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I'm currently revising for a vibrations and waves module that I am taking as part of my physics degree.

One of the our final questions involved finding equations for the displacements of the two masses in this system as a superposition of their normal modes: Two Coupled Pendulums

I found the equations of motion for each mass to be: $$ \ddot{x_a} + \gamma\dot{x_a} + x_a(\omega_0^2+\omega_s^2) = x_b\omega_s^2\\\ddot{x_b} + \gamma\dot{x_b} + x_b(\omega_0^2+\omega_s^2) = x_a\omega_s^2\\Here: \omega_0^2 = \frac{g}{l}~~\omega_s^2=\frac{k}{m}~~\gamma=\frac{b}{m}$$ Here I let $ q_1 = x_a+x_b~and~q_2 = x_a-x_b: $ $$\ddot{q_1} + \gamma\dot{q_1} + q_1\omega_0^2=0\\\ddot{q_2} + \gamma\dot{q_2} + q_2(\omega_0^2+2\omega_s^2)=0 $$ From here I can't see where to go. I did attempt substituting in a general solution such as $q_1 = C_1 \cos(\omega t)$ but I get a mixture of sines and cosines and I can't solve it for anything useful.

Any help would be great as this is the last topic that I need to learn! Thanks, Sean.

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The problem boils down to solving a linear system of (first order) differential equations. I suppose you have not had this topic in your math classes yet, so I will not delve into the intricacies. You did the transformation from x and y variables to qs, and now you seem to have what is essentially two uncoupled damped oscillators. I would imagine that you have had the general solution to these explained in your classes. See Wikipedia for the formulae in case you have not: linear differential equations and damping.

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  • $\begingroup$ Ah I didn't think of solving it like that, I've done that maths so I'll be able to solve it now. $\endgroup$
    – Vielbein
    Commented May 11, 2014 at 23:24
  • $\begingroup$ I just reattempted the question this morning by applying methods I'd use for linear second order differential equations and I got to the solution. Thanks again, I wouldn't have thought of using that method otherwise. $\endgroup$
    – Vielbein
    Commented May 12, 2014 at 11:14
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The oscillations are damped. You need to have a sine/cosine times a trig function with a non imaginary exponential. This should get you where you need to go.

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