# SAGE : Undamped Forced Vibrations

## Case I: $\omega_0\ne\omega$: Beats

Suppose the solution to an undamped forced-vibration system leads to the solution $u(t)=4\sin(0.05t)\sin(0.9t)$. The plot of the soultion is:

Graphing the "envelope functions" $E(t)=\pm 4\sin(0.05t)$ along with the actual solution we get:

## Case II: $\omega_0=\omega$: Resonance

Solve the inititial value problem: $u''(t)+16u(t)=10\cos(4t)$ and $u(0)=0$ and $u'(0)=0$.

An example of resonance: Galloping Gertie, The Tacoma Narrows Bridge