Computational gearing mechanics

This is an expository report summarizing the research efforts and results under NASA Grant NSG-3188 to the University of Cincinnati. Since the grant has now ended this report also serves as a final report for the grant. The focus of the research has been computational gearing mechanics. Research on gear geometry, gear stress, and gear dynamics is discussed. Current research and planned future efforts are also discussed. A comprehensive bibliography is presented

A new procedure for calculating contact stresses in gear teeth

A numerical procedure for evaluating and monitoring contact stresses in meshing gear teeth is discussed. The procedure is intended to extend the range of applicability and to improve the accuracy of gear contact stress analysis. The procedure is based upon fundamental solution from the theory of elasticity. It is an iterative numerical procedure. The method is believed to have distinct advantages over the classical Hertz method, the finite-element method, and over existing approaches with the boundary element method. Unlike many classical contact stress analyses, friction effects and sliding are included. Slipping and sticking in the contact region are studied. Several examples are discussed. The results are in agreement with classical results. Applications are presented for spur gears

Mesh refinement in finite element analysis by minimization of the stiffness matrix trace

Most finite element packages provide means to generate meshes automatically. However, the user is usually confronted with the problem of not knowing whether the mesh generated is appropriate for the problem at hand. Since the accuracy of the finite element results is mesh dependent, mesh selection forms a very important step in the analysis. Indeed, in accurate analyses, meshes need to be refined or rezoned until the solution converges to a value so that the error is below a predetermined tolerance. A-posteriori methods use error indicators, developed by using the theory of interpolation and approximation theory, for mesh refinements. Some use other criterions, such as strain energy density variation and stress contours for example, to obtain near optimal meshes. Although these methods are adaptive, they are expensive. Alternatively, a priori methods, until now available, use geometrical parameters, for example, element aspect ratio. Therefore, they are not adaptive by nature. An adaptive a-priori method is developed. The criterion is that the minimization of the trace of the stiffness matrix with respect to the nodal coordinates, leads to a minimization of the potential energy, and as a consequence provide a good starting mesh. In a few examples the method is shown to provide the optimal mesh. The method is also shown to be relatively simple and amenable to development of computer algorithms. When the procedure is used in conjunction with a-posteriori methods of grid refinement, it is shown that fewer refinement iterations and fewer degrees of freedom are required for convergence as opposed to when the procedure is not used. The mesh obtained is shown to have uniform distribution of stiffness among the nodes and elements which, as a consequence, leads to uniform error distribution. Thus the mesh obtained meets the optimality criterion of uniform error distribution

Computational gearing mechanics

The Final Report on computational gear mechanics is presented. This is an expository report summarizing the research efforts and results. Research on gear geometry, gear stress, and gear dynamics is discussed. Current research and planned future efforts are also discussed. A comprehensive bibliography is presented

Contact stresses in gear teeth: A new method of analysis

A new, innovative procedure called point load superposition for determining the contact stresses in mating gear teeth. It is believed that this procedure will greatly extend both the range of applicability and the accuracy of gear contact stress analysis. Point load superposition is based upon fundamental solutions from the theory of elasticity. It is an iterative numerical procedure which has distinct advantages over the classical Hertz method, the finite element method, and over existing applications with the boundary element method. Specifically, friction and sliding effects, which are either excluded from or difficult to study with the classical methods, are routinely handled with the new procedure. Presented here are the basic theory and the algorithms. Several examples are given. Results are consistent with those of the classical theories. Applications to spur gears are discussed

Computer-aided design of bevel gear tooth surfaces

This paper presents a computer-aided design procedure for generating bevel gears. The development is based on examining a perfectly plastic, cone-shaped gear blank rolling over a cutting tooth on a plane crown rack. The resulting impression on the plastic gear blank is the envelope of the cutting tooth. This impression and envelope thus form a conjugate tooth surface. Equations are presented for the locus of points on the tooth surface. The same procedures are then extended to simulate the generation of a spiral bevel gear. The corresponding governing equations are presented

Contact stresses in meshing spur gear teeth: Use of an incremental finite element procedure

Contact stresses in meshing spur gear teeth are examined. The analysis is based upon an incremental finite element procedure that simultaneously determines the stresses in the contact region between the meshing teeth. The teeth themselves are modeled by two dimensional plain strain elements. Friction effects are included, with the friction forces assumed to obey Coulomb's law. The analysis assumes that the displacements are small and that the tooth materials are linearly elastic. The analysis procedure is validated by comparing its results with those for the classical two contacting semicylinders obtained from the Hertz method. Agreement is excellent

On dynamic loads in parallel shaft transmissions. 1: Modelling and analysis

A model of a simple parallel-shaft, spur-gear transmission is presented. The model is developed to simulate dynamic loads in power transmissions. Factors affecting these loads are identified. Included are shaft stiffness, local compliance due to contact stress, load sharing, and friction. Governing differential equations are developed and a solution procedure is outlined. A parameter study of the solutions is presented in NASA TM-100181 (AVSCOM TM-87-C-3)

On dynamic loads in parallel shaft transmissions. 2: Parameter study

Solutions to the governing equations of a spur gear transmission model, developed in NASA TM-100180 (AVSCOM TM-87-C-2), are presented. Factors affecting the dynamic load are identified. It is found that the dynamic load increases with operating speed up to a system natural frequency. At operating speeds beyond the natural frequency the dynamic load decreases dramatically. Also. it is found that the applied load and shaft inertia have little effect on the dynamic load. Damping and friction decrease the dynamic load. Finally, tooth stiffness has a significant effect on dynamic loading; the higher the stiffness, the lower the dynamic loading. Also, the higher the stiffness the higher the rotating speed required for dynamic response