METHODS ARE INVESTIGATION OF SYSTEMS WITH THE IMPULSE ACTIONS.

Pilipchuk V.N., Volkova S.A., Belozerov A.V.

INTRODUCTION

The impulsive actions are reduced to the following directions:

1. The impulsive actions are simulated so, that the coordinates and velocities meet requirements in neighborhoods of points of localization of impulses. For example assignment of jump of velocities at the moment of activity of impulses.

2. The second direction is based on the theory of generalized functions. In this case impulsive actions are simulated by means of introduction in to the equation of the singular terms such as d -functions.

3. The impulsive actions are simulated by means of the second generalized derivative saw-tooth function, which represents a sequence d - impulses.

The basic virtue of the first expedient of model operation is, that the differential equations, featuring system as well as at the absence of impulses (Samoilenko A.M., Perestuk N.А., Akhmetov M. U.). However, it is necessary to considered these equations separately on each of intervals between impulses and, thus, it is necessary to solve a sequence of problems. The second expedient of model operation yields a uniform set of equations on all the time interval without introduction of the above mentioned requirements on variable, but the relevant analysis should be carried out correctly within the framework of the theory of generalized functions (Vladimirov V.S., Ketch V., Teodoresku P., Омеlianov G.А., Ivanov V.К., Масlоv . V.P.), in the demanding padding mathematical justifications in nonlinear cases.

The third approach based on non-smooth transformation of time (Pilipchuk V.N. [1], [2]) also allows, on the one hand, to construct a boundary-value problem which does not contain of delta-functions to receive its solution as uniform analytical expression on all the time interval.

NON-SMOOTH TRANSFORMATION OF ARGUMENT

The key stage of this method consists in transformation of time. Dimensional coordinate x(t) is representable in the expressing shape by means of reference steams of non-smooth periodic functions - sawtooth sine and rectangular cosine.

Such representation has a lot of convenient mathematical properties concerning the basic algebraic operations, and also differentiation and integration and it is offered by Pilipchuk V.N. [2] on the basis of the formulated lemma for arbitrary periodic functions. This method is possible to apply a combination with idea of average. Particular ("singl-component") case is explored in, where the relevant treatment from the point of view of asymptotic strongly nonlinear (vibroimpacted) of oscillators is given.

The possible generalizations and modifications of a method were considered in the articles. The enunciating of a method with the appendices can be found in the monographies [2]. A series of the publications, since the paper of [5] is devoted to the appendices of this method, where the "singl-component" variant of a method was utilized for the analysis of systems reverting in an extreme case in the vibroimpacted.

Other idea bound with application of non-smooth functions for model operation of nonlinear systems, is offered by Zhuravlev V.F. [3], [4]. The question is goes about the method of elimination of not retained connections in systems with "interior" impacts.

Method Zhuravlev V.F. evidently usable only to the model operation vibroimpact systems and systems with hard not retained connections. Let's note, in this method coordinate x(t), instead of explanatory variable (argument) exposes to transformation.

In operation [5] the building-up of mathematical model for systems with impulse actions is offered by representation of coordinate x(t) with Hevisaid’s functions.

The similar shape of the solution will be utilized for description of propellented tearing up in the theory of waves . In these representations time does not also expose to any transformation.

The method of non-smooth transformation of time is usable at study of concrete mechanical systems such as one-dimensional network on the non-linear - elastic basis . In these works in particular it is shown, that the introduction of non-smooth transformation of time allows to take into account the fact of periodic of viewed process, thus, essentially to expand a gang of functions for building-up the relevant solutions by iteration methods. Different appendices of a method and useful calculations are contained the present in works [1]-[5].

In the articles [5] the modified variant of non-smooth transformation considerably expanding opportunities of the appendices is constructed and is shown that the introduction of non-smooth argument by means of special identities given a additional means of the analysis of one-dimensional constructions of periodic structure and model operations of linear systems with impulse actions.

BIBLIOGRAPHI

1. Pilipchuk V.N. To calculation strongly nonlinear Systems close to vibration // PMM. - 1985. - Vol. 49, No. 5. - P. 744-751.

2. Pilipchuk V.N. About transformation oscillatory systems with the help steams of rough periodic functions // АN USSR. S. А. - 1988. - Vol. 4. - P. 37-40.

3. Zhuravlev V.F. Examination some vibroimpact of systems through higher transcendental functions // AN USSR, Ser. mechanics solid bodies. - 1976. - Vol. 2. - P. 30-34.

4. Zhuravlev V.F. Examination certain vibroimpact system by a method rough transformations // Mechanics of a Solid Body. -1977. - Vol. 12, No. 6. - P. 24-28.

5. Pilipchuk V.N. On the form of periodic solution representation (non-smooth transformations of arguments, the corresponding algebraic structure, and applications // Intern. Congr. of Mathematicians. Zurich. 1994. Short Communications. P.202.