Over Damped

Examples of Over Damped in the following topics:

• Damped Harmonic Motion

  • Over time, the damped harmonic oscillator's motion will be reduced to a stop.
  • A door shutting thanks to an over damped spring would take far longer to close than it would normally.
  • Sometimes, these dampening forces are strong enough to return an object to equilibrium over time .
  • $\gamma^2 > 4\omega_0^2$ is the Over Damped case.
  • We see that for small damping, the amplitude of our motion slowly decreases over time.

• 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 $x(t) = x_0 e^{-\frac{\gamma}{2} t}$ .

• Forced Vibrations and Resonance

  • 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.
  • The damping decreased when support cables broke loose and started to slip over the towers, allowing increasingly greater amplitudes until the structure failed.

• Power

  • (an equation derived by taking a time average of power, P(t) = I(t)V(t), over a period.
  • The circuit is analogous to the wheel of a car driven over a corrugated road, as seen in .
  • 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.

• 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

• 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.
  • \label{damping} }$
  • Square of the amplitude factor for forced, damped motion near a resonance $\omega_0$.

• Simple harmonic oscillation

  • That means the frequency must go as one over the square root of the mass!
  • Thus the total force on the mass (spring + gravity, but no damping for now) is mg $- k(\Delta l + x)$.
  • So far we have not accounted for the damping of the spring.

• Period of a Mass on a Spring

  • If the shock absorbers in a car go bad, then the car will oscillate at the least provocation, such as when going over bumps in the road and after stopping.
  • (e) In the absence of damping (caused by frictional forces), the ruler reaches its original position.

• 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.
  • 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?