基于参数展开的同伦分析法在强非线性随机动力系统中的应用

将基于参数展开的同伦分析法(PE-HAM)进行了推广，使之适用于谐和激励与随机噪声联合作用下的强非线性随机动力系统. 通过构造合适的同伦映射，将对强非线性随机动力系统响应的求解转化为对一组线性随机微分方程的求解. 进一步研究了受到谐和与Gauss白噪声激励的强非线性Duffing振子，由PE-HAM得到了该系统的解过程和稳态概率密度的解析表达式. 数值模拟的结果说明了PE-HAM方法的精确性. 关键词： PE-HAM方法 强非线性随机动力系统 稳态概率密度 解过程 随机激励     

随机参数Duffing系统中的随机混沌及其延迟反馈控制

研究了谐和激励下含有界随机参数Duffing系统(简称随机Duffing系统)中的随机混沌及其延迟反馈控制问题.借助Gegenbauer多项式逼近理论，将随机Duffing系统转化为与其等效的确定性非线性系统.这样，随机Duffing系统在谐和激励下的混沌响应及其控制问题就可借等效的确定性非线性系统来研究.分析阐明了随机混沌的主要特点，并采用Wolf算法计算等效确定性非线性系统的最大Lyapunov指数，以判别随机Duffing系统的动力学行为.数值计算表明，恰当选取不同的反馈强度和延迟时间，可分别达到抑制或诱发系统混沌的目的，说明延迟反馈技术对随机混沌控制也是十分有效的. 关键词： 随机Duffing系统 延迟反馈控制 随机混沌 Gegenbauer多项式     

窄带随机噪声作用下Duffing振子的双峰稳态概率密度

研究了Duffing振子在窄带随机噪声激励下的系统响应的双峰稳态概率密度问题.用多尺度法 分离了系统的快变项，得到了系统慢变项满足的随机微分方程.用线性化方法求出了双峰稳 态概率密度的表达式.数值模拟表明本文提出的方法是有效的. 关键词： Duffing振子 双峰稳态概率密度 多尺度法 线性化方法     

一类非线性相对转动动力系统的平衡稳定性及组合谐波近似解

建立一类含广义非线性弹性力的两质量相对转动系统的非线性动力学方程. 应用变量梯度法构造李雅普诺夫函数,研究相对转动非线性自治系统的稳定性. 应用摄动法求得相对转动非线性非自治系统在两种不同频率谐波共同激励下的组合谐波响应的近似解. 关键词： 相对转动 非线性动力系统 稳定性 组合谐波响应     

谐和与噪声联合作用下Duffing振子的安全盆分叉与混沌

研究了软弹簧Duffing振子在确定性谐和外力和有界随机噪声联合作用下，系统安全盆的侵蚀和混沌现象.推导出系统的随机Melnikov过程，根据Melnikov过程在均方意义上出现简单零点的条件给出了系统出现混沌的临界值，然后用数值模拟方法计算了系统的安全盆分叉点.结果表明，由于随机扰动的影响，系统的安全盆分叉点发生了偏移，并且使得混沌容易发生. 关键词： Duffing振子 安全盆 分叉 混沌     

随机Duffing单边约束系统的倍周期分岔

通过引进平均约束面和平均跃变方程对随机约束系统的约束条件进行处理，把研究随机光滑系统倍周期分岔的Chebyshev多项式逼近的方法运用到随机非光滑系统中，数值研究表明随机Duffing单边约束系统同样存在丰富的倍周期分岔现象，Chebyshev多项式逼近是研究带有约束的随机非光滑动力系统的有效方法. 关键词： 非光滑动力系统 随机Duffing系统 Chebyshev多项式 倍周期分岔     

一类谐波激励相对转动非线性动力系统的稳定性与近似解

建立具有一般非线性弹性力、广义摩阻力和谐波激励的一类相对转动非线性动力系统的动力学方程. 对相对转动非线性自治系统进行定性分析,通过构造Lyapunov函数研究自治系统奇点的稳定性. 运用多尺度法求解谐波激励下非自治系统在几种不同共振响应下的近似解,同时分析了主振系统稳态运动的稳定性. 关键词： 相对转动 非线性动力系统 Lyapunov函数 稳定性     

不同周期信号激励下分数阶线性系统的响应特性分析

研究了含分数阶导数阻尼的一类线性系统在不同周期信号激励下系统的响应问题.首先在简谐信号的激励下,利用谐波平衡法得到了系统响应的近似解,这一结果和已有文献(申永军,杨绍普,邢海军2012物理学报61 110505)的结果完全相同,但本文的求解过程大为简化,而且本文进一步扩展了分数阶导数阻尼微分阶数的取值范围.接着,利用傅里叶级数展开法和线性系统的叠加原理,求得了一般周期信号激励下系统响应的近似解,并以周期方波信号和周期全波正弦信号为例进行了说明.本文的结果表明,分数阶导数阻尼的微分阶数影响系统响应中各阶谐波的共振频率和共振振幅.系统响应的幅值与分数阶导数阻尼的微分阶数之间的单调关系主要受外激信号频率的影响.除解析分析外,本文还用数值模拟对相关结论进行了验证,两种结果符合良好,表明本文的分析方法是可行的.     

一类线性阻尼振子的随机共振研究

针对随机有色噪声参数激励和周期调制噪声外激励联合作用下的线性阻尼振子,利用Shapiro-Loginov公式推导了系统响应的一、二阶稳态矩的解析表达式.发现这类系统存在传统的随机共振、广义的随机共振和“真正”的随机共振;当乘性噪声强度和调制噪声强度的比值大于等于1时,系统出现随机多共振现象.通过数值计算的系统响应功率谱,验证了理论分析结果. 关键词： 随机共振 周期调制的噪声 线性阻尼振子     

带碰撞双稳态压电俘能系统的俘能特性研究

双稳态俘能系统的运动常常会陷入单个势能阱中, 导致俘能效率降低. 为了解决这个问题, 本文提出了一类带碰撞的磁斥力双稳态压电振动能量采集系统. 建立了该碰撞双稳态系统的机电耦合方程, 分析了碰撞对双稳态系统动力学特性的影响. 研究了确定性激励和低强度随机激励下碰撞对系统响应特性和俘能效率的影响. 结果表明: 简谐激励下, 碰撞能够使得原双稳态系统的单阱小幅周期运动转变为双阱间的大幅运动, 从而有效地提高输出功率. 得到了低强度随机激励下, 不同碰撞间隙对系统动力响应特性和输出功率的影响规律. 对一个给定的随机激励, 存在一个最优的碰撞间隙, 此时碰撞能够将原双稳态系统单阱内的随机运动转化为频繁的双阱跳跃, 出现大幅值运动, 从而大幅提高了系统的俘能效率.     

Response of a strongly non-linear oscillator to narrowband random excitations

The principal resonance of a van der Pol-Duffing oscillator subject to narrowband random excitations has been studied. By introducing a new expansion parameter the method of multiple scales is adapted for the strongly non-linear system. The behavior of steady state responses, together with their stability, and the effects of system damping and the detuning, and magnitude of the random excitation on steady state responses are analyzed in detail. Theoretical analyses are verified by some numerical results. It is found that when the random noise intensity increases, the steady state solution may change form a limit cycle to a diffused limit cycle, and the system may have two different stable steady state solutions for the same excitation under certain conditions. The results obtained for the strongly non-linear oscillator complement previous results in the literature for weakly non-linear systems.     

窄带随机噪声作用下单自由度非线性干摩擦系统的响应

研究了单自由度非线性干摩擦系统在窄带随机噪声参数激励下的主共振响应问题.用Krylov-Bogoliubov平均法得到了关于慢变量的随机微分方程.在没有随机扰动情形,得到了系统响应幅值满足的代数方程.在有随机扰动情形,用线性化方法和矩方法给出了系统响应稳态矩计算的近似计算公式.讨论了系统阻尼项、非线性项、随机扰动项和干摩擦项等参数对于系统响应的影响.理论计算和数值模拟表明,当非线性强度增大时系统的响应显著变小,系统分岔点滞后;随着激励频率的增大系统响应变大,而当激励频率小于一定的值时,系统响应为零;增加干 关键词： 单自由度非线性干摩擦系统 主共振响应 Krylov-Bogoliubov平均法     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

Response of nonlinear random business cycle model with time delay state feedback

The steady state response and bifurcation of nonlinear random business cycle model to random narrow-band excitation with time delay state feedback are studied in this paper. The method of multiple scales is used to determine the business cycle model of modulation of amplitude and phase. The effects of delay, detuning, bandwidth and magnitude of random excitation on dynamics of the business cycle system are investigated. The results show that the complex dynamics such as bifurcation, jump domain and so on are induced by time delay and the phenomena that multiple solution or bifurcation is induced by noise.     

一类准周期参激非线性相对转动动力系统的稳定性与时滞反馈控制

建立了一类含准周期参数激励和时滞反馈的相对转动非线性系统的动力学方程. 采用多尺度法求解1/2亚谐波主参数共振下的分岔响应方程,并分析了系统的稳定性. 在求解非受控系统的定常解的基础上,通过讨论系统的动力学特性,研究了准周期参数激励对系统响应的影响. 采用时滞反馈控制的方法对系统分岔和极限环(域)进行控制,数值模拟的结果表明通过改变时滞参数可以实现对系统分岔的控制,并能有效地控制极限环(域)的幅值和稳定性. 关键词： 相对转动 准周期参激 时滞反馈 极限环     

Non-trivial effect of fast vibration on the dynamics of a class of non-linearly damped mechanical systems

The effect of very high-frequency excitation on the slow dynamics of a class of non-linearly damped mechanical oscillators is considered. Two different models of damping namely, piecewise linear and pth power damping are considered. Fast excitation is modelled as triangular, sinusoidal and random base excitation. The effect of fast excitation is theoretically analyzed using the method of direct partition of motion (MDPM) and direct simulation. The method of numerical averaging is also used, where damping characteristics or excitations are not amenable to analytical techniques. Fast excitation has the non-trivial effect of increasing and decreasing the low-velocity damping of hard and soft dampers, respectively. The effect of fast excitation on the transient and steady state slow dynamics of the system is investigated by direct numerical integration of the equation of motion.     

Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling

The response of a two-degree-of-freedom system with autoparametric coupling under the action of broad band random excitation is investigated. The system corresponds to the autoparametric vibration absorber and is also typical of many common structural configurations. A method based upon the Markov vector approach, together with an approximate treatment of third and higher statistical moments, is used to derive a set of fourteen coupled non-linear equations for the first and second moments of the system responses. A numerical integration procedure is used to obtain quantitative results for the system mean and mean square responses over a range of system parameters.The results show that large random motions of the coupled system may occur when the internal detuning parameter is close to the principal internal resonance, and that these motions may give rise to a suppression effect on the random motions of the main system. A feature of the results is that under conditions of internal resonance the random motions are found to be quasi-stationary, with steady oscillatory terms in the response moments. This suggests the possibility of entrainment of regular harmonic responses by the system random motions.     

强Van der Pol振子的谐波与修正算法

引入谐波平衡近似解的符号运算与同伦延拓法,获得了强Van der Pol振子的解析近似极限环与稳定响应.为提高其近似精度,给出了谐波解的修正算法,并与数值模拟比较.结果表明,该算法非常精确可靠,是研究强非线性动力学系统的有效分析方法.     

STATISTICAL SEISMIC RESPONSE ANALYSIS AND RELIABILITY DESIGN OF NONLINEAR STRUCTURE SYSTEM

Industrial structure systems may have non-linearity, and are also sometimes exposed to the danger of earthquake. In the design of such system, these factors should be accounted for from the viewpoint of reliability. This paper proposes a method to analyze seismic response and reliability design of a complex non-linear structure system under random excitation. The actual random excitation is represented to the corresponding Gaussian process for the statistical analysis. Then, the non-linear system is subjected to this random process. The non-linear structure system is modelled by substructure synthesis method (SSM) procedure. The non-linear equations are expanded sequentially. Then, the perturbed equations are solved in probabilistic method. Several statistical properties of a random process that are of interest in random vibration applications are reviewed in accordance with the non-linear stochastic problem. The system performance condition in the design of system is that responses caused by random excitation be limited within safe bounds. Thus, the reliability of the system is considered according to the crossing theory. Comparing with the results of the numerical simulation proved the efficiency of the proposed method.     

随机激励下二自由度碰撞振动系统的响应分析

研究了一类二自由度碰撞振动系统在随机噪声激励下的响应问题.展示了这种非光滑系统在倍周期分岔通向混沌的道路中存在的擦边分岔行为.通过定义一种随机响应的度量,讨论了随机噪声对于系统响应的影响,并发现在某些参数条件下,随机噪声对于系统响应的影响是明显的,甚至改变了运动的性质.数值模拟表明此法是研究噪声激励下非光滑系统响应的一种有效方法. 关键词： 非光滑系统 二自由度碰撞振动系统 擦边分岔 倍周期分岔