Construction of Approximate Analytical Solutions to Strongly Nonlinear Coupled van der Pol Oscillators

摘要

Using nonlinear theory to research vibration model of engineering system has important theoretical and practical significance. Multi-degree-of-freedom (MDOF) coupled van der Pol oscillator is a typical model in the nonlinear vibration; many complex dynamic problems in practical engineering can be simplified as this model to be solved in the end. This paper discusses a class of two-degrees-of-freedom (2-DOF) coupled van der Pol oscillator, which was divided into three parameters of different situations alpha(1) not equal alpha(2), beta(1) not equal beta(2), and gamma(1) not equal gamma(2) to discuss. Employing symbolic software such as Mathematica for those problems, the explicit analytical solutions of frequency omega and displacements x(1) (t) and x(2) (t) are well formulated. Results showed that the homotopy analysis method (HAM) can effectively deal with this kind of parameter of different coupled vibrators, just request the values of some parameters are not too big. Finally, we got four important theorems to simplify the solution of the nonlinear system.