1、93FTM3A Rayleigh-Ritz Approach toDetermine Compliance and Root Stressesin Spiral Bevel Gears Using Shell Theoryby: Sathya Vaidyanthan, Henry Busby and Donald HouserOhio State UniversityAmerican Gear Manufacturers AssociationTECHNICAL PAPERA Rayleigh-Ritz Approach to Determine Compliance andRoot Stre
2、sses in Spiral Bevel Gears Using Shell TheorySathya Vaidyanthan, Henry Busby and Donald Houser, Ohio State UniversityThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconslz_edasanofficialactionoropinionof the American GearManufacturersAssociation.ABSTRACT:In this paper,a new m
3、athematicalmodel is proposed topredict deflectionsand root stresses in spiral bevel gears. Thetooth shape is modeled as a segment of a shear flexible thick shell of variablerigidity and height corresponding to thespiralbevel gear orpinion tooth dimensions. Theshell segmentis cantileveredalongthe cir
4、colaredge. The shellmodel ismore representative of the spiralbevel tooth geometrycompared toa beam or plate model. In addition, the model couldbe usedfor both face milled and face hobbed geomelrics. Thecompliance computationsbased on the shell model canbereadily integrated into existingcomputercodes
5、 for bevel geardesign to determine the load distribution, transmissionerror, and root stresses. The analysisis performed using the Rayleigh-Ritz approach with algebraic polynomial trialfunctions. The resultsare verifiedwithfinite element solutions. Theresults are obtainable on a personal computer an
6、dthe procedure is computationailymuch more efficient than the finite element method.Copyright 1993American Gear ManufacturersAssociation1500 King Slreet, Suite 201Alexandria, Vtrginia, 22314October, 1993ISBN: 1-55589-596-4A Rayleigh-Ritz approach to determine compliance and root stressesin spiral be
7、vel gears using shell theorySathya Vaidyanathan*Research and Development EngineerCone Drive TextronTraverse City, MI 49684Donald R. HouserProfessorDepartment of Mechanical Engineering206 West 18th Avenue, Columbus, Ohio 43210.Henry R. BusbyAssociate ProfessorDepartment of Mechanical Engineering206 W
8、est 18thAvenue, Columbus, Ohio 43210.INTRODUCTIONBevel gears are commonly used to transmit motion and this paper presents a new model for the bendingbetween angularly disposed shafts. They can be broadly compliance calculations in the tooth contact analysis programsclassified into straight bevel gea
9、rs, spiral bevel gears and and outlines its potential for bending strength calculations.hypoids. Straight bevel gears have teeth that radiate from thepitch apex while spiral bevel gears have oblique teeth on which The design of gears includes both the strengthcontact begins gradually and continues s
10、moothly from end to determination and a contact analysis which involves theend. Spiral bevel and hypoid gears are favored over straight location and movement of the contact zone under load. In spurbevel gears-in high performmaee transmissions in automobile, and helical gears,_the earliest gear stren
11、gth c-ai_mlationsweremarine and aviation industries because their curved teeth made using beam theory. Subsequently, plate models haveprovide smoother and quieter operation along with greater been successfully used in predicting deflections and stressesbending resistance. In this paper the ability o
12、f the circular 26-29 The use of the annular sector plate to predict deflectionscylindrical shell model to represent the salient features of the and stresses in straight bevel has been recently demonstrated byspiral bevel tooth geometry to determine tooth compliance and the authors.30 In spiral bevel
13、 gears however, the longitudinalroot stresses is demonstrated.curvature is significant and it is not possible to use beam orplate theory to adequately represent their flexural behavior.The early works on the meshing of bevel gears and the Hence a model based on shell theory is developed to model the
14、basic relationship of hypoid gears can be found in Wfldhaber. I- geometry and obtain solutions to the deflections and stresses in2 The work discusses in great detail the method of generation, spiral bevel gears. The spiral bevel tooth geometry can betooth profile curvature calculations and gear conj
15、ugate action, convenieiatly represented as a segment of a thick cylindricalThe geometrical characteristics and nomenclature of spiral shell by taking into account the rigidity variation along thebevel gears have been documented by AGMA (American Gear tooth height and facewidth. The effects of shear
16、deformationManufacturers Association) and others.3-5 The Gleason Works can become quite significant for small radius to thickness andPublication provide guidelines for the installation, assembly length to thickness ratios, which are both true for spiral beveland inspection of bevel gears.6-7 The geo
17、metry of spiral bevel gears. Hence the flexural behavior of spiral bevel gears isgears depends on the method of manufacture and various modeled employing shear deformation theories 31-32. Theinvestigators have attempted to characterize the geometric inclusion of variable rigidity and shear deformati
18、on effects inshape of the tooth surface and develop methods to determine the computations poses considerable mathematical difficultiesthe machine settings used to cut the gear.8-17 to solve for the deflections and stresses in closed form andnumerical solutions are sought.When computers were introduc
19、ed to the gearing industryin the 1950s, bevel and hypoid gear calculations were among The application of finite element methods in determiningits first applications. Programs were written to calculate stresses and evaluating compliance in spiral bevel gear designmachine settings, undercut and to ana
20、lyze unloaded tooth has been demonstrated by Wilcox 33. A loaded tooth contactcontact. Since then, computers have played an increasing role analysis based on the actual spiral bevel tooth surface geometryin the design, analysis, manufacture and inspection of bevel has been demonstrated by Vijayakar
21、et al.34 The proceduregears. 18-22 The loaded tooth contact analysis and the finite uses a combination of surface integral techniques and finiteelement stress model 23-25are important advances in this f_eld element methods in the contact analysis. A simulation of the1machine kinematics of the gear g
22、erierator is carried out based account the rigidity variation along the face width, theon the precomputed machine settings chosen to cut the gear. A lengthwise curvature, as well as the tooth height taper along thetheoretical tooth surface is thus calculated in the mesh face width.generator, which i
23、s then used to build the finite element model.While this procedure makes the finite element model veryaccurate, any small changes in the blank dimensions duringpreliminary design would require a recomputation of machinesettings and recalculation of the surface coordinates. Thismakes the finite eleme
24、nt method in its present formunattractive in an iterative design process. In this paper theproposed model and the method of solution is shown to haveconsiderable advantages. First, the model is based on the toothand blank dimensions rather than the machine settings. Thecomputations are based on the
25、Rayleigh-Ritz approach whichdoes not require a mesh generator or large computation times xo_zand cost. It is thus well suited to iterative computations due toits simplicity and computational efficiency. Finally the use ofshell theory to model the flexural behavior of the spiral bevelgear is shown to
26、 have comparable accuracy to the finiteelement solution.METHOD OF MANUFACTUREMost spiral bevel and hypoid gears are manufacturedeither by the face milling process or the face hobbing process.The two methods typically use different blank designs, cutting Fig.1 Spiral bevel tooth nomenclature GWPSD615
27、1Btools and contact pattern control for producing their respectivetooth geometries. Each process has its advantages and Fig. 2 shows the spiral bevel tooth shape modeled as adisadvantages and the choice of the preferred process is not .segment of a thick cylindrical shell, along with the chosenclear
28、 cut. A detailed discussion of the two types of processes coordinate system. The curved length of the shell segmentand their respective tooth geometries can be found in corresponds to the face width of the gear and the height andKrenzer35. thicknessof the shellcloselyapproximatethe beveltoothdimensi
29、ons. The shell segment is clamped along the bottomThe two manufacturing processes produce two very circular edge. Due to the small radius to thickness and lengthdifferent tooth geometries. The face milling process employs a to thickness ratios, the flexural behavior of the spiral bevel gearcircular
30、face mill cutter. In the formate process the cutter is set is modeled using shear deformation theories. The number ofinto position relative to the work such that it cuts the correct terms to be used in the Ritz series approximation is determinedspiral angle and pressure angle at the calculating poin
31、t and through a convergence study of the free edge deflection curvesweeps out the tooth form of the gear as it rotates about its axis. and the results of the static analysis verified using an ANSYSThe lengthwise tooth form is thus a circular arc of curvature finite element model. Two types of shear
32、based theoriesequal to the curvature of the cutter. The gear tooth has straight developed by Mindlin and Bhimaraddi, are used to model thesides in the normal plane. The pinion is generated conjugate to flexural behavior of the shell model and the deflections andan imaginary gear called the crown gea
33、r. The blanks are stresses compared to the finite element model.designed with tapering depth such that the tooth depth is afunction of the distance from the pitch apex. The face hobbingprocess is a conjugate generation method which employs t _!1. _continuous indexing. The lengthwise tooth curve is a
34、nextended epieycloid with teeth are of constant depth.The two processes thus produce different toothgeometries, one with varying depth and circular curvature andthe other with constant depth with the lengthwise tooth curvebeing an epicy.cloid. The model shown in this paper describes _ _/ _ vz Athe f
35、ace milled geometry. The face hobbed gear geometry can V _toot_heightbe modeled by assuming constant depth and incorporating theradius of curvature change along the face width. (o,_,_ _“ _MODEL DEFINITION AND ASSUMPTIONSThe shape and the nomenclature of a face milled spiralbevel tooth is shown in Fi
36、g. 1. It can be seen that the height ofthe tooth continually increases from the toe to the heel and thetooth is curved along the lengthwise direction. The tooth basethickness increases from the toe to the heel and the thicknessdecreases along the tooth height at any point on the face width.Hence, an
37、y model to determine the compliance should take into Fig. 2 Shell model of bevel tooth2The height h and the thickness t are assumed to vary 3W DUlinearly according to the equations 7= = _ Dz (8)(hho_ -h_) Two levels of shearbased theorieshavebeen usedin thish = h_e+ *0* rc (1) study. Thefirstset of
38、displacementassumptionsusedto modelf theshell flexuralbehavioris basedonthe itigherordersheartheory developed by Bhimaraddi.xt= t_o_+_(tap -t_)+ (thea;t_)*0*r_* 1- (2) ( 4z2 DWEqn. 1 describes accurately, the height variation of the pinion U = z_l - -:-3h_j13_(9)Z Dxand gear tooth with face width. I
39、n spiral bevel gears the gear isoften cut FORMATE (no generation) and all the profile (To+z (1 4z2- zDWcurvatureisputon thepinion.Theassumedthicknessvariation V = z - I_e (10)in Eqn. 2 is thus exact for FORMATE gears. For the pinion, _T) _ _) r, D0the assumption of the linear variation of thickness
40、with toothheight, providescomputationaladvantagesover a more W=W (11)accurate higher order assumption. The model can howeverincorporate any mathematically defined thickness variation in These assumptions allow for a parabolic variation of thethe compliance calculations, transverse shear strain throu
41、gh the thickness with zero values attop and bottom surfaces. From the displacement assumptions itcan be seen that the strain expressions will include thederivatives of the thickness with respect to the spatialSALIENT FEATURES OF THE SHELL MODEL coordinates which must be accounted for in the strain e
42、nergyexpression. The shell model includes the tooth height taper from the The second set of displacement assumptions is based ontoe to the heel. The radius of curvature along the face the Mindlin type of shear theory defined aswidth is assumed to be a constant equal to the radius ofDWthe cutter. The
43、 tooth heights at the toe and heel are U = z13_- z Dxevaluatedfromtheinputblankdimensions. (12) The rigidity (thickness) variation in both the . (r,+z_, z DWcircumferential and axial directions is taken into account v = (-_-jzpo (13)by using the actual thickn_ess at the toe and heel base and -_- r_
44、D0at the tip. A linear variation in thickness from the base tothetip is assumedalongwitha linearvariationin base W =W (14)thickness along the face width. This has worked well inother cases27. The Mindlin type of shear theory assumes a constant shearstrain through the thickness. It is however simpler
45、 toimplement as it does not have the thickness h or its derivatives Shear deformation is modeled either using Mindlin plate appearing in the strain energy expression.theory or Bhimaraddis higher order theory. TheRayleigh-Ritz solution is based on the strain energy The Rayleigh-Ritz method is used to
46、 obtain an expressionequation of the shell. The model is clamped along the for the transverse deflection and shear rotation from which thebottom circular edge. normal and shear stresses are evaluated. This method involvessetting the first variation of the total potential energy to zero.The potential
47、 energy is defined as the difference between theMATHEMATICAL FRAMEWORK strain energy of the shell and work done by the external force.Straindisplacementequation: V=SE-WF (15)The transverse deflections and shear rotations are assumed inIn the ease of the circular cylindrical shell we have the form of
48、 a finite linear combination of algebraic polynomialtrial functions which satisfy the geometric boundaryDU conditions. The assumed trial functions are substituted in the_= _-x (3) above equation. Each term of the series solution is a product oftwo expressions, one a function of the axial position x
49、and the1 (_V _ “_ other of the angular position 0. Specifically,e=- _+W(To+z)_D0 ) (4)W = _A_j_i(x)ei(0 ) (16)DW _ Je_ = _z (5) I_ = _ B_jt_(x)ej(0) (17)i JDV 1 DU7,0 = -_-x4 (r_+z) DO (6) 13o = _C_j,(x)ej(O) (18)i j1 DW DV V7_o= _ (7) To ensure convergence, the functions so chosen must satisfy(To+z) D0 Dz (To+z) thefollowingrequirements:31) (hi(x) and ej(0) must satisfy in their actual form, theessential boundary conditions associated with W, 3_and 13o.2) _)i(x) and _j(0) must be continuous (as specified by thevariational principle) and be linearly independent
