On the dynamics of high-order beams with vibration absorbers

In this work, a general method is proposed to solve a wide class of high-order bound-ary value problems governing the dynamics of beams equipped with vibration absorbers. The method relies on one-dimensional beam theories and theory of generalized func-tions 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 solu-tions of boundary-value problems of order n governing the frequency response of beams with absorbers, under concentrated/distributed loads. Remarkably, the solutions are de-rived for any order n of the boundary-value problem, providing the exact frequency re-sponse of beams of relevant engineering interest as composite beams, twisted beams, cou-pled 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 asym-metric 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 com-parison with an exact classical approach demonstrates correctness and advantages for ap-plications with multiple absorbers. (c) 2022 Elsevier Inc. All rights reserved.