Analytical Wave Propagation Method for Free and Forced Transverse Vibration Analysis of a System of Multiple Elastically Connected Beams

Abstract

An analytical wave propagation approach is developed in this paper for the free and forced vibration of a system of multiple elastically connected beams for the first time. The beams of the system are continuously joined by a massless, linear, elastic layer which can be regarded as continuous spring. The coupled partial differential equations governing the vibration of the multi-beam system are established and decoupled by using a technic developed based on matrix theory. For the decoupled equations, a general “vibration” state is introduced into the symplectic dual system. By solving the symplectic eigenproblem and utilizing the wave propagation theory, the general “vibration” state can be analytically described in symplectic space. By using these analytical expressions and satisfying the physical boundary conditions of the system, the natural frequencies, mode shapes and forced responses can be obtained analytically and explicitly. In the numerical examples, free and forced transverse vibration of the two- and three-beam system with various combinations of boundary conditions are considered. The effectiveness of the present method is validated by comparing the present results with the analytical results from the literature and the results calculated by the finite element method.