VIBRATIONS OF ONE-DIMENSIONAL STRUCTURE WITH ARBITRARY CONSTRAINTS

An analytical scheme is developed for the free and forced vibrations o f 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 fo r a discrete medium, the vibration motion is interpreted as a wave mot ion. The structure is treated accordingly as a waveguide, composed of side-by-side subwaveguides including intermediate constraints, boundar ies, and uniform sections without apparent constraints. Reflection and transmission matrices are derived to characterize wave scattering phe nomena for individual subwaveguides. Similar matrices for a union of m ultiple subwaveguides are obtained through a composition rule. The qua ntitative descriptions of a wave-scattering mechanism make it possible to effectively determine the characteristic equation for the free-vib ration and response solutions for the forced vibration. Because the su bwaveguides are treated equally in this approach, their numbers and ch ange patterns from one section to the other are immaterial in the anal ysis.