May 7, 2015

Reh-Livebot

... als Ersatz für den ReLiveBot. Mitschnitte von xenim zum runterladen, inkl. Pre- und Postshow. Zum Reh-LiveBot.

April 17, 2015

Howto: Access/Convert HAMEG *.mes file from Linux

We have a Hameg HM5014-2 spectrum analyzer in our lab, which can be controlled via RS232 from a windows box.
For this purpose, the software "AS100E V305 spectrum analyzer software" from www.hameg.com is installed on a Win XP laptop and analyzer is accessed via a USB-to-RS232 dongle.

So far, so good. But the stupid thing is that you can not save/export your data in standard formats (e.g. *.dat, *.csv, *.txt), which can be opened with any text editor. Instead, you only can save a *.mes file.

This file turns out to be a Microsoft JET Database (http://en.wikipedia.org/wiki/Microsoft_Jet_Database_Engine), which is proprietary and also flagged as depricated by MS. To access such an database, you could use "MS Access", the database program from Microsoft (office package).

Fortunately, there is a way to dump the data using free software: MDB-TOOLS (https://github.com/brianb/mdbtools). You can compile it from github source, but it also should be available via the package managing source of your distro (apt-get, pacman, etc).

I have a test measurement called "hameg.mes".

To see which tables are stored inside this file, one can use the tool mdb-tables:

>mdb-tables hameg.mes            
 Measurement Limitlines Settings
Now one can dump one specific table using mdb-dump:

> mdb-export hameg.mes Measurement

Frequency,Sample_dBmW,Average_dBmW,Maxhold_dBmW,QPeak_dBmW,memory_dBmW,Ambient_H_dBmW,Ambient_V_dBmW,V_memory_dBmW,H_memory_dBmW,H_Average_dBmW,H_Maxhold_dBmW,H_QPeak_dBmW,V_Average_dBmW,V_Maxhold_dBmW,V_QPeak_dBmW
2.50000000e+01,-5.04000015e+01,0.00000000e+00,0.00000000e+00,0.00000000e+00,0.00000000e+00,,,,,,,,,,
2.51000004e+01,-5.04000015e+01,0.00000000e+00,0.00000000e+00,0.00000000e+00,0.00000000e+00,,,,,,,,,,
2.52000008e+01,-
....
As visible, the dump is in the CSV format which can be easily piped into a csv-file:

> mdb-export hameg.mes Measurement > hameg_measurement.csv


PS: the our version of the HM5014-2 creates databases in the  JET3-format.
>mdb-ver hameg.mes
JET3
I was able to view/dump/edit this version with mdbtools version 0.7.1 (clone from github)


June 7, 2014

Numpy / Scipy / Matplotlib - How did I do ... again?




I almost completely switched my computing/plotting tasks from octave to numpy / scipy  and matplotlib.

Here's a little list about things i constantly keep forgetting


Matplotlib

  • manually set the ticks on colorbars:

    from matplotlib import pyplot as plt
    from matplotlib.ticker import FixedLocator
    ...
    cbar = plt.colorbar()
    locs = [0,-10,-20,-30]
    cbar.locator = FixedLocator(locs)
    cbar.update_ticks()

  • define position and size of the colorbar (place it freely somewhere)


    fig = plt.figure(1)
    ...
    is1 = plt.imshow( .... )
    cbaxes = fig.add_axes([0.9, 0.05, 0.02, 0.3])
    cb = fig.colorbar(is1, cax = cbaxes)) 
    cb.ax.tick_params(labelsize=7)


    sources: http://python4mpia.github.io/plotting/advanced.html

  • set position of ticks and labels from left to right axis:

    ax1 = plt.subplot(111)
    ax1.yaxis.tick_right()
    ax1.yaxis.set_label_position("right")


  • reduce the size of the legend

    plt.rcParams['legend.fontsize']=8
    plt.rcParams['legend.borderpad']=0.3
    plt.rcParams['legend.handlelength']=0.1

  • get a second y-axis: ax1a = ax1.twinx() 
  • create a plot on a machine that has no X (running):
    •  
      import matplotlib
      
      # Force matplotlib to not use any Xwindows backend.
      
      matplotlib.use('Agg')

  • Legend location codes:
  • 3d plot minimal example with mplot3d
  • #from plottingtools import *
    from matplotlib import pyplot as plt
    %matplotlib inline
    plt.rcParams['figure.figsize']=(5,4)

    x = np.linspace(-1.5,1.5,501)
    [xx,yy]=np.meshgrid(x,x)
    # some function
    f = np.exp(- (xx**2+yy**2)/0.5**2) \
    + 0.4*np.exp(- ((xx-1)**2+(yy+0.2)**2)/0.5**2) \
    + 0.8*np.exp(- ((xx+1.2)**2+(yy-0.5)**2)/0.5**2)


    from mpl_toolkits.mplot3d import Axes3D
    fig = plt.figure()
    ax = Axes3D(fig)#fig.gca(projection='3d')
    surf = ax.plot_surface(xx,yy,f,
        #cmap=sunrise(),
        antialiased=True,
        linewidth=1,
        rstride=10,
        cstride=10,
        shade=False)
    ax.view_init(75,-80)

    fig2 = plt.figure()
    ax2 = Axes3D(fig2)
    ax2.plot_wireframe(xx,yy,f,
       rstride=10,
       cstride=10)
    ax2.view_init(75,-80)

Numpy

December 11, 2013

List of useful linux tools whose names i keep forgetting

  • arping - ping IP and get the MAC adress 
  • engauge (aka engauge-digitizer)    get data points from images of plots
  • gpicview - an image viewer 
  • pdfposter - a tool to scale pdfs and print (tiled) poster
  • ripit  most easy-peasy cd ripper and mp3 encoding script
  • sitecopy a tool to sync folders via ftp 
  • tr    - can be used to translate characters of a text file - or delete them 

December 1, 2013

Xenim streamrecorder.

Als Podcasthörer stellte sich mir folgendes first-world-problem: Es gibt live-Podcasts, die interessante Post-/Preshows haben, die nachher nicht im Podcast sind. Weiterhin dauert es mir manchmal zu lange bis zur Veröffentlichung.

Ich habe ein wenig mit Python rumgespielt: Herausgekommen ist ein Skript, welches automatisiert die API von xenim abfragt und gegebenfalls mit VLC einen Stream mitschneidet. Das Skript sollte auch mit Streamabbrüchen umgehen können (Wiederaufnahme).

Vielleicht kann jemand was damit anfangen.

https://github.com/xmhk/xstreamrecorder

Das Ding läuft unter Linux (bei mir auf dem Raspberry Pi), braucht Python 2.(6?), und das Pythonpaket 'requests' (Debian, Ubuntu: sudo apt-get install python-requests, Arch Linux: pacman -S python2-requests).

Mit dem Programm 'screen' kann das Skript im Hintergrund laufen.

Anpassung auf den Mac sollte relativ einfach machbar sein, Windows eher: mehhh.


November 2, 2013

Run Docketport 467 Scanner under Linux

The Problem


I always wanted to get rid of my flatbed scanner and bought a DocketPORT 467, a tiny sheet fed scanner for small money on ebay.
But it didn't work out of the box under Linux (Ubuntu 12.04).
 Using the sane software tools, i was able to see that the scanner at least is detected:

>sane-find-scanner
found USB scanner (vendor=0x1dcc [Document Capture Technologies Inc.], product=0x4812 [DocketPORT 467], chip=GL842?) at libusb:001:003
  # Your USB scanner was (probably) detected. It may or may not be supported by
  # SANE. Try scanimage -L and read the backend's manpage.


When i try

 scanimage -L

i get

 No scanners were identified. If you were expecting something different,
check that the scanner is plugged in, turned on and detected by the
sane-find-scanner tool (if appropriate). Please read the documentation
which came with this software (README, FAQ, manpages)


basically the same happens when i start xsane (a graphical scanning programm).

What is wrong: it is the permissions! When i tried

sudo  xsane

the scanner was found and worked fine.
Because the permission situation, one is not allowed to use all usb devices available. 

Solutions

1. lazy bum version 

just use "sudo xsane" to access the scanner. When you save the images, maybe you have to adjust the file user and group of scanned images afterwards.

2. make a udev rule

the permissions for usb devices can be adjusted writing rules for the udev (device daemon). You can place them in the folder /etc/udev/rules.d.

The file name must have the ending .rules and can be for example 90scanner.rules. It can be important that the number at the beginning is higher that the number of the other filenames, because the system reads the files according to the alphanumeric ordering.

Our file contains the following:


#usb scanner readable by me#
SUBSYSTEMS=="usb", ATTRS{idVendor}=="1dcc", ATTRS{idProduct}=="4812", OWNER="THISisYOU"


(you have to change the owner (here THISisYOU) to your own username). Hint: Everything from "Subsystem" up to the end has to be put in one single line.

Now i was able to access and scan without sudo.


UPDATE:

another variation of udev rules is described on the bottom of this page (paragraph "Permission problem"): https://wiki.archlinux.org/index.php/Scanner
 

August 27, 2012

How to get rid of old google chrome versions on your Mac

Google chrome is a great browser, which automatically keeps itself up to date. For some reasons, it keeps all older versions which eat up disk space. This becomes obvious when your hard drive storage is not *that* big (as on my Macbook Air with a 64 GB SSD).

I wondered why my free disk space constantly shrinks. On linux, baobab is a great tool to visualize the file sizes on you system. I googled for alternatives on mac and found a java program called jdiskreport, which does the job.

In my case, the google chrome app folder was about 2 gigs big, growing since about 1.5 years of usage. It turned out that over 90 percent of that usage was taken by old versions of chrome which aren't useful anymore.

Where are the versions located?

the standard path is
/Applications/Google Chrome.app/Contents/Versions

How to get rid of the older versions? 

You can simply delete them.
Just type

cd /Applications/Google Chrome.app/Contents/Versions

into a terminal and use ls to see which versions are installed.

now you should  delete all the versions but the actual version (with the highest number), e.g.

sudo rm -fr  21.0.1180.79

to delete the version 21.0.1180.79.
The command remove (rm) has to be used with sudo to gain administrative rights.

Another, even simpler possibility ist to just delete chrome from the /Application folder and download and reinstall chrome.

July 23, 2012

octave quickies: change the position of the y-axis from left to right

Sometimes (e.g. when you have multible plots on one page) you want the y-ticks  and y-labeling on the right side of the plot (default is left).

This can be done by changing the current axis properties:

set ( gca(), "yaxislocation","right")



May 20, 2012

high quality plots in octave

When you're about to publish in a scientific journal, one problem appears: how do I create high quality plots for my paper?


You can use all kinds of expensive programs (Origin, for example), but free software is as well capable of doing this. For print, you definitely want to use some vector based image formats, of which EPS (encapsulated postscript) is the most common.

Well let's look at a simple eps, created with octave:

generic EPS plot. we have to work on this.
 code for this example is below
This looks quite poor:

  • the proportions of data points, axis, ticks are not good
  • the fonts look messy
  • there is no color
The good news is: we definitely can improve this. The bad news: i tried the example on different machines and depending on the octave version and the installed postscript drivers / conversion tools, the results differ. So, as always the parola is: play with the given values :)

A good idea is to look whether the journal you're intending to publish in has any style guidelines. Often, detailed information about the size, font, and axis labels can be found there.

A list of task that can be done to improve the above plot:

  • we'll use colors
  • we'll use another font: a sans-serif is standard in scientific plots (e.g. Helvetica or Arial). The font size will be changed.
  • we'll change the proportions of the image. The data points and lines are the most important issue on our plot, so we'll make them quite big and fat
  • the outer proportions of the image are changed. In a two-column layout which is common in many publications, often a width-to-height ratio of 2:1 is demanded
  • we'll  make the axis a little thicker. In some cases, it is desirable to have the ticks outside
  • Often, the axis are heavily overloaded, there is too much information distracting from the plot content. This can be reduced by leaving some ticklabels out.
The result looks more nicely:

an improved version.


I hope this is a good starting point. The above pictures where created on an Ubuntu 10.04 machine with octave 3.6.x. Now the code:


clear all;
more off;
%
% create some data: polynoms with random noise
%
points = 15
x = linspace(-5,5,points);
y1 = 0.3*x.^2 + 5 *(rand(1,points)-.5)+15;
y2 = 0.1*x.^3 + 5 *(rand(1,points)-.5);
%
% fit some curves to the noisy data
%
[p1,s1] = polyfit(x,y1,3);
[p2,s2] = polyfit(x,y2,3);
tx = linspace(min(x),max(x),500)
%
% create a generic plot
%
figure(1)
plot(  x,y1,'x',
  x,y2,'o',
  tx,polyval(p1,tx),
  tx,polyval(p2,tx))
xlabel 'X'
ylabel 'Y'
legend('y1',
'y2',
'fit1',
'fit2')
print('generic.eps')


%
% create a more advanced version of the plot
figure(2)
%
% in the plot command
% - define markersize
% - define linewidth
% - assign colors
%
plot(  x,y1,'x'
,'linewidth',6
,'color',[1,0,0]
,'markersize',6
  ,x,y2,'o'
  ,'color',[0,0,1]
  ,'markerfacecolor',[0,0,1]
  ,'markersize',6,
  tx,polyval(p1,tx),
  'linewidth',3,
  'color',[1,0,0],
  tx,polyval(p2,tx),
  'linewidth',3,
  'color',[0,0,1])
%
% now let's adjust the axis:
% - make the line thicker
% - make the ticks outside
% - adjust ticklength
% - use custom ticks
% - apply custom ticklabels
%
set(gca(),
'linewidth',2,
'tickdir','out',
'ticklength',[0.005,0.005],
'xtick',[-6:6],
'xticklabel',{'','-5','','','','','0','','','','','5',''},
'ytick',[-20:10:30],
'ylim',[-20,30],
'yticklabel',{'','-20','','0','','20',''})   
xlabel 'X'
ylabel 'Y'
%
% the legend shall appera on the bottom right of the plot
%
legend('y1','y2','fit1','fit2','location','southeast')

% let's print this thing to a eps.
% we use a custom size of 900x450,
% a defined Font and Size
% and the depsc2 driver (EPS 2 color) driver
print('test2.eps','-S900,450',
'-depsc2',
'-F:Helvetica:12',
'-tight'
)


I guess this is a good starting point for creating printable plots that look good. When you're not sure what can be done: Have a look at the attributes of your graphic and axis objects:

get( gcf())   %get current figure
get( gca())  %get current axis

you can set it with the set command

set( gca(), 'property','value')

Have a look at the octave manual. The mathworks website can also be useful, sometimes ;-)

April 21, 2012

octave quickies: customizing the colorbar

When you create colormap or 3D-plots, you may wish to customize the axis of your plot (e.g. change the linewidth, etc). One more difficult task  is to change the colorbar of the plot.

you should customize the colorbar to your needs!


As in matlab, in octave things like lines, plot points, axis and so on are basically graphical objects (see the octave manual). To change the properties of some object, you have to know its handle (although for some, e.g. xlim, there are special function to do so).

One thing I stumble across regularly is the colorbar, where i want to change the linewidth and the ticks. The octave manual is not very verbose at this point, but you basically can use the matlab doc.

Example: we create a blank plot with a colorbar. Then we aquire the colorbar handle and play around with the colorbar properties:


figure(1)
  caxis([0,50])            %change the range
  colorbar()
  colormap( summer(250))   %use some different map than jet


                           %get the  colorbar handle
  cbh = findobj( gcf(), 'tag', 'colorbar')
                          
  set( cbh,                %now change some colorbar properties
    'linewidth', 2,
    'tickdir', 'out',
    'ticklength',[0.005,0.005],
    'ytick', [0, 10, 23, 42, 50],
    'yticklabel',{'zero','ten','23','42','fifty'})


As for all objects, you get get a complete list of the actual colorbars properties with the get function:
get( cbh) 

Example : you can change the dimensions (position, height, width) of the colorbar using the position property:

%get the standard value
octave-3.4.2:8> get(cbh,'position')
ans = 0.827500   0.110000   0.046500   0.815000
%set new value
set(cbh,'position',[0.8275 0.11, 0.02325 0.81500]) 
%(makes a slicker colorbar)









March 29, 2012

simple algorithm for 2D-Peakfinding

When it comes to analyzing data, a common task is to identify peaks in an x-y dataset. There are several ways to do that. Maybe the first idea that would come into one's mind is to look for zeros in the derivatives. This works fine for smooth theoretical curves but fails when noise is present (as it is certainly all kinds of experimental data).
To get rid of noise you can apply a low-pass filter on your data, but in some cases this is not desired. A simple approach is to use an algorithm that can does

  1. find peak candidates that are above some user-defined threshold level
  2. find out for each candidate whether it is the local maximum of a user defined environment
  3. return the valid peaks
the algorithm is able to find peaks even in noisy data


I've implemented such piece of code for octave that does the job. The code below created the picture shown. The function expects three input parameters: the environment for maxima, the data list and a threshold value.



function peaklist = SimplePeakFind ( environment, data,  thresh)
%
% ----------------------------------------------------------
%
% function peaklist = SimplePeakFind (  environment, data, thresh)
%
% finds the positions of peaks in a
% given data list. valid peaks
% are searched to be greater than 
% the threshold value.
% peaks are searched to be the maximum in a certain environment
% of values in the list.
%
%  ----------------------------------------------------------

  listlength = length( data );  
  peaklist = zeros(listlength,1); % create blank output
  SearchEnvHalf = max(1,floor( environment/2));
  %
  % we only have to consider date above the threshold 
  %
  dataAboveThreshIndx = find (data >= thresh);
  for CandidateIndx = 1 : length(dataAboveThreshIndx)
    Indx = dataAboveThreshIndx(CandidateIndx);  
    %
    % consider list boundaries
    %
    minindx = Indx - SearchEnvHalf;
    maxindx = Indx + SearchEnvHalf;
    if (minindx < 1)
      minindx = 1;
    end
    if ( maxindx>listlength)
      maxindx = listlength;
    end
    %
    % data( CandidateIndx ) == maximum in environment?
    %
    if (data(Indx) == max( data( minindx : maxindx)))
      peaklist(Indx) = Indx;      
    end
  end 
  % finally shrink list to non-zero-values
  peaklist = peaklist( find( peaklist));
end


In many cases, this works fine for me. Let's look at a simple test case: we numerically create a number of gaussian pulses with random widths, positions and heights. Then we add noise. 


clear all;
more off;
N = 1500;
T = linspace(-20,20,N);  %x-vector
% create some random gaussian peaks
Npeaks = 9
ppos = T(round(rand(1,Npeaks)*N))
pheight = rand(1,15)*1;
pdur    = rand(1,15)*1;
data = zeros(1,N);
for i = 1:Npeaks
  data = data + pheight(i) * exp(- ((T-ppos(i))/pdur(i)).^2);  
endfor
% apply noisenoise = rand(1,N)*0.1;
datanoise = data+noise;
% Find peaks that are maxima in an
% environment of 25 data points and that
% bigger than 0.15 (noise level)
peaklist = SimplePeakFind(25, datanoise, 0.2);
%Plot result
figure(1)
plot(T,datanoise,"linewidth",2, 
     T(peaklist), datanoise(peaklist),
     "o","markersize",5,"markerfacecolor",[1,0,0])
legend("noisy data","found peaks")
xlabel "x"
ylabel "y"
set(gca(),"tickdir","out","linewidth",2,"ticklength",[0.005,0.005])

The result is shown in the picture above: all 9 peaks are identified correctly. Depending on your data, you'll have to play around with the parameters environment and thresh. Having the the peak positions, further data analysis (like curve fitting etc) is much easier.


Links: of course people already thought of this problem:



March 26, 2012

simulating nonlinear oscillators using the julia language

In my post 'simulating nonlinear oscillators using octave' i showed you how to ... -- i guess you can read. Today i discovered, that a package similar to the odepkg also is included in julia. You just have to find it.


Basically it works quite similar, with some little tweaks.
There is a file called 'ode.jl' in the 'extras' folder located in the julia source folder. We just have to load this thing and have some ode solvers. The drawback is that up to now there  the only way (i found) to adapt the accuracy is to change the lines 


    rtol = 1.e-5
    atol = 1.e-8

in the function ode23
and
    tol = 1.0e-7
in the definition of oderkf{T}, which is a quite dirty method.

Another issue is that the only arguments of the ode solvers are up to now

(F::Function, tspan::AbstractVector, y_0::AbstractVector)


which means we have to hide additional parameters in our solution vector (and make its derivatives zero).

A code example, that works for me (asymmetric nonlinear potential)



# add the path to your juliasource/extras folder
load("/Users/user/juliadir/extras/ode.jl")


function  rhs_asym(t, x)
  #
  # position and velocity are stored in x[1] and x[2]
  #
  xp = x[1]
  vx = x[2]  
  #
  # the parameters are stored in x[3] and x[4]
  #
  k=x[3]
  alpha =x[4]
  dxp = vx
  dv = - (k * xp - alpha * xp^2)
  dx = [dxp; dv;0;0]   # the two zeros ensure that k and alpha  
                       # will not change
end




# problems can occur when you forget to give these values 
# explicitly as Floats ...
k=1.
alpha =0.2
tend=42.


initialDisplacement = 0.0
initialVelocity     = 1.
xini = [initialDisplacement; initialVelocity; k; alpha]


#
# to get higher accuracy, change the values of rtol atol or tol 
# in the ode.jl-file (or better in a copy of it).
#
# now let's integrate
(T, X) = ode45( rhs_asym, [0; tend], xini)



# we store the whole data in one matrix and save it, 
# so we can load and plot with e.g. octave
#
M=[T1 Xlin]
csvwrite("oscil.csv",M)

March 22, 2012

Simulating nonlinear oscillators using octave

Hi there,

numerics are usefull especially when nonlinear problems are considered. Here, I want to show how to calculate the trajectory for some simple one-dimensional oscillators.
You can imagine such an oscillator as a little mass attached to a spring. There is a point of rest, where all forces vanish. When the oscillator is displaced, a force appears originating from the spring.

When energy is put in the system, the mass oscillates. Depending on the form of the potential (which basically is the integral of the force with respect to the displacement)  we can distinguish harmonic and anharmonic oscillators.

From textbooks (or wikipedia) we know the following items, which we want to check using numerics:
  • for the harmonic (linear) oscillator, the trajectory is a pure sine wave of some frequency
  • for the anharmonic (nonlinear) oscillator, more frequency components appear:
    • there are only odd components for the symmetric nonlinear potential
    • there are odd and even components for the asymmetric nonlinear potential


1. The linear (harmonic) oscillator
This is a well known textbook example. The potential energy of the harmonic oscillator is simply a parabula:

Epot = 1/2 k x^2
harmonic potential. The potential energy grows
quadratically with the displacement.

The resulting force is the negative derivative of the potential:


F = - dE/ds = - k x

This is Hooke's law which is a simple model for springs.

Force acting in the harmonic potential
It is also the reason why the harmonic oscillator is called the linear oscillator: the force is directly proportional to the displacement (linear dependency).
The equation of motion is can be derived from the force

d^2 x /dt^2 = F/m = -k/m x

where x is the displacement and k the spring constant.  We can easily give the solution for the equation:

x(t) = A1 cos( k^2/m^2   t   + A2) 

where A1 and A2 are two constants that have to be adapted to the initial values of the problem. The numerical solution using ODEs can be done following the steps described in one earlier post. As the linear oscillator is a special case of the more general problem i'll give the code later in this text.

After integration, we get a trajectory looking as the following  (k=1, initial position =0, initalspeed = 2)
trajectory of the harmonic oscillator
This looks like a nice, pure sine wave (as expected). To get more insight, we use the fourier fransform
of the trajectory x(t).  You have to be careful using results from octaves ode solvers, because in most cases the output points are not equally spaced in time. Using the interpolation function interp1 can help to linearize the data.
The fourier spectrum looks like the following:


fourier components for the harmonic oscillator:
 there is only one frequency present
As expected, we only see one single frequency component, which means we indeed have a harmonic oscillation here.

2. The anharmonic / nonlinear oscillator
Considering the example with the spring, Hooke's law is valid for small displacements. When the displacement gets bigger, additional terms can appear. We want to consider two cases: a symmetric and an antisymmetric nonlinear potential. We give the potential energy:


Epot,sym = 1/2 k x^2 + 1/3 alpha |x|^3

Epot,asym = 1/2 k x^2 + 1/3 alpha x^3


here alpha is a nonlinearity constant. The curves of the potential are shown below:

Two aharmonic potentials, compared with the
harmonic one. They differ in their symmetry
Again, the forces occuring are the derivatives of the potential with respect to the displacement:


Fsym = - dEpot,sym/ds = - (k x + alpha |x| x)
Fasym= - dEpot,asym/ds = - (k x + alpha x^2)


We see additional dependencies between the force and the displacement  which are of the order of x^2. This is why these oscillators are called nonlinear oscillators. The difference to the linear case becomes clear when we look at the force plot:

Forces derived from the potentials above. Note that the
asymmetric force is equal to the symmetric force
for positive displacements, but differs for negative
displacements.


For small displacements, the forces are almost identical. They begin to differ  considerably, when the displacement grows. For the symmetric potential, the absolute value of the force is equal for -x and x. For the asymmetric potential, the force for negative displacements is smaller than for positive values.

The equations of movements can be written as

(symmetric)
d^2 x /dt^2 = F/m = - (k/m x + alpha/m |x| x)

(asymmetric)
d^2 x /dt^2 = F/m = - (k/m x + alpha/m x^2)

Again, we can easily solve this with the odepkg (see code sample below)I used the same input energy as for the harmonic example resulting in  the following trajectories:

Trajectories for the nonlinear oscilators (two kinds), compared with the one
 of the harmonic oscillator


The two new trajectories basically also show  sine osciallations. As the force for the symmetric potential is stronger for big displacements, the maximum amplitude is smaller. This also results in a higher frequency for the oscillation. For the asymmetric potential, the oscillation is shifted  slightly towards negative values in the mean.
We can get the fourier components of this oscillations when analyzing the time traces x(t) of the signals.
We have a closer  look on the symmetric harmonic potential:
spectrum for the symmetric nonlinear oscillator.
Only odd multibles of the base frequency are generated
Besides the fact, that the base frequency is higher than for the harmonic oscillator, we see an additional frequency component! Its value is three times the one of the base frequency (third harmonic). Also, energy is converted into the fifth harmonic.

A similar behavior can be seen for the asymmetric potential:




spectrum for the asymmetric nonlinear oscillator.
Here, even and odd multibles of the base frequency are generated
Here we have generated additional frequencies at two, three and four times the base frequency. The conversion efficiency is higher than for the symmetric potential. 

As a result, the simulations show the expected behavior. We can see that nonlinear oscillators are capable of generating multiples of the base frequency.  This is extensively used in nonlinear optics, where light is sent through nonlinear crystals. The crystal structure is responsible for the potential, in most cases they have asymmetric potentials.
When e.g. light from a laser (at a wavelength of 1064 nm) of high intensity passes such a crystal, green light (532 nm) can be generated.

see 


  • https://en.wikipedia.org/wiki/Second-harmonic_generation



  • https://de.wikipedia.org/wiki/Frequenzverdopplung (german)


  • for further details.


    3. Code to solve the linear and nonlinear oscillators using octave
    clear all;
    more off;


    %
    % right hand side of the ODE for the symmetric harmonic potential
    %
    function dx = rhs_sym(t, x, k, alpha,tend)
      t/tend     %just prints the progress 
      xp = x(1);  % position
      vx = x(2); % velocity
      dxp = vx;   % position change
      dv = -(k * xp + alpha * xp * abs(xp) );  %velocity change
      dx = [dxp, dv]; %give the derivative
    endfunction




    %
    % right hand side of the ODE for the asymmetric harmonic potential
    %
    function dx = rhs_asym(t, x, k, alpha,tend)
      t/tend
      xp = x(1);
      vx = x(2);  
      dxp = vx;
      dv = -(k * xp + alpha * xp^2);
      dx = [dxp, dv];
    endfunction




    % program parameters
    k =1;
    alpha =0.2;
    initialDisplacement = 0;
    initialVelocity     = 2;
    xini = [initialDisplacement, initialVelocity];




    tend = 15;  %integrate up to that time
    % set integration options
    options = odeset( 'InitialStep',tend/1e3,
    'MaxStep',tend/1e3,
    'RelTol',1e-6,
    'AbsTol',1e-6,
    'NormControl','on')




    % linear case, using alpha = 0
    [T1, Xlin] = ode45( @rhs_sym, [0, tend], xini, options, k, 0, tend);


    %NL case, symetric potential:
    [T2, XnlS] = ode45( @rhs_sym, [0, tend], xini, options, k, alpha,\
    tend);
         
    %NL case, symetric potential:  
    [T3, XnlA] = ode45( @rhs_asym, [0, tend], xini, options, k, alpha,\    tend);


    % plot the result
    plot (T1, Xlin(:,1),'color',[0.5,0.5,0.5],
          T2, XnlS(:,1),'.','color',[0,0,1],
          T3, XnlA(:,1),'.','color',[1,0,0]);
    xlabel 'time'
    ylabel 'displacement'




    Hint: for the analysis of the frequency components shown above, the integration time has to be set to a higher value and the odeoptions have to be changed to higher accuracy...

    March 20, 2012

    How to solve ODEs using octave.

    The odepkg package for octave is a great tool to solve ordinary differential equations (ODEs). To solve some problem we only have to bring it into a form octave understands. We can do that following four steps:

    1. determine the variables of the problem
    2. figure out which variables change and how they do it
    3. if higher order derivatives do appear, reduce the order of the system
    4. write a function that can gives the first derivatives for all values considered and solve it with octave's ode solvers
    First example: Shooting a paperball using a slingshot (no friction).

    problems like the trajectory of a sphere fired by a slingshot can be solved using ODEs

    This is a very basic example. In fact, we can give the analytical solution (parabula). The paperball is accelerated by the catapult and flies some distance until it hits the groud. 
    1. The variables of the problem are
    • the position of the paperball in space r(t) = ( x(t), y(t) )
    • the velocity    v(t) = ( vx(t), vy(t) )
    • the acceleration a(t) = (ax(t), ay(t) )
    Here we use the fact that we can split each of the variables of motion into two independent parts, showing in the x or and in the y-direction.

    2. Now we have to consider how our variables change:
    • the change in position is proportional to the velocity at a certain moment:   dr/dt = (vx, vy)
    • the velocity is changed by the acceleration: dv/dt = (ax, ay)
    • for our case, the acceleration is constant. We consider the gravitational force: F=m a where a = g = 9.81 m/s^2 is the gravitational acceleration. It only acts on the y-component of the velocity (perpendicular to the ground)
    As we figured out which variables change in what way, we consider how our result should look like. Octave expects the problem to form a vector (1xN components). What we want to know is the position of the paperball and its velocities for all times t.

    The result should be of the form   result(t) = [rx, ry, vx, vy]

    What we now have to do is to give octave a vector containing the derivative of our result vector. We do this by defining a function.  [drx, dry, dvx, dvy] = deriv_result( time,position)

    3. if we did not already do so, we have to formulate the problem as a system of first order ordinary differential equations.

    What does this mean? Instead of writing the second derivative of some value, we define some additional variable that is  the first derivative of the initial value and consider its changes.

    example: 
    • the acceleration is the second derivative of the position  d^2r / dt^2 = a
    • we introduce the velocity. it is the first derivative of the position dr/dt = v
    • the change of the velocity is given by the acceleration dv/dt = a
    • now we have to solve the problem both for the position and the velocity and such the second order ODE has been reduced to two first order ODE
    4. solving the problem in octave

    We define derivative function

    function dr = dr_gravi(t,r)
      g  = 9.81; %m/s^2
      vx = r(3);
      vy = r(4);
      ax = 0;
      ay = -g;  
      dr = [vx,vy,ax,ay];
    endfunction

    In the program we have to set some initial value. For our problem this is the initial position r0=(x0,y0) and the initial speed v0 = (vx0,vy0)

    X0 = 0
    Y0 = 0
    VX0 = 2
    VY0 = 3

    initialVector = [ X0, Y0, VX0, VY0]

    Also,  some integration window is necessary: for our problem, this is the starting time and the ending time for time integration:

    StartT= 0 %s
    StopT = 0.7 %s

    then we let octave do the job:

    [T,Result] = ode45( @dr_gravi,[StartT, StopT], initialVector)

    Here we used the ode45  solver, but there are many more. See the odepkg documentation for details.
    Our result looks like the following

    T =

       0.00000
       0.07000
       0.14000
       0.21000
       0.28000
       0.35000
       0.42000
       0.49000
       0.56000
       0.63000
       0.70000
       0.70000

    Result =

       0.00000   0.00000   2.00000   3.00000
       0.14000   0.18597   2.00000   2.31330
       0.28000   0.32386   2.00000   1.62660
       0.42000   0.41369   2.00000   0.93990
       0.56000   0.45545   2.00000   0.25320
       0.70000   0.44914   2.00000  -0.43350
       0.84000   0.39476   2.00000  -1.12020
       0.98000   0.29231   2.00000  -1.80690
       1.12000   0.14179   2.00000  -2.49360
       1.26000  -0.05679   2.00000  -3.18030
       1.40000  -0.30345   2.00000  -3.86700
       1.40000  -0.30345   2.00000  -3.86700

    T is the vector that gives us the time in seconds for some row. Results contains the position and velocity components for the times stored in T.  Using this data, we can plot the trajectory and other dynamic information.

    The following code plots the trajectory of the ball and the time dependence of the velocity components:

    figure(1)
      plot(Result(:,1),Result(:,2),'o');
      xlabel 'x position / m'
      ylabel 'y position / m'

    figure(2)
      plot(T, Result(:,3),'.',
         T, Result(:,4),'x');
      xlabel 'time / s'
      ylabel 'velocity / (m /s)'
      legend('vx','vy')



    Is there a way to influence the calculation? Of course there is. We can do that using odeoptions

    options = odeset( 'RelTol',1e-4,
    'AbsTol',1e-4,
    'InitialStep',StopT/1e3,
    'MaxStep',StopT/1e3)
    [T,Result] = ode45( @dr_gravi,[StartT, StopT], initialVector , options )

    There are many options to play around with, of which most are self-explanatory . You have to find a tradeoff between accuracy and calculation time. Again, see the odepkg documentation.

    A more complex example: slingshot with friction.
    Additional to the gravity, let's consider the drag force influencing the flight of our paperball. It is a force proportional to the velocity  v of an object.

    F = -1/2 rho Cd A v^2

    Additional parameters appear, which we set here: 
    • rho, the density of air ( rho = 1.2 kg / m^3)
    • Cd - the drag coefficient . We consider a sphere here, which has Cd = 0.45
    • A is the crosssection of the sphere
    Again, we can split the force into two components: Fx and Fy. To calculate the acceleration components, we also need the mass of our object, as F=ma which leads to a = F/m.

    We can give the derivative function as follows with additional parameters (area and mass of the sphere):


    function dr = dr_gravi_friction(t,r,area,mass)
      g = 9.81; %m / s^2
      cdSphere = 0.45;
      rhoAir = 1.20; %kg / m^3
      frictioncoefficient = 1/2 * rhoAir * cdSphere * area / mass;  
      
      vx = r(3);
      vy = r(4);
      ax = - frictioncoefficient * vx^2;          % only friction

      ay = -( g + frictioncoefficient * vy^2 ) ;  % friction and  
                                                  %  gravitation
      dr = [vx,vy,ax,ay];
    endfunction

    Now we can try out how friction affects the trajectory.  We calculate this for a relative heavy sphere, a light sphere and without friction. The radius of the spheres shall be r=2 cm


    mass1 = 0.1 %kg
    mass2 = 0.001 %kg
    area1 = pi * 0.02^2   %m^2
    area2 = area1;
    options = odeset( 'RelTol',1e-4,
      'AbsTol',1e-4,
      'InitialStep',StopT/1e3,
      'MaxStep',StopT/1e3)

    % trajectory of heavy sphere
    [T1,Result1] = ode45( @dr_gravi_friction,[StartT, StopT], \
        initialVector , options, area1, mass1  )

    % trajectory of light sphere
    [T2,Result2] = ode45( @dr_gravi_friction,[StartT, StopT], \
        initialVector , options,area2, mass2 )
    % trajectory without friction
    [T3,Result3] = ode45( @dr_gravi,[StartT, StopT], \
        initialVector , options)

    When we look at the results, we see the following: The relative impact of the friction force on the light sphere is quite strong. It considerably gets slowed down and hits the ground earlier. The trajectory deviates more and more from the parabula shape while propagating. Contrarily, the heavy sphere almost acts as if  friction was absent.  This basically is what we expected.

    trajectories of spheres