My attention has been drawn in recent times to certain odd
and apparently inexplicable occurences in the operation of
control-line model aircraft; particularly, but not only, in high-
performance racing models.
In F2B, Brian Eather observed to me that electric F2B ships
have been found to hold out better on the hourglass bunts if the
prop is running in reverse rotation! The reason?. Gyroscopic yaw
outward on down elevator. So much for slow ships.
Now onto F2C and F2A. The word is that very stiff carbon
F2C models are slower than their balsa counterparts. So
much for hi-tech materials. Not only that, but the same applies
to F2A speed models. Thumbing thru Grant Lucas' collection of the Speed Times, I came across an article by Paramon, dated
in the late 1980's, wherein he described the materials used in his
model construction. The material was some sort of plastic that
read like a vibration damper. Of course, wood is famed for that
very property.
Then in ACLN for January (2010), there is a report of APC
prop failures in Class2 T/R. Fred's response was to make the
hub thicker, which is logical IF the failure is tensile. I have had
F3D prop failures in that region: the stresses could be read
before the prop failed, which was revealing. There was wrinkling
on the airfoil upper surface at the root. The wrinkling was
concentrated at the airfoil high point. I took this to be a
compression failure due to high impulsive torque loading
from the firing cycle. The cure was not to thicken the root, but
to flatten out the upper surface of the airfoil to distribute the
compression loads more evenly over the "surface fibres".
But now I'm not so sure. I have just read a dissertation on the
gyroscopically induced vibrations associated with 1 and 2 bladed
props. Whoops, this was new territory for me. I had the data, but
my mind was not prepared to receive it. Nothing beats having a
prepared mind. In fact, if you care to read this article many times
you may find understanding finally dawns: "En repetito et
studentum". But having studied the strange forward C/G location
of C/L speed models in terms of gyroscopic moments, I was
quite alert to this "new" phenomenon. Well, new to me, anyway.
I also found the gyro vibrations described in a book dated 1944!
So now its time to move on and have a closer look at these
vibrations, their strength and their source. This was a little tricky,
as I found errors in both my source materials. Normally, I
don't describe material that has not been fully explored by others.
Timid, I guess, but my main achievement in Physics was always
getting the wrong answers! But this time I found a weakness in
the texts where single-bladed propeller operations were described:
mainly in the lack of emphasis on single bladed props.
Not too strange a lack of emphasis, I suppose. Not many full size
planes use just one prop blade. In fact, the only one I can think
of was the pre-WW II Everel light aircraft prop. One blade was
used as it was easier to mount the blade to provide variable pitch.
Now lets get back to the notion of propeller as gyroscope. A true
gyroscope is pretty much like a flywheel. The mass is distributed
as a disc, Frisbee like. This disc spins on its axis, which lies thru
the C/G of the disc. The fact that the disc is spinning, giving it
rotational inertia, means that this axis does not want to change its
orientation. If a force is applied to make this axis change
orientation, say thru an angle of 10 degrees, then the gyro
surprises the unwary. Rather than just move to its new orientation,
the axis instead starts to wobble.
Now don't get too comfortable. You are sitting on a gyroscope
called planet Earth. And it is wobbling, with a period of 440 days.
What is more, the sun and moon apply gravitational forces which
cause, with a period of 26000 years, more wobbles which are
called precession!
Back to reality (propellers, that is). Now propellers are not discs,
so are not true gyroscopes. But they do have the rotational inertia,
also called angular momentum, that causes all the weirdness with
gyroscopes. So what does happen with the quasi-gyroscopic
propellers? Consider a single blade prop, as it moves around one
revolution.
If we push on the prop rotation axis when the rotating prop is lined
up with the yaw displacement, then the whole length of the blade
has to move, which suggests a large force. If the blade is vertical to
the push, then that suggests a small force is needed to twist a little to
the new orientation. Well, that sounds right. But what really happens
is that when the blade is vertical, the blade wants to tilt forward out
of the plane of rotation. The reaction to the push takes place ninety
degrees later in the direction of rotation. Gyroscopes are tricky.
It follows that the force needed to re-orientate the prop varies right
around the rotation. With a little bit of thought, it appears that the blade
has high inertia at 2 locations, as the blade is aligned with the push force
twice. Not good. The changes in inertia imply changes in acceleration: just the conditions for vibration to occur.
So with a single bladed prop, just because the airplane wants
to change direction (consequently changing the axis of rotation of the
prop) we will have an onset of vibration, with a frequency double the
shaft speed. The change of direction is about an axis we call the axis
of precession. In the case we will consider, the axis of precession
for a control-line speed model is vertically down thru the prop disc.
Alternatively, the axis is down thru the pilots head to his heels; same
axis.
Now to apply this thinking to props with more than one blade. Below
are the facts of the matter.
The quantities below are:
M : the gyroscopic moment
I : the moment of inertia
w : the propeller rotation angular velocity, in radians per second
W : the precession angular velocity, about the axis of precession
x : the angle of the prop blade starting at 90 degrees to the precession axis
Don't be too alarmed by this algebra stuff, I will illustrate with examples.
We have:
For a single bladed prop, M = I.w.W.Sin^2(x)
For a 2 bladed prop, M = 2. I.w.W.Sin^2(x)
For a 3 bladed prop, M = 3.I.w.W/2
For a 4 bladed prop, M = 2.I.w.W
For a 5 blades prop, M = 5.I.w.W/2
For an n bladed prop, M = n.I.w.W/2 n>2
Now some of this is weird. The gyroscopic moment for 1 blade and 2 blade
props depends on the position of the blade as it rotates (that is the meaning
of Sin^2(x)). But for props with more than 2 blades, the position of the prop
blades does not matter. In fact, propellers with more than 2 blades run
smoothly, by not producing unbalanced gyroscopic forces at the shaft.
Understand what is meant here by gyroscopic moment. It is the load
transmitted to the crankshaft by all the blades acting together. The bending
moment of each blade at the blade root (or shank) is just the same for each
blade as for a single blade propeller.
But the single and two bladed props do produce vibrations as a result of
their gyroscopic character: this vibration is due to the forces created by
the blades, acting at the shaft. In terms of the mean taken over the angle
of rotation, from 0 to 2.pi, the average gyroscopic moments are:
For a single bladed prop, Mav = I.w.W / 2
For a 2 bladed prop, Mav = I.w.W
By now you are, if anything like me, rather confused as to what these
symbols are about. To flesh them out, consider an F2A speed model in flight,
with RPM 40000 and airspeed 290 kph (or thereabouts). We have:
w = propeller angular velocity
= 40000 . 2. pi / 60
= 4188.8 radians per second.
Now W is the precession angular velocity, which is the angle swept out by
the model every second. Well, this is easy, the pilot needs to go thru one
lap (2.pi radians) in about 1.4 seconds, so:
W = 2.pi / 1.4
= 4.488 radians per second
The value of I, the propeller moment of inertia, is a bit more tricky, as it
depends on the shape of the blade. But first, what on earth is it? If you
throw a brick, you find it resists motion due to its mass. The need to
throw hard, is the meaning of inertia. Also, once the brick is moving, it
doesn't want to stop: we say the brick has inertia. Well, rotating bodies
also have inertia. They don't like being spun up or slowed down. We call
this rotational inertia: instead of referring to mass, we refer to the rotating
body's moment of inertia. Why not just use mass, as for the brick?
Well, its harder to get long bodies to spin than short bodies. So rotational
inertia depends not only on mass, but how that mass is shaped. We still
use mass, but have to add a new idea called radius of gyration. This radius
of gyration takes into account the shape of the body: then the product of the
radius of gyration and mass is called the moment of inertia, I.
But now for the trick. We wanted I, the moment of inertia, for a propeller
blade which is an irregular shape. Because most solid prop blades have
much the same shape, we can use an average value for the radius of
gyration. This value is 0.55 times the blade length (single blade, not
the whole prop length). So if our single prop blade is 90mm long (.09 M)
and weighs 2 grams (.002 Kg) then we have:
I = (0.55 * .09)^2 * .002
= 0.0000049 M.Kg
Then the thing we were after, the gyroscopic moment M, for a
single blade, is, on average:
M = 0.0000049 * 4188.8 * 4.488 / 2
= 0.046 N.M
The peak value is greater, double the above, or 0.092 M.Kg
Well, is this a large number, or an insignificant force we can neglect?
Compare it to the torque required to turn the prop. The engine in an
F2A ship gives something like 1.6 HP (1,200 watts) at 40000 RPM.
Then since power is torque times revs:
Prop torque = power / revs per second
= 1200 / (40000/60)
= 1.8 N.M (Newton.Metres)
Now torque and moment are the same thing. The names are different: torque is usually used for rotation of shafts, moment for lever arms.
They both refer to a twisting action, whether the prop blade is trying to
bend the blade against the shank, or the crankshaft is trying to rotate
the propeller around.
For the peak gyroscopic moment, we have 0.092 N.M, and for the average
engine torque we have 1.8 N.M. Now we have a handle on this moment
number business, and it is a worry. The gyroscopic imbalance acting on the
blade shank is about 1/20th of the average torque rotating the prop!
Definitely not negligible!. And remember, the gyroscopic moment varies from
zero to maximum twice every revolution. You want vibration? We got vibration
in spades!
This gyro moment is trying to rip the blade off, not by centrifugal action, but
by bending the blade out of the plane of rotation! [note that I have ignored the crankshaft/spinner gyroscopic moment.. I am only interested in the moment
acting on the prop]
Note also, the gyro moment only exists when the airplane is changing direction.
Thus flying in a circle (control-line), looping, or turning around a pylon, adds
a stress to the prop that is not present in straight and level flight. The propeller
blade shank, whatever the number of blades, must be made strong enough
to resist this bending.
Now this is non trivial. Just making the shank thicker won't do. Bending loads
are felt at the shank surface. The surface fibres are more heavily loaded than
material deeper into the prop shank: the surface fibres will fail first, followed by
those underneath, which have become the new surface fibres!
The greatest resistance to failure in the torque plane is thus when the
side of the shank is flat, thereby placing more fibres into a state of compression,
than, say, in a round shank. Similarly, for bending forward out of the plane of
rotation, the front of the shank should be flat.
Time to conclude. The mere fact that a controline model is flying in a circle
generates vibration of a non-negligible magnitude. The frequency of this
vibration is double the number of revs per second. If the number of blades is
greater than 2, then this vibration is balanced, the only vibration being that
inherent in the engine itself. However, no matter how many blades there are,
the blade roots (shanks) experience a bending moment out of the plane of rotation
which must be countered by the strength of the blade root.
This still leaves us with a problem. Most control-line racing models have either
one or 2 blades. Experience seems to indicate that the construction of the model
needs to have a vibration damping character. But there is another way of handling vibration, other than damping. The other method is called decoupling.
If the blade is hinged near the shaft, so that it free to move fore and aft
independently of the shaft, then the vibration will not be transmitted into the shaft.
This hingeing has been used for years in F1C free-flight power models, with the
intention of allowing the prop to fold, thereby reducing drag on the glide.
Decoupling can also be achieved, in part, if the propeller is flexible. A prop needs
to be rigid in torsion, but not so much in bending.
Let's not conclude just yet, not while we are having fun, after having stuck this
far into the weird world of the gyroscope.
This fact rang a bell: the gyro induced vibration occurs at twice the frequency of
rotation. Where have I heard that before? The vibration of a single cylinder engine
has primary and secondary components: the secondary components also run at
twice the frequency of rotation. Would it not be nice if these sources of vibration
were to cancel each other out, at least to some degree?
At first glance, there is a problem. The force involved in the gyro varies from zero
to a maximum value: it does not reverse direction. The secondary component of
engine vibration does reverse direction, so complete cancellation is unlikely. But
beggars can't be choosers: what is left? Partial cancellation is possible, if we can
just get the vibrations in antiphase.
The engine secondary vibration is a maximum at top dead centre (TDC), in line
with the piston motion axis. We need to position the prop so that its gyro moment
is then at a minimum, which occurs when the blade is in the plane of rotation of the
aircraft. So, rotate the engine to TDC, and move the prop horizontal (at least for
F2C team race). This should be the elusive "sweet spot" for propeller orientation.
Now we have the F2A airplane nodding its nose, about 1300 times per second, with
the engine either upright or inverted. Regretably, most F2A speed models have the
engine in the plane of precession of the model about the pilot (sidewinder), so that the secondary engine vibrations are perpendicular to the nodding nose; there is then
no possibility of cancellation.
There you have it!.
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