(An) efficient technique for calculating sensitivities of natural frequencies and mode shapes

Abstract

An efficient sensitivity analysis method with guaranteed numerical stability is developed for calculation of the derivatives of vibration natural frequencies and the corresponding mode shapes of the undamped systems, as well as of the damped ones, with both distinct natural frequencies and multiple ones. For the distinct natural frequencies, the natural frequency and mode shape derivatives of both structural systems with structural parameters and mechanical systems with lumped parameters can be obtained consistently by solving algebraic equations with symmetric coefficient matrix whose order is (n+1)×(n+1), where n is the number of equations, for the multiple natural frequencies, by solving the coefficient matrix of order (n+m)×(n+m), where m is the number of multiplicity of a multiple natural frequency. In this case the adjacent mode shapes must be first calculated, which lie adjacent to the m distinct mode shapes which appear when design parameters vary, and then can be used in the algebraic equation defined in the proposed sensitivity analysis method as side conditions. Some datum are represented to prove the efficiency of the proposed method and its accuracy; analysis time, sensitivity analysis results of some eigenpairs and their errors. The analysis time of the proposed method for calculating the eigenpair sensitivities of the systems with distinct natural frequencies is compared with that of Nelson``s method. For multiple natural frequencies, the analysis time of the proposed method is compared with that of Dailey``s method. The analysis time of the proposed method can be reduced dramatically. Furthermore, the method can be saved the computer space (when the number of design parameters is one, Dailey``s method needs the space for the matrices K, M, K``, M`` , K" and M" whereas in the proposed method space for only the matrices K, M, K`` and M`` is needed). In conclusion, the proposed sensitivity analysis method is an analytic one. The algorithm of the...