复合摆和耦合摆运动的可视化

利用视频跟踪技术和Tracker软件,通过运动时序图和李萨如图展示了不同数量复合摆构成的耦合摆的多种运动形式,包括两个复合摆的同振幅同方向振动、同振幅反方向振动、拍振以及3个和4个摆之间的能量转移运动.这种实验演示方法有助于学生加深对耦合摆运动规律的认识.     

一种新的"弹簧振子-单摆"耦合摆研究

介绍了一种新发现的"弹簧振子-单摆"耦合摆,采用拉格朗日方程推导出了耦合摆的运动微分方程.进一步根据耦合摆的实际参数,利用Matlab软件进行了数值计算,发现耦合摆的弹簧振子振幅与单摆的摆幅此消彼长,系统的能量在两者之间不断交换,呈周期性变化,与实际符合很好.通过作出耦合摆的轨迹图,发现与实际一致.最后,分析了耦合摆实现的条件.     

A gas pendulum

It is shown that a periodic two-dimensional isentropic motion of a gas exists and it is described by an exact solution of the equations of gas dynamics. A polytropic gas that fills a circular cylinder rotates and oscillates (in the radial direction) simultaneously under the action of periodically changing external pressure. The solution obtained belongs to the class of solutions with a velocity field that is linear in the coordinates (with homogeneous deformation). Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 115–119, September–October, 2000.     

Vibrations of the elasto-plastic pendulum

A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge-Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge-Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.     

一种提高冷镦件模具寿命的设计方法

 针对采用整模结构的冷镦模具疲劳寿命极低的现状，提出以三层圆筒组合模具结构 替代整体模具, 改变模具受力方式, 使冷镦件模具达到较高寿命的设计方法. 以螺栓 圆头的冷镦试验结果为依据，以力学的应力分析方法为基础，应用力学理论，导出了生产实 践中实用的组合模具设计原理与经验公式，并举例说明了该方法的应用.     

可倒摆的优化调节

根据优选学理论结合可倒摆运动规律来优化调节可倒摆,不公能够迅速、准确地找到其正、倒挂摆动周期的共轭点,从而有效地克服大学物理可倒摆实验的难点,实验也能从优选的过程中享受到实验的乐趣并拓展发散性思维。     

On a toroidal pendulum

We consider the problem of motion of a heavy particle on the surface of a torus with horizontal axis of rotation.     

The magnetic spherical pendulum

This article gives two formulae for the rotation number of the flow of the magnetic spherical pendulum on a torus corresponding to a regular value of the energy momentum mapping. One of these formulae is nonclassical and is based on an idea of Montgomery.

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Sommario In questo lavoro si danno due formule per il numero di rotazione del flusso del sferico pendolo magnetico su un toro che corrisponde a un valore regolare dell'applicazione energia-momento. Una di queste formule è nonclassica ed è basata su un'idea di Montgomery.

     

On oscillations of a frictional pendulum

Conditions are established under which a standard limit cycle occurs in the system under consideration, or the trajectory closes under the influence of a stagnation domain. It is pointed out that when the solution falls into the stagnation domain it makes no sense to use the asymptotic method because of a large error __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 104–112, February 2006.     

Dynamics of the nearly parametric pendulum

Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency of excitation where rotations are possible increases with the ellipticity. Second, the resonance tongues, which are the most characteristic feature of the classical bifurcation scenario of a parametrically driven pendulum, merge into a single region of instability.     

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.     

A damped simple pendulum of constant amplitude

A simple pendulum acted on by gravity and subjected to a resistance proportional to the velocity of the bob is considered. If the length of the string and the mass of the bob are held constant, the amplitude of the bob decreases gradually because of the damping. We want to keep the maximum swing of the bob constant for all time; this we achieve by varying the length of the string, the mass of the bob or both. The key to the solution of our problem is a second-order nonlinear differential equation having arbitrary nonlinearity and an arbitrary coefficient function, for which we give the exact integral. We also give an application of this differential equation to a boundary-value problem for a nonlinear generalization of a hypergeometric equation.     

Time-optimal swing-up feedback control of a pendulum

Time-optimal feedback control is obtained that brings a pendulum to the upper unstable equilibrium position. The solution is based on the maximum principle and involves analytical investigations combined with numerical computations. As a result, the switching and dispersal curves that bound the domains in the phase plane corresponding to different values of the optimal bang-bang control are constructed for various values of the maximal admissible control torque.     

Controlled pendulum on a movable base

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.     

Geometric aspects of the parametrically driven pendulum

The behaviour of the parametrically driven pendulum is very complex. Therefore, a global study is carried out to cover all possible situations. The study is mainly numeric, though primary bifurcations of subharmonic motions, as well as the homoclinic intersection of the hilltop saddle, are evaluated according to the Melnikov theory. Extended use is made of the cell-to-cell mapping algorithm to evaluate attracting basins of the various periodic motions. Heteroclinic intersections are always present, independently of the excitation intensity, so that the boundaries of attracting basins are always very complicated, even below the homoclinic tangency of the hilltop saddle. The oscillator exhibits various kinds of rotating and oscillating motions. All these motions lead to chaos after a period doubling cascade. It is shown that chaos usually occurs at a much greater excitation level than at that which produces homoclinic tangency of the hilltop saddle; the greater the damping, the greater the difference. The oscillatory chaotic motion is associated with the first change in the period two Birkhoff signature.     

The dynamics of an omnidirectional pendulum harvester

The pendulum applied to the field of mechanical energy harvesting has been studied extensively in the past. However, systems examined to date have largely comprised simple pendulums limited to planar motion and to correspondingly limited degrees of excitational freedom. In order to remove these limitations and thus cover a broader range of use, this paper examines the dynamics of a spherical pendulum with translational support excitation in three directions that operate under generic forcing conditions. This system can be modelled by two generalised coordinates. The main aim of this work is to propose an optimisation procedure to select the ideal parameters of the pendulum for an experimental programme intended to lead to an optimised pre-prototype. In addition, an investigation of the power take-off and its effect on the dynamics of the pendulum is presented with the help of Bifurcation diagrams and Poincaré sections.