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Coupled Oscillator

  1. Jan 10, 2012 #1

    I've been trying to solve this question from Goldstein's Classical Mechanics.

    The picture I have of the question is from a later edition and the hint was removed from the question, the hint was let



    What I have done is first let each particle be represented by a displacement x1...x5,

    Then wrote out T = 1/2m([itex]x^{2}_{1}+x^{2}_{3}+x^{2}_{5}[/itex]) + 1/2M([itex]x^{2}_{2}+x^{2}_{4}[/itex])

    and V = k/2 *( [itex] x_{i}-x_{j}-b[/itex] ) i = 2..5, j = 1..4 i≠j

    so V = k/2 *( [itex]x_{2}-x_{1}-b[/itex] ) + k/2(....) up to i = 5 j = 4

    then Since η = x - dx the system is at equilibrium when

    b = dx2 - dx1 = dx3 - dx2 = .... up to i = 5 j = 4

    then V = 1/2k (η2 - η1) + ... up to i = 5 j = 4

    Then I subbed in the hints it provided and also as one of the hints says treat the normal co-ords of 2 and 4 as symetric I let η22=-η4

    Some stuff cancelled and I ended up with

    V = k/2 *( [itex]ζ^{2}_{1}+ζ^{2}_{2}+ζ^{2}_{5}-2\sqrt{2}ζ_{2}ζ_{5}[/itex] )

    I turned it into a matrix which was (this is where I start stuffing up I think)

    ...............1.....0......0 (sorry had to use the ... to make the matrix look kind of like a matrix)
    V = k/2.....0.....4....sqrt2

    Then since there were only varibles of 1, 2 and 5 I turned T into

    ..............m 0 0
    T = 1/2...0 M 0
    ..............0 0 m

    Then did the usual thing for eigenvalues |V-ω2T|=0

    One was pretty ugly, one was sqrt(k/m), and the last one I had trouble finding because it was a mess of a cubic.

    I decided to put the question at the bottom so the add didn't squish it,

    http://img193.imageshack.us/img193/7747/asdasddh.jpg [Broken]

    Is what I did alright? The part I'm not confident about at all is when i turn V into a matrix, and the book they also drop the 1/2 for both V and T which I didn't really understand why.

    Thanks in advanced for any help.
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jan 11, 2012 #2
    So one problem was that symetric meant n2=n4, so i did it again,
    its a 5 variable quadratic form so its a 5x5 not a 3x3,
    I got a block diagnal matrix with a 3x3 as the first and a 2x2 as the second, the 2x2 had determinant kmw^2, so that shows one of the frequencies is 0,

    but yea im still not sure if thats right,

    Anyone have any ideas?
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