On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass

The non-linear period, for each of the first four modes, of planar, flexural large amplitude free vibrations of a slender, inextensible cantilever beam with a flexible root and carrying a lumped mass at an intermediate position along its span is investigated theoretically. With shear deformation and rotary inertia assumed to be negligible, but with account taken of axial inertia and non-linear curvature, two different, simple, approaches-for comparison purposes-are used to formulate the equation of motion. In the first approach, the governing partial differential field equation of motion is obtained by using Hamilton's principle, following closely the analysis presented in reference [1], which does not take into account the inextensibility condition. By retaining non-linear terms up to order five and using the single mode approximation in conjunction with the Rayleigh-Ritz method, the field equation is reduced to a non-linear, single mode, Duffing type temporal problem. In the second approach, an assumed single mode Lagrangian method, with account taken of the inextensibility condition, is used to form directly the fifth order non-linear unimodal temporal problem. Because of the particular non-linear terms in the temporal problem in both formulations, the time transformation approach [2] is used to obtain an approximate solution to the period of oscillation. Results in non-dimensional forms are presented graphically for the effects of the base stiffness, position and magnitude of lumped mass on the variation of period of oscillation with amplitude. Comparison of the present models with some of the existing ones, and comparison of the time transformation results with those of the harmonic balance, and existing ones are presented. (C) 1997 Academic Press Limited.