The modified Poincare-Dulac method in analysis of autooscillations of nonlinear mechanical systems

Abstract

A dynamic equation of a mechanical system with one degree of freedom with functionally odd nonlinear recover forces and small dissipative forces is considered. An improved method of transformation and integration of the equation, based on a method of normalization of Poincare-Dulac is developed. The improvement of the method is in application of the Chebyshev's economization to high-order nonlinear terms.

Mathematical modeling of the system of drilling rig

“…In contrast to the methods of calculation which proposed in [16], following method of polynomial transformations is applied [15]. We write the system of equations in matrix form: 1 Z Z R    .…”

“…We write the system of equations in matrix form: 1 Z Z R    . According to the method [15] to make substitution of variables:…”

“…According to the method of polynomial transformations [15] was found the coefficients of the transformed for autonomous system:…”

“…Among the known methods [10,13,14]: the method of averaging, the small parameter method, the method of Krylov-Bogolyubov, method of polynomial transformation, the method of Van der Pol, the method of harmonic balance, the method of Poincare. In the calculation of non-linear system with six degrees of freedom applied the method of polynomial transformations [15]. As a result of polynomial transformation the original system of nonlinear differential equations is transformed to the autonomous form.…”

On the equation of fourth order with quadratic nonlinearity

“…For the construction of solutions of nonlinear differential equations in partial derivatives [6][7][8][9][10][11][12] is used different analytical and numerical methods: the perturbation methods, the small parameter method, the separation of variables method, the linearization method, the averaging method, the method of the stretched coordinates, the method of composite expansions, grid methods -the method of finite differences and the finite element method [13][14][15][16][17][18][19].…”