Determening the Period of Coupled Oscillators

[omertech]

Hello everyone,
I was wondering how could you determine the period of the motion of two or more coupled oscillators. For example, two oscillators have the state variable equations:
$$x_1=A_1\cos{(\omega_1t+\phi_1)}+A_2\cos{(\omega_2t+\phi_2)}$$
$$x_2=A_1\cos{(\omega_1t+\phi_1)}-A_2\cos{(\omega_2t+\phi_2)}$$
Thanks!

 

[mfb]

It depends on your setup. Do you have equations of motion for your oscillators?
Can you transform them to get two independent equations? Solve those, and you get ω₁ and ω₂

 

[omertech]

I want to have a general solution, so let's assume ω₁ and ω₁ are known, as well as all of the other quantities. How can you determine the period in such general case?

 

[mfb]

ω₁ and ω₂ ARE the periods.
You can re-write the sum as product of two oscillations, but that won't give you a single frequency either. With weak coupling, you get the product of a quick oscillation and a slow one (half of the sum and the difference of ω₁ and ω₂ iirc), where the slow one can be interpreted as amplitude modulation of the quick one.

If you want a repetition period: This exists if ω₁ and ω₂ have a rational ratio, and corresponds to the least common multiple of them.

 

[omertech]

Thanks for the answer. As far as I know ω₁ and ω₂ are the angular frequencies. They are related to the periods T₁ and T₂ by:
$$T_1=\frac{2\pi}{\omega_1}$$
$$T_2=\frac{2\pi}{\omega_2}$$

What I am looking for is indeed the repetition period. I know about the common multiple thing, but isn't there any general solution for any oscillation? Because I know that there is a harmonic repetition in coupled oscillations, the question is in what period?

Thanks again

 

[mfb]

Oh sorry, least common multiple of the corresponding T_i, of course.
I think I answered this in my previous post, so I have no idea what to add.

 

[omertech]

Yes your answer is suitable if the ratio between them is indeed rational. But this is not always the case. If it's not rational than would it be the product of T₁ and T₂? Or perhaps there is a smaller answer?

 

[nasu]

Mathematically, if the ratio of the periods is not rational, there is no common period.
No matter how many periods of the first one you take, you can never fit an integer number of periods of the second one. The two periods are "incommensurable".

In practice, you may find some approximate period.
You measure the time and the period with some finite precision so the motion will repeat after some time, within experimental error (or will look like it's repeating).
If the two modes go through a maximum within 10 ns and you measure time with 1 μs, you cannot tell that they did not do it simultaneously.

 

[omertech]

mfb and nasu thanks a lot of clarifying that