Design and Manufacture of Spiral Bevel Gears with Reduced Transmission Errors

A method for the determination of the optimal polynomial functions for the conduction of machine-tool setting variations in pinion teeth finishing in order to reduce the transmission errors in spiral bevel gears is presented. Polynomial functions of order up to five are applied to conduct the variation of the cradle radial setting and of the cutting ratio in the process for pinion teeth generation. Two cases were investigated: in the first case the coefficients of the polynomial functions are constant throughout the whole generation process of one pinion tooth-surface, in the second case the coefficients are different for the generation of the pinion tooth-surface on the two sides of the initial contact point. The obtained results have shown that by the use of two different fifth-order polynomial functions for the variation of the cradle radial setting for the generation of the pinion tooth-surface on the two sides of the initial contact point, the maximum transmission error can be reduced by 81%. By the use of the optimal modified roll, this reduction is 61%. The obtained results have also shown that by the optimal variation of the cradle radial setting, the influence of misalignments inherent in the spiral bevel gear pair and of the transmitted torque on the increase of transmission errors can be considerably reduced. INTRODUCTION The traditional cradle-type hypoid generators are evolved into computer numerical control (CNC) hypoid generating machines, as are the Gleason Phoenix series and the Klingelnberg universal spiral bevel gear generating machine. These new CNC hypoid generators have made it possible to perform nonlinear correction motions for the cutting of the face-milled spiral bevel and hypoid gears. The following references summarize the proposed free-form cutting methods. Local synthesis of spiral bevel gears with localized bearing contact and predesigned parabolic function of a controlled level for transmission errors was proposed by Litvin and Zhang [1]. The theory of modified roll, the variation of cutting ratio in the process for pinion teeth generation was introduced. In the paper published by Argyris et al. [2], a computerized method of local synthesis and simulation of meshing of spiral bevel gears with pinion tooth-surface generated by applying a third-order function for modified roll was presented. Fuentes et al. [3] and Litvin and Fuentes [4] have developed an integrated computerized approach for the design and stress analysis of low-noise spiral bevel gear drives with adjusted bearing contact. The predesigned parabolic function of transmission errors was achieved by the application of modified roll for pinion tooth generation. Ref. [5] covered the design, manufacturing, stress analysis and results of experimental tests of prototypes of spiral bevel gears with low levels of noise and vibration and increased endurance. Three shapes of blade profile and a modified roll for pinion tooth generation were applied. Based on the grinding mechanism and machine-tool settings for the Gleason modified roll hypoid grinder, a mathematical model for the tooth geometry of spiral bevel and hypoid gears was developed by Lin and Tsay [6]. Chang et al. [7] proposed a general gear mathematical model simulating the generation process of a 6-axis CNC hobbing machine. The so-called Universal Motion Concept (UMC) was developed by Stadtfeld [8]. In the UMC eight correction mechanisms were introduced into the calculation of machine settings for gear cutting. In order to develop a gear geometry that reduces gear noise and increases the strength of gears, the Universal Motion Concept was applied for hypoid gear design by Stadtfeld and Gaiser [9]. Higher order kinematic freedoms up to the 4 order were applied to achieve the best possible result in noise, sensitivity, and adjustability. Linke et al. [10] presented a method for taking any additional motions mapped in the process-independent mathemati1 Copyright © 2008 by ASME cal model of the generating process into account in the simulation of the manufacturing process of bevel gears. It was demonstrated, how a specific influencing of the meshing and stress conditions can be achieved by such additional motions. A mathematical model of universal hypoid generator with supplemental kinematic flank correction motions was proposed by Fong [11] to simulate virtually all primary spiral bevel and hypoid cutting methods. The supplemental kinematic flank correction motions, such as modified generating roll ratio, helical motion, and cutter tilt were included into the proposed mathematical model. Wang and Fong in Ref. [12] proposed a methodology to improve the adjustability of the spiral bevel gear assembly by modifying the radial motion of the head cutter in the machine plane of the hypoid generator. A method to synthesize the mating tooth-surfaces of a face-milled spiral bevel gear set transmitting rotation with a predetermined fourth-order motion curve and contact path was presented by Wang and Fong [13]. By Achtmann and Bär [14], modified helical motion and modified roll were applied to produce optimally fitted bearing ellipses. The paper published by Fan [15], presented the theory of the Gleason face hobbing process. In Ref. [16], a generic model of tooth-surface generation for spiral bevel and hypoid gears produced by face-milling and face-hobbing processes conducted on free-form computer numerical control (CNC) hypoid gear generators was presented. Fan et al. [17] described a new method of tooth flank form error correction, utilizing the universal motions and the universal generation model for spiral bevel and hypoid gears. Shih and Fong [18] proposed a flank modification methodology for face-hobbing spiral bevel gear and hypoid gears, based on the ease-off topography of the gear drive. Gear generation with supplemental spatial motions (helical motion, tilt motion), particularly interesting for gear generation with modern free-form cutting machines, was presented by Di Puccio et al [19]. The application of modified roll and basic machine root angle variation in spiral bevel pinion finishing was introduced. In Ref. [20] published by Cao et al., third order function for the modified roll was applied in order to achieve a predesigned parabolic function of transmission errors and to improve the contact pattern with the desired shape of contact path in spiral bevel gears. Liu and Wang [21] presented a method for realizing and improving the conventional gear cutting, associated with a traditional machine-tool upon a CNC free-form gear cutting machine. The truly conjugated spiral bevel gears are theoretically with line contact. In order to decrease the sensitivity of the gear pair to errors in tooth-surfaces and to the mutual position of the mating members, carefully chosen modifications are usually introduced into the teeth of one or both members. As a result of these modifications, the spiral bevel gear pair becomes “mismatched”, and a point contact of the meshed teeth-surfaces appears instead of line contact. In practice, these modifications are usually introduced by applying the appropriate machinetool setting for pinion and gear manufacture. The generation of tooth-surfaces of the pinion and the gear in mismatched spiral bevel gears was described in Ref. [22]. In this paper a method is presented for the determination of the optimal polynomial function for the conduction of machine-tool setting variation in pinion teeth finishing in order to reduce the transmission errors in mismatched spiral bevel gears. NOMENCLATURE c = sliding base setting for pinion finishing, mm n e = composite manufacture and alignment error, mm p e = basic radial for pinion finishing, mm f = machine center to back, mm g = blank offset for pinion finishing, mm gp i = velocity ratio in the kinematic scheme of the machinetool for the generation of pinion tooth-surface 2 1 N , N = numbers of pinion and gear teeth p = distance of the initial contact point from pinion apex, mm max p = maximum tooth contact pressure, Pa 1 T r = pinion finishing cutter radius, mm s = geometrical separation of tooth-surfaces, mm T = transmitted torque, m N ⋅ 1 α = pinion finishing blade angle, deg. a ∆ = pinion offset, mm b ∆ = displacement of the pinion along its axis, mm c ∆ = displacement of the pinion along the gear axis, mm F ∆ = concentrated load, N n y ∆ = composite displacement of contacting surfaces, mm 2 φ ∆ = angular displacement of the driven gear, deg. h ε = horizontal angular misalignment of pinion axis, deg. v ε = vertical angular misalignment of pinion axis, deg. 2 1 ,φ φ = rotational angles of the pinion and the gear, deg. 20 10 ,φ φ = initial rotational angles of the pinion and the gear, deg. 1 γ = machine root angle for pinion finishing, deg. ( ) c ω = angular cradle velocity, 1 s 1 ψ = angle of pinion rotation during its generation, deg. cp ψ = cradle angle rotation for pinion finishing, deg. 0 cp ψ = initial cradle angle setting for pinion finishing, deg. THEORETICAL BACKGROUND The Spiral Bevel Gear Pair. A Gleason type spiral bevel gear pair with generated pinion and gear teeth is treated (see Fig. 1). The pinion is the driving member. The convex side of the gear tooth and the mating concave side of the pinion tooth are the drive sides. The modifications are introduced into the pinion tooth-surface by applying machine-tool setting variations in pinion tooth generation. As a result of these modifications the spiral bevel gear pair becomes mismatched and a point contact of the meshed teeth-surfaces appears instead of line contact. The relative position of the pinion and the gear is defined by the following equation (based on Fig. 1): 01 v h v h v h h v h v h v