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Related Concept Videos

Mechanical Systems01:22

Mechanical Systems

879
Mechanical systems are analogous to to electrical networks where springs and masses play similar roles to inductors and capacitors, respectively. A viscous damper in mechanical systems functions similarly to a resistor in electrical networks, dissipating energy. The forces acting on a mass in such systems include an applied force in the direction of motion, counteracted by forces from the spring, a viscous damper, and the mass's acceleration. This interplay of forces is mathematically...
879
One-Degree-of-Freedom System01:24

One-Degree-of-Freedom System

997
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...
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Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
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Damped Oscillations01:07

Damped Oscillations

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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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Constraints and Statical Determinacy01:26

Constraints and Statical Determinacy

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In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...
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Relation between Mathematical Equations and Block Diagrams01:20

Relation between Mathematical Equations and Block Diagrams

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In a spring-mass-damper system, the second-order differential equation describes the dynamic behavior of the system. When transformed into the Laplace domain under zero initial conditions, this equation can be effectively analyzed and manipulated. The transformation into the Laplace domain converts differential equations into algebraic equations, simplifying the process of isolating the output.
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Nondeterministic dynamics of a mechanical system.

Robert Szalai1, Mike R Jeffrey1

  • 1Engineering Mathematics, University of Bristol, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 13, 2014
PubMed
Summary

This study introduces a mechanical system with a nondeterministic singularity, where friction causes unpredictable chaotic dynamics. The system exhibits recurrent, uncertain behavior due to a local loss of determinism.

Area of Science:

  • Mechanical Engineering
  • Chaos Theory
  • Friction Dynamics

Background:

  • Deterministic systems typically have predictable future states.
  • Friction, particularly Coulomb friction, can introduce complex behaviors in mechanical systems.

Purpose of the Study:

  • To present a mechanical system exhibiting a nondeterministic singularity.
  • To analyze the emergence of chaotic dynamics from this singularity.

Main Methods:

  • Modeling a wheel on a turntable with Coulomb friction.
  • Analyzing system dynamics near points of non-unique friction force determination.

Main Results:

  • Identified a nondeterministic singularity where forward time trajectories become nonunique.

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  • Demonstrated recurrent, unpredictable behavior (nondeterministic chaotic dynamics) due to repeated transitions through the singularity.
  • Observed extreme sensitivity to initial conditions.

    • Conclusions:

    • A local loss of determinism in friction can lead to complex global dynamics and chaos.
    • The phenomenon is robust and expected to persist with advanced friction models.