The Centre of Gravity
by Alasdair Sutherland
from RC Model World

The Centre of Gravity is the balance point, the point through which weight acts. If you are a stickler for accuracy, it is the point about which the weight has no moment (rotation effect), so that the CG is the point at which you can suspend or support your model without it rotating, or falling over. It is a three dimensional point, but it should be near enough on the centreline of a symmetrical aircraft, the height doesn't much matter, so all we are concerned about here is its fore and aft position.

How do you find the CG of a model?

From the above definition of the CG, you find the point at which you can support the model suspended from string or balanced on finger tips, the edge of a ruler, or the blunt ends of two pencils. Or you could use a purpose made device like the "Great Planes CG Machine" which has been carefully thought out (but rather over-priced by the importers).

Why does the CG matter?

The position of the CG determines how much stability the aeroplane will have, and also how much control over it you will have. Stability and Control are opposites;- the more of one you have the less of the other you get, so CG position is a compromise. You are trying to get just the right mix of Stability and Control.

What is Stability?

A stable object goes back where it started. Put a ball in the middle of a level concave surface, a dish, and it will try to remain there. If you move it away from its position in the bottom of the dish it will return because it is a stable arrangement. The steeper the sides of the dish the greater the stability and so the quicker it returns to its original state.

If you carefully balance the ball at the top of a convex surface it will stay there unless it is disturbed. Move it slightly from its balanced position and it will keep going and roll right off at ever increasing speed. The steeper the slope the more unstable the situation is.

Of course there is a compromise situation in between the two. You put the ball on a perfectly flat level surface and it stays where you put it. You move it and it stays in its new position because it is neither stable nor unstable. Its stability is Zero, or Neutral.

We are concerned only with Elevator Control here, the rudder and ailerons do not come in to CG position. We want the elevator to pitch the nose of the aeroplane up and down in a reliable and predictable fashion, so that we have the full range of control without it being over-sensitive.

As you move the CG forward from its ideal position the aeroplane becomes more stable. That's OK, but you start run out of CONTROL on elevator. You gradually lose the ability to spin, stall, fly inverted, loop, and even get the nose up for a flared out landing. The elevator trim becomes less effective, requiring more and more trim movement when you change from high speed trim to low speed trim.

As you move the CG rearward the aircraft becomes less and less STABLE. You have to make constant corrections to the flight path, the elevator trim becomes sensitive to small movements making it difficult to trim the aeroplane in level flight, and a small movement of the elevator produces a large control response. If the CG is too far back an aeroplane becomes unstable, its flight path persistently diverges, it cannot be trimmed, and a small up elevator movement produces a gut wrenching, wing folding, loop.

The CORRECT position for the CG is that which suits YOU the pilot. Your CG might differ from someone else's, but if it is right for you, it is the right CG. The CG from a formula or the plan is a safe recommended starting point, then with experience you can experiment. You perform various Flight Tests.

The popular "Dip" test can be used on virtually any model. You trim it out at some comfortable speed - in level flight or on the glide - and push a little down elevator to dip its nose and let it build up speed. Then you release the up elevator and watch carefully for its reaction. If the nose rears up quickly it could indicate too much stability and too forward a CG. If the nose stays down in the dive the CG may be too far aft (or you may have a structural tuck under problem). What you want is a gentle recovery to level flight. Remember we are testing for stability here, and it is the stability which pulls it out of the dive, trying to get back to its original trimmed speed. Too much stability will make it rear up too quickly, the cure for which is to take weight out of the nose to move the CG back (strange as it may seem at first).

You can of course do the opposite:- gently hold the nose up to let some speed bleed off. When released the model should gently lower its nose and recover, not remain nose up (aft CG) or dive eagerly (forward CG).

Most models will be capable of a loop. If the CG is too far aft it will loop tightly with very little up elevator and will be prone to screwing out of the loop. With a forward CG the loop will be large, or it may not have enough up elevator authority to get around a loop at all. The same goes for outside loops (bunts) though heavily cambered wings may not be able to bunt.

My favourite check is inverted flight, though heavily cambered wings again may not be capable. Half-loop (or half-roll) the model and apply down elevator to hold inverted flight. If you need less than about 3 mm of down movement then I suggest a further forward CG - you have plenty control but insufficient stability. If you need more than half the down movement to fly steady inverted then the model has more than enough stability and the CG could be moved aft a little.

Now try a stall. Trim the model to a nice fast cruise, close the throttle (if you have one) and keep gently feeding in enough up elevator as the model slows down to maintain level flight or a slight climb. If the stability is small, with an aft CG, then very little up will allow the model to slow right down until it stalls, still with plenty of up stick left, which is wasted. What you want is for the model to stall when you have almost full up elevator applied (which will also depend on the control throw). Often on trainers, and sometimes on sport models, the CG and elevator movement are arranged such that full up will not quite stall the model. If the CG is too far forward then full up may allow the aircraft to fly nose down, at well above its stalling speed, leaving you without enough control to land the model properly. You need to move the CG further aft (or possibly use more control throw).

I am sure there are many other useful flight tests for CG position, but finally for me is the spin. A model with an aft CG will spin easily while with a forward CG it will not spin at all. Adjust it how you like it.

Popular Misunderstandings

If a new model "keeps wanting to climb", do you put lead in the nose to move the CG forward? No, you add down trim. You adjust the elevator trim to get the model flying straight and level 'hands off' and then perform some of the tests above.

You find after landing that the elevator needs to be down for trimmed flight, do you move the CG to remove the trim requirement? No! It means that the wing (or tail) is on at the wrong angle. Pack the wing's TE up or the LE down to correct. Or angle the tailplane itself more leading edge up (and vice versa for up elevator) if that is easier. The rigging angles of the wing and tail are nothing to do with stability, they only get the model to fly in trim.

"I've changed the wing section from semi-symmetrical to a less stable, flat bottomed, section:- should I change the CG?" No! There is no such thing as a stable or unstable wing section. The wing section does not significantly affect the stability, but you may have to change the wing or tail rigging angle.

"If I change to a lifting section tailplane should I move the CG to compensate?" No! Again, changing the tail's section doesn't affect stability -- and it does NOT make it lift!! In the following sections you will notice that the wing or tail sections or rigging angles do not matter significantly in stability and so are not involved in the rules and formulae for CG position.

"In the dip test the model keeps diving, so the nose must be too heavy?" No! There is not enough stability to recover from the dive so move the CG forward with added nose weight. (It is also possible that structural flexibility is causing a Tuck Under, but a more forward CG helps that as well. Try the inverted flight test, and stiffen the structure.)

For a kit or plan built model start where the designer says, though personally I would check first having seen a few wrong ones. The CG will be marked on the plan by a symbol like those in Figure 2. Start with this position and do the flight tests described above to see how the designer's CG suits YOU, and adjust if necessary.

Now, suppose that your model comes without a suggested CG position, or you have designed your own, where do you start? Where is a good CG position? Do you start with a forward CG? And what IS a forward CG, or an aft CG? You must be wondering when I mention a forward or aft CG, Forward or Aft relative to WHAT?

You often find the CG marked on the fuselage, but if you move the wing the CG must go with it because it's the wing that matters. The CG may be marked on the root chord or tip chord of the wing, but if you change the sweep of the wing (see Wee Annie photos) the CG has to change. The CG is referenced to the MEAN CHORD, but it could be at 50%, or 30% or 15% or even ahead of the wing mean chord because its position depends upon the WHOLE AEROPLANE, not just the wing.

I will now bring the Centre of Pressure into the discussion, but only to boot it straight out again. It is a floating point of no relevance to anything, certainly not the CG.

The NEUTRAL POINT is the reference point for the CG. The NP IS a CG position:- the NP is defined as the CG position at which you get Zero, or Neutral, Stability (a ball on a pool table). An aft CG means back close in front of the NP (a ball in a shallow dish), and a forward CG means well ahead of the NP (a ball in a deep dish).

The first step in CG calculations must be to find the mean (another word for average) chord. The simplest average chord is the Area divided by Span, usually called the "Geometric Mean Chord" (GMC) or the "Standard Mean Chord" (SMC). On a tapered wing it is also 1/2(root c + tip c), and it is found half way between root and tip. Incidentally, the root chord is usually taken as being on the centreline of the aircraft

For example on a wing with a 10" root and a 7" tip the GMC is 1/2(10 + 7) = 8.5".

The Mean Aerodynamic Chord (MAC from now on) is the technically correct reference length for calculating aerodynamic forces and moments, and hence CG positions, but the MAC is defined in terms of complicated mathematics. For a tapered wing this simplifies to

MAC = root chord x

where t is the taper ratio, tip chord/root chord.

If the root chord is 10" and the tip chord is 7" then taper ratio t is 0.7 and

MAC is 10x2(1+.7+.49)/3(1+.7) = 8.588".

The MAC is always bigger than the GMC, but only very slightly bigger, unless the wing is sharply tapered.

You will probably have heard of the graphical method of finding the mean chord, which is illustrated in Figure 3. You extend the root chord forward (or aft) by the length of the tip chord, and extend the tip chord aft (or forward) by the length of the root chord (whichever fits your paper). Then you join the points just marked with a diagonal line. Where either of these diagonals crosses the 50% chord line marks the position of the mean chord. You can of course draw both diagonals and miss out the 50% chord line. The mean chord which this method finds is the MAC. This method finds the location of the mean chord which you then measure, and project onto the centreline.

For most models it matters not the slightest which mean chord you use, because there is next to no difference. On deltas and sharply tapered wings I use the MAC, found as in Figure 3, but normally I just use whatever is easiest, drop the formal capital letters, and call it the mean chord or average chord.

When you find the location of the mean chord it is important to transfer it accurately to the side view of the fuselage, in the correct fore and aft position as indicated on Figure 3. While there, mark the quarter chord point:- that's a quarter of the mean chord from the front.

Having found the mean chord, suppose you are told to put the CG at 30% chord. You mark your mean chord on the side view in the correct location, and measure back 30% of 8,5", that's 2.55" back from the front of the mean chord.

If your model is a flying wing you need go no further than work out the mean chord. Put the CG at 15% of the mean chord to start with, and rig some elevon reflex for trim. However most aircraft have a tailplane or, as they say in America, a horizontal stabiliser. As you may guess it affects the stability and you will have to make allowances for it.

Stability depends on the whole aeroplane, and we usually take account of the fuselage by using the wing's gross area, but the main stabilising effect is from the tailplane. Just how effective is your horizontal stabiliser? The stability it contributes depends on how big it is compared to the wing, and how far behind the wing it is. A small tail with a long leverage arm will be as effective as a large tail near to the wing. It is human nature to want to evaluate things with numbers, but how can we use numbers to put a value on the tail's effectiveness?

Tail Volume Ratio (or Coefficient) is conventionally written in textbooks as a capital V with a bar over it, but as my word processor will not do that I shall write it as it is said, V-bar. It is a measure of the effectiveness of the tailplane. You will find it cropping up in all the best CG formulae and design criteria (though sometimes it is disguised and hard to spot). Basically it is the tail area, as a fraction of the wing area, times the tail arm as a multiple of the wing mean chord. It therefore has no dimensions because it is, as it says, a ratio.

V-bar

The top and bottom of the equation each consists of an area times a length, hence the name of tail volume ratio. It is usual to use gross wing area (i.e. including the area inside the fuselage) and net tail area (i.e. only the area out in the airflow). Tail arm is measured between the quarter chord points of the wing and tail mean chords.

A typical value is 0.4 to 0.7 for normal RC models, maybe down to 0.3 for gliders while some free flight and vintage models, and the odd scale model, can have V-bar over 1. A simple way to calculate Tail Volume is to use my Nomogram, Figure 4. Just measure up the aeroplane, mark the values and draw a line to get an answer on the centre scale.

You can't go wrong with Figure 5, a nomogram which makes working out the CG easier than "join the dots". Just mark the wing's Aspect Ratio (span/chord ratio) and the V-bar: the suggested CG is on the middle line. The Nomogram is based the formula in my "Basic Aeronautics" book

CG posn = 0.1 + 0.25 x 4Ö AR x V-bar
(note square root sign, with superscript 4 which makes it fourth root)

[or CG posn = 0.1 + 0.25*(AR)^0.25*V-bar] [or = 0.1+ 0.25*(AR).25*V-bar ]

where 4Ö AR [or(AR)^0.25 or (AR).25] is the fourth root of the wing Aspect Ratio.

It is also dead simple to work out on a calculator with square roots. Take the square root of the square root of the wing Aspect Ratio. Multiply by 0.25 and the Tail Volume Ratio, and add 0.1. That gives the CG position of a conventional monoplane as a fraction of the mean chord. Multiply by 100 for a percentage.

For example if the wing span is 99" and the mean chord is 11" the Aspect ratio is 9. The square root of 9 is 3 and the square root of 3 is 1.73. If the tail volume is 0.6 this gives a CG position of 0.1 + (0.25 x 0.6 x 1.73) = 0.36. The CG should be at 36% of the mean chord, or 3.96" aft of the LE of the MEAN chord, to start with anyway.

The formula, like any serious formula, estimates the NP position and puts the CG a fixed distance in front, called the Stability Margin, which in my case is 15% mean chord. If you like twitchy models replace the 0.1 with 0.15 or even 0.2 to push the CG back some more. Many competition glider fliers like an aft CG, but I suggest starting off with the original formula for the first flights. Note that this formula, and the resulting nomogram, are intended only for conventional aeroplanes whose tail is behind the wing and is much smaller than it.

P.S. Another popular CG formula which has been around for many years and which you will often see quoted is:-

CG distance aft of LE of mean chord = chord/7 + 3xTail area x tail arm/8xwing area

To get the CG position as a fraction of the chord (as in my formula) divide every term by the chord to get

CG posn = 1/7 + 3/8 V-bar = 0.14 + .375 V-bar

so you can see the formula is very like mine, and in the example gives almost the same answer.

This Arcticle was originally published in RC Model World
and is being used with thanks and their kind permission.