Alfonso Fuentes Faydor L Litvin Gear Research Center Department of Mechanical Engineering University of Illinois at Chicago Chicago IL 60607 7022 Baxter R Mullins Ron Woods Bell Helicopter Textron Inc Forth Worth TX 76101 Robert F Handschuh U S Army Research Laboratory NASA Glenn Research Center Cleveland OH 44135 Design and Stress Analysis of Low Noise Adjusted Bearing Contact Spiral Bevel Gears An integrated computerized approach for design and stress analysis of low noise spiral bevel gear drives with adjusted bearing contact has been developed The computational procedure is an iterative process requiring four separate steps that provide a a para bolic function of transmission errors that is able to reduce the effect of errors of align ment and b reduction of the shift of bearing contact caused by misalignment Applica tion of fi nite element analysis permits the contact and bending stresses to be determined and the formation of the bearing contact to be investigated The design of fi nite element models and boundary conditions is automated and does not require intermediate CAD computer programs A commercially available fi nite element analysis computer program with contact capability is used to conduct the stress analysis The theory developed is illustrated with numerical examples DOI 10 1115 1 1481364 1Introduction Spiral bevel gears have found broad application in helicopter and truck transmissions Design and stress analysis of such gear drives is a recent topic of research that has been performed by many scientists 1 10 Reduction of noise and stabilization of bearing contact of misaligned spiral bevel gear drives is still a very challenging topic of research although machine tool manu facturers have developed analysis tools and outstanding equip ment for manufacture of such gear drives The phenomenon of design of spiral bevel gears is that the machine tool settings of spiral bevel gears are not standardized and have to be specially determined for each set of parameters of design to guarantee the required quality of the gear drives A new approach is proposed for the solution of this problem that is based on the following considerations 1 The gear machine tool settings are considered as given adapted for instance from the Gleason summary The to be determined pinion machine tool settings have to provide the ob servation of assigned conditions of meshing and contact of the gear drive 2 Low noise of the gear drive is achieved by application of a predesigned parabolic function of transmission errors of a limited value of maximum transmission error of 6 8 arcsec A prede signed parabolic function of transmission errors is able to absorb almost linear discontinuous functions of transmission errors caused by errors of alignment that are the source of high noise and vibration 4 11 The authors concept of reduction of noise has been confi rmed by test of prototypes of existing design and pro posed design accomplished at NASA Glenn Research Center The test showed that the level of noise was reduced by 18 decibels of the whole amount of 90 decibels and the vibration was reduced by 50 at the spiral bevel gear meshing frequencies and its harmon ics 4 3 The provided orientation of the bearing contact has to re duce its shift caused by the errors of alignment of the gear drive 4 The developed procedure of design is an iterative process based on simultaneous application of local synthesis and TCA Tooth Contact Analysis The local synthesis provides assigned conditions of meshing and contact at the mean contact point of tangency of pinion gear tooth surfaces The TCA computer pro gram can simulate the conditions of meshing and contact for the entire meshing process Finite element method is used for stress analysis and the inves tigation of the bearing contact A model of three contacting teeth complemented with the boundaries conditions is applied for fi nite element analysis FEA A general purpose fi nite element analysis computer program 12 has been used to conduct the stress analy sis The design of contact models is automated and does not re quire application of CAD computer programs Computer programs for synthesis analysis and automation of FEA are based on application of the same programming language Graphic representation of results of computation is obtained by application of a commercially available graphical program 2Basic Ideas of Developed Approach Local Synthesis The mean contact point M is chosen on gear tooth surface S2 Fig 1 Parameters 2a h2 and m128are taken at M and represent the mayor axis of the instantaneous contact el lipse the tangent to the contact path on gear tooth surface and the derivative of the gear ratio function m125v 1 v 2 wherev 1 andv 2 are the angular velocities of the pinion and gear rota tions The program of local synthesis enables the pinion machine tool settings to be determined considering as known the gear machine tool settings and parameters a h2 and m128 11 The program requires solution of ten equations for ten unknowns but six of the ten equations are represented in echelon form The algorithm of local synthesis includes relations between principal curvatures and directions proposed in 11 13 15 TCA Tooth Contact Analysis The computer program al gorithm is based on conditions of continuous tangency of pinion gear tooth surfaces and is illustrated in Fig 2 The TCA program enables the function of transmission errors Df2 f1 to be deter mined and the bearing contact to be obtained for each iteration whereas the input variable parameters a h2 and m128of the re spective iteration are applied The computational procedure is divided into four separately applied procedures performed as follows Contributed by the Mechanisms and Robotics Committee for publication in the JOURNAL OFMECHANICALDESIGN Manuscript received December 2000 Associ ate Editor M Raghavan 524 Vol 124 SEPTEMBER 2002Copyright 2002 by ASMETransactions of the ASME Procedure 1 The purpose of the procedure is to obtain the as signed orientation of the bearing contact The procedure is accom plished by the observation of the following conditions a The local synthesis and TCA are applied simultaneously whereas the variable parameter is m128and parameters a andh2 are taken as constant The orientation ofh2is initially chosen as a longitudinally oriented bearing contact The errors of alignment are taken equal to zero b Using the output of TCA it becomes possible to obtain numerically the path of contact on gear tooth surface S2and determine its projection LTon plane T that is tangent to S2at M Fig 1 c The goal of the iterative process accomplished by simul taneous application of local synthesis and TCA is to obtain LT n as the straight line for the process of meshing of the cycle 2p N1 f1 p N1 This goal is achieved by variation of m128 and the sought for solution is obtained analytically as follows i i The numerically obtained projection LT i is represented by a polynomial function yt xt m128 i 5b0 m128 i 1b1 m128 i xt1b2 m128 i xt 2 1 ii Variation of m128 i i51 2 3 n in the iterative process based on simultaneous application of local synthesis and TCA enables such a path of contact to be obtained whenb250 and LT n becomes a straight line Figure 3 shows various lines LT 1 LT 2 and the desired shape L T n Fig 3 iii The iterative process is directed at obtainingb2 m128 i and is based on the secant method 16 that is illustrated with Fig 4 Procedure 2 Procedure 1 is accomplished obtaining LT n as a straight line However the output of the TCA for the function of the transmission errors function Df2 n f 1 where 2p N1 f1 p N1 is of unfavorable shape and magnitude The goal is to transform Df2 n f 1 into a parabolic function and limit the mag nitude of maximum transmission errors This goal is achieved by application of modifi ed roll for pinion generation Modifi ed roll means that during the pinion generation the angles of rotation of the pinion and the cradle of the generating machine c1andcc1 respectively are related as follows c1 j c c1 5m1ccc12b2cc1 2 2b3cc1 3 2 where m1c is the fi rst derivative of functionc1 cc1 atcc150 that is obtained by application of local synthesis 4 The super script j in Eq 2 indicates that the j th iteration is considered The purpose of procedure 2 is the transformation of the func tion of transmission errors Df2 n f 1 obtained at the fi nal step of Procedure 1 This goal is achieved as follows i The local synthesis and TCA computer program are ap plied simultaneously and the errors of alignment of the gear drive are taken equal to zero ii The function of transmission errors Df2 n f 1 is the out put of TCA obtained at the n th iteration of Procedure 1 and is represented numerically We represent Df2 n f 1 as a polynomial function of the third order designated as Df2 j f 1 5a0 j 1a 1 j f 11a2 j f 1 21a 3 j f 1 3 2 p N1 f1 p N1 3 Fig 1Illustration of parametersh2andaapplied for local synthesis Fig 2Tangency of tooth surfaces of a gear drive Fig 3Projections of various path of contactLTon tangent planeT Journal of Mechanical DesignSEPTEMBER 2002 Vol 124 525 The designation j51 2 k means that an iterative process for modifi cation of Df2 n f 1 is considered Function Df2 1 f 1 Df2 n is obtained at the fi nal iteration of Procedure 1 iii The goal of Procedure 2 is to transform the function of transmission errors and obtain Df2 k f 1 52a2 k f 1 2 2 p N1 f1 p N1 4 uDf2 k f 1 umax5a2 k Sp N1D 2 5DF 5 iv The goals mentioned above are obtained by variation of coeffi cients b2 j and b3 j of the function of modifi ed roll The secant method 16 is applied for this purpose whereas variations of b2 j and b3 j are performed separately and illustrated in Fig 5 Procedure 3 Procedures 1 and 2 discussed above enable to ob tain i a longitudinally oriented projection LTof the path of contact and represent LTis a straight line ii a parabolic function of transmission errors with the assigned level of maximum trans mission errors However these results have been obtained for an aligned gear drive The goal of Procedure 3 is to reduce the shift of the bearing contact caused by errors of alignment and this is achieved by the proper change of orientation of LTassigned initially in Procedure 1 The procedure is based on the fl ow chart shown in Fig 6 Errors of alignment of the gear drive will cause the shift of the bearing contact but they will not affect the obtained function of transmission errors since it is a parabolic function that is able to absorb the linear functions of transmission errors caused by mis alignment 11 13 15 Procedure 3 is performed as follows i Computer programs developed for local synthesis and TCA are again applied simultaneously but the expected errors of align ment are simulated Fig 4Illustrationofcomputationsfordeterminationof b2 m128 Fig 5 Illustration of variation of coeffi cientsb2andb3of modifi ed roll 526 Vol 124 SEPTEMBER 2002Transactions of the ASME ii The effect of all errors of alignment on the shift of the bearing contact is investigated separately The sensitivity of the shift of LTis reduced by the proper choice of parameterh2 Fig 1 Thus the variable parameter of the local synthesis in Proce dure 3 ish2 iii Application of procedure 3 causes that fi nally LTwill be chosen as the deviated from the longitudinally one The deviation depends on the design parameters of the gear drive particularly on the gear ratio m12 see Section 6 and on the errors of align ment that are applied Procedure 4 After completion of Procedures 1 2 and 3 the ob tained pinion machine tool settings guarantee that the designed gear drive is indeed a low noise gear drive with reduced sensitiv ity to errors of alignment see numerical examples of design in Section 6 The goal of Procedure 4 is the stress analysis and the investi gation of formation of the bearing contact whereas the contact ratio is mc 1 see Section 5 The approach developed for fi nite element analysis FEA has the following advantages 1 The computer language applied for automation of FEA is the same as applied for the synthesis and analysis of the gear drive 2 The contacting model formed by three teeth of the pinion and the gear and the boundary conditions is determined automati cally There is no need in application of intermediate CAD com puter programs to develop the fi nite element models for applica tion of FEA see details in Section 5 3Derivation of Equations of Gear Tooth Surfaces Remember at this point that the machine tool settings for the gear are considered as given and the to be derived equations al low the gear tooth surfaces to be determined The head cutter is provided with blades that are rotated about the Zg axis of the head cutter Fig 7 during the process of generation Both sides of the gear tooth are generated simultaneously The profi le of the blade consists of two parts Fig 7 i of a straight line and ii of a fi llet formed by two circular arcs connected by a straight line The blades by rotation about the Zg axis form the head cutter generating surfaces Applied Coordinate Systems Coordinate system Sm2 Sa2 Sb2 are the fi xed ones and they are rigidly connected to the cutting machine Fig 8 The movable coordinate systems are S2and Sc2 rigidly connected to the gear and the cradle respectively Coordi nate system Sgis rigidly connected to the gear head cutter It is considered that the head cutter is a cone and the rotation of the head cutter about the Zg axis does not affect the process of gen eration The head cutter is mounted on the cradle and coordinate system Sgis rigidly connected to the cradle coordinate system Sc2 The cradle and the gear perform related rotations about the Zm2 axis and the Zb2 axis respectively Anglescc2andc2are related and represent the current angles of rotation of the cradle and the gear The ratio of gear roll is designated as m2c2and is determined as m2c25 v 2 v c2 5 dc2 dt 4dcc2 dt 6 The installment of the tool on the cradle is determined by pa rameters Sr2and q2 that are called radial distance and basic Fig 6Flow chart for procedure 3 Fig 7Blade and generating cones for gear generating tool a illustration of head cutter blade b y c generating tool cones for concave and convex sides Journal of Mechanical DesignSEPTEMBER 2002 Vol 124 527 cradle angle Parameters DXB2 DEm2 DXD2 andgm2represent the settings of the gear Figs 8 a and 8 b show the installment of the head cutter for right hand and left hand gears respectively Procedure of Derivation The head cutter generating surface is represented in coordinate system Sgby the vector function rg Sg ug where sgandugare the surface parameters The family of generating surfaces is represented in coordinate system S2rigidly connected to the gear by the matrix equation r2 sg ug c2 5M2g c2 rg sg ug 7 wherec2is the generalized parameter of motion The equation of meshing is determined as f2g sg ug c2 50 8 determined as 11 14 S r2 ug 3 r2 sgD r2 c2 50 9 or as 11 13 14 Ng ng g2 50 10 Here Ng sg ug is the normal to the head cutter surface repre sented in coordinate system Sgandng g2 is the relative velocity represented in Sg Equations 7 and 8 determine the gear tooth surface by three related parameters 4Derivation of Pinion Tooth Surfaces The two sides of the pinion tooth surfaces are generated sepa rately The machine tool settings applied for generation of each tooth side are determined separately by application of Procedures 1 2 and 3 mentioned above Profi le blades of pinion head cutters are represented in Fig 9 Applied Coordinate Systems Coordinate systems applied for generation of pinion are shown in Fig 10 Coordinate systems Sm1 Sa1 Sb1 are the fi xed ones and they are rigidly connected to the cutting machine The movable coordinate systems S1and Sc1 are rigidly connected to the pinion and the cradle respectively Systems S1and S2are rotated about the Zm1 axis and Zb1 axis respectively and their rotations are related by a polynomial func tionc1 cc1 wherein modifi ed roll is applied see below The ratio of instantaneous angular velocities of the pinion and the cradle is defi ned as m1c c1 cc1 5v 1 cc1 v c The magni tude m1c c1 atcc1is called ratio of roll or velocity ratio Pa rameters DXD1 DXB1 DEm1 gm1are the basic machine tool settings for pinion generation Fig 8Coordinate systems applied for gear generation a and b illustration of tool installment for generation of right and left hand gears c illustration of corrections of machine tool settings Fig 9Blades and generating cones for pinion generating tool a convex side blade b convex side generating cone c concave side blade d concave side generating cone Fig 10Coordinate systems applied for the pinion generation a and b illustration of tool installment for generation of right and left hand gears c illustration of corrections of machine tool settings 528 Vol 124 SEPTEMBER 2002Transactions of the ASME Procedure of Derivation The pinion head cutter surface is represented by vector function rp sp up Fig 9 where sp up are the surface parameters The family of head cutters is represented in coordinate system S1rigidly connected to the pinion by the matrix equation r1 sp up cc1 5M1p cc1 rp sp up 11 Unlike the case of gear generation modifi ed roll is applied for the generation of the pinion and functionc1 cc1 relates the angles of rotation of the pinion and the cradle of the pinion gen erating machine by a polynomial but not linear function see Eq 2 The equation of meshing is determined as fp1 sp up cc1 50 12 that is derived by application of two following alternative ap proaches 11 13 14 S r1 up 3 r1 spD r1 cc1 50 13 or Np np p1 50 14 where Npis the normal to pinion head cutter surface Spthat is represented in Sp Equations 11 and 12 represent the pinion tooth surfaces by three related parameters 5Application of Finite Element Analysis Application of fi nite element analysis permits the following 1 Investigation of the bearing contact when multiple sets of teeth may be in contact under load simultaneously 2 Dete