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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.

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...

Second Order systems II

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.

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...

Limits with Oscillating Discontinuities

Limits with Oscillating Discontinuities

An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the...

Time-Domain Interpretation of PD Control

Time-Domain Interpretation of PD Control

Proportional-Derivative (PD) control is a widely used control method in various engineering systems to enhance stability and performance. In a system with only proportional control, common issues include high maximum overshoot and oscillation, observed in both the error signal and its rate of change. This behavior can be divided into three distinct phases: initial overshoot, subsequent undershoot, and gradual stabilization.
Consider the example of control of motor torque. Initially, a positive...

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

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Anomalous Critical Behavior of Driven Disordered Systems Beyond the Overdamped Limit.

Giuseppe Petrillo¹, Eduardo Jagla², Eugenio Lippiello³

¹Earth Observatory of Singapore (EOS), Nanyang Technological University, Singapore, Singapore.

Physical Review Letters

|April 25, 2026

Summary

Inertial effects in driven elastic interfaces create system-spanning "kings," causing bumps in avalanche size distributions. Dissipation is key, with genuine kings appearing in 2D systems and "ghost kings" in 1D or short-range systems.

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Area of Science:

Physics
Condensed Matter Physics
Statistical Mechanics

Background:

Elastic disordered interfaces exhibit intermittent avalanches.
In the overdamped limit, avalanche sizes follow a power-law distribution.

Purpose of the Study:

Investigate how inertial and dynamical terms affect avalanche behavior beyond the overdamped approximation.
Analyze the influence of interaction range, dimensionality, and dissipation on system dynamics.

Main Methods:

Large-scale numerical simulations of driven elastic interfaces.
Analysis of avalanche size distributions under varying parameters.

Main Results:

Inertial terms generate system-spanning events ('kings') in the presence of dissipation, creating a bump in avalanche size distributions.
Two mechanisms identified: genuine kings (2D, long-range elasticity) and ghost kings (1D or short-range interactions).
Ghost kings are finite, ballistically spreading avalanches from failed synchronization.

Conclusions:

Clarifies the origin of bump features in avalanche statistics.
Establishes conditions for the emergence of true system-spanning events in dissipative systems.