Numerical-analytical transformation method of analyzing nonlinear mathematical models with polynomial structure

Abstract

The article considers a scientific problem of investigating mathematical models with polynomial nonlinearity, which are represented by systems of nonlinear differential equations. There is presented a numericall-analytical transformation method for investigating nonlinear dynamical systems with many degrees of freedom of polynomial structure. As opposed to the analogues, this method allows solving a wide range of problems for nonlinear systems of general polynomial structure while reducing the computational resource intensity. An algorithm of the method of transformations for investigating the nonlinear dynamic systems with m degrees of freedom is given. The research of stability of solutions of nonlinear dynamic systems with m degrees of freedom is carried out. Algorithmic formulas of transformation method for the solution of systems with nonlinearity of the fourth degree are given. Computational experiments on solving systems of differential equations with a small nonlinear part in the form of a multinomial of the sixth degree using the transformation method show fourth order accuracy in computation. A nonlinear mathematical model of the vibration protection system is investigated by the presented method of transformations. The theorem on determination of the stationary state for systems of differential equations of polynomial structure by transformation method is proved. Algorithmic formulas for computation are presented. A general matrix form for vector indices is presented. Formulas for economical computation of right-hand sides of polynomial structure are presented and it is proposed to apply Pan's scheme with coefficient preprocessing. This method allows studying the different modes of nonlinear models dynamics, for example, to determine such extreme modes as resonance, sub-harmonic, and polyharmonic modes. As an example of the method application, the vibration protection problem of the tower from external periodic influences was solved. The transformed solution takes into account all nonlinear polynomial components. The method allows to investigate the dynamics of a wide range of nonlinear systems with the necessary accuracy