Free vibrations of a gravity-loaded clamped-free beam

Abstract

For the free vibrations of a gravity-loaded clamped-free Euler-Bernoulli-beam (flexible pendulum) no exact analytical solutions are known in the literature. Approximate analytical closed-form solutions are determined by use of the Ritz-Galerkin method with gravity-free beam eigenfunctions in the series expansion. A comparison with experimentally obtained modal data on a heavy beam with strong gravity influence shows good agreement and justifies the necessity of modelling gravity effects. Since in many vibration problems of engineering one of both influences of gravity or flexural rigidity is dominating, further approximated solutions are determined by applying regular perturbation theory (influence of bending moment dominates) and singular perturbation theory by the method of the “matched asymptotic expansions” (influence of gravity dominates: heavy rope with small bending stiffness), which creates a boundary layer problem at the clamped end. Within a wide range of values of the perturbated parameters the lower eigenfrequencies agree well with the Ritz-Galerkin solutions.