On the impulse response of a vibrator with a band-limited hysteretic damper

The ideal linear hysteretic damper possesses a non-causal impulse response precursor (IRP). This has been proved in theory and confirmed numerically. Yet the mathematical conciseness of this model is so attractive that several modifications have been tried to restrain this pitfall. In this paper, the modification proposed by Bishop et al., the band-limited hysteretic damper was studied in detail. The non-causal and causal parts of the unitary impulse response function h(t) were investigated systematically by the residue theorem. The closed form of h(0) was derived, then its property was studied numerically. The main conclusions are as follows: Firstly, the Bishop’s model is still non-causal. The asymptotic rate of the IRP is of an order t−1 as time approaches negative infinity. Secondly, the IRP is not monotonic, but a damped sinusoidal wave. The frequency of this wave relates to the lower bound of the band-limited damper. Thirdly, h(0) can be zero by appropriately combining the lower and upper bounds of the pass band.