Vibrations, energies, and power flows in complex systems.

Abstract

A general numerical method, the so-called Fourier spectral element method, is described for the dynamic analysis of complex structures. When applied to structural problems, this method treats a complex structure as an assembly of fundamental structural components such as beams and plates. Over each component, the basic displacement is sought directly in the form of a modified Fourier series which is mathematically guaranteed to converge absolutely at every field or boundary point. The Fourier coefficients are considered as the generalized coordinates and determined using the Rayleigh–Ritz method. Since this method does not involve any assumption or simplification, it is broadly applicable to the whole frequency range usually divided into low-, mid-, and high-frequency regions. Additionally, because the system model is mesh- and grid-free, a smooth transition between the different frequency regions can be automatically accomplished by switching on/off the statistical processes and/or the spatial- and frequency-averaging features. This method is particularly suitable for mid-frequency dynamic analysis of complex systems. The relationships are investigated between component energies and power flows. It is shown that the spatial- and frequency-averaging processes may not be appropriate for the mid-frequency analysis because the important dynamic characteristics of a system tends to be eliminated.