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Impulse effects on an overdamped vibrating system derivation

  1. Feb 12, 2013 #1
    1. The problem statement, all variables and given/known data

    For a zero initial deflection and for a finite initial velocity, the time dependence of the vibration response of an overdamped system to an impulse is given by:

    (1) θ(t)=[I/(mω)√(ζ^2-1)][e^-(ζωt)][cosh(ω√(ζ^2-1))t]

    which for large values of time becomes:

    (2) θ(t)=[I/(2mω)√(ζ^2-1)][e^-(ζ+√(ζ^2-1))t]

    How could equation (2) be derived from equation (1)?


    2. Relevant equations

    I=FΔt
    θ(t)=[I/(mω)√(ζ^2-1)][e^-(ζωt)][cosh(ω√(ζ^2-1))t]
    θ(t)=[I/(2mω)√(ζ^2-1)][e^-(ζ+√(ζ^2-1))t]

    3. The attempt at a solution

    I am trying to derive equation (2) from equation (1).

    As t becomes large, [e^-(ζωt)] approaches 0.

    I checked the chapter on impulse response functions in my vibrations text book, but couldn't seem to find either of these equations.

    Also, I tried a few arbitrary values with a large values for time in each equation, but was coming up with completely different answers. Are these equations viable?

    Thanks!
     
  2. jcsd
  3. Feb 12, 2013 #2
    I'm too brain dead to be of much help at this point, but have you tried writing cosh in terms of exponential functions and seeing if anything resolves itself that way? http://en.wikipedia.org/wiki/Hyperbolic_cosine

    I haven't seen this problem before, but that'd be my first try, and the 2 in the denominator of those expressions for cosh seems like it might be a friendly sign.
     
  4. Feb 13, 2013 #3
    Thanks! I didn't even think of converting the cosh to an algebraic function. I think I got it now!
     
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