Two coupled harmonic oscillator, damping each other

[dperkovic]

The problem is:
Two damped harmonic oscillator are coupled. Both oscillators has same natural frequency \omega_0 and damping constant \beta.
1st oscillator is damped by 2nd oscillator. Damping force is proportional to velocity of 2nd oscillator. And, vice versa, 2nd oscillator is damped by 1st oscillator, by a force proportional to velocity of 1st oscillator.
Find the positions (of both oscillator) as a function of time.

I started with this:

\ddot{x_1} + \frac{\beta}{m}\dot{x_2} + \omega_0^2(x_1-x_2) = 0 ! EDITED !
\ddot{x_2} + \frac{\beta}{m}\dot{x_1} + \omega_0^2(x_2- x_1) = 0

Is that O.K. ? If answer is yes ... what is the next step ? I would really appreciate it if somebody could give me just a hint !

 

[|squeezed>]

Addition and subtraction of the two eqs is a standard practice. This doesn't seem to work here. Check the last term of your eqs - why do you have omega*(x2-x1) in both eqs ?

 

[dperkovic]

Check the last term of your eqs - why do you have omega*(x2-x1) in both eqs ?

Ouch ! Mea culpa ! Must be omega*(x1-x2) in first eq !

 

[dperkovic]

Addition and subtraction of the two eqs is a standard practice.

Do I need to supstitute x_n, with standard eq for harmonic oscillator ( A_n\cos(\omega_n t+\phi_n)), before addition and substraction ?

 

[mikeph]

I don't think so.

Adding and subtracting will give you independent equations in two new variables, X1 = x1 + x2 and X2 = x1 - x2. Try to solve these now in X1 and X2.