This python script solves the equation of motion for a Pendulum with an Oscillating Support, as given by eq. (3.11) in Vibration and Stability - JJT.

This script requires the python packages: numpy, matplotlib and PyDSTool. The first two are common and usually already installed, whereas PyDSTool needs to be installed manually - more on that later.

There are several ways to solve a system of ODE's numerically in python. The easiest(and most MATLAB like) is to include scipy and use the function scipy.integrate.odeint. However - like using MATLAB's ode45 - this is quite slow. To demonstrate a way faster approach, we will use the library PyDSTool which is generally considered essential in modelling dynamic systems. This library is also able to do continuation and thus make bifurcation diagrams very efficiently.

To solve the EOMs, they need to be recasted into a system of first order equations. Equations(here: theta=..., v=...), parameters(here: q, OMEGA, omega0, beta) and initial conditions(here: q0, theta0) are set in DSargs.pars, DSargs.varspecs and DSargs.ics respectively.

PyDSTool gains it speedup by compiling the code containing the ODE's into pure c-code(and then machine code). A low level compiled language like c is generally way faster than high level interpreted language like python and MATLAB. For installing PyDSTool` we refer to the projects homepage, where installation guides can be found for Linux, Windows and OS X(mac).

Calculating the FFT needs a uniform time interval. This is achieved by setting precise=True in the sample call:

traj = ode.compute('polarization')  # integrate ODE
pts = traj.sample(dt=timestep, precise=True) 

When setting the PYTHONPATH variable, you must include the parent directory of PyDSTool, eg:

$PYTHONPATH=$PYTHONPATH:$HOME/src/python/lib:$HOME/src/python/lib/PyDSTool/:$HOME/src/PyDSTool/tests/

manual and this discussion

Remove the -m32 flags from the files PyDSTool/Generator/{Dopri_ODEsystem.py/Radau_ODEsystem.py}. These are around line 794 of Dopri and 935 of Radau. Remember to remove them from both extra_compile_args, extra_link_args.

It might also be necessary to install the following 32 bit libs:

sudo apt-get install ia32-libs libc6-i386 libc6-dev-i386 build-essential

The important parameters when calling the animation.FuncAnimation class are:

• interval=20 - 20 ms between frames. To get the "correct" interval(eg. the pendulum moves in real time), use: interval=t[-1]/len(t)*1000 - gives the correct time in ms

• blit=True - only redraw the pieces of the plot which have changed. Remember that func and init_func should return an iterable of drawables to clear.

• fps=50 - fps should match the interval, eg if there's a frame for each 20 ms then fps = 1000/20 = 40

Example

def init():
    line.set_data([], [])
    return line,

\# animation function. This is called sequentially
def animate(i):
    x = np.linspace(0, 2, 1000)
    y = np.sin(2 * np.pi * (x - 0.01 * i))
    line.set_data(x, y)
    return line,
anim = animation.FuncAnimation(fig1, animate, init_func=init,
frames=len(t)-25, interval=20, blit=True)

\# save the animation
anim.save('basic_animation.mp4', fps=50, extra_args=['-vcodec', 'libx264'])
plt.show()

Note the comma(,) ín the return statement! This means that the function returns a tupple which is needed by the animate class.

The extra_args ensure that the x264 codec is used, so that the video can be embedded in html5 (and gives better quality and smaller file). This requires ffmpeg or mencoder to be installed. In order to install ffmpeg + x264 codec use this PPA(simplest) or compile from source using this guide. It might be necessary to set the LDPATH to the directory of x264 - especially if there's multiple version installed: LD_LIBRARY_PATH=/path/to/my/compiled/x264/library before configuring ffmpeg.

If anim.save throws an error, try to run it with --verbose-debug to see what went wrong(missing encoder library probably). To see if x264 installed run:

ffmpeg -codecs | grep 264
EV    libx264         libx264 H.264 / AVC / ...  # output

In order to run a script with matplotlib on a machine without X11(headless), use another backend. To do this, have the following PyDSTool and general matplotlib stuff are imported:

import matplotlib
matplotlib.use('Agg')

Now matplotlib uses the Agg backend.

It is also possible to forward X by using thinlinc or ssh -X [email protected]. This can be set up in ~/.ssh/config using the option ForwardX11 yes.

On linuxsh it is possible to create virtual environments. add --no-site-packages if you want to isolate your environment from the main site packages directory (which you probably won't)

mkdir virtualenv
virtualenv ~/virtualenv/myEnv
source ~/virtualenv/myEnv/bin/activate

update the needed packages(remember to this on the platform you want to use the env in. Eg. HPC or APP nodes):

pip install --update numpy matplotlib

To exit the virtualenv just type deactivate.

Another way would be to create a clean virtualenv and link the packages that you need:

virtualenv --no-site-packages foo
source foo/bin/activate
ln -s /usr/lib/python2.7/dist-packages/PIL* $VIRTUAL_ENV/lib/python*/site-packages

Speedup compared to matlab:

I've had to do something similar to what you need, and I found the following example from the Gallery quite helpful: http://matplotlib.org/examples/pylab_examples/multicolored_line.html I think the second plot in particular is pretty close to what you want; however, you'll need to set the alpha values manually. This is what I've done for line collections, scatter plots, etc.

import numpy as np 
import matplotlib.pyplot as plt 

norm_data = np.random.rand(20) 
xs = np.random.rand(20) 

# Pick a colormap and generate the color array for your data 
cmap = plt.cm.spectral 
colors = cmap(norm_data) 
# Reset the alpha data using your desired values 
colors[:,3] = norm_data 

# Adding a colorbar is a bit of a pain here, need to use a mappable 
fig = plt.figure() 
plt.scatter(xs, norm_data, c=colors, s=55) 
mappable = plt.cm.ScalarMappable(cmap=cmap) 
mappable.set_array(norm_data) 
fig.colorbar(mappable) 
plt.show()