Oscillatory systems with two degrees of freedom and van der Pol coupling: Analytical approach

In this paper steady-state vibrations of the two-degrees-of-freedom oscillatory systems with van der Pol coupling are investigated. The model is a system of two differential equations with weak nonlinearity. A new solving procedure based on D′Alembert's method and the method of time-variable amplitude and phase is developed. The main advantage of the method in comparison to others is that it gives the solution of the system of two coupled weak nonlinear equations in the form that is simple to analyze, as it has the same form as the solution of the corresponding system of linear equations. In the paper two types of systems are considered: one, a two-mass system with two degrees of freedom, and second, the one-mass system with two degrees of freedom. The torsional vibrations of a two-mass system and vibrations of a Jeffcott rotor with two-degrees-of-freedom are analyzed. Analytically obtained results are numerically tested. It is obtained that the difference between analytic and numeric results is small and almost negligible. As the accuracy of the analytic solution is high, it is suitable for application in technics and engineering. Conclusions about steady-state self-sustainable oscillators, orbital, and limit cycle motions are given.