NASA AD-A242 332Technical I .

Paper "3096

AVSCOMTechnicalReport91 -C-020 A Method for Determining

Spiral-Bevel Gear ToothAugust 1991 Geometry for Finite

Element Analysis

Robert F. Handschuh

and Faydor L. Litvin

US AR M Y V * '"IAVIATION qnIlmJftd iSYSTEMS COMMANDAVIA11ON ,T A(I;V'.,

NASA 91- 74967I:I II I I I~ l lri: ;r I '. ... U

NASATechnicalPaper3096

AVSCOMTechnicalReport91 -C-020 A Method for Determining

Spiral-Bevel Gear Tooth1991 Geometry for Finite

Element Analysis

Robert F. HandschuhPropulsion Directorate - .U.S. Army-AVSCOM Tri 0 TIELewis Research Center L.s,..

Cleveland, Ohio

Faydor L. Litvin Istribkti•/

University of Illinois at Chicago Availabtity Cgoqg

Chicago, Illinois ive - i 1adorDist Speeial

National Aeronautics andSpace Administration

Office of Management

Scientific and TechnicalInformation Program

Summary elment modeling of parallel axis ,ears to determine the stress

field. Loads are typically applied at the point of highest single

An analytical method has been developed to determine gear- tooth contact, and then the stress in the fillet re•ion is

tooth surface coordinates of face-milled spiral bevel gears. The examined. Computer programs that perform this type of

method uses the basic gear design parameters in conjunction analysis are usually twAo dimensional in nature and ha\e

with the kinematical aspects of spiral bevel gear manufacturing computer storage requirements that are small enough formachinery. A computer program entitled 'SURFACE- was personal computers. These attributes make them very popular

developed to calculate the surface coordinates and provide and attractive to designers. However, a limited number of

three-dimensional model data that can be used for finite researchers (refs. 16 and 21) have in\estigated finite element

clement analysis. Development of the modeling method and analysis of spiral bevel gears.an example case are presented in this report. This method of Parallel axis components (involute tooth geometr\ ) ha\ e

anal. sis could also be applied in gear inspection and near-net- closed-form solutions that determine surface coordinates.shape gear forging die design. These coordinates can be used as input to finite element

mnethods and other analysis tools. Spiral bevel gears. on theother hand, do not have a closed-fornm solution to describe their

Introduction surface coordinates. Coordinate locations must be solvednumerically. This process is accomplished by modeling theSpiral bevel -,cars are currently used in all helicopter power kinematics of the cutting or grindingz machinerx and the

transmission systems. This type of gear is required to turn the -

-cncr type baicLardai

corner from a horii.ontal engine to the vertical rotor shaft. metry of the basic gear design.These (,ears carry large loads and operate at high rotational The objective of the research reported herein was to de\elop

a method for calculating spiral bevel gear-tooth surfacespeeds. Recent research has focused on understanding manys b -coordinates and a three-dimensional model for finite elementaspects of spiral bevel gear operation, including gear geometrv(refs. I to 12). gear dynamics (refs. 13 to 15). lubrication a Accomplishment of this task required a basic(ref. 16). stress analysis and measurement (refs. 17 to 21). understanding of the gear manufacturing process. \%hich is

described herein by use of differential geometr.n technique,,misali-nment (refs. 22 and 23). and coordinate measurements d r h(refs. 24 and 25). as well as other areas. (ref. I). Both the rmanufacturitin machine settings and the basic

Research in gear geometry hits concentrated on gear design data were used in a numerical anal\sis procedurethat yielded the tooth surface coordinates. After the toothunderstanding the meshing action of spiral be,,el gears (rels. 8 ufcs(rv n os idsIwr ecieatreC, surfaces (drive and coast sides) ,.'ere described. at three-

to II). This meshing action often results in much vibrationand noise due to an inherent lack of conjugation. Vibration deonam o For the th sembled A ompterstudies (ref. 26) have shown that in the frequency spectrum program. SURFACE, was developed to automate thc

"calculation of the tooth surface coordinates, and hence, theofan entire helicopter transmission, the highest response canZ71 coordinates fo~r the gear-tooth three-dimensional finite elementbe that from the spiral bevel gear mesh. Therefore if noise

model. The development of the analytical model is explained.reduction techniques Lire to he imiplemiented effectively. the

meshing action of spiral nevel gears must be understood. and an example of he finite element method is presented.

Also. investigators (refs. 18 and 19) have found that typicaldesign stress indices for spiral bevel gears can be significant]ydifferent from thoseC measured experimentall,,. In addition to Determination of Tooth Surfacemaking the dcsign process one of trial and error (forcing one Coordinatesto rcly ti past experience). this incotnsistencv makes

extrapolating oxera ,,ide range ofesizes difficult. and an overly The spiral gear machining process describc,d in this paper iscon,,crx ati, c design can result. that of he ftace-milled tx pc. Spiral bevel gears, man, lctured in

Research has been ongoing in an attenmpt to predict stresses this wax, are used Cxtensixe,,l in aerospace porxcr transnissionswi.e . hending and contact) bx using the finite clement method. (i.e.. helicopter imain'tail rotor transm,,isions) to transmit ploscrA grcat deal of \ýork (rels. 27 tw 30) has g-,one into finite bctx\ecn hori/ontal gas turbine engines and the \crtlical rooir

shaft. Because spiral bevel gears can accommodate various shaft figure 2. The head cutter (holding the cutting blades or theorientations, they allow greater freedom for overall aircraft grinding wheel) rotates about its own axis at the proper cuttinglayout. speed. independent of the cradle or workpiece rotation. The head

In the tollowing sections the methox of determining gear-toxoth cutter is connected to the cradle through an eccentric that allowssurface coxrdinates will be described. The manufacturing process adjustment of the axial distance between the cutter center andmust first be understood and then analytically described, cradle (machine) center, and adjustment of the angular positionEquations must be developed that relate machine and workpiece between the two axes to provide the desired mean spiral angle.motions and settings with the basic gear design data. The The cradle and workpiece are connected through a system ofsimultaneous solution of these equations must be done gears and shafts, which controls the ratio of rotational motionnumerically since no closed-form solution exists. A description between the two (ratio of roll). For cutting,. the ratio is constant.of this procedure follows, but for grinding. it is a variable.

Computer numerical controlled (CNC) versions of the cutting

and grinding manufacturing processes are currently beingG;ear Manufacture C

developed. The basic kinematics, however, are still maintainedSpiral bevel gears are manufactured on a machine like the one for the generation process: this is accomplished by the CNC

shown in figure 1. This machine cuts away the material between machinery duplicating the generating motion through point-to-the concave and convex tooth surfaces of adjacent teeth point control of the machining surfhce and location of thesimultaneously. The machining process is better illustrated in workpiece.

Cradle \ Cradle housing

\ /

-- .- Work offset

-Machine

center-to-back

. Root angle

Machin,e/ ". .\center

Sliding basei Work base

F iur,.e I M achinIh e t,.d to gener ate spiral hcel Lcar-rtooth ,,urlaic

Cradle axis -\ , Cutter axis Xc\ / X

ra le houisinig A

r- Head cutter

oc Z{ §Wokpiece V-b B e]ý . ycf" rc

Top view Inside blade (convex side);u =IA1hosnCradleA

Cradle houin Workpiece xxc

OcCW- Man•" contact r- r" -

"- Spiral angle"- Cutter circle o A

Machine / I\'

center_ Cutter centerFront view Outside blade (concave side);

h:Iurc 2.--Orieniation of t orkpiece to generation mnachiners. u =IABI

Coordinate Transformations Figure 3.-Head-cuLtter cone ,urtace,.

The surface of a generated gear is an envelope to the family blades cut the convex and conca,,e sides of the gear teeth.of surfaces of the head cutter. In simple terms this means that respectively. A point on the cutter blade surface is determinedthe points on the generated tooth surface are points of tangency by the following:to the cutter surface during manufacture. The conditionsnecessary for envelope existence are given kinematically by theequation of meshing. This equation can be stated as follows: the r cot i u cos -'

normal of the generating surface must be perpendicular to therelative velocity between the cutter and the gear-tooth surface ui sin • si. '3

at the point in question (ref. 1). r= (1)it sin " os 0

The coordinate transformation procedure that will now bedescribed is required to locate any point from the head cutter Linto a coordinate system rigidly attached to the gear beingmanufactured. Homogeneous coordinates are used to allowrotation and translation of vectors simply by multiplying the where fixed value r i, ie radius of the blade at x, = 0. andmatrix transformations, The method used for the coordinate is the blade angle. : ararneters u and 0 locate a point in systemtransformation can be found in references I. 5, and 8 to II. S, and are unkrowns whose value will be deternined.

Let us begin with the head-cutter coordinate system S, shown The head-cutter coordinate system S, is rigidly connected toin figure 3. This report assumes that the cutters are straight sided coordinate ,)stem S, (fig. 4). System S, is rigidly connected to(not curved as commonly used on the wheel for final grinding), the cralle that rotates about the x,,, axis of the machineSurface coordinates u and 0 determine the location of a current coordinate system S,,. Coordinate .;ystem S,, is a fixedpoint on the cutter surface as well as the orientation of the current coordinate system and is connected to the machine frame.point with respect to coordinate system S,. Angles ý,/, and ý,,, To reference the head cutter in coordinate system S,. theare the inside and outside blade angles. The inside and outside following transfornation is necessary:

YS Ym

zs= 0 cos 6, ±~sin o, ()Zl,, (3)(0 4:sin 0, cos 0, (

Coordinate systemn S, locates the machine center. andcoordinate systemi S~,, orients the pitch apex of' the -cear heing

o Oc nmanufalctured. The transformiation fromt coordinate s'~stenI .5,.to coordna~e S~ sterilSp, requires the mcietool settinos L

and E,, alono wkith dedendumn anefle 6 fromn the comnponentdesig-n (see figs. 5 and 6). Machine tool settingls L,, and E,, canhe found f'romi the summnary sheet that t),picalI\ accompanies a

kC ___cear. or the methods in reference 8 can he used. Relcrence 8Left-hand member; s OcOs converts, standard machine tool settines lir the slidinu, base. the

offtset, and the machine center-to-hack into settings L,, and E,.YC z as showkn iii table 1. The transf'ormation mnatrix is -,i\cn h\

____ p Om y C

Right-hand member, S OcOs

Top viev!

0 cos q 4: s in I/ 4:F s in1( r<S (2)

I) s i ~ c o s (/ N o sY p , Y a . Y w Y m 0 / w

0,0j. Th upe2q oZ

Coriaesse . nSoiisChere q is tie cradle angi-le and s~ is the distance between the Op. a. Ow ZIP Za,

and lowecr signs preceding the \arious termis in this matrix 0AL Wzmtranst'Ornnat in ( andl the rest ut the paper) pertain to left- and right - -L mhand gecars, respectik elk .

No%% . to transfoirm fromn S, to the fixed cooirdinatc systefn Sj Front view

the ro ll angle of' the cradle (.) is used. This trans h lrrlat ion is I irc5 ( irdtinaic '\ I ,kIi lIcniationI 'ii, _,nci ýiic i muit haind ,caj mi j ~ tcc

"given by 0 i1,il ~h ew Iir

4

X, r Cos 6 0 - sin"6 Lqi6

=P 0

OP. -a, O, 0 m y Y OcZ

WX This is shown in figure 5 fbr a right-hand mecmber and in figure 6lior a lefl-hand member. Figure 7 is given to clarity' thle orientationot' the coordinate Systems and machine tool settings (I11.11,1).

Za. Z, Thle next transformation involves rotation of' systemi .S, to S,.

Top viow Thle common origin for coordinate system S,, and S,, locates theapex of the geatr under consideration with respect to Coordinatesystemn S,,,. This requires rotation about v,, by the pitch angle

YP, Y. YWym i (see fig. 7). This is given by

Om~co 0 p.OS 0 Sinl it 0

0 0 0 0P, a. W MOP(5)

S -Sin it 0 Cos J( 0

coordinate System 7,,,, which is fixed to thle component beingmianuflactured. A rotation about the z:,,-axis thiroug~h an angle 6,,.is requtired. Angle n/i,,., Shown iti figure. 8. is thle worklpiece

Front viow rotation angle; it is directly related to thie angle of' rotation of'I 1,Ic0. C'l( wMIiiC * NýIm oncniial it', gilcrak' a lei- huts) paL~r N11ti aL'e the cradle 0,. (this relationship wvillI be described in thle next

section). T .. ,,, to Si,,. coordinate transformiation isgvnb

TAtti t.--StUN CO NVINti( NS OFt NACHINIf N-TO0l. SEir7NGSI Eromt ret', 8,1

Selitntg Sign Right-thand mneiher Lett-harnd nt.',nhcr

('tullc utigle. 1/ 4- Counierelockwike (CCWI Clockwisec (C\V)- Clonckwise (CW) Counteclockwise ('C\\')

Mauchtining ollsei. E" - Atboye machtine center Btelow mactine centerB Ietow machine center Ahove mactine center

miachtine center-lo-hack. + Wonrk withdrawal Work wiflhdrawal-I/ Wonrk advance Work advance

SlIding Nise .. A-,j Work wididtrtwat Work \withdrawalVi ik advance Work advaince

Vector %tln orX,, X511. -, XIA(f. - Nff +; .Xjfwt,rid X~,11 - Xg. -XIcl. + +v, :~,.4

xm Xa

OP• Oa, Ow Z,, . za, Zw

Plane n

yp YmnO

(a) Ya •

Xa YwXP Plane 7c Left-hand member

I'Xw. \--Planet

OWZP Xa xw

(OW "bi-

Ya, Yp Za, w

Yw 0~:a. Owz,(b)yw$•Z

(a) Machine ,ettings, and orientation. Yw

(hi Plane 7r and orientation of generated gear coKrdinate,,. YaFigure 7. -()nenLation o" machine settingr and generated gear coordinate ,,yser,,. Right-hand me.inber

Figure 8.-Rotation of .orkpiece for left- and right-hand gears during hooth-

ssurface generation.cos 'p,, ±sin 0,, 0 0

T =s in cosO,, 0 00 0 (6) Tooth Surface Coordinate Solution Procedure

0 0 I 0In order to solve for the coordinates of a spiral bevel gear-

0 0 0 1 tooth surface, the following items must be used simultaneously:

the transformation process, the equation of meshing. and thebasic gear design information. The transformation process

Ustng these matrix transformations, we can determine the described previously is used to determine the location of a pointcoordinates in S,, of a point on the generating surface from on the head cutter in coordinate system S_. Since there are

three unknown quantities (u, 0. and •,), three equationsrelating them must be developed.

r,, = IMIMII,,IIM,,,,,IIM,,,,I IM,Ir, (7) Values for u. 0, and 0, are used to satisfy the equation of

or meshing given by references I and 9:

r" = IM,,,fl, ) I IM,",, [M,,,I IM,,,, (, )1 IM,, Ir, (u,0) (8) n V = 0 (10)

or

r, = r, (u.0,,) (9) where n is the normal vector to the cutter and workpiecesurfaces at the specified location of interest, and V is the

This transformation describes the location of a point in the relative velocity between the cutter and workpiece surfacesgear fixed coordinate system based on machine settings (L,,, at the specified location. From the reference 9 equation ofE,,. q. s. r. and ý). parameters (u. 0. and 0,), and gear meshing for straight-sided cutters with a constant ratio of rolldesign information (, and 6). between the cutter and workpiece, equation (10) is defined as

6

(1i - r cot V/ cos k)cos It sin rXa, XW

+sI(mn -sin -y)cosý sin0 =F cos -y sin ý sin (q - X,)P p CXm

SPoint P± E,, (cos sin f- sin I cos cos r)

zp

-Lsin-ycosý sin =0 (11) Zm

where -y is the root angle of the component being rmanufactured, and

T (0 :F q+ ) (12) View A

and Yw Ya

n ,,, - (( 1 3 ) w

where in,,, is the ratio of angular velocity of the cradle to that \ Xa

of the workpiece. Since the ratio of roll in this report is View Aassumed to be constant, equation (13) can be written as Figure 9.-Orientation of gear io be generaled. ,Aih a,,unied po,,ition, rand

WO) The axial position must match the value found fromtransforming the cutter coordinates S, to workpiece

and coordinates S_. This is satisfied by the following (fig. 9):

LO t, =do(, = do,_ Z,, - z = 0 (16)

(it dtor

therefore

t , I f 2(u , O. 0 , ) = 0 ( 17 )

tu Finally the radial location from the work axis of rotation mustor be satisfied. This is accomplished by using the magnitude of

the location in question in the x,,-v,, plane (see fig. 9):

= I(14)In,.,,

,. - (x;, + vy;,(V = 0 (18)

Equation (14) is the relationship between the cradle andworkpiece for a constant ratio of roll and is used directly inequation (6). 0, 0, 0 (19)

Gear design information is then used to establish anallowable range of values of the radial (r) and axial (z) Now a system of three equations (eqs. (15). (17). and (19))positions that are known to exist on the gear being generated. is s ys te of for the t e p1 m t. u,7) . andThis is shown in figure 9. is solved simultaneously for the three parameters u, . and

First the equation of meshing must be satisfied. This was 0,. for a given gear design with a set of machine toolshown earlier to be settings. These are nonlinear algebraic equations that can be

solved numerically with commercially available mathematicalsubroutines. These equations are then solved simultaneously

n -V = 0 for each location of interest along the tooth flank, as shownin figure 10. In the SURFACE program a 10 by 10 grid of

or points is used on each side of the tooth. From the surface grids.

the active profile (working depth) occupied by a single tooth/]~(u, 6, O, ) = 0 (15) is defined.

.7

Whole Convex side

Cl~arance

Dededum ~ \~( 10 PointsDedendum 10 Points ),- Points

-£-Dedendum

Fhienr_ 10)--Calculation points (1 h%) 1~0 g~rid,. i.e.. 101) points each side)to r co ncas e and ci ns e \iIS iesfi tooth su rt~ace.

Application of Solution TechniqueAn application of the techniques previously discussed will

flo\X be presented. The component to be modeled was fromthe NASA Lewis Spiral Bevel Geai Test Facility. A photo- C74g raph of the spiral bevel gear mesh is shown in figure II., andthe design data for the pinion member are shown in table 11.zr I-NS prlbee er n cipret

The gear design data were used along with the methods ofreference 9 to determine the machine tool settinos for straiaht- of the :oncave and convex sides of the gear tooth (fig. 10).sided cutters (see table 11). These values were then used as orients the surfaces such that the top land is of the properinput to SURFACE. This program calculates the coordinates width, and then generates the required data for the three-

TABLE II. EXAMPLE CASE OF SURFACE COORDINATE GENERATION(a) Pinion dcsiszn data

Number of teeth

pinion ............. ............................................ ....... ....... t12,cear .................................. _............................ ....... 36

Dedendurn angle. d~g .................. .. ...................................... 1.0Addendum anglc. dc ........................................... ........... 3.883Pitch anule. deg . .. ..... . ... .... . ... .. .. ... ... . 18.433Shaft am! Ic. dc- 90~.0

M ean spiral angle. deg . ... ... ... .. .... .. ... ... .. .. ....3 .

[-ace wkidth. mmrr (in.) ...- I.. ..... ........ 25.4 (l.0tMean cone distance. mmn (in.)ý............ ......... 81.05 (3.191)Inside radiusJ o) gear blank. mmn (in.) ...-.... ..... 15.3 o),6094)

T o p land thickness. m m (in .) .. .. ... ... ... ... ... .... 2.0 )32 (0 . 1)80 )

Cl earance. mmi (in.) ..........- (..(... .......... .762 (0.t0301

bt) Generation machine settines

f Concax e on

Radius of cutter. i. miin (in.) 75.2212 (2.96 15) 78.,1329 (3.0761i

Blade anele. ,'. dee 161.358 2.3vector sumi. I.,,. tmi (in.) 1.0)363 ((1.04(18) -1.4249) -0.01561i1Machine offset. I6,. min ( in.) 3.9802, 0. 1567) -4,4856 I-0(.1 7066Cradle to cutter distance. s. nun (in.) 74,839 (2.9646) 71,_4 2.(5)Cradle angle. (1, &L 64.0)1 53.82Ratio of roll. mi 2t.O 31:8462 (1.321767Initial cutter lengtht. it. turti (in. 2395 9.43i I81. 1 (7. 131Initial Cutter orientatiin. (4, de, 120)).)11

Initial cradll irientation, o, . (lei! j()

g

dimensional modeling program PATRAN (ref. 31). The detailsof the procedure are described in the following paragraphs. Xt-urfa grdS• \ calculation grid

Xm \-Heel

Surface C'oordinate CalculationTo

Using figures 10 and 12 as retfrences, we will describe thecalculation procedure for surface c,,ordinates. First. the To.-Zrconcave side of the tooth is completely defined before mov ing - Clearanceto the convex side. These points are calculated hy starting at 'othe toe end and at the lowest point of active profile. Nine steps r, cof equal distance are used from the beginning of the activeprofile to the face angle (addendum) of the gear tooth, andthen back to the next axial position (see fig. 12). The procedureis repeated until the concave side is completely described. Thenthe same procedure is followed fc r the convex side.

In the discussion of surface coordinate determination, thecutting blades were described as straight sided. The point iur 2 C( ,, •,~C'dii~l O1 C.dlCtl.tll~l eI h!Y i |1 , lt ,1 .n,'L':. !'II) LH

radius r (eq. ( I )) is the radius the cutter would have if it were n ., . 'ij iiiril ,tiwlc. ,c ii', dc r'ijijx , I. n k

proiected down to the y,v-z, plane (ligs. 3. 5. and 6). Actuall\.the blade has a theoretical point width and corner radii thatgenerate the portion of the tooth from the working depth to

sx stem S. This is done b\ checking the tooth thickness at thethe root cone (see fig. 13). In the section C,,ordinate Transfor-Sface angle oo the toe end of the gear tooth. I Also. in the casemations, this part of' the cutting blade was not modeled. so It -

I d asno t..51 of a gecar, remember that a g-iven Cutting operation using Cutter,,the current analysis by SURFACE either assumes a full filletradius between the lowest point of active profile on ad.jacent aIs shown in fig. 3 actually cuts adiacent tee,h ,i the cone\

teeth or sets the fillet radius equal to the clearance (fig. 14). and concaxe sides sinmltaneously .) The distance betmeen thesetwo locations must correspond to the top land width. Fhe

Concave and Convex Orientation con\ex surface is then rotated according to the angle deter-mined bý the points at the face angle at the toc position. This

Since both sides of the gear-tooth surface are not analv. _d is shown in figure 15. (Note that this same procedure couldsimultaneously, for proper alignment of the surfaces. their have been done by considering the tooth mean circularorientation relative to each other must be established by thickness instead of the tooth thickness at the face angle a,determining the amount of rotation in the fixed coordinate "ias done in this report.ý

tActive profile

Inside~tiv blderdis ilpofl

bbginninngng

Cuttertr adges

f\ rradius--

I tci .i-.... (1,,,, width

-= Inside blade radius profile \i Job beginning _3

'--* Outside blade radius

Cutter axisof rotation

--- 2./N Yw

Working

depthpth

W aClearance I •i• P,

M, Xw

Fil.t radius jri L Constant Figure 15.-Orientation of concase and comncx sides of gear-tooth surlace,

equal to clearance --J 1 fillet and root to attain proper top lesel %&idth h% rotation of cotses side of tooth: P1 .radius concaxe side location of face angle point at toe end of tooth: P, and P".

initial and final con\ ex side locations, respeclis.el% ot face angle point attoe end of tooth: '. angle through which P, is rotated: and % .,and wI,.

[igure 14. -'No t\ pc, of fillet and root radius regions used h\ SURFACE desired and initial top level thicknesses. rcspcchel\

prograli: j . cimstant fillet and root radius: p, fillet radius equal to theclearance .V'. numbher of teeth: r,. inside radius of Lear blank.

x

z

Figure I -- Hidlen line pl•i of ine-to•th midel gener:ated bh the SURFACE: program using PATRAN (constant fillet and root radiu mnod,,l)

I')

(;eneration of Three-Dimensional Model Data Example Model and Results

Once the Surlace coordinates are described and properly From the one-tooth model described earlier the analysisoriented, then the data necessary for PATRAN have been techniques can be demonstrated. The model showvn in figureproduced. Currently. the model in SURFACE can have two 16 is that for a constant fillet and root radius (a! , called fulldifferent fillet and root radii configurations. The fillet radius fillet) model. The fillet and root radius on the convex side hascan be constant at the cross section between adjacent teeth or been added along with the tooth section (without the tooth)be equal to the clearance w ith a flat between the fillet radii to make the model symmetric about the tooth centerline. Figureot the adjacent teeth at the root angle. Also a constant inside 16 shows a hidden line plot of the finite element mesh %+ ithradius of the gear blank is used in this modeling method. These eight-noded isopararnetric three-dimensional solid continuumaSumptions are depicted in figure 14. elements. This model has 765 elcments and 1120 nodes. The

At this point, we have produced a one-tooth model for use boundary conditions are shown in figure 17. A 1724-MP-1in PATRAN. Now, the analyst must determine how complex (250 ksi) constant pressure load was applied normal to thethe model need be for a given application. If a complete gear tooth surface of nine elements, and the two edge surfaces ofiS required. simply rotate the one-tooth model in PATRAN. the gear rim had all degrees of freedom constrained.

Once the required number o"teeth have been described, then The results were calculated by MSC/NASTRAN and werethe finite element mesh density and the boundary condition subsequently displayed by PATRAN. Figure 18 shows theinformation are generated within the PATRAN environment, principle stresses, and figure 19 shows the total displacementsPATRAN produces the bulk data deck for MSC/NASTRAN for the boundary conditions shown in figure 17.and mani other computer codes. The example given in thisreport used MSC/NASTRAN for the static analysis.

1724 MPa (250 ksi)pressure loadnormal to gearsurface-'

'- Fixed surface

x

"-Fixed surface

z

y

hL ir 17 Botujndmgrý wnd~itiiu;1 lot ko n'iati fillet miid root rtadliti iiscl il'r ihe cs.viniplc aipplicatio.n

/I

Stress,ksi MPa

120 827

90 -

80- - 552

70 -

60

50

40 276

30

20

10

0 0

-10

x-20

zYXý -30 -207

-i ure I S. - .% lijlr piunoipal qre , Ilr tOundair\ c'ondilion', •pecified in figuic 17.

Displacement,in. gm

4.08xl-3

3.84 104

3.54 -

3.27 -

2.99 - 76

2.72 -

2.45

2.18

1.91 49

1.63

1.36

1.09

.82 21

.54x

.27z

y 0 0

IieurC V) - l t~il dIplde•eC)lIlCn hor the boundir% coindiuin,, specified in ignure 17

SuninarNy of Results I ~I .11.e LNi eirtnNeht. tSPLIT Flelkal. arid 51mW'

Be.cl (;car.. NAS 1S\1 tt8562. AAVSCOMt IR Sh ( 27 19M,~

A Ilthi d has been presenlted th.It LItSeS dtie ren~t id UWICi nte r I L\aci I )cteri I Hilah MH oI I Ith SILTracec, [,I Spirali Be', c andL H% p'II i( ; CaIr

tech niq CILs to Cafe Uk latehe su r taýCeCi to r-di natel Of 'I tCe- i I~ll ed Glao kkork . R 'cI.icstr- NY. ALI", I9-0

spiral bevel gear teeth. Tihe coordi mItes mu11.st be sot ed 1for 1 Mm Tit .1) Anak I, CR 4l9,. Vihri F\ 987iAimL -omS

tIIltCIitterIclN h\ al sitituftane)ous1 solu.tionl of' nonlinear algebraic 14. KýIniimmir A\ . ci il.. t)\ attle Aiiait..j. id Geaired Rolir. 6h\ I mile

equations., These equaltions' relate the kilnematics, of' nlanu- Lictiletilt.. PIroeetdii] ill thle I199 title[naiitioal RIt..cr I 1111"tl

taIcturc to the gear design parameters. Coordinates for a grid anid (carigi Cnteiiccne. .5th. N ul Iý ASMtL, I'M8. Pp 375- IS

of points are dete rminited for hot h the coneaie and cit n eX xSides. S )rg.R.n i ra.l ii a ' eR. hn ep

of (ite _cear toot h. These ci ( rd it mes are then comi tu ned to for tnt at Ico i tee 'i\8 Till [AtRic W; 1) Finite ElementMethode. Presnte

theC ettlTosed suirf'ace of one uear tooth. A compu~ter programl. 10 Cho ( .BC.i Bauter. %1t arid CltIICIe-. HýS A5 N L1PIttiptr SuliitIoni 11r

S LIR FACE1£. T as de~ eloped to sot v e r the gea r-toi t h sirt'ra~ce the I). niaii C t Loiad - ithrieatt [i Vit thicknHess.. and Su iace t ettipera(LTC

Coostrdinates, and pro\~ ide input to a three-dimensional geomuetric ITT Spirail Bet ci ( ica r. Ad anced P er irn .i 'it j*-i hi

mtodeling programt (i.e.. PATRAN). This enables anl analy.sis NASA CP 221I0. .AVRAt)(ONITIRS82 C 10. (;.K. t-t..cicr.ctd . t')81.

h\ý thle finite clemtent method. Anl example of' the technique ppte. 345 an Pal I iiliic ' 04i. ~.p~ccici. Iemci

Was presýented. Pini5 Hand (ear un t''th RooltSlrc'..... Spiral Btcci Ccat. J .tCtLII

Flan.. .-Nutlra,'it D~C.. \illi 107. 110 I. Ntar. 1985. pp 43 48iS. ib'A.:Id. 1:43. (;ear Fm''tth Sire-.. NtCA'itrItteVIIIt.,i Ollile LH 6H\.

1.x\i i,, Research Center Helicotpter Trainsmission..tt .NA.SA tP 209S. 198)'.(9., Pti//ati. (i.N. and tr'.RI. Somite t''tc.it [tue .N..tvmic

National Aeronaultics an1d S pace .Admtinistratiitn t~itall oI 1"oi Roo't iaid Hill * Sitc''I inI tihtxLim% t.Clevemlan. Ohtiit. Ieceniher 19. 1~990 Raittitied(iear.. A-GNI.- t'.per '292'1,2 Aititc~iitearl (; tat[utattitict..

Pattern Shape arid P01111011ti. (Gear TeciIT0O'et. \'Id 2. Iit' I, .JTii Fell1985. pp. 9 (5.23 (ailso. A(;NMA Paper 2- 1.25. (1982.

References 'I. tDraiii RI.; and tippaiuri. B.R.: tarue Rol'trcratt trati..i1-iti..tiIteelitiil I e Dc.. iiipmtitii ProLieair. %oI 1. t . echit-iie Report

(It hii. I-,.t..t iicllr\ oI (icaritie. NASA RI' 1212. 19)81). t)211t- 11172 - -NOt. t1, B'ctite Ncrtl' Co' NASAN ('titraci

2 Iit..it.1F.- i ' 1Sia e.c c'itt. tdC;a ri rc.t't NAS3 -221431 NASA CR-I 68(16. 1983-Nd..aviecd lloatct havt.i~im'.iii to lchionlouex . Ž.NASA. Ct'1- 22(1 22 B~tner. NI. t.. Jr.ý VIled t )I i..alietuirietrt 0tt t1olth -Netio o111 B1...ci anid

.\VR..\tC( t V R 82 U 10. Gi.K. [IerCIC :d.. 1981 . pp. 335 -344. H~ poid Gears. ASM.\S paper 61 \It) -201 Alar. 1961Htustoii. R.. atid CUO.1.. N Ba..j.. lor the A~d.\NNti.t. 0 Suriflace (iol eitict.. 23. Spear. GMýt. anid Baster. Nit... Adijustmnett Chlta.lietert'tt.. -I Spiral

,it Spiral Bc..ci CGears. *Nd..aneed I' Pit.. r It'dalit..iii.iit tecitillle. FCII(1 . Be I cI atd l\ piid (;ear.. ASNIF Paper 66 Nt [H I7. 0,1 )tIk966

\.NASA. (1' 22i0. NVIRAD~COM t R 2 C~ 10. G.K. ti.cltcr. etl.. ('181, 24. Isrint. H.: A1.to0itated Ins.pecttion aind Pceci.i'i (;Irtdtite Lit Spiral Betel

pp 1i7 33;4. Gear.. NA.SA- CR 4083.AASCONI FR 87 -C' 1 1,I98-14 ((taint. SB H. =HLIi.ititl. R, I.-. and (IJ. . ('01ittiputer -NidedL D~esign tif 2.tti. F. L. .ci ail. : Coitpuieri/cd fn.pcetion of (kira 1-001i SLiirl,kC.

Be ci, (ie,ii_001 SLTIrCe... PrllceedItie. it (the (98')9 ttterttaititnal Pit..er NASA TMt 10 l(395. ASCC)N TR 89. C i)l H, 19S9).IraId 1iiit.1 'Sltut 1nt (etr1 (lMInerenee Silt. \'l. 2. ASMtK 198'). 26 len icki. t..G. andi Co\, .J 1..VihrTitttio C'haracteris.tics It O1 H5NPP 58S 59, Htelicopter Mlaini Rotoir tran..ti-io..n. NASA 1t) 27115. A .NSC( (Nt IR

[,It\. 1ii. 1... ei al, Nietlid 1h1 irCiciteatin Lit Spiral Betel Gear.. W%'itl 86 ('-42. 1987.C'1'iiiiietiie Gear 11)11th SL1irCe.. J, Nltlel. Iran... Niiiiiiitat tDC,_ -1 7 (ak.ta.a. A:Cardini . iandl Dhtlt. C; : i-.tiuaioitii of Stre....e. altd

till 10'). tlil. 2. Jtlitt 19i87 . pp. (63, 1701 tetlectiniuit Spur mtid Helical Gear.. Is. li Bti art1,1d. tIleitIeilt Nltili xl.

06. ,IT Y. C.:ý anld C'liii. P.C , Silrlace Geiiiitetr. 'I StraiLght lintl Spiril A.SM1 ilpaper X4-DET:- 169. Oct. 1984

Betel Gear... j NCIec trait.. Nutittlat. Dc... .o iIlt'). ito 4. 28. Baret. C'.. ci al.. Stre.. Path Ailiny tire l-ace...ith Iin Splir Gear.. I (let

D~c- 1987. pp. 441344'). h\. 3t) P [Ni Models.. Proteedine.._ of thte 19S'1) Itntcrnatil'nail Pllttc

7 Ciutier1. I..:aid ~I~CI..eItit C.: Kiticitatte ANii~i-ixt. BceteiGear.. -NSMiI t1rari..ttt...ti .inicamL C oeriie lerenc. 5th. VIi. 1. A.SMIF. 191481). pp

paper 84. DF F 177, CXI l'984. 173 17')8. it.Ixr. 1: 1. I I..iitt WIT; intl hIe. 11 1 Genterationt oI Spitri Betel 29. Stil. H.. CI at. :('iti1paritit Ill B1'Uidar\ Hictiet lanl -Iitt m siI eltictit

Clear. Wihl (l01iiiir2ite Footh Stitiace..an itil h ilit i (ltiact ANialtv M..ethoids Iit Spur GePar Root Siress. .. nalt ..i.. Pt'tecitltii.s Iil tile I(List

N-ASA CR 4018S. SCONI FR 87 C 22. l')87. tittertational Potter Tran..niitt,.ioii miid Gcjariilt-, (nlerenec 4.1t1. Ni 1.

oISpirai Betel (cear.. With t'P.icdcieLlc Pai.6olit Fritct in.. oi 301 )..Kui. NI. three D~imiens.ional Finite [iciticrit Sire-.. Preictiti.. llt SPurTI I itl-tiii...ili tLrrllr.. N-NSA CR 425'). .. VSCOM IFR -XII C 01-14. I'M' (Cear.. CiiiipairedI 6(ea I ainieti Ri! Nica.uretiint.. AI N \ AtPaiper

10 týIli F I clit. ran mr io EtTo , id ic i-, ('11,0 iSp r. 8') 291 ,8 Jul. 198')

tticitkil and~ Spiral Betecl Clears. SNI. Paper 88121)4. Sept '1988 31 PA I RA.N t c.r.. MNtiaiiii PDt- tLneinmctmiii. Costt t.a. C'N Vuit\ liQ

Report Documentation Page1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.

NASA TP-3096 AVSCONI TR-91-C-020(

4. Title and Subtitle 5. Report Date

A Method for Determining Spiral-Bevel Gear Tooth Geometry for Finite August 1991

Element Analysis 6. Performing Organization Code

7 Author(s) 8. Performing Organization Report No.

Robert F. Handschuh and Fador L. LitvinE-5837

10. Work Unit No.9. Performing Organization Name and Address 505-63-51

NASA Lewis Research Center IL162211 A47ACleveland. Ohio 44135-3191and 11. Contract or Grant No.

Propulsion DirectorateU.S. Army Aviation Systems CommandCleveland, Ohio 44135-3191 13. Type of Report and Period Covered

12. Sponsoring Agency Name and Address Technical Paper

National Aeronautics and Space AdministrationWashington, D.C. 20546-0001 14. Sponsoring Agency Code

andU.S. Army Aviation Systems CommandSt. Louis. Mo. 63120-1798

15. Supplementary Notes

Robert F. Handschuh. Propulsion Directorate. U.S. Army Aviation Systems Command: Faydor L. Litvin.University of Illinois at Chicago, Chicago. Illinois.

16. Abstract

An analytical method has been developed to determine gear-tooth surface coordinates of face-milled spiral bevel

(ears. The method uses the basic gear design parameters in conjunction with the kinematical aspects of spiralbevel gear manufacturing machinery. A computer program entitled -SURFACE- was developed to calculate thesurface coordinates and provide three-dimensional model data that can be used for finite element analysis.Development of the modeling method and an example case are presented in this report. This method of analysiscould also be applied in gear inspection and near-net-shape gear forging die design.

17 Key Words (Suggested by Author(s)) 18. Distribution Statement

Spiral bevel gcars Unclassified - UnlimitedSurface coordinates Subject Category 37Gear geometryFinite element analysis

19. Security Classif. (of this report) 2.Security Classif. (of this page) 2.No. of pages 2.Prc

Unclassified Unclassified 20 A03

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