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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 |
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*[[Signals and systems|Signals and Systems]] |
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'''Data:''' |
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==What is a Fourier Transform?== |
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* Outside tire diameter = 36.5" or up to 40.5" due to tire expansion |
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* Inside tire diameter = 16" |
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* Width of tire = 17" |
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<b>Unfortunately, the Fourier Transform isn't a Transform-er. </b> |
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<br>(This handsome fellow is named Roolbar.)<br> |
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[[Image:transformer_roolbar.jpg]]<br> |
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<br>Check any of the other pages on this site to find fifty different ways to explain what a Fourier Transform is. If you already know what it is, or you're too lazy to look at the other pages, here's my super trite description: A Fourier Transform is a bunch of sinusoids of different frequencies and time offsets added together create a just about any function you want. Also, you can say that a Fourier Transform is the way of representing a function in the frequency domain instead of the time domain. |
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* Air pressure in tire = 7 psi |
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Instead of describing the Fourier Transform itself once again, the following pages describe how Fourier Transforms are analyzed and utilized in a computer. |
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* Volume of Tire = 9.82 ft^3 **(you have to add about 1.5ft^3 to account for possible expansion)** |
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* Fastest quarter mile time = 4.428 sec |
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== Fourier Transform Applications == |
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===The "Fast" Fourier Transform=== |
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* Fastest quarter mile speed = 337.58 |
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'''Equations:''' |
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<b>What is a Fast Fourier Transform? (FFT)</b><br> |
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Volume = [(pi)*(R^2)(h)-(pi)*(r)(h)] |
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y = .075/(Vmax-Vmin) **(this is the number of inches that the tire should expand given the current velocity.)** |
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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. |
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An intuitive brute force way of computing a Fourier Transform is just adding the function up one at a time starting at n=0 or n=1 like this: |
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:<math>\sum_{i=m}^n x_i = x_m + x_{m+1} + x_{m+2} +\cdots+ x_{n-1} + x_n. </math> |
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What an FFT does is rearrange 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. |
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<br><b> Cooley-Tukey Algorithm </b> |
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Three popular algorithms are the Cooley Tukey algorithm (radix-r algorithms), the Good-Thomas Algorithm, and the Winograd FFT algorithm. The Cooley Tukey Algorithm Radix-2 algorithm is the simplest and most popular of all. The algorithm separates a given discrete Fourier transform of size <math> N </math> into two equal transform of size <math> N/2 </math>. Here is the definition of the algorithm given on wikipedia's page. |
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:<math> X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} nk}</math> |
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where <math>k</math> is an integer ranging from <math>0</math> to <math>N-1</math>. |
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===Related Links=== |
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*[[http://en.wikipedia.org/wiki/Fast_Fourier_transform Wikipedia:Fast Fourier Transform]] |
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*[[http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm Wikipedia:Cooley-Tukey Algorithm]] |
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*[[http://www.relisoft.com/Science/Physics/fft.html Fast Fourier Transforms]] |
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*[[http://etd.lib.fsu.edu/theses/available/etd-06262004-162018/unrestricted/Divya_MS_Thesis.pdf FTT's for FPGA's Thesis]] |
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Latest revision as of 20:23, 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.)**