Free and forced vibration analysis of general multiple beam systems

Abstract

As typical structural elements, multiple beam systems exist widely in many practical engineering applications. Existing research usually makes some simplifications in the systems and cannot analyze the actual structures very precisely. This paper focuses on studying both free and forced transverse vibrations of general multiple beam systems, which are a set of parallel Euler-Bernoulli beams joined by viscoelastic connections. In particular, the to-be-solved models are very generalized and it enables one to simultaneously consider (1) arbitrary number of beams, (2) any boundary conditions, (3) different beam lengths, (4) variable cross sections, (5) axial loads, and (6) various types of viscoelastic connections. Single uniform beams are introduced to modify the original governing equations to a set of inhomogeneous differential equations. Defining some mode-shape coefficients and state variables, a state-space approach is proposed to transfer these differential equations into a state-space equation. Natural frequencies and forced vibration responses are obtained through the state-space equation. With the obtained natural frequencies and a modal Rayleigh-Ritz method, the differential equations about mode shapes are transformed into several algebraic equations that are easily solved. The accuracy of the proposed methods is validated by numerical examples from previous literature and the finite element method. Finally, the requirements for the mode number in the developed methods are presented. This study is of great significance in analyzing and designing multiple beam systems in engineering practices.