Harmonic Oscilator (driven-and-damped) problem

• Redsummers
In summary, the author tries to solve an equation of motion for a critically damped oscillator, but does not seem to have success using Fourier series. They eventually figure out the equation using an alternate method.
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.

Last edited:
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.

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.

1. What is a harmonic oscillator (driven-and-damped) problem?

A harmonic oscillator (driven-and-damped) problem is a mathematical model that describes the motion of a particle under the influence of both a driving force and a damping force. This problem is commonly used in physics and engineering to study the behavior of systems such as springs, pendulums, and electrical circuits.

2. What are the components of a harmonic oscillator (driven-and-damped) problem?

The components of a harmonic oscillator (driven-and-damped) problem include a mass or particle, a spring or elastic element, a damping force, and a driving force. These components work together to create a system that oscillates or vibrates at a specific frequency.

3. How does a driving force affect the motion of a harmonic oscillator?

A driving force is an external force applied to a harmonic oscillator that causes it to oscillate at a specific frequency. The amplitude and frequency of the driving force can greatly affect the motion of the oscillator, causing it to behave in different ways such as amplifying or damping the oscillations.

4. What is the role of damping in a harmonic oscillator (driven-and-damped) problem?

Damping refers to the dissipation of energy in a system, and in a harmonic oscillator (driven-and-damped) problem, it is the force that slows down the motion of the oscillator. Damping can be either internal or external and can greatly affect the amplitude and frequency of the oscillations.

5. How is a harmonic oscillator (driven-and-damped) problem solved?

A harmonic oscillator (driven-and-damped) problem is typically solved using differential equations. The specific equations used depend on the components of the system and the forces acting on it. These equations can be solved analytically or numerically to determine the motion of the oscillator over time.

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