On Stability of the Inverted Pendulum Motion with a Vibrating Suspension Point

Under study is the stability of the inverted pendulum motion whose suspension point vibrates according to a sinusoidal law along a straight line having a small angle with the vertical. Formulating and using the contracting mapping principle and the criterion of asymptotic stability in terms of solvability of a special boundary value problem for the Lyapunov differential equation, we prove that the pendulum performs stable periodic movements under sufficiently small amplitude of oscillations of the suspension point and sufficiently high frequency of oscillations.     

Stability condition for vertical oscillation of 3-dim heavy spring elastic pendulum

Equations of motion for 3-dim heavy spring elastic pendulum are derived and rescaled to contain a single parameter. Condition for the stability of vertical large amplitude oscillations is derived analytically relating the parameter of the system and the amplitude of the vertical oscillation. Numerical continuation is used to find the border of the stability region in parameter space with high precision. The stability condition is approximated by a simple formula valid for a large range of the parameter and of the amplitude of oscillation. The bifurcation responsible for the loss of stability is identified.      

On the instability of a periodic system in the case of internal resonance

The stability of periodic motion is studied in the critical case of n pairs of purely imaginary characteristic indices. It is shown that in the case of resonance, when the ratio of the modulus of one of the characteristic indices to the frequency of the unperturbed motion is an integer, instability usually occurs. The results obtained are used to study the free oscillations of an autonomous quasilinear system when the Andronov-Witt criterion /1/ cannot be used. The instability of free oscillations of the Froude pendulum at the bifurcation point is proved.     

Capture into resonance and escape from it in a forced nonlinear pendulum

We study the dynamics of a nonlinear pendulum under a periodic force with small amplitude and slowly decreasing frequency. It is well known that when the frequency of the external force passes through the value of the frequency of the unperturbed pendulum’s oscillations, the pendulum can be captured into resonance. The captured pendulum oscillates in such a way that the resonance is preserved, and the amplitude of the oscillations accordingly grows. We consider this problem in the frames of a standard Hamiltonian approach to resonant phenomena in slow-fast Hamiltonian systems developed earlier, and evaluate the probability of capture into resonance. If the system passes through resonance at small enough initial amplitudes of the pendulum, the capture occurs with necessity (so-called autoresonance). In general, the probability of capture varies between one and zero, depending on the initial amplitude. We demonstrate that a pendulum captured at small values of its amplitude escapes from resonance in the domain of oscillations close to the separatrix of the pendulum, and evaluate the amplitude of the oscillations at the escape.     

Forced oscillations of a non-linear system with a repulsive positional force

Non-linear systems with one degree of freedom, in which the positional force is directed away from the equilibrium position of the system, are considered. The existence of forced periodic oscillations, their Lyapunov stability, and the behaviour of amplitude-frequency characteristics are investigated. It is shown that stable periodic oscillations are possible in the case when the positional force has non-monotonic properties. Forced oscillations of a pendulum with respect to the upper equilibrium position are considered as an example.     

A General Theory of Resonance Between Weakly Coupled Oscillatory Systems

Resonant oscillations are investigated between several fullynon-linear oscillatory systems which are subject to a weak coupling.The equations of motion are obtained from a Hamiltonian whichis first expressed in terms of angle action variables, and thenaveraged. It is shown that this results in a reduced set ofequations, the reduction depending on the number of resonancesbetween the various systems. For example a resonance in a systemwith two degrees of freedom results in equations which are mathematicallyequivalent to those for one degree of freedom. The theory isillustrated by application to the forced oscillations of a simplependulum and to the resonant interaction between the two modesof oscillation of a double pendulum.     

The dynamics of a spherical pendulum with a vibrating suspension

The motion of a spherical pendulum whose point of suspension performs high-frequency vertical harmonic oscillations of small amplitude is investigated. It is shown that two types of motion of the pendulum exist when it performs high-frequency oscillations close to conical motions, for which the pendulum makes a constant angle with the vertical and rotates around it with constant angular velocity. For the motions of the first and second types the centre of gravity of the pendulum is situated below and above the point of suspension, respectively. A bifurcation curve is obtained, which divides the plane of the parameters of the problem into two regions. In one of these only the first type of motion can exist, while in the other, in addition to the first type of motion, there are two motions of the second type. The problem of the stability of these motion of the pendulum, close to conical, is solved. It is shown that the first type of motion is stable, while of the second type of motion, only the motion with the higher position of the centre of gravity is stable.     

非自治二阶Hamilton系统的周期解

本文推广了Willem的一个结果,Willem研究的受迫周期振动要求受迫势能关于空间变量是周期的,而本文只要求受迫势能对时间变量积分后关于空间变量是周期的,该结果包括了单摆的受迫振动.本文将用直接变分最小方法和Rabinowtz的鞍点定理来研究当势函数对时间变量积分后是周期时受迫摆方程的周期解.     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

On loss of stability of slow motions in stiff oscillatory systems

In stiff oscillatory systems often a reduction of the order of the system is possible by splitting the motion into an essential motion on a nearby slow manifold and neglecting the fast motion. However, if the system is conservative the question of stability of the slow motion is a delicate problem. For various spring pendulum systems we, first, perform numerical simulations showing that if the stiffness of the springs is gradually reduced the slow motion looses stability. For a single spring pendulum we give an explanation of this loss of stability. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear Oscillations of a Spring Pendulum at the 1 : 1 : 2 Resonance: Theory,Experiment, and Physical Analogies

Nonlinear spatial oscillations of a material point on a weightless elastic suspension are considered. The frequency of vertical oscillations is assumed to be equal to the doubled swinging frequency (the 1 : 1 : 2 resonance). In this case, vertical oscillations are unstable, which leads to the transfer of the energy of vertical oscillations to the swinging energy of the pendulum. Vertical oscillations of the material point cease, and, after a certain period of time, the pendulum starts swinging in a vertical plane. This swinging is also unstable, which leads to the back transfer of energy to the vertical oscillation mode, and again vertical oscillations occur. However, after the second transfer of the energy of vertical oscillations to the pendulum swinging energy, the apparent plane of swinging is rotated through a certain angle. These phenomena are described analytically: the period of energy transfer, the time variations of the amplitudes of both modes, and the change of the angle of the apparent plane of oscillations are determined. The analytic dependence of the semiaxes of the ellipse and the angle of precession on time agrees with high degree of accuracy with numerical calculations and is confirmed experimentally. In addition, the problem of forced oscillations of a spring pendulum in the presence of friction is considered, for which an asymptotic solution is constructed by the averaging method. An analogy is established between the nonlinear problems for free and forced oscillations of a pendulum and for deformation oscillations of a gas bubble. The transfer of the energy of radial oscillations to a resonance deformation mode leads to an anomalous increase in its amplitude and, as a consequence, to the break-up of a bubble.     

Noisy and chaotic conditions of a pendulum motion

The effect of noise on the rotational mode of a pendulum which is excited kinematically in vertical direction has been analyzed. We have applied the multifractal analysis to distinguish chaotic and noisy solutions in transitions from the oscillations to rotations motion of a pendulum. During increasing the noisy disturbance of the system we analyzed the basic multifractals criteria of the system as correlation and complexity. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

The stability of the equilibrium of a pendulum for vertical oscillations of the point of suspension

The motion of a pendulum, the point of suspension of which is subject to vertical harmonic oscillations of arbitrary frequency and amplitude, is considered. A complete rigorous solution of the non-linear problem of the stability of the relative positions of equilibrium of the pendulum along the vertical is given.     

An alternative approach for chaos: A plane pendulum with oscillating torque

The shooting method is applied to obtain chaotic motions for a pendulum with a oscillatory torque excitation on its support. It shows that if the pendulum is placed at certain spots, the corresponding motion will become chaotic. It proves the coexistence of uncountably many non-periodic motions and countably many periodic motions of the pendulum.     

Non-linear oscillations of a Hamiltonian system with 2:1 resonance

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

A case of plane rotations of an elastic pendulum

The motion of a point mass, suspended on a spring in a uniform gravity field, is investigated. The spring is assumed to be weightless and to possess linear elasticity. Motion occurs in a specified fixed vertical plane. It is shown that a pendulum motion exists in which the angle, made by the axis of the spring and the vertical, varies uniformly with time. The problem of the orbital stability of this motion is solved.     

Small Motions and Normal Oscillations of a Double Pendulum with Cavities Containing a Viscous Incompressible Liquid

The linear hydrodynamic problem involving the small motions and normal oscillations of a double pendulum with cavities completely filled with a liquid is examined. The problem is solved using the methods of functional analysis. An existence theorem is formulated for the solutions to the Cauchy problem and the properties of the normal oscillations are described.     

On the stability of a spring-pendulum

The stability of a linearized spring-pendulum system is analysed. It is shown that when the natural frequencies of the two degrees of freedom are equal, no resonance occurs. Analytical expressions are given for the boundaries of the unstable region when the frequency of the spring oscillations is about twice that of the pendulum, and compared to results of numerical investigation.

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Resumé L'analyse de la stabilité d'un système linéarisé d'un pendule-ressort est présenté. On démontre qu'il n'y a pas de résonance possible lorsque les deux fréquences naturelles sont les mêmes. On présente les expressions analytiques des frontières de la région instable, lorsque la fréquence des oscillations du ressort est le double de celle du pendule, et on les compare aux résultats obtenus par une étude numérique.