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The most suitable model that idealizes random sequences of shock and impacts on vibratory systems is that of a random train of pulses (or impulses), whose arrivals are characterized in terms of stochastic point processes. Most of the existing methods of stochastic dynamics are relevant to random impulsive excitations driven by Poisson processes and there exist some methods for Erlang renewal-driven impulse processes. Herein, two classes of random impulse processes are considered. The first one is the train of impulses whose interarrival timesare driven by an Erlang renewal process. The second class is obtained by selecting some impulses from the train driven by an Erlang renewal process. The selection is performed with the aid of the jump, zero-one, stochastic process governed by the stochastic differential equation driven by the independent Erlang renewal processes. The underlying counting process, driving the arrival times of the impulses, is fully characterized. The expressions for the probability density functions of the first and second waiting times are derived and by means of these functions it is proved that the underlying counting process is a renewal (non-Erlang) process. The probability density functions of the interarrival times are evaluated for four different cases of the driving process and the results obtained for some example sets of parameters are shown graphically.
The advantage of modeling the interarrival times using the class of non-Erlang renewal processes analyzed in the present dissertation, rather than the Poisson or Erlang distributions is that it is possible to deal with a broader class of the interarrival probability density functions. The non-Erlang renewal processes considered herein, obtained from two independent Erlang renewal processes, are characterized by four parameters that can be chosen to fit more closely the actual data on the distribution of the interarrival times.
As the renewal counting process is not the one with independent increments, the state vector of the dynamic system under a renewal impulse process excitation is not a Markov process. The non-Markov problem may be then converted into a Markov one at the expense of augmenting the state vector by auxiliary discrete stochastic variables driven by a Poisson process. Other than the existing in literature (Iwankiewicz and Nielsen), a novel technique of conversion is devised here, where the auxiliary variables are all zero-one processes. In a considered class of non-Erlang renewal impulse processes each of the driving Erlang processes is recast in terms of the Poisson process, the augmented state vector driven by two independent Poisson processes becomes a non-diffusive Markov process.
For a linear oscillator, under a considered class of non-Erlang renewal impulse process, the equations for response moments are obtained from the generalized Ito’s differential rule and the mean value and variance of the response are evaluated and shown graphically for some selected sets of parameters.
For a non-linear oscillator under both Erlang renewal-driven impulses and the considered class of non-Erlang renewal impulse processes, the technique of equations for moments together with a modified closure technique is devised.
The specific physical properties of an impulsive load process allow to modify the classical cumulant-neglect closure scheme and to develop a more efficient technique for the class of excitations considered. The joint probability density of the augmented state vector is expressed as sum of contributions conditioned on the ‘on’ and ‘off’ states of the auxiliary variables. A discrete part of the joint probability density function accounts for the fact that there is a finite probability of the system being in a deterministic state (for example at rest) from the initial time to the occurrence of the first impulse. The continuous part, which is the conditional probability given that the first impulse has occurred, can be expressed in terms of functions of the displacement and velocity of the system. These functions can be viewed as unknown probability densities of a bi-variate stochastic process, each of which originates a set of ‘conditional moments’. The set of relationships between unconditional and conditional moments is derived. The ordinary cumulant neglect closure is then performed on the conditional moments pertinent to the continuous part only. The closure scheme is then formulated by expressing the ‘unconditional’ moments of order greater then the order of closure, in terms of unconditional moments of lower order.
The stochastic analysis of a Duffing oscillator under the the random train of impulses driven by an Erlang renewal processes and a non-Erlang renewal process R(t), is performed by applying the second order ordinary cumulant neglect closure and the modified second order closure approximation and the approximate analytical results are verified against direct Monte Carlo simulation. The modified closure scheme proves to give better results for highly non-Gaussian train of impulses, characterized by low mean arrival rate. |
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