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Related Concept Videos

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...
Types of Damping01:20

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...
Damped Oscillations01:07

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...
One-Degree-of-Freedom System01:24

One-Degree-of-Freedom System

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.
A one-degree-of-freedom system is defined by an independent variable that determines its state and behavior. One example of a one-degree-of-freedom system is a simple harmonic oscillator, such as a...
Limits with Oscillating Discontinuities01:19

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 most...
Forced Oscillations01:06

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.

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Related Experiment Video

Updated: May 15, 2026

Data Acquisition Protocol for Determining Embedded Sensitivity Functions
07:46

Data Acquisition Protocol for Determining Embedded Sensitivity Functions

Published on: April 20, 2016

SENSITIVITY ANALYSIS FOR OSCILLATING DYNAMICAL SYSTEMS.

A Katharina Wilkins1, Bruce Tidor, Jacob White

  • 1Department of Chemical Engineering, 77 Massachusetts Avenue, Cambridge MA 02139 ; Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge MA 02139.

SIAM Journal on Scientific Computing : a Publication of the Society for Industrial and Applied Mathematics
|January 9, 2013
PubMed
Summary
This summary is machine-generated.

This study introduces boundary value methods for precise sensitivity analysis of oscillating systems. These novel techniques efficiently compute sensitivities of oscillation characteristics like period and amplitude.

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Area of Science:

  • Dynamical Systems
  • Computational Mathematics
  • Nonlinear Oscillations

Background:

  • Sensitivity analysis is crucial for understanding oscillating systems.
  • Existing methods for computing sensitivities can be computationally intensive and lack precision.
  • Oscillatory systems are prevalent in various scientific and engineering domains.

Purpose of the Study:

  • To develop exact and efficient boundary value formulations for sensitivity analysis of oscillating systems.
  • To compute sensitivities of key oscillation quantities like period, amplitude, and phase.
  • To provide a general framework applicable to different classes of oscillatory systems.

Main Methods:

  • Formulation of boundary value problems for sensitivity analysis.
  • Development of methods for computing sensitivities of derived quantities (period, amplitude, phase).
  • Novel decomposition of state sensitivities for intuitive analysis of parameter influence.
  • Extension of methods to general oscillatory systems and numerical solution techniques.

Main Results:

  • Accurate computation of sensitivities for limit-cycle oscillators and other systems.
  • Demonstration of a novel three-part decomposition for state sensitivities.
  • Quantification of the influence of time reference (phase locking) on sensitivity solutions.
  • Numerical results verified against finite difference approximations, showing superior efficiency and precision.

Conclusions:

  • Boundary value formulations offer an exact and efficient approach to sensitivity analysis for oscillating systems.
  • The proposed methods provide superior computational efficiency and numerical precision compared to existing techniques.
  • The developed framework is general and applicable to a wide range of oscillatory phenomena.