In designing a plane, is there a formula for tail and nose moments?

From : Don Stackhouse

Eric, there is no cut-and-dried answer to that (if there was, all airplanes would look alike!), but there are some rules of thumb that are a good starting point.

Tail moment and tail area are not independent items. Generally speaking, for a given amount of pitch or yaw stability, there is a linear relationship between static stability (how strongly the plane wants to return to its trimmed position) and either tail area or tail moment. That means that if you double the tail are or double the tail moment, you get double the static stability.

For dynamic stability (the ability to damp out oscillations), the relationship for tail area is linear (as above), but there is a squared relationship for tail moment arm. In other words, if you double the tail moment arm you get FOUR times the dynamic stability.

You can trade these off against each other; i.e: if you increase the tail area you can reduce the tail moment and vice versa.

So we should make all our models with enormous tail moments and tiny tail surfaces, right? Well, not quite. As you make the tail longer, the weight of the tail surfaces decrease, but the weight of the tail boom increases. the same is true for whetted area of the tail surfaces vs. the tailboom. Of particular importance to models, as the tail surfaces get smaller, so do their Reynolds numbers and/or span, hurting both their drag and their effectiveness. BTW, this is one of the lesser-recognized advantages of V-tails. By concentrating the total area into two surfaces instead of three, the span and/or chord of those surfaces is improved.

Longer tails also move mass further from the C/G, which hurts control and stability. This may also mean making the nose longer in order to minimize weight required to balance the model. Since surface area ahead of the C/G is de-stabilizing, you end up paying double in this regard for the longer tail.

As far as numbers go, there are quite a few methods, most of them complicated. The simplest on is probably the method of "volume coefficients".

Imagine that you had many, many years of experience designing models of a certain type. Because of that experience you could estimate with a fair degree of accuracy what size tail assembly a new model would require, right? Now, what if there was a way to attach numbers to all the major factors affecting stability, and a formula to combine those numbers into an overall "effectiveness" result. You could quickly calculate the "effectiveness" numbers for existing successful designs, and use those to determine the appropriate size tail for your new designs. In effect, the formula would allow you to utilize the results of the experience of all designers in the accumulated history of that type of aircraft.

The method of volume coefficients is a way to do exactly that. We take the measure of the dominant parameters influencing pitch or yaw stability through a formula, the result of which is a measure of that model's relative tail proportions. Since the numbers just happen to have cubic dimensions the way the formula works out, we call them "volume" coefficients.

For the horizontal tail (pitch stability), the pertinent parameters are the Mean Aerodynamic Chord (MAC) of the wing, the wing area, the horizontal tail area, and the tail moment arm as measured from the aerodynamic center (AC) of the wing to the AC of the tail, parallel to the fuselage. For our purposes the MAC is the chord of the surface is where the area of the panel outboard of the MAC equals the area inboard of that chord. You can assume that the aerodynamic center (AC) is located on the MAC, 25% of the chord back from the leading edge. Since more moment arm and more tail area makes the model more stable, we multiply those together. Since more wing area and more wing chord make the model less stable, we divide by those. The resulting formula for horizontal tail volume coefficient (Vht) is:

           (horizontal tail area x hor. tail moment arm) 
     Vht = ---------------------------------------------
                       (wing area x wing MAC)

For vertical tail volume coefficient (Vvt) the formula is similar, except we use semispan of the wing (i.e.: half the wingspan) instead of MAC. The formula is:

           (vertical tail area x vert. tail moment arm) 
     Vvt =  ------------------------------------------
                   (wing area x wing semispan)

Find some models with good stability and handling similar to the one you're working on, calculate their volume coefficients, then use those as a guideline for designing your model. V-tails are a more complicated than this, but volume coefficients are a starting point.

You have to use these numbers with some care, since there can be some other factors involved. For example, poly 2-ch vs. ailerons, the use of flaps vs. no flaps, relatively heavy wingtips vs. light wingtips, large model vs. small model, multi engine vs. single engine vs. no engine, can all effect the final results. The effects of local airflow during certain types of maneuvers can also complicate the picture. Be careful when comparing models that are not very similar in their design and intended use to the one you're designing.

As far as tail moment arm, the ratio of tail moment to tail area used in the baseline models you compared yours to are a good starting point. Ultimately the length of the tail is a structural engineering question. If you can design a long tail while still keeping mass and whetted area of the tailboom low, a long tail moment can be very effective. The Monarch series is one very successful example of this.

Don Stackhouse @ DJ Aerotech