Wave reflection, transmission and propagation in structural waveguides

Abstract

A generalised wave approach based on reflection, transmission and propagation of waves is presented for the analysis of one-dimensional structural waveguides. The state vector in the physical domain is transformed to the wave domain using the displacement and internal force matrices. The wave amplitudes at one point are then related to those at another point by the (diagonal) propagation matrix, which is true for deterministically varying waveguides as well as uniform waveguides. The response to external excitation, reflection and transmission at a point discontinuity, reflection at boundaries, the spectral element and the energy flow associated with waves are described in a systematic way using the matrices. Numerical results of the wave approach are always well conditioned since the positive- and negative-going wave motions are separated. The wave approach is illustrated for longitudinal and bending motions of deterministically varying straight beams, based on elementary theories such as Euler-Bernoulli theory. The energy transport velocity is derived using the relationship between power and energy. In contrast to that for uniform structures, the energy velocity for deterministically varying structures depends on position as well as frequency. The in-plane motion of uniform curved beams, in which longitudinal and bending motions are coupled, is studied as well. The energy flow associated with waves is described explicitly in terms of the wavenumbers. Numerical results for the power transmission through a U-shaped structure are presented. In conjunction with the piecewise approach, the exact results can be used in an efficient way for arbitrarily varying structures and, eventually, built-up structures. Employment of deterministically varying elements, rather than uniform elements, could lead to rapid convergence at low computational cost, especially when the non-uniformity of the structure becomes severe. The modal behaviour of a linearly tapered curved beam with clamped-free boundaries is studied and its asymptotic behaviour related to the pure bending and pure extensional motions is revealed.

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