Piednoir Can-Formed Helical Propeller Blades

From: <dgbj_at_aol.com>
Date: Sun, 3 Dec 2006 07:06:14 EST

Piednoir Can-Formed Helical Propeller Blades
 
Gary Hinze, May 2005
 
(Note: * is multiplication, ^ is exponentiation.)
 
Model airplane propeller blades made of thin sheets of wood may be twisted
by soaking them in water and binding them to the side of a cylindrical can,
with the blade radius aligned with a line offset a small angle from a vertical
line on the side of the can. This produces a blade twist angle that is
linearly related to the distance along the radius. This differs from the ideal
helical prop angle distribution, which lies along a tangent curve.
 
It is also possible to soak a rectangular sheet of balsa in water, twist
the ends and let it dry. This also produces a twist angle that is linearly
related to the distance along the radius.
 
Both of these methods have a linear distribution of blade angle along the
radius. Either method can produce the same distribution of blade angle plotted
against blade radius. Neither will produce a helical pitch, although various
approximations are possible.
 
Imagine you have a can of radius r and you cut it along a baseline parallel
with the axis, unroll it and flatten it out. Now draw an offset line at angle
T (theta) to the baseline and mark off a blade radius length R along the
offset line from the point where it intersects the baseline. Drop a
perpendicular from the end of this radius line to the baseline. That
perpendicular has length
a = R*sin(T). The point where the baseline intersects the offset line is at
the axis of the propeller and the point R up the offset line is the blade tip.
 Now roll the sheet back into the can. A tangent to the can at the tip
intersects a projected can diameter passing through the tip end of the baseline at
a tip blade angle A (alpha). The two radii passing through the blade tip and
the baseline tip form an angle C. The angle C is also the total blade twist
from the axis to the tip. At the propeller axis the blade angle is 90 degrees
and the twist is zero degrees. Note that A+C = 90 degrees. The dropped
perpendicular wrapped on the can is now an arc with angle C at its center. The
length of the arc is related to the can radius and central angle by a =
2*pi*r*C/360 (for angles in degrees). We may now set the two equations for arc length
a equal and find that R*sin(T) = 2*pi*r*C/360.
 
This immediately tells us two things. The equation for the offset angle
needed when you have the blade radius, the total twist and the can radius is T =
arcsin(2*pi*r*C/360*R). It also tells us that the rate of twist along the
radius is uniform, because C = R*sin(T)*360/2*pi*r. This is valid for any
fraction of R, so you can see the twist is linearly related to radius, since the
other terms in this equation form a constant for any given can radius and offset
angle. The rate of change of blade angle with radius is C/R =
sin(T)*360/2*pi*r. A plot of blade angle against radius will be a straight line. This
will not be a helically pitched prop. The plot of blade angle against radius
for a helically pitched prop is an arctangent curve. Also, if you take a
rectangular piece of sheet balsa by the ends and twist it, you will get a uniform
rate of twist. This means the two methods of forming prop blades can produce
the same distribution of twist. The can formed blade radius will be shortened
slightly because it lies along a curved helix. You can get it exactly thesame
by lengthening the radius of the layout line. Note that the radius of the
can formed prop blade should angle forward from the plane of rotation at the
prop hub. This is not the way they are usually constructed.
 
Since the can formed blade does not have a constant pitch, a choice must be
made regarding what blade radius or radii will be used to set the pitch of the
 propeller. This will be illustrated in an example, below.Jean-Marie
Piednoir developed a method of calculating a curved offset line that produces a
helical prop. The straight offset line imposes a constant twist rate on the
blade, but the curved line allows a varying twist rate as required by a true
helical prop. This was described on an Internet site which is no longer
available. There was a computer program, but no explanation of the theory. I
developed an algorithm that is a little different, because I started with a slightly
different set of assumptions. This note explains the theory and method of
constructing an offset curve and gives examples of such a calculation based on
the dimensions of a George Perryman prop and a Pennyplane prop.
 
The twist C of a can formed prop blade is linearly related to the angle T
(theta) of the offset line at radius R from the axle according to the equation C
 = R*sin(T)*360/2*pi*r where C is the twist angle and r is the radius of the
can. The rate of twist is dC/dR = sin(T)*360/2*pi*r which is constant for
fixed radius and offset line angle. Since the twist angle C is the complement of
the blade angle A, A+C = 90 degrees, the rate of change of the blade angle
is the opposite of the rate of change of the twist angle.
 
 We want a curve that makes a different angle T at each radius R such that
the rate of twist corresponds to the rate of twist of a helical prop blade.
The blade angle for a helical prop is given by A = arctan(P/2*pi*R) where P is
the constant pitch of the prop. When the offset angle T is small, the values
of R along the radius perpendicular to the prop axis and R along the curve are
 close. Since A and C are complementary angles, the tangents are reciprocal
and therefore C = arctan(2*pi*R/P). We also see that the change, da, in
perpendicular offset from the baseline to the offset line per change in radius,
dR, is da = dR*sin(T) = dR*2*pi*r*C/360*R. Substituting arctan(2*pi*R/P)
for C gives us da = dR*2*pi*r*arctan(2*pi*R/P)/360*R. If we define the
distances along the baseline as x, we get dx = SQRT(dR^2-da^2). (We can also use dx =
dR*cos(arcsin(2*pi*r*arctan(2*pi*R/P)/360*R)) to calculate dx, but the other
is simpler to calculate and gives identical results.) These differential
equations could perhaps be integrated to give an exact equation for the offset
curve, but I chose an approximate calculation method. I stepped along the
offset radius curve in fixed small increments of the radius, DR, calculating the
a increment, Da, and x increment, Dx, accumulating each to get successive
values of a and x as I went along to the tip, as shown in the following
examples.
 
A George Perryman Propeller
 
“The drawing for the Great Speckled Bird that was in the 1992 NFFS Sympo
shows prop blades about 10 inches long and says that they are formed on a 6 inch
diameter form at an angle of 15 degrees which gives about 45 degrees total
twist. The prop is supposed to be 26 inches diam by 28 inches pitch.” From
an Internet discussion.
 
This reports the method of setting the can formed blade to achieve
approximately the desired helical P/D with the example of a prop attributed to George
Perryman. Later, the coordinates of a Piednoir curve are calculated to lay
out a curved blade radius producing an exact helical pitch.
 
The propeller has a diameter of 26" and a pitch of 28". The blade has a
length of 10", extending from a radius of 3" to the tip at 13". The nominal P/D
is 28/26 = 1.077. For a helical prop, the blade angle at the tip would be
arctan(28/26*pi) = 18.92 degrees. The blade angle at the root would be
arctan(28/6*pi) = 56.05 degrees. The difference in these angles is 37.13 degrees, not
45 degrees. This is the total twist along the 10" length of the blade. The
blade twists at a uniform rate of 3.713 degrees per inch. The required offset
angle for a 6" diameter can is arcsin(2*pi*3*37.13/360*10) = 11.21 degrees.
(Note this is not the 15 degrees stated in the original description. More on
that later.) This blade may be set so that both ends have blade angles as stated
above, corresponding to a pitch of 28" at both the root and tip of the
blade. From the uniform rate of twist, the blade angle at each 1" station along
the blade may be calculated, as well as the local pitch and P/D. I find that
the pitch has a maximum value of 38.56 at 8" radius and a P/D of 1.48. The
average pitch is 34.43", corresponding to an average P/D of 1.32. This is not the
prop we thought we were going to get. Rather than setting this prop at the
tips, let's set it to match a helical blade angle at the 70% radius, 9.1".
The required blade angle is arctan(28/18.2*pi) = 26.09 degrees. Again, from the
uniform twist rate, we can calculate the blade angles, local pitches and
P/D's for each 1" station along the blade radius. The blade angle at the tip is
11.61 degrees, with a pitch of 16.78" and P/D of 0.645. The root blade angle
is 48.71 degrees with a pitch of 21.46 and P/D of 0.826. The highest pitch is
29.52" at 7" radius and the P/D is 1.135. The average pitch is 25.24" for an
average P/D of 0.971, about 10.9% low. Note that there are two points along
the radius where pitch is 28". The can formed blade angle line crosses the
helical blade angle curve in two points.
 
It is possible that another set point will produce a closer average pitch
and nominal P/D.
 
I tried calculating a blade formed at a 15 degree offset angle on a 6"
diameter can and found that it is not possible to set it so the root and tip blade
angles match any helical prop. The blade will have 49.43 degrees of twist
from root to tip, at a uniform rate of 4.943 degrees per inch. There is too much
 twist on this blade to fit any helical curve between the root and tip radii.
 
The blade formed at a 15 degree offset angle may be set to match the helical
blade angle at 70% radius. Again, the 28/26 P/D helical blade has a blade
angle of 26.09 degrees at a 9.1" radius. The uniform rate of twist allows
calculation of blade angle, pitch and P/D, as before. The blade angle at the root
is 56.24 degrees, the pitch is 28.20 and the P/D is 1.085. The blade angle at
the tip is 6.81 degrees, the pitch is 9.75 " and the P/D is 0.375. The
maximum pitch was 33.25" at 6" radius, with P/D of 1.279. The average pitch is
26.23" for an average P/D of 1.009.
 
Although it may be possible, with judicious selection of can radius, offset
angle and set points, to closely approximate a desired average pitch, there
are significant departures from the desired P/D, pitch and blade angles. What
do these significant variations do to the efficiency of the prop? It seems
that much of the prop will not be operating at an efficient blade angle.
There is a way to vary the rate of twist to more closely approximate the blade
angle curve for a helical prop. It utilizes an offset curve with a varying
angle to the baseline.
 
The coordinates of the Piednoir curve for this prop, 26" diameter, 28"
pitch, formed on a 6" diameter can, were calculated. Using DR = (1) simplifies
the calculations. The numbers are calculated from Da = (1)*
2*pi*r*arctan(2*pi*R/P)/360*R and Dx = SQRT(1-Da^2). The values of a and x are the cumulative
sums of Da and Dx. R is the distance along the radius curve, x is the
distance along the baseline and a is the perpendicular offset from the baseline to
the radius curve.
R Da a Dx x
1 0.6622 0.6622 0.7493 0.7493
2 0.6328 1.2950 0.7743 1.5236
3 0.5925 1.8875 0.8056 2.3292
4 0.5486 2.4361 0.8361 3.1653
5 0.5057 2.9418 0.8627 4.0279
6 0.4660 3.4078 0.8848 4.9128
7 0.4302 3.8381 0.9027 5.8155
8 0.3985 4.2365 0.9172 6.7327
9 0.3703 4.6069 0.9289 7.6616
10 0.3455 4.9523 0.9384 8.6000
11 0.3234 5.2758 0.9463 9.5463
12 0.3038 5.5796 0.9527 10.4990
13 0.2863 5.8658 0.9581 11.4571
 
Note that the straight line connecting the end points of this curve, plotted
flat, has a length of 12.87", about 1% short of the intended 13" blade
radius. The swept diameter will not be the full 26". The angle between the
baseline and the line connecting the endpoints is 27.11 degrees, the angle which
the radius curve chord should make to the plane of rotation at the hub.
 
Piednoir Curve for a Pennyplane Propeller.
 
I calculated the x and a coordinates of the Piednoir curve for laying out a
helical prop of 12" diameter and P/D = 1.9 on a 4" diameter can. I used 1/4"
increments along the curve and calculated the angle of the curve at the
midpoints of each increment to improve the accuracy of the calculation. The
coordinates are:

x a
0.00" 0.00"0.21 0.14
0.42 0.28
0.63 0.41
0.84 0.55
1.05 0.68
1.26 0.81
1.47 0.94
1.69 1.07
1.91 1.19
2.13 1.32
2.34 1.43
2.57 1.55
2.79 1.66
3.01 1.78
3.24 1.88
3.46 1.99
3.69 2.09
3.92 2.19
4.15 2.29
4.38 2.39
4.61 2.48
4.85 2.57
5.08 2.66
5.32 2.75
 
The 45 degree blade angle occurs at 3.63" radius.
 
This produces a curve which departs only 0.2" from a straight line drawn
with an offset angle of 27.32 degrees. It might seem that there would be little
difference between a blade laid out on this curve and one laid out along a
27.32 degree straight line. That is not correct, because a 0.2" difference on a
2" radius can makes an angular difference of 5.73 degrees, enough to push an
efficient blade attack angle close to a stall. The difference is even more
important for lower P/D props because there is an even greater difference in
blade angle along their radius.
 
This produces a curve which has a 5.98" chord plotted flat and 5.89" when on
the can.
 
To use this curve to form a prop, plot x and a on a piece of paper with x
going to the right and a going up. This makes a right handed prop. Plot the prop
 chords parallel with the a axis, centered on the curve. Connect the dots to
form the perimeter of the blade. The leading edge is the top curve. Use this
shape to make a cardboard template for cutting out blades. The blades are
made from 1/32" sheet balsa. Orient the x axis parallel with the grain. Mark the
45 degree chord on both sides of each blade. Make as many paper patterns as
you want blades. Wrap the paper patterns around the can(s) with the x axis
parallel with the centerline of the can and the a axis circumferential. You may
want to cover the patterns with plastic film to protect them from the we
tblade.
 
The blades on a 12" prop are typically 5" long with an inch of balsa dowel
projecting inward from the root. The dowel is a tight friction fit in a paper,
plastic or aluminum tube hub. It will be necessary to place the dowel from
the prop center point at an angle of 27.32 degrees to the chord of the radius
curve and projecting about an inch into the blade. A slot is cut in the blade
for the dowel to fit inside. The slot should be incorporated into the
cardboard blade pattern. When the dowels are fit into the tube hub, the chords
marked at the 45 degree radii can be used to set the angles of the blades in the
hub. Set the marked chords so they make a right angle with each other and the
prop shaft bisects one of the right angles. You can judge this very closely
by eye. You also can lightly strap bits of 1/16" square balsa to the blades
at the 45 degree chords with lose rubber bands, so the sticks project beyond
the blades. You can sight along the diameter of the prop and set the sticks
to form the proper angles.


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Received on Sun Dec 03 2006 - 04:12:40 CET