Design procedure of first-order perturbations for collisions trajectories with a perturbing body

The article is devoted to the determination of firstand secondorder perturbations in rectangular coordinates and velocity components of body motion. Special differential equation system of perturbed motion is constructed. The right-hand sides of this system are finitesimal polynomials in powers of an independent regularizing variate. This allows constructing a single algorithm to determine firstand second-order perturbations in the form of finitesimal polynomials in powers of regularizing variates that are chosen at each approximation step. Following the calculations results with the developed method use, the coefficients of approximating polynomials representing rectangular coordinates and components of the regularized body speed were obtained. Comparison with the numerical results of the disturbed motion equations shows their close agreement. The developed method make it possible to calculate any visa point of body motion by the approximating polynomials.


Introduction
One of the crucial tasks associated with trajectory measurements is the determination of the partial derivatives of rectangular coordinates that make up the body motion speed with respect to the initial conditions. Auxiliary functions, which are degree series with respect to the additional variable, were introduced in the article [1]. In addition, operating procedures of universal variables in a number of mechanics problems to determine disturbances by the constants variation method were presented in the article [2]. Whereby, initial values components of the radius-vector and velocity were considered as osculating variables. New methods of disturbances determination keep standard features of classical ones, the small parameter method is used while disturbances calculation, which makes it possible to obtain solution asymptotic decomposition. Recently, Picard method for integration of differential equations is more commonly used, which leads to a convergent process of successive approximations that gives a differential equations system solution.
The error of solution depends on the initial approximation accuracy of the perturbation function. General development principles of perturbation theory in coordinates were studied the article [2] as the use of regularizing variables for path-planning calculation. Research reveals that the regularizing variables use increases the calculations accuracy and significantly reduces the calculation time.
A crucial task of mechanics is to approximate the rectangular coordinates that make up the body speed and time in case of disturbed motion by algebraic polynomials of the lowest degree with respect to the auxiliary variable with a predetermined degree of accuracy.
One of the important problems in mechanics is the approximation of rectangular coordinates, the regularizing velocity constituents and the body motion time by the algebraic polynomials in power of the lowest degree additional variable with a prescribed degree of accuracy.
This article describes a differential equation system of the perturbed body motion. This system is ntegrated through successive approximations method, which gives insight into rectangular coordinate and body motion constituents in the form of polynomials in powers of some specially introduced additional variable that appears at each approximation step.

Mathematical model
Equations that integrate rectangular coordinates, constituents of regularized velocity and time with additional variables were found in works [3][4][5] as follows: (1) , where: -is partial solutions of equations in variations of regularized equations of a problem with 2 bodies. In order to find partial solutions it is necessary to differentiate the general solution of regularized equations of an unperturbed problem with 2 bodies, by initial data .
In work [5] the following equations for defining first-degree perturbations in values were found:  2  3  3  0  1  0  1  1  1  1  2  3  3  1  1  1  1  (9) Then, expressions (8) will have the following form: (10) Integrals are easy to calculate: Thus, first-degree perturbations in additional variables have the form: where:   Where as Thus by replacing a number of the independent first-degree perturbation variable in rectangular coordinates, regularized velocity constituents and bodies motion time, are represented as finitesimal polynomials of comparatively low degree and relatively regularizing variable with a sufficiently high accuracy degree.
Operating analysis diagram of first-order perturbations by analytical method consists of the following main blocks. The first block is concluded in building-up polynomials which show the perturbing body coordinates, coordinates, regularized velocity constituents and of unperturbed motion time of the studied body in powers of standardized regularizing variable . u Calculation results demonstrate respective polynomials coefficients.
The second block consists in representing the mutual distance square between the studied body and the perturbing body

Summary
Thus, by replacing a number of independent first-degree perturbation variable in rectangular coordinates, constituents of regularized velocity and of bodies motion time are represented as finitesimal polynomials of comparatively low degree with respect to 2 1  The suggested method may be used at simulating the deformation of metal-ridged outer shell plating of panels and interior shell plating made of aluminum foil. At developing the procedure for calculation of multi-layer construction structures which are subjected to the impact of shock-waves, as well in studying construction structures, in particular, crossbending of cantilevers, at simulating tensions and deformations in construction elements effected by static loads of various value and configuration. As well, the procedure may be used at calculation of the structure in terms of seismic effect.