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Characteristic modes of oscillation of three masses on a hoop

  1. Nov 12, 2009 #1
    1. The problem statement, all variables and given/known data


    Three beads of equal mass m are constrained to lie on a circular hoop of radius a = 1. The beads are connected by identical springs of spring constant k. The equations of motion for displacements are (I am going to use x, y, and z, where x = theta 1, y = theta 2, z = theta 3, because I suck at inserting greek symbols and respective subscripts. But just so you know, these are, I think, angular displacements??)

    m(dx/dt) = k(y+z-2x)
    m(dy/dt) = k(z +x-2y)
    m(dz/dt) = k(x+y-2z)

    Assume there are oscillatory solutions x(t) = xe^(iwt) (the real part of that) and so on. What are the characteristic frequencies of oscillation? What are the oscillation patterns? Because of the high degree of symmetry of the problem, you may be able to guess some of the characteristic modes, but show that you can choose modes that are orthogonal.


    2. Relevant equations



    3. The attempt at a solution

    so I solved for the eigenvalues like this:

    x -2 1 1
    y . [tex]\lambda[/tex] = 1 -2 1
    z 1 1 -2

    I got the Eigenvalues of -3 and 0

    I think these are supposed to be orthogonal... since the matrix is symmetric

    Since I am given that x = e^(iwt) is a solution...
    i tried

    (1/k)(iw)e^(iwt) = -3

    but... i don't really know how to isolate the W frequency. Also I have no idea if I am on the right track, am I/


    Thanks in advance for your time =)
     
  2. jcsd
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