On Parametrically Excited Flexural Motion of an Extensible and Shearable Rod with a Heavy Attachment

A simple Cosserat model is used to explore the coupled planar flemural and arial vibrations of a slender rod clamped at one end with a heauy attached mass free to move at the other. By assuming that the inertia of the rod is small compared to that of the attached mass it is shown how the equations of motionreduce to a dynamical system. The effects of grauity on the rod can be incorporated within this frame-work and the linearised stability of the system discussed in terms of solutions to the Mathieu-Hill equation