PHYS 210 · Lecture Notes

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

$$
m \frac{d^2 x}{dt^2} + c \frac{dx}{dt} + kx = 0 \quad \text{(6.1)}
$$

Substituting $x(t) = e^{rt}$ gives the characteristic equation $mr^2 + cr + k = 0$ , whose roots are

$$
r = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m} \quad \text{(6.2)}
$$

When $c^2 < 4mk$ the system is underdamped and oscillates with a decaying amplitude:

$$
x(t) = A e^{-\gamma t} \cos(\omega_d t + \varphi), \quad \gamma = \frac{c}{2m}, \quad \omega_d = \sqrt{\omega_0^2 - \gamma^2} \quad \text{(6.3)}
$$

where $\omega_0 = \sqrt{k/m}$ is the natural frequency. The energy of the oscillation decays as

$$
E(t) = \frac{1}{2} k A^2 e^{-2\gamma t} \quad \text{(6.4)}
$$

and the quality factor of the oscillator is

$$
Q = \frac{\omega_0}{2\gamma} = \frac{\sqrt{mk}}{c} \quad \text{(6.5)}
$$

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

<table>
  <thead>
    <tr>
      <th>Quantity</th>
      <th>Symbol</th>
      <th>Value</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Natural frequency</td>
      <td> $\omega_0$ </td>
      <td>20.0 rad/s</td>
    </tr>
    <tr>
      <td>Decay rate</td>
      <td> $\gamma$ </td>
      <td>0.80 s $^{-1}$ </td>
    </tr>
    <tr>
      <td>Damped frequency</td>
      <td> $\omega_d$ </td>
      <td>19.98 rad/s</td>
    </tr>
    <tr>
      <td>Quality factor</td>
      <td> $Q$ </td>
      <td>12.5</td>
    </tr>
  </tbody>
</table>

6-4