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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on March 21, 2008
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbn007
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© The author 2008. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

OSCILLATIONS OF A LOAD SUPPORTED BY INCOMPRESSIBLE, ISOTROPIC LIMITED ELASTIC SHEAR MOUNTS

Millard F. Beatty{dagger} {ddagger}

( Department of Engineering Mechanics, University of Nebraska-Lincoln, PO Box 910215, Lexington, KY 40591-0215, USA )

{dagger} < mbeatty2{at}unl.edu>

Received 12 November 2007. Revise 5 February 2008.
  Abstract

The small-amplitude, free vibrational motion of a load supported symmetrically by incompressible, isotropic and homogeneous rubber-like shear mounts is studied for a general class of materials having limiting molecular chain extensibility. The oscillational frequency of the small motion of the load superimposed on a finite static shear is determined for all materials in this class of limited elastic materials. It is shown that, for the same static shear, the normalized vibrational frequency of a load on neo-Hookean shear mounts is a lower bound for its normalized vibrational frequency on any limited elastic shear mounts in the class. Formal relations are derived for the finite-amplitude oscillations of the load. Specific exact results for both small- and large-amplitude horizontal motions are provided for the limited elastic Gent material model, including a specific upper bound on the period of the finite motion, and the effects of limited extensibility are described both analytically and graphically. The intermediate amplitude solution is given explicitly in terms of a Jacobian elliptic function, and the corresponding second-order solution for the inclined motion of the load in terms of elliptic functions is noted.

{ddagger} In honour of Professor Bernard Coleman, Engineering Science Medalist 2007.


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