Design Optimization of Gas Turbine Blades With Geometry and Natural Frequency Constraints

Abstract

In this paper an automated procedure is presented to obtain the minimum weight design of gas turbine blades with geometry and multiple natural frequency constraints. The objective is achieved using a combined finite element-sequential linear programming, FEM-SLP technique. Thickness of selected finite elements are used as design variables. Geometric constraints are imposed on the thickness variations such that the optimal design has smooth aerodynamic shape. Based on the natural frequencies and mode shapes obtained from finite element analysis an assumed mode reanalysis technique is used to provide the approximate derivatives of weight and constraints with respect to design variables for sequential linear programming. The results from SLP provide the initial design for the next FEN-SLP process. An example is presented to illustrate the interactive system developed for the optimization procedure. NOMENCLATURE A. element area C blade cross sectional chord length f. natural frequency K global stiffness matrix K. element stiffness matrix J K^: element stiffness matrix in modal coordinate Presented at the Gas Turbine and Aeroengine Congress j = 1, ...., ND (2) J J J with aj = t o (3) J 2A i=$. ( 2K -Ai aM ) 4i (9) Da. 2a. aa. J J J Since the bending modes are mostly affected by the thickness variations, the stiffness and mass matrices [5] for the jth design variable can be written as K. = a3 K? (10) J J ]M. = a.MJ°.Therefore, equation (9) can be reduced to2a . J 13 13 J where V. . and T. . are strain and kinetic energies associated with l he ith mode and the jth design variable V.. =½$TK^ ^i (13) 13 1 T.. = ½ A. 0T M^ ^i (14)Using the first order Taylor series expansion W and X. of the current design can be approximated as i OA Downloaded from https://asmedigitalcollection.asme.org/GT/proceedings-pdf/GT1988/79221/V005T14A004/2397925/v005t14a004-88-gt-105.pdf by guest on 08 July 2020