I'm looking for an explicit numerical example of a parametric surface (u, v) |-> (x(u, v), y(u, v), z(u, v)) that models a fan blade like that in the attached picture.
Googling results in NURBS/PDE approaches but with no numerical explicit data.
I would like to find a more simple model to be used in a engineering class for upper high school students.
Thanks, Humberto.
For propellers, one usually has blade reference lines called generator lines. They can be parametrically defined in terms of blade number, skew, rake, hub dia, tip dia. If the generator lines are the midpoint of the blade chord, they are called blade center lines BCL. Next, you have a parametric expression for the pitch progression along the generator (often the deviation from a nominal pitch progression). Next, you have the blade area distribution along the generator. Basically, this is the blade chord associated with each diameter (or projected arc length, choose one). Finally, you have a foil development for each diameter (this has to be consistent with the pitch definition). So it's a bit more complicated than what you envision. The point of parametric expressions in engineering is to be able to isolate performance variables and manufacturing variables and cost variables.
So a fan is built by choosing the number of blades, hub diameter, and tip diameter.
Next, define the blade generator - you need Skew(D), Rake(D).
Next, define the geometric pitch expression, P(D)
Next, develop the chordlines on the generator using (projected) blade arc and P(D).
Next, define the blade foil's camber shape as a series of offsets from the chordline.
Finally, find the surfaces by applying a thickness expression T(C,D) to the foil's camber line.
You usually use cylindrical coordinate system for props - X (axial), R (radial), and Theta (angle from reference point)
Michigan Wheel PDF of terms and geometry systems. http://navalex.com/downloads/Michigan_Wheel_Propeller_Geometry.pdf
So for your fan, lets say rake is zero. You need an expression for the skew(D) to create a generator. Let this be your "u". Skew(D) = Theta = f(r).
Take geometric pitch to be constant (meaning uniform geometric advance along the blade span.) Pitch angle(r) = Arctan(pitch/2*Pi*r). Given a uniform thickness, you now have everything needed to define face and back surfaces with respect to the generator ( X, R, Theta in terms of u(r), constant pitch, and constant thickness).
The arc length of the chord at each point is your expression "v". chord length(r) = g(r). This completes the boundary of the surface (along with hub dia, tip dia, if chord is nonzero at those radii.) The expression for chord length can be given in Expanded Coordinates, which is a shell of a cylinder unrolled into 2D and conventionally expressed in dimensions X and Z'. Or it can be expressed in the length projected onto the propeller plane and then unrolled. The two are related by the cosine of the local geometric pitch angle. So the trick now is you have a boundary value problem. One of your variables defines the surface and the other defines the boundary. This is why I don't think this isn't a very good example for your purpose.
