Taylor problem 5.36 (with Example 5.3 graphs)

last revised: 15-Jan-2019 by Dick Furnstahl [furnstahl.1@osu.edu]

This notebook generates the plots in Example 5.3 in Taylor. For problem 5.36, add an additional curve (do not replace the Example 5.3 curve) for the initial conditions in the problem.

%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

# Maybe this should all be in a class.  How would you do that?

def A_eval(omega0, f0, omega, beta):
    """Amplitude function A for damped, driven oscillator."""
    return np.sqrt(
             f0**2/((omega0**2 - omega**2)**2 + 4. * beta**2 * omega**2)
           )

def delta_eval(omega0, omega, beta):
    """Phase function delta for damped, driven oscillator."""
    return np.arctan(2.*beta*omega/(omega0**2 - omega**2))

def B_coeffs(x0, v0, omega0, omega, beta):
    """B1 and B2 from initial conditions and amplitude and phase functions.
        Formulas from Taylor (5.70).
    """
    A = A_eval(omega0, f0, omega, beta)
    delta = delta_eval(omega0, omega, beta)
    omega1 = np.sqrt(omega0**2 - beta**2)
    B1 = x0 - A * np.cos(delta)
    B2 = (v0 - omega*A*np.sin(delta) + beta*B1)/omega1
    return B1, B2

def damped_driven(t, omega0, f0, omega, beta, B1, B2):
    """Solution to dampled driven harmonic oscillator."""
    omega1 = np.sqrt(omega0**2 - beta**2)
    A = A_eval(omega0, f0, omega, beta)
    delta = delta_eval(omega0, omega, beta)
    return A * np.cos(omega*t - delta) + np.exp(-beta*t) * \
           (B1 * np.cos(omega1 * t)  +  B2 * np.sin(omega1 * t))


def driving_force(t, f0, omega):
    """External driving force as a function of time t."""
    return f0 * np.cos(omega*t)

omega = 2.*np.pi
omega0 = 5.*omega
beta = omega0/20.
f0 = 1000.

print(rf'A = {A_eval(omega0, f0, omega, beta):.5f}')
print(rf'delta = {delta_eval(omega0, omega, beta):.5f}')

t_pts = np.arange(0., 5., 0.01)

x0 = v0 = 0.
B1, B2 = B_coeffs(x0, v0, omega0, omega, beta)
#B1 = -1.05
#B2 = -0.0572
#print(B1, B2)
x_pts_1 = damped_driven(t_pts, omega0, f0, omega, beta, B1, B2)

### Add the initial conditions for 5.36 and find the B coefficients and
###  x_pts_2.
x0 = v0 = 0.  # change this
B1, B2 = B_coeffs(x0, v0, omega0, omega, beta)
#B1 = -1.05
#B2 = -0.0572
#print(B1, B2)
x_pts_2 = damped_driven(t_pts, omega0, f0, omega, beta, B1, B2)

fig = plt.figure(figsize=(6,6))

ax_drive = fig.add_subplot(2,1,1)  # driving force
ax_drive.plot(t_pts, driving_force(t_pts, f0, omega), color='black')
ax_drive.set_xlabel('t')
ax_drive.set_ylabel('driving force f(t)')

ax = fig.add_subplot(2,1,2)
ax.plot(t_pts, x_pts_1, color='blue', linestyle='dashed')
ax.set_xlabel('t')
ax.set_ylabel('x(t)')
ax.set_ylim(-2., 2.)
ax.axhline(0., color='black', alpha=0.3)  # light black line for reference
### add another curve to ax for the 5.36 conditions (make it red and solid)
ax.plot(t_pts, x_pts_2, color='red', linestyle='dashed')

fig.tight_layout()