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The designing of tails and wings :
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Tails and Wings :
The horizontal tail of a plane allows the weight to move forward and aft more while remaining stable and controllable. Where a plane balances if it were supported at only one point is called the Center of Gravity (CG). The CG can move further forward or aft due to different passenger and cargo loadings, and due to fuel burn (most jets carry about half their empty weight in fuel). All airplanes become unstable if the CG moves aft of a point called the Neutral Point. As the CG moves forward of the neutral point, the plane gets progressively more stable, and progressively needs more up elevator. Elevators on tails can be more effective than elevators on the back of wings, so planes with tails can have a greater CG range than planes without tails. With paper airplanes their CG does not move, so they are fine without a tail.
A tail is also needed to balance the pitching moment (tendency to make the plane rotate nose up or down) caused by flaps. Flaps are the control surfaces on the back edge of the wing which are deflected down to allow the plane to takeoff and land slower. Paper airplanes do not need to fly any slower, so they do not need flaps, or the tail needed to balance the flaps.
The tail of a real plane usually also has a vertical tail. The vertical tail acts like the fins of an arrow to keep the nose of the plane pointed in the direction its headed, this is called positive directional stability. The Fuselage (center body of a plane, on paper airplanes its the part you hold for throwing) acts like the vertical stabilizer of real airplanes. Sometimes bending the wingtips up on paper airplanes also helps to add directional stability. The combination of the fuselage and wingtips on paper airplanes allows them to have positive directional stability without a vertical tailWing Shape :
Paper airplanes usually have short "stubby" wings, called "low aspect ratio" wings. The distance from wing tip to wing tip is called wing span, and the distance from the front to the back of the wing is called the chord. The ratio of wing span to average chord is called "aspect ratio", and is an important characteristic of wings. For subsonic (less than the speed of sound) airplanes wing drag is reduced by increasing wing span and decreasing wing chord, both increase the aspect ratio. For that reason aspect ratio is a good indicator of overall wing drag. Notice that sailplane(glider) designers are extremely concerned with wing drag, and use high aspect ratio (big wing span, narrow chord) wings. Getting back to paper airplanes, or more correctly paper gliders, notice their wing shape is much different from real gliders because they have low aspect ratio wings. There are several good reasons for this difference .
Weight Forward is Good :
As mentioned in section before where a paper airplane balances is called the Center of Gravity (CG), and there is a specific CG position known as the Neutral Point which provides neutral pitch stability. If the airplane has a CG ahead of this point, the plane is stable, if its behind this point its unstable. Naturally all airplanes without computer assisted flight controls need a CG ahead of their neutral point. For rectangular wings the neutral point is ¼ of the distance from the nose to the tail. For delta wings (such as the common dart paper airplane) the neutral point is ½ of the distance from the nose to the tail.
Stability means the plane, if disturbed, will return to its original state. For pitch stability it means the plane will seek a single airspeed. A plane which is unstable in pitch will either pitch up into a stall, or nose dive, but won't settle out anywhere in between. A stable airplane will tend to oscillate up and down a few times, but converge on a steady flight speed. Many typical paper airplane designs are stable, but just barely. As a plane becomes more and more stable, it wants to fly faster and faster. To counter this tendency, up elevator must be used to produce a
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What about the airfoil shape?
Most people who are reading this know that airplane wings are "Cambered" which means they have generally a curved shape, with the top of the airfoil rounded and the bottom fairly flat. As explained before, paper airplane wings must be thin to work well. In addition, they need very little camber, and generally any curvature is limited to the front portion of the wing. . Since most paper airplanes are flying wings, only small amounts of camber are practical, as large amounts of camber create nose down pitching moments which need tails to balance. Generally I do use a little curvature at the leading edge of the wing. I have noticed that paper airplane performance is not noticeably degraded with flat, uncambered airfoils. The reason for this is likely due to low Reynolds numbers. Remember that a large portion of the boundary layer across the front of the wing is laminar flow, but for high lift we need a turbulent boundary layer. The use of a flat uncambered wing produces a large pressure gradient at the leading edge, which likely aids the transition to a turbulent boundary layer, which could likely be the reason for little camber in insect wings. Also, swept wings with uncambered leading edges promote vortex flow just behind the leading edge on the upper surface. Although lift coefficients at these Reynolds numbers aren't large enough to promote a large amount of vortex lift(vortex lift increases exponentially with lift coefficient), any vortex flow likely helps the transition to a turbulent boundary layer.
Ascent
A major reason why the world record plane is successful is the ascent. During the ascent the plane's angle of attack is near zero, resulting in near zero lift and allowing the plane to go virtually straight up. This is crucial for two reasons. In slow flight the plane is adjusted to produce a lift coefficient of about 0.7. If the plane were rigid, it would trim to the same lift coefficient at all speeds, with a sharp pull up into a loop at speeds higher than 10 mph. At the speed I launch it, it should enter into a 40 g loop, but it doesn't. The second reason zero lift is important is because of drag. If the plane stayed at its 0.7 lift coefficient, it would more than double the drag during the ascent and not allow the plane to climb high enough for a record flight (roughly 50% of the kinetic energy from the throw is used to overcome drag, the other 50% is converted into potential energy in the form of altitude). The plane does not go to exactly zero lift, and spirals a bit during the ascent to maintain a near vertical trajectory. Sometimes I have to add some rudder deflection to aid the spiraling to improve the ascent. I have also experimented with introducing intentional asymmetries into the plane to aid spiraling.
So why and how does the plane go to near zero lift? I'm not really certain, but I think I have the answer. As I said, it would trim to a 0.7 lift coefficient and enter a 40 g loop if it were rigid, but it isn't rigid. I suspect the reflexed section (the up elevator) to pushes the rear portion of the wing down, producing a more curved airfoil which wants to pitch the nose down and trim at a lower lift coefficient. Also the weight of the fuselage at the middle of the plane results in a large root bending moment as the plane pulls g's, so that the wings flex upward (added dihedral) which effectively lowers the angle of attack and lift coefficient the plane ascends at, with the wings returning to their original dihedral as the plane slows. I need to take some high speed video to analyze what happens during the launch.
The airfoil of the plane also affects the launch. I have tried using highly cambered airfoils optimized for slow gliding, but they tend to degrade the ascent. I wrote a computer program to reproduce the flight of the world record paper airplane to learn what parameters were most important for a long flight.
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One of the most important things I learned was that Cdo, zero lift wing drag, is more important in the ascent than it is in the descent. The airfoil optimized for slow gliding is not optimized for zero lift, and produces extra drag during the ascent. What is needed is an airfoil which produces low drag during slow, high lift flight, but more importantly has low drag during the ascent. I believe a nearly flat, uncambered airfoil does this. Certainly a flat airfoil is ideal for low drag at zero lift, but it can work at higher lift coefficients also. The flat wing at high lift results in a steep pressure gradient near the front of the wing on the upper surface, which likely aids transition to a turbulent boundary layer which is needed for low drag at high lift. I plan to do more airfoil tests during the spring of '97 to help find the best airfoil for long flight.
I have found pitch stability to be important also. The plane not only needs to be stable, but it needs to have just the right amount of stability. Pitch stability is controlled by how nose heavy the plane is, and that is controlled by the size and number of folds down the sheet of paper. The flexibility effects apparently only produce a small change in pitching moment, so the stability must be fairly weak to allow a significant change in trim angle of attack. Too little stability results in erratic gliding flight, with frequent stalls as the plane drifts slower than the desired angle of attack. One way to improve gliding stability is to tighten the turn radius. As a plane circles in flight it introduces a pitch rate. Natural pitch damping tends to try to nose the airplane down with positive pitch rate. As pitch rate increases with angle of attack, so does the nose down pitching moment due to pitch rate, thus providing added pitch stability to the plane. The tighter the circling, the better the stability. A drawback to this scheme is the increased load factor, and degraded gliding performance as the plane circles more tightly. Many times I set the circle size, by adjusting the rudder deflection, just small enough to keep the plane from porpoising (pitching up and down) into a stall. Generally circles less than 20 or 30 feet in diameter noticeably increase sink rate
Gliding Flight
The goal for gliding flight is to descend vertically as slowly as possible. This represents the lowest rate of change of potential energy(power) which is the minimum product of drag times velocity. Generally the minimum sink rate for gliders is just above stall, and that's true for paper airplanes as well. For those interested in the details and math, finding the minimum power required involves taking the equation for powered required, differentiating with respect to velocity, and setting this equal to zero (standard calculus procedure for finding the minimum or maximum of a function. Starting with the basic parabolic drag curve;
D=.5 * rho * v*v * S * ( Cdo + Cl*Cl/(pi*e*AR))
D=drag in pounds
rho=air density (slugs per cubic foot, .002377 at sea level)
v=paper airplane velocity (ft/sec)
S=wing area (square ft, .234 for world record plane)
Cdo=Drag coefficient at zero lift (about .07)
Cl=lift coefficient (about .7 for minimum sink)
pi=3.1415
e=span efficiency factor, estimate .7
AR=span/average chord=7.5"/4.5"=1.67
Converting Cl in terms of v (cl=2*wt/(rho*v*v*S)) wt=weight (lb, .01 for a sheet of paper)
and multiplying times v yields
Power=P=.5*rho*v*v*v*S*Cdo + 2*wt*wt/(pi*e*b*b*rho*v) (ft-lb/s)
b=span, ft
Differentiate, set equal to zero, yields Cl=sqrt(3*Cdo*pi*e*AR) and therefor v=sqrt(2*wt/(rho*S*sqrt(3*Cdo*pi*e*AR)))
This gives a lift coefficient and airspeed for minimum sink rate of about .7, and 8.4 ft/s (6 mph)
Substituting the minimum sink results into the power equation, and knowing that vertical velocity is power/weight, gives the following:
Min Vert Velocity=Vvmin=1.05*(rho**-.5)(f**.25)(wt**.5)(e**-.75)(b**-1.5) (ft/sec)
f=Cdo*S
This equation
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Interesting Things about Wings
These interesting facts about wings are useful in developing a more detailed understanding of how they work.
Wing Shape
The "standard" airfoil shape that we examined above is not the only shape for a wing. For example, both stunt planes (the kind that fly upside down for extended periods of time at air shows) and supersonic aircraft have wing profiles that are somewhat different than you would expect
The upper airfoil is typical for a stunt plane, and the lower airfoil is typical for supersonic fighters. Note that both are symmetric on the top and bottom. Stunt planes and supersonic jets get their lift totally from the angle of attack of the wing.
Angle of Attack
The angle of attack is the angle that the wing presents to oncoming air, and it controls the thickness of the slice of air the wing is cutting off. Because it controls the slice, the angle of attack also controls the amount of lift that the wing generates (although it is not the only factor).
Flaps
In general, the wings on most planes are designed to provide an appropriate amount of lift (along with minimal drag) while the plane is operating in its cruising mode (about 560 miles per hour, or 901 km per hour, for the Boeing 747-400). However, when these airplanes are taking off or landing, their speeds can be reduced to less than 200 miles per hour (322 kph). This dramatic change in the wing's working conditions means that a different airfoil shape would probably better serve the aircraft.
To accommodate both flight regimes (fast and high as well as slow and low), airplane wings have moveable sections called flaps. During takeoff and landing, the flaps are extended rearward and downward from the trailing edge of the wings. This effectively alters the shape of the wing, allowing the wing to turn more air, and thus create more lift. The downside of this alteration is that the drag on the wings also increases, so the flaps are put away for the rest of the flight.
Slats
Slats perform the same function as flaps (that is, they temporarily alter the shape of the wing to increase lift), but they are attached to the front of the wing instead of the rear. They are also deployed on takeoff and landing.
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