An Asymptotic Study of the Linear Vibrations of a Stretched Beam with Concentrated Masses and Discrete Elastic Supports

Abstract

An asymptotic analysis based on the homogenization technique in the framework of linear dynamics for an arbitrary range of frequencies has been applied to an infinite one-dimensional (1D) system which consists of elastically supported discrete masses, linked by beams. Three scale regions of eigenfrequencies are found. The first one corresponds to the continuum approach, when the system studied can be described as an effectively continuous homogeneous beam and the corrections are of a higher order of magnitude. The second region corresponds to an antiphase mode of neighboring masses vibrating with slowly varying amplitudes. The highest range of frequencies reflects the short beams vibration between neighboring masses, which are immobile in the first term approach. The completeness of the spectrum analysis is shown. Dispersion relations and peculiarities of the corresponding eigenmodes have been discussed. The system studied admits generalizations and may itself serve as an adequate model for various technical applications: civil engineering, ship building etc.