Harmonic Oscilator (driven-and-damped) problem

[Redsummers]

Homework Statement

An Oscillator with free oscillation period $$\tau$$ is critically damped and subjected to a periodic force with the 'saw-tooth' form:

$$F(t)=c(t-n\tau)$$ , $$(n-1/2)\tau<t<(n+1/2)\tau$$

for integer n with c constant. Find the ratios of the amplitudes of oscillations at the angular frequencies:

$$\frac{2\pi n}{\tau}$$

Homework Equations

Basically for the critically damped thing, we have that

$$\gamma = \omega_0$$.

the other formulas are just the ones used with oscillator under forces. (i.e. driven and damped oscillators.)

The Attempt at a Solution

Okay, so first I took the Fourier series (or at least tried to) of the force that is driving the oscillator:

$$F_n=1/\tau \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} c(t-n\tau)e^{(inwt)}dt$$

then, what I did was to plug the

$$\frac{2\pi n}{\tau}$$

into the omega above.

Then I got,

$$F_n= \frac{c\tau}{i2\pi n^2}$$

According to this, now my equation of motion (ordinary linear differential non-homogeneous eq.form) is:

$$mz'' +\lambda z' + kz = \frac{c\tau}{i2\pi n^2} e^{(i\omega t)}$$

where I use here z just because the complex force. Also, note that

$$\omega_0^2 = k/m$$

and,

$$\gamma=\lambda/2m$$

Trouble now is that I would need a general solution for that, and I don't really know if it's the good way to think, if I assume that the equation of motion will look like this:

$$x(t)= Ae^{(i\omega t)}$$

with A being,

$$A= \frac{F_n/m}{\sqrt((\omega_0^2-\omega^2)^2+(2\gamma\omega)^2)}$$

And if that is the case, is really the A what the question is asking for? Because they ask for the ratio of amplitudes, and that's just the amplitude.. Not quite sure.

I haven't tried plugging my F_n and other values because I don't think it will give me any substantial result, and I assume I must have done something wrong behind.

(I know that the answer should be:

$$a_n= c/m\omega^2 n(1+n^2).$$

according to Kibble.)

If you could spot what am I missing in my assumptions I would much appreciate.

P.S.: I always seem to have problems when previewing the post... as it seems now, none of my latex scripts are well written (according to the preview post), but I am pretty sure that they are... just that the preview is acting weirdly. So if there are flaws in my latex I will try to fix it asap.

 

[Redsummers]

Okay, I have figured something out, but it is without using Fourier series. I guess I should've considered going through the easier path first.

But anyway, I am still interested on solving this problem on Fourier Series, if anyone has got any ideas on what am I doing wrong or what should I do next, I would appreciate.

 

[Redsummers]

Now, trouble is to actually compute the ordinary linear differential non-homogeneous equation.

One guess would be

$$(-mn^2 \omega^2 + i\lambda n \omega + k)A_n=F_n$$

You may as well see above what is A_n. (Note that I am precisely looking for a_n.)

I also know now that the solution should somehow look like this:

$$z = z (t) = A_n e^{i n \omega t}$$

With

$$A_n = a_n e^{-i \theta_n \omega}$$.

Any thoughts??

P.S. Sorry for the double-triple post.