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Particular Solution of A Coupled and Driven Oscillator

  1. Oct 24, 2016 #1
    1. The problem statement, all variables and given/known data

    Consider two masses m connected to each other and two walls by three springs with spring constant k. The left mass is subject to a driving force ## F_d\cos(2 \omega t) ## and the right to ## 2F_d\cos(2 \omega t) ##

    2. Relevant equations

    Writing out the coupled equations:

    $$ m_1 x_1''+2kx_1-kx_1 = F_d \cos (2\omega t) $$
    $$ m_2 x_2''-kx_1+2kx_2 = 2F_d\cos (2 \omega t) $$

    3. The attempt at a solution


    Assume a solution
    $$ x_1 = A_1 \cos (\omega t) \rightarrow x_1'' = -A_1 \omega^2 \cos (\omega t) $$
    $$ x_2 = A_2 \cos (\omega t) \rightarrow x_2'' = -A_2 \omega^2 \cos (\omega t) $$

    Sub this into our original equation and write it as a matrix equation

    $$ -\omega^2 \cos (\omega t) \left( \begin{array}{c} A_1 \\ A_2 \end{array} \right) + \omega^2 \left( \begin{array}{cc} 2 & -1 \\ -1 & 2 \end{array} \right) \left( \begin{array}{c} x_1 \\ x_2 \end{array} \right) = \frac{F_d}{m} \cos (2 \omega t) \left( \begin{array}{c} 1 \\ 2 \end{array} \right) $$

    where ##\omega^2 = k/m ##. This is where I get stuck, before we would form an eignevalue problem and solve the characteristic polynomial to get the eigenvalues then the constants ##A_1## and ##A_2## however we have too many terms to do this.

    A suggestion on where to go from here would be greatly appreciated!
     
    Last edited: Oct 24, 2016
  2. jcsd
  3. Oct 24, 2016 #2

    TSny

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    The frequency ω in the driving forces is not necessarily equal to ##\sqrt{k/m}##. You should use another notation, such as ##\omega_0^2 = k/m##.

    Since the driving forces have frequency ##2 \omega##, you might be better off with a trial solution where ##x_1## and ##x_2## vary with frequency ##2 \omega##.
     
  4. Oct 24, 2016 #3
    Thanks for the reply! That was my bad actually in copying the question, it does specify in the problem that the frequency of the driving for is equal to ##\sqrt{k/m}##. I believe I managed to get the solution after a couple hours of work by adding the two equations and subtracting the two equations and making a substitution of variables:

    Adding the Two
    $$(x_1''+x_2'')+\omega^2(x_1+x_2)=\frac{3F_d}{m} \cos (2 \omega t) $$
    Set ##z=x_1+x_2 \rightarrow z''=x_1''+x_2''##
    Gives
    $$z''+\omega^2 z = \frac{3F_d}{m} \cos (2 \omega t)$$

    Similar argument for subtracting the two. Is it possible to solve this question using a matrix equation? I don't see any simple way of doing so but may not be seeing it.

    Otherwise thanks for the help!
     
  5. Oct 25, 2016 #4

    TSny

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    Yes. Proceed as in your first post, but assume ##x_1## and ##x_2## vary as ##\cos2\omega t##.
     
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