Critically Damped

Examples of Critically Damped in the following topics:

• Damped transient motion

  • The fluid will damp out the motion, more or less depending on whether it has the viscosity of water or honey.
  • This looks like the equation of a damped sinusoid.
  • First if $\frac{\gamma}{2\omega_0} < 1$ , corresponding to small damping, then the argument of the square root is positive and indeed we have a damped sinusoid.
  • In this case the motion is said to be "over-damped" since there is no oscillation.
  • The borderline case $\gamma = 2 \omega_0$ is called critical damping, in which case $x(t) = x_0 e^{-\frac{\gamma}{2} t}$ .

• Damped Harmonic Motion

  • A door shutting thanks to a critically damped spring would simply shut and stay closed.
  • Let the damping force be proportional to the mass' velocity by a proportionality constant, b, called the vicious damping coefficient.
  • This expression can be positive, negative, or equal to zero which will result in overdamping, underdamping, and critical damping, respectively.
  • $\gamma^2 > 4\omega_0^2$ is the Over Damped case.
  • $\gamma^2 < 4\omega_0^2$ is the Under Damped case.

• Exercises

  • For the characteristic frequency you estimated above, what is the minimum damping required to ensure that the mass does not oscillate if you pull it down and let it go.
  • Answer: $\gamma _ {\rm critical} = 2 \omega _0 = 2 \times 2 \pi \times 2 {\rm s^{-1}} \approx 25 {\rm s^{-1}}$ .
  • With this minimum (or "critical") damping, how long will it take for the mass to come to rest?
  • (a) What is the damping constant ( $\gamma$ ) for the circuit?

• Forced motion with damping

  • The causes of damping are extremely subtle.
  • Try extending a damping piston of the sort used on doors.
  • where $\gamma$ is a constant reflecting the strength of the damping.
  • The fact that we get the right answer in the end depends critically on the equations being linear.
  • \label{damping} }$

• Forced Vibrations and Resonance

  • At first the finger is held steady, and the ball bounces up and down with a small amount of damping.
  • In real life, most oscillators have damping present in the system.
  • It is interesting that the widths of the resonance curves shown in depend on damping: the less the damping, the narrower the resonance.
  • The more selective the radio is in discriminating between stations, the smaller its damping.
  • The narrowest response is also for the least amount of damping.

• Driven Oscillations and Resonance

  • Driven harmonic oscillators are damped oscillators further affected by an externally applied force.
  • If a frictional force (damping) proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator.
  • Driven harmonic oscillators are damped oscillators further affected by an externally applied force F(t).
  • \omega_0$, and the damping ratio $\!
  • Describe a driven harmonic oscillator as a type of damped oscillator

• Back EMF, Eddy Currents, and Magnetic Damping

  • Back EMF, eddy currents, and magnetic damping are all due to induced EMF and can be explained by Faraday's law of induction.
  • Eddy currents can produce significant drag, called magnetic damping, on the motion involved.
  • A common physics demonstration device for exploring eddy currents and magnetic damping.
  • (c) There is also no magnetic damping on a nonconducting bob, since the eddy currents are extremely small.
  • Explain the relationship between the motional electromotive force, eddy currents, and magnetic damping

• Applications of Second-Order Differential Equations

  • In real oscillators, friction (or damping) slows the motion of the system.
  • In many vibrating systems the frictional force $Ff$ can be modeled as being proportional to the velocity v of the object: $Ff = −cv$, where $c$ is called the viscous damping coefficient.
  • Driven harmonic oscillator: Driven harmonic oscillators are damped oscillators further affected by an externally applied force $F(t)$.
  • A solution of damped harmonic oscillator.
  • Curves in different colors show various responses depending on the damping ratio.

• The Revolutionary Army at Valley Forge

  • The snow that collected around the camp was too sparse to be melted into water, and the damp conditions that resulted allowed disease to fester and spread easily.
  • Undernourished, poorly clothed, and living in crowded, damp quarters, the army was ravaged by sickness.
  • Criticism of Washington’s leadership were at an all-time high in light of the harsh conditions experienced by the Continental Army during the winter of 1777-78.

• Power

  • The shock absorber is analogous to the resistance damping and limiting the amplitude of the oscillation.
  • The forced but damped motion of the wheel on the car spring is analogous to an RLC series AC circuit.
  • The shock absorber damps the motion and dissipates energy, analogous to the resistance in an RLC circuit.