislratlon

1. Report No. NASA CR-185200 2. Government Accession No. 3. Recipient's Catalog No.

AVSCOM TM 90-C-003

4. Title and Subtitle

Computer Simulation of Gear Tooth Manufacturing Processes

7. Author(s)

Dimitri Mavriplis and Ronald L. Huston

9. Performing Organization Name and Address

University of Cincinnati 11.

Department of Mechanical and Industrial Engineering

Cincinnati, Ohio 45221-0072

12. Sponsoring Agency Name and Address

Propulsion Directorate

U.S. Army Aviation Research and Technology Activity--AVSCOM

Cleveland, Ohio 44135-3127

and

NASA Lewis Research Center

Cleveland, Ohio 44135-3191

5. Report Date

February 1990

6. Performing Organization Code

8. Performing Organization Report No.

None

10. Work Unit No.

1L 162209A47A

505-63-51

13.

14.

Contract or Grant No.

NSG-3188

Type of Report and Period Covered

Contractor Report

Final

Sponsoring Agency Code

15. Supplementary Notes

Project Manager, Fred B. Oswald, Propulsion Systems Division, NASA Lewis Research Center.

16. Abstract

This report discusses the use of computer graphics to simulate gear tooth manufacturing procedures. An analytical

basis for the simulation is established for spur gears. The simulation itself, however, is developed not only for

spur gears, but for straight bevel gears as well. The applications of the developed procedure extend from the

development of finite element models of heretofore intractable geometrical forms, to exploring the fabrication of

nonstandard tooth forms.

17. Key Words (Suggested by Author(s))

Computer-aided design; Computer graphics; Spur gears;

Bevel gears; Gear tooth profile

18. Distribution Statement

Unclassified- Unlimited

Subject Category 37

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21, No. of pages

Unclassified Unclassified 52

NASA FORM "1626OCT86 *For sale by the National Technical Information Service, Springfield, Virginia 22161

22. Price*

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