A Multimode Approach to Geometrically Non-linear Forced Vibrations of Euler–Bernoulli Multispan Beams

Abstract

The geometrically non-linear free and forced vibrations of a multi-span beam resting on an arbitrary number of supports and subjected to a harmonic excitation force is investigated. The theoretical model developed here is based on the Euler–Bernoulli beam theory and the von Karman geometrical non-linearity assumptions. Assuming a harmonic response, the non-linear beam transverse displacement function is expanded as a series of the linear modes, determined by solving the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton’s principle, the problem is reduced to a non-linear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure deflection function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence are determined. Considering the non-linear forced response, an approximate multimode approach has been used in the neighbourhood of the predominant mode, to obtain numerical results, for a wide range of vibration amplitudes. The effects on the non-linear forced dynamic response of the support number and locations, the excitation frequency and the level of the applied harmonic force (a centered point force or a uniformly distributed force) have been investigated and illustrated by various examples.