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1、Computerized design, simulation of meshing, and contact and stress analysis of face-milled formate generated spiral bevel gears Faydor L. Litvin a,*, Alfonso Fuentesa, Qi Fana, Robert F. Handschuhb a Department of Mechanical Engineering, Gear Research Center, University of Illinois at Chicago, Chica
2、go, IL 60607-7022, USA b US Army Research Laboratory, NASA Glenn Research Center, Cleveland, OH 44135, USA Received 11 April 2001; accepted 4 December 2001 Abstract A new approach for design, tooth contact analysis (TCA) and stress analysis of formate generated spiral bevel gears is proposed. The ad
3、vantage of formate generation is the higher productivity. The purposes of the proposed approach are to overcome diffi culties of surface conjugation caused by formate generation, develop a low noise and stabilized bearing contact, and perform stress analysis. The approach proposed is based on applic
4、ation of four procedures that enable in sequence to provide a predesigned parabolic function of transmission errors with limited magnitude of maximal transmission errors, a bearing contact with reduced shift of contact caused by misalignment, and perform stress analysis based on application of Finit
5、e Element Method. The advantage of the approach developed for fi nite element analysis (FEA) is the automatic generation of fi nite element models with multi-pairs of teeth. The stress analysis is accomplished by direct application of ABAQUS. Intermediate auxiliary CAD computer programs for developm
6、ent of solid models are not required. The theory developed is illustrated with an example of design and com- putation. ? 2002 Elsevier Science Ltd. All rights reserved. 1. Introduction Design and generation of spiral bevel gears is a subject of intensive research of scientists, designers and manufac
7、turers 110. The phenomenon of spiral bevel gears, generated or for- mate cut, is that the machine-tool settings are not standardized and the quality of the gears *Corresponding author. Tel.: +1-312-996-2866; fax: +1-312-413-0447. E-mail address: fl itvinuic.edu (F.L. Litvin). 0094-114X/02/$ - see fr
8、ont matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0094-114X(01)00086-6 Mechanism and Machine Theory 37 (2002) 441459 Nomenclature ag profi le angle of gear parabolic blade at mean point M ap profi le angle of pinion straight blade ci(i 1;2) pitch cone angles of pinion i 1 and gear
9、i 2, respectively cmi(i 1;2) machine root angles for pinion i 1 and gear i 2, respectively cri(i 1;2) root cone angles for pinion i 1 and gear i 2, respectively cshaft angle gi(i 1;2) angle of tangent to the path of contact on the pinion i 1 and gear i 2 surface, respectively hp, hgsurface parameter
10、s of pinion and gear head-cutters, respectively kf, kw surface parameters of pinion and gear fi llet parts of the head-cutter /i(i 1;2) angle of rotation of the pinion i 1 or gear i 2 in the process of meshing wc1angle of rotation of the cradle in the process for generation of the pinion w1angle of
11、rotation of the pinion in the process for generation qf, qw fi llet radii for the pinion and gear blades Ri(i 1;2) pinion i 1 or gear i 2 tooth surface Rk(k p;g) Pinion k p or gear k g generating surface DH,pinion axial displacement Em1 blank off set of pinion XB1sliding base of pinion XD1machine ce
12、nter to back of pinion XGmachine center to back of gear D/2/1 function of transmission errors ac parabolic coeffi cient of gear head-cutter blade bi coeffi cients of modifi ed roll for pinion generation i 1;2;3 m021second derivative of transmission function /2/1 m12gear ratio Mmean contact point nk
13、i , Nk i unit normal and normal to surface Rkrepresented in coordinate system Si Ni(i 1;2) tooth number of pinion i 1 or gear i 2 q1installment angle of pinion head-cutter Rp, Rgradial dimensions of the head-cutter at mean point for the pinion and gear Rugear-cutter radius riposition vector in syste
14、m Si(i 1;2;b1;b2;h;m1;m2;p;g) sppinion surface parameters Sicoordinate system i 1;2;b1;b2;h;m1;m2;p;g Sr1radial setting of pinion head-cutter uggear tooth surface parameter vp1 m1 relative velocity at contact point in coordinate system Sm1 Xf, Zu center distances of fi llet circular arcs of pinion a
15、nd gear, respectively 2alength of major axis of contact ellipse 442F.L. Litvin et al. / Mechanism and Machine Theory 37 (2002) 441459 (reduced noise and improved bearing contact) depends on the developed method of synthe- sis. Synthesis of formate generated gears is complicated because the gear toot
16、h surface is gen- erated as the copy of the tool surface but not as an envelope for the purpose of high productivity. The authors have proposed a new approach for the development of machine-tool settings of formate generated spiral bevel gears that enables one to keep the main advantage of the metho
17、d, its high productivity, and substantially improve the quality of the gears by reduction of stresses and improvement of the bearing contact. The proposed approach is based on application of: (i) a parabolic profi le for blades of gear head-cutter or the gear grinder in case of grinding; (ii) mismat
18、ch of piniongear generating sur- faces; (iii) application of modifi ed roll for pinion generation; (iv) simultaneous application of local synthesis and tooth contact analysis (TCA). The developed approach enables one to localize the bearing contact, adjust its orientation for a misaligned gear drive
19、 and reduce the noise and vibration. The bearing contact is localized by application of mismatched piniongear generating surfaces. Therefore the piniongear tooth surfaces are in point contact at every instant. The bearing contact is oriented longitudinally for an aligned gear drive and this enables
20、to increase the contact ratio. The theoretical bearing contact is adjusted for a misaligned gear drive by properly assigned deviation from the longitudinal direction. This enables to reduce the shift of bearing contact caused by errors of alignment. The formation of bearing contact is analyzed by fi
21、 nite element analysis (FEA) and compared with results obtained by TCA. Reduction of noise is obtained by application of a parabolic function of transmission errors and limitation of maximal transmission errors. Such a function of transmission errors is able to absorb linear discontinuous functions
22、of transmission errors caused by misalignment that are the main source of noise. A parabolic function of transmission errors is obtained by application of modifi ed roll for pinion generation. The contents of the paper cover: (1) Computational procedures of proposed design that provide an adjusted a
23、lmost longitudinal bearing contact, a predesigned parabolic function of transmission errors, the ability to inves- tigate the infl uence of misalignment, and perform stress analysis. (2) The approach to stress analysis by application of FEA is modifi ed as follows: (a) The solid model of multi-pairs
24、 of teeth with boundary conditions is generated automatically, application of intermediate CAD computer programs is not required. (b) The stresses are determined by direct application of ABAQUS computer program. (3) Algorithms of computer programs developed for synthesis, analysis of meshing and con
25、tact of gear drives are represented. (4) Bearing contact obtained by application of FEA and TCA are compared for confi rmation of the obtained output. The developed theory is illustrated with an example of computerized design of a formate cut gear drive. F.L. Litvin et al. / Mechanism and Machine Th
26、eory 37 (2002) 441459443 2. Derivation of gear tooth surface Applied coordinate systems. Coordinate systems Sm2, Sg, and S2(Fig. 1) are rigidly connected to the cutting machine, the head-cutter and the gear, respectively. Generating tool surface. A head-cutter or a grinder is applied as the tool for
27、 gear generation. The blades of the gear head-cutter are of a parabolic profi le (Fig. 2). A parabolic profi le is applied for the grinder as well. The tool surface is a surface of revolution and its rotation about the Xg-axis does not aff ect the process of generation. Installment parameters. The t
28、ool installment is determined by parameters H2and V2(Fig. 1) called horizontal and vertical settings. Parameters XGand cm2(Fig. 1) represent the gear settings. The working part of the tool surface and the tool fi llet are determined in coordinate system by Sgby the following vector functions: ra g u
29、g;hg;1 rb g kw;hg:2 Here ug;hg and kw;hg are the respective surface parameters (Fig. 2). The unit normals to tool generating surfaces Ra g and Rb g are represented by the equations: na g ug;hg Na g jNa g j ;Na g ora g ohg ? ora g oug ;3 nb g kw;hg Nb g jNb g j ;Nb g orb g ohg ? orb g okw :4 The gear
30、 tooth surface R2is a copy of the gear tool surface and can be represented in S2by the application of coordinate transformation. Fig. 1. Machine-tool settings for gear generation: (a) for left-hand gear; (b) for right-hand gear. 444F.L. Litvin et al. / Mechanism and Machine Theory 37 (2002) 441459 3
31、. Derivation of pinion tooth surface Applied coordinate systems. Fixed coordinate systems Sm1, Sa1, Sb1are rigidly connected to the cutting machine (Figs. 3 and 4). The movable coordinate systems S1and Sc1are rigidly connected to the pinion and the cradle, respectively. They are rotated about the Zm
32、1-axis and Zb1-axis, re- spectively, and their rotations are related with a polynomial function w1wc1 , if modifi ed roll is Fig. 2. Illustration of gear-cutter blades and generating surfaces. Fig. 3. Machine-tool settings applied for installment of the to-be-generated pinion. F.L. Litvin et al. / M
33、echanism and Machine Theory 37 (2002) 441459445 applied. The ratio of instantaneous angular velocities of the pinion and the cradle is defi ned as m1cw1wc1 x1wc1=xc. The magnitude m1cw1 at wc1 0 is called ratio of roll. Parameters XD1, XB1, Em1, cm1are the basic machine-tool settings for pinion gene
34、ration (Fig. 3). Head-cutter surfaces. The pinion generating surfaces are formed by surfaces Ra p and Rb p generated by straight-line and circular arc parts of the blades (Fig. 5). Surfaces Ra p and Rb p are represented in coordinate system Spby vector functions: Fig. 5. Illustration of pinion-cutte
35、r blades and generating surfaces. Fig. 4. Machine-tool settings for installment of pinion head-cutter: (a) for right-hand pinion; (b) for left-hand pinion. 446F.L. Litvin et al. / Mechanism and Machine Theory 37 (2002) 441459 ra p sp;hp;5 rb p kf;hp:6 The unit normals to Ra p and Rb p are determined
36、 by equations similar to Eqs. (3) and (4). The head-cutter is mounted on coordinate system Sc1called the cradle of the cutting machine and its installment is determined by settings Sr1and q1(Fig. 4). Pinion tooth surfaces. The pinion tooth surface Ra 1 is the envelope to the family of head-cutter su
37、rface Ra p and is represented in coordinate system S1as follows 811: ra 1 sp;hp;w;f a 1p sp;hp;w1 0:7 Here vector function ra 1 represents the family of tool surfaces Ra p ; f a 1p 0 is the equation of meshing that may be determined as na m1 ? vp1 m1 f a p sp;hp;w1 0:8 Vectors na m1 and vp1 m1 of to
38、ol unit normal and relative velocity are represented in coordinate system Sm1 . Wherein the modifi ed roll is applied in the process of generation, the rotation angles w1and wc1of the pinion and cradle are related as w1 m1cwc1 b2w2 c1 b3w3 c1; 9 where b2, b3 are the modifi ed roll coeffi cients. The
39、 pinion fi llet surface is derived similarly. 4. Basic ideas of developed approach Local synthesis. The mean contact point M is chosen on gear tooth surface R2(Fig. 6). Pa- rameters 2a, g2, and m012are taken at M and represent the major axis of the instantaneous contact ellipse, the tangent to the c
40、ontact path on gear tooth surface, and the derivative of the gear ratio function m12 x1=x2, where x1and x2are the angular velocities of the pinion and gear rotations. The program of local synthesis enables one to determine the pinion machine-tool settings considering as known the gear machine-tool s
41、ettings and parameters a, g2, and m012. The program requires solution of 10 equations for 10 unknowns but six of the 10 equations are represented in echelon form. The algorithm of local synthesis includes relations between principal curvatures and directions proposed in 810. TCA. The algorithm of th
42、e computer program is based on conditions of continuous tangency of piniongear tooth surfaces and application of nonlinear equation solver 810,12. The output of TCA enables one to determine the function of transmission errors D/2/1 and the bearing contact obtained for each iteration whereas the inpu
43、t variable parameters a, g2, and m012of the respective iteration are applied. The computational procedure is divided into four separately applied procedures performed as follows: F.L. Litvin et al. / Mechanism and Machine Theory 37 (2002) 441459447 Procedure 1. The purpose of the procedure is to obt
44、ain the assigned orientation of the bearing contact. The procedure is accomplished by the observation of the following conditions: (a) The local synthesis and TCA are applied simultaneously whereas the variable parameter is m012and parameters a and g2are taken as constants. The orientation of g2is i
45、nitially chosen as for 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 R2and determine its projection LTon plane T that is tangent to R2at M (Fi
46、g. 7). (c) The goal of the iterative process (accomplished by simultaneous application of local synthesis and TCA) is to obtain Ln T as the straight line for the process of meshing of the cycle ?p=N16/16p=N1. This goal is achieved by variation of m012and the sought-for solution is obtained analytica
47、lly as follows: (i) The numerically obtained projection Li T is represented by a polynomial function ytxt;m 0i 12 ? b0m012i b1m012 ?ix t b2m012 ?ix2 t: 10 (ii) Variation of m0i 12 (i 1;2;3;.;n) in the iterative process based on simultaneous applica- tion of local synthesis and TCA enables one to obt
48、ain such a path of contact whereas b2 0 and Ln T becomes a straight line. Fig. 7 shows various lines L1 T , L2 T and the desired shape Ln T . (iii) The iterative process is directed at obtaining b2m012i 0 and is based on the secant method 13. Fig. 6. Illustration of parameters g2and a applied for local synthesis. 448F.L. Litvin et al. / Mechanism and Machine Theory 37 (2002) 441459 Procedure 2. Procedure 1 is accomplished with obtaining Ln T as a straight line. However, the output of the TCA for the function of the transmission