Forced Damped Oscillator frequency independent quantaties

[mbigras]

Homework Statement

For the forced damped oscillator, show that the following are frequency independent.
a) displacement amplitude at low frequencies.
b) the velocity amplitude at velocity resonance.
c) the acceleration amplitude at very high frequencies

Homework Equations

$$A(\omega) = \frac {F_{0}} {m} \frac {1} {\left(\left(\omega_{0}^{2}-\omega^{2}\right)^{2}+\left(\frac{\omega_{0}}{Q} \omega\right)^{2}\right)^{1/2}} \\ x = A \cos(\omega t + \alpha)\\ \\ Q = \omega_{0}/\gamma\\ \omega_{0} = \sqrt{\frac{k}{m}}$$

The Attempt at a Solution

To find the velocity and acceleration we can take the derivative and second derivatve of $x$. The amplitudes are then (for displacement, velocity and acceleration):
$$A(\omega)\\ -\omega A(\omega)\\ -\omega^{2} A(\omega)$$
Then the question asks me to show that the equations are frequency indepedent. What I'm most curious about is how to approach these questions. Every time I get one of these questions I experience a deer in the head lights effect. My general guess is that the structure of the question is:

Given an equation, a statement that you can use to make an approximation and a statement that you can use to find a point of interest:
Expand the equation about the point of interest, make an approximation and you'll see that the equation actually behaves in some useful way, or at least a way that makes a calculation easier.

Is this on the right track?

So part a, would I expand about 0, and use the approximation that $\omega << \omega_{0}$? For part b, would I expand about $\omega_{0}?$

What does it mean to be frequency independent? Does that mean the frequency term will drop out?

This question has me confused, obviously, but what I think is more important is that it's the whole process that confuses me. I'd like some tips for how to approach these types of questions. Specifically, when to expand, what type of expansions to use, how to choose where your expanding about, and how to understand what your looking for.

 

[rude man]

You have the right idea in computig A, wA and w^2 A for displacement, velocity and acceleration respectively. But then you over-complicate the problem.

a) What is A(w) if w goes to zero? Use your formula you derived.
b) wht is wA when w approaches w0?
c) what is w^2 A when w approaches infinity?