Derivation of New Transcendental Member Stiffness Determinant for Vibrating Frames

Abstract

Transcendental dynamic member stiffness matrices for vibration problems arise from solving the governing differential equations to avoid the conventional finite element method (FEM) discretization errors. Assembling them into the overall dynamic structural stiffness matrix gives a transcendental eigenproblem, whose eigenvalues (natural frequencies or their squares) are found with certainty using the Wittrick–Williams algorithm. This paper gives equations for the recently discovered transcendental member stiffness determinant, which equals the appropriately normalized FEM dynamic stiffness matrix determinant of a clamped ended member modelled by infinitely many elements. Multiplying the overall transcendental stiffness matrix determinant by the member stiffness determinants removes its poles to improve curve following eigensolution methods. The present paper gives the first ever derivation of the Bernoulli–Euler member stiffness determinant, which was previously found by trial-and-error and then verified. The derivation uses the total equivalence of the transcendental formulation and an infinite order FEM formulation, which incidentally gives insights into conventional FEM results.