Eigen-analysis using optimization with Rayleigh's quotient

doi:10.1016/S0045-7949(97)80862-0

1997-11-01

PRE

Loukas Papadopoulos

E. Garcı́a *

DOI：10.1016/S0045-7949(97)80862-0

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An eigen-analysis is usually required if it is desired to obtain the natural frequencies and mode shapes (i.e. eigen-parameters) from a finite element mass and stiffness matrix. In certain instances, however, only the lowest and/or largest natural frequencies are of interest. It is shown that determining the lowest and largest eigen-parameters can be formulated as an unconstrained optimization with Rayleigh's quotient as the objective function. The Powell method and a heuristic search technique, called a genetic algorithm, are used in the optimization. In addition, four initial starting point estimators are discussed for the Powell method. The effectiveness of the proposed optimization is shown using a 6 and 8 degree-of-freedom system, and a 42 degree-of-freedom finite element model of a cantilevered Euler-Bernoulli beam. The lowest and largest eigen-parameters were shown to be extremely well identified. Furthermore, Powell's Method produced better results than the genetic algorithm in terms of efficiency and accuracy. (C) 1997 Elsevier Science Ltd.