(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The first part of the problem is just finding the Lagrangian for a system with 2 d.o.f. and using small angle approximations to get the Lagrangian in canonical/quadratic form, not a problem. I am given numerical values for mass, spring constants, etc. and am told to find the normal modes of oscillation and the normal coordinates.

2. Relevant equations

3. The attempt at a solution

I find the eigenvalues (eigenfreq, [tex]\lambda = \omega^{2}[/tex]) by diagonalizing: |V-[tex]\lambda[/tex]T| = 0. (V and T are matrices, also called M and K, they're the matrix of coefficients of the velocities and coordinates from the canonical Lagrangian respectively)

I get 2 ugly numbers for my eigenfrequencies, approximately 0.3 and 15.7.

Now I sub the first one into V-[tex]\lambda[/tex]T * (a11, a21) = 0

to solve for the eigenvector (a11, a21) which are the amplitudes of oscillation of both of my 2 d.o.f.(1 and 2) for this normal mode (1).

Question: plugging them in gives me a system of 2 equations of 2 variables. I am told that necessarily, one of the equations is a multiple of the other, but it doesn't seem to be true (using approximate values might have to do with this but I'm not sure). Is this correct?

If I believe what I am told, I pick one of the equations (the end result depends on which one I pick!) and solve for a11 as a function of a21, then use the orthonormalization condition: (a11,a21)(transposed)*T*(a11,a21) = 1 which allows me to find one of the amplitudes, which I then use to go back and solve for the other.

I am also very worried about approximating everything, at this stage of the problem I've already approximated decimal points on at least 3 occasions.

Can someone PLEASE help clarify this to me? Am I on the right track?

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# (Small oscillations) Finding Normal modes procedure.

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