ON THE DERIVATION OF THE EQUATIONS OF MOTION FOR A PARAMETRICALLY EXCITED CANTILEVER BEAM

One of the most readily assimilated mechanical structures, in which both forced and parametric vibration phenomena can occur, is the base excited cantilever beam. The paper by Cartmell (1990) was an attempt to unify the necessary kinematics and dynamics for a simple vertical beam with a lumped end mass which is undergoing a single frequency harmonic excitation in the stiff y direction. The motivation for the present study is the enduring usefulness of simple structural systems for the investigation and interpretation of complex vibrational phenomena both for the researcher and the educator. This paper strengthens the geometrical proof of the important equation for the small displacement, v(l), of the top of the beam in the y direction. A rigorous analysis is presented to show that the modal co-ordinates for the bending u in the x direction do decouple in the expression for the kinetic energy, as shown in Cartmell. Finally, it is shown that, unfortunately, the modal co-ordinates for the bending u in the x direction do not decouple in the expression for the potential energy.