Simple Harmonic Oscillator

Undamped

$\ddot{x}+{\omega }^{2}x=0$

Solutions

• $x(t)=C_1{e}^{i\omega t}+C_2{e}^{-i\omega t}$
• $x(t)=C_1\cos (\omega t)+C_2\sin (\omega t)$
• $x(t)=A\cos (\omega t+phi)$

Damped

Underdamped

Overdamped

Critically Damped