Influence of Impulse Disturbances on Oscillations of Nonlinearly Elastic Bodies

Abstract

A method for studying the effect of impulse perturbation on the longitudinal oscillations of a homogeneous constant cross-section of the body and the elastic properties of a material which satisfies the essentially nonlinear law of elasticity has been developed. A mathematical model of the process is presented, which is an equation of hyperbolic type with a small parameter at the discrete right-hand side. The latter expresses the effect of impulse perturbation on the oscillatory process. As for the boundary conditions considered in the work, they are classic of the first, second and third genera. The methodology is based on: the principle of oscillation frequency in nonlinear systems with many degrees of freedom and distributed parameters; basic provisions of asymptotic methods of nonlinear mechanics; the idea of using special periodic Ateb-functions to construct solutions of some classes of nonlinear differential equations; properties of completeness and orthonormality of functions that describe the forms of oscillations of undisturbed motion. In general, the above allowed to obtain relations that describe for the first approximation the defining parameters of the oscillations of an elastic body. Their peculiarity is that even for undisturbed motion, the natural frequency of oscillations depends on the amplitude, and therefore, under the action of a periodic (over time) pulse force on the elastic body, both resonant and nonresonant processes are possible in the latter. It, in contrast to an elastic body with linear or quasilinear elastic properties of the body is determined not only by its basic physical and mechanical properties, but also by the amplitude of oscillations. As a special case, the oscillations of the body under the action of a constant periodic momentum perturbation are considered. It is shown that for the nonresonant case for the first approximation it does not affect the laws of change of amplitude and frequency of the process. As for the resonant is the amplitude of origin through the main resonance significantly depends not only on the speed but also on the points of action of the pulsed perturbation. Moreover, the closer the point of application of the pulsed force to the middle of the elastic body under boundary conditions of the first kind is greater (for boundary conditions of the second kind closer to the end).