Exact eigensolution of a class of multi-level elastically connected members

Abstract

Attention is given to determining the exact natural frequencies and modes of vibration of a class of structures comprising any number of related parallel members that are connected to each other, and possibly also to foundations, by uniformly distributed elastic interfaces of unequal stiffness. The members themselves are considered to have a uniform distribution of mass and stiffness and account can be taken of additional point masses and spring supports. The formulation is general and applies to any structure in which the motion of the component members is governed by a second order Sturm-Liouville equation. Closed form solution of the governing differential equations leads either, to a series of exact substitute systems that are easy to solve through a stiffness approach and which together yield the complete spectrum of natural frequencies and corresponding mode shapes of the original structure, or to simple exact relationships between the natural frequencies corresponding to coupled and uncoupled motion that enable hand solution of the more standard problems to be achieved. An appropriate form of the Wittrick-Williams algorithm is presented for converging on the required natural frequencies to any desired accuracy with the certain knowledge that none have been missed. Examples are given to confirm the accuracy of the approach and to indicate its range of application.