On the dynamics of high-order beams with vibration absorbers

Abstract

In this work, a general method is proposed to solve a wide class of high-order boundary value problems governing the dynamics of beams equipped with vibration absorbers. The method relies on one-dimensional beam theories and theory of generalized functions to model the reaction forces of the absorbers. The key step consists in deriving, via Laplace transform, the Green’s function of a linear differential operator of any order n, which involves even-order terms and constant coefficients. Next, the Green’s function, obtained in exact and elegant analytical form, serves as a basis to derive the exact solutions of boundary-value problems of order n governing the frequency response of beams with absorbers, under concentrated/distributed loads. Remarkably, the solutions are derived for any order n of the boundary-value problem, providing the exact frequency response of beams of relevant engineering interest as composite beams, twisted beams, coupled bending-torsion beams and multiple-beam systems including any number of beams; moreover, they are readily implementable for any number/positions of absorbers and loads. To illustrate the method, a case study is selected: a boundary value problem of order n=10 governing the frequency response of a coupled bending-torsion beam with asymmetric cross section and no warping effects. For this problem, an additional validation of the proposed analytical framework is obtained by the theory of distributions, while a comparison with an exact classical approach demonstrates correctness and advantages for applications with multiple absorbers.