Stick balancing with reflex delay in case of parametric forcing

The effect of parametric forcing on a PD control of an inverted pendulum is analyzed in the presence of feedback delay. The stability of the time-periodic and time-delayed system is determined numerically using the first-order semi-discretization method in the 5-dimensional parameter space of the pendulum’s length, the forcing frequency, the forcing amplitude, the proportional and the differential gains. It is shown that the critical length of the pendulum (that can just be balanced against the time-delay) can significantly be decreased by parametric forcing even if the maximum forcing acceleration is limited. The numerical analysis showed that the critical stick length about 30 cm corresponding to the unforced system with reflex delay 0.1 s can be decreased to 18 cm with keeping maximum acceleration below the gravitational acceleration.     

The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

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.     

Crane Dynamics with Modulated Hoisting

For many types of cranes commonly used in technical applications, the reduction of payload pendulations is an important design issue. Especially for cranes with variable cable length, oscillations are boosted by the hoisting of the payload due to nonlinear effects. Most of the techniques for active damping are based on a control input that displaces the support of the hoisting mechanism perpendicularly to the direction of the pendulum. However, controlled motion of the carrying structure might not be suitable or even impossible for some applications. The possibility to influence and reduce pendulations by means of feedback controlled variations of the cable length is hardly used in crane technology. A control strategy based on the phenomenon of autoparametric resonances in nonlinear dynamical systems is presented that manipulates the desired hoisting velocity by superposition of a suitably modulated motion in order to reduce amplifications of the pendulations, in particular in absence of other effective control inputs. Experimental results for a simple pendulum setup are presented. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trajectory tracking control of variable length pendulum by partial energy shaping

This paper concerns a trajectory tracking control problem for a pendulum with variable length, which is an underactuated mechanical system of two degrees-of-freedom with a single input of adjusting the length of the pendulum. We aim to study whether it is possible to design a time-invariant control law to pump appropriate energy into the variable length pendulum for achieving a desired swing motion (trajectory) with given desired energy and length of the pendulum. First, we show that it is difficult to avoid singular points in the controller designed by using the conventional energy-based control approach in which the total mechanical energy of the pendulum is controlled. Second, we present a tracking controller free of singular points by using only the kinetic energy of rotation and the potential energy of the pendulum and not using the kinetic energy of the motion along the rod. Third, we analyze globally the motion of the pendulum and clarify the stability issue of two closed-loop equilibrium points; and we also provide some conditions on control parameters for achieving the tracking objective. Finally, we show numerical simulation results to validate the presented theoretical results.     

Periodic orbit of the pendulum with a small nonlinear damping

We study the pendulum with a small nonlinear damping, which can be expressed by a Hamiltonian system with a small perturbation. We prove that a unique periodic orbit exists for any initial position between the equilibrium point and the heteroclinic orbit of the unperturbed system, depending on the choice of the bifurcation parameter in the damping. The main tools are bifurcation theory and Abelian integral technique, as well as the Zhang''s uniqueness theorem on Li\''enard equations.     

Numerical optimization of tuned mass absorbers attached to strongly nonlinear Duffing oscillator

We investigate the dynamics of the vertically forced Duffing oscillator with suspended tuned mass absorber. Three different types of tuned mass absorbers are taken into consideration, i.e., classical single pendulum, dual pendulum and pendulum-spring. We numerically adjust parameters of absorbers to obtain the best damping properties with the lowest mass of attached system. The modification of classical case (single pendulum) gives the decrease of Duffing system amplitude. We present strategy of parameters tuning which can be easily applied in a large class of systems.     

Synthesis of controllers for damping oscillations in dynamical systems under uncertainty

We review the modern approaches to the synthesis of robust H _∞ controllers that ensure optimal damping of oscillations in dynamical systems under uncertainty. In the synthesis method based on Riccati equations, these many-parameter equations can be solved only when the parameters are contained in a bounded parallelepiped with given boundaries. The synthesis of a robust H _∞ output control for systems with unknown bounded parameters is reducible to the solution of an optimization problem constrained by a system of linear matrix inequalities. The proposed controller synthesis algorithms are implemented using standard MATLAB procedures. The efficiency of the proposed methods and algorithms is demonstrated in application to optimal damping of oscillations in a parametrically excited pendulum. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 87–104, 2004.     

A mixing-length model for magnetohydrodynamic flows in channels and ducts with wall-parallel magnetic field

A spanwise magnetic field leads to turbulent drag reduction in channel flow of a conducting liquid due to the selective Joule damping of certain flow structures. This effect can be captured by a simple modification of Prandtl's classical mixing-length idea. The mixing length over which a turbulent fluctuation loses its momentum is not only constrained geometrically but also by magnetic damping. We therefore introduce a magnetic damping length that is proportional to friction velocity and the Joule damping time. The limitation of mixing length is implemented by using the harmonic mean between wall distance and this damping length. By combining this ansatz with the van-Driest model for turbulent stresses in channel flow we obtain a satisfactory prediction for the mean velocity distribution in magnetohydrodynamic channel flow with spanwise field for different Reynolds and Hartmann numbers. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A vertically acting tuned liquid column damper

The U-shaped tuned liquid column damper (TLCD) increases the effective structural damping of horizontal vibrations similar to the classical tuned mechanical pendulum type damper (TMD). The pipe-in-pipe TLCD applies to vertical vibrations, likewise to the spring-mass-dashpot TMD. When sealed, the gas-spring effect extends the frequency range of application to about five Hertz. The geometric analogy between the novel TLCD and the TMD still exists, making the first step in the tuning procedure “classical”. Subsequent fine tuning in state space when the TLCD is split into smaller ones in parallel action, renders an even more robust passive action. The experimentally observed averaged turbulent damping of the relative fluid flow and the weakly nonlinear gas-spring render the TLCD insensitive to overloads and parametric forcing. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalized perturbed pendulum

The simple pendulum is a paradigm in the study of oscillations and other phenomena in physics and nonlinear dynamics. This explains why it has deserved much attention, from many viewpoints, for a long time. Here, we attempt to describe what we call a generalized perturbed pendulum, which comprises, in a single model, many known situations related to pendula, including different forcing and nonlinear damping terms. Melnikov analysis is applied to this model, with the result of general formulae for the appearance of chaotic motions that incorporate several particular cases. In this sense, we give a unified view of the pendulum.     

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.     

Stabilization of a quasi-conservative system subjected to high frequency excitation

The existence and stability conditions for the steady motions and equilibrium positions of non-linear quasi-conservative systems with fast external perturbations having quasi-periodic and random components are investigated. A change of variables is proposed which reduces Lagrange's equations of the system to standard form. It is shown the averaged system of the first approximation has a canonical form and the effect of fast perturbations (not necessarily potential) is equivalent to a change in the system's potential. This leads to stabilization of unstable equilibrium positions and to the appearance of additional stationary points different from the equilibrium positions of the unperturbed system. The approach used is illustrated by examples; the stability of a pendulum on an elastic suspension when there is suspension point, and the steady motion of a sphere subjected to a high-frequency load. The critical loading of a double pendulum loaded by a pulsating tracking force is estimated. A form of wide-band random perturbations capable of stabilizing the system is considered.     

On Dispersion in the Mathematical Model of Poroelastic Materials with the Balance Equation for Porosity

This article concerns the nonlinear asymptotic analysis of the mathematical model of a multi-component poroelastic medium with the porosity balance equation. It is proved that there exist soliton-like solutions for porosity and kink-like solutions for phase velocities under conditions similar to the entropy one. It is shown that solutions appear that describe a regime of stable displacement (so-called piston displacement) and a regime similar to the Saffman–Taylor instability. It is established that the diffusion force of interaction between phases of the inner friction type causes a decrease in the amplitudes of the discontinuities. This leads to the damping of initial perturbations of stable states. Such regimes have some properties similar to those of the pendulum model in the vicinity of a stable equilibrium point.     

Stable periodic solutions in the forced pendulum equation

Consider the pendulum equation with an external periodic force and an appropriate condition on the length parameter. It is proved that there exists at least one stable periodic solution for almost every external force with zero average. The stability is understood in the Lyapunov sense.     

Singularity formation for the inhomogeneous nonlinear wave equation with damping

It is well known that the damping term will give more smooth effect to obtain global solutions. In this paper, we consider the effect of damping term on the solutions to system of inhomogeneous wave equation with damping term. We can obtain the singularity that will be formed in finite time for some large initial data. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Diffusion dynamics near critical bifurcations in a nonlinearly damped pendulum system

We numerically study the diffusion dynamics near critical bifurcations such as sudden widening of the size of a chaotic attractor, intermittency and band-merging of a chaotic attractor in a nonlinearly damped and periodically driven pendulum system. The system is found to show only normal diffusion. Near sudden widening and intermittency crisis power-law variation of diffusion constant with the control parameter ω of the external periodic force f sin ωt is found while linear variation of it is observed near band-merging crisis. The value of the exponent in the power-law relation varies with the damping coefficient and the strength of the added Gaussian white noise.     

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.     

A transverse spinning double pendulum

A transverse spinning double pendulum is introduced. This pendulum is of interest as a simple mechanical system with two degrees of freedom with rotation which is autonomous. In addition to having physical origins, the pendulum is constructable for experimental observation. Our main interest in introducing and analyzing this system is that it is the simplest physical system with the codimension two singularity – in the linearization about the trivial solution – associated with coalescence of four zero eigenvalues. It is the dynamics of the nonlinear system in the neighbourhood of this singularity that is of interest. We study this problem using normal form theory. An algorithm for the Cushman–Sanders normal form is constructed and analyzed. A representative model for the truncated normal form is presented. This truncated normal form has seven parameters; it is not integrable in general and it is predicted that the dynamics associated with this model will be quite complex.