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.     

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.     

Stability of high-frequency periodic motions of a heavy rigid body with a horizontally vibrating suspension point

The motion of a heavy rigid body one of whose points (the suspension point) executes horizontal harmonic high-frequency vibrations with small amplitude is considered. The problem of existence of high-frequency periodic motions with period equal to the period of the suspension point vibrations is considered. The stability conditions for the revealed motions are obtained in the linear approximation. The following three special cases of mass distribution in the body are considered; a body whose center of mass lies on the principal axis of inertia, a body whose center of mass lies in the principal plane of inertia, and a dynamically symmetric body.     

On the motion of a heavy dynamically symmetric rigid body with vibrating suspension point

The motion of a dynamically symmetric rigid body in a homogeneous field of gravity is studied. One point lying on the symmetry axis of the body (the suspension point) performs high-frequency periodic or conditionally periodic vibrations of small amplitude. In the framework of approximate equations of motion obtained earlier, we find necessary and sufficient conditions for the stability of the body rotation about the vertical symmetry axis and study the existence and stability of regular precessions of the body in the coordinate system translationally moving together with the suspension point.     

On the motion of a reversible double simple pendulum with tracking force

Stability domains of a pendulum in the presence of tracking force and viscoelastic elements are constructed. It is shown that the boundary of the stability domains consists of sections of two hyperbolas. The effect of the pendulum parameters on the configuration of the stability domains is considered. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 108–112, July, 1999.     

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.     

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     

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     

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     

Parametric instability of a magnetic pendulum in the presence of a vibrating conducting plate

Nonlinear Dynamics - A pendulum with an attached permanent magnet swinging in the vicinity of a conductor is a typical experiment for the demonstration of electromagnetic braking and Lenz’...     

Nonlinear wave motions of a liquid in a vibrating axisymmetric cavity

The problem of the nonlinear wave deformation of the free surface of a liquid due to the translational motions of the containing vessel is examined. Bogolyubov's averaging method is used to investigate the characteristics of the wave motions of the liquid in the resonance zones in the case of a cylindrical vessel. Relations are obtained characterizing the variation of the amplitude of the circular wave with the frequencies of the external perturbations in the steady-state wave process; the conditions of occurrence and stability of such processes are analyzed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 120–125, May–June, 1989.     

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 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.