高斯噪声激励下分段线性双稳系统的稳态特性研究

研究了关联乘性和加性高斯白噪声激励下分段线性双稳系统的稳态特性。通过随机等价变换方法得到了该系统稳态概率密度函数的表达式,讨论了噪声关联性和噪声强度对系统稳态概率密度函数的影响,并通过数值计算发现该系统出现了一些新的随机现象。研究结果表明,乘性噪声强度、加性噪声强度、噪声互关联强度都能使稳态概率密度函数曲线峰的数目发生改变,这说明系统出现了相变现象,且两关联高斯白噪声对系统稳态概率密度具有相同的影响作用。     

STOCHASTIC P-BIFURCATION OF TRI-STABLE VAN DER POL-DUFFING OSCILLATOR

分析了乘性和加性噪声作用下三稳态Van der Pol-Duffing振子的随机P分岔. 首先用随机平均法得到系统的随机微分方程,求得系统响应幅值的稳态概率密度函数. 然后应用分岔分析的奇异性理论,求得随机P分岔发生的临界参数条件,得到多种定性不同的稳态概率密度曲线. 讨论了2种激励噪声强度和系统阻尼对响应稳态概率密度曲线峰的个数、各峰值相对大小的影响. 通过Monte-Carlo数值模拟对理论计算结果进行了验证.该方法可用于其他系统的随机P分岔分析.     

两种不同类型噪声驱动下的非对称双稳系统的随机共振

研究了由色关联的乘性色噪声和加性白噪声联合激励下的非对称双稳系统的随机共振现象,运用两态模型理论和统一色噪声近似理论,在绝热近似条件下得到了信噪比的表达式.信噪比是乘性色噪声强度,加性白噪声强度,噪声耦合强度,乘性噪声自关联时间和噪声互关联时间的非单调函数,所以在该双稳系统中产生了随机共振. 在系统的偏度不是太大的情况下调节加性白噪声强度比调节乘性色噪声强度更容易控制随机共振,并且以信噪比作为噪声之间耦合强度的函数时可以观察到二次随机共振现象.     

双稳系统中随机共振与相同步时间的相关性

研究了双稳系统的随机共振(SR)与输入输出之间的相同步相关性. 首先得到系统的输出信噪比(SNR)与输入输出之间的平均相同步时间的表达式,然后讨论了随机共振与输入输出之间的平均相同步时间之间的关系. 结果表明:(1)系统出现了随机共振现象,且平均相同步时间对噪声是敏感的;由于加性与乘性噪声的相互影响,随加性噪声强度的增加,输出信噪比及平均相同步时间曲线上首先出现抑制现象,然后出现峰值,并且, 减小乘性与加性噪声强度比率,可提高输出信噪比和增长平均相同步时间. (2)系统的随机共振与平均相同步时间达到最大值不同步出现,但平均相同步时间对输出信噪比是敏感的. 该结论为信号传输中利用随机共振原则改变系统工作环境提供了依据.      

多稳态动力系统中随机共振的研究进展

非线性随机动力学是力学、数学、工程等多个领域关注的热点,在航空航天、机械工程、生物生态等领域有广泛的应用.多稳态动力系统作为其最重要的研究对象,在随机扰动下具有丰富的动力学行为,如随机分岔、随机共振等,尤其是随机共振,已经被应用于机械故障诊断、微弱信号检测和振动能量俘获等工程实际问题中.本文主要综述了多稳态动力系统中的随机共振理论、方法及工程应用.首先,通过几类典型的非线性随机动力学系统,介绍了随机共振的经典理论和度量指标;其次,重点阐述了多稳态动力学系统,尤其是三稳态和周期势系统,在各类噪声激励下的随机共振现象,分析了其诱发机理、演化规律和研究方法;最后,介绍了多稳态动力系统中随机共振的几类应用实例,并进一步给出了随机共振当前面临的难题和未来的发展趋势等开放性问题.     

Lévy噪声和高斯白噪声共同激励的FHN神经元系统的动力学特性

研究了Lévy噪声和高斯白噪声共同激励下的一维FHN神经元系统的动力学特性。利用Janicki-Weron算法产生Lévy噪声,并采用四阶Runge-Kutta算法模拟出方程的稳态概率密度函数;然后通过稳态概率密度函数图像进一步对FHN神经元系统进行了稳态分析。通过数值仿真发现:乘性噪声强度D、加性噪声强度Q、稳定性指标α、偏斜参数β这些参数都可以诱导系统产生相变现象;乘性噪声强度D和稳定性指标α的增大使得FHN神经元系统停留在激发态的概率逐渐升高;加性噪声强度Q和偏斜参数β的增大使得神经元系统逐渐从激发态转变到静息态;乘性噪声强度D和加性噪声强度Q的改变对系统的作用正好相反。     

含时滞反馈与涨落质量的记忆阻尼系统的随机共振

针对具有记忆效应的欠阻尼系统, 存在时滞反馈与涨落质量, 本文主要研究了其输出稳态响应振幅的随机共振效应. 首先通过引入新变量和运用小时滞近似展开理论, 将具有非马尔科夫特性的原系统转化为等价的两维马尔科夫线性系统, 再利用Shapiro-Loginov公式和Laplace变换获得了系统响应的一阶稳态矩和稳态响应振幅的解析表达式. 结果表明: 当系统参数满足Routh-Hurwitz稳定条件时, 稳态响应振幅随质量涨落噪声强度、周期驱动信号频率以及时滞的变化均存在随机共振现象, 其中随机多共振现象也被观察到. 在适当范围内, 通过控制时滞反馈, 系统的随机共振效应随着时滞的增大而增强, 而较长的记忆时间及增大阻尼参数均对共振行为呈现抑制作用.有效调控时滞反馈与记忆效应的变化关系将有助于增强系统对周期驱动信号的响应强度. 最后, 通过数值模拟计算验证了理论结果的有效性.      

外共振耦合Duffing-van der Pol系统的首次穿越

研究了谐和力与宽带噪声激励下二自由度强非线性Duffing-van derPol系统的首次穿越问题. 在外共振情形, 应用随机平均法将系统动力学方程化为关于振幅与角变量的Itô随机微分方程. 然后建立了系统的可靠性函数满足的后向Kolmogorov方程以及平均首次穿越时间满足的Pontryagin方程. 在一定的边界条件和初始条件下, 用有限差分法求解了这两个高维偏微分方程, 得到系统的条件可靠性函数、平均首次穿越时间以及平均首次穿越时间的条件概率密度. 讨论了不同参数对系统可靠性以及平均首次穿越时间的影响. 用Monte Carlo数值模拟验证了理论方法的有效性.     

Duffing振子在窄带随机激励下分岔现象的研究

本文运用数值模拟方法深入研究了Ｄｕｆｆｉｎｇ振 窄带随机激励下的稳态响应。第一全面研究了产生跳跃现象的参数区域附近Ｄｕｆｆｉｎｇ振子的分岔现象。研究发现，位移、速度的二维联合概率密度函数能全面，地表征系统的运动状态，得到了系统响应的位移、速度的二维联合概率密度的各种分岔模式，因而对该问题有了较全面的认识。同时分析了大量不同参数下系统的响应，发现激励参数系统决定着系统的运动状态，尤其激励强度Ｄ的影响     

加性非平稳激励下结构速度响应的FPK方程降维

结构在随机激励下的非线性响应分析是具有高度挑战性的困难问题. 对于白噪声或过滤白噪声激励,求解FPK方程将获得结构响应 的精确解. 遗憾的是,对于非线性多自由度系统,FPK方程难以直接求解. 事实上,其数值解法严重受限于方程维度,而解析求解 则仅适用于少数特定的系统,且多是稳态解. 因此,将FPK方程进行降维,是求解高维随机动力响应分析问题的重要途径. 本文针 对幅值调制的加性白噪声激励下多自由度非线性结构的非平稳随机响应分析问题,将联合概率密度函数满足的高维FPK方程进行降 维. 针对结构速度响应概率密度函数求解,通过引入等价漂移系数,原FPK方程可转化为一维FPK型方程. 建议了构造等价漂移系数 的条件均值函数方法. 进而,采用路径积分方法求解降维FPK型方程,得到速度概率密度函数的数值解答. 结合单自由度Rayleigh 振子、十层线性剪切型框架和非线性剪切型框架结构在幅值调制的加性白噪声激励下的非平稳速度响应求解,讨论了本文方法的精 度和效率,验证了其有效性.      

Path integral solution of vibratory energy harvesting systems

A transition Fokker-Planck-Kolmogorov(FPK) equation describes the procedure of the probability density evolution whereby the dynamic response and reliability evaluation of mechanical systems could be carried out. The transition FPK equation of vibratory energy harvesting systems is a four-dimensional nonlinear partial differential equation. Therefore, it is often very challenging to obtain an exact probability density. This paper aims to investigate the stochastic response of vibration energy harvesters(VEHs)under the Gaussian white noise excitation. The numerical path integration method is applied to different types of nonlinear VEHs. The probability density function(PDF)from the transition FPK equation of energy harvesting systems is calculated using the path integration method. The path integration process is introduced by using the GaussLegendre integration scheme, and the short-time transition PDF is formulated with the short-time Gaussian approximation. The stationary probability densities of the transition FPK equation for vibratory energy harvesters are determined. The procedure is applied to three different types of nonlinear VEHs under Gaussian white excitations. The approximately numerical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulation(MCS).     

Stochastic resonance in coupled weakly-damped bistable oscillators subjected to additive and multiplicative noises

With coupled weakly-damped periodically driven bistable oscillators subjected to additive and multiplicative noises under concern,the objective of this paper is to check to what extent the resonant point predicted by the Gaussian distribution assumption can approximate the simulated one.The investigation based on the dynamical mean-field approximation and the direct simulation demonstrates that the predicted resonant point and the simulated one are basically coincident for the case of pure additive noise,but for the case including multiplicative noise the situation becomes somewhat complex.Specifically speaking,when stochastic resonance(SR) is observed by changing the additive noise intensity,the predicted resonant point is lower than the simulated one;nevertheless,when SR is observed by changing the multiplicative noise intensity,the predicted resonant point is higher than the simulated one.Our observations imply that the Gaussian distribution assumption can not exactly describe the actual situation,but it is useful to some extent in predicting the low-frequency stochastic resonance of the coupled weakly-damped bistable oscillator.     

Trichotomous noise induced stochastic resonance in a linear system

We investigate stochastic resonance in an underdamped linear system subjected to multiplicative trichotomous noise. We carry out the Shapiro?CLoginov formula to find the exact expression of output amplitude gain, and the impacts of the input signal frequency and noise parameters will be observed, such as noise switching rate or noise correlation time, noise amplitude and noise flatness. Then one can find the stochastic resonance for the proposed linear system.     

Stochastic resonance in a harmonic oscillator with damping trichotomous noise

In this paper, the phenomenon of stochastic resonance (SR) in a prototype fluctuating damping harmonic oscillator with trichotomous Markovian noise is investigated. The exact expression of output amplitude gain has been calculated using the well-known Shapiro–Loginov formula. The phenomenon of SR has been found in a broad sense—that is, the non-monotonic behavior of output amplitude gain as a function of noise parameters. Then the influences of noise amplitude, noise switching rate, and noise flatness on the output amplitude gain have also been discussed. Finally, the reverse resonance phenomenon has been presented.     

Stationary response of Duffing oscillator with hardening stiffness and fractional derivative

The stationary response of Duffing oscillator with hardening stiffness and fractional derivative under Gaussian white noise excitation is studied. First, the term associated with fractional derivative is separated into the equivalent quasi-linear dissipative force and quasi-linear restoring force by using the generalized harmonic balance technique, and the original system is replaced by an equivalent nonlinear stochastic system without fractional derivative. Then, the stochastic averaging method of energy envelope is applied to the equivalent nonlinear stochastic system to yield the averaged Itô equation of energy envelope, from which the corresponding Fokker–Planck–Kolmogorov (FPK) equation is established and solved to obtain the stationary probability densities of the energy envelope and the amplitude envelope. The accuracy of the analytical results is validated by those from the Monte Carlo simulation of original system.     

Stochastic P-bifurcations of a noisy nonlinear system with fractional derivative element

Acta Mechanica Sinica - This paper investigates the stochastic P-bifurcation (SPB) of a fractionally damped oscillator subjected to additive and multiplicative Gaussian white noise. Variable...     

Probability density function for stochastic response of non-linear oscillation system under random excitation

Probability density function (PDF) of stochastic responses is a critical topic in uncertainty analysis. In this paper, orthogonal decomposition technique was extended to discuss non-stationary response of non-linear oscillator under random excitation. The PDF of stochastic reponses is represented by a set of standardized multivariable orthogonal polynomials. According to the Galerkin scheme, the original problem, which has to solve the Fokker-Planck-Kolmogorov (FPK) equation, was converted to a first-order linear ordinary differential equation, in terms of unknown time-dependent coefficients. Then, stationary and non-stationary PDFs of uncertainty responses were obtained. In numerical examples, first-order and second-order non-linear systems exposed to the Gaussian white noise were considered. Finally, the accuracy of the proposed method was demonstrated through appropriate comparisons to Monte-Carlo simulation and analytical results.     

EPC procedure for PDF solution of non-linear oscillators excited by Poisson white noise

The stationary probability density function (PDF) solution of the responses of non-linear stochastic oscillators subjected to Poisson pulses is analyzed. The PDF solutions are obtained by the exponential-polynomial closure (EPC) method. To assess the effectiveness of the solution procedure numerically, non-linear oscillators are analyzed with different impulse arrival rates, degree of oscillator non-linearity and excitation intensity. Numerical results show that the PDFs obtained with the EPC method yield good agreement with those obtained from Monte Carlo simulation when the polynomial order is 4 or 6. It is also observed that the EPC procedure is the same as the equivalent linearization procedure under Gaussian white noise in the case of the polynomial order being 2.     

Crossing Rate Analysis with a Non-Gaussian Closure Method for Nonlinear Stochastic Systems

The mean upcrossing rate of the stationary responses of nonlinear stochastic system excited by white noise is analyzed based on the assumption that the probability density function (PDF) of the responses is a linear superposition of basic functions. The Gaussian PDFs are used as the basic functions of which the coefficients are the reciprocal of the number of the basic functions. The Gaussian closure method is a special case of the proposed method. Based upon the approximate PDF, the explicit expression for the mean upcrossing rate (MCR) is given. Numerical results show that the approximate MCRs approach the exact ones in the tails of the MCR curves as the number of the basic functions increases.