PHYS 210 $\cdot$ Lecture Notes $\quad$ Week 6

# 6.2 The damped harmonic oscillator

A mass $m$ on a spring of stiffness $k$, with a damping force proportional to its velocity, obeys
$$m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0 \eqno(6.1)$$
Substituting $x(t) = e^{rt}$ gives the characteristic equation $mr^2 + cr + k = 0$, whose roots are
$$r = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m} \eqno(6.2)$$
When $c^2 < 4mk$ the system is underdamped and oscillates with a decaying amplitude:
$$x(t) = Ae^{-\gamma t}\cos(\omega_d t + \varphi), \quad \gamma = \frac{c}{2m}, \quad \omega_d = \sqrt{\omega_0^2 - \gamma^2} \eqno(6.3)$$
where $\omega_0 = \sqrt{k/m}$ is the natural frequency. The energy of the oscillation decays as
$$E(t) = \frac{1}{2}kA^2e^{-2\gamma t} \eqno(6.4)$$
and the quality factor of the oscillator is
$$Q = \frac{\omega_0}{2\gamma} = \frac{\sqrt{mk}}{c} \eqno(6.5)$$

**Example.** For $m = 0.50\text{ kg}$, $k = 200\text{ N/m}$ and $c = 0.80\text{ kg/s}$:

| Quantity | Symbol | Value |
| :--- | :---: | :--- |
| Natural frequency | $\omega_0$ | $20.0\text{ rad/s}$ |
| Decay rate | $\gamma$ | $0.80\text{ s}^{-1}$ |
| Damped frequency | $\omega_d$ | $19.98\text{ rad/s}$ |
| Quality factor | $Q$ | $12.5$ |

<br>
<center>6-4</center>