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Underdamped Harmonic oscillator with applied force

  1. Jan 31, 2012 #1
    1. The problem statement, all variables and given/known data

    An underdamped harmonic oscillator with mass m, spring constant k, and damping resistance c is subject to an applied force F0cosωt.
    (a) [analytical] If, at t = 0, x = x0 and v = v0, what is x(t)?

    2. Relevant equations

    Ωinitial = √(k/m)

    3. The attempt at a solution

    Fnet = -kx - cv + F0cos(Ωt) = ma
    x(t) = xh(t) (homogeneous) + xi(t) (inhomogeneous) so we will only be left with the inhomogeneous part
    x(t) = Acos(Ωt - ∅)

    A = (F0/m) / ((√(Ωinitial^2 - Ω^2)^2 + 4gamma*Ω^2))

    ∅ = tan-1((2*gamma*Ω)/(Ωinitial^2 - Ω^2))

    I have all that but I am confused where x = x0 amd v = v0 come into play and how to plug it all into a x(t) equation
  2. jcsd
  3. Jan 31, 2012 #2


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    Gold Member

    Last edited: Jan 31, 2012
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