Direct Gear Design: Bending Stress Minimization

Abstract

Introduction Bending stress evaluation in modern gear design is generally based on the more-than-onehundred-year-old Lewis equation. This equation, applied with the stress concentration factor Kf, defines the bending stress geometry factor J for traditionally designed standard or close-to-standard gears. The stress concentration factor Kf is defined from photoelastic experiments (Ref. 1) for rack-generated gears with standard tooth profile proportions. The Direct Gear Design method (Ref. 2) is not constrained by a choice of gear tooth profiles based on standard tool parameters and uses nonstandard tooth shapes to provide required performance for a particular custom application. This makes finite element analysis (FEA) more preferable than the Lewis equation for bending stress definition. This paper does not describe the FEA application for comprehensive stress analysis of the gear teeth. It presents the engineering method for bending stress balance and minimization. Tooth Modeling The Direct Gear Design method defines parameters of the gear mesh to provide complete geometry of the involute profile of the teeth, including the base diameter, form diameter, outside diameter, tooth thickness, tip radii, etc. The fillet profile initially is defined as a trace of the tip of the mating gear tooth. This kind of fillet profile is used for plastic molded gears (Ref. 3). The 2-D FEA model in Figure 1a presents a gear tooth profile that is limited from the sides and bottom by a constrained border with stationary nodes. All other nodes on the tooth profile and inside the tooth contour are movable. The fillet portion of the tooth profile (where maximum bending stress is expected) has equally spaced nodes with higher density (number of nodes per unit of profile length) than the rest of the tooth profile. The nodes on the involute profiles and the top land are located to have higher density close to the fillets and lower density in the top part of the tooth. The number of tooth profile nodes and the node density coefficient (ratio of the fillet profile node density to an average node density of the involute and top land profiles of the tooth) are selected. Fewer tooth profile nodes and lower node density coefficients yield less accurate stress calculations. Selection of larger numbers of tooth profile nodes and high node density coefficients provides a more accurate result, but increases calculation time. In most cases, 80–100 tooth profile nodes and node density coefficients of 1.75–2.5 were used. The tooth load distribution problem is considered to define a value, a set of application point coordinates, and the direction of the force resulting in maximum bending stress. The friction effect at the contact point has been ignored. The Direct Gear Design: Bending Stress Minimization