1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Coupled Oscillator

  1. Jan 10, 2012 #1
    Hey,

    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
    η33

    η1=[itex]\frac{ζ_{1}+ζ_{5}}{\sqrt{2}}[/itex]

    η5=[itex]\frac{ζ_{1}-ζ_{5}}{\sqrt{2}}[/itex]

    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
    ...............0..-sqrt2..1

    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?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Coupled Oscillator
  1. Coupled oscillation (Replies: 0)

  2. Coupled Oscillator (Replies: 1)

  3. Coupled Oscillators (Replies: 1)

Loading...