What are the simple modes and frequencies for weakly coupled oscillators?

[Zanatto]

Homework Statement

Two simple pendulums os equal length L=1m are connected with spring with a spring constant K=0,05 Mg/L. The pendulums are started by realeasing one of them from a displaced position. The subsequent motion is characterized by an oscillatory energy exchange between the pendulums. What is te period of this transfer?

Homework Equations

The Attempt at a Solution

In this situation, as a pendulum is displaced and the another is static, I suppose that when the amplitude of pendulum 1 is a maximum, the amplitude of pendulum 2 is minimum, because the result is displaying "beat" frequency. That's correct?

 

[BvU]

Yes. Why ask ?

 

[Orodruin]

Yes. When the amplitude of one of the pendula is zero, the amplitude of the other is maximal. This follows directly from energy conservation.

 

[Zanatto]

Well, just to get rid of the doubt.

Them, starting from the equation of position, and I found that the maximum values of the amplitude for the pendulum 1 occurs when:

επ(t/T) = nπ ---> t/T = n/ε

The time between maxima is T/ε, inversely proportional to the coupling spring constant.

And for the pendulum 2 when:

επ(t/T) = (2n +1)* π/2 ---> t/T = (2n+1)/2ε

Them, I stuck in here. Can I relate that with period of transfer in the oscillatory energy exchange in some way?

 

[BvU]

Good. So what's this exercise ? In a chapter on differential equations, on Larangian mechanics, a lab instruction preparation perhaps ?

The motion described is asymmetric and one can expect the pendulum that's initially immobile to start swingning too.
There are simple modes possible where there is no transfer; can you guess which ? Wht are their frequencies ?