Geometrically nonlinear forced vibrations of fully clamped multi-span beams carrying multiple masses and resting on a finite number of simple supports

Abstract

Geometrically nonlinear forced vibrations of fully clamped multi-span beams resting on multiple simple supports and carrying multiple masses may be encountered in many mechanical and civil engineering applications. The theoretical model developed here is based on the Euler-Bernoulli beam theory and the Von Karman geometrical non-linearity assumptions. Harmonic motion is assumed and the nonlinear beam transverse displacement function is expanded as a series of the linear modes, determined by solving first 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 nonlinear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure non-linear response function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence is determined. Numerical results are given in the neighbourhood of the predominant nonlinear mode shape, based on the single mode approach, for a wide range of vibration amplitudes, showing the effect of the added masses and their locations, as well as the applied uniformly distributed harmonic force on its non-linear dynamic response.