An inverted pendulum on a fixed and a moving base

The plane motions of a controlled single-link pendulum with a fixed suspension point and a pendulum with its suspension point located at the centre of a wheel which rolls without sliding along a flat horizontal surface are considered. The control torque, applied to the pendulum at the suspension point, is bounded in absolute magnitude. A controllability domain is constructed in the linear approximation for the one and the other pendulum, from all points of which the pendulum can be brought into the upper unstable equilibrium position without oscillations about the lower equilibrium. It is shown that the domain of controllability is greater for a pendulum mounted on a wheel, as a result it is more easily stabilizable. Control laws are constructed, under which the domain of attraction is identical to the controllability domain and is thereby the largest possible domain.     

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.     

Chelomei's problem of the stabilization of a statically unstable rod by means of a vibration

Chelomei's problem of the stabilization of an elastic, statically unstable rod by means of a vibration is considered. Formulae for the upper and lower critical frequencies for the stabilization of the rod are obtained and analysed. It is shown that, unlike the high-frequency stabilization of an inverted pendulum with a vibrating suspension point, a rod is stabilized by frequencies of a periodic force of the order of the fundamental frequency of the transverse oscillations of the uncompressed rod lying in a certain range.     

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.     

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.     

Critical stability of solutions to linear ordinary differential equations with large high-frequency terms

The stability problem is considered for certain classes of systems of linear ordinary differential equations with almost periodic coefficients. These systems are characterized by the presence of rapidly oscillating terms with large amplitudes. For each class of equations, a procedure for analyzing the critical stability of solutions is constructed on the basis of the Shtokalo-Kolesov method. A verification scheme is described. The theory proposed is illustrated by using a linearized stability problem for the upper equilibrium of a pendulum with a vibrating suspension point.     

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.     

The stability of an inverted pendulum when there are rapid random oscillations of the suspension point

The stability of the upper equilibrium position of a pendulum when the suspension point makes rapid random oscillations of small amplitude, is investigated. A class of random oscillations that make the system stable with unit probability for small friction is indicated. It is shown that, if there is no friction, there is no stability, which, as is well known, is not the case for harmonic oscillations of the suspension point. Some general results concerning the impossibility of stochastic stabilization of Hamiltonian systems are proved.     

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.     

Degenerate cases of stability loss of an elastic fluid-conveying tube

The two-dimensional non-linear dynamics of a liquid-filled tube is considered. The tube is clamped at the upper end, a point mass is fixed to its free lower end and laterally it is supported by two springs. The uniform flow velocity of the fluid, the end mass, the spring constant and the vertical position of the springs are considered as the distinguished parameters of the problem. A linear stability analysis shows that the (degenerate) case of a Takens-Bogdanov-Hopf bifurcation exists, which is associated with a high frequency flutter movement superimposed on a low frequency flutter around a statically buckled state of the tube. We account for this degenerate case by indicating the parameter regime necessary for its occurence and and give the bifurcation diagram for the trivial equilibrium position of the tube. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of pendulum oscillations with variable parameters

Two pendulum control problems are considered, in which oscillations are excited by changing the length (or the position of the center of mass) of the pendulum or by displacing the suspension point. The control objective is approaching a certain invariant set in the state space of the system. The solution approach is based on the speed gradient method. The obtained results make it possible to determine the domain of initial data and parameters on which the system possesses the desired properties.     

Bifurcation of periodic solutions of a system close to a Lyapunov system

Bifurcation of 2π-periodic solutions (2π-ps) of a system of second-order differential equations close to a Lyapunov system is investigated. The case of principal resonance, when an eigenfrequency of the linear oscillations of the unperturbed system is close to the frequency of the perturbing impulse, is considered. It is shown that, at certain values of the problem parameters, bifurcation of the 2π-ps that are generated from an equilibrium position, occurs. A constructive method is proposed for finding the bifurcation curve, as well as 2π-ps on it. The examples considered are bifurcation of 2π-ps in the problem of the oscillations of a mathematical pendulum with a horizontally vibrating suspension point, and in the problem of the planar oscillations of an artificial satellite in a weakly elliptical orbit. The bifurcation curves for these examples are constructed and the corresponding 2π-ps are found.     

Stabilization by vertical vibrations

We consider a general version of the classical problem of the stabilization of the inverted pendulum by a vertical periodic vibration of the point of suspension. Instead of the usual harmonic motion, we propose an oscillatory control with a piecewise‐constant acceleration, obtaining explicit conditions over the frequency or the amplitude leading to the desired stabilization. Copyright © 2008 John Wiley & Sons, Ltd.     

The precession of a parametric oscillation pendulum with Cardan suspension

The precession of a pendulum with a Cardan suspension point oscillating under an external periodic force is investigated. The parameters of precession versus the pendulum geometric characteristics and the external force frequency are studied both theoretically and numerically.     

Modelling of the reversed compound pendulum with a nonperiodically oscillating suspension

The vertical position is stable for a reversed compound pendulum provided its suspension is submitted to properly changing vibrations which can have periodic or weak almost periodic character. An abstract mathematical model describing phenomena of this kind by means of differential equations with weak almost periodic terms (wap functions) is considered. The results are of the homogenization type in the sense that the solutions of some nonautonomous systems of differential equations converge to the solutions of the homogenized autonomous (with respect to periodic variable) systems. The abstract general theorem is specified in some particular cases and some physical interpretation is given.     

Subharmonic Solutions of a Pendulum Under Vertical Anharmonic Oscillations of the Point of Suspension

We prove the existence of subharmonic solutions in the dynamics of a pendulum whose point of suspension executes a vertical anharmonic oscillation of small amplitude.     

An elliptic averaging method for harmonically excited oscillators with a purely non-linear non-negative real-power restoring force

Harmonically excited oscillators with a purely non-linear non-negative real-power restoring force are considered in this paper. The solution for motion is assumed in the form of a Jacobi cn elliptic function, the frequency and the parameter of which are obtained from the exact period of motion calculated from the energy conservation law. The parameter of the elliptic function is found to be negative for under-linear restoring forces and positive for over-linear restoring forces. By taking into account the time variation of the parameter of the elliptic function, a new elliptic averaging method is developed. The use of the equations derived is illustrated on the examples of oscillators with van der Pol damping and linear viscous damping with various integer and non-integer powers of the restoring force. New insights into dynamics of these oscillators are engendered. Numerical confirmations of analytical results are provided.     

The non-linear oscillations of a pendulum of variable length on a vibrating base

A generalized scheme for averaging a system with several small independent parameters is described: equations of the first and second approximations are obtained, and an estimate is made of the accuracy of the approximation and the value of the asymptotically long time interval. The problem of the oscillations of a pendulum of variable length on a vibrating base for high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and its suspension point is considered. Averaged equations of the first and second approximations are obtained, and the bifurcations of the steady motions in the equations of the first approximation, and also in the second approximation for 1:2 resonance, are obtained. One of the possible bifurcations of the phase portrait in the neighbourhood of 1:2 resonance is described based on a numerical investigation. It is shown that a change in the resonance detuning parameter from zero to a value of the first order of infinitesimals in the small parameter leads to stabilization of the upper equilibrium position through a splitting of the separatrices for the resonance case; the splitting of separatrices is accompanied by the occurrence of a stochastic web in the neighbourhood of this equilibrium, its localization, and subsequent contraction to an equilibrium point and the formation of a new oscillation zone.     

A damped bipedal inverted pendulum for human–structure interaction analysis

The bipedal inverted pendulum with damping has been adopted to simulate human–structure interaction recently. However, the lack of analysis and verification has provided motivation for further investigation. Leg damping and energy compensation strategy are required for the bipedal inverted pendulum to regulate gait patterns on vibrating structures. In this paper, the Hunt–Crossley model is adopted to get zeros contact force at touch down, while energy compensation is achieved by adjusting the stiffness and rest length of the legs. The damped bipedal inverted pendulum can achieve stable periodic gait with a lower energy input and flatter attack angle so that more gaits are available, compared to the template, referred to as spring-load inverted pendulum. The measured and simulated vertical ground reaction force-time histories are in good agreement. In addition, the dynamic load factors are also within a reasonable range. Parametric analysis shows that the damped bipedal inverted pendulum can achieve stable gaits of 1.6 to 2.4 Hz with a reasonable first harmonic dynamic load factor, which covers the normal walking step frequency. The proposed model in this paper can be applied to human–structure interaction analysis.