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X-t plot of underdamped motion

  1. Feb 4, 2012 #1
    Here is the x-t plot of underdamped motion:

    http://www.google.com/imgres?q=unde...2&tbnw=151&start=0&ndsp=20&ved=1t:429,r:2,s:0

    At the intersection of the red curve and the blue curve, the red curve has a slope of 0 but the blue curve has a nonzero slope.

    Therefore, does it mean that these two curves actually intersect at some other points very close to the intersections already show in the plot?
     
  2. jcsd
  3. Feb 4, 2012 #2

    AlephZero

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    Don't try to read too much into the plot. It's not very accurately drawn. You can see some "kinks" in the top blue curve at about t= 0.2 and t= 0.4 which shouldn't be there.

    Most likely it's a plot of [itex]e^{-at}[/itex] and [itex]e^{-at}\cos(bt)[/itex] (you can guess the values of a and b for yourself).

    EDIT: first comment was wrong!

    When [itex]\cos(bt) = 1[/itex], the slope of both curves is [itex]-a e^{-at}[/itex] so the two curves touch at one point.

    But the curves in your link are probably drawn by fitting a "smooth curve" through a set of points, and so they will not necessarily have the correct tangents and the two curves look as if they intersect twice.
     
    Last edited: Feb 4, 2012
  4. Feb 4, 2012 #3
    But mathematically there shouldn't be any reason that the equation exp(-at)=exp(-at)cos(bt) have 2 solutions at each peak.
     
  5. Feb 4, 2012 #4
    Or it might be the case that the red curve does not have a zero slope at the intersection. (Because exp(-at)cos(bt) and cos(bt) reach max/min at different t)
     
  6. Feb 4, 2012 #5

    AlephZero

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    Sorry, my first comments in post #2 were wrong - now fixed.
     
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