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Simple Harmonic Oscillator

  1. Jan 24, 2012 #1
    1. The problem statement, all variables and given/known data


    Show that the underdamped oscillator solution can be expressed as x(t)=x[itex]_{0}[/itex]e[itex]^{-γt}[/itex][cos(Ω't+((v[itex]_{o}[/itex]+γx[itex]_{o}[/itex])/(x[itex]_{o}[/itex]Ω')sinΩ't] and demonstrate by direct calculation that x(0)=x[itex]_{o}[/itex] and [itex]\dot{x}[/itex](0)=v[itex]_{o}[/itex]
    2. Relevant equations

    The underdamped oscillator solution is
    x(t)=ae[itex]^{-γt}[/itex]cos(Ω't+[itex]\alpha[/itex])

    3. The attempt at a solution
    This problem completely overwhelms me so my solution may be a little lacking...
    I took the general form
    Acos(ω[itex]_{o}[/itex]t)+Bsin(ω[itex]_{o}[/itex]t)
    Where
    A=acos([itex]\alpha[/itex]) and B=-asin([itex]\alpha[/itex])
    Which according to what I read in the book should yield
    x(t)=a[cos(ω[itex]_{o}[/itex]t+[itex]\alpha[/itex])]
    So im thinking that the equation ae[itex]^{-γt}[/itex]cos(Ω't+[itex]\alpha[/itex]) can be transformed into a more useful form using the same method

    and that is sadly as close as I could get

    Any input would be appreciated. Thanks.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jan 24, 2012 #2
    You wrote,

    x(t)=x0e−γt[cos(Ω't+((vo+γxo)/(xoΩ')sinΩ't]

    I think you are missing some ")" somewhere?
     
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