Dynamics and stability of support excited beam-columns with end mass

The dynamics and stability of beam-columns with support motion is discussed, and the general equations of motion for a translating beam-column are derived along with the associated boundary conditions. The response of a cantilevered beam-column supporting a large mass at its free end is considered where the support of the structure is excited harmonically in both the transverse and longitudinal directions. The free transverse vibration problem is solved yielding a frequency equation and set of mode shapes which are then used to generate an orthonormal set of functions corresponding to an equivalent problem, where the point mass is considered to be part of the beam-column. Employing an eigenfunction expansion, the stability of the structure is assessed, for the case of longitudinal excitation, by the determination of the boundedness of the solutions to the resulting system of differential equations governing the amplitudes of the eigenfunctions.