Historic, Recent and Ongoing Applications of the Wittrick-Williams Algorithm

Abstract

Eigenvalues (i.e. natural frequencies or buckling load factors) of structures are usually found by the finite element method, which involves a linear eigenproblem for which many solution methods exist. This paper covers the much harder transcendental eigenproblem which occurs when finite element discretisation errors are avoided by using the exact member equations yielded by solution of their governing differential equations. It outlines the Wittrick-Williams (W-W) algorithm, which was developed to obtain the eigenvalues of such transcendental eigenproblems, summarising its development history, its many applications and the development of associated mode finding methods. Key underlying concepts are indicated. Working at a trial value of the eigenparameter, the W-W algorithm computes the value of J, the number of eigenvalues of the structure between zero and the trial value, by summing the 'sign count' property of the transcendental overall stiffness matrix of the structure with Jm for all members of the structure, where Jm is the number of clamped/clamped member eigenvalues between zero and the trial value. An overview is given of member stiffness matrices, and associated values of Jm, for a wide range of member types. These include: Bernoulli-Euler beams and beam-columns; Timoshenko beams and beam-columns; and plates within prismatic plate assemblies which are isotropic or anisotropic (to cover both metal and laminated composite plates) and which may include out-of-plane shear deformation. Extensions of the algorithm include: cases for which the transcendental overall stiffness matrix is complex and Hermitian instead of being real and symmetric; the use of Lagrangian multipliers; very fast solutions for structures which are repetitive along one, two or three Cartesian directions; rotationally repetitive structures with members along the axis of rotational periodicity; multi-level substructuring; modes computed to the near machine accuracy to which the eigenvalues have always been found; design optimisation of stiffened wing panels using the W-W algorithm to obtain sensitivities to design changes; modal density plots; and locating pass bands and stop bands. Some non-structural areas of application of the W-W algorithm are indicated, including using a structural analogy in mathematics both to enable second and fourth order Sturm-Liouville problems to be solved competitively and also to obtain unique results for effectively infinite level trees of Sturm-Liouville equations.