Studies on structural safety in stochastically excited Duffing oscillator with double potential wells

The effects of the Gaussian white noise excitation on structural safety due to erosion of safe basin in Duffing oscillator with double potential wells are studied in the present paper. By employing the well-developed stochastic Melnikov condition and Monte–Carlo method, various eroded basins are simulated in deterministic and stochastic cases of the system, and the ratio of safe initial points (RSIP) is presented in some given limited domain defined by the system’s Hamiltonian for various parameters or first-passage times. It is shown that structural safety control becomes more difficult when the noise excitation is imposed on the system, and the fractal basin boundary may also appear when the system is excited by Gaussian white noise only. From the RSIP results in given limited domain, sudden discontinuous descents in RSIP curves may occur when the system is excited by harmonic or stochastic forces, which are different from the customary continuous ones in view of the first-passage problems. In addition, it is interesting to find that RSIP values can even increase with increasing driving amplitude of the external harmonic excitation when the Gaussian white noise is also present in the system. The project supported by the National Natural Science Foundation of China (10302025 and 10672140). The English text was polished by Yunming Chen.     

高斯色噪声和谐波激励共同作用下耦合SD振子的混沌研究

耦合SD振子作为一种典型的负刚度振子, 在工程设计中有广泛应用. 同时高斯色噪声广泛存在于外界环境中, 并可能诱发系统产生复杂的非线性动力学行为, 因此其随机动力学是非线性动力学研究的热点和难点问题. 本文研究了高斯色噪声和谐波激励共同作用下双稳态耦合SD振子的混沌动力学, 由于耦合SD振子的刚度项为超越函数形式, 无法直接给出系统同宿轨道的解析表达式, 给混沌阈值的分析造成了很大的困难. 为此, 本文首先采用分段线性近似拟合该振子的刚度项, 发展了高斯色噪声和谐波激励共同作用下的非光滑系统的随机梅尔尼科夫方法. 其次, 基于随机梅尔尼科夫过程, 利用均方准则和相流函数理论分别得到了弱噪声和强噪声情况下该振子混沌阈值的解析表达式, 讨论了噪声强度对混沌动力学的影响. 研究结果表明, 随着噪声强度的增大混沌区域增大, 即增大噪声强度更容易诱发耦合SD振子产生混沌. 当阻尼一定时, 弱噪声情况下混沌阈值随噪声强度的增加而减小; 但是强噪声情况下噪声强度对混沌阈值的影响正好相反. 最后, 数值结果表明, 利用文中的方法研究高斯色噪声和谐波激励共同作用下耦合SD振子的混沌是有效的.本文的结果为随机非光滑系统的混沌动力学研究提供了一定的理论指导.      

Effect of noise on erosion of safe basin in power system

We study the effect of Gaussian white noise on erosion of safe basin in a simple model of power system whose safe basin is integral in the absence of noise. The stochastic Melnikov method is first applied to predict the onset of basin erosion when the noise excitation is present in system. And then the eroded basins are simulated according to the necessary restrictions for the system’s parameters. It is found that for the noisy power system when the noise intensity σ is greater than a threshold, basin erosion occurs and as σ is further increased basin erosion is aggravated. These studies imply that random noise excitation can induce and enhance the basin erosion in the power system.     

Stochastic response and reliability of electromagnetic energy harvester with mechanical impact and Coulomb friction

Nonlinear Dynamics - Combining the complex factors of mechanical impact, Coulomb friction, harmonic excitation, and Gaussian white noise, an electromagnetic energy harvesting system is analytically...     

Noise-induced chaotic motions in Hamiltonian systems with slow-varying parameters

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.     

Complex fluctuations of flexible plates under longitudinal loads with account for white noise

This paper describes the influence of intensity of external additive white noise on the nonlinear dynamics of rectangular plates as mechanical systems with an infinite number of degrees of freedom. A new scenario is discovered, which is a combination of the classic Feigenbaum and Pomeau–Manneville scenarios. The classical methods of nonlinear dynamics and wavelet transforms were used to reveal the peculiarities of a modified scenario. The noise-induced transitions are investigated, and it is shown that the noise exposure is accompanied with the transition to chaotic fluctuations with a lower amplitude of the driving load. It is determined that the presence of external fluctuations does not affect the scenario of transition from harmonic to chaotic fluctuations.     

Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance

In this paper, we investigate nonlinear dynamical responses of two-degree-of-freedom airfoil (TDOFA) models driven by harmonic excitation under uncertain disturbance. Firstly, based on the deterministic airfoil models under the harmonic excitation, we introduce stochastic TDOFA models with the uncertain disturbance as Gaussian white noise. Subsequently, we consider the amplitude–frequency characteristic of deterministic airfoil models by the averaging method, and also the stochastic averaging method is applied to obtain the mean-square response of given stochastic TDOFA systems analytically. Then, we carry out numerical simulations to verify the effectiveness of the obtained analytic solution and the influence of harmonic force on the system response is studied. Finally, stochastic jump and bifurcation can be found through the random responses of system, and probability density function and time history diagrams can be obtained via Monte Carlo simulations directly to observe the stochastic jump and bifurcation. The results show that noise can induce the occurrence of stochastic jump and bifurcation, which will have a significant impact on the safety of aircraft.     

Wada分形吸引域边界上的混沌鞍

应用广义胞映射图论（Generalized Cell Mapping Digraph)方法，数值地研究Thompson的逃逸方程在最佳逃逸点附近的分岔。发现了嵌入在Wada分形吸引域边界上的混沌鞍，混沌鞍是状态空间不稳定（非吸引）的混沌不变集合。Wada分形吸引域边界是具有Wada性质的边界，即吸引域边界上的任意点也同时是至少两个其它吸引域的边界点，称为Wada域边界。我们证明Wada域边界上的混沌鞍导致局部鞍结分岔具有全局不确定性结局，研究了Wada域边界上混沌鞍的形成与演化，证明最终的逃逸分岔是混沌吸引子碰撞混沌鞍的边界激变。     

加性和乘性三值噪声激励下周期势系统的动力学分析

周期势系统是一类在机械工程、物理、化学、神经生物等领域应用十分广泛的系统,其随机动力学特性的研究是非线性科学的一个热点和难点问题.三值噪声是真实噪声的典型模型,不仅包含二值噪声和高斯白噪声情形,而且能更好地描述自然界中随机环境扰动的多样性,本文研究了由加性和乘性三值噪声驱动的周期势系统中概率密度的演化和随机共振.通过计...     

Noise influenced elastic cantilever dynamics with nonlinear tip interaction forces

In this work, the authors study the influence of noise on the dynamics of base-excited elastic cantilever structures at the macroscale and microscale by using experimental, numerical, and analytical means. The macroscale system is a base excited cantilever structure whose tip experiences nonlinear interaction forces. These interaction forces are constructed to be similar in form to tip interaction forces in tapping mode atomic force microscopy (AFM). The macroscale system is used to study nonlinear phenomena and apply the associated findings to the chosen AFM application. In the macroscale experiments, the tip of the cantilever structure experiences long-range attractive and short-range repulsive forces. There is a small magnet attached to the tip, and this magnet is attracted by another one mounted to a high-resolution translatory stage. The magnet fixed to the stage is covered by a compliant material that is periodically impacted by the cantilever’s tip. Building on their earlier work, wherein the authors showed that period-doubling bifurcations associated with near-grazing impacts occur during off-resonance base excitations of macroscale and microscale cantilevers, in the present work, the authors focus on studying the influence of Gaussian white noise when it is included as an addition to a deterministic base excitation input. The repulsive forces are modeled as Derjaguin–Muller–Toporov (DMT) contact forces in both the macroscale and microscale systems, and the attractive forces are modeled as van der Waals attractive forces in the microscale system and magnetic attractive forces in the macroscale system. A reduced-order model, based on a single mode approximation is used to numerically study the response for a combined deterministic and random base excitation. It is experimentally and numerically found that the addition of white Gaussian noise to a harmonic base excitation facilitates contact between the tip and the sample, when there was previously no contact with only the harmonic input, and results in a response that is nominally close to a period-doubled orbit. The qualitative change observed with the addition of noise is associated with near-grazing impacts between the tip and the sample. The numerical and experimental results further motivate the formulation of a general analytical framework, in which the Fokker–Planck equation is derived for the cantilever-impactor system. After making a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and numerically solved. The resulting findings support the experimental results and demonstrate that noise can be added to the input to facilitate contact between the cantilever’s tip and the surface, when there was previously no contact with only a harmonic input. The effects of Gaussian white noise are numerically studied for a tapping mode AFM application, and it is shown that contact between the tip and the sample can be realized by adding noise of an appropriate level to a harmonic excitation.     

非高斯随机激励下滞迟系统响应与首次穿越失效

针对由有界噪声、泊松白噪声和高斯白噪声共同构成的非高斯随机激励,通过Monte Carlo数值模拟方法研究了此激励作用下双线性滞迟系统和Bouc-Wen滞迟系统这两类经典滞迟系统的稳态响应与首次穿越失效时间。一方面,分析了有界噪声和泊松白噪声这两种分别具有连续样本函数和非连续样本函数的非高斯随机激励,在不同激励参数条件下对双线性滞迟系统和Bouc-Wen滞迟系统的稳态响应概率密度、首次穿越失效时间概率密度及其均值的不同影响;另一方面,揭示了在这类非高斯随机激励荷载作用下,双线性滞迟系统的首次穿越失效时间概率密度将出现与Bouc-Wen滞迟系统的单峰首次穿越失效时间概率密度截然不同的双峰形式。     

Duffing-van der Pol系统的随机分岔

应用广义胞映射图论方法(GCMD)研究了在谐和激励与随机噪声共同作用下的Duffing-van der Pol系统的随机分岔现象. 系统参数选择在多个吸引子与混沌鞍共存的范围内. 研究发现, 随着随机激励强度的增大,该系统存在两种分岔现象: 一种为随机吸引子与吸引域边界上的鞍碰撞, 此时随机吸引子突然消失; 另一种为随机吸引子与吸引域内部的鞍碰撞, 此时随机吸引子突然增大. 研究证实, 当随机激励强度达到某一临界值时, 该系统还会发生D-分岔(基于Lyapunov指数符号的改变而定义), 此类分岔点不同于上述基于系统拓扑性质改变所得的分岔点.     

Controlling erosion of safe basin in nonlinear parametrically excited systems

This paper considers the dynamical behavior of a Duffing-Mathieu type system with a cubic single-well potential during the principal parametric resonance. Both the cases of constant and time-dependent excitation amplitude are used to observe the variation of the extent and the rate of the erosion in safe basins. It is evident that the appearance of fractal basin boundaries heralds the onset of the losing of structural integrity. The minimum value of control parameter to prevent the basin from erosion is given along with the excitation amplitude varying. The results show the time-dependence of excitation amplitude can be used to control the extent and the rate of the erosion and delay the first occurrence of heteroclinic tangency. The project supported by the National Natural Science Foundation of China and PSF of China.     

FIRST-PASSAGE FAILURE OF PREISACH HYSTERETIC SYSTEMS

The first-passage failure of a single-degree-of-freedom hysteretic system with non- local memory is investigated. The hysteretic behavior is described through a Preisach model with excitation selected as Gaussian white noise. First, the equivalent nonlinear non-hysteretic sys- tem with amplitude-dependent damping and stiffness coefficients is derived through generalized harmonic balance technique. Then, equivalent damping and stiffness coefficients are expressed as functions of system energy by using the relation of amplitude to system energy. The stochastic aver- aging of energy envelope is adopted to accept the averaged It5 stochastic differential equation with respect to system energy. The establishing and solving of the associated backward Kolmogorov equation yields the reliability function and probability density of first-passage time. The effects of system parameters on first-passage failure are investigated concisely and validated through Monte Carlo simulation.     

Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries

In the present study, the large-amplitude vibrations and stability of a perfect circular cylindrical shell subjected to axial harmonic excitation in the neighborhood of the lowest natural frequencies are investigated. Donnell's shallow shell theory is used and the shell spatial discretization is obtained by the Ritz method. An efficient low-dimensional model presented in previous publications is used to discretize the continuous system. The main purpose of this work is to discuss the use of basins of attraction as a measure of the reliability and safety of the structure. First, the nonlinear behavior of the conservative system is discussed and the basin structure and volume is understood from the topologic structure of the total energy and its evolution as a function of the system parameters. Then, the behavior of the forced oscillations of the harmonically excited shell is analyzed. First the stability boundaries in force control space are obtained and the bifurcation events connected with these boundaries are identified. Based on the bifurcation diagrams, the probability of parametric instability and escape are analyzed through the evolution and erosion of basin boundaries within a prescribed control volume defined by the manifolds. Usually, basin boundaries become fractal. This together with the presence of catastrophic subcritical bifurcations makes the shell very sensitive to initial conditions, uncertainties in system parameters, and initial imperfections. Results show that the analysis of the evolution of safe basins and the derivation of appropriate measures of their robustness is an essential step in the derivation of safe design procedures for multiwell systems.     

拟可积哈密顿系统中噪声诱发的混沌运动

研究拟可积哈密顿系统在谐和与噪声激励联合作用下的混沌运动。通过对噪声性质的假定,并利用动力系统理论,给出了高维梅尔尼科夫方法应用于随机拟可积哈密顿系统的推广形式。根据这种推广的方法,研究了谐和与高斯白噪声激励联合使用下两自由度拟可积哈密顿系统 同宿分岔,得出了系统发生混沌运动的参数阈值,并由此讨论了噪声对系统的混沌运动的影响。蒙特-卡罗方法模拟、李雅普诺夫指数等数值结果表明,这种推广的方法是有效的。     

STOCHASTIC DISCRETE MODEL OF TWO-STAGE ISOLATION SYSTEM WITH RIGID LIMITERS

The possible intermittent impacts of a two-stage isolation system with rigid limiters have been investigated. The isolation system is under periodic external excitation disturbed by small stationary Gaussian white noise after shock. The maximal impact Poincare map is proposed based on the multi-body dynamics with unilateral constrains. Then in the period after shock, the zero order approximate stochastic discrete model and the first order approximate stochastic model are developed. The real isolation system of an MTU diesel engine is used to evaluate the established model. After calculating of the numerical example, the effects of noise excitation on the isolation system are discussed. The results show that the property of the system is complicated due to intermittent impact. The difference between zero order model and the first order model may be great. The effect of small noise is obvious. The results may be expected useful to the naval designers.     

Response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control

The response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control subject to Gaussian white noise excitation is studied by using the stochastic averaging method. First, a quasi-Hamiltonian system with delayed feedback bang–bang control subjected to Gaussian white noise excitation is formulated and transformed into the Itô stochastic differential equations for quasi-integrable Hamiltonian system with feedback bang–bang control without time delay. Then the averaged Itô stochastic differential equations for the later system are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution of the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both nonresonant and resonant cases. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback bang–bang control on the response of the systems.     

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.     

Numerical simulations of chaotic dynamics in a model of an elastic cable

The finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities.The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed.Part of this work was presented at the XVIIth Int. Congr. of Theor. and Appl. Mech., Grenoble, August 1988.