Non-liner vibrations of a beam-mass system under different boundary conditions

An Euler-Bernoulli beam and a concentrated mass on this beam are considered as a beam-mass system. The beam is supported by immovable end conditions, thus leading to stretching during the vibrations. This stretching produces cubic non-linearities in the equations. Forcing and damping terms are added into the equations. The dimensionless equations are solved for five different set of boundary conditions. Approximate solutions of the equations are obtained by using the method of multiple scales, a perturbation technique. The first terms of the perturbation series lead to the linear problem. Natural frequencies and mode shapes for the linear problem are calculated exactly for different end conditions. Second order non-linear terms of the perturbation series appear as corrections to the linear problem. Amplitude and phase modulation equations are obtained. Non-linear free and forced vibrations are investigated in detail. The effects of the position and magnitude of the mass, as well as effects of different end conditions on the vibrations, are determined. (C) 1997 Academic Press Limited.