Damped oscillator

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Underdamped, critically damped and overdamped motion of a mass on a spring.

The displacement obeys $\ddot{x} + 2\zeta\omega_0\dot{x} + \omega_0^2 x = 0$ . The damping ratio $\zeta$ decides between the three regimes. This example is a matplotlib figure, so the page shows a PNG, and the script prints the decay rate of the underdamped case.

Source code

import matplotlib.pyplot as plt
import numpy as np

omega0 = 2.0 * np.pi
t = np.linspace(0.0, 4.0, 800)


def displacement(zeta: float) -> np.ndarray:
    """x(t) for x(0) = 1, x'(0) = 0."""
    if zeta < 1.0:
        wd = omega0 * np.sqrt(1.0 - zeta**2)
        return np.exp(-zeta * omega0 * t) * (
            np.cos(wd * t) + zeta * omega0 / wd * np.sin(wd * t)
        )
    if zeta == 1.0:
        return np.exp(-omega0 * t) * (1.0 + omega0 * t)
    root = omega0 * np.sqrt(zeta**2 - 1.0)
    a, b = -zeta * omega0 + root, -zeta * omega0 - root
    return (b * np.exp(a * t) - a * np.exp(b * t)) / (b - a)


fig, ax = plt.subplots(figsize=(8, 4.5))
for zeta, label in [(0.1, "underdamped"), (1.0, "critical"), (2.0, "overdamped")]:
    ax.plot(t, displacement(zeta), lw=2, label=f"$\\zeta = {zeta}$ ({label})")
ax.axhline(0.0, color="0.7", lw=0.8)
ax.set(xlabel="t", ylabel="x(t)", title="Damped harmonic oscillator")
ax.legend(frameon=False)
fig.tight_layout()

print(f"Envelope decay rate of the underdamped case: {0.1 * omega0:.3f} per unit time")
plt.show()

Printed output

Envelope decay rate of the underdamped case: 0.628 per unit time