Insight into the dynamic behaviour of the Van der Pol/Raleigh oscillator using the internal stiffness and damping forces

The van der Pol oscillator is an archetypal nonlinear oscillator that has been studied for many years. It is a self-sustaining oscillator that vibrates in a limit cycle, and has the characteristic that it generates energy in the part of the cycle when the displacement is small and dissipates energy in the part of the cycle when the displacement is large. Almost all analyses for this type of oscillator have been conducted in a strict mathematical framework using the displacement and velocity of the mass to describe the motion in the phase plane. Physical insight into the behaviour is then generally only possible for very small or for very large damping nonlinearity. In this paper a fresh approach is taken. The internal forces of the Rayleigh oscillator are studied rather than van der Pol's equation as the key damping force is a function of only velocity. Simulations are presented which show how the stiffness and damping forces vary when the system is vibrating in a steady-state limit cycle.