Transcendental Eigenvalue Problem and Its Applications

Abstract

The algebraic eigenvalue problem is frequently encountered when analyzing the behavior of a multi-degree-of-freedom dynamic system. The characteristic equation associated with the algebraic eigenvalue problem is a polynomial that defines the eigenvalues by its roots. Dynamics and stability of distributed parameter systems are characterized by transcendental eigenvalue problems, with transcendental characteristic equations. By the use of finite element or finite difference methods, the transcendental eigenvalue problem is transformed to an algebraic problem. Because the behavior of a finite dimensional polynomial is fundamentally different from a transcendental function, such an approach may involve an inaccurate solution, which is attributed to a discretization error. The main idea is to replace the continuous system with variable physical parameters by a continuous system with piecewise uniform properties. The matching conditions between the various parts of the continuous model are expressed as a transcendental eigenvalue problem, which is then solved by the Newton's eigenvalue iteration method. Some classical problems in structural dynamics and stability are solved to demonstrate the method and its application.