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pendulum.py
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pendulum.py
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###############################################################################
#
# CSCI 4446 - Chaotic Dynamics
#
# File: pendulum.py
# Author: Ken Sheedlo
#
# Pendulum problems for Problem Set 4.
#
###############################################################################
from __future__ import division
import getopt
import matplotlib.pyplot
import numpy
import rungekutta
import sys
from utils import split_dict, ufunc
PHASE_POINTS = (
(3.0, 0.1),
(2.6, 0.1),
(0.1, 0.0),
(0.2, 0.0),
(-2*numpy.pi, 30.0),
(-2*numpy.pi, 22.0),
(2*numpy.pi, -30.0),
(2*numpy.pi, -22.0),
(numpy.pi/2, 0.0),
)
def pendulum(mass, length, damping, ampl=0.0, freq=0.0):
'''
Creates a pendulum function.
Parameters (all types are 64-bit floating point unless specified otherwise):
mass The mass of the simulated pendulum.
length The pendulum's length.
damping The damping coefficient.
ampl The drive amplitude (defaults to 0.0).
freq The drive frequency.
Returns: a callable pfunc(t, x) that returns the value of the derivative at
t, x.
'''
def _pendulum(t, xvec):
'''
Pendulum callable function.
'''
theta = xvec[0]
omega = xvec[1]
omega_dot = (ampl*numpy.cos(freq*t) - damping*length*omega -
mass*9.8*numpy.sin(theta)) / (mass*length)
return numpy.array([omega, omega_dot], dtype=numpy.float64)
return _pendulum
def mod2pi(theta):
modulus = int(numpy.floor(theta / (2*numpy.pi)))
return theta - (modulus*2*numpy.pi)
def render_plot(ts, xs, *args, **kwargs):
'''
Renders a plot to the screen or to a file.
'''
figure = matplotlib.pyplot.figure()
axes = figure.gca()
opts, plot_args = split_dict((
'xlabel', 'ylabel', 'mod2pi', 'title', 'file_prefix'
), kwargs)
title = opts.get('title', '$x(t)$')
file_prefix = opts.get('file_prefix')
modulo = opts.get('mod2pi', False)
axes.plot(ts, xs, *args, **plot_args)
axes.set_xlabel(opts.get('xlabel', 't'))
axes.set_ylabel(opts.get('ylabel', 'x'))
axes.set_title(title)
if modulo:
axes.set_xbound(0, 2*numpy.pi)
axes.set_xticks((
0,
numpy.pi/2,
numpy.pi,
3*numpy.pi/2,
2*numpy.pi
))
axes.set_xticklabels((
'0',
r'$\frac{\pi}{2}$',
r'$\pi$',
r'$\frac{3\pi}{2}$',
r'$2\pi$'
))
if file_prefix is None:
figure.show()
else:
figure.savefig('{0}.png'.format(file_prefix), dpi=220)
def make_phase_portrait(pfunc, *args, **kwargs):
'''
Constructs a phase portrait of the system.
'''
figure = matplotlib.pyplot.figure()
axes = figure.gca()
opts, plot_args = split_dict(('title', 'file_prefix'), kwargs)
file_prefix = opts.get('file_prefix')
title = opts.get('title')
for (theta, omega) in PHASE_POINTS:
_, xs = rungekutta.rk4(
pfunc,
0.0,
numpy.array([theta, omega], dtype=numpy.float64),
0.005,
2000
)
axes.plot(xs[0,:], xs[1,:], *args, **plot_args)
axes.set_xlabel(r'$\theta$')
axes.set_ylabel(r'$\omega$')
axes.set_xbound(-3*numpy.pi/2, 3*numpy.pi/2)
axes.set_xticks((
-3*numpy.pi/2,
-numpy.pi,
-numpy.pi/2,
0,
numpy.pi/2,
numpy.pi,
3*numpy.pi/2
))
axes.set_xticklabels((
r'$-\frac{3\pi}{2}$',
r'$-\pi$',
r'$-\frac{\pi}{2}$',
'0',
r'$\frac{\pi}{2}$',
r'$\pi$',
r'$\frac{3\pi}{2}$'
))
axes.set_title('Phase Portrait' if title is None else title)
if file_prefix is None:
figure.show()
else:
figure.savefig('{0}.png'.format(file_prefix), dpi=220)
def make_phase_portrait_mod2pi(pfunc, *args, **kwargs):
'''
Makes a phase portrait modulo 2*pi.
'''
figure = matplotlib.pyplot.figure()
axes = figure.gca()
opts, plot_args = split_dict(('title', 'file_prefix'), kwargs)
file_prefix = opts.get('file_prefix')
title = opts.get('title')
for (theta, omega) in PHASE_POINTS:
_, xs = rungekutta.rk4(
pfunc,
0.0,
numpy.array([theta, omega], dtype=numpy.float64),
0.005,
2000
)
xs[0,:] = numpy.array([
mod2pi(theta) for theta in xs[0,:]
], dtype=numpy.float64)
axes.plot(xs[0,:], xs[1,:], *args, **plot_args)
axes.set_xlabel(r'$\theta$')
axes.set_ylabel(r'$\omega$')
axes.set_xbound(0, 2*numpy.pi)
axes.set_xticks((
0,
numpy.pi/2,
numpy.pi,
3*numpy.pi/2,
2*numpy.pi
))
axes.set_xticklabels((
'0',
r'$\frac{\pi}{2}$',
r'$\pi$',
r'$\frac{3\pi}{2}$',
r'$2\pi$'
))
axes.set_title('Phase Portrait' if title is None else title)
if file_prefix is None:
figure.show()
else:
figure.savefig('{0}.png'.format(file_prefix), dpi=220)
def plot_pfunc(pfunc, *args, **kwargs):
'''
Convenience function for experimenting with pendulum functions.
'''
opts, plot_args = split_dict(('tstep', 'theta0', 'omega0', 'nsteps'), kwargs)
theta0 = opts.get('theta0', 3.0)
omega0 = opts.get('omega0', 0.1)
tstep = opts.get('tstep', 0.005)
nsteps = opts.get('nsteps', 2000)
_, xs = rungekutta.rk4(
pfunc,
0.0,
numpy.array([theta0, omega0], dtype=numpy.float64),
tstep,
nsteps
)
fix_domain = lambda x: mod2pi(x) if kwargs.get('mod2pi', False) else x
xs[0,:] = numpy.array([
fix_domain(theta) for theta in xs[0,:]
], dtype=numpy.float64)
render_plot(xs[0,:], xs[1,:], *args, **plot_args)
def main(argv=None):
if argv is None:
argv = sys.argv
file_prefix = None
try:
options, args = getopt.getopt(argv[1:], 'f:')
for opt, arg in options:
if opt == '-f':
file_prefix = arg
except getopt.GetoptError as err:
print str(err)
return 2
suffixed = lambda s, suf: None if s is None else '{0}{1}'.format(s, suf)
pfunc = pendulum(0.1, 0.1, 0)
ts, xs = rungekutta.rk4(
pfunc,
0.0,
numpy.array([3.0, 0.1], dtype=numpy.float64),
0.005,
2000
)
render_plot(xs[0,:],
xs[1,:],
title='Undriven, Undamped Pendulum',
file_prefix=suffixed(file_prefix, '_2a'),
xlabel=r'$\theta$',
ylabel=r'$\omega$'
)
ts1, xs1 = rungekutta.rk4(
pfunc,
0.0,
numpy.array([0.01, 0.0], dtype=numpy.float64),
0.005,
2000
)
render_plot(xs1[0,:],
xs1[1,:],
title='Undriven, Undamped Pendulum',
file_prefix=suffixed(file_prefix, '_2b'),
xlabel=r'$\theta$',
ylabel=r'$\omega$'
)
make_phase_portrait(
pfunc,
'b',
title=r'Phase Portrait ($m=0.1, l=0.1, \beta=0)$',
file_prefix=suffixed(file_prefix, '_3')
)
pfunc2 = pendulum(0.1, 0.1, 0.25)
make_phase_portrait(
pfunc2,
'b',
title=r'Phase Portrait ($m=0.1, l=0.1, \beta=0.25)$',
file_prefix=suffixed(file_prefix, '_4')
)
make_phase_portrait_mod2pi(
pfunc2,
'b.',
title=r'Phase Portrait ($m=0.1, l=0.1, \beta=0.25)$ Mod $2\pi$',
file_prefix=suffixed(file_prefix, '_5'),
markersize=0.6
)
return 0
if __name__ == "__main__":
sys.exit(main())