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Forced Oscillations

Forced Oscillations

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.

Time and frequency -Domain Interpretation of Phase-lag Control

Time and frequency -Domain Interpretation of Phase-lag Control

Phase-lag controllers are widely used in control systems to improve stability and reduce steady-state errors. A dimmer switch controlling the brightness of a light bulb serves as a practical example of phase-lag control, gradually adjusting the bulb's brightness. Mathematically, phase-lag control or low-pass filtering is represented when the factor 'a' is less than 1.
Phase-lag controllers do not place a pole at zero, but instead influence the steady-state error by amplifying any finite,...

Stability

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...

Oscillations about an Equilibrium Position

Oscillations about an Equilibrium Position

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 because...

Time and frequency -Domain Interpretation of Phase-lead Control

Time and frequency -Domain Interpretation of Phase-lead Control

Phase-lead controllers are commonly used in various control systems to enhance response speed and stability. Adjusting the brightness on a television screen offers a practical example of phase-lead control. When contrast is enhanced, a phase-lead controller is employed. Mathematically, phase-lead control is identified when the first parameter is smaller than the second.
The design of phase-lead control involves the strategic placement of poles and zeros to balance steady-state error and system...

Pole and System Stability

Pole and System Stability

The transfer function is a fundamental concept representing the ratio of two polynomials. The numerator and denominator encapsulate the system's dynamics. The zeros and poles of this transfer function are critical in determining the system's behavior and stability.
Simple poles are unique roots of the denominator polynomial. Each simple pole corresponds to a distinct solution to the system's characteristic equation, typically resulting in exponential decay terms in the system's response.

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Related Experiment Video

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Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

Shaping the Amplitude and Phase of Laser Beams by Using a Phase-only Spatial Light Modulator

Published on: January 28, 2019

Pattern formation through phase bistability in oscillatory systems with space-modulated forcing.

Germán J de Valcárcel¹, Kestutis Staliunas

¹Departament d'Optica, Universitat de València, Dr. Moliner 50, 46100 Burjassot, Spain, EU.

Physical Review Letters

|September 28, 2010

Summary

This summary is machine-generated.

We introduce a new forcing method for self-oscillating systems that creates phase bistability and associated dissipative structures. This technique uses time-periodic, spatially modulated forcing, proving effective in complex Ginzburg-Landau equations.

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Area of Science:

Nonlinear dynamics
Complex systems science
Mathematical physics

Background:

Self-oscillatory systems are fundamental in various scientific fields.
Controlling phase dynamics and dissipative structures is crucial for understanding system behavior.
Existing forcing techniques may lack spatial modulation capabilities.

Purpose of the Study:

To develop a novel forcing technique for spatially extended self-oscillatory systems.
To investigate the excitation of phase bistability and dissipative structures.
To demonstrate the effectiveness of spatially modulated forcing.

Main Methods:

Analytical investigation of self-oscillatory systems.
Numerical simulations using the complex Ginzburg-Landau equation.
Application of time-periodic, spatially modulated forcing.

Main Results:

The proposed forcing technique successfully excites phase bistability.
Dissipative structures associated with phase bistability are observed.
Both spatially periodic and spatially random drives are effective.

Conclusions:

Spatially modulated forcing is a powerful tool for controlling self-oscillatory systems.
The complex Ginzburg-Landau equation serves as a valid model for demonstrating this phenomenon.
This technique offers new possibilities for manipulating complex systems.