Dimensionless Solution to the Optimal Design of Spur Gear Sets

In some earlier papers (Savage, Coy, and Townsend, 1982; Carroll and Johnson, 1984), the design of spur gear sets based on minimum size has been addressed considering the interaction of bending and contact stress constraints. In this paper, we present a new approach to the spur gear problem. The new method makes use of some newly defined dimensionless parameters. In the resulting design space, the optimal dimensionless design (which defines the optimal tooth geometry) is independent of load and speed requirements of the gear set. However the optimum is dependent on the physical properties of the materials used. We introduce a new quantity called the Material Properties Relationship Factor, C_MP. In the problem formulation presented here, we show that the optimum will always be constraint bound and it will occur at one of three possible constraint intersections. C_MP is used to identify which of three possible constraint intersections is the correct one. After the dimensionless optimum is found, we present an example which shows how to transform the solution back into the real design space considering the load and speed requirements of the gear set along with discrete value constraints on the number of teeth and the diametral pitch. Tabulated optimal dimensionless designs are included for some standard sets of tooth proportions.