Vibrations of One-Dimensional Structure with Arbitrary Constraints

Abstract

An analytical scheme is developed for the free and forced vibrations of a one-dimensional structure with arbitrary intermediate constraints and boundaries. Upon applying eigenanalysis to governing differential equations for a continuous medium or a conventional transfer matrix for a discrete medium, the vibration motion is interpreted as a wave motion. The structure is treated accordingly as a waveguide, composed of side-by-side subwaveguides including intermediate constraints, boundaries, and uniform sections without apparent constraints. Reflection and transmission matrices are derived to characterize wave scattering phenomena for individual subwaveguides. Similar matrices for a union of multiple subwaveguides are obtained through a composition rule. The quantitative descriptions of a wave-scattering mechanism make it possible to effectively determine the characteristic equation for the free-vibration and response solutions for the forced vibration. Because the subwaveguides are treated equally in this approach, their numbers and change patterns from one section to the other are immaterial in the analysis.