Response Of Stochastic Systems Research Articles

Experimental Investigation of Stochastically Forced Rijke-Type Supercritical Thermoacoustic Systems

Intense thermoacoustic oscillations may lead to severe deterioration due to the induced intolerable damage to combustors. A better understanding of unstable behaviors is important to prevent or suppress these oscillations. Active thermoacoustic coupling in practical combustors is caused primarily by two approaches: inherent turbulent fluctuations and the flame response to acoustic waves. Turbulent fluctuations are generally characterized by random noise. This paper experimentally expands on previous analytic studies regarding the influence of colored disturbances on the thermoacoustic response near the supercritical bifurcation point. Therein, a laboratory-scale Rijke-type thermoacoustic system is established, and both supercritical and subcritical bifurcations are observed. Then, Ornstein–Uhlenbeck (OU)-type external colored noise is introduced near the supercritical bifurcation point, and the effects of the corresponding correlation time τc and noise intensity D are studied. The experimental results show that these variables of the colored noise significantly influence the dynamics of thermoacoustic oscillations in terms of the most probable amplitude and autocorrelation properties. A resonance-like behavior is observed as the noise intensity or the autocorrelation time of the colored noise is continuously varied, which means that the coherent resonance occurs in the thermoacoustic system. Finally, when the system is configured closer to the stability boundary, the extent of the coherence motion is intensified in the stochastic system response. Meanwhile, the signal-to-noise ratios (SNRs) of the colored-noise-induced response are found to become more distinguished, the optimal colored noise intensity decreases, and the optimal autocorrelation time increases. These findings provide valuable guidance to predict the onset of thermoacoustic instabilities.

Efficient non-stationary random vibration analysis of vehicle-bridge system based on an improved explicit time-domain method

Based on the recently developed explicit time-domain method (ETDM), this paper presents a new efficient computational framework for non-stationary random vibration analysis of the vehicle-bridge system (VBS) subjected to random track irregularity excitation. In this framework, the time-varying VBS equation is discretized using the time-stepping integration method, leading to an explicit expression for the stochastic system response in terms of the track irregularity excitation. Based on this expression, the response moments of the VBS can be easily evaluated in a direct way; the response probability density can also be obtained by using the Gaussian characteristic of the irregularity excitation. To further improve the computational efficiency, an efficient spectral representation-based formulation is proposed to calculate the excitation covariance matrix required in the analysis, which avoids tedious Fourier integrals required in the traditional ETDM. In the numerical application, the accuracy and efficiency of the proposed method are validated by the solutions obtained using the Monte Carlo and pseudo-excitation methods; the effects of several parameters on the stochastic responses of the VBS are investigated.

The most probable response of some prototypical stochastic nonlinear dynamical systems

Abstract The response of the stochastic system hardly provides useful information for researching its characteristic. Therefore, the paper is aimed at utilizing a deterministic tool, namely, the most probable response, to explore the stochastic nonlinear system. Firstly, one defines the most probable response of the stochastic system. Then, its analytical solution is derived by incorporating the extremum theory into the associated Fokker–Planck equation. Finally, two examples are given to illustrate respectively the implication of this method. Meanwhile, it can conclude that the large the intensity of multiplicative noise is, the more obvious the impact is on the response of nonlinear system. However, no matter how the intensity of additive noise changes, the most probable response does not change. The numerical method verifies the validity of analytical solution.

Stochastic dynamic analysis of composite plate with random temperature increment

During the service life, laminated composites may be subject to some random thermal environment. Quantification of the uncertainty in static and dynamic response of the composites under such condition is still a challenging issue. This work presents a stochastic dynamic response analysis of a graphite-epoxy composite plate using generalized polynomial chaos (gPC) expansion due to random mean temperature increment. A stochastic finite element method (SFEM) based on the first-order shear deformation theory (FSDT) is used to describe the free and forced vibration response of the graphite-epoxy composite plate under a uniform distribution of the temperature throughout the plate. Newmark’s time integration scheme is used to predict the time-dependent displacement response under dynamic loading. The collocation-based non-intrusive gPC expansion method is used for stochastic dynamic analysis of the graphite-epoxy composite plate. The increment in the temperature is considered as an uncertain parameter and presented by the truncated gPC expansion. The stochastic system response of the plate is projected to the deterministic solver by using the stochastic Galerkin method. The statistical response of eigen frequencies and dynamic displacements of the composite plate at incremental random mean temperature are investigated, and are compared with the results of the Monte Carlo simulation. The numerical studies show a reduction in amplitude of the dynamic mean displacements with the increment in the time and it increases with the increment in the random mean temperature. The characteristics of loading have also significantly influenced the uncertainty in the time-dependent displacement response.

Nonlinear Stochastic Optimal Control of MDOF Partially Observable Linear Systems Excited by Combined Harmonic and Wide-Band Noises

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

Transient responses of stochastic systems under stationary excitations

Stochastic systems to stationary random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. It has been a challenge to obtain exact transient response properties. A novel approximate technique for determining non-stationary probability density function (PDF) of randomly excited nonlinear oscillators is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. Based on the results from statistical linearization, residual error of the FPK equation associated with proposed approximation solution can be treated by weighted residual method. As a result, nonlinear ordinary differential equations are produced. Numerical method is adopted to solve these equations and approximate PDF solutions are then obtained. In order to verify the efficiency of the proposed procedure, four examples of stochastic vibrating systems with additional excitations or/and parametric excitations are considered. It is shown that the results obtained by the proposed procedure agree well with those from Monte Carlo simulation.

Design criteria for dissipative devices in coupled oscillators under seismic excitation

Design criteria for a visco-elastic connection of two simple oscillators under seismic excitation are proposed. The dissipative connection is described by Kelvin–Voigt rheological model. The features of the selected criteria are explored for the system parameter set based on the analytical solutions of the associated eigenvalue problem and to the closed-form solutions of the stochastic system response. Design maps are drawn in the nondimensional space of the relevant system parameters in which a direct comparison of the obtained design values is possible. Furthermore, performance indices are used to assess the response of the two simple coupled oscillators subject to natural excitation to demonstrate the robustness with respect to nonstationary excitation.

Improved Neumann Expansion Method for Stochastic Finite Element Analysis

This paper presents an improved Neumann expansion method for calculating statistical moments of either a structural response or system inverse matrix under stochastic uncertainties in system properties and loads. The essentials of the proposed method are the successive matrix inversion and partial bivariate dimension reduction method that enable efficient multidimensional probability integration. The successive matrix inversion computes an exact realization of a stochastic system response, or an inverse matrix, by using the concept of inverse Neumann expansion. In characterizing the stochastic response, multidimensional probability integration is solved for statistical moments and covariance matrices. The existing univariate decomposition method is highly efficient to approximate the multidimensional integration. However, due to the missing interaction effects, the univariate decomposition method could lead to a huge error. In this study, a partial bivariate decomposition method is proposed to capture bivariate interaction effects approximately with a marginal increase in computational cost. Several numerical examples, including representative structural problems, are discussed to demonstrate the accuracy and efficiency of the proposed method compared with other existing methods. It is found that the proposed method provides better accuracy with significant computational cost savings when random fluctuations make locally limited changes in a system.

Nonlinear analysis of a rub-impact rotor with random stiffness under random excitation

Stochastic bifurcation and chaos of a rub-impact rotor system with random stiffness under random excitation are studied in this article. Due to the irrational and fractional expressions existing in the denominator of rub-impact force, the integral process is very complicated. Taylor series expansion is used to expand the irrational and fractional expressions into a series of polynomials. Chebyshev polynomial approximation method is applied to reduce the system equations with random parameter to its equivalent deterministic one, and the responses of stochastic system can be obtained by numerical methods. Numerical simulations show that random parameters have a significant effect on the rub-impact rotor system. It may promote the nonlinear response when the rotational speed is near the 1/2 first-order critical speed and may suppress the nonlinear response when the rotational speed is over the first-order critical speed.

Role of Roots of Orthogonal Polynomials in the Dynamic Response of Stochastic Systems

This paper investigates the fundamental nature of the polynomial chaos (PC) response of dynamic systems with uncertain parameters in the frequency domain. The eigenfrequencies of the extended matrix arising from a PC formulation govern the convergence of the dynamic response. It is shown that, in the particular case of uncertainties and with Hermite and Legendre polynomials, the PC eigenfrequencies are related to the roots of the underlying polynomials, which belong to the polynomial chaos set used to derive the polynomial chaos expansion. When Legendre polynomials are used, the PC eigenfrequencies remain in a bounded interval close to the deterministic eigenfrequencies because they are related to the roots of a Legendre polynomial. The higher the PC order, the higher the density of the PC eigenfrequencies close to the bounds of the interval, and this tends to smooth the frequency response quickly. In contrast, when Hermite polynomials are used, the PC eigenfrequencies spread from the deterministic eigenfrequencies (the highest roots of the Hermite polynomials tend to infinity when the order tends to infinity). Consequently, when the PC number increases, the smoothing effect becomes inefficient.

An adaptive spectral Galerkin stochastic finite element method using variability response functions

SummaryA methodology is proposed in this paper to construct an adaptive sparse polynomial chaos (PC) expansion of the response of stochastic systems whose input parameters are independent random variables modeled as random fields. The proposed methodology utilizes the concept of variability response function in order to compute an a priori low‐cost estimate of the spatial distribution of the second‐order error of the response, as a function of the number of terms used in the truncated Karhunen–Loève (KL) expansion. This way the influence of the response variance to the spectral content (correlation structure) of the random input is taken into account through a spatial variation of the truncated KL terms. The criterion for selecting the number of KL terms at different parts of the structure is the uniformity of the spatial distribution of the second‐order error. This way a significantly reduced number of PC coefficients, with respect to classical PC expansion, is required in order to reach a uniformly distributed target second‐order error. This results in an increase of sparsity of the coefficient matrix of the corresponding linear system of equations leading to an enhancement of the computational efficiency of the spectral stochastic finite element method. Copyright © 2015 John Wiley & Sons, Ltd.

A simplified method for the analysis of the stochastic response in discrete coordinates

Nowadays, the problem of the computational cost vs. the accuracy of the results is an important field of research in engineering, particularly for the analysis of the response of a stochastic system. By using a set of discrete coordinates, this response can become computationally challenging, especially when using a modal representation. Many dynamic load cases, significant for engineering applications, have random and convective properties such as wall pressure fluctuations due to the turbulent boundary layer (TBL). In this work, a new method for the evaluation of the dynamic response of a linear system excited by a TBL like load is proposed: it is named as frequency modulated pseudo-equivalent deterministic excitation (PEDEM) and it is based on the pseudo-excitation method (PEM). The latter is an exact representation since it uses the modal decomposition of the cross-spectral density matrix of the load. Unfortunately, the extraction of the eigensolutions of the load matrix is required at each frequency step and thus can become computationally unacceptable when a large number of eigensolutions is required. PEDEM tries to overcome this computational bottleneck by introducing some approximations, which are based on the analysis of the eigensolutions of the dynamic load matrix vs. frequency. Three different approximations are proposed and thus modulated for the three frequency ranges wherein the dynamic matrix of the considered random and convective load has different characteristics. A criterion to identify the above-mentioned frequency ranges is also proposed by introducing a dimensionless representation of the frequency. The results show that the proposed approximations combine a good accuracy in representing the stochastic system response with a 40 per cent of reduction of the computational costs if compared to a full stochastic response. The method is successfully applied to a simple 1D configuration, a chain of oscillators, which still presents the main features of a generic system. The extension to 2D systems is also analysed considering a preliminary test case with a flexural plate.

A hybrid spectral and metamodeling approach for the stochastic finite element analysis of structural dynamic systems

A novel approach for uncertainty propagation and response statistics estimation of randomly parametrized structural dynamic systems is developed in this paper. The frequency domain response of a stochastic finite element system is resolved at randomly sampled design points in the input stochastic space with an infinite series expansion using preconditioned stochastic Krylov bases. The system response is expressed in the eigenvector space of the structural system weighted with finite order rational functions of the input random variables, termed spectral functions. The higher the order of the spectral functions, the more accurate is the order of approximation of the stochastic system response. However, this increased accuracy comes at a computational cost. This cost is mitigated by using a Bayesian metamodel. The proposed approach is used to the analyze the stochastic vibration response of a corrugated panel with random elastic parameters. The results obtained with the proposed hybrid approach are compared with direct Monte Carlo simulations, which have been considered as the benchmark solution.

Transient Response of Structural Dynamic Systems with Parametric Uncertainty

The time-domain response of a randomly parameterized structural dynamic system is investigated with a polynomial chaos expansion approach and a stochastic Krylov subspace projection, which has been proposed here. The latter uses time-adaptive stochastic spectral functions as weighting functions of the deterministic orthogonal basis onto which the solution is projected. The spectral functions are rational functions of the input random variables and depend on the spectral properties of the unperturbed system. The stochastic system response can be accurately resolved even when using low-order spectral functions, which are computationally advantageous. The time integration required for the resolution of the transient stochastic response has been performed with the unconditionally stable single-step implicit Newmark scheme using a stochastic integration operator. A semistatistical hybrid analytical and simulation-based computational approach has been utilized to obtain the moments and probability density functions of the solution. The simulations have been performed for different degrees of variability of the input randomness and different dimensions of the input stochastic space and compared with the direct Monte-Carlo simulations for accuracy and computational efficiency.

Variability response functions for stochastic systems under dynamic excitations

The concept of variability response functions (VRFs) is extended in this work to linear stochastic systems under dynamic excitations. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of linear stochastic systems under static loads, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is assumed. This assumption is here validated with brute-force Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). The same integral expression can be used to calculate the mean response of a dynamic system using a Dynamic Mean Response Function (DMRF) which is a function similar to the DVRF. These integral forms can be used to efficiently compute the mean and variance of the transient system response together with time dependent spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability system response.

Variability response functions for effective material properties

The variability response function ( VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.

A top-down approach to calibration, validation, uncertainty quantification and predictive accuracy assessment

This paper describes a “top-down” uncertainty quantification (UQ) approach for calibration, validation and predictive accuracy assessment of the SNL Validation Workshop Structural Dynamics Challenge Problem. The top-down UQ approach differs from the more conventional (“bottom-up”) approach in that correlated statistical analysis is performed directly with the modal characteristics (frequencies, mode shapes and damping ratios) rather than using the modal characteristics to derive the statistics of physical model parameters (springs, masses and viscous damping elements in the present application). In this application, a stochastic subsystem model is coupled with a deterministic subsystem model to analyze stochastic system response to stochastic forcing functions. The weak nonlinearity of the stochastic subsystem was characterized by testing it at three different input levels, low, medium and high. The calibrated subsystem models were validated with additional test data using published NASA and Air Force validation criteria. The validated subsystem models were first installed in the accreditation test bed where system response simulations involving stochastic shock-type force inputs were conducted. The validated stochastic subsystem model was then installed in the target application and simulations involving limited duration segments of stationary random vibration excitation were conducted.

Analysis of stochastic bifurcation and chaos in stochastic Duffing–van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing–van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing–van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random field case

This paper is the second of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. The concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in the first paper for the special case of material property variations modeled by random variables to more general problems involving random fields. Specifically, a hierarchy of spectral- and probability-distribution-free upper bounds on the mean, variance, and exceedance values of the response of stochastic systems is established when only the coefficient of variation and lower limit of the stochastic material properties are known. Furthermore, a hierarchy of probability-distribution-free upper bounds on these quantities is established when the spectral density function describing the stochastic material properties is known in addition to the coefficient of variation and the lower limit.

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.