Thursday, March 20, 2014

EJS Resonance Frequency vs Amplitude Curve Model

EJS Resonance Frequency vs Amplitude Curve Model by Wolfgang and lookang.

based on a conversation 2008 here http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=821.0
the original can be found in EJS examples by wolfgang, such as \source\users\davidson\wochristian\osc\SHOResonance.xml

EJS Resonance Frequency vs Amplitude Curve Model by Wolfgang and lookang.
http://weelookang.blogspot.sg/2014/03/ejs-resonance-frequency-vs-amplitude.html
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejs_model_SHOResonancewee.jar
author: Wolfgang and lookang

Contextualization of spring mass system:

refer to another model here
EJSS simple harmonic motion model with x vs t, v vs t and a vs t graphs
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxva/SHMxva_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxva.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre


Physics Model of spring mass system with amplitude vs driving frequency graph showing the concept of resonance :

The following ordinary differential equations are used:

$ \frac{\delta x}{\delta t} = v $

$ \frac{\delta v}{\delta t} = -\frac{k}{m}x -\frac {b}{m}v + \frac{A cos( \omega t)}{m} $

where
$ -\frac{k}{m}x $ is the restoring acceleration component from simple harmonic motion

$ -\frac {b}{m}v $ is the damping acceleration as a result of the viscous fluid the spring mass system in experiencing

$ +\frac{A cos( \omega t)}{m} $ is the driving acceleration component due to an external driving force agent.

The key to determining the maximum amplitude is from the energy equation

since


$ TE= \frac{1}{2}mv^{2} +  \frac{1}{2}kx^{2}$


it can be shown that the various maximum amplitudes happens at $ v = 0 $, thus,

$ X_{max}= \sqrt{\frac{2TE}{k}}$

by stepping through 50 transientCounter, the $ X_{max} $ can be determined and plotted by plotting by $ \delta f $, the corresponding $ X_{max} $ can be found. 

The equation is used to determine the natural frequency and natural angular velocity of the spring mas system

$ f_{o}=\frac{1}{2} \pi \sqrt \frac{k}{m} $

$ \omega_{o}= \sqrt \frac{k}{m} $


Levels of damping

the following assumption are made for modeling the damping factor 

$ \tau = \frac{b}{2 \sqrt{mk}}$

very_light_damping when $ \tau = 0.05$
very_light_damping when $ \tau = 0.05$


light_damping when $ \tau = 0.1$
light_damping when $ \tau = 0.1$


moderate_damping when $ \tau = 0.2 $
moderate_damping when $ \tau = 0.2 $



critical_damping when $ \tau = 1.0$
critical_damping when $ \tau = 1.0$


heavy_damping when $ \tau = 2.5$
heavy_damping when $ \tau = 2.5$


for the corresponding damping coefficient  $ b $ values to show correctly, the equation is use

$ b = 2 \sqrt{mk} \tau $

Changes made by lookang


  1. added dropdrop menu with ease of learning thanks to fu-kwun many examples
  2. added dotted line for visualization of natural frequency $ f_{o} $ thanks to paco for sharing how
  3. modified the trail instead of trace for color change thanks to paco
  4. modified the existing object oriented programming style to draw thank to wolfgang
  5. added pause when $ f >= 2f_{o}$ for plotting 2 twices the x size consistently
  6. layout to my usual design
  7. added m and k for contextualization of the spring mass system


Physics i don't understand


strangely which a different driving force component, the graph can show maximum curve characteristics but it starts at (0,0) instead of (0,A).


start at (0,A) which i am not is correct, but has maximum curve characteristics (which i believe is correct)


$ +\frac{A sin( \omega t)}{m} $ is the driving acceleration component due to an external driving force agent.