Global Dynamics of an Autoparametric System With Multiple Pendulums

2005 ◽  
Vol 1 (1) ◽  
pp. 35-46 ◽  
Author(s):  
Ashwin Vyas ◽  
Anil K. Bajaj
Keyword(s):  
Initial Conditions ◽  
Hamiltonian Formulation ◽  
Forced Response ◽  
Global Dynamics ◽  
Equilibrium Solutions ◽  
External Excitation ◽  
Poincaré Sections ◽  
Poincare Sections ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

The Hamiltonian dynamics of a resonantly excited linear spring-mass-damper system coupled to an array of pendulums is investigated in this study under 1:1:1:…:2 internal resonance between the pendulums and the linear oscillator. To study the small-amplitude global dynamics, a Hamiltonian formulation is introduced using generalized coordinates and momenta, and action-angle coordinates. The Hamilton’s equations are averaged to obtain equations for the first-order approximations to free and forced response of the system. Equilibrium solutions of the averaged Hamilton’s equations in action-angle or comoving variables are determined and studied for their stability characteristics. The system with one pendulum is known to be integrable in the absence of damping and external excitation. Exciting the system with even a small harmonic forcing near a saddle point leads to stochastic response, as clearly demonstrated by the Poincaré sections of motion. Poincaré sections are also computed for motions started with initial conditions near center-center, center-saddle and saddle-saddle-type equilibria for systems with two, three and four pendulums. In case of the system with more than one pendulum, even the free undamped dynamics exhibits irregular exchange of energy between the pendulums and the block. The increase in complexity is also demonstrated as the number of pendulums is increased, and when external excitation is present.

Fractional Hamilton’s equations of motion in fractional time

Open Physics ◽  
2007 ◽  
Vol 5 (4) ◽  
Author(s):  
Sami Muslih ◽  
Dumitru Baleanu ◽  
Eqab Rabei
Keyword(s):  
Fractional Derivatives ◽  
Equations Of Motion ◽  
Hamiltonian Formulation ◽  
Mechanical Systems ◽  
Hamilton's Equations ◽  
Hamilton’S Equations

AbstractThe Hamiltonian formulation for mechanical systems containing Riemman-Liouville fractional derivatives are investigated in fractional time. The fractional Hamilton’s equations are obtained and two examples are investigated in detail.

Hamiltonian Chaos in a Model Alveolus

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
F. S. Henry ◽  
F. E. Laine-Pearson ◽  
A. Tsuda
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Lyapunov Exponent ◽  
Fine Particles ◽  
Hamiltonian Dynamics ◽  
Maximal Lyapunov Exponent ◽  
Poincaré Sections ◽  
Pulmonary Acinus ◽  
Hamiltonian Chaos ◽  
Poincare Sections

In the pulmonary acinus, the airflow Reynolds number is usually much less than unity and hence the flow might be expected to be reversible. However, this does not appear to be the case as a significant portion of the fine particles that reach the acinus remains there after exhalation. We believe that this irreversibility is at large a result of chaotic mixing in the alveoli of the acinar airways. To test this hypothesis, we solved numerically the equations for incompressible, pulsatile, flow in a rigid alveolated duct and tracked numerous fluid particles over many breathing cycles. The resulting Poincaré sections exhibit chains of islands on which particles travel. In the region between these chains of islands, some particles move chaotically. The presence of chaos is supported by the results of an estimate of the maximal Lyapunov exponent. It is shown that the streamfunction equation for this flow may be written in the form of a Hamiltonian system and that an expansion of this equation captures all the essential features of the Poincaré sections. Elements of Kolmogorov–Arnol’d–Moser theory, the Poincaré–Birkhoff fixed-point theorem, and associated Hamiltonian dynamics theory are then employed to confirm the existence of chaos in the flow in a rigid alveolated duct.

scholarly journals Nonlinear Behavior of a Magnetic Bearing System

1995 ◽  
Vol 117 (3) ◽  
pp. 582-588 ◽  
Author(s):  
L. N. Virgin ◽  
T. F. Walsh ◽  
J. D. Knight
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Nonlinear Behavior ◽  
Analytical Techniques ◽  
Magnetic Bearing ◽  
Forced Response ◽  
The Mathematical Model ◽  
Two Degree Of Freedom ◽  
Sensitivity To Initial Conditions ◽  
Bearing System

This paper describes the results of a study into the dynamic behavior of a magnetic bearing system. The research focuses attention on the influence of nonlinearities on the forced response of a two-degree-of-freedom rotating mass suspended by magnetic bearings and subject to rotating unbalance and feedback control. Geometric coupling between the degrees of freedom leads to a pair of nonlinear ordinary differential equations, which are then solved using both numerical simulation and approximate analytical techniques. The system exhibits a variety of interesting and somewhat unexpected phenomena including various amplitude driven bifurcational events, sensitivity to initial conditions, and the complete loss of stability associated with the escape from the potential well in which the system can be thought to be oscillating. An approximate criterion to avoid this last possibility is developed based on concepts of limiting the response of the system. The present paper may be considered as an extension to an earlier study by the same authors, which described the practical context of the work, free vibration, control aspects, and derivation of the mathematical model.

Numerical Study of an Impact Oscillator

1995 ◽  
Author(s):  
Kannan Marudachalam ◽  
Faruk H. Bursal
Keyword(s):  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Initial Conditions ◽  
Numerical Study ◽  
Harmonic Excitation ◽  
General Purpose ◽  
Constrained Systems ◽  
Global Dynamics ◽  
Impact Oscillator ◽  
Stick Slip

Abstract Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.

scholarly journals Characterization of the Transient Response of Coupled Optimization in Multidisciplinary Design

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Erich Devendorf ◽  
Kemper Lewis
Keyword(s):  
Optimization Problems ◽  
Initial Conditions ◽  
Systems Design ◽  
Discrete Systems ◽  
Product Development Process ◽  
Multidisciplinary Design ◽  
Distributed Design ◽  
Equilibrium Solutions ◽  
Acceptable Solution ◽  
Number Of Iterations

Time is an asset of critical importance in a multidisciplinary design process and it is desirable to reduce the amount of time spent designing products and systems. Design is an iterative activity and designers consume a significant portion of the product development process negotiating a mutually acceptable solution. The amount of time necessary to complete a design depends on the number and duration of design iterations. This paper focuses on accurately characterizing the number of iterations required for designers to converge to an equilibrium solution in distributed design processes. In distributed design, systems are decomposed into smaller, coupled design problems where individual designers have control over local design decisions and seek to achieve their own individual objectives. These smaller coupled design optimization problems can be modeled using coupled games and the number of iterations required to reach equilibrium solutions varies based on initial conditions and process architecture. In this paper, we leverage concepts from game theory, classical controls, and discrete systems theory to evaluate and approximate process architectures without carrying out any solution iterations. As a result, we develop an analogy between discrete decisions and a continuous time representation that we analyze using control theoretic techniques.

Free and Forced Response of an Axially Moving String Transporting a Damped Linear Oscillator

1993 ◽  
Author(s):  
W. D. Zhu ◽  
C. D. Mote
Keyword(s):  
Initial Conditions ◽  
Numerical Algorithms ◽  
Time History ◽  
Coupled System ◽  
Interaction Force ◽  
Forced Response ◽  
Linear Oscillator ◽  
Response Analysis ◽  
History Of ◽  
Axially Moving

Abstract The transverse response of a cable transport system, which is modelled as an ideal, constant tension string travelling at constant speed between two supports with a damped linear oscillator attached to it, is predicted for arbitrary initial conditions, external forces and boundary excitations. The exact formulation of the coupled system reduces to a single integral equation of Volterra type governing the interaction force between the string and the payload oscillator. The time history of the interaction force is discontinuous for non-vanishing damping of the oscillator. These discontinuities occur at the instants when transverse waves propagating along the string interact with the oscillator. The discontinuities are treated using the theory of distributions. Numerical algorithms for computing the integrals involving generalized functions and for solution of the delay-integral-differential equation are developed. Response analysis shows a discontinuous velocity history of the payload attachment point. Special conditions leading to absence of the discontinuities above are given.

Perturbing variables in poincaré sections to suppress chaos

2008 ◽  
pp. 339-340
Author(s):  
J. M. Gutiérrez ◽  
A. Iglesias ◽  
J. Güémez ◽  
M. A. Matías
Keyword(s):  
Poincaré Sections ◽  
Poincare Sections
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