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Related Concept Videos

Transient and Steady-state Response01:24

Transient and Steady-state Response

In control systems, test signals are essential for evaluating performance under various conditions. The ramp function is effective for systems undergoing gradual changes, while the step function is suitable for assessing systems facing sudden disturbances. For systems subjected to shock inputs, the impulse function is the most appropriate test signal.
These test signals are integral in designing control systems to exhibit two key performance aspects: transient response and steady-state response.
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Linear time-invariant Systems01:23

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
Double Resonance Techniques: Overview01:12

Double Resonance Techniques: Overview

Double resonance techniques in Nuclear Magnetic Resonance (NMR) spectroscopy involve the simultaneous application of two different frequencies or radiofrequency pulses to manipulate and observe two distinct nuclear spins. One important application of double resonance is spin decoupling, which selectively suppresses coupling with one type of nucleus while observing the NMR signal from another nucleus, simplifying the spectrum and enhancing resolution.
Spin decoupling is usually achieved by...
Stability01:28

Stability

The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
The stability of an LTI system is determined by the roots of its characteristic equation, known as poles. A system is stable if it produces a bounded...
Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...

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Microwave Photonics Systems Based on Whispering-gallery-mode Resonators
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Published on: August 5, 2013

Nearly resonant multidimensional systems under a transient perturbative interaction.

Antonia Ruiz1, José P Palao, Eric J Heller

  • 1Departamento de Física Fundamental y Experimental, Electrónica y Sistemas and IUdEA, Universidad de La Laguna, La Laguna, Spain.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 7, 2010
PubMed
Summary

Researchers identified precise adiabatic invariants in classical systems with multiple degrees of freedom under resonance conditions. This work estimates the domain of intersecting resonances, with applications in molecular collisions.

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Area of Science:

  • Classical mechanics
  • Nonlinear dynamics
  • Chemical physics

Background:

  • Systems with multiple internal degrees of freedom can exhibit complex dynamics under resonance conditions.
  • Understanding transient interactions and their effect on system stability is crucial in various physical phenomena.

Purpose of the Study:

  • To analyze the response of classical systems with N>=2 internal degrees of freedom to perturbative transient interactions.
  • To identify and characterize adiabatic invariants under approximated resonance conditions (R<=(N-1)).
  • To estimate the domain of intersecting resonances.

Main Methods:

  • Analysis of a classical system with N>=2 internal degrees of freedom.
  • Application of resonance conditions (R<=(N-1)) and external perturbative transient interactions.
  • Derivation of precise N-R adiabatic invariants under specific assumptions on internal frequencies and coupling.

Main Results:

  • Demonstration of precise N-R adiabatic invariants for the studied systems.
  • Estimation of the span of the domain defined by intersecting resonances.
  • Illustration using a system of three anharmonic oscillators with time-dependent coupling.

Conclusions:

  • The study provides a theoretical framework for understanding adiabatic invariants in complex classical systems.
  • The findings have direct implications for modeling low-energy vibro-rotationally inelastic molecular collisions.
  • The methodology can be extended to other systems exhibiting transient interactions and resonance phenomena.