Back from vacation. Boy, does hanging out at an airport help when trying to understand aircraft!
To recap: Much wrangling over tail size, worried about flat spin recovery procedures. Spent empty time imagining an aircraft in a spin, the forces and the possible ways out of the situation. Yep, that was me at the airport with my hand outstretched, wing-fingertips falling and twisting through space, watching the plane fly in a curved spiral. I get it this way:
The airplane is falling in a nose down attitude, flying down in a tight circle around a central vertical line. Gravity and drag vectors combine with a centripetal force vector on one side of the equation, but lift from the high wing keeps everything in equilibrium. The force vector from the high wing supports the vertical component and centrifugal components, and the wing's attitude, along with windage against the tail determine the radius and rate of the spiral. Its a falling maple seed, the 'helicopters' we watch falling from trees. If the spiral becomes too tight, then the vertical fall rate becomes large enough to provide lift to both wings and we're in a spiraling dive, which should be easy to recover from with ailerons and elevator, but this maneuver will burn up altitude. It is the slower rate of descent that keeps the inside wing stalled and perpetuates the condition. We're flying, but we're also out of control.
This shows a hypothetical situation, a top-down view. The radius of the spiral, to the aircraft CG is 15 feet. I assume a 2 second rotation period and a 20 ft/sec descent rate to get some starting numbers. The velocity of the wind across the high wing, Vh, at a radius of 20', combined with descent velocity is about 45MPH, while the low wing at a radius of 10' has a Vl of only 23 MPH, with the vector coming from a much steeper angle because the descent air velocity is proportionately large; the lower wind speed and the higher angle of attack combine to fully stall the inner wing. The g force outward is about 1 g, making the total g force seen by the pilot of about 1.4g. The airspeed across the tail is not insignificant, and we can imagine that if a nose down force can be generated by the tail, a spiral dive could be achieved that would give lift to both wings, a condition we should be able to recover from with ailerons and an up elevator pull-out.
The outward-rudder approach to recovery, followed by down elevator to increase airspeed seems like the recovery technique that would give the least loss of altitude. This does not mean that down elevator first to cause a rolling dive followed by ailerons and up elevator to recover won't work, it just uses up more altitude. We can see how ailerons won't do much initially, but to slightly change the radius of the spiral. The inner wing aileron doesn't have much air speed to affect things. The outer wing can be made to have more lift with down aileron, but this just changes the conditions of the spin, it doesn't stop the spin. Up aileron on the outer wing might make the spiral radius increase, but it surely would make the descent rate increase, as the wing will have lower lift, and is not flying around the spiral as effectively. -This does not mean that up aileron on the outer wing, giving more potential lift to the inner one (when air speed becomes sufficient) is not a bad idea, it's only a partial solution; ailerons won't fly us out of the spin.
The NASA research didn't include the prop gyroscopic action, which I think is a serious flaw. The torque provided by the prop, heavily depending on engine RPM, is one of the forces that I haven't included in the drawing, but is important nonetheless... Most single engine aircraft rotate the prop in a clockwise direction, when viewed from the cockpit. This means that when you're at the top of a stall, slowing in airspeed, the torque from the engine will roll the aircraft to drop the left wing. This is why single engine tractor aircraft enter a counter-clockwise spin (when viewed from above) in every case of level, power-on stall. Additionally, when the nose drops, the gyroscopic forces of the prop cause the aircraft to yaw to the left, and bingo, we're in an uncontrollable spin. Because the aircraft in the spin condition is flying about a central point, the gyroscopic action of the prop causes a nose-down torque to the airframe. -In this case, powering up with down elevator and left pedal should be the best course to obtain a sharp vertical descent from which recovery is possible, but only after burning up tremendous altitude! -The gyroscopic action of the prop will force the aircraft to eventually point down the spiral's axis.
The spin is a condition that has several potential exits, it's just that most of them burn so much altitude that you're in the dirt before the recovery is complete. The trick to recovery is to do it with a minimum of altitude loss. This, and with a pilot that is bug-eyed and soiling his pants!
The tail however (which is what this is about), doesn't need to be out on a long arm; it would seem that this would only cause the spiral to have a slightly larger radius. If the tail is out on a very long boom, it may have the ability to actually prohibit a spin condition, but this would require a very long and sluggish airplane! What is important is that the rudder be effective to cause a sufficient yaw torque to the aircraft, causing the inner wing to pick up sufficient speed to get us out of the spiral condition. The aircraft CG has a momentum determined by its velocity of about 50 ft/s, and if the aircraft yaw attitude can be abruptly changed to make use of this momentum to sharply increase the velocity across the inner wing, lowering the velocity across the outer one, then the spin is abruptly terminated. Further, in a generally nose-down condition of the spin, we exit at an angle that is ideal for air speed recovery. What I'm getting at is that in an aircraft that has a small moment of inertia, one with short stubby wings and a short tail, the exit strategy for a spin is to use these properties to 'snap' the airplane in yaw with a large surface rudder. -After all, we have a reasonable airspeed along the center of the aircraft, and therefore momentum from the mass of the airplane, and we just need to transfer this momentum to a yaw component, countering the spin. Note: This is only possible if the engine RPM is at idle, as a quick right-yaw action to the airframe will cause the gyroscopic action of a high speed prop to pitch the nose up, losing the present angle of descent we need for recovery, and robbing energy from a yaw change by translating it to a pitch change that we don't need.
For our 'cute' little airplane, we don't need a large broad side cross-section to the fuselage, or necessarily a high elevator, but we do need a rudder that has extreme control, and not shielded by the elevator. The fact that our tail and wings are 'stubby' works to our advantage if we have a strong, large rudder, coupled to the aircraft with strong support, because our aircraft's moment of inertia is low, and can be 'snapped'...
I like this. To make it work, we need control over the rudder that is much stronger than that required to just fly the airplane in a peaceful, straight line. In fact, if we are in a counterclockwise spin-stall and just slowly push the right pedal, we will probably just change the conditions of the spin, but not really change the fact that we have no control. This is because the only way to exit is to transfer forward momentum to a yaw component, not just slowly change the yaw condition; I'm convinced that we could continue in a counterclockwise spin with full right rudder, if it is applied slowly, simply changing the spin condition, not removing it. To exit the spin we need to torque the entire aircraft in yaw abruptly, before the new spin condition can develop, which means the ability to apply a very strong force to the rudder controls immediately. They say that you should push on the rudder pedal that is stiffest; I say that you chose the pedal that's pushing back and kick the hell out of it! -This means the control cables must be very strong and the rudder attachments and control horns need to be robust.
You're pushing against a pedal that will put the rudder directly across the wind, and make use of that wind speed to kick the plane around to the right. If you take too long, the new attitude of the airplane will slow the airspeed across the rudder and make it's effect insufficient, especially in an aircraft that has a short tail. It's the rate of yaw that causes the dropped wing to gain lift quickly, giving the aircraft a new attitude that delivers more equal lift to the two wings. This clockwise torque to the aircraft requires a strong force. It is the force that accelerates the left wing and decelerates the right wing. We don't recover by changing the attitude of the airplane, we accelerate it in a clockwise direction, which requires tremendous force. We need to do it quickly too, which makes the short tail boom work to our advantage; the tail doesn't have to move very far as the airplane yaws to the right, as long as it applies the torque force quickly and forcefully.
There may be no possible way to statically fly your way out of a spin, depending on the aircraft, at least without taking the nose-dive, altitude reducing route, but there may be a way to dynamically recover, and very quickly!
In this hypothetical spin, the aircraft is rotating in yaw at a continuous 180 degree per second rate. We need to change this to zero degrees per second through transferring forward momentum to angular acceleration in yaw. If we can do this, we will have equal air speed across the two wings and we will be flying again, under control. -At least we soon will be, as the nose down attitude of the spin condition is ideal for air speed recovery once our yaw orientation is correct.
I calculate that our 470 pound aircraft, traveling at about 50 feet per second, has a momentum of 23,500 ft-lbs/s and a kinetic energy of 25,000 joules. The aircraft has a moment of inertia around the yaw axis of about 4,000 lbs-ft^2, or 125 slug-ft^2. In a static condition, we could rotate this angular mass around the required 180 degrees with a very small torque, simply taking time to do it. In this dynamic situation, we can't wait around, because the aerodynamic situation is constantly finding equilibrium, and if we apply a wimpy torque, the equilibrium will simply shift, and we won't affect any real rotation. How quickly do we need to affect the rotation then?
Let's try to accomplish the task in 1/4 spin-turn, or 1/2 second. The rudder needs to apply a force to the tail arm that torques the airframe to a 180 degree per second rotation rate (pi radians/s) in 1/2 second; we're calling this an acceleration, but really it is a deceleration, from the spin to straight. This will require an acceleration of 4pi radians per second squared, or to our moment of inertia we will need 1570 ft-lbs of torque for 1/2 second. At a lever arm of 6 feet, the force on the rudder will be about 262 pounds. That's a lot of force for a small plane, and I can't imagine the rudder connections or a human that could stomp that hard, but if we wanted to do the job in 1 second (1/2 normal spin turn) it would take only 66 pounds at the rudder, which seems more reasonable, but can we wait that long? Will the aircraft simply shift its spin equilibrium in this rather long time, leaving us with a continued spin at a different attitude? We need to know so we can proportion the rudder dimensions to provide the required force at the expected airspeed.
Further, if the sideways force on the rudder is equal to the axial force on the rudder, the craft will lose some forward velocity. A 262 pound decelerating force for 1/2 second on a 470 pound body will reduce our forward velocity by about 9 ft per second, leaving the wind speed across both wings at about 41 ft/s. If the operation is conducted over a full second of time, at only 66 pounds, the forward velocity will only be reduced to about 45 ft/s.
If the rudder force is applied at the 262 pound value for 1/2 second, the peak increase in lower wing velocity will be from 23MPH to about 40MPH and the the high wing will go from about 45MPH down to 28MPH. This says that the 1/2 second solution may be too extreme. In the 1 second, 66 pound case, the low wing goes from 23MPH to 33MPH and the high wing goes from 45MPH to about 35MPH. I'll expect that the 1 second, 66 pound side force on the rudder will do it, and go in search of a tail area that can provide this force at an air speed of 50 ft/s.
This means a minimum 5 square foot rudder at about 40 degrees of deflection on a 6 foot moment arm.
From yet another point of view, looking at the static conditions of spin, that is the attitude, forces and velocities for a given control configuration, our interest comes in comparing a neutral control spin to one that comfortably accepts a static hard right rudder condition. We're concerned here about the time it takes the aircraft to transition from the neutral case to the hard right rudder case, so we can estimate how quickly the torque must be applied to correct the spin situation before the airplane assumes the new hard rudder spin equilibrium.
With the rudder over hard, the aircraft will be flying slightly sideways. The drag will be much greater, as the rudder will prohibit a smooth airflow past the structure and this will translate into a greater descent rate and a steeper downward angle; the increased airspeed is required to generate the lift to support the weight and centripetal forces. Also, the air will no longer be impinging the high wing as directly; the airstream over the high wing will be slipping off it's wingtip. This all demands a higher airspeed to balance the equation of the spin model. Now, how long does it take for the airplane to go from the neutral rudder configuration? In freefall, a body would fall 16 feet in one second, and change its velocity by 32 feet per second. This condition is not freefall however, as the difference in the forces between the two stable conditions cannot be greater than gravity itself... I feel comfortable with the 1 second, lighter force solution now, but only if the pedal is shoved to the metal quickly!
Blah, Blah Blah... This problem is not going to be solved here, but I did find a reassuring reference here. The article indicates that slotted wings are very favorable to keep the aircraft from getting into a spin in the first place. I will simply go back to making the Flying Machine as 'cute' as possible, with a reasonable rudder, with stronger than I previously thought control cables. Done.
Jetlag is hell. New side view:
The elevator begins about 28 inches directly above the trailing edge of the flaperons. The Rudder is about 5 sq ft, and will be strongly hinged to the tail. The wing is still about 48" off the ground and the tail is just over 6 feet. 'Cuteness' comes by squashing the design and just fitting the plane around the pilot, minimizing wingspan, increasing the wing chord and setting the tail as close in as possible, making its large size almost comical. The fast rise on the bottom of the tail section allows for a high AOA on takeoff and landing. The stubby shape of the frame allows for rigid construction and light weight.
The spin issue may be unimportant; we have a strong tail to kick the plane into a new orientation and the low moment of inertia should make it a nimble flyer.
Have to complete the rudder pedal/toebrake design before I know about how much the engine can be crammed rearward. We need the pedals to have a high force capability, maybe with a preloaded spring force limit to keep connections from breaking. Would like really linear pedal motion, and a good long throw too. This means some kind of linear travel mechanism.
The tail however, can be detailed and then modified to fit the final pilot cage.
Received info from insurance agent today... It's OK to fly an ultralight! Feeling a great temptation to start with the wings, but I do need to finish out the tail first. This is a real problem, involving the stick and the rudder pedals. Although they are potentially lightweight, arranging the tail components so that they are strong, light and effective (and cute too...) has been driving me crazy. Rudderpost, bearings, elevator placement, how close to the main wing, etc.
Yes, its been a month, but an exciting one! Visited a local private airstrip, 35' by 2000' long, which is small, but they have an available hangar. During my visit, an instructor from another airport came by, and we had the opportunity to talk. He claims that this little strip is too small for flight testing, and after all, he had 20,000 hours of flight experience, he should know... I didn't inquire if he would be interested in applying his experience to flight testing my monstrosity though. He may have thought his advice was a good warning, but I consider it a great challenge!
Discovered a great resource for aircraft design, click here.
If the tail design wasn't enough, I'm leaning toward increasing the wingspan now. I want the thing cute and small, with low moments of inertia for quick maneuvers, but I just don't know if two 6 foot stubby wings with their clever wingtips will do. The main problem is slow flight speed, which must be under 27MPH to meet the rules. I still don't know how this is calculated. Also, I took a short walk down the runway, and even with a 5MPH crosswind my intuition started telling me that things could get dicey at a 27MPH landing speed. In a real airplane, you might touch down at 60MPH, and such light winds don't matter. For the Flying Machine I suppose we could keep the flaps off and land at 45MPH in case of a serious crosswind. The flaps might be just to make the FAA happy about part 103.
About the tail positioning... Here is a link that demonstrates tail position. The idea is that if the tail is close into the wing, then the range over which the CG can be located narrows. If the tail is far to the rear, then the airplane may respond sluggishly. Since the pilot and gas are close to the CG point in the flying machine, I'll expect the tail can be close in, very different from designs where the engine is in back and the pilot sits up front... In such a configuration, the pilot's weight will be a variable, and the tail needs to be further aft to allow for such variation.