Rotary motion of the parametric and planar pendulum under stochastic wave excitation

In this paper a rotary motion of a pendulum subjected to a parametric and planar excitation of its pivot mimicking random nature of sea waves has been studied. The vertical motion of the sea surface has been modelled and simulated as a stochastic process, based on the Shinozuka approach and using the spectral representation of the sea state proposed by Pierson–Moskowitz model. It has been investigated how the number of wave frequency components used in the simulation can be reduced without the loss of accuracy and how the model relates to the real data. The generated stochastic wave has been used as an excitation to the pendulum system in numerical and experimental studies. For the first time, the rotary response of a pendulum under stochastic wave excitation has been studied. The rotational number has been used for statistical analysis of the results in the numerical and experimental studies. It has been demonstrated how the forcing arrangement affects the probability of rotation of the parametric pendulum.     

两自由度参数激励摆旋转运动分析和实验

研究具有支撑参数激励摆系统的支撑结构振动对摆旋转的影响,其中支撑结构是受到扭簧约束的刚性悬臂梁,参数激励摆与刚性悬臂梁的悬臂段铰接．首先,通过拉格朗日方程建立了系统两自由度的动力学方程．其次,利用多尺度法对建立的模型进行理论分析,得到悬臂梁的振动与上摆不同运动形式的关系,从而得到上摆不同运动形式下的参数平面分类和悬臂梁在上摆转动时的振动频响．最后,通过建立实验装置,观察理论预测,实验结果验证了理论分析的正确性．实验与理论对照得到,当参数激励频率接近悬臂梁的一阶固有频率时,悬臂梁的振幅变大,会破坏摆的转动稳定性．     

A simple feedback control system: Bifurcations of periodic orbits and chaos

We consider a pendulum subjected to linear feedback control with periodic desired motions. The pendulum is assumed to be driven by a servo-motor with small time constant, so that the feedback control system can be approximated by a periodically forced oscillator. It was previously shown by Melnikov's method that transverse homoclinic and heteroclinic orbits exist and chaos may occur in certain parameter regions. Here we study local bifurcations of harmonics and subharmonics using the second-order averaging method and Melnikov's method. The Melnikov analysis was performed by numerically computing the Melnikov functions. Numerical simulations and experimental measurements are also given and are compared with the previous and present theoretical predictions. Sustained chaotic motions which result from homoclinic and heteroclinic tangles for not only single but also multiple hyperbolic periodic orbits are observed. Fairly good agreement is found between numerical simulation and experimental results.     

Complex response of an oscillating vertical cantilever with clearance

We study the dynamics of an elastic inverted pendulum with amplitude limiters excited horizontally. This particular model corresponds to a class of systems where a clearance is present naturally as an effect of imperfect clamping or it is included to tailor the response. We explore the complex responses of the system for a fixed value of amplitude clearance. The simulation and experimental results are analysed by a 0–1 test, Fourier, and wavelet transforms. The results show that the system can vibrate with subharmonic solution where the main response frequency of a flexible beam is 3 times lower than the excitaion frequency. We claim that an inverted pendulum with imperfect clamping of mechanical resonator can be used in broad frequency band energy harvesting.

     

Modeling, numerical analysis and application of triple physical pendulum with rigid limiters of motion

The mathematical model of a triple physical pendulum harmonically forced with rigid limiters of motion is presented together with the model of stability based on the Aizerman–Gantmakher theory. Some examples of numerical simulations are given for the system of three coupled rods with a horizontal barrier. The system behavior near the grazing incidence is presented; in particular the spectra of Lyapunov exponents are calculated using the presented model of stability.     

基于Gauss伪谱法的3D刚体摆姿态最优控制

研究Gauss伪谱法求解3D刚体摆姿态最优控制问题．针对其最优姿态控制问题,既要满足由任意位置运动到平衡位置姿态运动规划问题,又要满足系统含有动力学约束的力学模型问题,提出基于四元数来描述3D刚体摆的数学模型,建立3D刚体摆姿态的动力学和运动学方程,为了解决3D刚体摆在平衡位置处的姿态最优控制问题,设计基于Gauss伪谱算法的最优姿态开环控制器,得到了3D刚体摆的姿态最优控制轨迹,得到满足的可行解,通过仿真实验验证了其开环解在平衡位置的控制姿态最优性．     

Non-linear behavior in a discretely forced oscillator

The simulated and experimental responses of a rigid-arm pendulum driven by an external impactor are considered. Here, impact occurs if the trajectory of a rotating impactor intersects that of the pendulum. Using the rotation rate of the impactor as the control parameter, experimental trials have demonstrated much of the dynamic behavior predicted by numerical simulations. The system exhibits chatter (i.e., multiple impacts within a single forcing period), sticking (i.e., contact between the pendulum and the impactor for non-negligible amounts of time), high-order periodicity, and behavior suggestive of chaos. A new convention for classifying periodic motions as well as insights regarding the nature of the coefficient of restitution (COR) in an experimental impacting system are also presented.     

基于扭摆的微冲量测量方法及实验研究

超高微重力水平的卫星平台在空间引力波探测、地球重力场测量中发挥着重要的作用,脉冲微推力器可以帮助微重力卫星实现姿态控制.微冲量是评价脉冲微推力器性能的重要指标之一,常用的基于扭摆的微冲量测量方法有两种,方法一是根据单个冲量元瞬间作用于无阻尼扭摆后,扭摆转动最大角位移计算冲量,方法二是根据高固定频率的连续脉冲作用于有阻尼...     

Vibration energy harvesting for low frequency using auto-tuning parametric rolling pendulum under exogenous multi-frequency excitations

An electromagnetic parametrically excited rolling pendulum energy harvester with self-tuning mechanisms subject to multi-frequency excitation is proposed and investigated in this paper. The system consists of two uncoupled rolling pendulum. The resonance frequency of each the rolling pendulum can be automatically tuned by adjusting its geometric parameters to access parametric resonance. This harvester can be used to harvest the energy at low frequency. A prototype is developed and evaluated. Its mathematical model is derived. A cam with rolling follower mechanism is employed to generate multi-frequency excitation. An experimental study is conducted to validate the proposed concept. The experimental results are confirmed by the numerical results. The harvester is successfully tuned when the angular velocity of the cam is changed from 1.149 to 1.236 Hz.     

What does head movement tell about the minimum number of mechanical degrees of freedom in quiet human stance?

In this study, we checked experimentally whether anterior–posterior accelerations of the head during quiet human stance are usually below or above known thresholds of the otolith sensor. Thereto, we measured head kinematics with high spatial resolution. Furthermore, we used both these experimental data and computer simulations of two double inverted pendulum (DIP) models in order to verify the validity of DIP models in general. The results are clear cut. First, not only are acceleration thresholds regularly exceeded about once a second but also are velocity thresholds exceeded, albeit probably less frequently. Second, COM and head movement predicted by interwoven DIP model dynamics can not reproduce the mean measured amplitudes at once. Thus, neither the formerly promoted single inverted pendulum nor any DIP model can causally explain the dynamics of quiet human stance. Instead, we suggest to factor in at least three mechanical degrees of freedom. Due to a couple of reasons discussed, the triple inverted pendulum (TIP) model seems to be a promising abstraction implying potential to better understand the dynamics of quiet human stance.     

基于修正型罗德里格斯参数的三维刚体摆姿态控制

3D 刚体摆是研究地球静止轨道航天器的一个力学简化模型, 它绕一个固定、无摩擦的支点旋转, 具有3 个转动自由度. 文章给出基于修正型罗德里格斯(Rodrigues) 参数描述的3D 刚体摆的姿态动力学方程, 针对3D 刚体摆姿态和角速度稳定的非线性控制设计问题, 基于无源性控制理论利用能量法设计了3D 刚体摆的系统控制器, 并证明了系统满足无源性. 构造了系统的李雅普诺夫(Lyapunov) 函数, 利用能量法设计出3D 刚体摆的姿态控制律, 并由拉萨尔(LaSalle) 不变集原理证明了该控制律的渐近稳定性. 仿真实验给出了3D 刚体摆在倒立平衡位置的姿态和角速度的渐近稳定性, 仿真实验结果表明基于能量方法的3D 刚体摆姿态控制是有效的.     

CHARACTERISTICS OF AZIMUTHAL-RADIAL PENDULUM MOTION BASED ON ANALYTICAL MECHANICS1)

方位径向摆的研究对机械工程领域有非常重要的用途,基于此,本文采用分析力学的方法对影响方位径向摆运行轨迹的因素进行了理论和实验研究。理论上通过建立拉格朗日函数,得到此系统的运动微分方程并对方程进行数值求解,实验上通过Tracker 软件追踪得到方位径向摆的实际运动轨迹,最终发现理论分析得到的杆长、绳长及其初始位置等参数对方位径向摆运行轨迹的影响与实验结果吻合较好。     

Energy harvesting from pendulum oscillations

In this work we analyse the possibility of energy harvesting from the vibration of the environment. The investigations are performed using experimental rig, which consists of a parametrically forced pendulum and an energy harvester, and the mathematical model developed based on the experimental rig. Numerical studies focus on the oscillating motion of pendulum in 2:1 resonance and show good agreement with experimental results. We present that the energy harvesting is possible and is more efficient for shorter reduced length of the pendulum, as proved numerically and experimentally.     

Damping in a parametric pendulum with a view on energy harvesting

The present article addresses the quantification of damping in a parametric pendulum, with a view on further applications in the design of energy harvesting devices. Detailed new experimental data is obtained for such purpose, and a novel mathematical model is presented. Linear and quadratic viscous damping and also dry friction are taken into account. To introduce the dry friction component, the pendulum axis is mounted on ball bearings. This is considered as a very realistic situation of a harvester. Damping parameters are determined by minimizing the difference between numerical and experimental time histories. It is shown that the damping model here presented is more adequate to replicate experiments than commonly used linear models, which consider only a linear viscous damping term characterized by means of free decay tests. It is also pointed that linear models are not adequate for refined studies, since they can lead to erroneous predictions of rotation zones, and consequently to wrong considerations in the design of pendulum harvesters.     

On the Need for Control of Nonlinear Oscillations in Machine Systems

Oscillations in machines are invariably nonlinear. This is either because of inertial coupling effects between different motions of the moving components, material and constitutive phenomena giving rise to stiffness modifications, nonlinear dissipation mechanisms, large deflections, or, as is most likely, some sort of combination of all of these. The net effect of nonlinear vibrations is that at best the machine may well behave a little differently from the way the designer intended, or at worst, in a manner which renders it completely unsuitable for the job. The extent of such problems depends on the nature and the scale of the nonlinearities that are present but it is safe to say that nonlinear oscillations can rarely be completely overlooked in precision machinery analysis and design. The unifying theme in this paper is pendulum motion, firstly in the case of a mobile gantry crane for container stacking where we wish to minimise such motion and converge on a target, and then secondly in the case of a vibration absorber in which we choose to initiate pendulum motion within a special absorber, for the purposes of vibration minimisation. The third example involves the potential for pendulum motion at a very much larger scale and summarises the main control problem that is likely to be encountered in a fully deployed momentum exchange propulsion tether operating in space. The paper discusses the general mathematical issues that pertain to pendulum motion in each of the three cases. This motion is investigated initially in the context of the mobile gantry crane, in the form of a basic three dimensional dynamical model. A feedback linearised controller is shown to offer some advantages for the control of such a system and then a simulation based on data from a practical implementation of this within a real-time control system on a 1/10 laboratory scale model is discussed. It is recalled that the real-time effectiveness of the controller can be compromised by relatively slow sensing and data logging hardware but that despite this some useful performance gains can still be obtainable using this sort of control strategy. The second example comprises an autoparametric vibration absorber and here it is shown how even a simple hunting controller can exploit the mode-locking and wide detuning region effects inherent in autoparametric systems. Further experimental results are discussed in the case of a hunting controller for detuning a vertically oriented parametrically excited pendulum in order to exploit and enhance the powerful and persistent absorption available during autoparametric interaction. The paper concludes with a summary review of the third problem in which the theoretical attitude dynamics of a motorised momentum exchange space propulsion tether are summarised and it is shown that they need to be controlled for reliable and optimal payload velocity boost from both circular parking orbits and elliptical transfer orbits about the Earth.     

Bifurcations in the Mean Angle of a Horizontally Shaken Pendulum: Analysis and Experiment

A pendulum excited by high-frequency horizontal displacement of its pivot point will vibrate with small amplitude about a mean position. The mean value is zero for small excitation amplitudes, but if the excitation is large enough the mean angle can take on non-zero values. This behavior is analyzed using the method of multiple time scales. The change in the mean angle is shown to be the result of a pitchfork bifurcation, or a saddle-node bifurcation if the system is imperfect. Analytical predictions of the mean angle as a function of frequency and amplitude are confirmed by physical experiment and numerical simulation.     

Rotating a pendulum with an electromechanical excitation

The objective of this paper is showing investigation of pendulum rotations via vertical, non-linear electromechanical excitation generated using a RLC-circuit-powered solenoid, which is originally built for an electro-vibro-impact mechanism. Various non-linear phenomena of pendulum dynamics, namely period-1 rotation, period-1 oscillation and period-2 oscillation, have been observed experimentally from the proposed apparatus. A mathematical model has been developed for the experimental rig and the system parameters have also been identified for the mathematical model. Finally, numerical results have been generated using the developed mathematical model and identified parameters, and their correlations with experimental observations have been discussed.     

Experimental Investigation of Dynamics and Bifurcations of an Impacting Spherical Pendulum

Since Newton first considered the motion of a spherical pendulum over 200 years ago, many researchers have studied its dynamic response under a variety of conditions. The characteristic of the problem that has invited so much investigation was that a spherical pendulum paradigms much more complex phenomena. Understanding the response of a paradigm gives an almost multiplicative effect in the understanding of other phenomena that can be modeled as a variant of the paradigm. The spherical pendulum has been used to damp irregular motion in helicopters and on space stations as well as for many other applications. In this study an inverted impacting spherical pendulum with large deflection was investigated. The model was designed to approximate an ideal pendulum, with the pendulum bob contributing the vast majority of the mass moment of inertia of the system. Two types of bearing mechanisms and tracking devices were designed for the system, one of which had low damping coefficient and the other with a relatively high damping coefficient. An experimental investigation was performed to determine the dynamics of an inverted, impacting spherical pendulum with large deflection and vertical parametric forcing. The pendulum system was studied with nine different bobs and two different base configurations. During the experiments, the frequency of the excitation remained between 24.6 and 24.9 Hz. It was found that sustained conical motions did not naturally occur. The spherical pendulum system was analyzed to determine under what conditions the onset of Type I response (a repetitive motion in which the pendulum bob does not traverse through the apex. The bob strikes the same general area of the restraint without striking the opposite side of the restraint.), sustainable Type II response (this is the repetitive motion in which the pendulum bob traverses through the apex. The bob strikes opposite sides of the restraint.), and mixed mode response (motion in which the pendulum bob randomly strikes either the same area of the restrain or the opposite side of the restraint) occurred.     

带有新型涡激振动抑制罩的圆柱体的水动力特性

通过模型实验和数值模拟计算,研究了带有涡激振动抑制罩的圆截面柱体的水动力特性.模型实验主要测试了柱体上附加谐波型和类圆锥型涡激振动抑制罩的单摆结构在不同流速下发生涡激振动的性质;数值模拟则针对谐波型和圆锥型扰动,在雷诺数Re为102到105范围内,研究其水动力参数,如阻力、升力和涡脱落频率等,随扰动波长和波动强度的变化.模型实验结果表明,在直圆柱开始发生共振的流速下,带抑制罩的柱体的振幅显著降低,而在更高流速下则显著增大.数值模拟结果表明,谐波型和圆锥型扰动具有相似的水动力特性;且在不同Re时,阻力、升力和涡脱落频率具有相似的变化规律;随波动强度的增大,阻力一般逐渐增大,升力则在多数情况下先减小而后增大,而涡脱落频率一般逐渐减小.      

石英挠性加速度计二元调宽数字脉冲控制系统的研制

本文介绍了石英挠性加速度计二元调宽数字脉冲控制系统的研制及其应用。提出了通过提高系统采样频率来抑制石英挠性加速度计摆片摆幅，从而最大限度地减小其疲劳强度的方法。并基于这一思想设计了石英挠性加速度计二元调宽数字脉冲控制模型，对模型进行了模拟仿真，证明该方案是合理可行的。其已成功地应用于各新型导航产品中