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Damped Oscillations

Damped Oscillations

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...

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.

Types of Damping

Types of Damping

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...

Design Example: Underdamped Parallel RLC Circuit

Design Example: Underdamped Parallel RLC Circuit

Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
Starting with a fixed...

RLC Circuit as a Damped Oscillator

RLC Circuit as a Damped Oscillator

An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
Consider a series RLC circuit. Here, the presence of resistance in the circuit leads to energy loss due to joule heating in the resistance. Therefore, the total electromagnetic energy in the circuit is no longer constant and decreases with time. Since the magnitude of charge, current, and potential difference continuously decreases, their oscillations are said to be damped. This is...

Concept of Resonance and its Characteristics

Concept of Resonance and its Characteristics

If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano strings and many other musical instruments. Conversely, to achieve small-amplitude oscillations as in a car's suspension system, heavy damping is required. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies. Speed bumps and gravel roads prove that even a car's suspension system is not immune...

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Related Experiment Video

Updated: May 17, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Self-consistent Born theory for the Duffing oscillator.

Martín E Giuliano¹

¹Instituto Balseiro (UNCUYO), Centro Atómico Bariloche (CNEA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), C1425FQB Buenos Aires, Argentina and , - , 8400 San Carlos de Bariloche, Argentina.

Physical Review. E

|May 16, 2026

Summary

We developed a self-consistent Born approximation (SCBA) for nonlinear oscillators driven by noise. This method accurately predicts system behavior, outperforming standard approximations, especially for large amplitudes.

Related Experiment Videos

Last Updated: May 17, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Area of Science:

Nonlinear dynamics
Statistical physics
Field theory

Background:

Nonlinear oscillators driven by stochastic forces are common in physics.
Standard perturbative methods often fail to accurately describe these systems, especially in the nonlinear regime.

Purpose of the Study:

To develop a robust analytical framework for studying nonlinear oscillators under stochastic driving.
To accurately predict system observables like mean-square displacement and renormalized frequency.

Main Methods:

Development of a self-consistent Born approximation (SCBA) within a field-theoretic framework.
Construction of a self-consistent modal mean-field solution analogous to particle field theory.
Incorporation of dynamic correlations between noise-activated internal Fourier modes (NAIFMs).

Main Results:

The SCBA successfully renormalizes the oscillator's natural frequency and captures amplitude-frequency dependence.
Static and dynamic correlations among NAIFMs are effectively included.
The SCBA accurately reproduces the mean-square displacement (MSD) and renormalized frequency, agreeing well with simulations, unlike perturbative expansions.

Conclusions:

The SCBA provides a reliable analytical method for nonlinear oscillators subjected to stochastic driving.
This approach is highly relevant for understanding micro- and nanomechanical resonators.