Non-stationary Stochastic Response Research Articles

A joint time–space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems

A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.

Non-stationary stochastic response determination of vibro-impact system under combination harmonic and Gaussian white noise excitations

Vibro-impact system (VI-S) have demonstrated their potential to contribute to engineering practice throughout the last few years. Nonetheless, their scope of application are still very limited. One of a primary reason is that dynamic behaviors of VI-S under the combination of deterministic period excitations (deterministic period excitation) and random noises (random noises) are not entirely understood. Toward this, the radial basis function neural network (rbfnn) is used to analyze dynamical behaviors evolution of VI-S under deterministic period excitation combined with random noises. To illustrate the potential of the proposed scheme, two classical examples of Mathieu–Duffing oscillator and Duffing–Rayleigh oscillator are considered. The evolutionary of dynamics behaviors and stochastic P-bifurcation phenomenon are examined through a qualitative change of the probability distribution, which indicates that system parameters and noise intensity, respectively, can be treated as bifurcation-induced parameters. Besides, numerical results are also carried out to confirm the accuracy of semi-analytical prediction.

Determination of Nonstationary Stochastic Response of Linear Oscillators With Fractional Derivative Elements of Rational Order

Abstract In this paper, a technique is developed for determining the nonstationary response statistics of linear oscillators endowed with fractional derivative elements. Notably, fractional operators are particularly effective in modeling solid mechanics problems as they offer the option of influencing both the elasticity and the energy dissipation capacity of the system. In this paper, particular attention is devoted to the case of fractional derivatives of rational order that approximates reasonably well any real order model. The oscillators are subjected to stationary stochastic excitations, and the pertinent nonstationary response statistical moments are determined by first introducing a finite number of oscillator response related states; this is afforded by the rational number order of the fractional operator. Next, the technique involves proceeding to treating the problem in the Laplace transform domain. This leads to multiple convolution integrals determined by representing the transfer function of the oscillator in a partial fraction form by a pole-residue formulation. In this manner, the response evolutionary power spectral density of the fractional oscillator is derived in a closed form, while nonstationary second-order statistics can be obtained by mundane numerical integration in the frequency domain. Applications to oscillators comprising one or two fractional derivative elements are presented, considering the case of a white noise excitation and of a random process possessing the classical Kanai–Tajimi spectrum. Reliability of the developed technique is assessed by juxtaposing its analytical results with pertinent Monte Carlo simulation data.

A KDE-based non-parametric cloud approach for efficient seismic fragility estimation of structures under non-stationary excitation

With the development of performance-based earthquake engineering, the risk-informed assessment framework has received broad recognition over the world, of which the probability seismic fragility analysis is an important step. The classic seismic fragility adopts the lognormal assumption and forms a parametric derivation. With the development of fragility theory, researchers are hoping to seek out non-parametric approaches to express the intrinsic fragility in a pure analytical form without any distribution assumptions. Besides, how to keep the calculation efficiency (e.g., combining with cloud approach) and how to consider the non-stationary stochastic responses (e.g., combining with non-stationary stochastic excitation model) are critical aspects in fragility that deserve further attention of researchers. In this paper, a kernel density estimation (KDE) based non-parametric cloud approach is proposed for efficient seismic fragility estimation of structures under non-stationary excitation. First, the methodology framework of the efficient approach is illustrated. Then, the procedures of non-stationary stochastic seismic response of structures and KDE-based non-parametric cloud approach for efficient seismic fragility are demonstrated. After that, an application example via a three-span-six-story reinforced concrete frame is given for implementation, followed with a parametric analysis of critical factors. During the process, the classic parametric linear-regression based cloud approach (cloud-LR) and benchmark Monte-Carlo-simulation based cloud approach (cloud-MCS) are also incorporated for validation. In general, the analysis verifies the effectiveness of the non-parametric cloud-KDE approach without requiring more computation work (i.e., same as the parametric cloud-LR approach and much less than the benchmark cloud-MCS approach). Meanwhile, the non-parametric cloud-KDE approach indicates a comparable accuracy with the classic fragility approaches (i.e., less deviation than the parametric cloud-LR approach and much closer to the benchmark cloud-MCS approach), and with the increase of stochastic cloud-point number, the corresponding fitting degree of cloud-KDE approach is growing better. The research provides a new sight for the development of non-parametric seismic fragility approach, and the corresponding findings can be further combined with the probabilistic hazard and risk analysis for a non-parametric assessment procedure in performance-based earthquake engineering.

Nonstationary Stochastic Response of Hysteretic Systems Endowed With Fractional Derivative Elements

AbstractIn this paper, a computationally efficient approach is proposed for the determination of the nonstationary response statistics of hysteretic oscillators endowed with fractional derivative elements. This problem is of particular practical significance since many important engineering systems exhibit hysteretic/inelastic behavior optimally captured only through the concept of fractional derivative, and many natural excitations as seismic waves and atmospheric turbulence are both stochastic and nonstationary in time. Specifically, the approach is based on a statistical linearization scheme involving an equivalent system of augmented dimension. First, relying on a transformation scheme, the fractional derivative term is represented by a set of coupled linear ordinary differential equations. Next, the evolution of the system response statistics is captured by incorporating the statistical linearization technique in a nonstationary sense. This involves integrating in time a set of ordinary differential equations. Several numerical applications pertaining to classical hysteretic oscillators are considered, and the versatility of the proposed method is assessed via comparison with pertinent Monte Carlo simulations.

Nonstationary stochastic response determination of nonlinear oscillators endowed with fractional derivatives

Fractional calculus has been broadly used in diverse engineering applications. In this regard, vibrations of fractional oscillators subject to stochastic loads have attracted considerable attention. This paper proposes a semi-analytical approach to determine the nonstationary response statistics of nonlinear oscillators endowed with fractional derivatives. Specifically, the fractional derivative term is represented by introducing a discretization scheme and associated first-order differential equations. This leads to an augmented dimension dynamic system. Further, in conjunction with the statistical linearization technique, the evolution of the system response is captured by a set of coupled ordinary differential equations with time-dependent coefficients. Furthermore, solving the associated Lyapunov equation for the randomly excited dynamic system yields the nonstationary statistics of the oscillator response. The reliability of the proposed method is demonstrated by Monte Carlo simulations pertaining to classical nonlinear oscillators.

Investigation of the Nonstationary Stochastic Response of Functionally Graded Piezoelectric Material Plates with General Boundary Conditions

The functionally graded piezoelectric material (FGPM) has been widely used in various engineering cases due to its excellent mechanical properties. So far, the investigations on the stochastic dynamic behaviors of FGPM structures are few. This paper develops a unified solution for the nonstationary stochastic responses of the FGPM plate with general boundary conditions, in which the point, distributed and base acceleration random excitations can be considered. The first-order shear deformation theory (FSDT) is adopted to formulate the energy functional of the system and the governing equations are deduced by using the Hamilton's principle. The solutions of the system are obtained based on the modified Ritz method and pseudo excitation method (PEM). The accuracy of the proposed model is validated by comparing the obtained free vibration and random response results with those from the published literature. Finally, the effects of key parameters including boundary conditions, external electric voltage, material constituent and time modulated function on the stochastic response of the FGPM plate are investigated.

New Scheme for Estimation of the First and Senior Moment Functions for the Response of Linear Delay Differential System Excited by Additive and Multiplicative Noises

The paper describes the theoretical apparatus and the algorithmic part of a scheme for stochastic examination of systems of linear stochastic delay differential equations perturbed by additive and multiplicative white noises. The presence of an efficient method of analyzing such problems, i.e., a method that can compete with the method of statistical simulation (Monte Carlo), is important both from the theoretical and applied points of view. The systems are substantial for modeling various objects and processes and can be the results of linearization of nonlinear stochastic systems with delay and multiplicative fluctuations. The importance of developing a procedure for calculating just the senior moment functions for the state vectors comes from the presence of multiplicative perturbations implying the inability to restrict the first moment functions as in the case of linear systems with Gaussian white noises as the input. To obtain ODEs for the first and senior moment functions without delays, we apply a new modification of our scheme combining the classical method of steps and extension of the system state space with the addition of the use of multidimensional matrices for the compact recording of the algorithm. The scheme realized in the environment of Wolfram Mathematica software package is demonstrated by an example devoted to an estimation the moment functions until the 4th order for a non-stationary stochastic response of the system with one degree of freedom.

Wiener path integrals and multi-dimensional global bases for non-stationary stochastic response determination of structural systems

Abstract A novel approximate technique based on Wiener path integrals (WPIs) is developed for determining, in a computationally efficient manner, the non-stationary joint response probability density function (PDF) of nonlinear multi-degree-of-freedom dynamical systems. Specifically, appropriate multi-dimensional (time-dependent) global bases are constructed for approximating the non-stationary joint response PDF. In this regard, two distinct approaches are pursued. The first employs expansions based on Kronecker products of bases (e.g., wavelets), while the second utilizes representations based on positive definite functions. Next, the localization capabilities of the WPI technique are exploited for determining PDF points in the joint space-time domain to be used for evaluating the expansion coefficients at a relatively low computational cost. In contrast to earlier implementations of the WPI technique, the herein developed generalization and enhancement circumvents computationally daunting brute-force discretizations of the time domain in cases where the objective is to determine the complete time-dependent non-stationary response PDF. Several numerical examples pertaining to diverse structural systems are considered, including both single- and multi-degree-of-freedom nonlinear dynamical systems subject to non-stationary excitations, as well as a bending beam with a non-Gaussian and non-homogeneous Young’s modulus. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the technique.

Efficient Nonstationary Stochastic Response Analysis for Linear and Nonlinear Structures by FFT

AbstractBased on the fast Fourier transform (FFT), an efficient and accurate algorithm was developed for improving the computational efficiency of structural responses under nonstationary stochasti...

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

• A system of linear stochastic integro-differential equations is under consideration. • Gaussian nonstationary noises and random model uncertainties excite the system. • The problem is to find the mean and covariance functions of the state vector process. • Green matrix-valued functions are used to solve the problem. • As example, a response of a thermoviscoelastic system is estimated by our scheme. The paper describes a theoretical apparatus and an algorithmic part of application of the Green matrix-valued functions for time-domain analysis of systems of linear stochastic integro-differential equations. It is suggested that these systems are subjected to Gaussian nonstationary stochastic noises in the presence of model parameter uncertainties that are described in the framework of the probability theory. If the uncertain model parameter is fixed to a given value, then a time-history of the system will be fully represented by a second-order Gaussian vector stochastic process whose properties are completely defined by its conditional vector-valued mean function and matrix-valued covariance function. The scheme that is proposed is constituted of a combination of two subschemes. The first one explicitly defines closed relations for symbolic and numeric computations of the conditional mean and covariance functions, and the second one calculates unconditional characteristics by the Monte Carlo method. A full scheme realized on the base of Wolfram Mathematica and Intel Fortran software programs, is demonstrated by an example devoted to an estimation of a nonstationary stochastic response of a mechanical system with a thermoviscoelastic component. Results obtained by using the proposed scheme are compared with a reference solution constructed by using a direct Monte Carlo simulation.

Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation

SUMMARYNonlinear viscous dampers are supplemental devices widely used for enhancing the performance of structural systems exposed to seismic hazard. A rigorous evaluation of the effect of these damping devices on the seismic performance of a structural system should be based on a probabilistic approach and take into account the evolutionary characteristics of the earthquake input and of the corresponding system response.In this paper, an approximate analytical technique is proposed for studying the nonstationary stochastic response characteristics of hysteretic single degree of freedom systems equipped with viscous dampers subjected to a fully nonstationary random process representing the seismic input. In this regard, a stochastic averaging/linearization technique is utilized to cast the original nonlinear stochastic differential equation of motion into a simple first‐order nonlinear ordinary differential equation for the nonstationary system response variance. In comparison with standard linearization schemes, the herein proposed technique has the significant advantage that it allows to handle realistic seismic excitations with time‐varying frequency content. Further, it allows deriving a formula for determining the nonlinear system response evolutionary power spectrum. By this way, ‘moving resonance’ effects, related to both the evolutionary seismic excitation and the nonlinear system behavior, can be observed and quantified.Several applications involving various system and input properties are included. Furthermore, various response parameters of interest for the seismic performance assessment are considered as well. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed technique. Copyright © 2014 John Wiley & Sons, Ltd.

Comparison of Stochastic Responses of Asphaltic Concrete Core and Asphaltic Lining Dams with Clay Core Dams to Stationary and Nonstationary Excitation

This paper presents the advantages of using asphalt in lining dams and in asphaltic concrete core dams when comparing with clay core dams. For this purpose, the linear and nonlinear stochastic responses of a clay core dam, an asphaltic lining dam and an asphaltic concrete core dam subjected to stationary and non-stationary excitation including spatially varying ground motion (SVEGM) effects are determined by finite element method. The nonlinear response is based on equivalent linear method, which considers the nonlinear variation of soil shear moduli and damping as a function of shear strain. The results are compared with those of the responses computed using linear response for uniform ground motion and SVEGM. The spatial variability of ground motion is taken into account with incoherence and wave passage effects. Stationary as well as non-stationary stochastic response analyses are performed for three types of dam. A time dependent frequency response function is used throughout the study for non-stationary responses. It is observed from the results that it can be possible to use of asphaltic lining dams and an asphaltic concrete core dams instead of clay core dams.

Stationary and non-stationary stochastic response of linear fractional viscoelastic systems

A method is presented to compute the stochastic response of single-degree-of-freedom (SDOF) structural systems with fractional derivative damping, subjected to stationary and non-stationary inputs. Based on a few manipulations involving an appropriate change of variable and a discretization of the fractional derivative operator, the equation of motion is reverted to a set of coupled linear equations involving additional degrees of freedom, the number of which depends on the discretization of the fractional derivative operator. As a result of the proposed variable transformation and discretization, the stochastic analysis becomes very straightforward and simple since, based on standard rules of stochastic calculus, it is possible to handle a system featuring Markov response processes of first order and not of infinite order like the original one. Specifically, for inputs of most relevant engineering interest, it is seen that the response second-order statistics can be readily obtained in a closed form, to be implemented in any symbolic package. The method applies for fractional damping of arbitrary order α(0≤α≤1). The results are compared to Monte Carlo simulation data.

Nonstationary random analysis of structures with hybrid mass damper system

The building structure is modeled as a linear single-degree-of-freedom system, and an active mass damper (AMD) is also adopted as the control device in order to reduce the structural response due to the external random load. In addition, suppose that the input acceleration of earthquake can be modeled as the product of nonstationary envelope function and narrow-band stationary random process with Kanai-Tajimi spectrum. In order to obtain the nonstationary stochastic response of control system due to the narrow-band nonstationary random process, in consideration of exchanging the active control mode for the passive one, considerable efforts are generally required in the analytical formulation. This paper, therefore, proposes an analytical technique to acquire the response covariance matrix by utilizing the state transition matrix method in state space. The proposed technique is evaluated by comparing the analytical results with those from numerical simulations.

Non-stationary stochastic vibration analysis of fuzzy truss system

A new method (fuzzy factor method based on the fuzzy sets theory) for the dynamic response analysis of fuzzy truss system under non-stationary stochastic excitation is presented. Considering the fuzziness of the structural physical parameters and geometric dimensions simultaneously, the fuzzy correlation function matrix of the structural displacement response in time domain is derived using the fuzzy factor method and the optimisation method; then from the structural non-stationary stochastic response in the frequency domain, the fuzzy mean square values of the displacement and stress response are developed by the fuzzy factor method. The influences of the fuzziness of the physical parameters and geometric dimensions on the fuzziness of the mean square values of the structural displacement and stress response are illustrated via two engineering examples and some important conclusions are obtained.

Optimum frictional elements in sliding isolation systems

The optimum friction coefficient of a sliding system with a restoring force for the minimum acceleration response of a base-isolated structure under earthquake ground motion is investigated. The stochastic model of El-Centro 1940 earthquake which preserves the non-stationary evolution of amplitude and frequency content of the original record is used for the model of earthquake. The base-isolated structure consists of a linear flexible multi-storey structure supported on the sliding system. The sliding system is modelled to provide a friction force (ideal Coulomb-friction type) and a linear restoring force. The non-stationary stochastic response of the isolated structure is obtained using the time dependent equivalent linearisation technique as the force-deformation behaviour of the sliding system is highly non-linear. The response of the system is analysed for the optimum friction coefficient of the sliding base isolation system. The criterion selected for optimality is the minimisation of the root mean square top floor absolute acceleration. The optimum friction coefficient of sliding isolation system is obtained under important parametric variations such as: period and damping of the superstructure, ratio of the base mass to the superstructure floor mass, the damping ratio of the isolation system, the period of base isolation system and the intensity of earthquake excitation. It has been shown that the above parameters have significant effects on optimum friction coefficient of the sliding base isolation system.

OPTIMUM DAMPING IN A NON-LINEAR BASE ISOLATION SYSTEM

Optimum isolation damping for minimum acceleration of a base-isolated structure subjected to earthquake ground excitation is investigated. The stochastic model of the El-Centro1940 earthquake, which preserves the non-stationary evolution of amplitude and frequency content of ground motion, is used as an earthquake excitation. The base isolated structure consists of a linear flexible shear type multi-storey building supported on a base isolation system. The resilient-friction base isolator (R-FBI) is considered as an isolation system. The non-stationary stochastic response of the system is obtained by the time dependent equivalent linearization technique as the force-deformation of the R-FBI system is non-linear. The optimum damping of the R-FBI system is obtained under important parametric variations; i.e., the coefficient of friction of the R-FBI system, the period and damping of the superstructure; the effective period of base isolation. The criterion selected for optimality is the minimization of the top floor root mean square (r.m.s.) acceleration. It is shown that the above parameters have significant effects on optimum isolation damping.

Random Vibration of Beam under Moving Loads

Structural response to moving loads is an important area of study for highway bridge designers. In this paper, a new approach for the evaluation of nonstationary stochastic response of a bridge under moving loads of vehicular traffic is presented. The random nature of the force arrivals at bridge beams constitute a Poisson process of events. The proposed approach considers the input process as a filtered Poisson process--the response of a linear undamped oscillator excited by a superposition of two Poisson white noise processes. Through an extension of Ito's differential rule, the cumulant differential equations of every order of the response process are obtained. These equations are first-order linear differential equations with constant forcing function and a closed-form solution of the random stochastic response is provided. The results of a numerical application of the approach is offered in terms of cumulants up to the fourth order and a comparison with those obtained by other means is presented.

Nonstationary equivalent linearization of nonlinear continuous systems

This paper presents a technique for obtaining the nonstationary stochastic response of a nonlinear continuous system. The method of equivalent linearization is generalized to continuous systems subjected to nonstationary random excitation. This technique allows replacement of the original nonlinear system with a time-varying linear continuous system. A numerical implementation is also described. In this procedure, the linear replacement system is discretized by the finite element method. Application to systems satisfying the one-dimensional wave equation with a constitutive nonlinearity is discussed. Results are presented for nonlinear stress-strain laws of a strain-hardening type.