Don discusses the proper use of tail volume coefficients

From : Don Stackhouse

There seems to be quite a bit of confusion and misinformation floating around the list about the proper use of tail volume coefficients. There is quite a bit of info about these and how to use them in the "Ask Joe and Don" section of our website. Just go to our website, get into AJ&D, then type "tail volume" or "tail volume coefficients" into the search engine, and it will give you a whole list of links to articles in AJ&D that discuss this topic.

In a nutshell:

Imagine that you were the world's most experienced airplane designer, with literally thousands of both successful and unsuccessful designs to your credit. It's a pretty safe bet that you could tell just by looking at a new design whether the tail surfaces were about the right size or not.

Now imagine that you're NOT the world's most experienced airplane designer (a pretty easy thing for most of us), but that you had a method to attach a number of some sort to each and every airplane design you had basic dimensions for, that would give you an estimate of that tail's effectiveness. Armed with that information, which represents the collective experience of all the airplane designers for whose designs you have data, you could probably come pretty close to the ability of that super-experienced designer for estimating required tail sizes.

Tail volume coefficients are just such a measure of estimated tail effectiveness.

Essentially all we do to figure the tail volume coefficient is to measure all the things that are easy to measure (typically four items) that have something to do with tail effectiveness. We then multiply together all the things that help tail effectiveness, then divide by the things that make the tail's job more difficult. The resulting number is the tail volume coefficient.

It is customary to take the measurements from the Aerodynamic Centers ("AC") of the individual surfaces. If you summed up the little forces generated all over the flying surface and found a single force that equals the collective effects of all the distributed little forces, that single total force would act through the AC.

The AC is typically located at about the 25% point back from the leading edge of the Mean Aerodynamic Chord ("MAC"). The MAC is generally a little bigger and a little inboard of the average chord, since the inboard half of a tapered wing has more area than the outboard half. For a straight-tapered wing, the MAC is at the point where half the wing area is inboard of that chord, and the other half is outboard of there. There is more discussion in AJ&D on our website of how to calculate the MAC and AC of various surfaces.

So, OK, now we've found the place on each flying surface where the total aerodynamic force on that surface seems to act. What next?

Well, for pitch, it's the chord of the wing that's being rotated around the pitch axis when we try to control pitch. The chord length that most represents the wing's chord for this discussion is the MAC. If the MAC is bigger, then the horizontal tail's job is more difficult. Also, the bigger the wing area, the tougher the tail's job.

OTOH, if the tail has a longer moment arm (the distance from the wing's AC to the tail's AC, parallel to the aircraft's centerline) and/or the tail has more area, it will be more effective.

So, to find the volume coefficient for the horizontal tail ("Vht"), just multiply the tail area ("Aht") divided by the wing area ("Aw"), times the horizontal tail moment arm ("Lh") divided by the wing's MAC:

Vht = (Aht / Aw) x (Lh / MAC)

I notice some of the posters have discussed using other numbers such as the distance from leading edge to leading edge, and the root chord instead of the MAC. These are approximations, and as long as the chords of the wing and tail are fairly close to each other, you can probably get away with it. In most cases, the chord of the wing is larger than the tail chord, so the leading-edge-to-leading-edge method will probably give you a number for tail moment arm that is a little too big. OTOH, using the root chord of the wing on a tapered wing (or one with rounded tips) will give you a chord that is larger than the actual MAC, so those two errors together will tend to cancel each other out in at least some cases. However, it does compromise the accuracy of the volume coefficient, which does weaken its usefulness, particularly when comparing planes that have different planforms and general layouts.

Typical numbers for Vht of a single-engine aircraft are around .35 to .55, depending on a number of factors. Besides your desires for control and stability, there are also things like the use of flaps to consider (flaps dramatically increase the wing's pitching moment, which then increases the lift demands on the horizontal tail). In general, you should look at other airplanes that have the sort of stability and control effectiveness you'd like, and that have similar layouts and configurations to yours. The tail that works on a non-flapped thin-wing racer with a nearly symmetrical airfoil might not be adequate for a STOL airplane with enormous flaps and massive amounts of camber in the wing.

In the case of the vertical tail, it's the wing panels that are pivoting about the yaw axis, not the wing chord, so we use the either the span ("b") or the semispan (i.e.: half the wingspan, or "b/2") to calculate the Vertical Tail Volume Coefficient, or "Vvt":

Vvt = (Avt / Aw) x (Lv / b)

Note, I used the wing span in this case, but I've seen the semispan used as well. Either way works, just make sure that if you're comparing numbers for different airplanes that they were all calculated by the same method.

Vertical tail volume coefficients are a bit trickier to interpret than the coefficients for the horizontal, because of a lot more complicating factors. In general, multi-engine aircraft that are subject to failures of one engine (this could be a judgement call in the case of electrics) need about three times as much tail as an otherwise similar single-motor or centerline-thrust twin, in order to deal with the asymmetric thrust and drag when one engine is dead. Wing dihedral and vertical tail also have to be balanced with each other as well as keeping the airplane's moment of inertia (flywheel effect) about the yaw axis as small as possible, if you want to avoid dutch roll and/or spiral stability problems, but that's a whole 'nother subject in itself.

All of this refers to control authority and static stability (i.e.: if the airplane is disturbed in pitch or yaw, will it try to come back to the original attitude, or will it stay at the new position or even try to get worse). Dynamic stability (i.e.: the ability to damp out oscillations) is a little different. Dynamic stability is linearly proportional to changes in wing and tail areas, but proportional to the square of changes in chord or tail moment arm. In other words, if you double the tail area you get twice the dynamic stability, but if you double the tail moment arm, you get FOUR TIMES the dynamic stability.

Thus, for checking for adequate dynamic stability, one parameter I use is what I call "dynamic tail volume coefficients". These are the same as the static volume coefficients, except that I square the ratio that contains the tail moment arm:

Vdht = (Aht / Aw) x (Lh / MAC)^2

Vdvt = (Avt / Aw) x (Lv / b)^2

All of these formulas tend to get a bit tedious to work with by hand, but can be easily programmed into a spreadsheet such as Excel.

There is more discussion of all of this in AJ&D. Other good references are "Light Airplane Design" by L. Pazmany, and the series of articles in "Sport Aviation" magazine by John Roncz titled "Designing your Homebuilt", which ran from Feb. 1990 to Feb. 1991.

Don Stackhouse
DJ Aerotech