Integral representations and regularizations for a divergent series solution of a beam subjected to support motions

Abstract

Derived herein is the integral representation solution of a Rayleigh‐damped Bernoulli–Euler beam subjected to multi‐support motion, which is free from calculation of a quasi‐static solution, and in which the modal participation factor for support motion is formulated as a boundary modal reaction, thus making efficient calculation feasible. Three analytical methods, including (1) the quasi‐static decomposition method, (2) the integral representation with the Cesàro sum technique, and (3) the integral representation in conjunction with Stokes' transformation, are presented. Two additional numerical methods of (4) the large mass FEM simulation technique and (5) large stiffness FEM simulation technique are easily incorporated into a commercial program to solve the problem. It is found that the results obtained by using these five methods are in good agreement, and that both the Cesàro sum and Stokes' transformation regularization techniques can extract the finite part of the divergent series of the integral representation. In comparison with the Mindlin method and Cesàro sum technique, Stokes' transformation is the best way because it is not only free of calculation of the quasi‐static solution, but also because it can obtain the convergence rate as rapidly as the mode acceleration method can.