基于状态反馈参数激励系统的超谐共振分岔控制

设计了非线性参数控制器，改变了参数激励系统2 倍超谐共振时的稳态响应，减小了系统的响应幅值和消除了超谐共振时的鞍结分岔，从而消 除了跳跃和滞后现象.首先由多尺度法得到参数系统的近似频响方程，再进行分岔分析，从 而实现非线性控制的目标.通过数值模拟，说明状态反馈控制是可行的和有效的. 关键词： 参数激励 鞍结分岔 分岔控制 2倍超谐共振     

弹簧螺旋摆系统非线性振动的理论研究

运用牛顿第二运动定律对弹簧螺旋摆系统建立了模型方程,该方程为一组非线性微分方程,表明该系统具有复杂的非线性特征.理论分析表明,当系统固有的振动频率和摆动频率之比为2时,自由振动的弹簧螺旋摆系统存在内共振现象,数值求解结果证实了这一结论.并认为在一般情况下,弹簧螺旋摆系统的自由振动可能是准周期的.     

Langford系统Hopf分岔极限环幅值控制

系统地研究了Langford 系统Hopf分岔极限环幅值非线性反馈控制问题.根据中心流形定理和规范型降维理论, 推导含控制增益项的非线性曲率系数控制公式和幅值近似解,通过数值仿真验证并绘制极限环幅控关系曲线. 所推导的控制公式为Langford 系统极限环幅值控制提供了方便有效的控制方法.     

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     

一类相对转动系统Hopf分岔的非线性反馈控制

研究一类非线性摩擦阻尼力作用下的相对转动系统的Hopf分岔现象,给出系统产生Hopf分岔的充要条件,提出一种非线性反馈控制方法对系统的Hopf分岔点进行转移,并控制极限环的稳定性和幅值,数值模拟说明该方法对一类相对转动系统的Hopf分岔控制是有效的. 关键词： 相对转动 非线性反馈控制 Hopf分岔 极限环     

研究强非线性振动系统同宿分岔问题的规范形方法

以改进的规范形理论为基础，采用强非线性振动问题的分析方法，拓展了原有弱非线性振动系统同宿分岔判据的适用范围.首先在复规范形求解过程中引入待定固有频率，计算了一类单自由度强非线性振动系统的周期解.然后分别依据系统的待定固有频率趋于零和周期轨道趋近于鞍点两条途径获得了强非线性振动条件下系统同宿分岔的解析判据.最后通过与原有解析结果和数值结果相比较验证了本文方法的有效性. 关键词： 规范形 同宿分岔 强非线性 周期解     

非线性切换系统的动力学行为分析

建立了周期切换下的非线性电路模型,基于子系统平衡点及其稳定性分析,分别给出了其相应的fold分岔和Hopf分岔条件,讨论了子系统在不同平衡态下由周期切换导致的各种复杂行为,指出切换系统的周期解随参数的变化存在着倍周期分岔和鞍结分岔两种失稳情形,并相应地导致不同的混沌振荡,进而结合系统轨迹及其相应的分岔分析,揭示了各种振荡模式的动力学机理. 关键词： 周期切换 倍周期分岔 鞍结分岔 混沌     

乘性色噪声激励下三稳态van der Pol-Duffing振子随机P-分岔

研究了乘性色噪声作用下三稳态van der Pol-Duffing振子的随机P-分岔问题. 首先应用随机平均法得到系统振动幅值稳态概率密度函数的表达式, 进而应用奇异性理论, 得到刻画随机P-分岔发生的临界参数条件的转迁集以及系统存在的典型稳态概率密度曲线, 并通过Monte-Carlo数值模拟进行了验证. 以此为基础讨论了噪声强度、相关时间、系统线性阻尼系数对随机P-分岔和系统稳态响应行为的影响.     

试样激励下轻敲模式原子力声显微镜的非线性动力学特性

利用Euler-Bernoulli梁理论和DMT针尖-样品作用力模型建立了试样激励下轻敲模式原子力声显微镜(AFAM)系统的动力学方程,并应用非线性动力学分析方法对AFAM微悬臂梁的振动特性进行研究。通过合理改变超声激励幅值、超声激励频率和针尖-样品初始间距等模型参数模拟得到微悬臂梁的超谐波、次谐波、准周期和混沌振动现象,采用时间序列、频谱、相空间、Poincare截面和Lyapunov指数等方法对不同非线性振动特性进行表征。通过分析不同模型参数条件下微悬臂梁针尖-样品作用力特性,探索了微悬臂梁不同非线性振动现象的产生机制。此外,研究了AFAM微悬臂梁运动的分岔特性,发现当超声激励幅值和针尖-样品初始间隙连续变化时,周期、准周期和混沌运动交替出现。研究结果对AFAM系统非线性动力学行为分析和混沌振动控制提供了理论参考。     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响。     

Experimental control for initiating and maintaining rotation of parametric pendulum

In this paper, the authors have studied experimentally the control methods of a parametric pendulum excited harmonically to initiate and maintain a period one rotation – the most superior response for energy harvesting. For initiating the period one rotation inherent in the system, first the bang-bang method is applied. Then a new method where velocity is monitored is proposed and applied and finally the time-delayed feedback method with multi-switching is considered. Ultimately the problem of maintaining the rotation of the pendulum is addressed. For first time, robustness and sensitivity of the latter method to change of frequency and amplitude of excitation and added noise are studied. Finally, it has been demonstrated how the delayed feedback method can be applied in a system of two pendula to ensure synchronized rotation.     

弱阻尼非线性单摆的周期研究

用平均法研究了弱阻尼条件下非线性单摆的振动,得到周期比随角振幅和阻尼常量变化的表达式.     

用静电探针反馈控制KT—5C托卡马克边缘湍流

在ＫＴ－５Ｃ托卡马克上，利用两组边缘静电探针分析作为控制探针和探测探针进行了反馈控制边缘湍流的实验，在适当的回路增益下的，当系统相移为９０°时，反馈使湍动幅度明显地抑制，电子密度和电子温度扰劝均降低了约２５％，横向粒子输运通量也相应地降低了２５％，而相移为０°，１８０°和－９０°的反馈却使湍动幅度增强，这显示反馈是基于一种非线性机制而作用于湍动的，在一定意义上，这正体现了湍流本身的非线性特征。     

Response control for the externally excited van der Pol oscillator with non-local feedback

A non-local control force is introduced in such a way to obtain a third-order nonlinear differential equation (jerk dynamics) and to control nonlinear vibrations in an externally excited van der Pol oscillator. Two first-order nonlinear ordinary differential equations governing the modulation of the amplitude and the phase of solutions are derived and subsequently the performance of the control strategy is investigated. Excitation amplitude–response and frequency–response curves are shown. In certain cases when the excitation amplitude is very low an approximate analytic solution corresponding to a modulated two-period quasi-periodic motion can be obtained for the uncontrolled system. Uncontrolled and controlled systems are compared and the appropriate choices for the feedback gains are found in order to reduce the amplitude peak of the response and to exclude the possibility of quasi-periodic motion. Numerical simulation confirms the validity of the new method.     

Dynamics of a pendulum with a quasiperiodic perturbation

The dynamics of a damped pendulum with a quasiperiodic external perturbation is investigated. It is shown that in contrast to a pendulum with a periodic perturbation, a quasiperiodic perturbation leads to chaos in the weakly nonlinear limit when the peak-to-peak amplitude of the oscillations of the pendulum is small. This effect is attributed to the appearance of saddle states induced by the external perturbation. The analytical conditions for the appearance of chaotic oscillations are obtained by the method of running Lyapunov exponents and by the repeated-averaging technique. Zh. Tekh. Fiz. 67, 1–7 (October 1997)     

基础水平运动的弹簧摆的周期解与稳定性

针对基础水平运动的弹簧摆的非线性动力学响应进行研究,利用拉格朗日方程建立了系统的动力学方程.将离散傅里叶变换、谐波平衡法以及同伦延拓方法相结合,对系统的周期响应进行求解,避免了传统方法计算中使用泰勒展开引起的小振幅的限制,与数值计算结果的对比表明该求解方法具有较高的精确度.利用Floquet理论分析了周期响应的稳定性,给出了基础运动振幅和频率对系统周期响应的影响.研究发现:对应某些基础频率和振幅,系统的周期响应可能发生Hopf分岔;利用数值计算研究了Hopf分岔后系统响应随基础频率和振幅的变化,发现系统出现了倍周期运动、拟周期运动和混沌等复杂的动力学行为.研究表明系统进入混沌的主要路径是拟周期环面破裂和阵发性.     

一类耦合非线性相对转动系统的Hopf分岔控制

建立一类耦合非线性相对转动系统的动力学方程,研究系统在主共振和1∶1内共振情况下的Hopf分岔行为,设计非线性反馈控制器,控制系统Hopf分岔的发生、极限环的稳定性和幅值,数值模拟证明了该方法的有效性.     

单摆非线性振动的计算机辅助测量

介绍了用计算机辅助测量单摆非线性振动的实验方法,给出了非线性区单摆的振幅和周期和测量结果。     

Self-excited oscillation under nonlinear feedback with time-delay

Several important applications use nonlinear feedback methods for synthetically inducing self-excited oscillations in mechanical systems. The van der Pol and saturation function type feedback methods are widely used. The effects of time-delay on the self-excited oscillation of single and two degrees-of-freedom systems under nonlinear feedback have been studied in this paper. It is shown that a single degree-of-freedom oscillator with the van der Pol type nonlinear feedback can produce unbounded response in presence of time-delay. In general, an uncontrolled time-delay in the feedback changes the state of oscillations in an uncertain manner. Therefore, a bounded saturation type feedback with controllable time-delay is proposed for inducing self-excited oscillations. The feedback signal is essentially an infinite weighted sum of a nonlinear function of the state variables of the system measured at equal intervals in the past. More recent is the measurement, higher is the weight. Thus, the feedback signal uses a large amount of information about the past history of the dynamics. Such a control signal can be realized in practice by a recursive means. The control law allows three parameters to be varied namely, the time-delay, feedback and recursive gains. Multiple time scale analysis is used to plot amplitude vs. time-delay curves. Time-delay can be controlled to vary the amplitude of oscillation as well as to switch the oscillation from one mode to the other in a two degrees-of-freedom system. It is shown that a higher recursive gain can exercise a better and a more robust control on the amplitude of oscillation of the system. Analytical results are compared with the results of numerical simulations.