Finite Amplitude Vibrations of a Body Supported by Simple Shear Springs

Abstract

The exact solution of the problem of the undamped, finite amplitude oscillations of a mass supported symmetrically by simple shear mounts, and perhaps also by a smooth plane surface or by roller bearings, is derived for the class of isotropic, hyperelastic materials for which the strain energy is a quadratic function of the first and second principal invariants and an arbitrary function of the third. The Mooney-Rivlin and Hadamard material models are special members for which the finite motion of the load is simple harmonic and the free fall dynamic deflection always is twice the static deflection. Otherwise, the solution is described by an elliptic integral which may be inverted to obtain the motion in terms of Jacobi elliptic functions. In this case, the frequency is amplitude dependent; and the dynamic deflection in the free fall motion from the natural state always is less than twice the static deflection. Some results for small-amplitude vibrations superimposed on a finely deformed equilibrium state of simple shear also are presented. Practical difficulties in execution of the simple shear, and the effects of additional small bending deformation are discussed.