R. Iwankiewicz

Dynamic response of non-linear systems to poisson-distributed pulse trains: Markov approach

Abstract The dynamic response of non-linear systems with algebraic non-linearities to Poisson-distributed trains of impulses and general pulses is considered. The displacement and velocity response of the system form in that case a Poisson-driven Markov vector process. The differential equations governing the joint response moments are obtained by making use of a generalized Itos differential rule which is valid for this kind of problems. Two closure techniques are used to truncate the hierarchy of moment equations: an ordinary and a modified cumulant-neglect closure technique. Transient response statistics such as the mean value and the variance are evaluated numerically. Verification of the obtained approximate analytical results against Monte Carlo simulations shows that the ordinary cumulant-neglect closure technique is appropriate in the case of non-linear systems subject to Poisson-distributed impulses and general pulses if the mean arrival rate of impulses is not very low, i.e. if the departure of the excitation from the Gaussian process is not very large. Otherwise, i.e. in the case of a low mean arrival rate of impulses, the modified cumulant-neglect closure scheme provides better results.

Vibration of a Beam under a Random Stream of Moving Forces

ABSTRACT The problem of dynamic response of a beam to the passage of a train of concentrated forces with random amplitudes is considered. Force arrivals at the beam are assumed to constitute a Poisson process of events. Thus, the excitation process idealizes vehicular traffic loads on a bridge. An analytical technique is developed to determine the response of the beam. This technique is based on the introduction of two influence functions, one of which satisfies nonhomogeneous, the other homogeneous differential equations for beam response. Explicit expressions for expected value and variance of the beam deflection are provided. As an example, the response of a beam to a stationary stream of forces is determined for some practical situations and discussed. The extension of the approach presented to the case of two-axle vehicles is also outlined

Dynamic response of hysteretic systems to Poisson-distributed pulse trains

Abstract A single-degree-of-freedom hysteretic system subjected to a specific non-Gaussian random excitation in the form of a Poisson-distributed train of random pulses is considered. The hysteretic behaviour is described by the Bouc-Wen model of smooth hysteresis. The total hysteretic energy dissipation is assumed as a cumulative damage indicator. The state variables of the system together with the damage indicator form in that case a Poisson-driven, non-diffusive Markov vector process. Two equivalent systems are introduced by substituting the original non-analytical, non-algebraic non-linearities by equivalent linear and cubic forms in the pertinent state variables. Equations for mean responses are obtained by direct averaging of the governing equations, whereas the equations for second- and higher-order joint central moments are derived from the equivalent systems with the help of a generalized Itos differential rule. Appearing in the equations for mean responses and for equivalent coefficients, expectations of non-algebraic functions of state variables are performed with respect to a non-Gaussian joint probability density function assumed in the form of a generalized, bivariate Gram-Charlier expansion. In the case of equivalent cubic non-linearities, the equations for moments form an infinite hierarchy which is truncated with the help of a cumulant-neglect closure technique. Mean values and variances of the response variables are evaluated by the numerical integration of equations for moments for two equivalent systems. For the sake of comparison the excitation process is also substituted by a Gaussian white noise and the usual equivalent linearization technique, combined with the Gaussian closure, is applied. The cases of non-zero-mean as well as zero-mean excitation processes are included. The case of general pulses is dealt with by a suitable augmentation of the state vector. The accuracy of the analytical techniques developed is verified against Monte Carlo simulations.

Vehicle Moving Along an Infinite Beam With Random Surface Irregularities on a Kelvin Foundation

The paper deals with the stochastic analysis of a single-degree-of-freedom vehicle moving at a constant velocity along an infinite Bernoulli-Euler beam with surface irregularities supported by a Kelvin foundation. Both the Bernoulli-Euler beam and the Kelvin foundation are assumed to be constant and deterministic. This also applies to the mass, spring stiffness, and damping coefficient of the vehicle. At first the equations of motion for the vehicle and beam are formulated in a coordinate system following the vehicle. The frequency response functions for the displacement of the vehicle and beam are determined for harmonically varying surface irregularities. Next, the surface irregularities are modeled as a random process. The variance response of the mass of the vehicle as well as the displacement variance of the beam under the oscillator are determined in terms of the autospectrum of the surface irregularities. ©2002 ASME

Dynamic response of non-linear systems to renewal-driven random pulse trains

Abstract The moment equations technique is developed for the non-linear dynamical systems under random pulse trains driven by a class of renewal processes. Since the increments of the considered point process are not statistically independent, the direct application of the generalized Itos differential rule does not yield the explicit equations for moments. Hence, the approach is suitably modified. First, the excitation term is recast, for an ordinary renewal process with gamma-distributed, with k = 2, interarrivai times, as a transformation of a Poisson counting process, which allows to perform the averaging of the differential rule. Next, for the additional unknown expectations which consequently appear in the equations for moments, the differential equations in the form of the correlation splitting formulae are derived. The technique developed is applied to a linear oscillator and to a Duffing oscillator. In the latter case, suitable closure approximations are used in order to truncate the hierarchy of moment equations. The analytical results (transient response moments up to fourth order) are verified against the results of Monte Carlo simulations.

Stochastic Stability of Mechanical Systems Under Renewal Jump Process Parametric Excitation

A dynamic system under parametric excitation in the form of a non-Erlang renewal jump process is considered. The excitation is a random train of nonoverlapping rectangular pulses with equal, deterministic heights. The time intervals between two consecutive jumps up (or down), are the sum of two independent, negative exponential distributed variables; hence, the arrival process may be termed as a generalized Erlang renewal process. The excitation process is governed by the stochastic equation driven by two independent Poisson processes, with different parameters. If the response in a single mode is investigated, the problem is governed in the state space by two stochastic equations, because the stochastic equation for the excitation process is autonomic. However due to the parametric nature of the excitation, the nonlinear term appears at the right-hand sides of the equations. The equations become linear if the state space is augmented by the products of the original state variables and the excitation variable. Asymptotic mean and mean-square stability as well as asymptotic sample (Lyapunov) stability with probability 1 are investigated. The Lyapunov exponents have been evaluated both by the direct simulation of the stochastic equation governing the natural logarithm of the hyperspherical amplitude process and using the modification of the method wherein the time averaging of the pertinent expressions is replaced by ensemble averaging. It is found that the direct simulation is more suitable and that the asymptotic mean-square stability condition is not overly conservative.