Poynting oscillations of a rigid disk supported by a neo-Hookean shaft

Simultaneous axial and torsional oscillations of a rigid disk attached to an elastomeric shaft are investigated. Five cases are solved exactly. The uncoupled, small amplitude axial and torsional oscillations of the disk are investigated for neo-Hookean and Mooney-Rivlin shafts with static stretch. The finite torsional vibration of the load superimposed on a static stretch of the shaft is studied for the Mooney-Rivlin model. Solutions for both small and finite amplitude, uniaxial vibrations of the body superimposed on a pretwisted neo-Hookean shaft with static stretch are derived. Simple bounds on the period for the finite motion are provided; and various universal frequency relations for neo-Hookean and Mooney-Rivlin materials are identified.Finally, the main problem of finite, uniaxial vibrations accompanied by a small twisting motion is studied for the neo-Hookean model. The exact periodic solution for the axial response is obtained; and the coupled, small torsional motion is then determined by Hill's equation. A stability criterion for the Mathieu-Hill equation is used to obtain stability maps in a physical parameter space. Geometrical conditions sufficient for universal stability of the motion are read from this graph. Instability of the torsional oscillation, the beating phenomenon and exchange of energies, and the relation of the stability diagram to amplitude bounds on the uncoupled, linearized motion sufficient to assure universal stability predicted for small amplitude vibrations, are discussed and described graphically with the aid of a numerical model. It is shown that an unstable configuration may be stabilized by increasing the diameter of the disk.