Second order mode-finding method in dynamic stiffness matrix methods

Abstract

This paper addresses the transcendental eigenvalue problem, which arises for those structures, e.g., frames, for which dynamic member stiffnesses are available and are exact, in the sense that the appropriate differential equations have been solved. Thus it does not relate to traditional finite element vibration (or buckling) methods which are approximate and give a linear eigenvalue problem which can be solved by many excellent methods. The well-established Wittrick–Williams algorithm is a powerful, reliable and efficient means of obtaining the desired number of natural frequencies for transcendental eigenvalue problems but, except for one very recent paper involving explicit derivatives and a recursive method, the corresponding vibration mode computations suffer from various difficulties and far from match the elegance of the frequency computations. This paper presents a newly developed, mathematically elegant and computationally efficient algorithm for accurate and reliable computation of vibration modes. It uses standard inverse iteration with approximate natural frequencies of only first order accuracy to produce the corresponding vibration modes to second order accuracy. Extrapolation can then be used to improve the original natural frequencies to second order accuracy. Moreover, the algorithm automatically provides a helpful so-called “ μ ̃ -check” which enables the acceptability of the mode accuracy to be judged. Numerical examples presented include some demanding ones, e.g., with coincident natural frequencies. These show the excellent performance of the method. Two advantages over the recent recursive paper are that explicit derivatives are replaced by differencing and recursion is avoided. In combination, these two advantages make the new method much more suitable for retrofitting into existing computer programs.