Challenge to naval architects and marine engineers

I've laid low for a good while, now it's again time to irritate the experts. Texts on propellers follow wing theory and talk about camber. Sport and race boat prop shops and high performance marine prop designers talk about progressive pitch, the pitch measured over a small chord on the radial arc of a cambered blade. Here's the challenge: give me a formula that shows exactly how camber and progressive pitch are mathematically related. I have such a formula, it's simple, and I'll post it after I get some attempts at answers. Anyone who answers correctly wins a free copy of my book (the equation will appear in later issues).

 

Diagram, because "radial arc of a cambered blade" is imprecise and has way too many interpretations.

 

Wrong. An arc of constant radius on a surface is uniquely defined. Every pitch gauge makes a radial arc on a propeller blade.

 

This is a trick question. The definition of radial is: Radial refers to the pattern that you get when straight lines are drawn from the center of a circle to a number of points round the edge. There is no such a thing as a radial arc.

 

That is a narrow interpretation. As an arc is not necessarily an outcut of the circumference of a circle (e. g. parabolic arc), the term radial arc is referencing to an arc drawn with a constant radius, so a circular arc.

 

In that case, the question refers to really ancient design propellers. Modern propellers are usually an asymmetrical foil, so it is still a trick question.

 

It refers not to the curves defining the shape of a prop blade section but to the "line" along the section of the blade is cut. See below.

 

Exactly; it is not a circular section.

 

Like @jehardiman I find the callenge not clear enough worded. So I will make myself some assumptions. Camber is defined as a variable value along the chord line of a foil profile. It is the local distance between the chord line and the mean line, measured perpendicular to the chord line (see pic below).


If the foil in the picture above represents a prop blade section (along a cylinder surface, see post #7), the leading edge is on the right, the upper curve is the back, and the lower curve the face of the blade.
Talking about pitch we turn the pic by say 45 degree CCW (the way prop foils a shown normally). Going along the face curve of the blade (from right to left as the water passes) we will find no progressive pitch.

If we pick an appropriate blade section with an concave blade face we will find the progressive pitch but the camber is not easy to find. So the pragmatic solution should be to replace the chord line as reference line with the line we establish by siting a ruler against the section face (CF) and define the camber as the distance between this new reference line and the blade surface (face side) perpendiculat to CF (see pic below).


@sandhammaren05 what do you think about this understanding of the challenge?

 

I am confused also about "the pitch measured over a small chord on the radial arc of a cambered blade". What is a "small chord"? A chord is a line segment joining two point on a curve. I think the OP needs to moderate his smugness and put more effort in defining his question. I still think it is simply a trick question; probably a riddle.

 

I understand the "small chord" as an narrow spaced interval e. g. in my sketch between theta3 and theta4. The gauge shows for this angular values the ordinates x3 and x4 respectively. So you are able to determine the pitch angle as
atan((x4 - x3)/(theta4 - theta3))
The pitch is then
2 * pi * r ((x4 - x3) / (theta4 - theta3))

A gauge like this in the pic will deliver the local value of x as the depth of the vertical "measurement pin". Then you turn the prop by an certain small angle and measure again.


As for "chord": it is a linear function of the angle theta as you can see in my sketch in #9. No matter if you pick the replacement chord CF or the real chord CR.

 

The chord needs to be defined to have a meaningful discussion. Otherwise, nobody will know what the other is talking about. My understanding is that the pitch is the maximum chord. That is from one edge of the blade to the opposite.

 

No, chord lenght and pitch are independent. You can easily widen or shorten the chord line and have the same pitch if the angle between chord line and a plane perpendicular to the propeller axis stays unaltered.

 

As has been shown by the discussions above, you still haven't defined what you want. Do you want to know how blade gauges are made, or do you want to know how blade sections are developed for arbitrary J,P,D and T? Yes for a SP or SC blade there can be a "shop secret" measured pitch over the face which has little to do with the geometric developed pitch of the section...with or without camber. If all you are interested in the apparent instantaneous pitch of the face without respect to the back, go ahead and use what you have.

 

Heimfried is correct, Gonzo as always is shooting from the hip without hitting anything. A circular arc can be inscribed about any axis on any surface. Were a blade helicoidal (i.e., with out camber) then taking the z-axis along the propshaft centerline, the arc length (infinitesimal chord) at constant radius r would obey ds^2=(r^2+(p/2pi)^2)dphi^2 where (r,phi,z) are polar coordinates and p is the pitch.

I have posed a well-defined question where I have already worked out the answer. One needs to understand how a pitch gauge works, and also know elementary analytic geometry.