色关联的乘性与加性噪声驱动下的幂函数型单势阱系统的稳态特征

研究了色关联的乘性与加性噪声作用下幂函数型单势阱系统的稳态问题。首先利用一致有色噪声近似方法,推导得到稳态概率密度函数(仅以b=1为例),分析色关联的乘性与加性噪声对于稳态概率密度的影响,并通过势阱结构参数分析了势阱的结构对于稳态概率密度的影响。结果表明:参数b=1条件下的互相关强度、自相关时间以及势阱结构参数a,b都会诱导非平衡相变。     

基于周期势函数的自适应二阶欠阻尼随机共振信号增强方法

针对滚动轴承微弱故障振动信号在噪声环境下故障特征难以提取的问题,提出一种基于周期势函数的自适应二阶欠阻尼随机共振信号增强方法。采用粒子群算法对系统参数和阻尼系数的自适应匹配,实现对多个拟增强频段的随机共振,更加适用于工程实际中多故障信号提取。数据库考题检验和工程实验验证表明:1)该方法明显提高了输出信噪比,故障特征频率处主峰突出,边带干扰少,方便故障的机器判读,误判率低;2)随着噪声强度的增加,虽然输出信噪比有所降低,但该方法的检测效果仍优于基于周期势函数的自适应一阶随机共振方法的检测效果;3)该方法对噪声的适应性更强,在噪声环境下对于微弱故障信号的提取有着明显优势。     

Stochastic resonance of a multi-stable system and its application in bearing fault diagnosis

This paper investigates the stochastic resonance and mean-first passage time of a quad-stable potential in the presence of Gaussian white noise and periodic forcing. The analytical expressions of mean-first passage time and spectral amplification are obtained, respectively. It is found that even small noise intensity can lead to noise-assisted hopping between two adjacent potential wells for the case of small damping coefficient. For large noise intensity, the escape process of Brownian particles is accelerated in an underdamped nonlinear system. Moreover, the curve of spectral amplification displays a typical resonant peak at an optimal noise intensity, suggesting the onset of stochastic resonance. Meanwhile, with the decrease of periodic signal frequency, the peak value of spectral amplification is enhanced. Especially, an optimal quad-stable potential structure exists to maximize the stochastic resonance effect. The proposed multi-stable stochastic resonance method is applied to the fault diagnosis of inner and outer race bearing, and the quantum particle swarm optimization algorithm is used to optimize the system parameters and damping coefficient. The good agreement between fault frequency and theoretical value validates efficiency of the proposed method. Compared with the overdamped tri-stable stochastic resonance method, the performance of fault diagnosis is enhanced substantially by the proposed method.     

Nonlinear random vibration of the slender deep-water pier under seismic excitation

The present study investigates the nonlinear random vibration of the deep-water pier exposed to horizontal seismic excitation. First of all, the stochastic dynamic model of the pier is formulated. During the process, the pier is simplified as a cantilever beam fixed on the rock foundation, the seismic excitation is treated as Gaussian white noise, the hydrodynamic pressure is described with the radiation wave theory, and the equation for the nonlinear kinematic of the pier is deduced by the means of Kane’s method. Then, with the application of the stochastic averaging (SA) technique, the Fokker–Plank–Kolmogorov (FPK) equation governing the transient probability density function (PDF) of the amplitude envelope and the backward Kolmogorov (BK) equation for the conditional reliability function (CRF) are derived, respectively. The closed-form stationary PDF can be yielded directly from the reduced FPK equation, while the CRF and the conditional PDF of first-passage time are given after solving the BK equation numerically. Numerical discussions are performed to illustrate the trend of excitation intensity, mass ratio, immersion ratio, and inner and outer hydrodynamic effect on the stationary response and first-passage failure are examined, respectively. It has been shown that increases in the excitation intensity, mass ratio, and immersion ratio can amplify the response and reduce the reliability of the deep-water pier system. The hydrodynamic effect also leads to an amplification of the system response and a reduction in the reliability of the system. Similarly, the presence of inner water can also exacerbate these effects, and this phenomenon becomes more pronounced with increasing immersed ratios. Additionally, the analytical solution is validated by the result obtained by pertained Monte Carlo simulations (MCS). It is noted that this work will be helpful for the optimal seismic design of deep-water piers.     

利用欠阻尼二阶线性随机共振抑噪的模型设计及实现

以经典力学模型——质量-阻尼-弹簧欠阻尼二阶线性系统为研究对象,当系统的阻尼系数和固有频率同时受乘性高斯噪声干扰时,利用此系统产生的随机共振来消除此类噪声.理论分析表明,欠阻尼二阶线性系统中存在随机共振现象,系统的平均输出幅度增益呈现非单调变化,不仅在一定条件下大于无噪声时的增益,而且调节适当的系统参数和噪声强度能够提高幅度增益.因此,采用可视化仿真软件SIMULINK建立仿真模型,并进行实例模拟.仿真进一步表明,通过调节适当的系统参数或噪声强度,使系统处于共振区域,就会把夹杂在噪声中的被测信号突现出来,从而实现了弱信号的检测,证实该方法消除乘性噪声的可行性和有效性.     

Approximate solution of the Fokker–Planck–Kolmogorov equation

The aim of this paper is to present a thorough investigation of approximate techniques for estimating the stationary and non-stationary probability density function (PDF) of the response of nonlinear systems subjected to (additive and/or multiplicative) Gaussian white noise excitations. Attention is focused on the general scheme of weighted residuals for the approximate solution of the Fokker–Planck–Kolmogorov (FPK) equation. It is shown that the main drawbacks of closure schemes, such as negative values of the PDF in some regions, may be overcome by rewriting the FPK equation in terms of log-probability density function (log-PDF). The criteria for selecting the set of weighting functions in order to obtain improved estimates of the response PDF are discussed in detail. Finally, a simple and very effective iterative solution procedure is proposed.     

Parameters of a noisy Lorenz system reconstructed using the method of proper coordinates

Parameters of a noisy Lorenz system were reconstructed from its time series using the method of proper coordinates. The effects of low-and high-frequency additive and multiplicative noise on the relative error of reconstruction of the system parameters have been studied. It is demonstrated that the method of proper coordinates provides for increased accuracy of reconstruction of the Lorenz system parameters in the presence of additive noise.     

Laser-induced thermal breakdown of a semiconductor in the presence of mutually correlated noises

The laser-induced thermal breakdown in a semiconductor plate under the joint action of an external multiplicative noise (laser radiation energy density fluctuations) correlated with an internal additive noise is studied theoretically. The correlated additive noise is demonstrated to increase the degree of bistability of the crystal temperature hysteresis.     

Correlation between upper and lower sidebands

Experimental measurements supported by a simple model show that the upper and lower phase modulation (PM) noise sidebands are always equal and 100% correlated, independent of the noise power originating from multiplicative or additive processes. Similarly, we show that the upper and lower amplitude modulation (AM) noise sidebands are also equal and 100% correlated, independent of the noise power originating from multiplicative or additive processes, Moreover, the single sideband (SSB) PM noise is always equal to one-half the total PM noise. The same is true for the AM noise. Although the upper and lower PM or AM noise sidebands are equal and correlated for broadband additive noise, the phase between the AM and the PM sidebands varies randomly with time. These conclusions still hold even when the RF noise sidebands are not symmetric about the carrier.     

First passage problem: a probabilistic dynamic analysis for hot aerospace components

Structures subjected to non-white random excitations with uncertain system parameters affected by surrounding environments are studied. Methods are developed to determine the statistics of dynamic responses such as time-varying mean, standard deviation and autocorrelation functions. Moreover, the first-passage problems with deterministic and stationary/nonstationary random barriers are evaluated. Time-varying (joint) mean crossing rate and the probability density function of the first-passage time for various random barriers are derived.     

Fluctuating noise drives Brownian transport

The transport properties of Brownian ratchet were studied in the presence of stochastic intensity noise in both overdamped and underdamped regimes. In the overdamped case, an analytical solution using the matrix-continued fraction method revealed the existence of a maximum current when the noise intensity fluctuates on intermediate timescale regions. Similar effects were observed for the underdamped case by Monte Carlo simulations. The optimal time-correlation for Brownian transport coincided with the experimentally observed time-correlation of the extrinsic noise in Escherichia coli gene expression and implied the importance of environmental noise for molecular mechanisms.     

基于光纤矢量水听器的浅海噪声矢量场特性研究

基于简正波理论分析了浅海噪声矢量场声压和质点振速的强度特性,仿真了浅海声压和质点振速的噪声强度在深度和频率上的变化特性,噪声矢量场强度特性仿真结果与实验测量结果一致。针对声场测量的有效性,给出了加速度通道自噪声谱级和灵敏度必须满足的关系式,并提出了降低自噪声对接收系统影响的两种措施。最后分析并对比了系统自噪声谱级和海洋环境噪声谱级,结果表明,声压通道的自噪声比环境噪声谱级低20 d B左右,Y通道和Z通道的自噪声比环境噪声低3.5 d B以上,X通道的自噪声谱级在200 Hz附近与环境噪声谱级最为接近,约比环境噪声低1.5 d B。     

时滞反馈对三稳态van der Pol系统稳态概率密度的影响

利用时滞反馈调节系统随机响应特征是随机动力学与控制重要的课题之一。为解决多稳态随机系统的控制器参数设计问题,针对加性噪声激励下的三稳态van der Pol系统,研究了时滞差分反馈对系统稳态概率密度的影响。利用随机平均法得到系统幅值稳态概率密度函数的解析表达式,随后分别讨论了给定噪声强度情况下时滞和反馈强度对系统稳态概率密度的影响。结果表明,反馈强度和时滞变化均能使系统稳态概率密度曲线的拓扑结构发生改变,从而调节系统稳态响应的幅值分布;转迁集计算的结果对控制参数选择有直接指导作用。     

基于矢量水听器的南海海洋背景噪声测试

矢量声压振速联合处理建立在信号的声压和振速相位基础上,海洋环境边界影响将改变矢量声场声压振速的幅度和相位特性．首先根据南海环境条件结合水下目标辐射噪声测量,采用声场矢量简正波理论估算海面非相干偶极子噪声和水下点声源矢量强度随深度的变化,然后设计了可用于深海海域噪声测量的矢量水听器测试系统,获取了南海海域典型深度上的背景噪声矢量强度并进行了特性分析．结果表明：深海背景噪声谱级在500Hz以下基本上不随深度变化,在500Hz～3kHz频段浅深度背景噪声谱级略高于较深深度的背景噪声;振速垂直分量的背景噪声要小于声压和振速水平分量的噪声谱级．     

Stochastic response of nonlinear system in probability domain

A stochastic averaging procedure for obtaining the probability density function (PDF) of the response for a strongly nonlinear single-degree-of-freedom system, subjected to both multiplicative and additive random excitations is presented. The procedure uses random Van Der Pol transformation, Ito’s equation of limiting diffusion process and stochastic averaging technique as outlined by Zhu and others. However, the equations are rederived in generalized form and arranged in such a way that the procedure lends itself to a numerical computational scheme using FFT. The main objective of the modification is to consider highly irregular nonlinear functions which cannot be integrated in closed form and also to solve problems where analytical expressions for probability density function cannot be obtained. The procedure is applied to obtain the PDF of the response of Duffing oscillator subjected to additive and multiplicative random excitations represented by rational power spectral density functions (PSDFs). The results are verified by digital simulation. It is shown that the procedure provides results which compare very well with those obtained from simulation analysis not only for wide-band excitations but also for very narrow-band excitations, which are weak (when normalized with respect to mass of the system.) This paper is dedicated to Prof R N Iyengar of the Indian Institute of Science on the occasion of his formal retirement.     

Finite element solution of Fokker–Planck equation of nonlinear oscillators subjected to colored non-Gaussian noise

Nonlinear oscillators subjected to colored Gaussian/non-Gaussian excitations are modelled through a set of three coupled first-order stochastic differential equations by representing the excitation as a first-order filtered white noise. A 3-D finite element (FE) formulation is developed to solve the corresponding 3-D Fokker Planck (FP) equations. The joint probability density functions of the state variables, obtained as a solution of the FP equation, are typically non-Gaussian and are used for computing the crossing statistics of the response – an essential metric for time variant reliability analysis. The method is illustrated through a noisy Lorenz attractor and a Duffing oscillator subjected to additive colored noise. The increase in state-space dimension when the Duffing oscillator is additionally excited with a parametric Gaussian noise is effectively handled by using stochastic averaging to reduce the state-space dimension. Investigations are carried out to examine the accuracy of the FE method vis-a-vis Monte Carlo simulations. The proposed method is observed to be computationally significantly cheaper for these three problems.     

泊松白噪声激励下斜拉索的面内随机振动

目前针对斜拉索非线性随机振动的研究已广泛开展,但仅限于高斯随机激励情形。然而,现实中大部分的随机扰动都是非高斯的。若使用高斯激励模型将产生较大误差。假设拉索所受非高斯激励为泊松白噪声,研究了泊松白噪声激励下斜拉索面内随机振动。推导了受泊松白噪声激励的斜拉索面内振动的随机微分方程,建立了支配系统平稳响应概率密度函数的广义FPK方程。提出迭代加权残值法求解了四阶广义FPK方程,得到了系统响应概率密度函数的近似稳态闭合解。考察了垂跨比、阻尼系数以及脉冲到达率对拉索面内随机振动响应的影响。结果表明:拉索的响应随着垂跨比的增大,响应呈现不对称现象愈加明显;随阻尼比增加,系统响应得到显著抑制;当脉冲到达率增大,拉索的响应也随之增大,并逐渐接近于高斯白噪声激励的情形。另外,获得的理论结果与蒙特卡罗模拟的结果吻合地非常好。     

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.     

随机噪声激励下轻敲式原子力显微镜动力学特性研究

针对有界随机噪声激励下轻敲式原子力显微镜 (AFM：Atomic force microscope)系统的非线性动力学问题,建立Lennard-Jones力场作用下针尖-样品的集总参数模型,应用现代微分方程和分岔理论,分析了随机扰动强度和弯月面接触角对AFM针尖-样品耦合系统动力学特性的影响。结果表明,轻敲式AFM耦合动力学系统中存在丰富的周期运动和混沌运动,表现出复杂的非线性行为,混沌特性随着随机扰动强度增大而增强,弯月面接触角越大混沌特性越明显,因此在轻敲式AFM优化设计中,随机噪声对AFM系统的影响不可忽视。     

抗噪声源时滞引起的尖峰噪声的抑制

针对抗噪声源时滞引起的尖峰噪声,分析周期信号与非周期随机信号不同频率和不同模拟延时情况下信号叠加后的降噪效果及尖峰噪声产生,并对因时滞产生的尖峰噪声进行处理,提出一种利用模糊控制抑制有源噪声控制中产生的尖峰信号的抑制策略,仿真验证利用模糊控制降噪及抑制尖峰噪声的效果。