How does dihedral effect the stability of a flying wing?
From : Don Stackhouse
Jeff, those are excellent questions.
It appears that your model has several problems.
It sounds like its pitch stability is very marginal, which is the result of
a too-far-aft C/G and/or not enough sweep.
On most flying wings, pitch stability comes from either a reflexed trailing
edge (where the reflexed portion is performing the function of the
stabilizer of a conventional model), or else by a combination of sweep plus
washout (where the washed-out wingtips perform the function of a
stabilizer). The amount of reflex and/or washout is analogous to the
relative incidence ("decalage") between the wing and the stabilizer of a
conventional model. In this sense, a flying wing isn't truly tailless, it
just has the tail combined integrally with the wing.
In a conventional model, as you move the C/G forward, pitch stability
increases but you have to increase decalage (make the wing incidence more
leading-edge-up and/or make the tail incidence more leading-edge-down) to
hold the same airspeed. The same thing happens in a flying wing; as you
move the C/G forward, pitch stability increases, but you have to increase
reflex and/or washout to hold the same airspeed.
The problem is, how to decide on a good starting C/G? What really matters
is the distance from the C/G to something called the "neutral point".
That's the point where if you located the C/G at that point, the stability
would be exactly neutral. Balanced at that point, wherever you point the
nose of the model, it will keep pointing that direction when you neutralize
the controls, with no tendency to return to any particular "trimmed"
attitude. If you shove the nose down it will stay down; if you pull the
nose up it will stay up.
If the C/G is ahead of the neutral point, then when you pull the nose up
and release the controls, the nose will tend to come back down.
If the C/G is behind the neutral point, the model will be unstable, or
what's called "statically divergent". If you pull the nose up and release
the controls, the model will want to continue pulling up, even to the point
of doing a back flip. The Curtiss "Ascender" experimental canard fighter
from WW II would do this quite spectacularly in certain flight regimes (it
may have been also due to the wing stalling before the canard), which led
to a change in the way the test pilots pronounced its name!
So where exactly is this mysterious "neutral point"? That's where things
start to get complicated. All kinds of factors enter into that
determination, even things like dihedral setup and fuselage shape. For a
wild approximation though, figure the aerodynamic center of the wing and
the aerodynamic center of the tail, then multiply the length of the tail
moment arm by
(tail area)/(tail area + wing area). Neglecting fuselage effects (don't
count on this method if you're building a scale "Super Guppy"!), the
neutral point will probably be close to this distance from the wing
aerodynamic center, toward the tail along the tail moment arm line. Put
your C/G forward of this point, the exact amount depending on how much
stability you want.
So how do you find the aerodynamic centers of the wing and tail? First find
the mean aerodynamic chord (MAC). For a straight taper, draw a line equal
to the root chord forward from the leading edge of the tip. Draw a line the
same size as the tip chord aft from the root trailing edge. Connect the
ends of these two lines with a diagonal line across the wing. Now draw a
line from the center (i.e.: 50% point) of the root chord to the 50% point
on the tip chord. The point where this line crosses the diagonal line is
the location of the MAC. There can be some complicating factors, but
normally it's safe to assume that the aerodynamic center (AC) of that wing
panel is on the MAC approximately 25% back from the leading edge. The tail
moment arm is the distance PARALLEL TO THE DIRECTION OF FLIGHT (usually
along the fuselage) from the wing AC to the tail AC.
For a flying wing, the same principle holds more-or-less true. You can look
at it from the standpoint of the tail area being zero and put the C/G
somewhere safely forward of the wing AC. Another approach that's a bit more
conservative is to estimate which portion of the wing is acting as a
stabilizer and figure it as the tail area, the rest of the wing as the
effective wing area, and calculate an approximate neutral point based on
the combination of the two, just as if they were wing and tail of a
conventional aircraft. Of course all of these are GROSS approximations, but
they're better than just guessing, testing and repairing over and over!
There are better, more precise (and far more complex) methods, but if you
want a quick "rule of thumb" and are willing to accept some inaccuracies in
order to get a fast answer, this method will get you in the ballpark. No
matter which method you use, even the complicated ones, expect to do some
flight testing to fine tune the final results. Also make sure you have a
precise way to check C/G. Just like any very short-coupled aircraft, flying
wings tend to be extremely sensitive to small C/G changes.
Now for the dihedral question. In a flying wing you have to deal with some
contradictory requirements. Often the toughest requirement is providing
sufficient yaw stability. Tip fins are one common solution. They act both
as fins and as winglets in many cases, supposedly providing yaw stability
and improving wing efficiency at the same time. In actual practice, I doubt
that very many of them are significantly helping wing efficiency, but they
do seem to work ok for the yaw stability requirement.
Another way to provide yaw stability in a flying wing is with sweep. If you
yaw a swept flying wing, the wing panel yawed forward will be sticking out
more perpendicular to the airflow than the aft-yawed wing. Presumably this
will give it more drag, which will tend to yaw the aircraft straight again.
The only catch here is that when you increase span you also decrease
induced drag. The forward-yawed wing will have a longer apparent span that
the aft-yawed wing, so even though it has more parasite drag, it may have
LESS induced drag. In most cases the parasite drag dominates and the
aircraft tends to straighten out, but it is possible that the induced drag
could dominate and make the yaw worse! It's things like this that make the
design of flying wings a tricky business.
In general, pitch stability of flying wings tends to be extremely sensitive
to C/G changes, and flying wings as a class tend to have a lot of problems
finding adequate yaw stability.
So can we use dihedral to help yaw stability? In a conventional tailed
design, dihedral is typically used to provide ROLL stability. This is NOT
the same as yaw stability. Although adding dihedral aft of the C/G can
increase yaw stability, adding equal amounts of it to the entire wing will
generally have NO effect on yaw stability. Your addition of dihedral to the
entire wing does NOT help satisfy the yaw stability requirements.
Making matters even worse is the effect of dihedral plus sweep. Sweep in a
wing has a similar effect as dihedral. The effect varies with angle of
attack; at low angles, with near zero lift coefficients, the effect is
negligible, but at high angles (such as during thermalling), the effect can
be quite strong. The general rule of thumb is that in low speed flight,
three degrees of sweep is roughly equivalent to one degree of dihedral.
Now we come to another problem: "dutch roll" vs. "spiral instability"
balance. Dutch roll is where the model wants to oscillate side-to-side in
both yaw and roll, sort of like a falling leaf. It's the result of having
too much dihedral and not enough fin. The opposite problem is spiral
instability, where in a turn, the model wants to overbank and tighten up
into a "graveyard spiral". It's the result of too much fin and/or not
enough dihedral. It's very difficult to develop a model that has just the
right balance to solve both problems, while still having good control
response and stability in both roll and yaw. BTW, one of the reasons for
the excellent handling of our Monarch 'D' is the HUGE amount of development
effort we put into getting that balance just right.
Flying wings get especially tricky on this issue. Yaw stability is
generally tough to come by for them. Lots of sweep is one way to do it, but
that also adds to dihedral effect at most airspeeds. The result can be a
bad dutch roll problem, from too much dihedral effect and not enough fin
effect to balance it. Tip fins add to yaw stability without significantly
adding to dihedral effect.
Another fix commonly seen is to add anhedral (like dihedral, only downward
instead of upward). This cancels out some of the dihedral effect of the
sweep, so that the yaw stability from the sweep (alone, or in combination
with a central fin or tip fins) is enough to keep the dutch roll/spiral
stability characteristics in balance. Make sure to just add enough anhedral
to balance the sweep's dihedral effect with the available yaw stability;
the object is to eliminate dutch roll, not to make the model
roll-divergent! The major drawback to this approach is that it doesn't work
inverted, so it's probably not appropriate for a slope combat wing. For a
thermal wing it might do just fine. Another drawback is that the amount of
anhedral required for good thermalling behavior might be enough to cause
roll instability at high speeds. For a hand-launch model, which must fly
well at both flight conditions, this could be a difficult problem.
In any case, a model with minimal sweep (weak pitch and especially weak yaw
stability), no tip fins (even weaker yaw stability), a battery taped to the
center section ahead of the C/G (acts like a fin on the FRONT of the model,
making yaw stability even worse), and gobs of extra dihedral (even stronger
roll stability, but no additional yaw stability) is likely to have a
monstrous dutch roll problem. If you haven't seen it yet, I'll bet it will
show up as soon as you get the C/G and pitch stability issues under
control. Yes, there is a reason why swept flying wings usually don't have
For a lot more on this general subject, I recommend a visit to the B2
Streamlines page (http://www.halcyon.com/bsquared/) and Andy MacDonald's
Flying Wing Page (http://www.csccs.com.au/~andy/FWING.HTM). They also have
additional links to other pages you might find helpful. In particular, the
B2 Streamlines page has a small program for Panknin's formula for
calculating the necessary twist and C/G for a swept flying wing.
Flying wings are not tremendously difficult to design and build, but they
are less forgiving of design and construction errors (you would be too if
your effective tail moment arm was often less than your wing chord!). If
you study the theory so you can understand what's happening, and if you do
your homework, you can get very good results.
Good luck, and please let me know how your experiments work out!
Don Stackhouse @ DJ Aerotech