A pendulum on a rotating and vibrating foundation

The motion of a pendulum is investigated in a uniform gravity field. The foundation of the pendulum is assumed to rotate around the vertical simultaneously accomplishing harmonic vibrations along it with a high frequency and small amplitude. With use of the classical perturbation theory and modern methods of analysis of nonlinear dynamical sets, the problem on the existence and stability of periodic motions of a pendulum with the period equal to that of the vibrations of the foundation is solved.     

New aspects of energy transfer from an alternating magnetic field to a pendulum with ferromagnetic suspension

New data are presented about energy transfer from an alternating magnetic field to a pendulum with ferromagnetic suspension. For each frequency of the alternating magnetic field there is a certain field intensity for which energy transfer is maximum.     

On mechanical motion damping of a magnetically trapped diamagnetic particle

We discuss eddy-current-induced limitations of the attenuation of mechanical motion of a diamagnetic particle trapped by an inhomogeneous magnetic field in the Earth's gravitational potential. We show that the mechanical frequency of the particle oscillation is independent on the particle properties and is proportional to the free fall acceleration constant, similarly to the classical mechanical pendulum. The frequency can be used to measure the gravity field. The eddy-current induced attenuation constant does not depend on the mass of the particle and reduces with the particle volume. The quality factor of the mechanical motion can be as high as 10⁹ and is comparable with the attenuation due to interaction of the particle with incompletely evacuated air. A possibility of usage of the particle as a quantum mechanical system is discussed.     

用微元法推导无阻尼单摆运动任意角度下的周期公式

对单摆运动进行了讨论。通过对单摆进行能量分析,得出了单摆角频率与角度的关系式,然后运用微元法,推导出了单摆的理论周期公式。计算结果表明,所得到的单摆运动周期的理论公式与展开公式符合得很好。     

Bi0.8Pb0.2SrCaCuOy超导体中磁通运动引起的内耗峰

在线性增加磁场的条件下,用变频倒扭摆测量了配比成份为(Bi_0.8Pb_0.2)SrCaCu₂O_y的试样在超导态时磁化过程中的低频内耗。测量温度为95K,试样处于零电阻状态,所用频率为0.5-5Hz.在内耗-场强H曲线上观测到内耗峰,它有如下特点:1)峰高随加场速率H的增大而增大,但随测量频率的增大而降低;2)峰值处的场强为4-6mT;3)当H由非零值突然改变至零时,磁化过程特征内耗消失;4)试样处于正常态时,上述特征内耗消失。认为超导体在磁化过程中的上述特征内耗是量子磁通的运动所引起来的。 关键词：     

The dynamics of the pendulum suspended on the forced Duffing oscillator

We investigate the dynamics of the pendulum suspended on the forced Duffing oscillator. The detailed bifurcation analysis in two parameter space (amplitude and frequency of excitation) which presents both oscillating and rotating periodic solutions of the pendulum has been performed. We identify the areas with low number of coexisting attractors in the parameter space as the coexistence of different attractors has a significant impact on the practical usage of the proposed system as a tuned mass absorber.     

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.     

An autoparametric energy harvester

This paper presents a numerical study of an autoparametric system composed of two elements: a pendulum and an excited nonlinear oscillator. Owing to an inertial coupling between the two elements, different types of motion are possible, from periodic to chaotic. This study examines a linear induction of an energy harvester depending on the pendulum motion. The harvester consists of a cylindrical permanent magnet mounted on a rotor and of four windings fixed to the housing as a stator. When the pendulum is rotating or swinging, the converter is generating energy due to magnetic induction. In this paper, a method utilizing parametrical resonance for harvesting energy from low frequency vibrations is studied. The authors compare energy induced by different types of pendulum motion: swinging, rotation and chaotic dynamics. Additionally, voltage values for different parameters of excitation are estimated.     

一种采用双换能器和摆式结构的宽频振动能量采集器

针对悬臂梁振动能量采集器在大振幅振动下梁容易断裂的缺点,本文提出了一种基于摆式结构的具有宽频和倍频特性的振动能量采集器,该采集器由两个Terfenol-D/PMN-PT/Terfenol-D磁电换能器和嵌有六个磁铁的旋转摆构成.文中建立了摆式结构的摆动方程,分析了采集器的频率响应特性以及谐振时的机-磁-电转换特性,并对采集器输出电压波形进行了频谱分析.理论和实验研究表明:该采集器具有宽频和倍频特性,采集器样机在1 g(1 g=9.8 m/s~2)有效值加速度振动下,向下扫频时的半功率带宽达到4.8 Hz,且能在f=16.9 Hz的振动下获得3.569 mW的负载功率.利用双换能器以及采集器的倍频和宽频特性,能有效地提高低频时采集器的输出功率.     

准周期激励与应变超晶格的动力学稳定性

假设超晶格锯齿形沟道对粒子的作用等效为形状相似的周期场作用. 在经典力学框架内,引入正弦平方势,把粒子运动方程化为具有阻尼项和双频激励项的摆方程. 用Melnikov方法对单频激励系统的分叉与混沌进行分析;用Lyapunov方法对双频激励系统的稳定性进行讨论. 结果表明：在弱非线性情况下,双频激励系统存在局域不稳定,且这种不稳定将向全局扩展,直至混沌的出现;导致混沌的双频激励强度远小于单频激励强度;外加一个适当的超声场可望将这种敏感钝化,使系统的稳定性得到改善. 关键词： 超晶格 准周期激励 混沌 稳定性     

Instabilities in the main parametric resonance area of a mechanical system with a pendulum

Vibrations of an autoparametric system, composed of a nonlinear mechanical oscillator with an attached damped pendulum, around the principal resonance region, are investigated in this paper. Approximate analytical solutions of the model are determined on the basis of the Harmonic Balance Method (HBM). Correctness of the analytical results is verified by numerical simulations and experimental tests performed on an especially prepared experimental rig. The influence of all essential parameters such as damping, excitation amplitude and frequency, nonlinear stiffness of the spring, on the localisation of the instability region and the system dynamics is presented in the work. Regions of regular system oscillations, chaotic motions, and full rotation of the pendulum are confirmed experimentally.     

不同控制参数下的弹簧摆

通过拉格朗日方程得出弹簧摆系统的动力学方程,并以频率比作为控制参数,利用Matlab软件对不同控制参数和初始摆角下的弹簧摆进行了数值模拟,从而直观地研究了弹簧摆的运动.     

Measurement and analysis of complex modulated motions in a weakly nonlinear system

The two-timing dynamics of a forced, weakly nonlinear system are considered. Experimental results on a roll-forced spherical pendulum verify the occurrence of slow-time, subharmonically and chaotically modulated oscillations in a frequency band near resonance; the modulatory behavior follows the occurrence of a Hopf bifurcation in the slow-time coordinates. A phase-sensitive detection method for experimentally isolating the slow-time behavior is described. A Lagrangian model for the pendulum is developed, and slow-time equations are presented. A two-thirds power scalling law relating roll angle to the appearance of the modulations is derived from the slow-time equations and tested with experimental results.     

Determination of the thermal noise limit in test of weak equivalence principle with a rotating torsion pendulum

Thermal noise is one of the most fundamental limits to the sensitivity in weak equivalence principle test with a rotating torsion pendulum. Velocity damping and internal damping are two of many contributions at the thermal noise, and which one mainly limits the torsion pendulum in low frequency is difficult to be verified by experiment. Based on the conventional method of fast Fourier transform, we propose a developed method to determine the thermal noise limit and then obtain the precise power spectrum density of the pendulum motion signal. The experiment result verifies that the thermal noise is mainly contributed by the internal damping in the fiber in the low frequency torsion pendulum experiment with a high vacuum. Quantitative data analysis shows that the basic noise level in the experiment is about one to two times of the theoretical value of internal damping thermal noise.     

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.     

Harvesting base vibration energy by a piezoelectric inverted beam with pendulum

We proposed a two-degrees-of-freedom inverted piezoelectric beam with pendulum to promote the performance of vibration energy harvesting. This configuration is composed of an inverted elastic beam and a pendulum attached to its free end. The electromechanical equations governing the nonlinear system were derived. The harmonic balance method(HBM)is applied to solve the equation and the results prove that there exists a 1:3 super-harmonic resonance. The simulation results show that owing to the particular nonlinearity, there appears a special bending effect in the amplitude-frequency response, i.e., bending right for the first natural frequency and left for the second natural frequency, which is beneficial for harvesting vibration energy. The HBM results are verified by the entity simulations. Furthermore, over a relatively wide range of power spectral density, it could reach a dense jumping and give a dense high pulse voltage.     

Experiments on the bifurcation behaviour of a forced nonlinear pendulum

We investigate a mechanical system (forced nonlinear torsion pendulum). The state diagram is given as function of both the external driving frequency and the damping parameter. A bifurcation diagram is measured showing period doubling, chaos and periodic windows. The results are in qualitative agreement with the recent theory.     

Stability of the parametrically excited damped inverted pendulum: theory and experiment

The parametrically driven, damped, inverted pendulum can be dynamically stabilized in particular regions of the parameter space. The impact of damping on dynamic stabilization can be stabilizing or destabilizing depending on the location in parameter space (i.e., drive frequency and amplitude). Floquet analysis and numerical simulations were used to determine the stable regions. An experiment was conducted that verifies the model. Physical explanations and simple bounding approximations are provided to summarize findings. The utility of the highly damped pendulum results are illustrated by drawing the analogy to dynamic stabilization of the Rayleigh-Taylor instability: it permits ready demonstration that dynamic stabilization is impossible in that system absent surface tension.     

Time-independent description of rapidly driven systems in the presence of friction: multiple scale perturbation approach

The dynamics of a classical system driven by a rapidly oscillating field (with frequency ω) in the presence of friction has been investigated using the multiple scale perturbation theory (MSPT). By exploiting the idea of separation of time scales, the slow motion has been computed in a systematic expansion in the inverse of ω to the order ω(-3). This perturbation series can be viewed as a generalization of the calculation presented by Landau and Lifshitz for Kapitza's pendulum (where the point of suspension is moved periodically) in the presence of friction. The radiation induced dynamics of the system is found to be described by an effective time-independent potential with friction that controls the slow motion. The explicit appearance of friction in our computed effective potential is a manifestation of the dynamical effect due to the fast motion. The present study demonstrates that MSPT can be used to understand and predict the classical dynamics of a driven system in the presence of friction.     

扭摆——探测低频引力波的一种可能的天线

本文研究了扭摆对引力波的响应和吸收截面,并对有关的问题进行了讨论,指出它可以作为探测低频引力波的机械共振天线。 关键词：