Recent discoveries related to the use of the Wittrick-Williams algorithm to solve transcendental eigenproblems

Traditional finite element methods yield linear eigenvalue problems, e.g. when the eigenvalues are undamped natural frequencies. In contrast, when exact elements are available the overall stiffness matrix is a transcendental function of the eigenparameter. The Wittrick-Williams (W-W) algorithm must then be used to solve the resulting transcendental eigenproblem. This paper reports how analogies with linear eigenproblems have recently been used for transcendental eigenproblems to: discover a new “member stiffness determinant” property; develop an inverse iteration based recursive second order transcendental eigensolver; understand why “frequency squared” should be used and; to explore exact inverse iteration substructuring methods. The W-W algorithm is also applied to solve the topical homogeneous tree problem of mathematics. The resulting transcendental eigenproblem involves over 1012 linked Sturm-Liouville equations, is solved with negligible computational effort and is well conditioned. Additionally, a deeper understanding is obtained by using the engineer’s physical understanding of the analogous structural problem.