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).

Asymptotic method and wave theory of motion in studying the effect of periodic impulse forces on systems characterized by longitudinal motion velocity

“…Several specific features of dynamic processes in strongly non-linear systems were established that are not characteristic of linear of quasi-linear systems. One of them is the fact that the frequency of self-oscillations depends on the amplitude, and thus specificities of resonance processes [10,11]. Another important class of non-linear one-dimensional systems with distributed parameters is made up by those systems, the unexcited (linear) analogues of which do not allow using the classical Fourier and D'Alembert methods for integration.…”

“…. )-functions that the addends of the right part of the relation (3) expressing the aforesaid component of the impulse action can be represented in the following form without damage to accuracy [11]:…”

“…We fail to clearly determine the unknown functions A 1 (a), B 1 (a) from equation (11). Therefore, let us impose an additional condition 2π 0 U 1 (a, x, ϕ, θ) cos ϕ sin ϕ dϕ = 0 on the function U 1 (a, x, ϕ, θ).…”

“…This condition equals selecting the amplitudes of their first modes for the amplitudes of the direct wave and the reflected wave. By making simple trigonometric transformations of coefficients of the right part of the relation (11) at A 1 (a) and B 1 (a) considering the aforesaid condition, we obtain the following:…”

Methodology of investigation the dynamics of longitudinally moving systems under the action of impulse perturbations

“…The work deals with a more complex and important problemthe problem of the dynamics of such systems under the action of a periodic impulse perturbations, which is modeled using the Dirac delta function. For the analytical study of the considered class of systems, the main idea of the works [1]- [5] is generalized to a new class of boundary value problems. This made it possible, using the properties of completeness and orthogonality of certain functions and mathematical transformations that do not change the accuracy of the mathematical model, to transform impulse actions on longitudinally moving systems to a form that allows spreading the basic ideas of asymptotic integration.…”