Effect of angular speed variations on the nonlinear vibrations of a rotational spring-mass system

A rotating spring-mass system is considered using polar coordinates. The system contains a cubic nonlinear spring with damping. The angular velocity harmonically fluctuates about a mean velocity. The dimensionless equations of motion are derived first. The velocity fluctuations resulted in a direct and parametric forcing terms. Approximate analytical solutions are sought using the Method of Multiple Scales, a perturbation technique. The primary resonance and the principal parametric resonance cases are investigated. The amplitude and frequency modulation equations are derived for each case. By considering the steady state solutions, the frequency response relations are derived. The bifurcation points are discussed for the problems. It is found that speed fluctuations may have substantial effects on the dynamics of the problem and the fluctuation frequency and amplitude can be adjusted as passive control parameters to maintain the desired responses.