-AD-A2 -42 709

-NASA, AVSQMTechnical Memnorandum 105150- Technicalk Reort 91- C-018

Howto- Determine Spiral Bevel

Gear Tooth Geometry for! F initeI Element Analysis

'RobertF. Handschuh-Propulon DirectorateD T CU.-S.-Armyn Aviation Systems-Commiand FLC IWLewis--Research- Center 4OV13199rCleveland, Ohio

and

~Faydor L. LitiviniUniversity of Illinois -at ChicagoChicago, Illinois

ApqiOyc4 fOr publc role=;

Prepared-for-theInternational Conference on Motion and-Power Transmissionsmsonsored -by the Japan- S ociety of Mechanical- Engineers- with the'particip ation-of ASME, I. Mech. E., VDiI.E.T., -CSME_ and other Societies

'Hiroshima, Japan, -Noveme242,19

US ARMYVAVIATIONV

N ASASYSTEMS COMMAND

191-14965

Acoession For

HOW TO DETER4INE SPIRAL BEVEL GEAR TOOTH GEOMETRY NTIS GRA&IFOR FINITE ELEMENT ANALYSIS DTIC TAB -

Unannounced 5Justitricatio

Robert F. HandschuhPropulsion Directorate

U.S. Army Aviation Systems Command D 4.trtbution/

Lewis Research Center Availability CodesCleveland. Ohio jAvai2. and/or

and Dist Special

Faydor L. Litvin IUniversity of Illinois at Chicago

Chicago, Illinois

.IJUTY

ABSTRACT bevel gear-tooth surface coordinates and a three- IN-cTEDdimensional model for finite element analysis.

Ar. analytical method has been developed Accomplishment of this task requires a basic in-to determine gear tooth surface coordinates of derstanding of the gear manufacturing proess.face-milled spiral bevel gears. The method com- which is described herein by use of differentialbines the basic gear design parameters with the geometry tec.hniques Ill. Both the manufactur:ngkinematical aspects for spiral bevel gear manu- machine settings and the basic gear design datafacturing. A computer-program was developed to were used in a numerical analysis procedure thatcalculate the surface coordinates. From this data yielded the tooth surface coordinates. After thea three-dimensional model for finite element ana- tooth surfaces (drive and coast sides) were de-lysis can be determined. Development of the scribed, a three-dimensional model for the toothmodeling method and an example case are presented. was assembled. A omputer program was developed

to automate the calculation of the tooth surfaceINTRODUCTION coordinates. and hence, the coordinate for the

gear-tooth three-dimensional finite element mouel.Spiral bevel gears are currently used in all The basic development of the analytical model is

helicopter power transmission systems. This type explained, and an example of the finite elementof gear is required to turn the corner from a method is presented.horizontal engine to the vertical rotor shaft.These gears carry large loads and operate at high DETERMINATION OF TOOTH SURFACE COORDINATESrotational speeds. Recent research has focused onunderstanding many aspects of spiral bevel gear The spiral gear machining process describedoperation, including gear geometry [1-121, gear in this paper is that of the face-milled type.dynamics (13-151, lubrication (161, stress analy- Spiral bevel gears manufactured in this way aresis and measurement (17-211, misalignment (22.231. used extensively in aerospace power transmissionsand coordinate measurements [24.251, as well as (i.e., helicopter mainitail rotor transmissions)other areas. to transmit power between horizontal gas turbine

engines and the vertical rotor shaft. BecauseResearch in gear geometry has concentrated on spiral bevel gears can accommodate various shaft

understanding the meshing action of spiral bevel orientations, they allow greater freedom for over-gears f8-111. This meshing action often results all aircraft layout.in much vibration and noise due to an inherentlack of conjugation. Vibration studies [261 have In the following sections the method of de-shown that in the frequency spectrum of an entire termining gear-tooth surface coordinates will beheli.opter transmission, the highest response can described. The manufacturing process must firstbe that from the spiral bevel gear mesh. There- be understood and then analytically described.fore if noise reduction techniques are to be im- Equations must be developed that relate machineplcmented effectively, the meshing action of and workpiece motions and settings with the basicspiral bevel gears must be understood. gear design data. The simultaneous solution of

these equations must be done numerically since no

Also, investigators f18.191 have found that closed-form solution exists. A description oftypical design stress indices for spiral bevel this procedure follows.gears can be significantly different from thosemeasured experimentally. In addition to making Gear Manufacturethe design process one of trial and error (forcing Spiral bevel gears are manufactured on aone to rely on past experience), this inconsis- machine like the one shown in Fig. 1, Thistency makes extrapolating over a wide range of machine cuts away the material between the concavesizes difficult, and an overly conservative design and convex tooth surfaces of adjacent teeth simul-can result. taneously. The machining process is better illus-

trated in Fig. 2. The head cutter (holding theThe objective of the research reported herein cutting blades or the grinding wheel) rotates

was to develop a method for calculating spiral about its own axis at the proper cutting speed,

1

Cradle axis- e- Cutter axis

Cradle\ Cradle housing

I-- orkpiece

I+ ,/--- Work offset Top view

Cradlehousing

-Machine .W'otpiece

7- Cra-ll poin

N,/ -. \ ,,Root angle

Machen . ,/center -"

Sli~ngbas ~Work base /' Cu,,:er cacte

center -Cutter cen:er

Figure 1.-Machine used for spiral bevel gear tooth surface generation. Front view

Figure 2.-Orjenta:ion of workpiece to generation mac.',ery.

about its own axis at the proper cutting speed, allow rotation and translation of vectors sim-vindependent of the cradle or workpiece rotation. by muliuplying the matrix trarsformations. TheThe head cutter is connected to the cradle through method used for the coordina:e t:ransformation canan eccentric that allows adjustment of the axial be found in Refs. 1. 5. 8 to 11. and 27.distance between the cutter center and cradle

(machine) center, and adjustment of the angular To begin the process we start witn the headposition between the two axes to provide the de- cutter (Fig. (3)). In this report it is assumiedsired mean spiral angle. The cradle and workpiece that the cutters are straight-sided. The para-are connected through a system of gears and meters u and B determine the location of ashafts, which controls the ratio of rotationalmotion between the two (ratio of roll). Forcutting, the ratio is constant, but for grinding, Xc

it is a variable. xC

AComruter numerical controlled (CNC) versions ,

of the-cutting and grinding manufacturing pro-cesses are currently being developed. The basickinematics, however, are still maintained for the

generation process; this is accomplished by the 0, ZCCNC machinery duplicating the generating motion Vb - 8through point-to-point control of the machining

k mt hp i t t o inter con trol lo ma ciin ng fo he4surface and location of the workpiece. Insde blade (convex side):

Coordinate Transformations, u.IBIThe surface of a generated gear is an •

envelope to the family of surfaces of the head xccutter. In simple terms this means that the 0,points on the generated tooth surface are points

of tangency to the cutter surface during manu- - -facture. The conditions necessary for envelope Ko0 >'"existence are given kinematically by the equation Yc ,B X- C

of meshing. This equation can be stated as fol- 0 0lows: the normal of the generating surface must Yc /be perpendicular to the relative velocity betweenthe cutter and the gear-tooth surface at the point

in question j1.V Vab A

The coordinate transformation procedure that ;/will now be described is required to locate anypoint from the head cutter into a coordinate Outside blade (concave side);system rigidly attached to the gear being manu- ufactured. Homogeneous coordinates are used to Figure 3-Head cutler cone surfaces.

2

current point -in the cutter coordinate system S . xP xcAngles fib and ob are the blades that cut the rse X/convex and concave sides of the gear tooth respec-tively. Thus r-- is given by the followingequation: W(c)

rcot -u Cos - _- u sinl sinO (1) 0

0. 0

'.w •Om- 10u sink-0cos 9

Once we have -C we then transform from one- coor-dinate system to the next for the coordinate sys- za.z.tems as shown in Figs. (4) and (5) J2I.. Top view

Using matrix transformations we can determinethe coordinates in S. of a point on the generat-ing surface by:

- (2) /)

Yp. Ya. Yw YM O0

Here (M. describes -the required homogeneous[t iiN, wcoordinate transfornation from system "j" to s/ 2system "i". Therefore Eq. (2) describes the 0 O' z"location-of a point in the gear fixed coordinate E I 2Wsystem based on machine settings L.. Ea. q. s. r,t. and gear design information p9 and- 6. At Lthis point the machine settings and the geardesign values -are known. Parameters u.- . and FrontviewOc are the unknown variables that are solved for FigureS.-Topardftontvievof ighthandgearsurace enecatcnnumerically. (oc . 0 shown here).

Tooth Surface Coordinate Solution ProcedureIn order to solve for the coordinates of a items must be used simultaneously: the trans-

spiral bevel gear-tooth surface, the following formation process, the equation of meshing, andthe basic gear design information. The trans-

y$ Ym formation process described previously is used todetermine the location of a point on the head cut-ter in coordinate system S,. Since there are

._.._zS three unknown quantities (u. 0. and 0c). threeequations relating them must be developed.

Values for u, 0, and 0 are used to satisfythe equation of meshing given by Refs. ! and 9.

; o i 0 (3)

where - is the normal vector to the cutter andworkpiece surfaces at the specified location ofinterest, and V is the relative velocity betweenthe cutter and workpiece surfaces at the specifiedlocation.

tC Gear design information is then used toLefthand member: s TOc establish an allowable range of values of the

radial (F) and axial (E) positions that are knownYc zC to exist on the gear being generated. This is

shown in Fig. 6.

Ym Y s 0 First the equation of meshing must be satis-fied. This is given as I]:

(. - rcot cos )cos sin(0 q _ 0,)q

-+ sf(j.) - siny)costsinO ; cos7sin sin(q -

Zs-hh membe . E(coo7oin + sincoocos( ; q tFigure 4.-OSenation of cutter cradle. and fixed Coordinate i q - 0

systems. Sc. Ss. and SM respectively. - (0encosn~

3

In suwnary the procedur is as follows.Xa. Xw Known loce ions E and i r4 the active profile

-o xC are used along with the equat.on of meshing toPoint P determine .unknown parameters u, 0, and 0.. The

parameter va;ues. machine tool -ettings, and geardesign values are used in the coordinate trans-_Zp formation shown earlier to find -. ? radial and

Zm axial positions in the gear fixed ,:nrdinatesystem S.. This procedure results in the solu-tion of three simultaneous nonlinear algetraicequations that are solved numerically.

ViewA APPLICATION OF SOLUTION TECHNIQUE

An application of the techniques previouslyP discussed will now be presented. The component to

be modeled was from the NASA Lewis Spiral BevelGear Test Facility. A photograph of the spiralbevel gear mesh is shown in Fig. 8. and the designdata for the pinion member are shown in Table i.

View AFigue 6.-Oientation of gear tc be genenaed with assumed postionsr and z.

The upper and lower signs preceding the 7-above terms pertain to left and right hand gearsrespectively.

The axial position must match the value found 7from transforming the cutter coordinates S, toworkpiece coordinates S.. Zhis is satisfied bythe following (Fig. 6): % V 14

Z. (4)

Finally the radial location from the workaxis of rotation must be satisfied. This isaccomplishe,. by using the magnitude of the loca-tion in question in the x, - y. plane (Fig. 6):

-x2 + 2)0.5 (

Now a system of three equations (Eqs. (3, to(5)) is solved simultaneously for the three para-meters u, 9. and 0.. for a given gear design with Aa set of machine tool settings. These are non-linear algebraic equations that can be solvednumerically with cmmercially available mathe- 717matical subroutines. These equations are then 0-7-117solved simultaneously for each location of in-terest along the tooth flank, as shown in Fig. 7. Figure8.-PhotographofNASAspralbevelFrom the surface grids, the active profile geartestrigcomponents.(working depth) occupied by a single tooth isdefined. Surface Coordinate Calculation

Using Figs. 7 and 9 as references, we willdescribe the calculation procedure for surfacecoordinates. First. the concave side of the tooth

Aung Addendum is completely defined before moving to the convexne /side. These points are calculated by starting at

depth- the toe end and at the lowest point of activeprofile. Nine steps of equal distance are usedfrom the beginning of the active profile to theface angle (addendtam) of the gear tooth, and then

learanc ," back to the next axial position (Fig. 9). Theconcave", procedure is repeated until the concave side iss'idelloPe 1% completely described. Then the same procedure isSedPndim-nt s followed for the convex side.10 Points-1 " '

-Oededum Examole Model and ResultsFigure 7.-Calculation points (100 each side) for concave and From the one-tooth modeL described earlier

convexsidesoltoothsurface. the analysis techniques can be demonstrated. The

4

TABLE 11 XAMPLE CASE FOR SURFACE COORDINATE GENERATION

Gear design-data J-

-Number of teeth (pinion, gear). ............... 12. 36-Dedendum- angle, 65. deg. ......................Addendum-angle, deg ..................... 3.883-Pitch angle. 1p. deg.. ................... 18.433-Shaft angle. deg .... ................... 90.0-Mean spiral angle. deg .... ................ 35.0Face width. mmn. (-in.) .................. 25.4 (1.0)-Mean cone distance. ar.n (in.) ........... .....05 (3.191)Inside radius of gear blank. mm (in.)........15.3 (0.6094)Top land thickness. mm (i;n.) ......... .... 2.032 (0.080)Clearance min (in. ) 0 . .762 (0.030)

4Generation machine settings

Concave -Convex -

Rudius of cutter. r, mnm (in.) 75.222 (2.9615) 78.1329 (3.0761)Blade angle. ~.deg 161.358 24.932Vector sum. La mm. in. 1.0363 (0.0408) -1.4249 (-!i.0561j-Machine offset. E. =m (in.) 3.9802 (0.1567) -4.4856 (--.. 766-Cradle to cutter distance.

s. rmn (in.) 74.839 -2.9646) 71.247 (2 3050,-Cradle angle. q. deg - 64.01 53.a2

X. Surface cocrcnate%calculation grid

Sm ~-Hel', -Tcp

Iland

r,:. cone-Izw

r- oot angle L7 75.- dedendurn angle 74- addendum angler,- inside radius of gCei lank~-

Figure 9 -Crcss section of calculaaion grid.

.724 LlPa (250 kst)pressure~ loao

normal to gstr

-Fixed surface

x' Fixed surface

Figure 10.-Boundary conditionis for the constant fileolrool radius model for the example application,

model shown in Fig. 10 is that for a constant shown in Fig. 10. A 1724-HPa (250 ksi) constantfillet and root radius (also called full fillet) pressure load was applied normal to the toothmodel. The fillet and root radius on the convex surface of nine elements, and the two edge sur-side has been added along with the tooth section fazes of the gear rim had all degrees of freedom(without the tooth) to make the-model symmetric constrained.about the tooth centerline. Figure 10 shows ahidden line plot of the finite element mesh-with The results were calculated by-MSCIfIASTRANeight-noded isoperLmetric three-dimensional solid and were subsequently displayed by PATR.AN. F.Ig-continuum elements. This model has 765 elements ure II shows -the major princ,ple stresses tor theand 1120 nodes. The boundary conditions are also boundary conditions shown in Fig. 10.

ps MPa

-34000 A -234-22000 B-10000 C

2000 D~ ~~fl ~14000 E

- ~26000 F :79~iL~*.~38000 G

50000 H~ 62000 1

744000 J 510Z. 86000 K

'O'-- 98000 LT-~i 110000 .1. 759

Figure 11..-Ma,r pdriple stress for te boundary cordizion, speclied in figure 10.

SU2ARY OF RESULTS 3. Huston, R. and Coy. j.. *A Basis for theAnalysis of Surface Geometry of Spiral Bevel

A method has been presented that uses dif- Gears.' Advanced Power Transmission Tech-ferential geometry techniques to calculate the nology. NASA CP-2210. AVRADCOM TR 82-C-16.surface coordinates of face-milled spiral bevel G.K. Fischer. ed.. 1981. pp. 317-33&.gear teeth. The coordinates must be solved fornumerically by a simultaneous solution of non- 4. Chang. S.11., Huston. R.L., and Coy. J.J..lint.ar algebraic equations. These equations "Computer Aided Design of Bevel Gear Toothrelate the kinematics of manufacture to the gear Surfaces.- Proceedings of the 1989 inter-design parameters. Coordinates for a grid of national Power Transmission and Gearingpoints are determined for both the concave and Conference. Sth. Vol. Z. ASME. 1989.convex sides of the gear tooth. These coordinates pp. 585-591. (Also, NASA TH-101-49).are then combined to form the enclosed surface ofone gear tooth. A computer program. was developed 5. Litvin. F.L., et al., 'Method for Generationto solve for the gear-tooth surface coordinates of Spiral Bevel Gears Wi:h Conjugate Gearand provide input to a three-dimensional geometric Tooth Surfaces.* J. Mech. Trans. Automat.modeling progran. ,i.e., PATRAN). This enables an Des.. vol. 109. no. 2. June 1987.analysis by the iinite element method. An example pp. 163-170.of the technique was presented.

6. Tsai. Y.C. and Chin, P.C.. 'Surface GeometryREFERENCES of Straight and Spiral Bevel Gears.* J.

1ech. trans. Automat. Des., vol. 109, no. 4.1. Litwin. F.L , "Theory of Gearing." NASADe.18.p.4-49RP-112. 989.Dec. 1987, pp. 443-449.RP-1212, 1989.

2. Litvin, F.L. and Choy, J. *Spiral Bevel Gee- 7. Clouiter. L. and Gasselin. C., Kinematic Ana-io 'lysis of Bevel Gears. ASHE Paper 84-DET-177.

mesry and Gear Train Precision.' Advan:ed Oct. 1984.Power Transmission Technology, NASA CP-2210.AVRADCOH TR 82-C-16, G.K. Fischer. ed., 1981.pp. 335-34i4.

6

8. Litvin, F L.. Tsung, W.J.. and-Lee. H.T.. 18. Oswald. F.B.. 'Gear Tooth Stress Measurements'Generation of Spiral Bevel Gears With on the UH-60A Helicopter Transmssion.' NASAConjugate Tooth Surfaces and Tooth Contact TP-2698. 1987.Analysis.'- NASA-CR-4088. AVSCOM-TR 87-C-22,1987. 19. Pizzigati. G.A. and Drago. R.J.. 'Some Pro-

gress in the ASccurate Evaluation of Tooth9. Litvin. F.L. and Lee. H .T., *Generation and Root and Fillet Stresses in Lightweight.

Tooth Contact Analysis of Spiral Bevel Gears Thin-Rizned Gears.'- AGMA Paper 229.21.With Predesigned Parabolic Functions of American Gear Manufacturers Association.Transmission Errors.'- NASA CR-4259. AVSCOM- Oct. 1980.TR-89-C-014. 1989.

20. Wilcox. L.E.. 'Analyzing Gear Tooth Stress as10. Litvin. F.L. * et al.. ' Transmission Errors and a Function of Tooth Contact Pattern Shape and

Bearing Contact of Spur. Helical and Spiral Position.' Gear Technology. vol. 2. no. 1.Bevel Gears.'* SAE Paper 881294, Sept. 1988. Jan./Feb. 1985. pp. 9-15.23 (also. AGIIA Paper

229.25. 1982).11. Litvin. F.L. * et al.. "New Generation Methods

for Spur. Helical. and-Spiral Bevel Gears." 21. Drago. R.J. and Uppaluri. B.R.. 'Large Rotor-NASA TM-88862. AVSCOM-TR-86-C-27. 1986. craft Transmission Technology Development

Program.' Vol. 1. Technical Report. (0210-12. Exact Determination of Tooth Surfaces for 11972-1-VOL-i. -Boeing Vertol Co.: NASA

Spiral Bevel and Hypoid Gears. Gleason Contract IIAS3-22143) NASA CR168116. 1983.Works. Rochester. NY. Aug. 1970.

22. Baxter. M.L. * Jr.. -Effect of Misalignment on13. Mark. W.D.. *Analysis of the Vibratory Excita- Tooth Action of Bevel and Hypoid Gears.'-

tion Arising From Spiral Bevel Gears.'- NASA ASME Paper 61-MD-20. Mar. 1961.CR-4081. 1987.

23. Spear. G.M. and Baxter. M.L.. 'Adjustmenc14.. Kahraman. A.. et al., *Dynamic Analysis of Characteristics of Spiral Bevel and Hypoid

Geared Rotors by Finite Elements.* Pro- Gears.* ASME Paper 66-M4ECH-17. Oct. 1966.ceedings of the !989 International PowerTransmission and Gearing Conference. 5th. 24. Frint. H.. *Automazed Inspection and PrecisionVol. 1. AS. 1989, pp. 375-382. (also. NASA Grinding of Spiral Bevel Gears.'- NASA CR-TM-102349). 4083. AVSCOM-TR-87-C-11. 1987.

15. Drago. PR. and Margasahayam. R.. -Analysis of 25. Litving. F.L.. et al.. 'Computerized inspec-the Resonant Response of Helicopter Gears tion of Gear Tooth Surfaces.- NASA TM-With the 3-D Finite Element Method.*- Pre- 102395. AVSCOM-TR-G9-C-011. 1989.sented at the 1988 MSCINASTRAN world User'sConference. Mar. 1988. 26. Lewicki. D.G. and Coy. J.J.. 'Vibration

Characteristics Of OH-58A Helicopter Main16. Chao. H.C.. Baxter. M., and Cheng. H.S., 'A Rotor Transmission.' NASA TP-2705. AVSCOM-

Computer Solution for the Dynamic Load. Lub- TR-86-C-2. 1987.ricat Film Thickness, and Surface Tempera-tures in Spiral Bevel Gears." Advanced Power 27. Handschuh. R.. and Litvin. F.L.. 'A Method forTrans-mission Tecl-nology. NASA CP-2210. AVPA- Determining Spiral Bevel Gear Tooth GeometryDCOM TR- 82-C-16. G.K. Fischer, ed.. 1981. for Finite Element Analyis. - NASA TP-3096.pp. 345-364.. USAAVSCOM-TR-91-C-020. 1991.

17. Winter, H. and Paul. H.. 'Influence of Rela-tive Displacements Between Pinion and Gear onTooth Root Stresses of Spiral Bevel Gears."-J. Mech. Trans. Automat. Des.. vol. 107.no. 1. Mar. 1985, pp. 43-48.

7

NASAReport Documentation PageSpace Adminiaration

1. Report No. _NASA TM-105150 2. Government Accession No. 3. Recipients Catalog No.

4. Title and Subtitle 5. Report Date

How to Determine Spiral Bevel Gear Tooth Geometry forFinite Element Analysis

6. Performing Organization Code

7. -Author(s) 8. Perfor ming Organization Report No.

Robert F. Handscliuh and Faydor L. Litvin E-6433

10. Wlork Unit No.9. -Performing Organization Name and Address 505-63-36

NASA Lewis Research Center IL16221IA47ACleveland, Ohio 44135-3191 ________________

and 11. Cortrator 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 Memorndum,National Aeronautics and Space Administration________________Washington, D.C. 20546-0001 14. Sponsoring Agency CodeandU.S. Army Aviation Systems CommandSt. Louis, Mo. 63120-1798

15. Supplementary Notes

Prepared for the International Conference on Motion and Power Transmissions sponsurcd by the-Japan Society ofMechanical Engineers with the participation of ASME, 1. Mech. E.,-VDI, I.E.T., CSME and other societies, Hiroshima,Japan, November 24 - 26, 199 1. Robert F. Handschuh, Propulsion Directorate, U.S. Army Aviation Systems Command;Faydor L. Litvin, University of Illinois at Chicago, Chicago, Illnois. Responsible pcrson, Robert F. Handschuh, (216)433-3969;-

16. Abstract

An analytical method has beenr developed to determine gear tooth surface .oordinatces of fac-milled:spiral bevel-years. Themethod combines the basic gear design parameters with the kinenlati-al aspects for spiral bevel gearmanufacturing. A computer program was developed to calculate tile surface coordinates. Fromt this data a three-dimensional model for finite element analysis can be determined. Development of the modeling method and anexample-case are presented.

117. Key Words (Suggested by Author(s)) 18. itbuonaamt

Gears Ucasfe niie

Mechanical drivesFinite element analysis

19. Secuirity Classif. (of tho report) 20. Security Classif. (of this page) 121. No. of pages 122. Price*

Unclassified Unclassified A

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