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Harmonic oscillator analytic vs. numerical

  1. May 17, 2012 #1
    1. The problem statement, all variables and given/known data

    trying to write a program in C++ to calculate the solution of a damped harmonic oscillator and compare with the exact analytic solution. i am using the classic 4th order Runge-Kutta, which i'm fairly sure is programmed right.


    2. Relevant equations

    m[itex]\ddot{x}[/itex] + c[itex]\dot{x}[/itex] + kx

    t = 0, x = 4, v = 0

    i'm interested in the underdamped case

    3. The attempt at a solution

    a solution of the form

    exp[rt] exists

    therefore i get exp[rt](r^2m + rc + k) = 0
    since exp[rt] ≠ 0
    then can solve the quadratic to get a value for r

    r = (- c [itex]\pm[/itex] [itex]\sqrt{c^2 + 4*m*}[/itex])/2m

    now we obtain two possible solutions
    for simplicity i will say that [itex]\alpha[/itex] = -c/2m and ω = sqrt{c^2 + 4*m*}[/itex])/2m

    since, this case is underdamped ω is complex. so i rewrite as iω'. where ω' = 4*m - c^2.

    my full solution then becomes

    A*exp(-\alpha*t)cos(w't) + B*exp(-alpha*t)sin(ω't)

    when i set the damping term, c = 0. the two solutions match. as the ω' just reduces down to \sqrt{k/m} which is the frequency for an undamped harmonic oscillator. also when i plot a graph for a simple harmonic oscillator analytically vs. my damped numerical solution to peaks shift as expected as the damping term affects the frequency of oscillation.
     
  2. jcsd
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