Algorithm of the Hybrid Transformation Method for Modeling Dynamic Systems

Abstract

The paper solves the scientific problem of developing a method algorithm for the study of nonlinear dynamic systems. To increase the accuracy and speed of calculations, the paper proposes an algorithm for the hybrid method of transformations for the study of nonlinear mathematical models of a polynomial structure. The paper presents a method of polynomial transformations for the study of systems with three degrees of freedom, a study of a nonlinear vibration protection system with three degrees of freedom is carried out. The paper presents a hybrid numerical-analytical method for the analysis of nonlinear mathematical models of a general polynomial structure, which makes it possible to study systems with controlled accuracy while reducing the resource intensity of calculations. The method introduces additional complex exponential variables, formulas for calculating the transformation coefficients and the transformed system are presented. An analytical solution is constructed for the transformed system in the resonant and nonresonant cases. For the economical calculation of the right parts of the polynomial structure, formulas are presented and it is proposed to apply Pan's scheme with preliminary processing of the coefficients. The developed algorithm of the method of polynomial transformations makes it possible to construct an approximate analytical solution, taking into account the nonlinear components of higher degrees of the polynomial. The proposed algorithm of the method makes it possible to study the dynamic characteristics of the object under study, special cases of subharmonic, polyharmonic regimes, determine extreme regimes, and resonance with controlled accuracy. The above algorithm is implemented in the created software package using the modern object-oriented programming language C#. The proposed method makes it possible to carry out a qualitative and quantitative analysis of models of dynamic systems.