Damped Oscillations

�  Typically, when something is oscillating, there is an opposing force (friction or drag) acting on the oscillation and causing it to slow down and come to a stop.

�  If you were to swing a ball attached to the end of a string hanging on the ceiling, you would eventually see that it comes to a stop. This is called damping!

Equations

�  Displacement as a function of time: x(t) = Ae-bt/2mcos(ωDt+φ) where

�  The first half of the function for displacement is an equation describing how the amplitude changes (shrinks as a result of damping): A(t) = Ae-bt/2m

�  The cosine half of the equation describes the oscillation. b = drag coefficient with units of kg/s So, the bigger b is, the bigger the drag force is.

Frequency

�  Mass on a spring:

�  Simple pendulum:

�  Drag affects frequency, so it’s different in damped oscillations:

Natural Frequencies

Frequency of Damped Oscillations

�  This is the equation for frequency in damped oscillations:

�  Where ωo changes depending on whether you’re working with a simple pendulum or a mass on a spring. You use one of those natural frequencies.

Question

�  Given a system where a mass of 120 g is oscillating on a spring, the spring constant k 97 N/m and the drag coefficient b is 0.18 kg/s, how long will it take for the amplitude to decrease to 25% of its original amplitude?

120 g

b=0.18 kg/s k=97 N/m

Solution

�  This is the equation for the displacement of a damped oscillator: x(t) = Ae-bt/2mcos(ωDt+φ)

�  But since the question is only asking about the amplitude, we only need to use this part of it: A(t) = Ae-bt/2m

Solution (cont’d)

�  The question asks when the amplitude will be 25% of the original, so we go from this: A(t) = Ae-bt/2m to 0.25Ao= Aoe-bt/2m

�  Here you can see that you don’t actually need to know the initial amplitude because there’s a term on both sides, so they cancel out and you get this: 0.25=e-bt/2m

�  Since we’re solving for time, we need to bring the power down. We do so by logging both sides.

Solution (cont’d)

�  Logging both sides gives you this: ln0.25 = (-bt/2m)(lne)

�  The natural log of e is just 1 so you get: ln0.25 = -bt/2m

�  Now you can isolate t and solve for time:

�  ln0.25 gives you a negative number, which cancels with the negative in front of the negative 2 to give you a positive number overall. Kg cancel out to give you time in seconds. Notice that k wasn’t used at all.

Don’t forget to change from g

to kg!