Characteristic modes of oscillation of three masses on a hoop

[yeahhyeahyeah]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

so I solved for the eigenvalues like this:

x -2 1 1
y . $$\lambda$$ = 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 =)

 

[Jhero]

It looks like you are on the right track! First, let's clarify a few things. The equations you have written down are indeed for angular displacements, and the eigenvalues you calculated are correct. However, the eigenvalues are actually -3k/m and 0, which have units of inverse time squared.

To find the characteristic frequencies of oscillation, we can use the formula w = sqrt(k/m), where w is the angular frequency and k/m is the spring constant divided by the mass. In this case, we have three identical springs, so the spring constant is just k. Therefore, the two characteristic frequencies are w = sqrt(3k/m) and w = 0.

As for the oscillation patterns, we can use the eigenvalues to find the corresponding eigenvectors, which will tell us the direction and magnitude of the oscillation for each mode. For the mode with eigenvalue -3k/m, the corresponding eigenvector is (1,1,1), meaning that all three beads will oscillate in unison with equal amplitude and in the same direction. This is known as the symmetric mode.

For the mode with eigenvalue 0, the corresponding eigenvector is (1,-1,0), meaning that the first two beads will oscillate in opposite directions with equal amplitude, while the third bead will remain stationary. This is known as the anti-symmetric mode.

These two modes are indeed orthogonal, as you can see by taking the dot product of their respective eigenvectors. The dot product of (1,1,1) and (1,-1,0) is 0, indicating orthogonality.

I hope this helps! Let me know if you have any further questions.