NASA

Technical Memorandum 105718

AVSCOM

Technical Report 92-C-003

Minimization of Deviations of GearReal Tooth Surfaces Determined

by Coordinate Measurements

,// j¢

f

EL. Litvin, C. Kuan, and J.-C. Wang

University of Illinois

Chicago, Illinois

R.E Handschuh

Propulsion Directorate

U.S. Army Aviation Systems Command

Lewis Research Center

Cleveland, Ohio

J. Masseth

Dana Corporation

Fort Wayne, Indiana

and

N. Maruyama

Nissan Motor Corporation

1 Natsushima-cho, Yokosuka, Japan

Prepared for the

1992 Power Transmission and Gearing Conference

Phoenix, Arizona, September 13-16, 1992

N/ A

4-

0 _ Lr_

I _ ,.1.

e_J U 0Or- C ,..-4

em

e_.

LL _.i

C; I--

7" Z

(O 3E

_-4 t,-- O

¢'t C? fTt _"_

7 _c:

Z L_A _2

P" _L. 'J'

t__ r- _-- t.,,

I 7 ._;

I t-- L) r:

S_MS COMMAND

https://ntrs.nasa.gov/search.jsp?R=19920021704 2018-08-02T01:03:43+00:00Z

MINIMIZATION OF DEVIATIONS OF GEAR REAL TOOTH SURFACES

DETERMINED BY COORDINATE MEASUREMENTS

F. L. Litvin, C. Kuan and J.-C. Wang

(University of Illinois at Chicago)

R.F. Handschuh (NASA Lewis Research Center)

J. Masseth (the Dana Corporation)

N. Maruyama (the Nissan Motor Co.)

tt'b

r-4

I

ABSTRACT

The deviations of a gear's real tooth surface from tim theoret-

ical surface are determined by coordinate measurements at the

grid of the surface. A method has been developed to transform

tile deviations from Cartesian coordinates to those along the nor-

real at the measurement locations. Equations are derived that

relate the first order deviations with the adjustment to the man-

ufacturing machine-tool settings. The deviations of the entire

surface are minimized. The minimization is achieved by appli-

cation of the least-square method for an overdetermined system

of linear equations. The proposed method is illustrated with a

numerical example for hypoid gear and pinion.

INTRODUCTION

Coordinate measurements of gear tooth surfaces coupled with

the ability to correct the initially applied machine-tool settings

is becoming a significant part of advanced gear technology. We

may consider two stages of this technique:

(i) Application of coordinate measurements of the manufac-

tured gears for numerical determination, in 3D space, of devia-tions of real tooth surfaces.

(ii) The goal of minimization of deviations can be achieved

by proper corrections of initially applied m_chine-tool settings.

The determination of corrected machine-tool settings is found

numerically.

The technological aspects of the problem to-be discussed areas follows:

(i) The deviations of real tooth surfaces are inevitable due to

surface distortion by heat-treatment, errors of initial machine-

tool settings, deflection by manufacturing, etc.

(ii) Application of an additional finishing operation for elim-

ination of the deviations would be too expensive in compari-

son with the approach based on corrections of initially applied

machine-tool settings. The advantage of this approach is the

possibility of using the same equipment to correct the deviations.

The disadvantage is that the approach will be successful only if

the deviations are repeatable.

(iii) The coordinate measurements must be performed with

high precision, which currently prohibits them from being per-

formed simultaneously with the manufacturing. Therefore, the

coordinate measurements are performed after manufacturing, but

only the first gear of the whole gear set to-be manufactured is

tested.

(iv) In some cases master-gears are used and the coordinate

measurements provide the information about the deviations from

the master-surface for the surface being tested. The authors

consider this approach less effective as compared to computerized

determination of surface deviations and corrections of machine-

tool settings.

The mathematical solutions to this problem are represented

in the Appendix to this paper. The technique described in the

paper has been developed in the response to the increasing re-

quirements of high quality gear transmissions. Minimizing thedeviations of real tooth surfaces results in a reduction in the

level of transmission errors that cause gear noise and vibration.

The proposed approach is applied to hypoid gear drives that

have found a wide application in transmissions [],2]. The con-

tents of the paper are complemented with a numerical example

for a hypoid pinion and gear to illustrate the effectiveness of the

proposed approach. The level of deviations of the pinion _urfacehas been reduced from 30 microns to the theoretical level of 2-3

microns.

1. OVERVIEW OF MEASUREMENT AND MOD-

ELLING METHOD

The approach developed in this paper enables the determi-nation of deviations of a real surface from the known theoretical

surface. This is accomplished by two steps: (i} coordinate mea-

surements for determination of surface deviations and (ii) min-

imization of the deviations through correction of the previously

applied machine-tool settings.

Thesurfacedeviationsobtainedinitiallyin Cartesian coor-

dinates are transformed into deviations along the normal to the

theoretical surface. The coordinate mcasurements are performed

by a machine with four or five degrees-of-freedom. In the case of

four degrees-of-freedom, the probe performs three translational

motions (fig. 1); the fourth motion, rotation, is perforrned by a

rotary table. The axis of rotational motion coincides with the

axis of the workpiece. In the case of a five degree-of-freedom ma-

chine, the fifth degree of freedom is used to provide the deflections

of thc probe in the direction of the normal to the thcoreticai sur-

face. The probe is provided with a changeable spherical surface

whose diameter can be chosen from a wide range.

CeLt

Bac.kface-B_Pl_e"/_ [

I

l

i i

I '\ IRota:y Table

(b)

Ce_ z_7t ProLe

Backfac.-Base Pique [, I

l t \1 ]I Rotary Table

Fig. i. Surface Measurement.

The motions of the probe and the workpiece by coordinate

measurements are computer controlled and therefore a grid com-

prising of the set of surface points to be measured must be chosen

(fig. 2). There is a reference point on the grid that is necessary

for the initial installments of the probe. There are two orienta-

tions of the probe installment that are used to measure a gear

(fig. l(a)) and a pinion (fig. l(b)), depending on tile angle of the

pitch cone.

Zf

¢

Fig. 2. Grid.

The mathematicM aspects of coordinate measurements will

now be described [2]: First, it is necessary to dcrive the equa-

tions of the theoretical surface. In many cases this surface ca,1

be derived as the envelope to the family of generating surfaces,

namely the tool surfaces. Next, the results of coordinate mea-

surements must be transformed into deviations of the rcal surface

represented in the direction of the surface normal. Then, the re-

lations between the surface variations and the corrcctions to tile

machine-tool settings must be determined. The surface dcvi-ations obtained from coordinate measurements and the surface

variations determined by the corrections of machine-tool settings

can be represented by an overdetermined system of lincar equa-

tions. The number of these equations, k, is equal to the number

of points of the grid, and the number of unknowns, m, is equal

to the number of corrections of machine-tool settings (m << k).

The optimal solution to such a system of linear equations results

in the determination of the machine-tool setting corrections.

2. EQUATIONS OF THEORETICAL TOOTH SUR-

FACE Et

Considering that the theoretical surface can be deterrnined

directly, we represent it in coordinate system St in two parametric

form as:

_,(_,,o), n,(_, 0) (1)

Here: rt and ne are the position-vector and the surface unit nor-

mal, respectively; (u,0) are the Gaussian coordinates (surfacc

coordinates).

For the case when surface Et is the envelope to the f_mily

of generating surface E¢, we represent surface Et and the unit

normal n, to Zt in S, as [3]

r, = Mt, r,(u¢, 0¢), f(u¢, O,, ¢) = 0 (2)

n, = L,,n,(u_, O,), f(u¢, O¢, ¢) = 0 (3)

Here: (u_, 0_) are the Gaussian coordinates of the generating sur-

face E_; ¢ is the generalized parameter of motion in the proccss

for generation, Thc equation of meshing is givcn by:

f(u,, 0o, ¢) = N (°) • v ('0 = 0 (4)

where N (*} is the normal to E,, v (a} is the relative motion for

a poiut of contact of E, and ,E,. The 4 x 4 matrix Mt, and

3 x 3 matrix Le, describe the coordinate transformation from S,

to St of a position vector and surface unit normal, respectively.

Position vectors in 3-D space are represented with homogeneouscoordinates.

3. COORDINATE SYSTEMS USED FOR COORDI-

NATE MEASUREMENTS

Coordinate systems Sm and St are rigidly connected to the

coordinate measuring machine (CMM) and the workpiece being

measured, respectively (fig. 3). The back face of the gear is in-

stalled flush with the base plane of the CMM. The distance l

between the origins O= and Ot is known but the parameter of

orientation 8 must be determined (see section ,t). The coordi-

natetransformationfromS_ to S,_ is represented by the matrix

equation

r,_ = M,_,r, (5)

L.

xm

.41t i

rm

r.

O=.... Ot

Fig. 3. Coordinate Transformation.

Z m z_

4. GRID AND REFERENCE POINT

The grid is a set of points on Zt chosen as points of contact

between the probe and Et (fig. 3). Fixing the value of z, for the

point of the grid, and the value of, say yt (or xt), we can obtain

the following equations

v,(u,,O,) = hi, _,(u,,O_)= t, (i = l,...,k) (6)

where k is the number of grid points.

We consider hi and li as given and solve equations (6) for

(Ul,01). Then we can determine the position vectors and the

unit normals for k points of the grid using the equations

r_ i) = [x,(ui,Oi) y,(ui, Oi) zt(ui,Oi)] r, (i = 1,... ,k) (7)

nl i) = [n_,(ui, Oi) nu,(ui, Oi) na(ui, Oi)] T, (i = 1,...,k) (8)

The position vector for the center of the probe, if the devia-

tions are zero, is represented by the equation

nl ') = rl ') + pn_ '1 (i = 1,...,k) (9)

where p is the radius of the probe tip.

The reference point

r_°1= [z,(u(°),0(°_) ,_,(uC°),0(°))z,(u¢°),O_°))]r (10)

is usually chosen as the mean point of the grid.

The center of the probe that corresponds the reference point

on Et is determined from equation (9) as

al°) = [X,(uC°),O(°)) _(u(°),0 (°)) Z,(uC°),0_°))lr (11)

Here: (u {°1, 0(°)) are known values.

The coordinates of the reference center of the probe are rep-

resented in coordinate system S., of the measuring machine by

the matrix equation

R_ = M_,(6) RI°_

Equation (12) yields

(12)

(13)

The three equations (13) contain four unknowns:6, z(°),y(°l,

z_). To solve these equations we may consider that one of the

coordinates of the reference point of the probe center, say y_),

may be chosen equal to zero. Then the system of equations

(13) allows the determination of 6, z_) and z_l [2]. Coordinates

x_), y(,0) = 0, z_ ) are necessary for the initial installment of the

center of the probe.

5. DEVIATIONS OF THE REAL SURFACE

The deviations of the real surface are caused by errors of man-

ufacturing, heat treatment, ctc. Vector positions of the center of

the probe for thc theoretical surface and the real surfacc can be

represented as follow

Rr_= r,,(u, 0) + pn,,(u,0) (14)

R_ = r,_(u,0) + )_n,_(u,0) (15)

Here: r,_ and n,, are the position vector and the unit normal

to the theoretical surface, respectively, that are represented in

coordinate system ,9., of the measuring machine; )_ determines

the real location of the probe center and is considered along the

normal to the theoretical surface; R= and R_,, represent in S,_

the position vector of the probe center for the theoretical and

real surfaces, respectively. Equations (14) and (15) yield

R: - 1_ = (A - p)n,, = Ann,_ (_6)

and

An = (R_, - R,_) -n,, (17)

The position vector R_ is determined by coordinate measure-

ments for points of the grid. Equation (17) determines numeri-

cally the function:

/x_i=Ani(ui,0_) (i=l,...,k) (18)

that represents the deviations of the real surface for each pointof the grid.

6. MINIMIZATION OF DEVIATIONS

The procedure of minimization of deviations can be repre-

sentedin twostages:(i) determinationof variations of theoret-

ical surface caused by changes of applied machine-tool settings,

and (ii) minimization of deviations of real surface by appropriate

correction of machine-tool settings.

We consider that the theoretical surface is represented in St

as

r, = r,(,,,O,a_) (i = 1.... ,m) (10)

where parameters d_ are the machine-tool setti,gs.

The surface variations are represented by

Oft. 0rt Or,

)=1

(20)

We multiply both sides of equation (20) by the surface unitOr, _r,

normal n, and take into account that _-. nt = _uu" n, = 0 since

Ort . Ore

frO-- and _ lie in the plane that is tangent to the surface. Thenwe obtain:

m OFt m

_r, . n, = (__,j=,-_j . n,)6d, = __a&d, (21)i=1

We can now consider a system of k linear equations in m

unknowns (m ,*: k) of the following structure

alt,bdx + ax2,bd 2 + ... + ax,_6d,,_ = bt ]

......................................................... / (22)ajd_bdt + aj,2,bd2 + ... + ak=6d_ = bk

Here:

(23)

where i designates the number of grid point; a,j (s = l, ..., k;

j = l, ..., m) represent the dot product of partial derivatives

0r---Land unit normal nt.

Od_Fhe-- system of linear equations (22) is overdetermined since

rn _:_ k. The essence of the procedure of minimization of de-

viations is determination of such unknowns 6dj (j = 1, ..., m)

that will minimize the difference between the left and right sides

of equations (22). The solution was accomplished by the least-

square method. The subroutine DLSQRR of IMSL MATH /

LIBRARY [4] was used for computerization of the procedure.

The success of minimization of deviations depends on the

number of parameters that may be varied (the number of ma-

chine-tool settings that may be corrected). The number of pinion

machine-tool settings is larger than for the gear. The minimiza-

tion of deviations can be performed for each pinion tooth side

separately. However, it must be performed simultaneously for

both sides of the gear tooth since the gear is cut by the duplex

method. For these reasons the minimization of deviations for the

pinion is more effective than for the gear (see below the numerical

examples).

7. APPLICATION TO INSPECTION OF FORMATE

HYPOID GEAR

Each tooth side of a formate face-milled gear is generated by

Ca)

Geuerating Cones _/\ CutterBlade

(b)

Fig. 4. Generating Cones for Format Face-milled Gear.

a cone and the gear tooth surface is the surface of the generating

cone. Two cones that are shown in fig. 4Ca) represent both sides

of the gear space. The following equations represent in coordinate

system S, gear surfaces for both sides and the unit normals to

such surfaces (fig. 4(b))

r C _-

--$G cos ol G

(r_ - so sin c_a)sin Oc

(r_ - so sin cro) cos Oc

1

(24)

n c

sinus

- cos otG sin 0c

-- cos e',,C cos 0 G

(25)

Here: r, is the position vector and n¢ is the surface unit normal;

r, is the cutter tip radius; ao is the cutter blade angle (an > 0

for the concave side and aG < 0 for the convex side).

Fig. 5 shows the installment of the generating cone on the

cutting machine. Coordinate systems So and St are rigidly con-

nected to the cutting machine and the gear being generated,

respectively. Systems S_, So and St are rigidly connected to each

other since the gear is formate cut (no relative motion between

the cutter and workpiece). To represent in St the theoretical gear

tooth surface Et and the unit normal to Et we use the following

matrix equations

r,(so, Oo,d)) = M,, ro(sc, 0_) (26)

Fig. 5.

where

Mtc

Hz

O+ "'.

/ ! _, ! /_o J I ' I /

"t1"y. Yt

Machine-tool Settings for Formate Face-milled Gear.

n,(sc, On,d_) = L,c no(so, 0a)

Mlo Moc

(27)

EE

o

@

eo

Z

Fig.

0.04 -

0.04-

0.0_

0.00-

-0.02-

-Q.O4

-0.0_l

6.

6 10 lS 20 26 30 35 40 45

no. of polnts

Deviations of Gear Real Tooth Surface (Driving Side).

o

.)rl

oZ

Fig.

0.06'

0.04'

-0,02 "_

-0,04

o 6 to m 20 20 30 s5 40 46no. o! points

7. Deviations of Gear Real Tooth Surface (Coast Side).

cos't._ 0 - sin 7-, 0

0 l 0 0

sinT., 0 cos 7,. -AX,_

0 0 0 1

i 0 0 0

o i o-v:

0 0 1 H2

0 0 0 l

28)

The surface Gaussian coordinates are so and Oa and dj (Tin,

V2, //2 and A.X,_) are the machine-tool settings.

The numerical example presented in this paper is based on

the experiment that has been performed at the Dana Corpora-

tion (Fort Wayne, USA). The initial deviations An for each side

of real tooth surface have been obtained by measurements on a

coordinate measuring machine (fig. 1). The grid for the mea-

surements is formed by nine sections along the tooth lcngth with

each section having five points. The number k of grid points is

• therefore 45 and the reference point is at the middle of the grid,

(i.e., the third point of the fifth section). In the measurement,

the coordinate y_) of the reference point is chosen to be zcro

and the alignment angle _ is determined from solving equation

system (13).

The minimization of deviations was performed in accordance

to the algorithm described in section 6 for the formate cut gear.The measurement of the initial and final deviations are shown

in figures 6-8. The machine-tool settings initially used and cor-rected are shown in Table I.

7. EQUATIONS OF PINION THEORETICAL TOOTHSURFACE

The pinion tooth surface is generated as the envelope to the

family of tool surfaces that are cone surfaces (fig. 9).

Henceforth, we will consider the following coordinate systems:

-- 0.00 "E

0.04

O" 0.02 -0

_1 0.00 -:!4Pr_ -0.02 -

a

-0.04 -O

'_' -0.01_

0 x _I0

Fig. 8.

(Otivirall: Side) (Co_.st $idr_

ts 20 20 30 _s 40 4s 5o st to et To rs to to ,o

no. of points

Minimized Deviations of the Gear.

Table 1: Results of Gear Minimization

M 4w-.hine Setting Initial Corrected

Pressure Angle oG

Cutter Diameter, rnm (in)

Point Width of Cutter, mm (in)

V2, mm (in)

H_, rnra (in)

3'm, radl.

AX,., mm (in)

21.25"

2_.s (9)

2.o31.o8)

t03.202_S0 (4.(_S06)

27.,,SSS (L0813S)

1.059816

0.00_6rT (.00038)

21.25'

_.os (.os)

103.25220 (4.0_SOS)

27.21603 ( 1.07150}

1.06437

J -0.53343 (-0.0210)

(i) the fixed ones, So (xo, yo, zo) and Sq (x+, yg, z+) that are rigidly

connected to the cutting machine (fig. 10 and fig. 11); (ii) the

movable coordinate systems S, and S r that are rigidly connected

to the cradle of cutting machine and the pinion, respectively;

(iii) coordinate system St that is rigidly connected to the head

cutter. In the process of generation the cradle with S, performs

rotational motion about the Zo-aXis with angular velocity w (_1,

and the pinion with S_ performs rotational motion about tile

zq-axis with angular velocity wl_) (fig. 11).

The tool (head-cutter) is rnounted on the cradle and performs

rotational motion with the cradle. Coordinate system St is rigidlyconnected to the cradle. To describe the installment of the tool

[Z-t

(a) Head-Cutter SurfacePararncters

Final Cutter

Orientation

g_ gt

(b) Coordluate System_ for Head-Cutter Tilt

Fig. 9. Pinion l lead-cutter Surface.

.---" Original Cutter

Orientation

i -"FiR Angle

Fig. I0.

Yo

Cutting biachine and Cradle Coordinate Systems.

with respect to the cradle we use coordinate system S# (fig. 9 andfig. 10). The required orientation of the head-cutter with respectto the cradle is accomplished as [ollows: (i) coordinate systems

S, and St are rigidly connected and then they are turned as one

rigid body about the z¢-axis through the swivel angle j = 27r-(fig. 10); (ii) then the head-cutter with coordinate system St istilted about the y_-axis under the angle i (fig. 9(b)). The head-cutter is rotated about its axis zt but the angular velocity in this

¢#(c}

J:o

V7 o- /

ABe.

7.q

Fig. Ii. Angular Velocities of Cradle and Pinion.

motion is not related with the generation process and depends

only on the desired velocity of cutting.It will be shown later that the deviations of real pinion tooth

surface can be minimized by corrections of parameters of install-

ment of the pinion and the head-cutter. These pinion settin_parameters are E,,- the machine offset, _'_- the machine-root

angle, AB- the sliding base, AA- the machinc center to back(fig. 11). The head-cutter settings parameters are: Sn- radial

setting, 0¢- initial value of cradle angle, j- the swivel angle(fig. 10), and i- the tilt angle (fig. 9(b)).

Tool Surface EquationsThe head-cutter surface is a cone and is represented in .5't

(_g. 9)

rt(s, 0) =

(r_ 4- s sin or)cos O

(r= 4- s sin a) sin 0

--S COS

I

(29)

Here: (s, 0) are the Oaussian coordinates, _ is the blade angleand ro is the cutter point radius. Vector function (29) with cxpositive and c_ negative represents surfaces of two head-cv.ttcrsthat are used to cut the pinion concave side and convex side,

respectively.The unit normal to the head-cutter surface is represented in

St by the equations

n,= [-cosacos0 - cos_sin0 - sin_]r (30)

The Family of Tool Surfaces is represented in Sp by the matrixequation

r_(s,0,_p) = M_, M,_ M.o Mo: M:b Mb, r,(s,0) (31)

Here: .5". is an auxiliary fixed coordinate system whose axcs par-allel to S° axes.

cosi 0 sini 00 1 0 0

-sini 0 cosi 0

0 0 0 1

Mcb

-sinj -cosj 0 .-qa

cosj -siaj 0 0

0 0 1 0

0 0 0 1

Moe _

cosq sinq 0 0

-sinq cos q 0 0

0 0 1 0

0 0 0 1

Mrto

I 0 0 0

0 1 0 E,_

0 0 1 -AB

0 0 0 1

M?_. -_"

cos 7-, O sin 7,. -AA

0 1 0 0

-sinT.) 0 cos %, O

0 0 0 1

Mpq

1 0 0 0

0 cosCp sinCp 0

0 -sine, cost, 0

0 0 0 I

g = 2r - j; q = Oc+ mo$_ where 0¢ is the initial cradle angle

and m_, = w(')/w Ip).

Equation of Meshing

This equation is represented as [3]:

n (pl. v (c") = N Iv). v (¢v) = f(s, 0, _bp)= 0 (32)

where n Ip) and N (v) are the unit normal and the normal to thetool surface, and v(V) is the velocity in relative motion.

Equation (32) is invariant with respect to the coordinate sys-tem where the vectors of the scalar product are represented.These vectors in our derivations have been represented in So as

follows,

n° = Lo_ Lob Lbt nt

Here:

ro = Mo, Mcb Mbt rt

-E,, /xB)

= -[ cos-r o sin-r 1T ; (1 = l)

Pinion Tooth Surface

Equations (31) and (32) represent the pinion tooth surfacein three-parametric form with parameters s,O and ¢_. However,

since equation (32) is linear with respect to s we can eliminates and represent the pinion tooth surface in two-parametric formas

rp(0,¢p, dj) (33)

Here: di (j = 1,...,8) designate the installment parameters:E._, 7,_, AB, AA, Sa, 0_, j and i.

The normal to the pinion tooth surface is represented as

n,(0, ep, dk) (34)

where d_ (k = 1,2,3,4) designate the installment parameters7-_, 0c, j and i.

Results of MinimizationFig. 12 and fig. 13 illustrate the initial deviations _bi of the

real surface, that have been obtained by measurements and cal-culations for the concave side and convex side, respectively. The

blank data, the basic machine-tools settings, the corrections of

machine-tool settings and the corrected machine-tool settings areshown in Table 2-3. Based on the corrected machine-tool set-

tings, we can manufacture a new surface that will optimally fitthe theoretical surface after the surface is distorted by manufac-

turing processes and heat treatment. The minimized deviationsbetween the new surface and the theoretical surface are shown in

fig. 14 and fig. 15. These figures confirm the elfectiveness of theproposed approach. The deviations of approximately 30 micronshave been reduced to 2-3 microns.

8. CONCLUSION

A general approach for computerized determination of deviations of a real surface from the theoretical one based on coordi-

nate measurements has been proposed. An algorithm for mini-mization of deviations by corrections of initially applied machine-tool settings through application of a least square approach hasbeen developed. The approach is illustrated with an example of

the tooth surface of a hypoid pinion and gear.

ACKNOWLEDGEMENT

This research has received financial support from the NASA-Lewis Research Center, Gleason Memorial Fund, the Dana Cor-

poration and the Nissan Motor Co.

REFERENCES

1. Gleason Works, 1987, "G-Age T.M. User's Manual ", for

the Gleason Automated Gear Evaluation System Used with

Zeiss Coordinate Measuring Machines.

2. Litvin, F.L., Zhang, Y., Kieffer, J., Handschnh, R. F.,

"Identification and Minimization of Deviations of Real Gear

Tooth Surfaces", ASME Journal of Mechanical Design, Mar.

1991, pp. 55-62.

3. Litvin, F. L., 1989, "Theory of Gearing", NASA publica-

tions; RP-1"212 (AVSCOM technical report; 88-c-035).

4. Dongarra, J. J., Bunch, J. R., Moler, C. B., and Stewart, G.

W., 1979, "LINPACK User's Guide", SIAM, Philadelphia.

5. Litvin, F. b., Zhang, Y., Kuan, C., and Handschuh, R.

F., "Computerized Inspection of Real Surfaces and Mini-

mization of their Deviations", Proceedings of International

Conference on Mctrology and Properties of Engineering

Surfaces, F_ngland, 10th -12th, April 1991.

Table 2: Blank Data of Hypoid Pinion

Number of Teeth = 13

Shaft Angle = 1.57079 tad.

Pitch Diameter = 88.22 mm

Outside Diameter = 103.96 mm

Pitch Angle = 0.32055 rod.

Face Angle = 0.,11480 rod.

Root Angle = 0.30136 rad.

Mean Spiral Angle = 0.84677 rad.

Face Width = 38.30 mm

Whole Width = 11.63 mm

Hand of Spiral : R. H.

0.011

0,04

002 _o

0.00

-0,02-0.04

0 -0.06 • + .... J .... ', , - - - _ - , .... J .... i ....

o s lo _0 20 =+ so _0 40 400o. of points

Fig. 12. Deviations of Pinion Real Tooth Surface (Concave Side).

/"--7

EE

o

>ot_w

oZ

Fig.

O.Oe

0.04-

0.02 :

0.00

-0.02 :

-0.04

-0.00

14.

0 s to =s 20 :s =0 ,,5 40 40no. of points

Minimized Deviations of the Pinion (Concave Side).

0.0ll --g,_ 0.04 '

0.02 -o

>ot-. -0.02

.010 4

oZ -0.01 I .... i .... f .... , .... i .... _ .... ,-. i , ....

6 10 15 20 25 30 36 40 45

no. of points

_e

9m>oQ

Ii

o

O.O(ll -

0.04 "

0.02"

0.00

-0.02-

-0.04

-0.06

Fig. 13. Deviations of Pinion Real Tooth Surface (Convex Side). Fig. 15.

" 6" t() .... 15"" 20" 2'5 '310 .... 3_5" 4'0'" 45

no. of points

Minimized Deviations of the Pinion (Convex Side),

Table 3: B<_sic, Currected, a,d Machine-Tool Setting Differences of the Pinion

(Unit: Length in mm; Angle in rod.)

Machine Setting Basic Mar.blue-Tool Settings Corrected Machlne-Tool Settings Sating DJfl'erence=

Convex Side Concave Side

B_-;c Tilt AnKle

Swivel Ande

Machine Root AnKle

Cradle Angle

l"Lltdial Setting

Sliding Base

Ma_'.hine Center to Back

Blank OffSet.

Cuttin K Ratio

Cutter Paint Radiul

Cutter Blade AnKle

Convex Side Concsve Side

0,3761599

5.766247

6.233"/36

4.846199

I 14.0236

23.87000

3.280000

-40.12000

0.3020446

114.9350

-0.5410521

0.4104054

6.000656

6._29372

1.$66173

109.6660

14.62000

-3.10CC_1

-34.58000

0.3230215

113.0300

0.2443461

0,371212S

5.768892

6.236861

4.8,15308

113.6155

23.87000

3.767510

-39.63248

0.30'20446

114.9350

-0.5410521

0.4360375

6.042021

6.202694

1.573228

110.4463

14.62000

-3.970493

-35.45049

0.3230215

113.0300

0.2443461

-0,4977365E_02

0.2644968E,-02

0.3125.239E-02

o0.6006187E-03

-0.3780939

O.O000000E+O0

0.5769¢174 i_.01

0.467S106

0.00000OOE+O0

O.O0000COE+0O

O.O000000E+O0

Convex Side Concave side

O.2563208E-.OI

O.4t 36530F_.,-0]

-0.2647799F_,-01

0.7054806F,,-02

0.7'803197

O.OIXXIOOOE+ 00

-0.$540259

-O.STO4OJ24

O.0000000E+00

0,0000000E+O0

0.0000000E+00

Form Approved

REPORT DOCUMENTATION PAGE OMB NO. 0704-0188

Public reportingburden for this collectionof information is estimated to average 1 hour per response, including the time for reviewinginstructions, searching existing data sources.gatheringand maintainingthe data needed, and complebngand reviewing the collectionof information. Send comments regardingthis burden estimateor any other aspect of thiscollectionof information,including suggestionsfor reducingthis burden, to WashingtonHeadquarters Services, Directoratefor information Operations and Reports,1215 JeffersonDavis Highway, Suite 1204, Arlington.VA 22202-4302, and to the Off.meof Management and Budget, PaperworkReductionProject (0704-0188), Washington, DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

September 1992 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Minimization of Deviations of Gear Real Tooth Surfaces Determined

by Coordinate Measurements

6. AUTHOR(S)

F.L. Litvin, C. Kuan, J.-C. Wang, R.F. Handschuh,

J. Masseth, and N. Maruyama

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Lewis Research Center

Cleveland, Ohio 44135-3191

and

Propulsion Directorate

U.S. Army Aviation Systems Command

Cleveland, Ohio 44135-3191

9. SPONSORING/MONITORING AGENCY NAMES(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, D.C. 20546.-4)001

and

U.S. Army Aviation Systems Command

St. Louis, Mo. 63120-1798

WU-505--63-36

1L162211A47A

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-7115

10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

NASA TM-105718

AVSCOM TR-92-C-003

11. SUPPLEMENTARY NOTES

Prepared for the 1992 Power Transmission and Gearing Conference, Phoenix, Arizona, September 13-16, 1992. F.L. Litvin, C. Kuan, and J.-C.

Wang, University of Illinois, Chicago, lllinois 60680. R.F. Handschuh, Propulsion Directorate, U.S. Army Aviation Systems Command. J. Masscth,

Dana Corporation. P.O. Box 750, Fort Wayne, Indiana 46825. N. Maruyama, Nis_n Motor Corporation, 1, Natsushima-cho, Yokosuka, 237 Japan.

Responsible person, R.F. Hand.huh, (216 t 433-3969.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Unclassified - Unlimited

Subject Category 37

13. ABSTRACT (Maximum 200 words)

The deviations of a gear's real tooth surface from the theoretical surface are determined by coordinate measurements

at the grid of the surface. A method has been developed to transform the deviations from Cartesian coordinates to

those along the normal at the measurement locations. Equations are derived that relate the first order deviations with

the adjustment to the manufacturing machine-tool settings. The deviations of the entire surface are minimized. The

minimization is achieved by application of the least-square method for an overdetermined system of linear equations.

The proposed method is illustrated with a numerical example for hypoid gear and pinion.

14. SUBJECT TERMS

Gear; Grid surface; Deviations of gear; Measurements

17. SECURITY CLASSIFICATION

OF REPORT

Unclassified

18. SECURITY CLASSIFICATION

OF THIS PAGE

Unclassified

19. SECURITY CLASSIRCATIONOF ABSTRACT

Unclassified

15. NUMBER OF PAGES

1016. PRICE CODE

A0220. UMITATION OF ABSTRACT

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18298-102

Nmtional Aeronautics and

Space Administration

Lewis P,esem_ Center

Cleveland, Ohio 44135

Ofllclel ludmme

pwalty fer I_m thin _

FOURTH CLASS MAIL

ADDRESS CORRECTION REQUESTED

I!1111

Postage and Fees PatO

Nahonal Aeronautics ancI

Soace Adrnlnlslratlon

NASA 451