Suspension point vibration parameters for a given equilibrium of a double mathematical pendulum

The motion of a double mathematical pendulum under the action of the gravity force and a vibration force whose frequency substantially exceeds the system natural frequencies is considered. An oblique vibration stabilizing the pendulum in an arbitrarily given position is sought. The domain of existence of the pendulum equilibrium points and the vibration parameters corresponding to a given equilibrium of the pendulumare obtained analytically. In the domain of existence of equilibrium points, the subdomain of their stability is distinguished.     

Global stabilization of a double inverted pendulum with control at the hinge between the links

We study the plane motion of a double pendulum with fixed suspension point. The pendulum is controlled by a single moment applied to the internal hinge between the links. The moment is assumed to be bounded in absolute value. We construct a feedback control law bringing the pendulum from the position in which both links hang vertically downwards into the unstable upper position in which both links are inverted. The same feedback ensures the asymptotic stability of the pendulum in the upper equilibrium position. Since the pendulum can be brought to the lower equilibrium position from any initial states, it follows that the constructed control law ensures the global stability of the inverted pendulum.     

Equilibrium states of a pendulum with an asymmetric follower force

The effect of the linear eccentricity of the follower force on the equilibrium states of an inverted pendulum is examined. Bifurcation points and catastrophes associated with changes in pendulum parameters and type of springs are analyzed. Phase flows are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 121–129, April 2007.     

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．     

Equilibrium of an Inverted Mathematical Double-Link Pendulum with a Follower Force

A discrete model of an elastic pendulum with a follower force is studied. This model is an inverted mathematical two-link pendulum with viscoelastic hinges. It is shown that divergent bifurcations are possible for some absolute values of the follower force and the stiffness of the restraint of the pendulum's upper end. As a result, the vertical position of the equilibrium becomes unstable and two new nonvertical stable equilibrium states (fork bifurcation) occur.     

一种特殊瞬态动力学现象释疑

引入瞬态平衡点的概念 ,来解释一种简单的摆式减振器模型中的瞬态动力学行为中的一种特殊现象 :即振子从开始小幅振动突然增大最后达到大幅振动 ,而减振摆也从铅垂位置附近的小幅振动过渡到绕悬挂点的圆周运动。分析表明 ,产生此现象的原因是系统瞬时平衡点的剧烈变化     

Various analogies for the equilibrium shapes of an elastic thread on two-dimensional surfaces

In accordance with the Kirchhoff analogy, the equilibrium equations of an elastic thread on a plane are equivalent to the equations of motion of a simple pendulum. This analogy is generalized to the case when the thread is situated on a smooth curved surface. The equilibrium equations for the threads in the general case and in the particular cases of flat, cylindrical, and spherical surfaces are derived. For these surfaces the Kirchhoff analogy is generalized to the case of a simple pendulum in an additional force field. There are also considered the electromagnetic and nonholonomic analogies for the equilibrium equations of an elastic thread.     

Parametric resonances in a base-excited double pendulum

Two parametrically-induced phenomena are addressed in the context of a double pendulum subject to a vertical base excitation. First, the parametric resonances that cause the stable downward vertical equilibrium to bifurcate into large-amplitude periodic solutions are investigated extensively. Then the stabilization of the unstable upward equilibrium states through the parametric action of the high-frequency base motion is documented in the experiments and in the simulations. It is shown that there is a region in the plane of the excitation frequency and amplitude where all four unstable equilibrium states can be stabilized simultaneously in the double pendulum. The parametric resonances of the two modes of the base-excited double pendulum are studied both theoretically and experimentally. The transition curves (i.e., boundaries of the dynamic instability regions) are constructed asymptotically via the method of multiple scales including higher-order effects. The bifurcations characterizing the transitions from the trivial equilibrium to the periodic solutions are computed by either continuation methods and or by time integration and compared with the theoretical and experimental results.     

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.     

Influence of the Nonlinearity of the Elastic Elements on the Stability of a Double Pendulum with Follower Force in the Critical Case

A generalized mathematical theory of a double mathematical pendulum with follower force is used to analyze the stability of the vertical equilibrium position of the pendulum with both linear and nonlinear (hard and soft) elastic elements in the critical case of one zero root of the characteristic equation. The influence of the parameters of these elements on the safe and dangerous sections of the stability boundary is demonstrated__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 133–142, April 2005.     

Output feedback stabilization of the inverted pendulum system: a Lyapunov approach

In this work, we present an output feedback stabilization method for the Inverted Pendulum Cart (IPC) system around its unstable equilibrium point. The pendulum is initialized in the upper-half plane, and the position of the cart and the pendulum angular positions are always available. Our strategy was accomplished introducing a suitable coordinate change to obtain a nonlinear version of the original system, which is affine in the unmeasured velocities state. This fact allows us to adapt an observer based controller devoted to render the closed-loop system to the origin. The proposed observer based controller was designed using the direct Lyapunov method. This allows estimating the corresponding attraction domain for the whole system, which can be as large or as small as desired. While the corresponding closed-loop stability analysis was made using the LaSalle Invariance Theorem. Convincing numerical simulations were included to show the performance of the closed-loop system.     

Asymptotic stability boundaries for the equilibrium positions of a double pendulum with vibrating point of suspension

The mechanisms whereby a double pendulum with vibrating point of suspension loses stability in equilibrium positions are studied. Stability conditions for the equilibrium positions in critical cases are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 120–133, July 2008.     

Numerical Study of a Forced Pendulum with Friction

We first describe the model of a forced pendulum with viscousdamping and Coulomb friction. Then we show that a unique local solutionof the mathematically well-posed problem exists. An adapted numericalscheme is built. Attention is devoted to the study of the nonlinearbehaviour of a pendulum via a numerical scheme with small constant timesteps. We describe the global behaviour of the free and forcedoscillations of the pendulum due to friction. We show that chaoticbehaviour occurs when friction is not too large. Lyapunov exponents arecomputed and a Melnikov relation is obtained as a limit of regularisedCoulomb friction. For larger friction, asymptotic behaviour correspondsto equilibrium.     

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.     

Influence of concurrent use of springs with characteristics of different types on the equilibrium of an inverted pendulum

The effect of the concurrent use of springs with characteristics of different types (hard, soft, or linear) on the equilibrium of an inverted simple pendulum is studied __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 107–113, July 2007.     

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.     

Open-plus-closed-loop control for chaotic motion of 3D rigid pendulum

An open-plus-closed-loop （OPCL） control problem for the chaotic motion of a 3D rigid pendulum subjected to a constant gravitationM force is studied. The 3D rigid pendulum is assumed to be consist of a rigid body supported by a fixed and frictionless pivot with three rotational degrees. In order to avoid the singular phenomenon of Euler＇s angular velocity equation, the quaternion kinematic equation is used to describe the motion of the 3D rigid pendulum. An OPCL controller for chaotic motion of a 3D rigid pendulum at equilibrium position is designed. This OPCL controller contains two parts： the open-loop part to construct an ideal trajectory and the closed-loop part to stabilize the 3D rigid pendulum. Simulation results show that the controller is effective and efficient.     

Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum

The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 115–128, August 2007.     

Dynamic Behavior of Multilink Pendulums under Follower Forces

The results from studies of the dynamic behavior of a double pendulum under the action of a follower force are analyzed. It is pointed out that bifurcations and catastrophes of equilibrium states may occur at some values of the parameters. Differential equations are presented which describe the plane-parallel motion of a pendulum with an arbitrary number of links and angular and linear eccentricities of the follower force whose orientation depends on one parameter. The basic problems of dynamics of pendulums with different number of links are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 3–35, June 2005.     

Theory of inverted pendulum with follower force revisited

The effect of the type of springs on the equilibrium states of an inverted pendulum is examined. The angular and linear eccentricities of the follower force are taken into account __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–137, June 2007.