STOCHASTIC DYNAMIC SENSITIVITY OF UNCERTAIN STRUCTURES SUBJECTED TO RANDOM EARTHQUAKE LOADING

The present study involves computation of stochastic sensitivity of structures with uncertain structural parameters subjected to random earthquake loading. The formulations are provided in frequency domain. A strong earthquake-induced ground motion is considered as a random process defined by respective power spectral density function. The uncertain structural parameters are modelled as homogeneous Gaussian stochastic field and discretized by the local averaging method. The discretized stochastic field is simulated by the Cholesky decomposition of respective co-variance matrix. By expanding the dynamic stiffness matrix about its reference value, the advantage of Neumann Expansion technique is explored within the framework of Monte Carlo simulation, to compute responses as well as sensitivity of response quantities. This approach involves only a single decomposition of the dynamic stiffness matrix for the entire simulated structure and the facility that several stochastic fields can be tackled simultaneously are basic advantages of the Neumann Expansion. The proposed algorithm is explained by an example problem.     

RELIABILITY EVALUATIONS OF 3-D FRAME SUBJECTED TO NON-STATIONARY EARTHQUAKE

The objective of the present work is to evaluate the integrated reliability of multistoried space frame subjected to random earthquake. The stochastic ground motion is described by fully non-stationary sigma-oscillatory model. The stochastic dynamic analysis is performed in the frequency domain to obtain the power spectral density function of random response. Finally, the reliability formulations are developed based on computed random response through the solution of first passage problem. A building frame idealized as a space frame in finite element modelling is considered for reliability analysis. Simple modal analysis is also performed for comparison of results.     

Vibration of a beam due to a random stream of moving forces with random velocity

The problem of dynamic response of a beam to the passage of a train of concentrated forces with random amplitudes and velocities is considered. Force arrivals at the beam are assumed to constitute the point stochastic process of events. Thus, the excitation process is an idealization of vehicular traffic loads on a bridge. An analytical technique is developed to determine the response of the beam. Explicit expressions for the expected value and the 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.     

Micro-vibration suppression of equipment supported on a floor incorporating magneto-rheological elastomer core

In this study, magneto-rheological elastomers (MREs) are adopted to construct a smart sandwich beam for micro-vibration control of equipment. The micro-vibration response of a smart sandwich beam with MRE core which supports mass-concentrated equipment under stochastic support-motion excitations is investigated to evaluate the vibration suppression capability. The dynamic behavior of MREs as a smart viscoelastic material is characterized by a complex modulus dependent on vibration frequency and controllable by external magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of the smart sandwich beam supporting mass-concentrated equipment is developed based on the Galerkin method and random vibration theory. First, the displacements of the beam are expanded as series of spatial harmonic functions and the Galerkin method is applied to convert the partial differential equations of motion into ordinary differential equations. With these equations, the frequency-response function matrix of the beam–mass system and the expression of the velocity response spectrum are then obtained, with which the root-mean-square (rms) velocity response in terms of the one-third octave frequency band can be calculated. Finally, the optimization problem of the complex modulus of the MRE core is defined by minimizing the velocity response spectrum and the rms velocity response of the sandwich beam, through altering the applied magnetic fields. Numerical results are given to illustrate the influence of MRE parameters on the rms velocity response and the response reduction capacity of the smart sandwich beam. The proposed method is also applicable to response analysis of a sandwich beam with arbitrary core characterized by a complex shear modulus and subject to arbitrary stochastic excitations described by a power spectral density function, and is valid for a wide frequency range.     

Chaotic versus nonchaotic stochastic dynamics in Monte Carlo simulations: a route for accurate energy differences in N-body systems

We present a method to efficiently evaluate small energy differences of two close N-body systems by employing stochastic processes having a stability versus chaos property. By using the same random noise, energy differences are computed from close trajectories without reweighting procedures. The approach is presented for quantum systems but can be applied to classical N-body systems as well. It is exemplified with diffusion Monte Carlo simulations for long chains of hydrogen atoms and molecules for which it is shown that the long-standing problem of computing energy derivatives is solved.     

Statistical equilibrium in simple exchange games I

Simple stochastic exchange games are based on random allocation of finite resources. These games are Markov chains that can be studied either analytically or by Monte Carlo simulations. In particular, the equilibrium distribution can be derived either by direct diagonalization of the transition matrix, or using the detailed balance equation, or by Monte Carlo estimates. In this paper, these methods are introduced and applied to the Bennati-Dragulescu-Yakovenko (BDY) game. The exact analysis shows that the statistical-mechanical analogies used in the previous literature have to be revised. An erratum to this article is available at .     

The vibration signature of chordal cracks in a rotor system including uncertainties

The aim of this paper is to investigate the effects of the presence of a transverse crack in a rotating shaft under uncertain physical parameters in order to obtain some indications that might be useful in detecting the presence of a crack in rotating system. The random dynamic response of the cracked rotor is evaluated by expanding the changing stiffness of the crack (i.e. the breathing mechanism) as a random truncated Fourier series. To avoid the use of the Monte Carlo simulations (MCS), an alternative procedure that is based on a combination of the Harmonic Balance Method and the Stochastic Finite Element Method (SFEM) using the Polynomial Chaos Expansion (PCE) is proposed. So the response of the Fourier components of the cracked rotor is expanded in the polynomial chaoses. The random dynamic response obtained by applying this procedure is compared with that evaluated through numerical integration based on the Harmonic Balance Method and the Monte Carlo simulations.     

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system

In this paper, the Chebyshev polynomial approximation is applied to the problem of stochastic period-doubling bifurcation of a stochastic Bonhoeffer--van der Pol (BVP for short) system with a bounded random parameter. In the analysis, the stochastic BVP system is transformed by the Chebyshev polynomial approximation into an equivalent deterministic system, whose response can be readily obtained by conventional numerical methods. In this way we have explored plenty of stochastic period-doubling bifurcation phenomena of the stochastic BVP system. The numerical simulations show that the behaviour of the stochastic period-doubling bifurcation in the stochastic BVP system is by and large similar to that in the deterministic mean-parameter BVP system, but there are still some featured differences between them. For example, in the stochastic dynamic system the period-doubling bifurcation point diffuses into a critical interval and the location of the critical interval shifts with the variation of intensity of the random parameter. The obtained results show that Chebyshev polynomial approximation is an effective approach to dynamical problems in some typical nonlinear systems with a bounded random parameter of an arch-like probability density function.     

Stochastic joint time–frequency response analysis of nonlinear structural systems

A novel approximate analytical approach for determining the response evolutionary power spectrum (EPS) of nonlinear/hysteretic structural systems subject to stochastic excitation is developed. Specifically, relying on the theory of locally stationary processes and utilizing a recently proposed representation of non-stationary stochastic processes via wavelets, a versatile formula for determining the nonlinear system response EPS is derived; this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets. Further, the nonlinear system non-stationary response amplitude probability density function (PDF), which is required as input for the developed approach, is determined either by utilizing a numerical path integral scheme, or by employing a time-dependent Rayleigh PDF approximation technique. A significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non-separable in time and frequency EPS as well. The hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in the numerical examples section. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the approach.     

The SU(2) chiral model in an external field: A complex stochastic process on a non-abelian group

The principal chiral model in an external field has a complex action. Conventional methods for performing Monte Carlo simulations are therefore useless for strong fields or weak coupling (i.e. large lattices). We apply the Langevin equation to this action and generate a complex stochastic process on a non-abelian group. Nice agreement with the pertubative expansion is observed for strong fields. A clear verification of the second-order phase transition for weak fields proved by Polyakov and Wiegmann seems to require larger lattices and more computer time than we have available, but should be straightforward using the complex Langevin equation.     

Deposition-evaporation stochastic systems in two and higher dimensions

We study the stochastic dynamics of deposition-evaporation cooperative processes of dimers, trimers, etc., in two- and higher-dimensional lattices. The dimer system in bipartite lattices allows for an exact solution of dynamic correlations and scaling functions by means of a quantum spin equivalence. Autocorrelations exhibit a diffusive asymptotic kinetics and crossovers of different dynamic regimes in highly anisotropic lattices. Monte Carlo simulations combined with finite-size scaling arguments support the validity of the diffusive picture in more general situations. Steady-state coverages and diffusion constants are obtained using mean-field approaches, spin wave calculations, and random walk analyses in nearly jammed configurations.     

An Analysis of Polynomial Chaos Approximations for Modeling Single-Fluid-Phase Flow in Porous Medium Systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.     

Stochastic differential equation analysis for sound scattering by random internal waves in the ocean

The scattering of a weakly divergent narrow sound beam by random inhomogeneities of a fluctuating ocean is considered in the coupled-mode approximation. The random index of sound refraction is described using the Garrett-Munk internal wave spectrum. The problem is solved using the stochastic differential equations for the first-and second-order statistical moments of the acoustic field. The equations are formulated according to the cumulant expansion method. The existence of weakly divergent narrow sound beams in long-range sound propagation was one of the last discoveries of L.M. Brekhovskikh, to which he attached much importance. The concentration of sound into narrow beams away from the axis of the underwater sound channel was first observed experimentally and then explained by Brekhovskikh and his former students Goncharov, Kurtepov, and Petukhov. In the present paper, the scattered field intensity of a sound beam is calculated for different frequencies and source depths. Analytical expressions are obtained for the coefficients of the differential equation. The intermode energy transfer that accompanies the long-range propagation of a weakly divergent sound beam is analyzed. A comparison with the conventionally used Monte Carlo simulation in the parabolic equation approximation is performed.     

First passage time on pattern formation in a non-local Fisher population dynamics

We use stochastic dynamics to develop the patterned attractor of a non-local extended system. This is done analytically using the stochastic path perturbation approach scheme, where a theory of perturbation in the small noise parameter is introduced to analyze the random escape of the stochastic field from the unstable state. Emphasis is placed on the specific mode selection that these types of systems exhibit. Concerning the stochastic propagation of the front we have carried out Monte Carlo simulations which coincide with our theoretical predictions.     

随机激励下双稳态压电俘能系统的相干共振及实验验证

随着压电晶体材料的迅速发展, 基于压电效应的能量采集系统是俘获环境中的宽带随机振动能量的一种有效途径. 研究了有限宽带随机激励作用下, 磁斥力双稳态压电俘能系统的相干共振俘能机理, 并进行了实验验证. 运用Euler-Maruyama方法求解了随机非线性压电振动耦合方程, 比较分析了相干共振发生前后系统的动力学特性和俘能效率, 然后基于Kramers逃逸速率解释了相干共振. 最后的随机振动实验结果验证了双稳态压电俘能系统的相干共振俘能机理. 并且观察到: 当相干共振发生时, 系统会在两个势能阱之间剧烈运动, 此时宽带随机振动能量会被转化为大幅值窄带低频振动响应, 从而极大地提高了宽带随机振动能量的俘获效率.     

First passage time of nonlinear ship rolling in narrow band non-stationary random seas

The generalized extended stochastic central difference (GESCD) method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non-stationary random seas. The GESCD method is based on a combination of the extended stochastic central difference method with a statistical linearization technique, modified adaptive time scheme, and time coordinate transformation. The extended stochastic central difference method is, however, an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non-stationary random disturbances. Approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined. It is concluded that the GESCD method is very accurate, simple and efficient to apply compared with Monte Carlo simulation. The proposed method is applicable to cases with large nonlinearities and intensive random excitations. The approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the Monte Carlo simulation. It is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature.     

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

In recent years, there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary/partial differential equations. To this end, the spectral stochastic finite element method (SSFEM) is the most popular method due to its fast convergence rate. Recently, the stochastic sparse grid collocation method has emerged as an attractive alternative to SSFEM. It approximates the solution in the stochastic space using Lagrange polynomial interpolation. The collocation method requires only repetitive calls to an existing deterministic solver, similar to the Monte Carlo method. However, both the SSFEM and current sparse grid collocation methods utilize global polynomials in the stochastic space. Thus when there are steep gradients or finite discontinuities in the stochastic space, these methods converge very slowly or even fail to converge. In this paper, we develop an adaptive sparse grid collocation strategy using piecewise multi-linear hierarchical basis functions. Hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region. Numerical examples, especially for problems related to long-term integration and stochastic discontinuity, are presented. Comparisons with Monte Carlo and multi-element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method.     

The non-Markovian quantum behavior of open systems

It is shown that the exact dynamics of a composite quantum system can be represented through a pair of product states which evolve according to a Markovian random jump process. This representation is used to design a general Monte Carlo wave function method that enables the stochastic treatment of the full non-Markovian behavior of open quantum systems. Numerical simulations are carried out which demonstrate that the method is applicable to open systems strongly coupled to a bosonic reservoir, as well as to the interaction with a spin bath. Full details of the simulation algorithms are given, together with an investigation of the dynamics of fluctuations. Several potential generalizations of the method are outlined.Received: 29 October 2003, Published online: 10 February 2004PACS: 03.65.Yz Decoherence; open systems; quantum statistical methods - 02.70.Ss Quantum Monte Carlo methods - 05.10.Gg Stochastic analysis methods (Fokker-Planck, Langevin, etc.)     

Sensitivity of footbridge vibrations to stochastic walking parameters

Some footbridges are so slender that pedestrian traffic can cause excessive vibrations and serviceability problems. Design guidelines outline procedures for vibration serviceability checks, but it is noticeable that they rely on the assumption that the action is deterministic, although in fact it is stochastic as different pedestrians generate different dynamic forces. For serviceability checks of footbridge designs it would seem reasonable to consider modelling the stochastic nature of the main parameters describing the excitation, such as for instance the load amplitude and the step frequency of the pedestrian. A stochastic modelling approach is adopted for this paper and it facilitates quantifying the probability of exceeding various vibration levels, which is useful in a discussion of serviceability of a footbridge design. However, estimates of statistical distributions of footbridge vibration levels to walking loads might be influenced by the models assumed for the parameters of the load model (the walking parameters). The paper explores how sensitive estimates of the statistical distribution of vertical footbridge response are to various stochastic assumptions for the walking parameters. The basis for the study is a literature review identifying different suggestions as to how the stochastic nature of these parameters may be modelled, and a parameter study examines how the different models influence estimates of the statistical distribution of footbridge vibrations. By neglecting scatter in some of the walking parameters, the significance of modelling the various walking parameters stochastically rather than deterministically is also investigated providing insight into which modelling efforts need to be made for arriving at reliable estimates of statistical distributions of footbridge vibrations. The studies for the paper are based on numerical simulations of footbridge responses and on the use of Monte Carlo simulations for modelling the stochastic nature of actions of a single pedestrian traversing various pin-supported single-span footbridges with frequencies in vertical bending in the range of 1.6-2.4 Hz.