Shifted impact oscillator: Tuned multiple resonances and step load

Abstract

Impact oscillator with non-zero bouncing point or shifted impact oscillator is a linear oscillator that only moves above a certain value of displacement. It is the simplest model of non-linear and non-smooth dynamic systems such as drill bit - rock system with long (and soft) drill string or materials with pre-opened internal cracks or delamination surfaces which considerably reduce the material effective moduli when open. Positive bouncing point (displacement shift) reduces the period of free oscillations, while the negative bouncing point increases it as compared to the classical (unshifted) impact oscillator. Furthermore, the ratio of amplitudes (or energies) of the first two harmonics of the free oscillator can be used for the determination of the bouncing point. We found two types of multiple resonances. Type 1 resonances involve the periods of free oscillations. The driving force producing resonances of this type is harmonic (with the frequency equal or multiple to the “eigenfrequency”) with a shift. As the resonance progresses (from cycle to cycle) the period increases such that both the driving frequency and the shift should be adjusted for each driving frequency. Type 2 resonances are based on the period (and “eigenfrequency”) of free oscillations of the unshifted impact oscillator. The driving force creating resonances is also harmonic and with a shift. The resonances coincide with the corresponding resonances of the unshifted impact oscillator after adding the displacement shift. Under step load the shifted impact oscillator shows periodic solutions similar to the free vibrations but with the amplitude and period depending upon the impact amplitude (both positive and negative).