Comparison of several methods for calculating vibration mode shape derivatives

Abstract

Four methods for the calculation of derivatives of vibration mode shapes (eigenvectors) with respect to design parameters are described. These are finite-difference method, modal method, a modified modal method and Nelson's method. The methods are implemented in a general-purpose commercial finite-element program and applied to the following test problems: a cantilever beam and a stiffened cylinder with a cutout. Design variables are a beam tip mass, a beam root height, and specific dimensions of the cylinder model. The methods are compared on the basis of central processor (CP) seconds required to obtain the derivatives, and two of the methods are also evaluated for the rapidity of convergence. Data is presented showing the amount of CP time used to compute the first four eigenvector derivatives for each example problem. A scalar measure of the error in the mode shape derivative is defined, and numerical results illustrating the rapidity of convergence of the approximate derivative to the exact derivative are presented. Results indicate an advantage in using Nelson's method because this method is exact and requires less CP time, especially when derivatives with respect to several design variables are computed.