On the stability properties of a damped oscillator with a periodically time-varying mass

In this paper the vibrations of a damped, linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be considered. Both the free and forced vibrations of the oscillator will be studied. For the free vibrations the minimal damping rates will be computed, for which the oscillator is always stable. The forced vibrations are partly due to small masses, which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator. Not only solutions of the oscillator equations will be constructed, but also stability properties for the free, and for the forced vibrations will be presented for various parameter values. For the external, harmonic forcing case an interesting resonance condition will be derived.