Stability of the stationary motion of a double pendulum interacting with a string

The equation of in-plane vertical motion of a double pendulum suspended at some point of a horizontal elastic string is derived using a hybrid model of this mechanical system. The conditions for the asymptotic stability of the stationary motion of the pendulum interacting with the string are established     

Limit cycles of a double pendulum subject to a follower force

It is shown that there is a magnitude of the follower force at which two limit cycles, stable and unstable, are born in the phase space of a double simple pendulum     

Bifurcation analysis of the motion of an asymmetric double pendulum subjected to a follower force: Codimension three problem

Local and global bifurcations in the motion of a double pendulum subjected to a follower force have been studied when the follower force and the springs at the joints have structural asymmetries. The bifurcations of the system are examined in the neighborhood of double zero eigenvalues. Applying the center manifold and the normal form theorem to a four-dimensional governing equation, we finally obtain a two-dimensional equation with three unfolding parameters. The local bifurcation boundaries can be obtained for the criteria for the pitchfork and the Hopf bifurcation. The Melnikov theorem is used to find the global bifurcation boundaries for appearance of a homoclinic orbit and coalescence of two limit cycles. Numerical simulation was performed using the original four-dimensional equation to confirm the analytical prediction.     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Limit cycles of a double pendulum with nonlinear springs

The limit cycles of a double pendulum with hard, soft, or linear springs subject to a follower force are drawn using computer simulation     

Bifurcation analysis of a double pendulum with internal resonance

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Estimating the chaos boundaries of a double pendulum

Loss of the orbital stability of a double pendulum is considered in terms of Lyapunov exponents. The boundaries of the domain of stochastic motion caused by bifurcational and chaotic processes are estimated     

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.     

Stability of the stationary motion of a spherical pendulum interacting with a string

The equation of motion of a spherical pendulum suspended at some point of a horizontal string is derived using a hybrid model of this mechanical system. The conditions for the asymptotic stability of the stationary motion of the spherical pendulum interacting with the elastic string are established     

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.     

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.     

The behaviour of a simple pendulum with uniformly shortening string length

Whilst watching the loading of a ship by crane it is often observed that the cargo shifts abruptly during the lifting stage. In order to try to understand this kind of behaviour the motion of a simple undamped pendulum, which is being shortened at a constant speed, is determined. It is found that the tension in the string always increases initially from its static value and that this increase in tension is very large even when the initial angular speed is small. Finally, it is proved that the linearised approximation is most unsatisfactory even when the initial amplitude of the swing and the shortening rate are small.     

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.     

Ultimate energy of a double pendulum undergoing quasiperiodic oscillations

The boundary of the phase domain of periodic solutions of a double pendulum is constructed and shown to be closed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 106–114, September 2007.     

Coupled flutter and divergence bifurcation of a double pendulum

The loss of stability of the equilibrium position of a double pendulum with follower force loading and elastic end support is studied. At a special parameter combination the linearized system is characterized by a zero root and a pure imaginary pair of eigenvalues. Therefore, the stability problem is a complicated critical case in the sense of Liapunov and requires a non-linear analysis. A complete post-bifurcation investigation of the coupled divergence and flutter motions is given by means of centre manifold theory, and bifurcation diagrams. Among the different types of motions even the appearance of chaotic behavior is shown.     

Parametric identification of a chaotic base-excited double pendulum experiment

The parametric identification of a chaotic system was investigated for a double pendulum. From recorded experimental response data, the unstable periodic orbits (UPOs) were extracted and then used in a harmonic balance identification process. By applying digital filtering, digital differentiation and linear regression techniques for optimization, the results were improved. Verification of the related simulation system and linearized system also corroborated the success of the identification algorithm.