Mathematical programming methods for the optimal design of turbine blade shapes

This paper describes some optimization techniques for the design of turbine blade profiles with a vibration constraint. The vibration characteristics were modelled by a Timoshenko beam with idealized boundary conditions permitting the system dynamics to be simulated by differential equations. Elliptical cross-sectional shapes were assumed, resulting in an optimization problem in a finite number of variables. The methods used were (1) a direct handling of the differential equations describing the system, in which penalty function transformations were used, and (2) a finite difference discretization with the system equations replaced by finite difference approximations. In the latter formulation the vibrational frequencies are the eigenvalues of the system while in the former case they are regarded as control parameters.This paper includes a numerical study of these methods and their implementation together with a discussion of results.