Solution of regular second- and fourth-order Sturm–Liouville problems by exact dynamic stiffness method analogy

Abstract

This paper treats all regular second- or fourth-order Sturm–Liouville (SL) problems as generalised vibration problems of non-uniform structural members on elastic foundations. Hence such SL problems can be solved by the Wittrick–Williams (WW) algorithm and the authors’ recursive second-order exact dynamic stiffness vibration method. The coefficients of the mathematical SL problems range more widely than for structural vibration problems, and so the method must account for additional possibilities, e.g. the equivalent of large negative stiffness continuous elastic supports. The method computes exact dynamic stiffnesses and their derivatives with respect to the eigenparameter accurately by solving the associated linear boundary-value problems using a standard adaptive solver. The difficulty of calculating the number of exact fixed-end eigenvalues below any trial eigenvalue needed by the WW algorithm is overcome by dividing the whole SL problem domain into an appropriate mesh, in which each element is guaranteed to have all fixed-end eigenvalues above the current trial eigenvalue. Results for second- and fourth-order SL problems shown in the literature to be particularly challenging demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.