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Homework Statement
I have to determine the frequencies of the normal modes of oscillation for the system I've uploaded.
Homework Equations
[/B]
I determined the following differential equations for the coupled system:
[tex]\ddot{x_A}+2(\omega_0^2+\tilde{\omega_0}^2)x_A\omega_0^2x_B = 0[/tex]
[tex]\ddot{x_B}+2\omega_0^2x_B\omega_0^2x_A = 0[/tex]
which I confirmed are correct.
The Attempt at a Solution
I assumed that the usual normal modes solutions are:
[tex]x_A=Ccos(\omega t)[/tex]
[tex]x_B=C'cos(\omega t)[/tex]
where C and C' are the amplitudes for each of the oscillating masses and [tex]\omega[/tex] is the associated normal mode frequency. Therefore, I proceeded by determining the ratio [tex]\frac{C}{C'}[/tex] for each equation, after substituting the solutions in the differential equations.
[tex]\frac{C}{C'}=\frac{\omega_0^2}{\omega^2 +2\omega_0^2+2\tilde{\omega_0}^2}[/tex] and [tex]\frac{C}{C'}=\frac{\omega^2+2\omega_0^2}{\omega_0^2}[/tex]
My problem now is that, when I try to determine the solutions for [tex]\omega[/tex], I arrive to 4 solutions of the form:
[tex]\omega=\pm \sqrt{\frac{(4\omega_0^2 2\tilde{\omega_0})\pm \sqrt{(4\omega_0^2 2\tilde{\omega_0})^2 4(4\omega_0^2 2\tilde{\omega_0})(3\omega_0^3 +4\tilde{\omega_0}\omega_0)}}{2}}[/tex]
which I think it's wrong, since with these solutions I don't even know how to schematically draw the normal modes with these kind of solutions. So is my solving method wrong or is this method only applicable to symmetric systems? Did I get the calculations wrong? Is there another way to solve this problem?
Thank you in advance!
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