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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
from scipy.integrate import odeint
from IPython.html.widgets import interact, fixed
The equations of motion for a simple pendulum of mass $m$, length $l$ are:
$$ \frac{d^2\theta}{dt^2} = \frac{-g}{\ell}\sin\theta $$When a damping and periodic driving force are added the resulting system has much richer and interesting dynamics:
$$ \frac{d^2\theta}{dt^2} = \frac{-g}{\ell}\sin\theta - a \omega - b \sin(\omega_0 t) $$In this equation:
When $a=0$ and $b=0$, the energy/mass is conserved:
$$E/m =g\ell(1-\cos(\theta)) + \frac{1}{2}\ell^2\omega^2$$Here are the basic parameters we are going to use for this exercise:
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g = 9.81 # m/s^2
l = 0.5 # length of pendulum, in meters
tmax = 50. # seconds
t = np.linspace(0, tmax, int(100*tmax))
Write a function derivs for usage with scipy.integrate.odeint that computes the derivatives for the damped, driven harmonic oscillator. The solution vector at each time will be $\vec{y}(t) = (\theta(t),\omega(t))$.
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def derivs(y, t, a, b, omega0):
"""Compute the derivatives of the damped, driven pendulum.
Parameters
----------
y : ndarray
The solution vector at the current time t[i]: [theta[i],omega[i]].
t : float
The current time t[i].
a, b, omega0: float
The parameters in the differential equation.
Returns
-------
dy : ndarray
The vector of derviatives at t[i]: [dtheta[i],domega[i]].
"""
dtheta = y[1]
domega = -g/l*np.sin(y[0])-a*y[1]-b*np.sin(omega0*t)
return [dtheta, domega]
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assert np.allclose(derivs(np.array([np.pi,1.0]), 0, 1.0, 1.0, 1.0), [1.,-1.])
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def energy(y):
"""Compute the energy for the state array y.
The state array y can have two forms:
1. It could be an ndim=1 array of np.array([theta,omega]) at a single time.
2. It could be an ndim=2 array where each row is the [theta,omega] at single
time.
Parameters
----------
y : ndarray, list, tuple
A solution vector
Returns
-------
E/m : float (ndim=1) or ndarray (ndim=2)
The energy per mass.
"""
try:
if y.shape[1]:
return g*l*(1-np.cos(y[:,0]))+.5*l**2*y[:,1]**2
except:
Epm = g*l*(1-np.cos(y[0]))+.5*l**2*y[1]**2
return Epm
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assert np.allclose(energy(np.array([np.pi,0])),g)
assert np.allclose(energy(np.ones((10,2))), np.ones(10)*energy(np.array([1,1])))
Use the above functions to integrate the simple pendulum for the case where it starts at rest pointing vertically upwards. In this case, it should remain at rest with constant energy.
atol and rtol arguments of odeint until $E/m$, $\theta(t)$ and $\omega(t)$ are constant.Anytime you have a differential equation with a a conserved quantity, it is critical to make sure the numerical solutions conserve that quantity as well. This also gives you an opportunity to find other bugs in your code. The default error tolerances (atol and rtol) used by odeint are not sufficiently small for this problem. Start by trying atol=1e-3, rtol=1e-2 and then decrease each by an order of magnitude until your solutions are stable.
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a=0
b=0
omega0=2
pend = odeint(derivs, [0,0], t, args=(a,b,omega0),atol=1e-3, rtol=1e-2)
f=plt.figure(figsize=(15,10))
ax = plt.subplot(311)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
plt.plot(t,pend[:,0]);
plt.title(r"$\theta$ vs. time")
plt.xlabel("Time")
plt.ylabel(r"$\theta$")
ax = plt.subplot(312)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
plt.plot(t,pend[:,1]);
plt.title(r"$\omega$ vs. time")
plt.xlabel("Time")
plt.ylabel(r"$\omega$")
ax = plt.subplot(313)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
plt.plot(t,energy(pend));
plt.title(r"Energy vs. time")
plt.xlabel("Time")
plt.ylabel("$Energy$")
plt.tight_layout()
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# YOUR CODE HERE
raise NotImplementedError()
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assert True # leave this to grade the two plots and their tuning of atol, rtol.
Write a plot_pendulum function that integrates the damped, driven pendulum differential equation for a particular set of parameters $[a,b,\omega_0]$.
atol and rtol even futher and make sure your solutions have converged.
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def plot_pendulum(a=0.0, b=0.0, omega0=0.0):
"""Integrate the damped, driven pendulum and make a phase plot of the solution."""
pend = odeint(derivs, [-np.pi+0.1,0], t, args=(a,b,omega0),atol=1e-11, rtol=1e-10)
f=plt.figure(figsize=(15,10))
ax = plt.subplot(111)
ax.spines['top'].set_visible(False)
ax.spines['right'].set_visible(False)
ax.yaxis.set_ticks_position('left')
ax.xaxis.set_ticks_position('bottom')
plt.plot(pend[:,0], pend[:,1]);
plt.xlim(-2*np.pi, 2*np.pi)
plt.ylim(-10, 10)
plt.title(r"$\theta$ vs. $\omega$")
plt.xlabel(r"$\omega$")
plt.ylabel(r"$\theta$")
Here is an example of the output of your plot_pendulum function that should show a decaying spiral.
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plot_pendulum(0.5, 0.0, 0.0)
Use interact to explore the plot_pendulum function with:
a: a float slider over the interval $[0.0,1.0]$ with steps of $0.1$.b: a float slider over the interval $[0.0,10.0]$ with steps of $0.1$.omega0: a float slider over the interval $[0.0,10.0]$ with steps of $0.1$.
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interact(plot_pendulum, a=(0.0,1.0, 0.1), b=(0.0,1.0,0.1), omega0=(0.0,10.0,0.1))
Use your interactive plot to explore the behavior of the damped, driven pendulum by varying the values of $a$, $b$ and $\omega_0$.
Describe the different classes of behaviors you observe below.
As we increase a with b and $\omega_0$ fixed, we see a spiral with increasing stability. Fixing a and varying b and $\omega_0$, we see that the stability of the spiral is lessened, but we never see an unstable spiral form. As well, circles and closed loops appear to for around the location of the spiral nodes, indicating a resonant frequancy of the pendulum.