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Damped Harmonic Oscillator

  1. Jun 29, 2011 #1
    1. The problem statement, all variables and given/known data

    The equation for a damped oscillator is d2x/dt2+2βdx/dt +ω02 x = 0. Let ω0=1.0 s−1 and β = 0.54 s−1. The initial values are x(0) = x0 and v(0)=0.
    Determine x(t)/x0 at t = 2π/ω0.


    2. Relevant equations

    the solution to equation is given by;

    x(t)=e-[itex]\betat[/itex](A1et[itex]\mu[/itex]+A2e-t[itex]\mu[/itex])

    where [itex]\mu[/itex]=[itex]\sqrt{\beta2-\omegao2}[/itex]

    3. The attempt at a solution

    A1=1/2(xo+(xo[itex]\beta[/itex])/[itex]\mu[/itex])
    A2=1/2(xo-(xo[itex]\beta[/itex])/[itex]\mu[/itex])

    The problem I am running into is that the parameter I defined as [itex]\mu[/itex] is imaginary for this case, which keeps throwing me off. My only guess is to ignore the term multiplied by A1 because it is not real, then use only the A2 term and its multiplier because of the -t in its exponent making -i =1. I do not know if this correct and also even the constants A1 and A2 have an i in them as wel.
     
  2. jcsd
  3. Jun 29, 2011 #2

    vela

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    The fact that μ is imaginary tells you you have an underdamped system.

    Apply the initial conditions to solve for A1 and A2. The coefficients will be complex values. Then use Euler's formula, [itex]e^{i\theta} = \cos \theta + i \sin \theta[/itex], to express the solution in terms of sines and cosines. You'll find everything works out so the i's cancel and x(t) is real.
     
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