
An indoor flying wing model by R. Eppler in 1942. 

A modern radio controlled F3B flying wing model of 1994. 
Like its full sized cousins, each model airplane should have a minimum amount of stability, i.e. it should be able to return to its trimmed flight condition after a disturbance by a gust or a control input. How much stability is required, depends on the pilots personal taste: contest pilots prefer a small stability margin, beginners like to fly with a large margin. Here, only a brief introduction into the topic will be given, which will make it possible to find a first guess for the center of gravity and a reasoneable combination of sweep and camber for a flying wing.
While the horizontal tailplane provides the necessary amount of longitudinal stability on a conventional plane, it is the wing, which stabilizes an unswept wing. In most cases, airfoils with reflexed (sshaped) mean lines are used on flying wing models to achieve a longitudinally stable model.
To understand, why a reflexed airfoil is able to provide longitudinal stability to a wing, two things are important:
The pressure forces, which act on the surface of each wing section, can be replaced by a single total force and a single total moment. Both act at the quarterchord point of the airfoil. When the angle of attack changes (e.g. due to a gust), the moment stays nearly constant, but the total force changes. Increasing the angle of attack increases the force.
Translations and rotations of "free floating" bodies are performed relative to their center of gravitiy. When the angle of attack of a plane changes, the plane rotates (pitches) around its center of gravity (c.g.).
Let's have a look at a trimmed flight condition, where all forces and moments are in equilibrium and let's compare a conventional, cambered airfoil with an airfoil with a reflexed camber line. The moments and forces for this trimmed state are denoted with an asterisk (*). The forces are the weight of the model m, multiplied with the gravity acceleration g (9.81m/s) and the aerodynamic lift L, which have to cancel out (sum of forces in vertical direction equals zero). The drag forces are neglected here. The sum of the moments around c.g. (caused by the airfoil moment M and the lift force L, acting at a distance from c.g.) must also be zero.
conventional airfoil with camber  airfoil with reflexed mean line 

Equilibrium State  
This airfoil has a nose heavy moment. As stated above, the center of gravity is also the center of rotation of the wing. When it is shifted behind the c/4 point, the air force L* in front of the c.g. counteracts the nose heavy moment M* to achieve equilibrium. The distance between c.g. and c/4 point is depending on the amount of M*. A symmetrical airfoil has M*=0, which means we have to place the c.g. at the c/4 point.  The reflexed camber line makes the moment coefficient positive, which means, that the moment around the c/4 point is working in the tail heavy direction. Therefore the center of gravity has to be located in front of the c/4 point to balance the moment M* by the lift force L*. The larger the moment (coefficient) of the airfoil, the larger the distance between c/4 and the c.g. for equilibrium. 
Disturbed State  
When the angle of attack is increased (e.g. by a gust), the lift force L increases. Now L>L* and the tail heavy moment due to the lift is larger than the moment around c/4, which still is M=M*. Thus the wing will pitch up, increasing the angle of attack further. This behaviour is instable and a tailplane is needed to stabilize the system.  Here, we have the air force acting behind the c.g., which results in an additional nose heavy moment, when the lift increases. With L>L*, the wing will pitch down, reducing the angle of attack, until the equilibrium state is reached again. The system is stable. 
As we learned above, an unswept wing with a reflexed airfoil is able to stabilize itself. Its c.g. must be located in front of the c/4 point, which is also called neutral point (n.p.). The distance between the neutral point (quarter chord point for an unswept wing) and the center of gravity is defining the amount of stability  if the c.g. is close to the n.p., the straightening moment is small and the wing returns (too) slowly into its equilibrium condition. If the distance c.g.  n.p. is large, the c.g. is far ahead of the c/4 point and the wing returns quickly to the equilibrium angle. You will require larger flap deflections to control the model, though. If the distance is too large, the wing may become overstabilized, overshooting its trimmed flight attitude and oscillating more and more until the plane crashes.
A measure for stability is the distance between c.g. and n.p., divided by the mean chord of the wing. Typical values for this number for a flying wing are between 0.02 and 0.05, which means a stability coefficient sigma of 2 to 5 percent. We can express the equlilibrium of moments around c.g. for our design lift coefficient by
,
which can be transformed to find the moment coefficient needed to satisfy a certain stability coefficient:
.
Example  We want to use an unswept flying
wing (a plank) for ridge soaring and decide to use a
target lift coefficient of =0.5. We want to have a
stability coefficient of 5% and are looking
for a matching airfoil. We walculate the necessary
moment coefficient Cm = 0.5 * 0.05 = +0.025. Searching through a publication about Eppler airfoils [28], we find, that the airfoils E 186 and E 230 could be used for our model. 

We have already learned, that the center of gravity must be located in front of the neutral point. While the n.p. of an unswept, rectangular wing is approximately at the c/4 point, the n.p. of a swept, tapered wing must be calculated. The following procedure can be used for a simple, tapered and, swept wing. First, we calculate the mean aerodynamic chord length of a tapered wing, which is independent from the sweep angle:
with the root chord lr,
the tip chord lt and the
taper ratio .
We can also calculate the spanwise location of the mean chord , using the span b,
.
The n.p. of our swept wing can be found by drawing a line, parallel to the fuselage center line, at the spanwise station y. The chord at this station should be equal to. The n.p. is approximately located at the c/4 point of this chord line (see the sketch below).
Geometric parameters of a tapered, swept wing.
Instead of using the graphical approach, the location of the neutral point can also be calculated by using one of the following formulas, depending on the taper ratio:
, if taper ratio > 0.375
, if taper ratio < 0.375.
The c.g. must be placed in front of this point, and the wing may need some twist (washout) to get a sufficiently stable wing.
The selection of the appropriate location of the c.g. is not a guarantee for equilibrium  it is only a requirement for longitudinal stability. As explanined above for unswept wings, the sum of all aerodynamic moments around the c.g. must be zero. Because we have selected the position of the c.g. already to satisfy the stability criterion, we can achieve the equilibrium of the moments only by airfoil selection and by adjusting the twist of the wing. On conventional airplanes with a horizontal stabilizer it is usually possible to adjust the difference between the angles of incidence of wing and tailplane during the first flight tests. On the other hand, flying wings have the difference built into the wing (twist), which cannot be altered easily. Thus it is very important to get the combination of planform, airfoils and twist right (or at least close) before the wing is built. Again, the calculation of these parameters is quite complex and shall not be presented here; the relations are shown in great detail in [27]. Here I will present a simple, approximate approach, which is based on two graphs, and can be used for swept, tapered wings with a linear airfoil variation from root to tip.
We start with the same geometric parameters, which we have used for the calculation of the n.p. above. Additionally, we calculate the aspect ratio (AR = b²/S, where S is the wing area) of the wing. The selection of the airfoil sections also defines the operating range of the model. Airfoils with a small amount of camber are not well suited for slow, thermalling flight, but good for F3B flight style and ridge soaring. We can design the twist distribution for one trimmed lift coefficient, where the wing will fly without flap deflections. This lift coeffcient will usually be somewhere between the best glide and the best climb performance of the airfoil. With the selected lift coefficient Cl of the airfoils, we can also find the moment coefficient Cm0.25 from the airfoil polars. If we plan to use different root and tip sections, we use the mean value of the moment coeficient of the two airfoils. The required twist of the wing can be combined from two parts:
Using graph 1, we enter the graph with the aspect ratio AR on the horizontal axis, and draw a vertical line upwards, until we intersect the curve, corresponding to the sweep angle of the c/4 line. Continuing to the axis on the left border, we find the standard value for the required twist angle.
This standard value is valid for a wing, which:
is trimmed at =1.0 and,
has a stability coefficient of sigma=10% (see above), and
uses airfoils with a moment coefficient of zero.
From the standard value we calculate the true, required twist angle, using the formula inset into the graph. Therefore, we calculate the ratio of our target lift coefficient to the standard lift coefficient (CL/) and the ratio of our desired stability coefficient to the standard . We see, that a reduction of the lift coefficient to CL=0.5 also reduces the required twist by 50%. Also, if we use a smaller stability marging, we need a smaller amount of twist.
Graph 1: Finding the required twist.
If we use different airfoils at root and tip, they may have different zero lift directions, which influences the equilibrium state. The geometric twist has to be reduced by the difference of the zero lift directions alfa0 of tip and root sections:
.
Using the same airfoil for both sections, we can set ß_alfa0 to zero.
The moment coeffcient of the airfoils contributes to the equilibrium, and has to be taken into account for the calculation of the twist. Graph 2 can be used to find the equivalent twist due to the contribution of Cm, which has to be subtracted from the required twist. If we use airfoils with positive moment coefficients, the contribution will be positive, which results in a reduction of the amount twist, highly cambered airfoils yield negative values ß_Cm, which force us to build more twist ino the wing. Similar to the previous graph, we enter with the aspect ratio, intersect with the sweep curve and read the value for ß*_Cm from the lefthand axis.
Graph 2: Finding the additional twist due to the airfoils
moment coefficient.
Again, the graph has been plotted for a certain standard condition, which is a moment coeficient of Cm*=0.05 (note: positive). We apply the ratio of the moment coefficients (Cm/Cm*) to find the contribution ß_Cm of the moment coefficient to the geometric twist. This contribution has to be subtracted from the required twist angle, too. Using cambered airfoils with negative moment coefficients will change the sign of the ratio Cm/Cm*, which results in negative ß*_Cm values. This means, that the subtraction from ß_req will actually be an addition, increasing the geometric twist angle. If we have different airfoils at root and tip, we can use the mean moment coefficient (Cm_tip+Cm_root)/2 to calculate the ratio Cm/Cm*.
Finally, we can calculate the geometric twist angle ß_geo, which ha to be built into the wing:
.
Example  As you have noticed, the graphs
contain an example, which is recalculated here. We
consider a flying wing model with the following data:
We calculate the wing area S: S = (l_r + l_t)/2 * b = 0.5085 m² and the aspect ratio AR = b²/S = 11.0 and the mean moment coefficient Cm = (Cm_r + Cm_t)/2 = 0.02 . Using graph 1, we find ß*_req = 11.8°, which has to be corrected to match our design lift coefficient and the desired stability coefficient:
The difference of the zero lift angle of tip and root section is . Now we read the twist contribution of the moment coefficient from graph 2, which is ß*_Cm = 5.8°, which has to be corrected for our smaller mean moment coefficient:
Finally, we calculate the geometric twist from
The negative value means, that we could use a small amount of washin, which means, that we have enough stability due to the selection of airfoils with reflexed camber lines. Since the calculated amount is very small, we can use the same angle of incidence for the root and tip ribs. Since the presented method is not perfect, we can assume an accuracy to 1 degree, which is also a reasoneable assumption for the average building skills. 

last modification of this page: 05.10.01
[Back to Home Page] [Index of all Pages] Suggestions?
Corrections? Remarks? e Martin Hepperle.
As I get quite a number of messages, it might take some time until you receive an
answer and in some cases I get lost in the flood and you may even receive no answer at
all. I apologize for this, and if you have not lost patience, you might want to send me a
copy of your email after a month or so.
Copyright 19862001 Martin Hepperle
You may use the data given in this document for your personal use. If you use this
document for a publication, you have to cite the source. A publication of a recompilation
of the given material is not allowed, if the resulting product is sold for more than the
production costs. Some names appearing on this web are trademarks or registered names whose rights belong to their owners, even if not explicitely stated. This document
can be found by navigating from the Web site http://www.MHAeroTools.de/