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.