He has good news. he's excited to tell us about electric currents! In particular, how a radios use resonance to selectively listen to a particular frequency:
where $\omega$ is called the angular frequency and equals $\frac{\sqrt{ 4mk-b^2 }}{2m}$
and where $Ae^{-bt/2m}$ is called the transient part of the equation (goes to 0 as t->$\infty$).
For particular solution, we use method of undetermined coefficients #mouc
We guess: $y_p=A_{1}\cos(\gamma t)+A_{2}\sin(\gamma t)$ where $\gamma\ne \omega$ because if $\gamma=\omega$ and b=0 then we would have to multiply our guess by t.
where we define $\mu(\gamma)=\frac{1}{\sqrt{ (k-m\gamma^2)^2+b^2\gamma^2 }}$ called the gain factor
and the general solution is $y(t)=y_{h}(t)+y_{p}(t)$
See how $y_h$ goes to zero as time progresses but $y_p$ stays? $y_p$ is the steady state part of the solution. If you were to graph $y(t)$ you would see a "beating" effect due to the sum of the two sins that eventually decays off.
If we make the value in the denominator of the gain factor small, the amplitude goes to a very high value, higher than $F_{o}$! This is the equivalent of tuning a radio receiving circuit to resonate to a certain transmitted frequency, the amplitude grows to a great value in the receiver when excited with the right frequency. This is resonance! as b (friction) is decreased to zero we get stronger and stronger resonance.
Lets find the maximum of the amplitude (resonance point)
By plugging in $\gamma_{r}$ into $\mu()$ we can know the maximum amplitude at resonance: $$\mu_{max}(\gamma_{r})=\mu(\gamma_{r})=\frac{2m}{b\sqrt{ 4mk-b^2 }}$$
$y_{p}=A\cos(\gamma t)$ we can guess the particular solution is a constant times $\cos$. There will be no $\sin$ term on the LHS as there's no first derivative (aka no friction)
Hmm, It's not very easy to visualize a cos minus cos term.
Use trig identity to make equation easier to visualize: $2\sin(\alpha)\sin(\beta)=\cos\left( \frac{{\alpha-\beta}}{2} \right)-\cos\left( \frac{{\alpha+\beta}}{2} \right)$
This is an amplitude modulated signal! also can be seen as a "beating frequency".
To recap, taking a frictionless mass spring system which is initially at rest, and applying a driving frequency that is strictly different from the system's natural frequency, results in a displacement that follows an AM modulated signal. How cool is that!
>Sometimes I feel this effect when I'm sitting in the back of the bus, where the bus is stopped at a red light. I wonder if this modulated vibration pattern is attributed to this exact phenomenon.
This simplifies our approach to solving a mass spring system. We could do it without this rearrangement, but it's more complex as the RHS has a sum of two terms. Either way works though, pick what you like.
