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Related Concept Videos

Forced Oscillations01:06

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When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
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Damped Oscillations01:07

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In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...
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If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
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Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke's law. If the net force can be described by Hooke's law and there is no damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position. To derive an equation for period and frequency, the equation of motion is used. The period of a simple harmonic oscillator is given...
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In mechanical engineering, one-degree-of-freedom systems form the basis of a wide range of electrical and mechanical components. Using these models, engineers can predict the behavior of various parts in a larger system, which gives them insight into how different forces interact with each other.
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Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so...
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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Stable dynamics in forced systems with sufficiently high/low forcing frequency.

M Bartuccelli1, G Gentile2, J A Wright1

  • 1Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom.

Chaos (Woodbury, N.Y.)
|September 3, 2016
PubMed
Summary
This summary is machine-generated.

Parametrically forced Hamiltonian systems exhibit stable dynamics across a wide range of forcing frequencies and amplitudes. The Kolmogorov-Arnold-Moser (KAM) theorem applies even with large amplitudes at high frequencies, and numerically, stability persists at very large amplitudes with low frequencies.

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Area of Science:

  • * Hamiltonian dynamics
  • * Perturbation theory
  • * Dynamical systems theory

Background:

  • * Hamiltonian systems with one-and-a-half degrees of freedom are fundamental in classical mechanics.
  • * Understanding stability under parametric forcing is crucial for predicting system behavior.
  • * The Kolmogorov-Arnold-Moser (KAM) theorem provides conditions for the persistence of quasi-periodic orbits.

Purpose of the Study:

  • * To investigate the stability of parametrically forced Hamiltonian systems.
  • * To explore the applicability of KAM theorem under high and low forcing frequencies.
  • * To determine the influence of forcing amplitude on system stability beyond perturbation regimes.

Main Methods:

  • * Analytical investigation of parametrically forced Hamiltonian systems.
  • * Application of Kolmogorov-Arnold-Moser (KAM) theorem under specific frequency conditions.
  • * Numerical simulations to explore stability at large forcing amplitudes.

Main Results:

  • * KAM theorem is applicable for sufficiently high forcing frequencies, even with large amplitudes.
  • * For low frequencies, KAM theorem applies with moderate amplitudes, extending beyond traditional perturbation regimes.
  • * Numerical evidence shows stable dynamics for very large forcing amplitudes when frequency is correspondingly low.

Conclusions:

  • * Parametrically forced Hamiltonian systems demonstrate robust stability.
  • * Forcing frequency significantly influences the applicability of KAM theorem and overall stability.
  • * Numerical results suggest broader stability regimes than predicted by analytical methods alone.