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Types of Damping01:20

Types of Damping

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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...
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Second Order systems II01:18

Second Order systems II

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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.
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Damped Oscillations01:07

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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...
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Parameters Affecting Nonlinear Elimination: Zero-Order Input, First-Order Absorption and Two-Compartment Model01:13

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Drugs administered through various routes can lead to nonlinear elimination, resulting in complex pharmacokinetic behaviors crucial to understanding efficacious drug dosing.
When a drug is administered through a constant intravenous infusion and eliminated via nonlinear pharmacokinetics, it follows zero-order input. For example, oral drugs undergo first-order absorption upon administration and are eliminated through nonlinear pharmacokinetics.
In the case of subcutaneously administered drugs,...
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Second-Order Circuits01:17

Second-Order Circuits

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Integrating two fundamental energy storage elements in electrical circuits results in second-order circuits, encompassing RLC circuits and circuits with dual capacitors or inductors (RC and RL circuits). Second-order circuits are identified by second-order differential equations that link input and output signals.
Input signals typically originate from voltage or current sources, with the output often representing voltage across the capacitor and/or current through the inductor. For example, in...
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Stability of structures01:14

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In mechanical engineering, the stability of systems under various forces is critical for designing durable and efficient structures. One fundamental way to explore these concepts is by analyzing systems like two rods connected at a pivot point, O, with a torsional spring of spring constant k at the pivot point. This system is similar in appearance to a scissor jack used to change tires on a car. In this case, the arms of the linkage (equivalent to the rods in this system) are entirely vertical,...
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Related Experiment Video

Updated: Jul 1, 2025

Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing
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Characterizing Dissipative Elastic Metamaterials Produced by Additive Manufacturing

Published on: June 28, 2024

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Second Order Splitting Dynamics with Vanishing Damping for Additively Structured Monotone Inclusions.

Radu Ioan Boţ1, David Alexander Hulett1

  • 1Faculty of Mathematics, University of Vienna, Vienna, Austria.

Journal of Dynamics and Differential Equations
|March 4, 2024
PubMed
Summary
This summary is machine-generated.

This study introduces a novel splitting system for finding operator zeros in Hilbert spaces. The method ensures convergence to solutions and fast velocity reduction, with applications in convex optimization.

Keywords:
Asymptotic stabilizationDamped inertial dynamicsLyapunov analysisMonotone inclusionsSplitting systemVanishing viscosity

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Area of Science:

  • Optimization Theory
  • Functional Analysis
  • Numerical Analysis

Background:

  • Addresses the challenge of finding zeros for the sum of maximally monotone and cocoercive operators.
  • Builds upon existing methods by introducing a novel second-order dynamical system with vanishing damping.

Purpose of the Study:

  • To analyze the asymptotic behavior of trajectories generated by a time-dependent forward-backward splitting system.
  • To establish weak convergence of trajectories to the solution set of the operator equation.
  • To demonstrate fast convergence of velocities to zero.

Main Methods:

  • Utilizes a second-order dynamical system with vanishing damping.
  • Employs a time-dependent forward-backward splitting operator.
  • Analyzes the system's behavior in a real Hilbert space framework.

Main Results:

  • Proves weak convergence of generated trajectories to the set of zeros of A + B.
  • Demonstrates that the velocities of the trajectories converge rapidly to zero.
  • Derives fast convergence rates for a specific convex optimization problem as a special case.

Conclusions:

  • The proposed splitting system is effective for finding zeros of sums of monotone and cocoercive operators.
  • The system offers theoretical guarantees for convergence and fast velocity reduction.
  • Numerical experiments validate the theoretical findings and demonstrate practical applicability.