Neutral-Point

Proper center-of-gravity (CG) location seems to be most often determined by trial and error. For sailplanes, best performance seems to be when the CG is as far rearward as possible. The plane pitches easily due to a thermal, the tail goes up, and the elevator becomes more sensitive.

It seems the CG is played with when the plane doesn't fly right. When this happens I've tended to add nose weight to presumably make thing more stable, often without success. With the help of experienced people, I've found that my poorly flying planes are too nose heavy. It seems that there is such a thing as too stable, or is it that there is simply a range for the CG and too far forward or back are both bad, except that too far back is much worse than too far forward.

There are several methods for determining proper CG, some more controversial than others. One is the dive-test. The (sail) plane is placed into a shallow dive (45 ^o) and watched to see if it noses up, down or continues straight. A nose heavy plane requires an elevator setting producing excessive down force. In a dive test, as the plane increases speed, the elevator produces greater down force pulling the plane out of the dive. Conversely, a tail heavy plane would have an elevator setting producing less down force. Such a plane test would presumably increase its dive angle.

I've also people fly inverted and observe the amount of elevator needed to maintain level flight. A nose heavy planes requires excessive elevator. Another is if a sailplane is thrown out of a thermal, the plane is presumably nose heavy. But wouldn't it be great if we could eliminate all this testing, and and simple calculate the proper location.

In his book Model Aircraft Aerodynamics, Martin Simons describes how to calculate the neutral-point of an aircraft. The neutral-point (NP) is the point around which all the aerodynamic forces are balanced. It is therefore a very good reference point for the CG. A conservative CG would be ahead of the NP. I've found that the final CG location of my planes is usually very close to the calculated NP location. The equation is

h _n = h ₀ + h _s V _s ( a_s / a_w) (1 - (de / da ))

where

h _n neutral-point
h ₀ aerodynamic center of the wing, typically 0.25
h _s stabilizer efficiency typcially 0.6 (0.9 for a T-tail)
V _s stabiliser volume coefficient
a_s lift curve slope of stabiliser
a_w lift curve slope of wing
de / da change in stabiliser downwash angle versus change in wing angle-of-attack, typically 0.5 to 0.33

The typical values for h ₀ , h _s and de / da are more than adequate in most cases. All that is needed are the stabiliser-volume and lift-curve values.

The tail-volume is calculated by multiplying the area of the stabiliser, including the elevator, by the distance between the tail and wing. This is called a volume because the results are in cubic inches or meters. The value is then divided by the area of the wing, and the average wing chord. Dividing by the wing area makes it relative to the wing area, a ratio to the wing area, and dividing by the wing chord makes the NP value relative to the chord, a percentage/fraction of the chord value.

V _s = (S_s L_s) / (S_w c)

where S_s is the stabiliser area, L_s is the tail arm length, S_w is the wing area, and c is the average chord of the wing.

Consider the Airbear as an example. It is a handlaunch glider described in Model Aviation. It is a simple design with both rectangular wing and stabiliser. The wing is 60" by 8.5" and the stabiliser is 18" by 5". The stabiliser if 24.6" behind the trailing-edge of the wing. For the Airbear, the tail-volume coefficient is

V _s = (18 * 5) * 24.6 / ((60 * 8.5) * 8.5) = 0.51

The lift-curve, a_s / a_w, also need to be calcuated. This relates how much the lift-coefficients of the stabiliser and wing change with an equal change in angle-of-attack. In other words, how much does the lift-coefficien of the tail change compared to the wing. But before this ratio can be calculated, estimates for the lift curve slopes are needed for both the wing and stabiliser. The lift curve slope is how much the lift coefficient changes per degree change in angle of attack.

The theoretical lift-curve slope for wings of infinite length is typically 0.11 per degree (a wing at an angle-of-attack of 9 ^o has a lift coefficient of 0.99). A wing of infinite length would have zero induced (or vortex) drag since there are no wing tips. A wind-tunnel measurement effectively measures a wing of infinite length because the test wing section fits between and abutts against the walls of the wind tunnel. The lift-curve can be estimated for a real wing knowing its aspect ratio. The aspect ratio is the wingspan divided by the average chord. The average chord of a rectangular wing is simply the chord value. The adjusted lift slope is calculated

a = A a ₀ / (A + 18.25 a ₀)

where A is the aspect ratio, and a₀ is the lift slope for the infinite wing (0.11 per degree). The aspect ratios for the wing and stabiliser of the Airbear are 7.0 (60/8.5) and 3.6 (18/5) respectively. The Airbear therefore has an adjusted lift-curve slope for the wing

a_w = (7.0 * 0.11) / (7.0 + 18.25 * 0.11) = 0.085

For smaller wings, such as a stabiliser, the theoretical value is 0.095 per degree

a_s = (3.6 * 0.095) / (3.6 + 18.25 * 0.095) = 0.064

Finally, putting all these values together in the original equation, and using a value in the middle of the range for de / da, (0.4), the NP value for the Airbear comes out to

h _n = 0.25 + ( 0.6 * 0.51 (0.064 / 0.085) * (1 - 0.4) = 0.37

In other words, the NP on the Airbear is located 37% behind the leading-edge of the wing, about a inch further behind the 1/4 chord point. The following table shows calculated values for a few other planes (gliders). The NP of all of them is greater than 33%. It also shows the affect of increasing the Airbear wing from 60" to a more common 2 meter wing, the NP moves forward. A longer tail or bigger stabiliser would similarly cause the NP to increase, move rearward.

I hope this note shows you how to calculate the neutral-point of your plane to compare it to what your current CG location is.

Wing Stabiliser Tail Arm Tail Volume Lift Slope Neutral Point (%) Plane
span chord area span chord area stab wing a_s/a_w
72.0 8.0 576 19.5 3.9 76.1 21.5 0.35 0.071 0.090 0.786 33.7 Drifter-2
99.0 9.4 931 24.0 5.3 127.2 26.4 0.38 0.069 0.092 0.744 33.9 Olympic-II
68.0 7.8 530 19.5 4.3 83.8 21.1 0.43 0.065 0.089 0.730 34.8 1-26
72.0 8.0 576 18.0 5.0 90.0 24.6 0.48 0.064 0.090 0.714 35.7 Airbear -2m
60.0 8.5 510 18.0 5.0 90.0 24.6 0.51 0.064 0.085 0.750 37.0 Airbear
49.2 9.1 450 19.7 5.5 108.7 22.0 0.58 0.064 0.080 0.798 39.5 Bantam

h _n	neutral-point
h ₀	aerodynamic center of the wing, typically 0.25
h _s	stabilizer efficiency typcially 0.6 (0.9 for a T-tail)
V _s	stabiliser volume coefficient
a_s	lift curve slope of stabiliser
a_w	lift curve slope of wing
de / da	change in stabiliser downwash angle versus change in wing angle-of-attack, typically 0.5 to 0.33

Wing			Stabiliser			Tail Arm	Tail Volume	Lift Slope			Neutral Point (%)	Plane
span	chord	area	span	chord	area	Tail Arm	Tail Volume	stab	wing	a_s/a_w	Neutral Point (%)	Plane
72.0	8.0	576	19.5	3.9	76.1	21.5	0.35	0.071	0.090	0.786	33.7	Drifter-2
99.0	9.4	931	24.0	5.3	127.2	26.4	0.38	0.069	0.092	0.744	33.9	Olympic-II
68.0	7.8	530	19.5	4.3	83.8	21.1	0.43	0.065	0.089	0.730	34.8	1-26
72.0	8.0	576	18.0	5.0	90.0	24.6	0.48	0.064	0.090	0.714	35.7	Airbear -2m
60.0	8.5	510	18.0	5.0	90.0	24.6	0.51	0.064	0.085	0.750	37.0	Airbear
49.2	9.1	450	19.7	5.5	108.7	22.0	0.58	0.064	0.080	0.798	39.5	Bantam