Dynamic response of linear systems to random trains of non-overlapping pulses with arbitrary shapes

摘要

The excitation considered is a train of non-overlapping pulses, defined as a train of alternate random pulse durations and time gaps between the consecutive pulses. As the time gaps are regarded as continuous random variables, hence the pulses are non-adjoining, i.e. they can only adjoin with probability zero. All durations are assumed to be characterized by one common probability distribution. Likewise, the time gaps are all identically distributed, but the probability distribution of the durations and of the time gaps are different. As the interarrival time is the sum of the duration and of the time gap, the arrival times of the pulses are driven by a renewal process. It is further assumed that the durations and time gaps are negative exponential distributed, with different parameters. Pulse shapes are given by an arbitrary, deterministic function and pulse heights are given by independent, identically distributed random variables. The mean value and autocorrelation function of the pulse train are evaluated analytically for "on" and "off' initial conditions and are found to be expressed in terms of the renewal densities pertinent to the renewal process driving the arrival times. It is shown that for rectangular pulses with equal, deterministic heights these expressions reduce to the ones obtained independently with the aid of another approach, where the mean value and autocorrelation function are given in terms of "on" probabilities (i.e. the probabilities of the pulse being on). The mean value and variance of the response of a linear oscillator are obtained via a time domain analysis. Illustrative numerical results are given for the train of pulses with both parabolic and rectangular pulse shapes.