Finite amplitude oscillations of a simple rubber support system

Exact solution of the nonlinear problem of undamped, finite amplitude, free vertical oscillations of a mass supported by a rubber spring made of a neo-Hookean material is presented for both suspension and compression supports. The motion in the special case of free fall of the mass from rest at the unstretched state is characterized in terms of elliptic integrals, and it is shown that the periodic time may be expressed universally in terms of the tabulated Heuman lambda-function. The finite amplitude, free vibrational frequency and the dynamic deflection of a neo-Hookean oscillator are compared with those for a linear spring oscillator having the same constant stiffness; and both upper and lower bounds on the ratio of these frequencies are presented. Numerical values for several cases are illustrated, and the physical results are described graphically. General solutions for the free vibrations with arbitrary initial data are obtained in terms of certain generalized lambda and beta-functions, and some transformation identities relating these functions are derived.