Finite amplitude waves in bounded media: nonlinear free vibrations of an elastic panel

Abstract

The equations governing the stretching of a nonlinear elastic panel of finite length are integrated in the limit when the strains are small but the strain rates are comparable with the natural frequency of the panel. Such a theory is needed to describe many small amplitude but nonlinear phenomena, such as near-resonant oscillations. It is shown that such a theory must be used when investigating the distortion and decay of free vibrations. These are discussed in detail. A brief discussion of the effect of friction on such vibrations is also given. The mathematical problem reduces to solving nonlinear difference equations and involves functions whose values repeat at a sequence of times which are separated by a time interval which depends on the value repeated. Although the problem is stated in the language of elasticity theory the results are directly applicable to nonlinear optical systems and gas flows in Kundt tubes.