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Driven, damped harmonic oscillator - need help with particular solution

  1. Nov 6, 2005 #1
    Driven, damped harmonic oscillator -- need help with particular solution

    Consider a damped oscillator with Beta = w/4 driven by
    F=A1cos(wt)+A2cos(3wt). Find x(t).

    I know that x(t) is the solution to the system with the above drive force.

    I know that if an external driving force applied to the oscillator then the total force is described by F = -kx - bx' + F0cos(wt).

    But in our case the driving force is A1cos(wt)+A2cos(3wt) so
    F=-kx-bx+A1cos(wt)+A2cos(3wt).

    Then our differential equation is mx''+bx'+kx=A1cos(wt)+A2cos(3wt).

    This can also be written as x''+2Betax'+(w^2)x=A1cos(wt)+A2cos(3wt).

    For the complementary solution, we set the right side of the equation equal to zero and solve for x. This is o.k.

    However, I am having trouble with the particular solution. Can someone tell me how I find a particular solution for this? I can find the particular solution for x''+2Beta x'+ (w^2)x = A cos (wt), but what about the particular solution when the driving force is not A cos (wt), as we have in this case?

    Any help GREATLY appreciated!
     
  2. jcsd
  3. Nov 7, 2005 #2

    dextercioby

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    Homework Helper

    Use Lagrange's method of varying constants. That is assume that the particular solution to the nonhomogenous ODE is

    [tex] x_{part}(t)=C_{1}(t)\cos\omega t+C_{2}(t)\cos 3\omega t [/tex].

    Daniel.
     
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