A Method of Selecting Grid Size to Account for Hertz Deformation in Finite Element Analysis of Spur Gears

A method of selecting grid size for the finite element analysis of gear tooth deflection is presented. The method is based on a finite element study of two cylinders in line contact, where the criterion for establishing element size was that there be agreement with the classic Hertzian solution for deflection. Many previous finite element studies of gear tooth deflection have not included the full effect of the Hertzian deflection. The present results are applied to calculate deflection for the gear specimen used in the NASA spur gear test rig. Comparisons are made between the present results and the results of two other methods of calculation. The results have application in design of gear tooth profile modifications to reduce noise and dynamic loads.

On the Kinematics of the Closed Epicyclic Differential Gears

The epicyclic differential gear has been known in modern times since 1575 when it appeared as a mechanism in a clock. Artifacts from an ancient shipwreck prove that it was known to the ancient Greeks at least 100 hundred years before Christ. The methods of kinematic analysis by either the relative angular velocity method or the instant center and linear velocity method, as given in the literature, are oriented toward specific solutions rather than general ones; they do not readily allow for parametric trend studies and they require a degree of imagination and intuition which may well be beyond the capabilities of those who are not practitioners of the art. The discussed methodology defines simple and compound epicyclic gears in terms of the overall ratio of a geometrically similar planetary gear. The kinematic analysis is derived in general terms for both the simple and compound epicyclic gear. It is shown that location of the point of zero tangential velocity of the velocity triangle relative to the system datum governs the characteristics of the gearset and whether it will perform as a differential gearset, or as a solar, star, or planetary gear. Simple mathematical relationships are given which define the proportions of the component gears, their speeds (rpm) and directions of rotations, and the resulting power splits. The formulas may be incorporated into simple computer programs oriented toward specific design requirements.

Optimal Tooth Numbers for Compact Standard Spur Gear Sets

The design of a standard gear mesh is treated with the objective of minimizing the gear size for a given ratio, pinion torque, and allowable tooth strength. Scoring, pitting fatigue, bending fatigue, and the kinematic limits of contact ratio and interference are considered. A design space is defined in terms of the number of teeth on the pinion and the diametral pitch. This space is then combined with the objective function of minimum center distance to obtain an optimal design region. This region defines the number of pinion teeth for the most compact design. The number is a function of the gear ratio only. A design example illustrating this procedure is also given.

Use of Figures of Merit in Computer-Aided Material and Manufacturing Process Selection

An interactive computer program to aid a designer in selecting candidate manufacturing process and material combinations for a part is described. The program uses a twelve-digit code to eliminate unsuitable combinations from consideration and to rank the remainder using numerical figures of merit.

Design Optimization of Hydraulic Networks Using Mixed Integer Linear Programming

The optimum design problem is formulated for the selection of pipe sizes in a hydraulic network such as a power plant service water or bearing cooling water system. The flows in each branch of the network are taken to be known, which makes the design problem linear in the variables. The optimization problem is formulated as a mixed integer linear programming problem. A design example is given. The role of this problem formulation and solution method in an interactive computer aided design (CAD) system is discussed.

A Minimum Strain Energy Approach for Obtaining Optimal Unbalance Distribution in Flexible Rotors

A method is developed to design for optimal unbalance distribution in a rotor system which has elements that are assembled on the shaft and operates above the first critical speed. This method can also be used for computing the optimal selection of balance weights in specified planes for a rotor with a known distribution of unbalance—the classic balancing problem. The method is an optimization problem where the strain energy of the rotor and its supports are minimized subject to the constraints of the equations of motion of the rotor system at a particular balancing speed. The problem is a quadratic program that has a unique minimum.