Thanks for posting that info, I made up my own spreadsheet and charts at bottom:
Quote:
Originally Posted by 3-Wheeler
After digging around for a mathematical solution to the posted equation by Mathematica, I came across a description of what is meant by "sin^m" listed above.
The math form below shows another way of handing the "sin^m", which is not solvable by normal means.
So now we had a way of properly solving the shape of a Tear Drop, and an excerpt is shown below. Here are the equations for X and Y:
X=COS(A10)
Y=SIN(A10)*SIN((A10/2)^$F$8)/$C$7
Where:
Increment = 0.1 (number of desired steps in shape)
A10 = cell A10
$C$7 = 1.925 (constant) used to create a shape with a 2.5:1 length/width ratio
$F$8 = M (constant)
And here is what the spreadsheet data looks like....
To solve for a shape where Y is positive, vary T from 0 to PI. To solve for a shape where Y is negative, vary T from PI to 2PI. In this case I am only interested in solving for Y in the positive state, so T is varied from 0 to PI.
Here is the original graph presented by SGT.
I'm not sure about the Y-axis units for this graph, as they do not give output of 2.5:1 for length/width.
The graph below however does indeed maintain AeroHead's 2.5:1 length/width ratio however. We have "two units" in the X-axis (-1 to +1) and one-half of the 0.8 units of width (0.4 units) shown. 2.0/0.8 = 2.5:1
SGT, thanks for posting your original graph of this shape, as it gave me the incentive to dig deeper for those who are math challenged, like myself.
Hope this helps, Jim. |
I did the chart with mirror image to see how uniform shape is, I think it looks pretty good, be good if someone can overlay the templates to compare, don't have the knack of doing overlays.
The shape is more aggressive at the start, bit like new template, then it tapers back and angle begins to reduce somewhere around 75% and ultimately goes back to 0 at 100%.
Although this is not in line with template theory, I actually thinks it makes more sense.
If an object is moving through free air and ultimate aero principles is to allow air to fall back in behind it, then it is better to have a trailing edge with minimum angle rather than one where two airstreams are meeting at 44°, ie 22 above and 22 below, either way it doesn't matter much as most will not go beyond 80% and you can see at 98% it is still at 13°.