The Non-Linear Response of a Slender Beam Carrying a Lumped Mass to a Principal Parametric Excitation: Theory and Experiment

The non-linear response of a slender cantilever beam carrying a lumped mass to a principal parametric base excitation is investigated theoretically and experimentally. The Euler-Bernoulli theory for a slender beam is used to derive the governing non-linear partial differential equation for an arbitrary position of the lumped mass. The non-linear terms arising from inertia, curvature and axial displacement caused by large transverse deflections are retained up to third order. The linear eigenvalues and eigenfunctions are determined. The governing equation is discretized by Galerkin\u27s method, and the coefficients of the temporal equation—comprised of integral representations of the eigenfunctions and their derivatives—are computed using the linear eigenfunctions. The method of multiple scales is used to determine an approximate solution of the temporal equation for the case of a single mode. Experiments were performed on metallic beams and later on composite beams because all of the metallic beams failed prematurely due to the very large response amplitudes. The results of the experiment show very good qualitative agreement with the theory