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*[[Signals and systems|Signals and Systems]]
For this homework assignment I wanted to try and see if I could find a correlation between a dragsters rear tire expansion in comparison to its velocity by using either the method of Laplace transform or the Fourier series. To help me with my model I will me using the Army's dragster for some of my data. If you would like to check it out you can find it at [http://www.goarmy.com/army-racing/nhra-top-fuel/dragster.html]. If you want to watch a video of dragster tires click here : http://www.youtube.com/v/V3yj_OGezWc?version=3
==Fourier Transform Introduction and focus on Applications==  




Unfortunately, the Fourier Transform isn't a Transformer. If it was, you would have seen it in the movie that came out lately. [[Image:transformer_roolbar.jpg]]
'''Data:'''
<br>One way to explain a Fourier Transform is to say it's a bunch of sinusoids added to create a just about any function you want. Another way to describe it is to say it's a way of representing a function in the frequency domain instead of the time domain.
* Outside tire diameter = 36.5" or up to 40.5" due to tire expansion
<br>For example, a particular square wave could be represented by the following discrete fourier transform:
<math>x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ((2k-1)2\pi ft \right )}\over(2k-1)} </math><br>


== Fourier Transform Applicaitons ==
* Inside tire diameter = 16"
===The "Fast" Fourier Transform===
====Cooley-Turkey Algorithm ====


<b>What is a Fast Fourier Transform? (FFT)</b><br>
* Width of tire = 17"


It's an algorithm that can compute the discrete Fourier transform faster than other algorithms. In digital systems, continuous Fourier Transforms are sampled, turning them into discrete Fourier Transforms which then can be computed and manipulated using Digital Signal Processing.  
* Air pressure in tire = 7 psi


An intuitive brute force way of computing a Fourier Transform means rearranging the the summation so that you don't compute the transform in sequential order - you group similar elements together and simplify before combining them. This cuts down the adding and multiplying, thus cutting computation time down by about 100 times.  
* Volume of Tire = 9.82 ft^3      **(you have to add about 1.5ft^3 to account for possible expansion)**


One of the most popular FFT algorithms is called the Cooley-Turkey algorithm. Which I will explain on Friday
* Fastest quarter mile time = 4.428 sec
 
* Fastest quarter mile speed = 337.58
 
'''Equations:'''
Volume = [(pi)*(R^2)(h)-(pi)*(r)(h)]
 
y = .075/(Vmax-Vmin)    **(this is the number of inches that the tire should expand given the current velocity.)**

Latest revision as of 21:23, 31 October 2010

For this homework assignment I wanted to try and see if I could find a correlation between a dragsters rear tire expansion in comparison to its velocity by using either the method of Laplace transform or the Fourier series. To help me with my model I will me using the Army's dragster for some of my data. If you would like to check it out you can find it at [1]. If you want to watch a video of dragster tires click here : http://www.youtube.com/v/V3yj_OGezWc?version=3


Data:

  • Outside tire diameter = 36.5" or up to 40.5" due to tire expansion
  • Inside tire diameter = 16"
  • Width of tire = 17"
  • Air pressure in tire = 7 psi
  • Volume of Tire = 9.82 ft^3 **(you have to add about 1.5ft^3 to account for possible expansion)**
  • Fastest quarter mile time = 4.428 sec
  • Fastest quarter mile speed = 337.58

Equations: Volume = [(pi)*(R^2)(h)-(pi)*(r)(h)]

y = .075/(Vmax-Vmin) **(this is the number of inches that the tire should expand given the current velocity.)**

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