Design optimization procedure using robustness for minimizing transmission error in spur and helical gears

This dissertation deals with the development of a procedure that incorporates manufacturing tolerances and operational variances in the design optimization stage to achieve designs with robust and optimal performance. The proposed procedure optimizes the expected value of a performance characteristic subject to a set of constraints and uses concepts from statistical design of experiments to approximate the expected value of a performance characteristic. The procedure incorporates uncertainties in design variables and variations in constraints due to uncertainty in design variables. This work discusses the following three methods to incorporate variations in constraints: (1) Method using heuristics that evaluates constraints at the worst combinations of design variables, (2) Method with built-in constraint variation that models constraints using first order Taylor expansion and (3) Method based on differentiating KKT optimality conditions. The methods developed in this dissertation enable engineers to design machine elements with optimum and robust performance. Design of spur and helical gears with minimum transmission error serves as the target application. A computer program called the Load Distribution Program (LDP) evaluates transmission error as a function of elastic properties of the gear mesh and errors in gear tooth profile. This dissertation characterizes transmission error by its peak to peak value. The two key gear design research issues addressed are: (1) Determination of optimal combination of geometric design variables like number of teeth, pressure angle that minimizes transmission error subject to constraints like minimum number of teeth to avoid undercut and maximum bending stress and (2) Finding the optimum tooth profile that not only minimizes transmission error but also is insensitive to manufacturing errors in tooth profile, shaft misalignment and torque variations. The statistical optimum shows considerable reduction in the sensitivity of transmission error to manufacturing errors, shaft misalignment and torque variation. This dissertation also identifies the effect of geometric design variables like number of teeth, pressure angle, helix angle, hob shift, diametral pitch and working depth on transmission error.