CG Placement, Part Deux

Last time, we learned that every wing has an aerodynamic center (AC) at the quarter chord point.  The AC is where a change in lift (ΔL) appears in response to a change in angle of attack (ΔAoA).  For a flying wing, the CG needs to be in front of this point for pitch stability.

But what if our model has more than one lifting surface (and most do)?

Figure 1 shows a wing and stabilizer, connected by a stick fuselage.  I've neglected the equlibrium forces (lift, drag, thrust, and weight) because, if the aircraft is trimmed, they just cancel each other out (as we saw for the single wing last time).  For stability purposes all that matters is how things change when AoA changes.

Now let's suppose this combination encounters an updraft, increasing AoA (ΔAoA).

The wing acts as if it were flying alone.  The stabilizer acts just like a little wing, with a change in lift at its own aerodynamic center.  So we have two incremental forces, ΔLw and ΔLt (wing and tail).  Because the wing and tail are connected by the fuselage, both of these changes will act in the same direction (as shown).

Suppose the CG is in between the two aerodynamic centers (quarter chord points).  Will the model nose up, or nose down?  That depends mostly on the strength of the two changes in lift (which depends on the surface areas of the wing and tail), and their lever arms (distances from the CG).  Lots of different numbers.  Is there a way to simplify the problem?

In Figure 2, we've combined ΔLw and ΔLt into one total change in lift, ΔL.  The point where this force acts is the aircraft aerodynamic center (target symbol).  If we can locate this point, we only have one change in lift to worry about and the relationship with the CG will immediately show whether the model is stable or unstable.

So, how can we find this AC?

Well, it turns out that for models with straight, constant chord wings and tails we can boil things down to a fairly simple formula.  Rather than frightening people with it, I'll show the results in the form of a graph:

 X_ac (aircraft aerodynamic center location) is shown on the vertical axis.  This is expressed as as a fraction of chord length, so the AC starts at the wing quarter chord (0.25) for zero "tail volume".

Tail volume - what's that?

V_bar (horizontal tail volume coefficient) is just the ratio of the surface areas multiplied by the ratio of the lever arms):

V_bar = (stab area/wing area) x (length from tail AC to wing AC/wing chord)

What's a typical tail volume for scale models?  Well, for example, the PZL P-11c I'm building has a V_bar of 0.62.  My quarter scale Cub V_bar is 0.49, and my Hobby King B-17 (hello Rob!) V_bar is 0.65.  Some old Free Flight models probably had V_bar large enough to move the AC off the wing chord (greater than 1), because they had to be balanced behind the trailing edge of the wing.

OK, fine; figure out V_bar and we can find this AC point lickety split.  Except for one thing - what the heck is this "efficiency" thingie on the right side of the graph?

Well - this is what engineers call a "fudge factor".  It reflects the fact that the tail will not generate lift as efficiently as the wing.  There are several reasons for this:

- The tail usually has a lower aspect ratio than the wing

- The wing wake takes some of the energy out of the flow before it reaches the tail (reduced dynamic pressure)

- The tail is flying in the wake of the wing, which causes an effect called "downwash", reducing the tail AoA.  This is especially important for short stubby wings and tails.

- The tail has a shorter chord than the wing (Reynolds Number effects)

Rather than worry about all these individually, we lump them together into one "efficiency factor".

So, how efficient is MY plane's tail?  You'll never know for sure without putting it in a wind tunnel.  But, you CAN be sure it will never be more than 100% efficient - and it will never even  approach that unless it's a sailplane with a T tail.  It's also unlikely to be as low as 50%, and won't get there unless it has fat, stubby wings with the tail set ridiculously close to them.  Somewhere between 70% - 85% is reasonable for most models.  If we can't get a precise answer, we call this "bounding the problem".

OK, great!  As long as my model has a constant chord wing and tail, I can figure this V_bar thingie, look at the graph, and find my AC!  And as long as my CG is ahead of it, I know my model will be stable!  Life is good!!

But wait - the plane I'm building has swept wings.  Or it's a Spitfire.  Or it's a biplane.  What about a Pitts Special - with one swept wing, and one straight.

What the heck do I do now?

Tune in next time...