Integro-differential Chapman-Kolmogorov equation for continuous-jump Markov processes and its use in problems of multi-component renewal impulse process excitations

In the present paper the applications of the integro-differential Chapman-Kolmogorov equation to the problems of pure-jump stochastic processes and continuous-jump response processes are discussed. The pure-jump processes considered herein are the counting Poisson process, a two-state jump process, and a multi-state jump process. The differential equations governing the Markov state probabilities are obtained from the degenerate, pure differential form, of the general, integro-differential Chapman-Kolmogorov equation, with the aid of the jump probability intensity functions. The continuous-jump response process is the response of a dynamic system to a multi-component renewal impulse process excitation. The excitation consists of a number of n statistically independent random trains of impulses, each of which is driven by an Erlang renewal process with parameters ν_j,k_j. Each of the impulse processes is characterized by an auxiliary zero-one jump stochastic process, which consists of k_j negative exponential distributed phases. The Markov states for the whole problem are determined by the coincidences of the phases of the individual jump processes. Thus the response probability distribution may be characterized by a joint probability density-discrete distribution of the state variables of the dynamic system and of the states of the pertinent Markov chain. The explicit integro-differential equations governing the joint probability density-discrete distribution of the response are obtained from the general forward integro-differential Chapman-Kolmogorov equation, after the determination of the jump probability intensity functions for the continuous-jump and pure-jump processes.     

Explicit solutions for the response probability density function of linear systems subjected to random static loads

In the present work a new version of the Probabilistic Transformation Method (PTM) has been reported for the study of linear systems subjected to static random loads. Even if this application could appear trivial, it allows to find some exact results, difficulty obtainable by other approaches. In particular, some interesting results have been obtained in the case of uniformly distributed random loads. For a generic vector of random loads this version of the PTM has allowed to obtain the characteristic function (cf) of any response elements in a very simple and effective way.     

Response of dynamic systems to renewal impulse processes: Generating equation for moments based on the integro-differential Chapman–Kolmogorov equations

In the present paper the method is developed for the derivation of differential equations for statistical moments of the state vector (response) of a non-linear dynamic system subjected to a random train of impulses. The arrival times of the impulses are assumed to be driven by a non-Poisson counting process. The state vector of the dynamic system is then a non-Markov process and no method is directly available for the derivation of the equations for response moments. The original non-Markov problem is converted into a Markov one by recasting the excitation process with the aid of an auxiliary, pure-jump stochastic process characterized by a Markov chain. Hence the conversion is carried out at the expense of augmentation of the state space of the dynamic system by auxiliary Markov states. For the augmented problem the sets of forward and backward integro-differential Chapman–Kolmogorov equations are formulated. The general, generating equation for moments is obtained with the aid of the forward and backward integro-differential Chapman–Kolmogorov operators. The developed method is illustrated by the examples of several renewal impulse processes.     

Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method

The development of high-speed railway has made it important to clarify the influence of random system parameters (i.e. vehicle load, elastic modulus, damping ratio, and mass density of bridge) on train-bridge dynamic interactions. The probability density evolution method (PDEM), a newly developed theory which is applicable to train-bridge systems, can capture instantaneous probability density functions of dynamic responses. In this study, PDEM is employed to implement random dynamic analysis of a 3D train-bridge system subjected to random system parameters. The number theory method (NTM) is employed to choose the representative point sets of random parameters, whose initial probability distribution is divided by Voronoi cells., MATLAB® software is prepared for calculation, the Newmark-β integration method and the bilateral difference method of TVD (total variation diminishing) are adopted for solution. A case study is presented in which the train travels on a three-span simply supported high-speed railway bridge. The calculation accuracy and computational efficiency of the PDEM has been verified and some conclusions are provided. Furthermore, the influence of train speed under various combinations of random parameters is beyond discuss.     

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.     

Response of linear dynamic systems to non-Erlang renewal impulses: Stochastic equations approach

Linear dynamical systems under random trains of impulses driven by a class of non-Erlang renewal processes are considered. The class considered is the one where the renewal events are selected from an Erlang renewal process. The original train of impulses is recast, with the aid of an auxiliary stochastic variable, in terms of two independent Poisson processes. Thus, by augmenting the state vector of the dynamic system with the auxiliary stochastic variables, the original non-Markov problem is converted to a Markov one.

The differential equations for the response statistical moments can then be derived from the generalised Ito's differential rule.

Numerical results obtained for a few different models and various sets of parameters, show that the present approach allows to account for a variety of inter arrival time's probability distributions. Transient mean value and variance of the response of a linear oscillator have been obtained from the equations for moments.     

A method for the evaluation of the response probability density function of some linear dynamic systems subjected to non-Gaussian random load

This paper deals with the characterization of the random response of linear systems subjected to stochastic load. It proposes a new method based on the new version of the Probabilistic Transformation Method (PTM) that allows obtaining, with a very low computational effort, the probability density function of the response. An important aspect of the proposed approach is the ability to join directly the pdfs of the input load with those of the response. Based on the step-by-step integration method, explicit solutions will be proposed for the random response of systems loaded by seismic and windy sampled inputs.     

Stochastic direct integration schemes for dynamic systems subjected to random excitations

In this work, a stochastic version of direct integration schemes is constructed, based on a general recursive state space formulation. The technique is applicable to evaluating the second order response statistics of systems subjected to non-stationary random excitations, and is potentially able to handle non-proportional damping. A specific group of these algorithms obtained using one of the most frequently used families of time stepping schemes is introduced. It is shown that statistics obtained for two multi-degree of freedom (MDOF) dynamic systems using the proposed method are in close agreement with those obtained from Monte Carlo simulation when a proper time step is used. Stability of the proposed scheme is systematically investigated and it is concluded that, for the response statistics to be bounded, the time step used in the algorithm has to lie within certain limits.     

Explicit solutions for the response probability density function of nonlinear transformations of static random inputs

This work is the second paper of two companion ones. Both of them show the use of a new version of the Probabilistic Transformation Method (PTM) for finding the probability density function (pdf) of a limited number of response quantities in the transformations of static random inputs. This is made without performing multi-dimensional integrals of the response total joint pdf for saturating the non-interested variables. While in the first paper the linear transformations have been considered, in the present one some nonlinear systems are taken into account. In particular, first the case when the loads on a linear structural system are a nonlinear combination of static random inputs is studied. Then the attention is placed on the case of nonlinear structural systems, for which the new version of the PTM allows to determine approximated, but accurate, results.     

An RKPM-based formulation of the generalized probability density evolution equation for stochastic dynamic systems

In the stochastic dynamic analysis, the probability density evolution method (PDEM) provides an optional way to capture the complete probability distribution of the stochastic response of general nonlinear systems. In the PDEM, the key point is to solve the generalized probability density evolution equation (GDEE), which governs the evolution of the joint probability density function (PDF) of the response and the randomness. In this paper, a new numerical method based on the reproducing kernel particle method (RKPM) is proposed. The GDEE can be approximated through the RKPM. By some particles in the response domain, the instantaneous PDF and its partial derivative with respect to response are smoothly expressed. Then, the approximated GDEE can be discretized directly at the collocation points in the response domain. At the same time, discretization in the time domain is achieved by the difference scheme. Therefore, the RKPM-based formulation to obtain the numerical solution of GDEE is formed. The implementation procedure of the proposed method is given in detail. The accuracy and efficiency of this method are illustrated with some numerical examples. Some details of parameter analysis are also discussed.     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

Reliability of dynamic systems under limited information

A method is developed for reliability analysis of dynamic systems under limited information. The available information includes one or more samples of the system output; any known information on features of the output can be used if available. The method is based on the theory of non-Gaussian translation processes and is shown to be particularly suitable for problems of practical interest. For illustration, we apply the proposed method to a series of relevant examples and compare with results given by traditional statistical estimators. It is demonstrated that the method delivers accurate results for the case of linear and nonlinear dynamic systems, and can be applied to analyze experimental data and/or mathematical model outputs.     

Analysis of multi-degree of freedom strongly non-linear mechanical systems with random input: Part II: equivalent linear system with random matrices and power spectral density matrix

We present a method for estimating the (power spectral density) PSD matrix of the stationary response of lightly damped randomly excited multi-degree of fredom mechanical systems with strong non-linear asymmetrical restoring forces. The PSD matrix is defined as the mean value of the PSD matrix response of an equivalent linear system (ELS) whose damping and stiffness matrices depend on non-linear vibration modes of the associated conservative system, the frequencies and modes shapes being amplitude dependent. Based on a generalized van der Pol transformation and using a stochastic averaging principle, as developed in a companion paper, a stationary probability density function for the amplitude process is derived to characterize the ELS fully. Some possible simplifications of the method, such as modal reduction and/or local linearization, are also discussed. The results obtained are in good agreement with those of direct numerical simulations taking two typical examples.     

Equivalent nonlinearization of nonlinear systems to random excitations

The broadest class of solvable reduced Fokker-Planck equation is given and a new equivalent nonlinear method is presented to obtain an approximate probability density function for the response of a nonlinear oscillator to Gaussian white noise excitations. The method is based on the least mean-square criterion and Euler equation. It is shown that this method, which is simpler and more reasonable, generalizes Caughey's method and gives the same results as Cai and Lin under purely additive excitations. Examples are given to show the applications of the method. In one of the examples, this method leads to a better approximation than that obtained from the energy dissipation criterion.     

Almost-sure stability of a class of distributed parameter systems subjected to random excitations

The stability of a class of distributed parameter systems subjected to stationary and non-stationary randomx excitations is studied. Several criteria regarding the almost-sure stabihty of the equilibrium state of the system are developed. A variational scheme for obtaining sharp stability criteria is formulated-Examples of flutter of a panel in a random wind and stability of a tall structure under a seismic load are solved to illustrate the applicability of the method.     

On deriving exact probability density functions for functionally graded exponential profile orthotropic plates driven by laterally random excitation

In this article, the exact probability density function for a nonlinear exponential functionally graded orthotropic plate under lateral random excitation is investigated. After that, a study on the probability density function (PDF) is done to examine the dynamic instability and bifurcation. Using nondimensional parameters, the results are justified for a wide range of plates. The outcomes are studied with respect to the variation of both mean value of lateral loads and in-plane forces. The famous Monte Carlo simulation is employed to validate the obtained analytical PDFs. An analogical study is done between the behaviors of homogenous plates with their corresponding functionally graded material ones. Finally, it is shown how the exponential profile functionally graded orthotropic material can affect the quality of instability and bifurcation.     

Reliability analysis for multi-state systems subject to distinct random shocks

This paper analyzes the competing and dependent failure processes for multi-state systems suffering from four typical random shocks. Reliability analysis for discrete degradation is conducted by explicitly modeling the state transition characteristics. Semi-Markov model is employed to explore how the system vulnerability and potential transition gap affect the state residence time. The failure dependence is specified as that random shocks can not only lead to different abrupt failures but also cause sudden changes on the state transition probabilities, making it easier for the system to stay at the degraded states. Reliability functions for all the exposed failure processes are presented based on the corresponding mechanisms. Interactions between different failure processes are also taken into account to evaluate the actual reliability levels in the context of degradation and distinct random shocks. An illustrative example of a multi-state air conditioning system is studied to demonstrate how the proposed method can be applied to the engineering practice.     

Model selection for a class of stochastic processes or random fields with bounded range

Methods are developed for finding an optimal model for a non-Gaussian stationary stochastic process or homogeneous random field under limited information. The available information consists of: (i) one or more finite length samples of the process or field; and (ii) knowledge that the process or field takes values in a bounded interval of the real line whose ends may or may not be known. The methods are developed and applied to the special case of non-Gaussian processes or fields belonging to the class of beta translation processes. Beta translation processes provide a flexible model for representing physical phenomena taking values in a bounded range, and are therefore useful for many applications. Numerical examples are presented to illustrate the utility of beta translation processes and the proposed methods for model selection.