Optimization of Gear Design and Manufacture. Vilmos SIMON *

Transcription

1 7 International Conference on Mechanical and Mechatronics Engineering (ICMME 7) ISBN: timization of Gear Design and Manufacture Vilmos SIMN * Budaest Universit of Technolog and Economics, Facult of Mechanical Engineering, Deartment of Machine and Product Design, Budaest, Műegetem rk. 3, Hungar *Corresonding author Kewords: Gears, Design, Manufacture, Machine tool setting, Tool geometr, timization. Abstract. In this stud a method is roosed for the otimization of the design and manufacture of gears. timal modifications are introduced into gear tooth surfaces in order to imrove the oerating characteristics of the gear air, namel, to reduce the tooth contact ressure and the transmission errors, and to decrease the sensitivit of the gear air to errors in tooth surfaces and to the relative ositions of the mating members. The otimal modifications of gear tooth surfaces are introduced b the alication of the adequate machine tool settings and tool geometr. An otimization methodolog is alied to sstematicall define otimal tool geometr and machine tool settings to simultaneousl minimize tooth contact ressures and angular dislacement error of the driven gear. The use of a CNC gear manufacture machine makes it ossible to erform nonlinear correction motions for cutting the otimized gears. Effectiveness of the method was demonstrated b using a siral bevel gear examle. Significant reductions in the imum tooth contact ressure and in the transmission errors were obtained. Introduction Numerous research works were directed towards the determination of otimal tooth surface modifications in different tes of gears and of the corresonding machine tool settings to introduce these modifications. The goal is to imrove load distribution, to reduce tooth contact ressure and transmission errors, and to avoid edge contact of tooth surfaces. Because of the limited length of the aer, onl art of these research works is referenced [-8]. To achieve imum life in a gear set, aroriate bearing attern location with low tooth contact ressure and low loaded transmission error must coexists. Loaded transmission error is the rimar source of noise and vibration. The imum tooth contact ressure and transmission error deend substantiall on tooth geometr. In order to reduce the tooth contact ressure and the transmission errors, and to decrease the sensitivit of the gear air to errors in tooth surfaces and to the relative ositions of the mating members, carefull chosen tooth surface modifications are usuall alied to the teeth of one or both mating gears. As a result of these modifications, a oint contact relaces the theoretical line contact of the full conjugated tooth surfaces. These modifications are introduced into the gear tooth surfaces b aling the aroriate machine tool setting for the manufacture of the inion and the gear and/or b using a tool with otimized geometr. The new CNC gear manufacture machines have made it ossible to erform varing correction motions during the cutting of gears. In this aer, a method is resented to determine otimal tool geometr and otimal olnomial functions for the conduction of machine tool setting variation in gear teeth finishing simultaneousl reducing imum tooth contact ressure and transmission errors. The develoed otimization rocedure relies heavil on the loaded tooth contact analsis for the rediction of imum tooth contact ressure and transmission errors. The load distribution and transmission error calculation method emloed in this stud was develoed b the author of this aer [9-]. The otimization is based on machine tool setting variation on the cradle-te generator conducted b otimal olnomial functions and on otimal tool geometr. In the second ste an algorithm is develoed for the execution of motions on the CNC hoid generator using the relations on the cradle-te machine. 59

2 Effectiveness of the method was demonstrated b using a siral bevel gear examle. Significant reductions in the imum tooth contact ressure and in the transmission errors were obtained. The timization Procedure An otimization method is alied to sstematicall define otimal head-cutter geometr and machine tool settings to simultaneousl minimize imum tooth contact ressure and angular dislacement error of the driven gear. The otimization roblem to be solved is as follows: ( m) φ ( m) min f ( m ) = min c + cφ () m m φ subject to C ( m ) = where C ( m) reresents the constraints, and φ are the imum tooth contact ressure and transmission error obtained for the initial values of manufacture arameters; c and c φ are non-negative weight coefficients, exressing the relative imortance of and φ, resectivel. The imum ressure and the imum dislacement error of the driven gear, m m, are functions of the manufacture arameters, ar i : ( ) and ( ) φ ( m ) ( ar, ar, ar... ar ) = 3 N ( m ) φ ( ar, ar, ar... ) φ = 3 arn () where N is the number of manufacture arameters. The new CNC gear manufacture machines have made it ossible to erform varing correction motions during the cutting of gears. The variations of the manufacture arameters on such a machine (excet the arameters describing the tool geometr) ma be conducted b olnomial functions of fifth-order: i i ( ψ ψ ) + c ( ψ ψ )... + c ( ψ ψ ) 5 ar = c + c (3) i i5 where ψ is the arameter of relative motion of the tool and the gear in tooth surface generation and i =...N mts, where N mts is the number of machine tool settings. The otimization roblem formulated according to Eqs. () is a nonlinear constrained otimization roblem. Functions f ( m) and C ( m) are not available analticall, the exist numericall through the load distribution calculation. Therefore, the comuter simulation of load distribution must be run, reeatedl, in order to comute the various quantities needed b the otimization algorithm. The load distribution calculation is based on a highl nonlinear sstem of equations. Because of that an aroximate and iterative technique is used to erform the load distribution calculation [9-]. This causes that the calculation of artial derivatives for gradient-based otimization algorithms to be quite imractical. For this reason, a nonderivative method is selected to solve this articular otimization roblem. ne of the direct search methods described in Ref. [3] can be adoted. Here, the Hooke and Jeeves attern search method [4] is used. 6

3 α ν ψ Examle: timization of the Design and Manufacture of a Siral Bevel Gear Pair A comuter rogram was develoed to imlement the formulation rovided above. B aling this rogram the otimal tool geometr and machine tool settings were calculated and the corresonding functions were develoed for the execution of motions on the CNC hoid generator using the relations on the cradle-te machine. The main design data of the examle siral bevel gear air used in this stud are: numbers of teeth 3 and 5, module 5 mm, face width 3 mm, mean siral angle 35 deg. m m Cutter T δf Gear α M r t θ r u M γ δ δk b f x x z m m ψ c e ω (c) m c x m z z g z ω () f x ψ x x z z Figure. Machine tool setting for inion teeth finishing. x T r α T r rof T h TM x r fil x T r t P M T r rof T T Figure. Head-cutter rofile for inion teeth finishing. 6

4 The tooth surface modifications are introduced into the inion b the following machine tool settings (Fig. ): sliding base setting, c, machine center to back, f, blank offset, g, machine root angle, γ, velocit ratio in the kinematic scheme of the machine tool for the generation of the inion tooth surface, i g, and b the tool arameters: the radii of the head-cutter blade rofile, r rof and r rof (Fig. ), and the difference in head-cutter radii for the manufacture of the contacting tooth flanks of the inion and the gear, rt. Therefore, the number of manufacture arameters N = 8 and the number of machine tool settings N mts = 5. The load distribution calculation was erformed for instantaneous ositions of the inion and the gear rolling through a mesh ccle. The tooth contact ressure distributions along the otential contact lines for instantaneous ositions and for all the adjacent tooth airs engaged for a articular osition of the mating members, for the case when no modifications are introduced into the inion teeth, namel straight-lined head-cutter rofile and the basic values of machine tool settings are alied, are shown in Fig. 3. In this case the inion and gear tooth surfaces are full conjugate. The obtained imum tooth contact ressure is 55 MPa and the imum angular dislacement error of the driven gear is.3 arcsec. The tooth contact ressure distribution for the case when the inion teeth are manufactured b the head-cutter of otimized geometr and b otimal variation in machine tool settings governed b equation (3), is shown in Fig. 4. It can be observed that the imum tooth contact ressure is reduced to = MPa and the imum transmission error to φ = 3.63arc sec. Figure 3. Tooth contact ressure distribution when the inion and gear tooth surfaces are full conjugate. Figure 4. Tooth contact ressure distribution when the inion teeth are manufactured b the head-cutter of otimized geometr and b otimal variation in machine tool settings. 6

5 Conclusions A method for the otimization of the design and manufacture of gears is roosed. The otimal tool geometr and machine tool settings are determined to introduce the otimal tooth modifications into the teeth of gears in order to reduce the tooth contact ressure and transmission errors. The method is based on machine tool setting variation on the cradle-te generator conducted b olnomial functions of fifth-order. An algorithm is develoed for the execution of motions on the CNC hoid generator using the otimal relations on the cradle-te machine. Effectiveness of the method was demonstrated b using a siral bevel gear examle. n the basis of the obtained results it can be concluded that b aling the head-cutter of otimal geometr and the otimal variation in machine tool settings, drastic reductions in the imum tooth contact ressure of 6% and in the transmission error of 7% were obtained. References [] V. Simon, timal Tooth Modifications for Sur and Helical Gears, ASME Journal of Mechanisms, Transmissions and Automation in Design, (989) [] E. Mermoz, J. Astoul, M. Sartor, J.M. Linares, A. Bernard, A New Methodolog to timize Siral Bevel Gear Toograh, CIRP Annuals Manufacturing Technolog, 6 (3) 9-. [3] J. Astoul, E. Mermoz, M. Sartor, J.M. Linares, A. Bernard, New Methodolog to Reduce the Transmission Error of the Siral Bevel Gears, CIRP Annuals Manufacturing Technolog, 63 (4) [4] J.K. Jiang, Z.D. Fang, High-rder Tooth Flank Correction for a Helical Gear on a Six-Axis CNC Hob Machine, Mechanism and Machine Theor, 9 (5) [5] R. Tan, B. Chen, C. Peng, General Mathematical Model of Siral Bevel Gears of Pure-Rolling Contact, Proceedings of the Institution of Mechanical Engineers Part C - Journal of Mechanical Engineering Science, 9 (5) [6] Z.H. Fong, G.H. Chen, Gear Flank Modification Using a Variable Lead Grinding Worm Method on a Comuter Numerical Control Gear Grinding Machine, ASME Journal of Mechanical Design, 38 (6) Art. No [7] M. Wasif, Z.C. Chen, S.M. Hasan, Determination of Cutter-Head Geometr for the Face-Milling of Hoid Gears, The International Journal of Advanced Manufacturing Technolog, Februar (6) -. [8] W. Guo, S. Mao, Y. Yang, Y. Kuang, timization of Cutter Blade Profile for Face-Hobbed Siral Bevel Gears, The International Journal of Advanced Manufacturing Technolog, 85 (6) 9-6. [9] V. Simon, Load and Stress Distributions in Sur and Helical Gears, ASME Journal of Mechanisms, Transmissions and Automation in Design, (988) 97-. [] V. Simon, Load Distribution in Hoid Gears, ASME Journal of Mechanical Design, () [] V. Simon, Load Distribution in Clindrical Worm Gears, ASME Journal of Mechanical Design, 5 (3) [] V. Simon, Load Distribution in Siral Bevel Gears, ASME Journal of Mechanical Design, 9 (7) -9. [3] T.G. Kolda, R.M. Lewis, V. Torczon, timization b Direct Search: New Persectives on Some Classical and Modern Methods, SIAM Rev., 45 (3) [4] R. Hooke, T.A. Jeeves, Direct Search Solution of Numerical and Statistical Problem, J. Assoc. Comut. Mach., 8 (96)