Exact substructuring in recursive Newton's method for solving transcendental eigenproblems

The well-established Wittrick-Williams (W-W) algorithm guarantees accurate convergence on natural frequencies or critical buckling loads of structures in the transcendental eigenproblems arising from the use of exact member stiffnesses, i.e. dynamic member stiffnesses for vibration. The associated mode calculations had no such guarantee until they were recently greatly improved by solving the transcendental eigenproblem exactly, by reducing it to a generalised linear eigenproblem which is solved by a guided recursive Newton method involving inverse iteration. The present paper demonstrates the benefits of using frequency squared, rather than frequency, as the eigenparameter. Next, exact substructuring is introduced into the recursive Newton method, with accuracy retained because the inverse iteration includes the substructure nodes. If member fixed end eigenvalues lie close to the sought eigenvalue they can cause inaccuracy, or even wrong results, and so they are removed efficiently by inserting interior nodes to create simple substructures. Numerical results for a moderately large structure show that exact substructuring reduces the transcendental eigensolution time without reducing accuracy. Simpler examples designed to be numerically ill conditioned in the absence of interior nodes show that such ill conditioning is almost removed by inserting interior nodes within substructures. Exact substructuring is also applicable to other inverse iteration and W-W algorithm-based methods.