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Homework Help: Problems with Critical Damping and Underdamping

  1. Jan 21, 2013 #1
    1. The problem statement, all variables and given/known data
    The concept of damping is new to me and the problems I have seen have had different known values than I see in the equations I have. Here's two I am working on.

    1) An automobile suspension is critically damped, and its period of free oscillation with no
    damping is 1s. If the system is initially displaced to a distance x0 and released from rest, find the displacement at t = 1 s.

    2) Show that the ratio of two successive maxima in the displacement of an underdamped
    harmonic oscillator is constant.

    2. Relevant equations

    Critical: x(t) = A e^ (-γ t ) + B e^ (-γ t) where A and B are constants and γ= (2c)/m

    Underdamping: x(t) = A0 e ^ (-γt) cos (ω't + θ0)

    I honestly don't know if these are the equations I need to find the answers.

    3. The attempt at a solution

    1. γ=ω0
    = ω0 = √k/m
    I found this Using the equation for the frequency of a spring (without damping)
    ∴ γ = 2π
    If I plug this into the equation I have, I am still left with those constants.... so I must have the wrong approach.

    2. I can't think of where to begin on this one... it seems like the variables wouldn't be constant if I set up a ratio like x2/x1
  2. jcsd
  3. Jan 22, 2013 #2
    1. The equation you have for the solution of a critically damped system in incorrect. What is the correct equation?

    2. Just do it: set up the ratio for two successive maxima and see where that gets you.
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