# Damping: Three Basic Questions

by mudbone
Tags: basic, damping
 P: 1 Hello All, I have some basic damping related questions that I am hoping you all can shed some light on or suggest some resources on to someone who has taken a few undergraduate level physics courses. i.) Can someone give me an intuitive explanation as to why linear damping (underdamped case) of a harmonic oscillator shortens the period? ii.) Must an oscillatory system with any kind of damping (including nonlinear damping) have a shorter period/higher frequency than the undamped system? iii.) Why does damping cause a broader peak in the power spectrum of the oscillatory time series (please try to give me an intuitive reason not a purely mathematical one)? Thank you in advance -Mudbone
 P: 754 Increasing the damping lengthens the period (lowers the frequency). The effect is very small unless the damping is very heavy; for example even if the damping halves the oscillation amplitude every cycle, the period is lengthened by less than 1 per cent. Why does damping lengthen the period? Well, a hand-waving explanation would be along the lines that damping impedes the motion... Your third question is perhaps the easiest to answer. Far from resonance the amplitude is low, so the speed of motion is low, so the damping forces are smaller. Thus extra damping has less effect on the amplitude far from resonance than it does nearer resonance, when the body is (on average) moving faster. So the resonance curve is depressed more in the middle than at the edges, that is, it is broadened.
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 Quote by Philip Wood Why does damping lengthen the period? Well, a hand-waving explanation would be along the lines that damping impedes the motion...
Sightly less hand-waving: Suppose you have an undamped oscillator that exchanges potential energy and kinetic energy in each cycle of vibration.

If you add some damping (of any type), you take energy out of the system. So if you start with the same amount of PE as the undamped system, by the time the PE has reduced to 0 the amount of KE in the system is smaller. So the velocity is smalller. So the system takes longer to get to the position where the PE = 0. So the frequency is lower.

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## Damping: Three Basic Questions

 Quote by mudbone iii.) Why does damping cause a broader peak in the power spectrum of the oscillatory time series (please try to give me an intuitive reason not a purely mathematical one)?
A sharp, narrow peak in the frequency spectrum corresponds to a single sine or (cosine) wave. Still has some width due to the fact that the signal is finite in time.
The damped oscillation is not a pure sine but is exponentially decreasing. How can you get that? One way would be to add many sine waves with various frequencies and amplitudes. These are represented by the wide peak in the frequency spectrum.
 P: 754 AlephZero. Got it! You're considering a quarter cycle at a time. Nice! [I was going to object that although the mean speed is less, so too is the amplitude, but if we consider each quarter of a cycle at a time, the objection disappears.]