Application of numerical integration in technical practice

In this contribution we present the methodology of the solution of ordinary differential equation by the Runge – Kutta numerical method of fourth order. Analysed function is continuous and it has derivatives at every point. Records of the centre of the gravity velocities of agricultural technological vehicle are function values of the derivatives. Algorithm of numerical integration was implemented by the help of programming language C# in MS Visual Studio Pro 2010 developing environment. Trajectory of the centre of the gravity movement in three-dimensional space is the result of the listed algorithm.


INTRODUCTION
When realising a measurement in agricultural conditions, technical exciting functions are usually measured continually with the constant increase of time.In this case, the use of Runge -Kutta method of fourth order, implemented into the algorithm in suitable programming language, is the most advantageous for numerical integration.Utilization of this method is extensive in science as well as in research.Solution of the system of algebraic differential equations in form: (1) is listed in [1].In [5] is introduced numerical integration of differential algebraic systems.The use of Runge -Kutta method to solve non-linear system of differential equations is presented in [3].When time increase (or other parameter used as a step) is variable, it is appropriate to use modified Runge -Kutta -Nyström method, which is modified for variable step of numerical integration.This issue is closely addressed in [2].Algorithmisation and implementation of numerical methods in programming language C++ was published in [6].In [4] the issues of stability of numerical integration of linear differential equations by use of Runge-Kutta method is described.Methods of numerical integration Runge-Kutta of higher order and solution of differential equations by these methods were published in [8].Methods were implemented into programming language C#.

MATERIAL AND METHODS
Algorithm of derivation of Runge -Kutta method is published by several authors, for example in [1].Implementation into C# language was presented in [7].The base of Runge -Kutta method is the formulation of difference between y values in points x n+1 a x n in form: where i w are constants and , ) , ( By writing of both sides of equation ( 2) into Taylor series for i=1..4 members of the series, we get the system of equations for i k and i n h .By comparing we obtain system of parameters: If we solve this system with two degrees of freedom we get: into equations (4), we obtain general form of Runge -Kutta method of fourth order as follows: where:    .
It is also possible to realize the solution when we use this relation: The value i y y  for i x , it holds that . The equation ( 7) can be written through the Taylor series into form: When we use known relation and substitute it into the equation ( 8) we get: By modification of equation ( 9) we obtain final form of the method, which is identical with the relation (5,6).

Technical functions
Technical functions were obtained from experimental manoeuvres of universal tool carrier MT8-222 Synona.Drive manoeuvre was the ride on the slope along the down-grade slope with 45 degree yaw angle with turning to the down-grade slope.The record of centre of gravity velocities of the vehicle is shown in Fig. 1 Fig. 1 Record of velocities.Source: Authors Implementation of Runge-Kutta 4-th order with C# Algorithm of numerical integration was implemented into computer language C#.We used standard models of object programming.Solitary function values with the needed parameters for evaluation are contained in the structure RK_DATA_STRUCT.There was chosen following step of numerical integration: 1 56 t  .
Dislocations of centre of gravity towards inertial coordinate system are the result of numerical integration of technical exciting function of translational velocity of centre of gravity.The resulting functions are depicted in Fig. 2. In the contribution we present analysis of the Runge Kutta method of fourth order for numerical integration of continuous technical functions with the constant step of numerical integration such as numerical solution of the system of simultaneous differential equations.We introduce algorithm of numerical integration implemented by programming language Microsoft Visual C#.As the demonstration of data processing we present the processing of real values of technical exciting functions of the centre of gravity velocities of off-road technologic vehicle and their visualisation.

Fig. 2
Fig. 2 Dislocation of centre of gravity.Source: Authors Solitary evaluation of numerical integration is included in function Runge_Kutta, where input and output parameter of the function is defined by structure RK_DATA_STRUCT.