Stochastic averaging for a class of single degree of freedom systems with combined Gaussian noises

We develop the stochastic averaging method to investigate the asymptotic stationary solution and the stochastic bifurcations of a class of single degree of freedom system with combined Gaussian white and colored noises, and to derive the Fokker–Planck–Kolmogorov (FPK) equations. The general stationary solutions will be obtained analytically under some suitable conditions. Theoretically, a general algebraic expression of the stationary probability density function of amplitude for the dynamical system is presented to consider the influences of correlation time of the noise and the noise intensity on stochastic bifurcations. Then, an example is given to illustrate the averaging method, and the effectiveness of the averaging method is verified via comparing the analytical results with those from Monte Carlo simulation. Finally, stochastic bifurcations are discussed through a qualitative change of the stationary probability distribution. It indicates that system parameters, noise intensity, and noise correlation time can be treated as bifurcation parameters, respectively.     

矩形薄板在面内随机参数激励下的随机分岔研究

建立了四边简支的矩形薄板在受面内随机激励时的振动模型,并用Galerkin法将该系统化简为二自由度常微分非线性动力学方程组。得出系统的广义能量（Hamilton函数）表达式后,又利用拟不可积Hamilton系统平均理论将方程等价为一个一维的Ito随机扩散过程,并通过计算该系统的最大Lyapunov指数来研究系统的局部随机稳定性,同时利用基于随机扩散过程的奇异边界理论研究了模型的全局稳定性,最后通过稳态概率密度函数的形状变化探讨了系统参数变化对系统随机Hopf分岔的影响。数值模拟结果验证了理论分析的正确性。     

形状记忆合金梁在随机激励下的随机分岔与首次穿越

根据形状记忆合金（SMA）的等应变拉压实验的数据,利用van der Pol环模型模拟了形状记忆合金在加载和卸载过程中的应力应变迟滞环特性。并且根据弹性理论和Galerkin方法建立了形状记忆合金简支梁在受轴向高斯白噪声激励时的振动模型。应用拟不可积Hamilton随机平均法将函数表示为一维扩散过程后,通过最大Lyapunov指数判断系统的局部稳定性,同时用奇异边界理论讨论了系统的全局稳定性。随后通过分析稳态概率密度和联合概率密度的图形特征,得到了此模型的随机Hopf分岔现象,并讨论了系统产生随机Hopf分岔的条件。最后,用数值法模拟了系统在特定初始条件和边界条件下可靠性函数和首次穿越时间的概率密度函数所满足的BK(Backward Kolmogorov)方程,并且分析了系统发生首次穿越的条件。     

Bifurcation and stability analysis of a hybrid energy harvester with fractional-order proportional–integral–derivative controller and Gaussian white noise excitations

Hybrid energy harvester (HEH) has received much attention in the field of energy harvesting. The HEH inevitably suffers from external stochastic disturbances in the working environment such as winds, waves, and ocean currents. Only few works deal with the stochastic dynamics of a hybrid energy harvester with fractional-order proportional–integral–derivative (PID) controller. This paper aims to investigate bifurcation control and stability analysis of a HEH with fractional-order PID controller subjected to Gaussian white noise excitations. An approximately equivalent dimensionally reduced system is formulated via the variables transformation method, and its approximate stationary solutions are derived through the stochastic averaging method. One example is worked out in detail to verify the effectiveness of the proposed procedure. It is shown that the analytical results provide a good approximation to the numerical simulation results. The influences of system parameters on the stochastic bifurcation and the asymptotic Lyapunov stability are also discussed.     

Design of feedback control of a nonlinear stochastic system for targeting a pre-specified stationary probability distribution

An innovative approach to designing feedback control of a nonlinear stochastic system for targeting a pre-specified stationary probability density function (SPDF) is proposed based on the techniques for obtaining the exact stationary solutions of nonlinear stochastic system. First, the techniques for obtaining the exact stationary solutions of nonlinear stochastic systems, including the stationary potential and the generalized stationary potential, are briefly reviewed. Then, the approach to designing the feedback control of nonlinear stochastic systems for targeting a pre-specified SPDF and the Lyapunov function method for proving that the transient PDF of the controlled systems do converge to the pre-specified SPDF are presented. Finally, two examples are worked out in detail to illustrate the proposed approach and its effectiveness.     

二自由度碰撞振动系统的随机响应

用拟不可积哈密顿系统随机平均法研究了二自由度磁撞振动系统的随机响应，先将二自由度随机激励的碰撞振动系统表示成随机激励的耗散的哈密顿系统形式，然后用拟不可积哈密顿系统的随机平均法得到了以系统哈密顿函数为基本变量的一维It^↑o随机微分方程，最后用数值方法求解与该方程相应的稳态FPK方程，得到系统响应的平稳概率密度。两个算例的结果与数字模拟结果的比较表明了随机平均法在二自由度磁撞振动系统的随机响应分析中的有效性。     

Multimodal nonlinear spectral response of a beam with impact under random input

A linearization procedure for estimating the spectral response of a randomly excited beam—stop system is proposed. The elastic stop is replaced by a spring with a stiffness depending on the amplitude of the deflection at the impact location. The probability density function of the amplitude is obtained using the stochastic averaging principle. Next, an estimate of the nonlinear response spectrum is derived providing the expectation of the spectral density function of the random spring linear system with respect to the probability density function of the amplitude response (assumed to be a random variable). The efficiency of the method is checked by comparing results with those of numerical simulations.     

Non-linear stochastic control via stationary response design

This paper presents a strategy for controlling externally excited stochastic systems with uncertain parameters. The control objective is to drive the system response from an arbitrary initial distribution to a prescribed stationary probability density function (PDF). This problem can be interpreted as the stochastic stabilization about a PDF. The control consists of a non-linear feedback part and a switching term inspired by robust sliding control concepts. The control of a one-dimensional stochastic process with a Gaussian target-PDF is used to illustrate the approach. The control performance is evaluated by studying the time evolution of the first and second order moments of the response. The dependence of the response on the number of feedback terms and the rate of convergence to the stationary PDF are studied numerically. Motivated by the fast convergence observed, a feedback controller with time-varying gains is applied to the problem of tracking a moving PDF. Monte Carlo simulations validate very well the control for both stabilization and tracking problems.     

Small noise expansion of moment lyapunov exponents for two-dimensional systems

We construct an approximation for the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response of a two-dimensional linear system driven by real or white noise. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noise. As an example, we study the moment stability of the stationary solution of nonlinear structural and mechanical systems subjected to real noise excitation. The usefulness of the moment Lyapunov exponent in predicting parameter values at which qualitative changes in the probability density function occur (stochastic bifurcation) is also illustrated.     

Identification of stochastic loads applied to a non-linear dynamical system using an uncertain computational model and experimental responses

The paper is devoted to the identification of stochastic loads applied to a non-linear dynamical system for which experimental dynamical responses are available. The identification of the stochastic load is performed using a simplified computational non-linear dynamical model containing both model uncertainties and data uncertainties. Uncertainties are taken into account in the context of the probability theory. The stochastic load which has to be identified is modelled by a stationary non-Gaussian stochastic process for which the matrix-valued spectral density function is uncertain and is then modelled by a matrix-valued random function. The parameters to be identified are the mean value of the random matrix-valued spectral density function and its dispersion parameter. The identification problem is formulated as two optimization problems using the computational stochastic model and experimental responses. A validation of the theory proposed is presented in the context of tubes bundles in Pressurized Water Reactors.     

随机地震激励作用下自复位结构的平稳响应

自复位结构是一种震后不需修复或仅需少量修复即可继续投入使用的新型抗震结构。目前有关自复位结构地震响应的研究多是在确定性激励下进行,鲜有涉及随机激励环境。假定地震动加速度过程为金井清过滤白噪声模型,研究了随机地震激励下单自由度自复位体系的平稳响应。应用广义谐波平衡技术分解旗帜形的恢复力,建立原系统的等效非线性随机系统。通过Stratonovich-Khasminskii极限定理作随机平均,得到关于幅值的近似一维扩散过程。建立并求解对应的FPK方程,得到关于幅值的稳态概率密度函数并进行参数分析。数值结果表明,能量耗散系数与屈服位移的减小能降低系统响应,同时,随着这两个参数的变化,系统会出现随机P分岔现象。通过蒙特卡罗数值模拟法验证了解析解的有效性。     

Importance sampling for stochastic systems under stationary noise having a specified power spectrum

We develop an importance sampling simulation scheme for estimating an extremely small probability of system failure with respect to a time-dependent stochastic system excited by stationary random noise having a specified power spectrum. First, we construct a system of random differential equations driven by a Wiener process, which can approximately give such a stationary random noise by the use of an extended version of the well-known Ornstein–Uhlenbeck process. Next, we suppose a stochastic response system driven by the constructed stationary random noise. Next, formulating the probability of system failure, we give an importance sampling scheme through the probability measure transformation based upon the Girsanov theorem, where multi design times are introduced to cope with stationary or almost stationary behavior of the system. Finally, we give numerical examples to demonstrate the efficiency of the proposed scheme.     

一类受随机激励的强非线性包装振动系统的随机平均

对一类受有界噪声激励作用的强非线性包装振动系统使用随机平均法和MLP方法,给出了此类系统的随机平均方程和FPK方程.利用路径积分方法,初步得出了一些关于系统稳态响应的联合概率密度的模拟结果.     

Imperfection sensitive variation of critical loads at hilltop bifurcation point

Variation of critical loads due to initial imperfections at the hilltop bifurcation point is described by elastic stability theory. We derive a system of bifurcation equations for a potential system expressing local behavior at this bifurcation point, which is a double critical point occurring as a coincidence of a simple pitchfork bifurcation point and a limit point. The piecewise linear law of imperfection sensitivity of critical loads in Thompson and Schorrock [J. Mech. Phys. Solids 23 (1975) 21] is revised by extending initial imperfections to be considered in the bifurcation equations. Based on this sensitivity law, a procedure to determine the most influential (worst or optimum) initial imperfection is formulated. As the most essential development of this paper, under the assumption that initial imperfections are subject to a multi-variate normal distribution, we derive the probability density function of critical loads that follows a Weibull-like distribution. The validity of theoretical developments is assessed through its application to elastic truss structures.     

Nonlinear Dynamic Characteristics and Optimal Control of Giant Magnetostrictive Laminated Plate Subjected to In-plane Stochastic Excitation

In this paper, nonlinear dynamic characteristics and optimal control of giant magnetostrictive laminated plate (GMLP) subjected to in-plane stochastic excitation were studied. Von del Pol nonlinear item was introduced to interpret the hysteresis phenomenon of the strain–magnetic field intensity curve of giant magnetostrictive material, and the nonlinear dynamic model of GMLP subjected to in-plane stochastic excitation was developed. Local and global stochastic stabilities were analyzed according to largest Lyapunov exponent theory and singular boundary theory. The functions of steady-state probability density and joint probability density were obtained, and the condition of stochastic Hopf bifurcation was analyzed. The reliability function was solved from backward Kolmogorov equation, and the probability density of the first-passage time was obtained. Finally, the optimal control strategy was proposed in stochastic dynamic programming method. Numerical simulation shows that the stability of the solution varies with parameter, and stochastic Hopf bifurcation appears in the process; the reliability of the system was improved by optimal control, and the first-passage time was delayed. The result is helpful to engineering applications of GMLP.     

Joint distribution of peaks and valleys in a stochastic process

General expressions and numerical results are presented pertaining to the occurrence of two local extrema of a stochastic process at prescribed time values. The extrema may be either peaks or valleys and the process may be either stationary or nonstationary. General formulas are presented for the rates of occurrence, the joint and conditional probability distributions, and the moments of the extreme values. These formulas are relatively simple multiple-integral expressions, but the integrands involve the joint probability density function for six random variables. The procedures are then applied for the special case of a stationary mean-zero Gaussian process for which the calculations are greatly simplified. Numerical results for three different spectral density functions demonstrate that conditioning on either only the existence or both the existence and the value of one peak can have a very significant effect on both the rate of occurrence and the probability distribution of a second peak.     

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.     

巨子型有控结构体系中附加柱的连接方式

附加柱是巨子型有控结构体系（Mega-sub controlled structural system,即MSCSS）中的必要组成部分,附加柱的连接方式对MSCSS的动力特性有较大影响。结合概率密度演化理论及切球选点方法,通过对巨子型有控结构体系在结构参数及地震激励均为随机变量情况下进行附加柱应力响应计算,得到了附加柱应力响应的概率密度演化曲面,对不同附加柱与巨型梁连接方式下的附加柱应力进行了研究。结果表明,释放附加柱的水平约束,可以有效地降低附加柱应力响应的均值及标准差,整体提高巨子型有控结构体系的抗震能力;而附加柱应力概率密度的不均匀性表明,即使巨子型有控结构体系受到平稳随机地震作用激励,其附加柱应力响应也将表现出明显的非平稳性。     

Some aspects of chaotic and stochastic dynamics for structural systems

In this paper, the bifurcation behaviour of an externally excited four-dimensional nonlinear system is examined. Throughout this paper, a two-degree-of-freedom shallow arch structure under either a periodic or a stochastic excitation will be considered. For the case when the excitation is periodic, the local and global behaviour is examined in the presence of principalsubharmonic resonance and1:2 internal resonance. The method of averaging is used to obtain the first order approximation of the response of the system under resonant conditions. A standard Melnikov type perturbation method is used to show analytically that the system may exhibit chaotic dynamics in the sense of Smale horseshoe for the 1:2 internal resonance case in the absence of dissipation. In the case of stochastic excitation, the stability of the stationary solution is examined by determining themaximal Lyapunov exponent andmoment Lyapunov exponent in terms of system parameters. An asymptotic method is used to obtain explicit expressions for various exponents in the presence of weak dissipation and noise intensity. These quantities provide almost-sure stability boundaries in parameter space. When the system parameters lie outside these boundaries, it is essential to understand the nonlinear behaviour. The method of stochastic averaging is applied to obtain a set of approximate Itô equations which are then examined to describe the local bifurcation behaviour.     

Optimum design criteria for elastic structures subject to random dynamic loads

A new stochastic optimization method, which makes use of a constraint on structural reliability, is proposed for structures subject to dynamic random loads. A minimum weight problem is posed, in which a constraint condition imposes that the failure probability must be smaller than a given admissible level. The failure is determined by the first crossing outside the safe domain of a suitable structural response vector. The method is used to find the optimal shape of an elastic vertical column supporting a fixed mass positioned on the top, subject to a Gaussian filtered stationary stochastic horizontal acceleration process. The column, with variable annular cross-section, is described by a deterministic elastic multi-degree-of-freedom system. It is assumed that failure is reached when its lateral displacement exceeds an acceptable threshold value. Under this constraint, the structural weight is minimized and the optimal shape is determined for different structural conditions.