Bivariate 'Cut-Glue' Approximation of Strongly Nonlinear Mathematical Models Based on Experimental Data

摘要

Researchers meet the difficulties of experimental and computer modeling of a statics and dynamics of aircrafts connected with their essential nonlinearity. This is due to the fact that the aerodynamic effects of the interaction complex aircraft designs or their models with air environment generate abrupt changes of the character of the some dependencies. Aerodynamic coefficients in the model of interaction can be obtained only or by full-scale tests or by computer simulations. Therefore, the construction of mathematical models of the objects is associated with the mathematical processing of the points of the experimental data. In this case, the experimentally obtained dependence is usually essentially nonlinear up to piecewise, or even discontinuous nature. Approximation of such dependencies, even with the use of spline functions, is very difficult and is associated with large errors. The solution to this problem was proposed by the author and was reported to the ASME Congress in November 2014 and published in the Proceedings of the Congress in its final form. In that work possibility of "approximating&multiplicative&additive" processing of dot experimental data for creation of unified mathematical model of the studied object or the phenomenon completely is mathematically proved. The offered method is called "Cut-glue" approximation as it is based on "cutting" of the well approximated intervals of the modelled dependence and their "gluing" in the one analytical function. However, this problem has been solved only for the univariate functions case. In this paper the author presents the solution of a problem of the bivariate 《Cut-glue》 approximation, which significantly expands the scope of application of the method. The creation examples of mathematical models are given. Fragments of the flying devices using the aerostatic flight principle are modeled. Examples show that models even piecewise dependences represent the unified analytical functions. However we can approximate their forms to piecewise so, how it is necessary for the accuracy of the description of experimental data. It is shown that combined application of the "Cut-glue" method of approximation and the piecewise description of separate intervals of the modelled experimental dependence by methods of the regression analysis considerably increases the accuracy of the mathematical dependence description in general. For bivariate models the effect of application of a method becomes stronger, because the error of the description of the significantly nonlinear bivariate dependences by regression methods much more, than univariate dependences.