Three-dimensional forced harmonic vibrations of a uniform cantilever beam with -n- concentrated masses

Abstract

In this investigation an exact solution for the problem of three-dimensional forced harmonic vibrations of a straight uniform cantilever beam with rotational and translational springs at its support, carrying n concentrated masses, is presented. The effects of rotatory inertias, as well as of transverse shear deformations and of a constant compressive force, acting on the free end of the beam, have also been included in the analysis. The natural frequencies and the eigenfunctions of the system are calculated, based on the Laplace transform and on generalized functions. Furthermore, the general solution of the linear partial differential equation of the fourth order with respect to x and time t, derived from this analysis, which governs the dynamic equilibrium of the beam, was achieved. Finally, a special case is examined and the resulting solutions are in agreement with existing special solutions of this particular problem.