Mathematical Modelling and Resonance Analysis in Impact Oscillators

A variety of mechanical phenomena can be simulated by modeling bilinear oscillators which have different stiffness in pressure and tension. In this paper, bilinear oscillators with unlimited stiffness in compression say impact oscillators, and the eigenset which is homogenous solution of equation without external load are investigated. The results show that this set and the corresponding subsets are stable with respect to variation in initial conditions. In addition, among all periodic collections of impact times which are proportional to the period of external load, only the eigenset can support resonance, especially the multi-harmonic resonance. The rest of the resonances should produce the nonperiodic impact times. This phenomenon shows that the usual assumption that the times between impacts are proportional to the period of external load is not always confirmed. Furthermore it is shown that in half frequency of the main resonance (the first sub-harmonic resonance), the impact times are close to the eigenset and unlike linear increase of multi-harmonic resonances, the envelope of the oscillations increases as a square root of time.