PHYS1004 - Chapter 1

The equation of motion for a damped harmonic oscillator is given by:

$$\frac{d^2 z}{dt^2} + 2\gamma \frac{dz}{dt} + \omega_0^2 z(t) = 0, \quad z(t) \in \mathbb{C}.$$

Here, $z(t)$ is a complex-valued function of time, $\gamma$ is the damping coefficient, and $\omega_0$ is the natural frequency.

The general solution to the equation of motion can be written as:

$$z(t) = C_1 e^{\omega_+ t} + C_2 e^{\omega_- t},$$

where the roots $\omega_\pm$ are given by:

$$\omega_\pm = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}.$$

The constants $C_1$ and $C_2$ are determined by the initial conditions.

In the underdamped case, the roots are complex:

$$\omega_\pm = -\gamma \pm i \sqrt{\omega_0^2 - \gamma^2}.$$

This leads to oscillatory motion:

$$z(t) = e^{-\gamma t} \left( A \cos(\omega t) + B \sin(\omega t) \right),$$

where $\omega = \sqrt{\omega_0^2 - \gamma^2}$.

In the overdamped case, the roots are real and distinct:

$$\omega_\pm = -\gamma \pm \sqrt{\gamma^2 - \omega_0^2}.$$

The motion is non-oscillatory and can be expressed as:

$$z(t) = C_1 e^{\omega_+ t} + C_2 e^{\omega_- t}.$$

In the critically damped case, the two roots coincide:

$$\omega_\pm = -\gamma.$$

The motion is described by:

$$z(t) = (C_1 + C_2 t) e^{-\gamma t}.$$

The total energy of the oscillator decreases exponentially in the underdamped case:

$$E(t) \propto e^{-2\gamma t}.$$