Resonances of a forced van der Pol equation with parametric damping

This work entails an analysis of secondary resonances in the parametrically damped van der Pol equation, with and without external excitation. A potential application of this system is a vertical-axis wind-turbine blade, which can have cyclic damping, aeroelastic self-excitation, and direct excitation. We analyze the system using the method of multiple scales and numerical solutions. For the case without external excitation, the analysis reveals nonresonant phase drift (quasiperiodic responses) and subharmonic resonance with possible phase drift or phase locking (periodic responses). The case of external excitation consists of a constant load and a harmonic load with the same frequency as the parametric term. Hard excitation is treated for nonresonant conditions and secondary resonances. Subharmonic and superharmonic resonances show possible phase drift and phase locking. Primary resonance is observed but not analyzed here.

Automated summary

This work analyzes secondary resonances in the parametrically damped van der Pol equation, both with and without external excitation. It focuses on a potential application in vertical-axis wind-turbine blades, which experience cyclic damping, aeroelastic self-excitation, and direct excitation. The system is studied using the method of multiple scales and numerical solutions. The analysis reveals various responses, including nonresonant phase drift, subharmonic resonance, and potential phase locking.