Why is the static margin expressed as percentage of the MAC?

Question

Most textbooks use the static margin (the distance between the aircraft's centre of mass and its aerodynamic centre) to assess the longitudinal static stability.

Instead of using an absolute value for the allowable range of the static margin, it is expressed as a percentage of the Mean Aerodynamic Chord (MAC) - for example between 5 and 30%. Why is this so?

For a given aircraft, the higher its aspect ratio for a given wing area the shorter will the chord become. Hence the MAC will also be smaller. This will in turn reduce the absolute value of the allowable range for the static margin.

Is this simply a convention or is there an aerodynamic or flight mechanics reason why the allowable range for the centre of mass is smaller for smaller MACs or shorter wing chords?

Answer 1

The static margin is the distance between the center of gravity and the neutral point. By itself, this is a dimensional value. This makes it hard to compare between aircraft, so it makes sense to divide it by some other length. That the mean aerodynamic chord has been chosen has several reasons:

• The MAC is the reference length for all parameters of longitudinal motion.
• The wing's pitching moment scales with the MAC.
• The wing's pitch damping changes with the square of the MAC. The biggest contribution to pitch damping, however, is the lever arm of the horizontal tail.

Normally, the tail lever arm is also chosen as a multiple of MAC. A higher wing chord means more tail force is needed in order to set the correct angle of attack. If you compare different airplanes, you will notice that the relative size of the horizontal tail goes down with a higher wing aspect ratio. While GA airplanes have horizontal tails sized to between 20% and 25% of their wing surface, gliders have horizontal tails of around 10% to 15%. It must be said, however, that gliders need less flexibility in their cg location, which is another reason for reducing horizontal tail size.

If all linear dimensions are referenced to MAC and all areas referenced to the wing area, calculations become easier and similarities between different aircraft become obvious.

For lateral motion parameters the usual reference length is the semispan or the span of the wing.

Answer 2

I'd like to offer a different viewpoint. The wing's pitch damping contributes little to aircraft pitch damping, so that is not a reason to base the static margin on MAC. Believe it or not tail size is almost entirely based on trim requirements, not stability, so again basing tail size on wing area or MAC is not important to stability. Two aircraft can have widely different MAC's, yet have the same actual stability, so comparisons between aircraft using MAC is not really useful nor accurate. The ONLY reason MAC is referenced for static margin is for convenience for one particular plane only. If you look at various aircraft (military, commercial, private) you'll find static margins that range from 35% (literally in front of the leading edge - some lear jets) to near 0% for essentially the same flying qualities. In my (and others) opinion, there is no true relationship between MAC and stability, but I will soon publish a new theory on stability that mostly solves the "how much stability is needed and how do you measure it" problem. It is interesting that every article/paper I've ever seen (lots of them) shows elaborate stability calculations that conclude nothing more than the CG should be in front of the neutral point - with nothing about how much in front (except the traditional 5-15% MAC).

Hope this peaks your curiosity.