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Lagranguan / Coupled Oscillator

  1. Apr 21, 2009 #1
    1. The problem statement, all variables and given/known data

    WIthin the framework of an idealised model, let a square plate be a rigid object with side "w" and mass "M", whose corners are supported by massless springs, all with a spring constant "k". The string are confined so they stretch and compress vertically with upperturbed length L.

    Look for solutions proportional to exp(-iwt). Show that three modes of oscillation exist with angular frequencies of
    Untitled-Scanned-03.jpg

    2. Relevant equations



    3. The attempt at a solution
    Ok ive tried this several ways but my working attached is the one i belive most to correct, however it differs from the answer the most!

    Ive assumed 3 general coordinates, one for the COM (center of mass) translation vertically "z", and two for rotation about the x-z "theta", and y-z "thi", with both angles being the angular rotation of the plate with respect to the axis of the plate in its equilibrium position (x,y,z)equilibrium = (0,0,0)

    My working attached go through how i figure it out, however im stuck with determining the angular frequency due to the "Mg" which has no "z" component. Im unable to move any further from this point if any1 can give me a hint.

    page 1
    Untitled-Scanned-01.jpg

    page 2
    Untitled-Scanned-02.jpg


    2nd ive also worked through this, which is basically identical except i ignored the gravitational PE on the Center of mass, and also the left velocity of each spring. SInce i had no "Mg" to deal with i was able to continue to get 3 modes
    My Lagranguan in this case is
    Untitled-Scanned-06.jpg


    This results nicely. i get w1, but w2=w3= ((2*k)/M)1/2 which doesnt match!

    If any1 can point out what ive done wrong or anything i have overlooked or looked too deeply when forming my Lagranguan thanks heeps
     
  2. jcsd
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