Three pendulums, two springs normal modes.

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I have a coupled oscillator system that is three pendulums attached together by two springs. The first part of the problems asks to argue, using symmetry arguments, that there are two "obvious" normal modes: one with w^2=g/l and another with w^2=g/l + k/m. I understand that these two frequencies correspond to a) all three pendulums moving simultaneously and b) the two outer pendulums swinging out of phase while the centre one stays still.

The second part of the question asks: The system is started with the initial conditions (x1, x2, x3) = (2a, a, 0). From the results of part 1, solve for the subsequent motion for x1. Under what condition is the motion periodic?

It occurred to me that possibly this question has to do with the normal coordinates of the system. I understand how to derive them in a two pendulum system, but I can't figure out how I would do that in a triple system.

However, perhaps I'm looking at this the wrong way. Does anyone have any suggestions as to how I would go about solving this?

Any advice would be appreciated.

Cheers,
W. =)



The Attempt at a Solution

 

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