Chapter 10

Visualizing resonance.

Friction effects on resonance.

Here we consider the following second-order IVP

\begin{align} y'' + 2\zeta y' + y = \operatorname{cos}(\omega t),\ y(0) = y'(0) = 0, \end{align}

where $\zeta \ge 0$ and $\omega \ge 0$. This is the driven damped harmonic oscillator discussed in this chapter, with a periodic driving force $f(t) = \operatorname{cos}(\omega t)$.

To play with the visulization yourself, change the damping ratio $\zeta$ or the frequency of forcing $\omega$, and see how the graph changes.

Questions to consider:

How does the undamped case ($\zeta =0$) behave? Does the graph look bounded at resonance $\omega = 1$?
For the damped case, does the graph look bounded for aribitrary $\omega$?
For $\omega = 1$, what happens to the plots as $\zeta$ increases to $1$? (critically damped). What happens beyond this?

Damping Ratio

$\zeta$:

Undamped: $\zeta =0$;

Underdamped: $0< \zeta < 1$;

Critically damped: $ \zeta = 1$;

Overdamped: $\zeta > 1$.

Frequency of Forcing

$\omega$:

Period: $T = \dfrac{2\pi}{\omega} = $

Graph