# Determening the Period of Coupled Oscillators

1. ### omertech

13
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!

### Staff: Mentor

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

3. ### omertech

13
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?

### Staff: Mentor

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.

5. ### omertech

13
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

### Staff: Mentor

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

7. ### omertech

13
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?

8. ### 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.

9. ### omertech

13
mfb and nasu thanks alot of clarifying that