Exact Transfer Function Analysis of Distributed Parameter Systems by Wave Propagation Techniques

Abstract

The vibrations of elastic structures such as strings, beams, and plates can be described in terms of waves traveling in waveguides (Cremer et al., 1973; Graff, 1975; Fahy, 1987). While the subject of wave propagation has been extensively studied in the fields of acoustics in fluids and solids rather than vibrations of elastic structures, wave analysis techniques have been employed to reveal physical characteristics associated with structural vibrations of elastic media (Argento & Scott, 1995; Kang & Tan, 1998). One of the advantages of the wave analysis technique, when applied to the structural vibration analysis, is its compact and systematic approach to analyze complex structures with discontinuities (Mace, 1984; Yong & Lin, 1989; Kang et al., 2003; Mei & Mace, 2005). Applying the concept of wave reflection and transmission, Mace (1984) obtained the frequency equations of Euler-Bernoulli beams including waves of both propagating and near-field types. By the phase-closure principle, also referred to as the wave-train closure principle (Cremer et al., 1973), Mead (1994) determined natural frequencies of Euler-Bernoulli beams. This principle states that if the phase difference between incident and reflected waves is an integer multiple of 2, then the waves propagate at a natural frequency and their motions constitute a vibration mode. Based on the same principle, Kang (2007) presented a systematic approach to the free and forced vibration analysis of multi-span beams. The classical method, known as the normal mode or eigenfunction expansion, of solving the forced vibration problem of a distributed parameter system involves expansion of the forcing function into the eigenfunctions of the associated free vibration problem. While this method is theoretically sound and powerful, the method is difficult to implement when the problem to be solved is a non-self-adjoint system typically due to complicating effects such as damping, discontinuities, or non-classical boundary conditions, for which case obtaining the exact eigensolutions is not often feasible. Although approximate eigensolutions may be used instead of exact ones, the problem still persists in the form of poorly convergent solution and/or significant error in the solution. As an alternative approach to solve forced vibration problems, Yang and Tan (1992) presented a method for evaluating exact closed-