Q-talk 100 - Tandem Wing Airfoil Selection Part 2
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- Published: Wednesday, 23 December 2009 16:24
- Written by David J Gall
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The Three Equations
There are three equations needed to design an airplane. The first relates the balance of forces: thrust must equal drag and lift must equal weight. This is commonly called the sum-of-forces, since the equal forces on opposite sides of the equals sign can all be moved to the left and "summed" (added or subtracted as need be):
Lift = Weight
becomes Lift - Weight = 0
The second equation relates the pitching moments: Those "forces" (moments) acting to pitch the airplane nose down must be balanced by moments acting to pitch the airplane nose up. This is called the sum-of-moments:
Pitch-up moments = Pitch-down moments
becomes
Pitch-up moments - Pitch-down moments = 0
You'll notice that the "summing" operation in each of the preceding formulas looks suspiciously like a minus sign instead of the plus sign you might have expected. I'll explain this in detail further on.
The third equation relates coefficients obtained from wind tunnel tests to the real world and takes the form:
Force or moment = q x S x coefficient of force or coefficient of moment
Here q is "dynamic pressure" and S is an area, usually wing area of the wing in question, q deserves further scrutiny: it is the product of air density times half the square of the airspeed:
usually written as q-p/2 x V2
q is a measure of the energy available to do work, and believe me, lifting an airplane is work The more energy available to do that work, the easier the job becomes for that thing that is doing the work, namely, the wing. So, at higher speeds, the wing can be operated at a lower coefficient of lift, whereas at lower speeds the wing will need to be operated at higher lift coefficients.
Who "operates" the wing? Why, we do, of course! As pilots, we manipulate the controls to operate the wing at the required angle to produce the required coefficient of lift, thus producing the needed lift to stay aloft (archaic: "sustentation"); as designers, we "operate" the design of the wing so that the pilot will have that capability.
Coefficients vs. Actual Forces and Moments
Now, why do I keep emphasizing the word "coefficient" in the last few paragraphs? Because this is an important word that frequently gets left out of our discussions and our thinking. That omission leads to no end of confusion, misconceptions, and debate. I think we all know what happens to lift and coefficient of lift when the airplane slows, but how many times have you read or heard the phrase that the wing must "make more lift" at slower speeds. What those speakers really mean is that the wing must operate at a numerically larger coefficient of lift, in order that the balance of forces between weight and lift (not its coefficient, but the actual force) will be maintained. This implies (correctly) that the lift is a constant although the coefficient of lift has changed.
We also all know that coefficient of lift is proportional to angle of attack (AOA), so we correctly conclude that at a lower speed we need a larger AOA (hence larger coefficient of lift) in order to maintain the balance of forces between lift and weight. Unfortunately, some of us have never had this pointed out and go on believing that they are somehow actually making more lift to fly slower. That would require that your airplane gets heavier as it slows down. Even Einstein would have a headache with that one!
Where the concept of coefficients really leaves most people completely confused, though, is with pitching moments. Examining a typical airfoil performance characteristics chart, the coefficient of moment is almost always a nearly horizontal line running across the chart from left to right just below the horizontal axis. In fact, what this is saying is that the coefficient of moment is (almost) constant for "all" (any) usable angles of attack. Pick an angle of attack; the coefficient of moment is always (almost) the same number. Coefficient of moment; what about the actual moment? Remember q, the dynamic pressure? Thus:
actual moment = q x S x coefficient of moment
What this says is that, since coefficient of moment and wing area S are constants, then, as we slow down, the actual moment will decrease because q decreases with decreasing speed (q = p!2 x V2). So, at stall speed, the actual moment is as small as it will ever get. At higher speeds, the actual moment gets larger; the coefficient of moment remains roughly constant.
Sign Conventions
As we have just seen, at higher speeds the actual moment gets larger even though the coefficient of moment remains constant. However, what we really mean when we say this is that the actual moment get larger in magnitude as speed increases: Because moment coefficients turn out to always be negative numbers for normal, non-reflexed airfoils, then the actual moment, too, is a "negative" amount. Negative-ness is not some imaginary quality but only a way of accounting for the direction of action. If positive is up, then negative is down; if positive is clockwise, then negative is counterclockwise. That is exactly correct for an airplane drawn in side view with the nose pointing to the left, as is standard. So, when I said earlier that:
Pitch-up moments - Pitch-down moments = 0
what I really meant was:
magnitude of pitch-up moments - magnitude of pitch-down moments = 0
Using the sign convention that nose-down pitching moments are given a negative sign, this translates into the correct sum-of-moments formula:
Pitch-up moments + Pitch-down moments = 0
where pitch-down moments are understood to be represented by negative numbers.
Notice now that "sum" really does mean "addition," and that the subtraction we saw earlier and that pervades the popular press on subjects aeronautical is actually an artifact of the sign convention being carelessly applied. The correct formulation of summations is to add-ALWAYS ? and let the sign convention dictate whether to add a negative or a positive value based on the direction of action. That, of course, means that popular expressions like "Lift equals Weight" are really incorrect statements, too: Lift must equal the negative of Weight, or:
Lift + Weight = 0
That addition results in a correct answer only if we remain cognizant of the sign convention: Up is positive and down is negative. So we write the number for Lift as a positive number and write the number for Weight with a minus sign in front of it, then add. If the magnitudes are equal the result is zero.
So, at stall speed the moment from the main wing and the moment from the canard are both trying to pitch the nose down. At faster speeds, they both try to pitch the nose down more. Remember: At stall speed the actual moment is of the smallest magnitude (in the negative direction) and it gets larger in magnitude (still in the negative or nose-down direction) as speed (q) increases. So, what keeps the plane from being a lawn dart that just pitches nose down and plants itself in the ground?
Longitudinal Balance
Conventional airplanes balance themselves on a wing. The majority of the lift comes from just one wing and a small balancing force provided by the tail keeps the airplane level. This all hinges (pardon the pun) on having the weight of the airplane slightly ahead of the place on the wing where the lift "acts." Then, treating that lift-point as the fulcrum of a teeter-totter, the tail is made to provide a down-force that "levers" the nose up. Maybe.
What the tail is actually doing might be just the opposite: the tail might be lifting, because the lift might be "acting" on the wing at a point forward of the center of gravity. That's exactly the way our tandem-winged planes are designed to operate. Consider the canard to be the "wing" and consider the main wing to be the "tail." Then the lift from the "wing" (canard, remember) is most definitely acting ahead of the center of gravity. Be this as it may, what's the difference between a lifting tail and a downward-pushing tail?
The difference is two-fold. First, the lifting tail may be called on to generate a lift force that is beyond its capability to provide. In this case, the stall of the tail or "aft wing" will allow the tail to fall, which will drive the front wing to a larger angle of attack. The larger angle of attack will accelerate the pitching-up of the airplane that was started by the aft wing stalling, thereby pushing the aft wing further into the stall and practically guaranteeing that there can be no recovery until the front wing also stalls or the airplane tumbles into an attitude where other forces bring it back into line.
(As a side note, the deliberate under-sizing of a down-pushing tail is one effective way to make an airplane "stall-proof." Fred Weick did this with the design of the Ercoupe; so long as you were loaded within the CG range, the airplane would run out of elevator power before it would stall. This same "feature" manifested itself unintentionally on the early Cessna Cardinals and T-tail Piper Lances, allowing them to boast the ability to stall the stabilator before the main wing when in the landing configuration. This usually happened just when you least wanted it, as you were trying to get the last little bit of flare for landing... Ouch! There went another nose gear and prop!)
Second, a more insidious condition called "instability" may manifest itself. The preceding paragraph describes an extreme condition of instability caused by the stall of the aft wing, but that stall need not occur for instability to be present. Consider that as the nose comes up as a result of some small disturbance, the aft wing is called on to "push" the tail up to get the nose back down to resume whatever trimmed condition existed before. If the aft wing is not equipped with a moveable elevator, how can it do this? Since the angle of attack of the front wing is increased by the nose coming up, and so is the angle of attack of the aft wing, then pushing the nose up or down will be the result of the battle of the pitching moments from the two wings. Let's look at this in more detail.
Earlier we noticed that typical airfoils make nose-down pitching moments; their pitching-moment coefficient is negative and nearly constant for usable angles of attack. Thus, each wing makes a nose-down pitching moment. However, now we're going to ignore those pitching moments for a little bit and look at a more obvious pitching moment from each wing.
To understand this obvious pitching moment requires that we accept that there is a magic place on the airplane called the center of gravity about which all the airplane's rotations takes place. I won't go into detail on why the center of gravity is such a magic point; suffice it to say that it has been observed by careful study that the center of gravity is the point about which all aircraft rotations take place, and it makes the math a whole bunch easier to do, too! We could arbitrarily select any point on the airplane as our point-of-reference for measuring and calculating rotations (indeed, in the early days of aviation the wing leading edge was often used, as was the firewall), but Leonhard Euler had already discovered long before that the math gets tremendously simpler if we just use the center of gravity, so CG it is! Thus, to calculate an airplane pitching moment caused by a wing, one need only multiply the value of the lift force from that wing by the distance of that wing from the center of gravity (CG).
Actually, it is a little more complicated than that. The paragraph above has a little hidden assumption about the location of the lift force in that it assumes that there is a single point that can be called the point at which the lift force of the wing acts. It's easy to say, "Measure the distance of the wing from the CG." It's another thing entirely to specify where on the wing to measure from. For this, we'll need to discover the "aerodynamic center." Continued in issue 101.
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