Application of reverse engineering techniques in the design process of face-milled spiral bevel gear drives

Abstract

A complete computerised integrated approach for design, generation and stress analysis of low-noise high-endurance face-milled spiral bevel gear drives manufactured by means of the Five-Cutting Process is presented. The proposed approach is based on inverse engineering techniques, and constitutes an improvement of the Local Synthesis Method developed by Faydor L. Litvin. Firstly, a desired ease-off topography is defined on pinion driving active tooth surface in order to provide favourable conditions of contact (an adjusted contact pattern with a parabolic function of negative transmission errors with limited maximum magnitude). Next, constrained and unconstrained optimization algorithms are applied to derive numerically both roughing and finishing machine-tool settings of the pinion member. A normalised design procedure based on AGMA Standards is implemented computationally with the main purpose of sizing both hypoid and spiral bevel gear drives. It makes use of the basic gear drive data and it let determine blank data such as pitch cone parameters and other dimensions of gearing elements, as well as checking if undercutting effect appears or not. A generalised mathematical model of computational generation of hypoid and spiral bevel gear drives is implemented. Fundamentally, it is based on the kinematical model of traditional cradle-style hypoid gear generators, and it includes both face-milling and face-hobbing cutting processes. A general purpose enhanced numerical algorithm based on an purely geometric approach and a matrix-type assembly model is proposed and applied for simulation of meshing and Tooth Contact Analysis (TCA) of face-milled spiral bevel gear drives. In essence, the independence of the type of bearing contact between mating surfaces (point, linear or edge contact), the high computational stability and the consideration of adjacent pairs of meshing teeth on contact pattern estimation constitute their main features. Furthermore, elliptic grid generation has been implemented for discretization of two-dimensional domains into boundary fitted uniform numerical grids and applied as an isolated stage in the proposed TCA algorithm, as well as in generation of finite elements models of spiral bevel gear drives. An approach for determination of finishing machine-tool settings corresponding to the gear member of hypoid and spiral bevel gear drives is presented. Both generating and non-generating processes are considered. Basically, it makes use of the gear drive geometry blank data previously determined. Machine-tool settings are calculated analytically by means of the concept of an imaginary generating crown gear. The proposed approach constitutes the starting point of synthesis methods for determination of both roughing and finishing machine-tool settings corresponding to the pinion member. A novel numerical approach based on the resolution of a constrained nonlinear optimization problema is proposed for determination of machine-tool settings applied in the rough-cutting operation of facemilled spiral bevel pinions by means of spread-blade grinders. The main objectives are the decrease of the machining cycle time and the maximization of the rough material without damaging the to-befinished surfaces. A computational approach for automatic and parametric generation of finite element models of lownoise high-endurance face-milled spiral bevel gear drives is presented as part of the aforementioned integrated approach for design, generation and stress analysis. Finite Element Analysis (FEA) is performed by application of a general purpose finite element computer program. Modelling of supporting components (shafts and bearings) corresponding to gearing elements is included in it, and, consequently, shaft deflections and torsional deformation caused by power transmission are taken into account. Lastly, an analytical procedure for determination of relative errors of alignment between gearing elements in face-milled spiral bevel gear drives is presented. It allows supporting shafts deflections caused by nominal torque transmission to be compensated through modification of the pinion driving active surface microgeometry. Several numerical examples are presented to validate the proposed algorithms.