Forced Harmonic Oscillation and Resonance

I was going to continue on from the previous post all the way to resonance, but if I tacked this on too it’d get super long, so

this one’s going to be a pretty short post but… let’s just go ahead and do it!

The main point: the driving force’s frequency

has some relationship with the oscillator’s natural frequency

such that at certain driving frequencies, a peculiar phenomenon happens.

Let’s unveil the secret of that peculiar phenomenon.​

step1. Start from the no-damping case, step by step (no resistance, yes external force)

So the relationship between w and A is like this, right?????????????????

Then this phenomenon will also make sense!!!!

Alright, so now, like we always do, let’s think about the resistance term too! hehe

It’s just one term added, right

In the no-damping case, when sinΦ=0 the imaginary term vanished so it was simple,

but now it’s a bit diff-y… two equations come ouuut.

Hint!!!!!

this will be the maximum amplitude.

Alright, so as mentioned above 1!!! let’s find the maximum of A(w).

Heads-up. The figures here are the original notes, slips and all. A few things to watch for:

• The note about transient vs. steady state says $A$ and $B$; those are the $A'$ and $B'$ from the figure right before it.
• In the damped equation of motion, the third line writes $iw$ where it should be $icw$ (the $c$ went missing), and the last line shows $e^{-i\varphi}$ where $e^{+i\varphi}$ is what actually follows from the line above.
• In the resonance figure the red line says $dA/dt$. It should be $dA/dw$: we’re hunting for the driving frequency that makes the amplitude biggest.
• In the last figure, the subscript “ma” is just “max” cut short, and the second denominator should have $4\gamma^4$, not $4\gamma^2$. The final answer is fine.