Generate plots for free damped oscillators

damped_oscillator_plot.py

import matplotlib.pyplot as plt
from numpy import exp, cos, pi, linspace
from scipy.optimize import minimize

eta = 0.5
omega = 2 * pi


def damped_os(t):
    return exp(-eta * t) * cos(omega * t)

t = linspace(0, 4, num=2000)
x = damped_os(t)
x_u = exp(-eta * t)
x_d = -exp(-eta * t)

fig, axes = plt.subplots()
axes.plot(t, x, 'black')
axes.plot(t, x_u, 'red')
axes.plot(t, x_d, 'red')

axes.set_xlabel('t')
axes.set_ylabel('x/A')

# plt.savefig('freedampedfull.png')

# Zooming
axes.set_xlim(0.9, 1.1)
axes.set_ylim(0.55, 0.65)


def neg_damped_os(t):
    return -damped_os(t)

result = minimize(neg_damped_os, 1)

# maximum arrow
x_max, y_max = result.x, -result.fun
dx, dy = 0.02, 0.03
plt.arrow(x_max - dx, y_max - dy, dx, dy,
          width=0.0001, length_includes_head=True)
plt.annotate("max", xy=(x_max - dx - 0.01, y_max - dy - 0.005))

x_tang, y_tang = 1.0, damped_os(1.0)
dx, dy = 0.03, 0.01
plt.arrow(x_tang + dx, y_tang + dy, -dx, -dy,
          width=0.0001, length_includes_head=True)
plt.annotate("tangent", xy=(x_tang + dx + 0.005, y_tang + dy))

plt.savefig('freedampedzoom.png')