Three pendulums, two springs normal modes.

[wakko101]

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

 

[Jhero]

:

Hello W. =),

Thank you for your post and for providing the initial conditions for the system. To solve for the subsequent motion for x1, we can use the normal coordinates of the system. In a triple pendulum system, the normal coordinates are the amplitudes of each pendulum's motion.

To find the normal coordinates, we can use the principle of superposition. We can break down the motion of the system into two parts: the first normal mode (where all three pendulums move simultaneously) and the second normal mode (where the outer pendulums swing out of phase while the center one stays still).

For the first normal mode, the amplitudes of each pendulum's motion will be equal. So if we let the amplitude of each pendulum's motion be A, then the normal coordinates for this mode would be (A, A, A).

For the second normal mode, the amplitudes of the outer pendulums' motion will be equal, but opposite in direction. So if we let the amplitude of each outer pendulum's motion be B, then the normal coordinates for this mode would be (B, -B, 0).

Now, for the given initial conditions (x1, x2, x3) = (2a, a, 0), we can see that the amplitudes would be (2a, a, 0), which does not match either of the normal coordinates we found. This means that the initial conditions do not correspond to a pure normal mode of the system.

However, if we let A = 2a and B = a, then the normal coordinates for the first normal mode would be (2a, 2a, 2a) and for the second normal mode would be (a, -a, 0). This means that the given initial conditions correspond to a combination of the two normal modes.

To determine if the motion is periodic, we can use the concept of resonance. The system will be in resonance if the frequency of the driving force (in this case, the frequency of the initial conditions) matches one of the natural frequencies of the system.

Since the natural frequencies of the system are w^2 = g/l and w^2 = g/l + k/m, we can see that the initial conditions will be in resonance with the second normal mode (w^2 = g/l + k/m) if g/l + k/m = (2