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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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==An Introduction to the Fourier Transform== |
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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]] |
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<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. |
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<br>For example, a square wave could be represented by: |
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<math>x_{\mathrm{square}}(t) = \frac{4}{\pi} \sum_{k=1}^\infty {\sin{\left ((2k-1)2\pi ft \right )}\over(2k-1)} </math> |
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'''Data:''' |
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That's alot of numbers seemingly out of the blue, at first observance. I bet your questions are: |
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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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<br> |
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=== 1. Telling me the difference between a transformer and a Fourier Transform hasn't helped me finish my assignment. What else can you tell me? === |
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Wait till Friday. That's when the deadline for this assignment is, so hopefully I'll panic before then and get something done on this page in addition to the spectacular transformers picture. |
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<br> |
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=== 2.Can you show me some really easy plug-in formulas, so I can get my homework done faster? === |
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Yes. <br> |
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=== 3.Why would I bother getting one? I don't know what to do with it. === |
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<br> Fourier Transforms are awesome! They allow continuous functions to be represented by ones and zeros, which means we can |
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implement functions on the computer. Fourier Transforms form the basis of signal processing. They allow us another way to transform |
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a math problem so it's alot easier to solve. |
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* Width of tire = 17" |
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==Some Useful Fourier Transform Pairs== |
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<math> |
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\mathcal{F}[\alpha(t)]=\frac{1}{\mid \alpha \mid}f(\frac{\omega}{\alpha}) |
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</math> |
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<br> |
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{| |
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|- |
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|<math>\mathcal{F}[c_1\alpha(t)+c_2\beta(t)]</math> |
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|<math>=\int_{-\infty}^{\infty} (c_1\alpha(t)+c_2\beta(t)) e^{-j2\pi ft}\, dt</math> |
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|- |
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| |
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|<math>=\int_{-\infty}^{\infty}c_1\alpha(t)e^{-j2\pi ft}\, dt+\int_{-\infty}^{\infty}c_2\beta(t)e^{-j2\pi ft}\, dt</math> |
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|- |
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| |
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|<math>=c_1\int_{-\infty}^{\infty}\alpha(t)e^{-j2\pi ft}\, dt+c_2\int_{-\infty}^{\infty}\beta(t)e^{-j2\pi ft}\, dt=c_1\Alpha(f)+c_2\Beta(f)</math> |
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|- |
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|} |
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<br> |
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<math> |
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\mathcal{F}[\alpha(t-\gamma)]=e^{-j2\pi f\gamma}\Alpha(f) |
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</math> |
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<br> |
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<math> |
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\mathcal{F}[\alpha(t)*\beta(t)]=\Alpha(f)\Beta(f) |
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</math> |
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<br> |
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<math> |
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\mathcal{F}[\alpha(t)\beta(t)]=\Alpha(f)*\Beta(f) |
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</math> |
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<br> |
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Some other usefull pairs can be found here: [[Fourier Transforms]] |
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* Air pressure in tire = 7 psi |
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==A Second Approach to Fourier Transforms== |
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*[[Fourier Transforms]] |
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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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* Fastest quarter mile speed = 337.58 |
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'''Equations:''' |
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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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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.)**