Fast-slow analysis of van der Pol-Duffing oscillator coupled nonlinear energy sink at different scales

Abstract

The van der Pol-Duffing oscillator is widely used in various linear motion mechanisms due to the nonlinear effects of spring force and friction, and is accompanied by self-excited oscillation. Continuous self-excited oscillation can cause serious damage to equipment and production life, so it is extremely necessary to suppress its self-excited oscillation. This article focuses on the vibration control problem of van der Pol-Duffing oscillators at different scales, and innovatively uses a coupled nonlinear energy sink (NES) to suppress unexpected oscillations of external excitation and self-excited systems to reduce oscillation amplitude. Apply the complex variable averaging method to approximate the analytical solution of the coupled system and compare it with the numerical solution of the fourth-order Runge Kutta method. In the primary resonance region, different scale coupling characteristics are exhibited based on the difference in structural mass scale. Using the fast-slow analysis method, the slow invariant manifold is explored to determine the excitation amplitude range that causes the strongly modulated response, and it is found that the vibration reduction effect is optimal at this time. In the non-primary resonance region, there are also coupling phenomena of different scales based on frequency level differences. The fast-slow analysis method is used to evaluate the vibration reduction effect of the coupled NES system and explain the vibration reduction mechanism, clarifying that the change in the balance point type of the autonomous system is the key factor in the vibration reduction of the non-autonomous system.