Damped Pendulum Research Articles

Slosh-Free Positioning of Containers with Liquids and Flexible Conveyor Belt

Slosh-Free Positioning of Containers with Liquids and Flexible Conveyor Belt This paper describes a method for slosh-free moving of containers with a liquid at which the conveyor belt is flexible. It shows, by means of experimental results, that a container filled with a liquid can be regarded as a damped pendulum. Upon parameter identification of the two single-mode subsystems, theoretical modelling of the complete system is described. With respect to industrial application, feedforward control is used to transfer the container in horizontal direction without sloshing. The applied method requires deterministic and hard real time communication between the PLC and the servo amplifier which is realized here with Ethernet POWERLINK. The principle structure, calculations, time duration and the robustness of the basic types of input shaper are described and compared with each other. Additionally the positioning results of the two-mode system are presented. Finally, a possibility of the principle software implementation is presented.

Bifurcations and chaos in simple dynamical systems

Chaos is an active research subject in the fields of science in recent years. It is a complex and an erratic behaviour that is possible in very simple systems. In the present day, the chaotic behaviour can be observed in experiments. Many studies have been made in chaotic dynamics during the past three decades and many simple chaotic systems have been discovered. In this work, the behaviour of some simple dynamical systems is studied by constructing mathematical models. Investigations are made on the periodic orbits for continuous maps and idea of sensitive dependence on initial conditions, which is the hallmark of chaos, is obtained. A small attempt has been made to find out the reasons / unknown conditions for the production of chaos in a system. This is explained through simple dynamical systems. (Why chaos is produced in a forced damped simple pendulum?) Besides, another attempt has been made to identify, algebraically simplest chaotic flow. These are the significance of this study - “Bifurcations and chaos in simple dynamical systems”. Accordingly, an analysis is done on different dynamical systems. The exact solution is obtained by solving the differential equation by using Runge-Kutta method; as a result, it is clear from the analysis that, period multiplication occurring in a forced damped simple pendulum can leads to chaos. This result is proving that it is possible to find out the reasons / unknown conditions under which chaotic behaviour exhibits in various systems. The importance of result is, “why chaos is produced in various systems? - may be identified in future. Key Words: Bifurcations, chaos, dynamical systems.

Stabilization of a 1/3-order subharmonic resonance using nonlinear dynamic vibration absorber

The paper proposes a stabilization method for a 1/3-order subharmonic resonance with a new type of nonlinear vibration absorber using nonlinear coupling between a main system and the absorber. The main system with nonlinear restoring force and harmonic excitation, i.e., subjected to a sinusoidally changed magnetic force, is introduced as a model which produces a 1/3-order subharmonic resonance. A damped pendulum, whose natural frequency is tuned to be in the neighborhood of twice that of the main system, is connected through a link to the main system as a nonlinear vibration absorber. Theoretical results using the method of multiple scales show that only a stable nontrivial steady state is changed into an unstable one due to the effect of absorber. In addition, we numerically confirm the validity of the proposed absorber using Runge–Kutta method.

Computer Assisted Proof of the Existence of Homoclinic Tangency for the Hénon Map and for the Forced Damped Pendulum

We present a topological method for the efficient computer assisted verification of the existence of the homoclinic tangency which unfolds generically in a one-parameter family of planar maps. The method has been applied to the Henon map and the forced damped pendulum ODE.

Monotonicity in the damped pendulum type equations

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincare map PT is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincare map has in the cylindrical phase space an invariant circle, on which PT is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of PT on this curve.

Mathematical simulation modeling of the drift of the zero point of a sea gravimeter

Mathematical simulation modeling of the elimination of the systematic error of discrete readings obtained by the “method of crossings” of the indications of two supercritical damped pendulums is performed. Modeling is carried out to filter the dynamic distortions of the indications of a sea gravimeter created by inertial noise. A simplified single-component model is used.

Instabilities in the main parametric resonance area of a mechanical system with a pendulum

Vibrations of an autoparametric system, composed of a nonlinear mechanical oscillator with an attached damped pendulum, around the principal resonance region, are investigated in this paper. Approximate analytical solutions of the model are determined on the basis of the Harmonic Balance Method (HBM). Correctness of the analytical results is verified by numerical simulations and experimental tests performed on an especially prepared experimental rig. The influence of all essential parameters such as damping, excitation amplitude and frequency, nonlinear stiffness of the spring, on the localisation of the instability region and the system dynamics is presented in the work. Regions of regular system oscillations, chaotic motions, and full rotation of the pendulum are confirmed experimentally.

Synchronization of Forced damped Pendulum via Active Control

In this paper active controllers are designed to synchronize two identical forced damped pendula. The performance of the controllers in the synchronization of the chaotic dynamics of the two pendula, resulting from different initial conditions, is investigated numerically and found to be effective. Transition from nonsynchronous state via both temporary phase lock (TPL) and intermittent synchronous states to complete synchronous state was observed. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 143-148

Stability problems for the mathematical pendulum

The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If the coefficient is a step function, then small oscillations are described by the equation $$ \ddot x + a^2 (t)x = 0,a(t): = a_k ift_{k - 1} \leqslant t < t_k ,k = 1,2,.... $$ Using a probability approach, we assume that (a k ) =1 ∞ is given, and {t k } =1 ∞ is chosen at random so that t k − t k−1 are independent random variables. The first problem is to guarantee that all solutions tend to zero, as t → ∞, provided that a k ↗ ∞ (k → ∞). In the problem of swinging the coefficient a 2 takes only two different values alternating each others, and t k − t k−1 are identically distributed. One has to find the distributions and their critical expected values such that the amplitudes of the oscillations tend to ∞ in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation $$ \ddot x + 10^{ - 1} \dot x + sinx = cost. $$ In 1999 J. Hubbard discovered that some motions of this simple physical model are chaotic. Recently, using also the computer (the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof.

A Computer-Assisted Proof of $\Sigma_3$-Chaos in the Forced Damped Pendulum Equation

The present paper is devoted to studying Hubbard's pendulum equation $\ddot{x} + 10^{-1} \dot{x} + \sin(x) = \cos(t)$. Using rigorous/interval methods of computation, the main assertion of Hubbard on chaos properties of the induced dynamics is raised from the level of experimentally observed facts to the level of a theorem completely proved. A special family of solutions is shown to be chaotic in the sense that, on consecutive time intervals $(2k\pi, 2(k+1)\pi)$ ($k \in \mathbb{Z}$), individual members of the family can freely “choose” between the following possibilities: the pendulum crosses the bottom position exactly once clockwise or does not cross the bottom position at all or crosses the bottom position exactly once counterclockwise. The proof follows the topological index/degree approach by Mischaikow, Mrozek, and Zgliczynski. The new feature of this paper is a definition of the transition graph for which the periodic orbit lemma—the key technical result of the approach mentioned above—turns out to...

RESHAPING-INDUCED CONTROL OF CHAOS AND CRISIS PHENOMENA IN A DAMPED, PARAMETRICALLY DRIVEN PENDULUM

The behavior of a damped pendulum parametrically excited by a periodic string of symmetric pulses of finite width and amplitude is investigated. Analytical (Melnikov method) and numerical (bifurcation diagrams) results show that chaos and crises are reliably controlled over a wide range of parameters by hump-doubling of a parametric excitation which is initially formed by a periodic string of single-humped symmetric pulses. In particular, the analysis reveals that the chaotic threshold amplitude when altering solely the pulse shape presents a minimum as a single-humped pulse transforms into a double-humped pulse, the remaining parameters being held constant. Additionally, the mechanism underlying the hump-doubling-induced crises is discussed with the help of a two-dimensional map.

Introduction to Nonlinear Dynamics of Electronic Systems: Tutorial

The study of chaos has generated enormous interest in exploring the complexity of the behavior in nature and in technology. Many of the important features of chaotic dynamical systems can be seen using experimental and computational methods in simple nonlinear mechanical systems or electronic circuits. Starting with the study of a chaotic nonlinear mechanical system (driven damped pendulum) or a nonlinear electronic system (circuit Chua) we introduce the reader into the concepts of chaos order in Sharkovsky's sense, and topological invariants (topological entropy and topological frequencies). The Kirchhoff's circuit laws are a pair of laws that deal with the conservation of charge and energy in electric circuits, and the algebraic theory of graphs characterizes these linear systems in terms of cycles and cocycles (or cuts). Here we discuss methods (topological semiconjugacy to piecewise linear maps and Markov graphs) to find a similar situation for the nonlinear dynamics, to understanding chaotic dynamics. Thus to chaotic dynamics we associate a Markov graph, where the dynamical and topological invariants will be seen as graph theoretical quantities.

Towards a computer-assisted proof for chaos in a forced damped pendulum equation

We report on the first steps made towards the computational proof of the chaotic behaviour of the forced damped pendulum. Although, chaos for this pendulum was being conjectured for long, and it has been plausible on the basis of numerical simulations, there is no rigorous proof for it. In the present paper we provide computational details on a fitting model and on a verified method of solution. We also give guaranteed reliability solutions showing some trajectory properties necessary for complicate chaotic behaviour.

A damped pendulum forced with a constant torque

The dynamics of a damped pendulum driven by a constant torque is studied experimentally and theoretically. We use this simple device to demonstrate some generic dynamical behavior including the loss of equilibrium or saddle node bifurcation with or without hysteresis and the homoclinic bifurcation. A qualitative analysis is developed to emphasize the role of two dimensionless parameters corresponding to damping and forcing.

Local effects of St∼O(1) on small particle motion in a linear flow field

Small particle motion in a linearized flow field is investigated analytically. The aim is to study the effects of relatively large Stokes numbers (St). It is assumed that the only dominant hydrodynamic force is the quasisteady Stokes drag where all other effects are neglected. An exact solution for a general, three-dimensional, unsteady flow field, including gravity, is obtained. Emphasis is placed on the role taken by the spatially varying flow field and the particle inertia. Although an immediate consequence of the linearization is that the long-term particle velocity is insensitive to initial conditions, no matter the value of St, the short-term behavior is quite intriguing. This behavior is found to be highly sensitive to the initial particle velocity when St exceeds critical values depending on the flow; this happens, for example, in shear flows. The exponential approach of the particle velocity to the fluid velocity, predicted by the asymptotic solution for St∼o(1), may under certain flow conditions transform into damped oscillations for critical values of the St number. An explanation is provided by showing the equivalence of the particle equation to that of a damped pendulum.

Another look at a damped physical pendulum

Abstract

Symmetry breaking in a driven and strongly damped pendulum

We examine the conditions for appearance of a symmetry breaking bifurcation in damped and periodically driven pendulums in the case of strong damping. We show that symmetry breaking, unlike other nonlinear phenomena, can exist at high dissipation. We prove that symmetry breaking phases exist between phases of symmetric normal and symmetric inverted oscillations. We find that symmetry broken solutions occupy a smaller region of the pendulum's parameter space in comparison to the statements made in earlier considerations [McDonald and Plischke, Phys. Rev. B 27, 201 (1983)]. Our research on symmetry breaking in a strongly damped pendulum is relevant to an understanding of the phenomena of dynamic symmetry breaking and rectification in pure ac driven semiconductor superlattices.

On the sensitivity of strongly nonlinear autonomous oscillators and oscillatory waves to small perturbations

Original asymptotic solutions are determined for two autonomous differential equations. The application of initial conditions for the energy, wave number and phase shift proves to be less complicated than in previous work. For the damped simple pendulum, explicit solutions demonstrate the dependence on the initial conditions. For strongly nonlinear wave packets of the Klein-Gordon equation, asymptotic solutions are compared. In both cases, the phase shift is shown to be highly sensitive to small perturbations in the initial conditions.

Modeling dynamic resource utilization in populations with unique constraints: Preadolescents with and without Down syndrome

In this study we used a damped inverted pendulum and spring with an escapement function model to compare the global levels of stiffness and forcing used by 12 preadolescents with Down syndrome (DS) and 12 with typical development (TD). Participants walked overground at their self-selected speed and on a treadmill at speeds slower and faster than overground. Children with DS, who are characterized as hypotonic with reduced capacity for producing muscle force, exhibited significantly higher levels of stiffness and forcing (angular impulse) when walking on the treadmill and higher forcing but not stiffness overground, than children with TD. Both groups adapted to imposed speed increases similarly by increasing their global stiffness and angular impulse. We propose children with DS increased stiffness in order to overcome their hypotonia and joint laxity, thus, optimizing on stability, rather than metabolic efficiency. Higher angular impulse values for children with DS may reflect higher energy cost associated with increasing stiffness and their inherent biomechanical and physiological characteristics that reduce efficiency. We conclude that the inverted pendulum and spring with escapement function model is a useful tool for uncovering solutions to movement problems-solutions that reflect the dynamic resources of the individual and ones that are discovered, rather than prescribed.

Chaos in a four-dimensional system consisting of fundamental lag elements and the relation to the system eigenvalues

Abstract A simple system consisting of a second-order lag element (a damped linear pendulum) and two first-order lag elements with piecewise-linear static feedback that has been derived from a power system model is presented. It exhibits chaotic behavior for a wide range of parameter values. The analysis of the bifurcations and the chaotic behavior are presented with qualitative estimation of the parameter values for which the chaotic behavior is observed. Several characteristics like scalability of the attractor and globality of the attractor-basin are also discussed.