EA _ _A AD-A266 723

AIAA-93-2296Contact Stress Analysis of Spiral Bevel GearsUsing Nonlinear Finite ElementStatic AnalysisG.D. BibelUniversity of North DakotaGrand Forks, NDA. Kumar and S. ReddyUniversity of Akron DTXCAkron, OH XT ELF-CTEand

R. HandschuhU.S. Army Vehicle Propulsion DirectorateNASA Lewis Research Center

Cleveland, OH93-14984

93 6 CC' 0 5 0 , l illIlll]IAIAA/SAE/ASME/ASEE29th Joint Propulsion 1pxovedforpbicr4Q

Conference and Exhibit 1June 28-30, 1993 / Monterey, CA

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics370 LEnfant Promenade, S.W., Washington, D.C. 20024

NASA Technical Memorandum 106176 Army Research LaboratoryAIAA-93-2296 ARL-TR-158

Contact Stress Analysis of Spiral Bevel GearsUsing Nonlinear Finite ElementStatic Analysis

G.D. BibelUniversity of North DakotaGrand Forks, North Dakota

A. Kumar and S. ReddyUniversity ofAkronAkron, Ohio

and

R. HandschuhU.S. Army Vehicle Propulsion DirectorateLewis Research CenterCleveland. Ohio

Prepared for the2Q:h Joint Propulsion Conference and Exhibitcosponsored by the A.AA, SAE, ASME. and ASEEMonterey, California, June 28-30, 1993

U.S. ARMY

NASANawrau,-1-cs w4,

5saxe Aa.$Tkrajo RESEARCH LABORI.TORY

CONTACT STRESS ANALYSIS OF SPIRAL BEVEL GEARSUSING NONLINEAR FINITE ELEMENT STATIC ANALYSIS

G. D. BibelUniversity of North Dakota Azces~on ForGrand Forks, North Dakota NTIS CRA&lIDTIC TAB _•

A. Kumar U TaBno=iiedS. Reddy i Jst,, coilon

University of AkronAkron, Ohio By.

R. HandschuhI '

[

Vehicle Propulsion Directorate Avaiiablhy CodesU.S. Army Research Laboratory Aajl at;d/o,

Lewis Research Center Dist Sp-cialCleveland, Ohio .,-/P

AbstractMuch effort bas focused on predicting stresses in gears

A procedure is presented for performing three- with the finite element method. Most of this v.ork hasdimensional stress analysis of spiral bevel gears in involved parallel axis gears with two dimensional models.mesh using the finite element method. The procedure Only a few researchers have investigated finite elementinv.olves generating a finite element model by solving analysis of spiral bevel gears'-.equations that idcntif tooth surface coordinates.Coordinate transformations are used to onentate the Finite element analysis has been done on a single spiralgear and pinion for gear meshing- Contact boundary bevel tooth using an assumed contact st.;-ess distribution .conditions are simulated wAith gap elements. A The research reported here will utilize the numericalsolution technique for correct onentation of the gap solution for spiral bev.el surface geometry to study geneelements is gi%e. Example models and results are meshing. Pinion tooth ana gear tooth surfaces -wl bepresented. developed based cc the gear mtmn iceuring kinematics.

The individual teeth are then rotated in space to create aIntroduction multi-tooth model (4 gear and 3 pinion teeth). The tooth

pair contact zones are modeled with gap elements. TheSpiral bevel gears are used to transmit power between model development procedure and finite element resultsinterWsAing shafts. One such application is in are presented.helicopter transmission systems. In this criticalapplication, the gears operate at relatively high Equations for Tooth Surface Coordinatesrotona spo e an transmit substantial power (i. e.1500 HP at 21.000 rpm). The system of equations, briefly summaried here, re-

required to define the coordinates of a face-milled spi-ralPrior research has focused on spiral bevel gear bevel gear surface were developed by Handschuh andgeometry" to reduce vibration and kinematic error. .itvin'.improve manufacturability and improve inspection.Stress analysis is another important area of ongoing A conical cutting head, attached to a rotating cradle,resear..h Accurate prediction ofconta stressc• and sings through the work piece. Parameters U and 8tooth rootfillet stresses are important to increase locate a point on the cutting head in coordinate sytem S,reliability and reduce% weight- attached to the cutting bead as shown in figure 1 and

I

described by the following equations, coordinates of a spiral bevel gear. The 3 parameters U,0 and 9, awe defined relative to the cutting head and cradle

cot - U cosI coordinate systems (S, and SJ respectively. TheseU si:, sineparameters can be transformed through a series of

cot • (1) coordinate transformationstoa coordinate system attachedU sin gr cosO to the work piece. Or U, 0. o, can be mapped into X_,

I J Y., Z,, in coordinate system S_ attached to the workpiece. These transformations, used in conjunction withtmo other geometric requirements, give the two additional

The roll angle of the cradle 0, is used to locate the equations.rotating cradle with respect to the fixed machinecoordinate system S.. Parameters U, 0 and 0., along The correct U. 0 and o. that solves the equation ofwith various machine tool settings can be used to meshing, must also, upon transformation to the '.orkcompletely define the location of a point on the piece coordinate system S., result in a axial coordinate •,cutting head in space. that matches with a preselected axial position Z. (See

figure 2)Since the kinematic motion of cutting a gear isequivalent to the cutting bead meshing with a Z - Z = 0 (4)

simulated crown gear, an equation of Meshing can bewrtten in terms of a point on the cutting head (i.e.. This equation along with the correct coordinatein terms of U, 0 and -pj. The equation of meshing transformations (see equation 11) result in a secondfc" straight-sided cutters with a constan ratio of roll equation of the formbetween the cutter and work piece is gien an"' as: (,0,)= o (5)

(U-rcot u,'cOs4ý)Cos-f si-r + S(Mt.-sin )cos 4 sin B : cos sin -y sin (q - J] A similar requirements for the radial locatioa of a point= E (cos f sin 1 -- sin f coscosr) on the work piece results in (See fig'.ure 2).- L, sin -y cos 5 sin 7 = 0 (2) ,-(x '- + = 0 (6)

"The upper and lower signs are for left and right handgears respectiely. The following machine toolsettings are defined -". The appropnate coordinate transformations (see equation

ix) will convert equation (6) into a function of U, 0, and41 cutting blade ;agle 9,7 (0 ": q ±2 p9l

q cradle angle f(eO%) = 0 (7)

'Y root anglh. of work pieceE. machining offset Equations (3). (5) and (7) form the system of nonlinear1_ vector sum of change of machMit equations necessary to define a point on the tooth surface.

center to back and the sliding basein,. ojo., the relionship betuven

the cradle and work piece for a Solution Techniqueconstant ratio of roll

U generating cone surface coordinate An initial guess U*. 0% *,* is used to start iterative solu-S radial location of cutting head in tion procedures. Newton's method is used to determine

coordinate system S. subsequent values of the updated wector (U'. 6%' o,'yr radius of generating cone surface

This is equivalent to: fUsII uk-l 1

f,(u-.-,4) 0 (3) : o - j yti- (S)

Because, there are 3 unknoows U. B. and o,; three L~1tlJ lrequations must be developed to solve for the surface

2

Where the '.ector Y is the solution of- numerical differentiation into X. - inc. Y_ - mnc -., -,-inc. Equations (4) and (6) are then used to evaluate the

aOA(Ul) afl(01-1) Bf (4i•') numerical differentiation. Function fi. f., f. and the

at au ak partial derivatives of f,, f, f3 requred to the Jacobianli.ii matrix are updated each iteration. The iteration continues

a2 ui-i) mfWe-I) af5 ('.) .. y~1

until the Y vector is less than a predetermined tolerance.au aiF ao, This completes the solution techniques for a single point

g iI) 04,(e -') af,(4"'•) 1113 on the spiral bevel gear surface.

aU au (9) The four comers of the active profile are identified fromthe tooth geometry plane as shown in figure 2. Point 1

A w-. ei i-f , on the surface is chosen to be the lowest point of theactive profile on the toe end. The initial guess to start the

- (fUi" e0''" Ie`?) procedure has to be sufficiently close to the correct(U.,t- .1-1 solution for convergence to occur. The soluton for the

'fist point proves to be an adequate initial guess for anysubsequent points on the surface.

The 3x3 matrix in the preceding equation is theJacobian matrix and must bei mierted each iteration Subsequent interior surface points are found byto solve for the Y vector. The equation of meshing. increrm-mng r = (X ,- Y-)'" and Z. By adjusting thefunction f, is numerically differentiated directly to increments used, a surface mesh of any density can befind the terms for the Jacobian matrix. Function fL calculated. The process is repeated four times for each ofand f, can not be directly differentiated with respect four surfaces, gear convex, gear concave, pinion convexto U, 0 and -p,, After each iteration U". 6ki, ,k I and pinion concave.(in the cutting head coordinate system Sc) aretransformed into the work piece coordinate system, Since aU four surfaces are generated independentl),S_ with the series of coordinate transformatons as additional matrix transformations are required to obtaingiven in equation 11. the appropriate orientation for meshing. The proper

convex and concave surface orientation (for both the gear

Y U co and pinion) is found by fixing the concave surface andJr cot i'i -1 U )5 rotating the convex surface until the correct tooth

U sin 4 sin 0 (10) thickness is obtained. The correct angle of rotation isU sin ct co e obtained y matching the tooth top land thick•ess ith the

Gear and Pinion Orientations Required For Mes-hing%here:

After generating the pinion and gear surface as describeoabove, the pinion cone and gear cone apex will meet at

[M1,,,] = the same point as shown in figure 3. This point is the

[M f(4,)) [1,.k] [M,,] [,,, I (,,] [4m J (II) origin of the fixed coordinate system attached to the workpiece being generated. To place the gear and pinion inmesh with each other rotations described in the followingexample are required:

Each matrix [M] above represents a transformationfrom on- coordinate system to another. (See I. The 19th tooth is selected for meshing. ThisAppendix I for the specific matrices.) corresponds to rotating the generated tooth 190" CW

about the 7.,, axis. (For this example, N, = 36 total teethFunctions f. and f, are eviluated b) starting with an on the gear.) For the gemeral case the gear tooth isinitial U. 0' and 0,'. performing the transformation rotated 360N, + I SIO CW about its axis of rotation (2,,).in equation II and evaluating equations (4j and (6). ThMs corresponds to selecting the ith tooth of the gear toThe numerical differentiation of f. and f4 is be in mesh wberei = N,/2- + 1.performed by transforming UL .' inc 64 + inc. 0,- mc (whiere ic is a small ncrement appropriate for 2. The pinion is rotated b) 90' CCW about the Y axis.

3

Note: rotation (1) corresponds to selecting a different This vector is written relative to S,, a coordinate systemtooth on the gear to be in mesh; however, rotation (2) fixed to the cutting head. To obtain the vector nt. relativecorresponds to physically rotating the entire pinion to the coordinate system fixed to the work piece, theuntil it meshes with the gear. following transformation must be performed.

3. Because the four surfaces are derived {n.) = [L{J(nj (13)independently, their orientation is random withrespect to meshing. The physical location of the gear Where [I_ is found by removing the 4th roA andand pinion after rotations (1) and (2) correspond to column from [,M.,].the gear and pinion in mesh with severe interference.To correct the interference the punion is rotated CW Each gap element is connected between a node on theabout its axis of rotation (Z,,) until surface contact pinion surface, hereafter designated node 1, and theoccurs. For this example the rotation was 3.56P. corresponding intersection node point on the gear surface,Figure 4 shows an example of a simulated gear pair designated node 2. The intersection point on the gearmeshing. The generated pinion tooth was copied and surface is found as follows:rotated '2 times and the generated gear tooth wascopied and rotated 36 times. Consider node 1. a point on the pinion with coordinates

P,.P., P., in S Let Q, Q, Qbe any point in spacesuch that (Q,. Q,. Qy = b(n,.n ,. n, where b is a scale

Contact Simulation factor, and n,. n, and n. are the components of the normalvector at the contact point. The intersection of the normal

The tooth pair mesh contact point can be located by vector from node I with the mating gear surface definesa method described by Litim6 or by a search node 2 and has to satisf1 the following three equations:technique. Pairs of finite element node points (oneon each tooth surface) are evaluated until the pair Q. = G, = X-b nwith smallest separation distance is obtained. (A Q, = G, = Y- b n (14)finer finite clement mesh would yield greater Q. = G. = _. + b n,resolution.) Once the contact point is established, a.ector normal to the surface at the contact point is Where G,. G,, G0 is a point on the gece surface- A pointcalculated. on the gear surface must also satisfy

The ntersectionof the normalector on the pimon at G r cos - U 00$

the contact point with the gear surface identifies the u in"second point required to define the gap element. G, (15)Additional gap elements are obtained by taking U sin ,o 0 Iadditional wectors from other pinion surface finite G.dement nodal points (parallel to the contact pointnormal sector), and calculating where they intersectthe mating gear surface. Finite element nodal points Equation 15 leads to a system of 3 nonlinear algebraicon the gear surface are located to the mterse.tion equations. These three equations, along with the equationpoints of the normal sectors and the gear tooth of meshing for the gear surface provide a s)stem of 4surface, equations and 4 unknowns (u. 0, q,, b). these equations

are solsed with Newion's method described earlier. TheThe %ector normal to the cutting surface is gisen in' intersection of the normal from node I on the pinion withas: the gear surface is now obtained.

This procedure is used to locate the intersection of

si*. 1[ normals from all points on the pinion surface (in the

(in.} = o�C s i• (12) contact zone) with the gear surface. The gear toothsurface is reineshed utilizing the inteasection points as

Cos Cos € T€c shown in figure 5. Gap elements am connected betweencorresponding nodal points on the pinion and theintersection points on the gear surface.

4

Finite Element Model concentration from the gap elements. These stress rangesbracket the estimated hertzian stresses for the gear set

An example analysts was performed using gears from under the same load conditions. Contact with only 4 gapthe NASA Lewis Spiral Bevel Gear Test Facility. In elements, along with large stress gradient among adjacentthis case, the left hand ptnion mates with the right nodes indicate .he need for a finer finite element mesh forhand gear. Counter clockwise rotation of the pinion improved stress prediction.results in the concave surface on the pinion matingwith the convex gear surface.

ConclusionsThe design data for the pinion and gear are given inTable I. The design data are used with methods A multi tooth finite element model was used to performgiven &n to determine the machine tool settings for three-dimeasional contact analysis of spiral bevel gears inthe straight sided cutter data given in Table 11. mesh. Four gear teeth and three pinion teeth are

generated by solving the equations, based on gearThe finite element gear pair model, shown in figure manufacturmig kinematics, that identify tooth surface6. contains 4 gear teeth and three pinion teeth. The coordmates. The gear and pinion are orientated formodel had 10.101 nodes (30,303 degrees of freedom) meshing with coordmate transformattions.and 7596 eight nc tad 3D brick elements.

Surface stresses are evaluated with gap elements. TheThe three pinion teeth are fixed in space with zero gear surface is remeshed with nodal points relocated todisplacements and rotations. Trus is done by setting idemtify the correct gap element orientation. Initial FEAx, y and z d:splicements -qual to zert, on the four stress results compare favorably with calculated Letcazocorner nodes af each ,im section. The gear is contact stresses. However; large stress gradients t..e'.vconstrained to .ctation about it axis of rotation The adjacent nodes in the contact zones indicate , nzed b:.-gear is lo3a•-' ;,. a tor2.Iue of 9450 in lbs on the greater finite element mesh refine-n"t.gear by at'py,, g 4- 125 lb force located 2.0976' fromthe gear..: of rctat u: References

At the orientationchosen beleen thepinionand gear L. Limin, F.L-. Theory of Geanng, NASA RP-two pairs of teeth werje in contact. One rair had 1212. 1989.contact near the middle region of the tooth and 7. Huston, R., azd Coy, J., A Basic for Analysis ofanother pair had contact n--: the too (i.e.. about to Surface Geocmetry of Spiral Bevel Gears.go out of mesh). Initially the model started w:th z Advanced Pcewr Transmission Technology,total of twenty one gap elements. For the tooth that NASA CP-2-210, AVRADCOM TR 82-C-16,is approximately midway through s,-. fii ' ýn zo G.K. Fischzr. ed., 1991, pp. 317-334.elements were used. For the tooth sb..- t " !cave 3 Tsai. Y.C., and Cairx. P.C.. Surface Geometrymesh, four gap elements were used. "I.,e analysis of S:uai.ht and Spiral Bevel Gears, J. Mech.starts with one gap dlements closed in ..,.. contact Tia, .v.,omat. Des., vol. 109, no.4, Dec.zone. Within the finite elemext ccd- an !itlat'.e 3;r7

-p; A.43-449.process is used to determine how ma=; gap eements 4 Clot'.:,r L.. and Gosselm, C., Kinematicmust close to reach static equilibriu.. The solution Aaz-!'.,-. of Bevel Gear. ASME Paper 84-DET-iterated four times before reachmg equilibnum with Vl7, •ct. 19S4.four gap element closed in the main contact region 5. Litvin, F.L, Tsuig. W.J., and Lee, H.T..a;nd one closed in the edge contact region. Stress Generation of Spiral Bevel Gears with Conjugatecontours for the tmo pinion teeth with contact are Tooth Surfaces and Tooth Contact Analysis.shown in Figures 7 an: g. NASA CR-4038. AVSCOM TR 97-C-22, 1987.

6. Litvin. F.L., and Lee. H.T., Generation andThe a,,erage nodal minmmum princpal stresses in the Tooth Contact Analysts of Spiral Bevel Gearsmain contact zone average -204,000 psi wit" a with Predesigned Parabolic Functions ofmaximum of -299.900 psi and a minimum of Transmission Errors. NASA CR-4259.-103,574 psi. The corresponding elemental stresses AVSCOM "IR S9-C-014, 19g9.average -103.500 psi with a maximum and mmum 7. Co.o. H.C., Baxter, M. and Cheng, H.S.. Aof -123.900 psi and -79.500 psi. respectively. Computer Solution for the Dynamic Load.Thenodal stresses are higher because of load Lubricant Film Thickness. and Surface

Temperatures in Spiral Bevel Gears.Advanced Power Transmission Technology. 11 0 0 0~NASA CP-2210, AVRADCOM TR-82-C- 0 cos4 ý ;sin 4, ( 016, G. K. Fischer, ed., 1981, pp. 345-364. (17)

8. Drago, R.J., and Uppaluri, B.R. Large 0 O siw$' cos4' 0

Rotorcraft Transmission Technology 10 0 0 1Development Program, Vol. 1, TechnicalReport. (D210-11972-1-VOL-1, Boeing Matrix [M.3 transforms coordinate system S. into systemVertol Co., NASA Contract NAS3-22143) S, vhich orientates the apex of the gear beingNASA CR-168116, 1983. manufactured.

9. Handschub, R.F., and Litvin, F.L., AMethod of Determining Spiral-Bevel Gear 6cos 0 -sin6 -L.sin6Tooth Geometry for Finite Element 0 1 0 -E IAnalysis, NASA TP-3096m AVSCOM TR (M,1) s]0(sc)91-C-020, 1991. (sS) 0 cos6 L.7s

10. Burden, R.L, and Faires, j.D., Numerical 0 0 0 1Analysis, 3rd edition, Prindle, Weber &

Schmidt. 1985.Matrix [M.v] transforms coordinate system S, into systemS, which located the apex of the gear being manufactures

Appendix I Vwth respect to S,

and translation require mixed matrix operations of i 0multiplication and addition. Matrix representation of [vf, = 1 cos4 0 (19)coordinate transformations will need only - 0 cots 0multiplication of matrices if position vectors a 0 0 0 1represented by homogeneous coordm2tes'. Thefollowing4 x 4 matrices are required to transform tht Matrix (M,, transforms coordinate Sa into system Swhomogeneous coordinates of a point on the cutting attached to the work piece.head to a point on the work piece. 4 0 01

Matrix [14] transforms coordinate system S,, ;I-~ cs,, 0 0attached to the cutting i'.ad, into system S,. rigidly [Mj = / n = 0 0 (20)connected to the cradle. 0 0 1 0

0 0 o 1]

1 0 0 010) cos = q smq isq (16) TABLE I - PINION AND GEAR DESIGN DATA0 ±sinq creq soosqJ S$•qMIMSO GEAR0 0 0 1 O"e• f %=% s ::iio 12 36

Dkadc.dus.k. e.it 1.5666 3.8533A&4-45.4= ,n.e d&Z 3 833 1.5666Pi:- agk, d1 S4333 1Z.,666Stift.Og•, eZ 900 90.0

Matrix [Mj transforms coordinate system S. into M= i:.U a.,nk. dsg 35.0 35.0sy.t- .F- I, idh .) 25.V(.01 23.4(1.01

sys.em 5 attached to the frame. M- .diure.= c,=. = ) 12.05(3 191) 31.05(3.191)

ts&r.'s of Z.,blNa.A.m =(.) 5.3 (0.6094) 3.0 (.3449)

Top hI..d ibikala = (,s ) 2 0M"- ( 030) 2.49 (.09M)Cak'•i .. :m C.) 0.762 (0.030) 0.92964 (0 0366)

6

TABLE II - GENERATION MACHINE SETTINGS

PINION GEAR

Concave Convex Concave Convex

Radius of cutter, r, in 2.9656 3.0713 3.0325 2.9675Blade angle, #, deg 161.9543 24.33741 58.0 22.0Vector sum, L., 0.038499 -0.051814 0.0 0.0Machine offset, E., in 0.154576 -0.1742616 0.0 0.0Cradle to cutter distance, s, in 2.947802 2.8010495 2.85995 2.85995Cradle angle q, deg 63.94 53.926 59.234203 59.234203Ratio of roll, M_ 0.30838513 0.32204285 0.950864 0.950864Initial cutter length, u, in 9.59703 7.42534 8.12602 7.89156Initial cutter orientation, 0, deg 126.83544 124.43689 233.9899 234.9545Initial cradle orientation, p., deg -0.85813 -11.38663 -0.35063 -12.3384

xcXcA A

I Xc

Yc tIinside blade (convex side).

AlA \

Yc IC

YYc

Outside blade (concave side).

Figure 1. Cutting head cone surfaces and attached coordinate systern.

7

lop land

Xw Suri•cce 9r~dHeel

Xfn Toe z"

Clecrcnice

Rba~ RO 0'c ne,

ZbarZ

Figure 2. Projection of gear tooth with XZ plane.

X Co-VEX xCONCAVEGEAR

-. --------------- -..(((6' CONCAVEj

CONVEX ' PNION----- -------- _------__

Sy \ Z

APEX

Gear after 190 deg

---- --- ---- - rotatior. about 2

P iGnion after -90 degrotation about Y

GEAR

Figure 3. Orientation of gear and pinion bzsed on solution of the system of equations, and afterrotations required for mesh.

Figure 4. Complete 3D model of gea.r and pinion in mesh.

Figure S. Distorted gear after connecting gap elements.

9

Figure 6- Seven tooth model used for finite element analysis of mesh.

-13711O

Figure 7. Stress contours in pinion tooth.

10

REPORT DOCUMENTATION PAGE o)VB No 0704-01B

1. AGENCY USE ONLY (Leýo b'k, 12. REPORT DATE 3. REPORT TYPE AND DATES COVEREDI May 1993 Technical Memorandum4TITLE AND SUBrITL.E S. FUNDING NUMBERS

Contact Stress Anal%: ts of Spiral Be%. el Gears Using Nonlinear Finite Element

Static AnalysisWVU-505-62-10

6. AUTHOR(S) 1L162211A47A

G.D. Bibel, A. Kumar, S Reddy. and R. HandSchLh

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) &. PERFORMING ORGANIZATIONNASA L.,. Rrmach Come. REPORT NUMBER

Ck, .rL• 05.a 4.4135-3191amdVzhtae PoP6ho. Dccoas. E-7976U.S A.-.) Rcssah Labemao)C"-rb.d. Oho 44!35-3191

9. SPONSORINGIMONITORL9G AGENCY NAME(S) AND AODRESS(ES) 10. SPONSORINGIMONITORING

haw," Aýro•.ý a•c d Spr.ý Ad•oinwalst~ AGENCY REPORT NUMBER

Wash-ot.. DC. 254&-6.001 NASA TM-106176

U.S Ans Reooch Laba.Ia ARL-TR-158Adcp. sh . h. 20783-1':45 AIAA-93-2296

Ii. SUPPLEMENTARY NOTES

Porpaod ft 2%1h Jo-ol Propolao Co-fo-raoo sod Eah.bs pna d by &z AIAA. SAE. A.SSE. s.d ASEE, %Wa.r-) Cad-s. Jo."co 2&-3019-3 G D Bbel. L-oroo, of Nowh Dakuto. Grd Fors. Nonh Dskoa 58201. A K).oo sod S Rod.. 14), a L of Ak=ac. A•.ro% 0, 44325.s.Od R H.•4, ho.! US Alas %'oh.cL oqk-opion DaIrrtoea. NASAL... R- r-cCootc Rasý t.bp-rýo.- R Hýs-odshi. (:16)433-3969

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13. ABSTRACT (MA.u. 200 Ords)

A procedure is presented for performin; khree-dimensional stress analsts of spiral be% el gears tI- mesh asing the finite

cieimentimiethod The proceduro. involesgcnoratinga finite element model b, solrSngeq.ations that jdentifý toothc surface

coordinatc.z CoordmatS. Uansformaiotr1S arc osed to orientate the gear and pinion for Sear meshing. Zj-o- -t boundam

conditions are simulated a.tth gap elements. A solution technique for correct orientation of the gap elements is g1 -a

Example models and results arc presented

14. SUBJECT TERMS IS. NUMBER OF PAGES

12Gears. Gear teeth: Mechanical drives [&. PRICE CODE

17. SECURITY CLASSIFICATION '&. SECURITY CLASSIFICATION 119. SECURITY CLASSIFICATION 20L ULMITATION OF ABSTRACTO.REPORT OF THIS PAGEI OF ABSTRACT

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