Finite amplitude, horizontal motion of a load symmetrically supported between isotropic hyperelastic springs

• Published in: International journal of non-linear mechanics Vol. 47; no. 2; pp. 166 - 172
• Main Authors: Beatty, Millard F., Young, Todd R.
• Format: Journal Article
• Language: English
• Published: Netherlands Elsevier Ltd 01.03.2012
• Subjects: Amplitudes; Elastic deformation; Exact lambda function solution; Finite amplitude nonlinear vibrations; Horizontal; Hyperelastic springs; Mathematical analysis; Mathematical models; Neo-Hookean model; Periodicity theorem; Springs; Springs (elastic); Vibration; Hyperelastic springs; Periodicity theorem; Neo-Hookean model; Exact lambda function solution; Finite amplitude nonlinear vibrations
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2011.04.004

The undamped, finite amplitude horizontal motion of a load supported symmetrically between identical incompressible, isotropic hyperelastic springs, each subjected to an initial finite uniaxial static stretch, is formulated in general terms. The small amplitude motion of the load about the deformed static state is discussed; and the periodicity of the arbitrary finite amplitude motion is established for all such elastic materials for which certain conditions on the engineering stress and the strain energy function hold. The exact solution for the finite vibration of the load is then derived for the classical neo-Hookean model. The vibrational period is obtained in terms of the complete Heuman lambda-function whose properties are well-known. Dependence of the period and hence the frequency on the physical parameters of the system is investigated and the results are displayed graphically. ► Two exact integrals for the finite motion of a load supported by hyperelastic springs are derived. ► Inertia of the springs is included, and a general relation for the small amplitude frequency follows. ► Periodicity for arbitrary finite motions is proved for all hyperelastic springs with monotone engineering stress. ► The exact closed form finite amplitude solution is obtained for the neo-Hookean material. ► The period is given by the Heuman lambda function, and the overall system behavior is discussed.