Forced vibrations of a body supported by viscohyperelastic shear mountings

Abstract

The damped, finite-amplitude forced vibration of a rigid body supported symmetrically by simple shear springs and by a smooth inclined bearing surface is studied. The spring material is characterized as a compressible or incompressible, homogeneous and isotropic viscohyperelastic material for which the shear response function in a simple shear deformation is a quadratic function of the amount of shear. The trivial case of constant shear response is included. The equation for the damped motion of the load is a nonlinear, ordinary differential equation of the forced Duffing type with a constant static shift term due to gravity, and for which an exact solution is unknown. An approximate solution is obtained by the method of harmonic balance. Results for the motion of the load relate the system design parameters to the amplitude-frequency response and to the amplitude-driving force intensity response of the system. Regions of stable motion are identified in terms of the amplitude of the motion, driving-force intensity, driving frequency, and system design parameters. Geometrical characterizations of the motion are related schematically to certain cross-sections through the full three-dimensional solution surfaces for the amplitude and for the phase of the motion. A simple diagram maps the loci of all bifurcation points against the static shear deflection, which serves as the system design parameter for the inclined motion. An infinitesimal stability analysis shows that the bifurcation points of the inclined motion fall on the stability boundaries of the numerical solution of a three-parameter Hill equation. The solution provides information that illustrates how the system design parameters affect the motion of the load and how these may be chosen to control the amplitude of the oscillations and the stability of the system. The results are valid for all compressible or incompressible, homogeneous and isotropic, viscohyperelastic materials in the aforementioned class.