Comments (damped harmonic motion)
Friday 26 February 2021
When someone leaves a comment on a page I always try to answer as quickly as possible, usually within a couple of hours. Sometimes you ask a difficult question that takes a bit more time, This week Yang Bing asked a difficult question.
"Hi Chris, for the equation of ,could you show me the derivation of that? I found it from your Book in topic of Q factor, damping harmonic motion."
Hmm, I have no idea but I will try to find out.
Google didn't help, all the maths was too difficult so I tried to work it out myself. It took some time but here it is.
From the definition :
The equation for the displacement of a body oscillating with damped harmonic motion is:
Sorry, there is a typo in the text book, I missed ot the 2 in -b/2m.
Energy is proportional to displacement2 so the energy at t = 0 is proportional to
If the time period is T the energy after one cycle is
But
So the energy after one cycle is
The energy lost is therefore
Substituting into the equation for Q
This is of the form
This is where my memory of A'level maths fails but I do remember something called a Laurent series which is used to expand functions and show the value they tend to when x is small. The Q value is only relevant for examples of light damping so this seems to be the way to go, unfortunately I don't remember how to do this but no problem Wolframalpha can do it for me.
Not sure what "series expansion at x = 0 means but let's go with it anyway.
This shows that for small values of x tends to 1/x.
Just to be sure we can plot the two functions in GeoGebra.
Looks good :-)
So