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Two coupled harmonic oscillator, damping each other

  1. Nov 10, 2009 #1
    The problem is:
    Two damped harmonic oscillator are coupled. Both oscillators has same natural frequency [tex]\omega_0[/tex] and damping constant [tex]\beta[/tex].
    1st oscillator is damped by 2nd oscillator. Damping force is proportional to velocity of 2nd oscillator. And, vice versa, 2nd oscillator is damped by 1st oscillator, by a force proportional to velocity of 1st oscillator.
    Find the positions (of both oscillator) as a function of time.

    I started with this:

    [tex]\ddot{x_1} + \frac{\beta}{m}\dot{x_2} + \omega_0^2(x_1-x_2) = 0[/tex] !!! EDITED !!!
    [tex]\ddot{x_2} + \frac{\beta}{m}\dot{x_1} + \omega_0^2(x_2- x_1) = 0[/tex]

    Is that O.K. ? If answer is yes ... what is the next step ? I would really appreciate it if somebody could give me just a hint !
     
    Last edited: Nov 11, 2009
  2. jcsd
  3. Nov 10, 2009 #2
    Addition and subtraction of the two eqs is a standard practice. This doesn't seem to work here. Check the last term of your eqs - why do you have omega*(x2-x1) in both eqs ?
     
    Last edited: Nov 10, 2009
  4. Nov 11, 2009 #3
    Ouch ! Mea culpa ! Must be omega*(x1-x2) in first eq !!!
     
  5. Nov 11, 2009 #4
    Do I need to supstitute [tex]x_n[/tex], with standard eq for harmonic oscillator ( [tex]A_n\cos(\omega_n t+\phi_n)[/tex]), before addition and substraction ?
     
  6. Nov 11, 2009 #5
    I don't think so.

    Adding and subtracting will give you independent equations in two new variables, X1 = x1 + x2 and X2 = x1 - x2. Try to solve these now in X1 and X2.
     
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