By Jo Ivens
Communicate with the Author by email at this address:
pj.ivens@virgin.net
Tailless Pusher
For nearly seventy years my
trouble has been a desire to produce unusual models, quite possibly to be a
“show off” at the field; also to find out things. Anyway, over that time-span,
with the usual intervals, I have tried out all sorts of unconventional designs,
such as tandem wing, canard, three-surface, tailless, Burnelli and Flying Flea.
The main objective is to get the design near enough right to survive the first
flight. For this, stability in pitch and yaw, and trim, must certainly be
addressed, which requires some calculations.
In 1986 an article by David
Fraser (Ref. 1) provided the means to locate the center of gravity of all
except the tailless. Then, the computer was beginning to be capable of taking
his algebra and running fast and responsive design programs, which became my
strong interest. Thus, for some twenty years, I have enjoyed designing all
sorts of model configurations, based on Fraser, with my additional programs –
written in BASIC - which is still a capable method.
Pitch stability.
For adequate stability, the
center of gravity (“CG”) must be suitably located. Every model flying
publication that I consulted over the years told one to base it on the wing
mean aerodynamic chord (“MAC”), which I will call the “popular” method; the
honest authors tell you that is valid only for conventional models. Some
recommend a static margin (“SM”) of their preferred fraction of the MAC. The SM
is the distance that the CG is ahead of the neutral point (“NP”). The NP is the
CG location corresponding to neutral stability, which requires some calculation
compounded with judgment – not usually a simple matter.
The idea was that stability should be “normalized”, i.e. expressed
as a dimensionless number, so that it is independent of model size, and
dividing the restoring couple by a dimension typical of the model did this;
which was taken to be the MAC.
That
was reasonable; however, there is no mathematical relation between pitch
stability and an MAC, so that the popular method is not correct. For
conventional models the popular method may locate the CG fairly well, by luck.
For unconventional designs it is useless, and it was necessary to find a
properly based method.
Fraser showed that CG location
ahead of the NP, for a chosen degree of pitch stability, is actually based on
the distance (usually written “Ls”) between the aerodynamic centres
(“AC”) of groups of flying surfaces, comprising wing(s) and tail(s); this I
will call the “proper” method. That statement is derived from his math analysis
based on elementary calculus, and it has been supported by my own trials. It
follows that a short model is more sensitive to a given change of CG location,
e.g. due to using fuel, than a long one; this is intuitively correct. The
popular method for CG location cannot detect this.
Also, the popular expression “Vbar” or “tail volume”, which is the
tail area times Ls divided by the wing area times MAC, can be seen
to be irrelevant. This is because the tail exerts its effect about the CG, not
about the wing AC, a fact seemingly ignored by all publications simply because
the error involved is not all that great for a conventional design. However,
that error can be very large for other layouts. Vbar should be ignored; it is
not a proper measure of stability.
A simple test is to locate the CG
of a tandem wing design with wings of different MAC. One should get the same
result whether the design is treated as conventional (wing-first) or
tail-first. Similarly, try the Burnelli twin, which has a large lifting
fuselage: what is the MAC? The popular method fails, and the proper method
succeeds. That is true for all the other unconventional designs that I have
been flying for years, so that there is plenty of evidence.
By developing Fraser’s concept we
arrive at the notion of a pitch stability coefficient, which I call “PC”. This
enables one to choose how stable the model is to be, and to design all sorts of
models of different layout and size with the same or any other chosen degree of
stability. The eminent aerodynamicist Glauert proposed his normalized stability
coefficient based upon Ls before 1930, but I have not found it in
more modern textbooks.
Fraser`s article explained
clearly that pitch stability is independent of airfoil pitching moment, angles
of incidence, steady-state downwash, and of weights and wing loading. However,
all these become effective when calculating trim settings.
I transposed his equations to
make them suitable for a computer; at that time (1986-90) the available
software was BASIC, venerable, but easy to use and actually quite
sophisticated.
At this point the concept of the
“reference line” must be introduced. It is a lateral line on the plan, from
which dimensions, such as the location of the AC of a surface, are to be
measured for the purpose of calculating pitch stability. Its location is an
open choice; I usually place it at the point where a main lifting surface meets
the fuselage, e.g. at a wing root chord LE.
Tailless, tractor, very low AR, parabolic.
Fraser dealt with location of the
NP by assigning a value (“E”) for the effectiveness of the trailing surfaces.
Thus, the NP is at the location of the weighted mean of the area times distance
from a reference line of all the lifting surfaces, with the trailing surface
areas modified by E. In other words, you find the “centre of lift” of the plan
of all the lifting surfaces, with the trailing surfaces area modified by the following
factors:
1) The ratio
of the lift slopes (dCL/da) of the trailing and leading surfaces,
which depends on their aspect ratio (“AR”). The expression I use, to suit low
Reynolds Numbers, is: lift slope = 0.1*AR/(AR+2). Thus, if the AR of the
trailing surfaces is less than the AR of the leading surfaces, then E<1.
2) The
expression (1 – de/da), where de/da = rate of change of downwash versus angle
of attack; this is usually less than 1, depending mainly on the vertical and
horizontal distances between the leading and trailing surfaces, and sometimes
on their spans. The value for a T-tail is close to 1. It is independent of the
actual downwash.
3) Loss of
dynamic pressure, which is proportional to the square of the ratio of the air
velocities downstream and upstream of the leading surfaces.
Since the value of E is obtained
by successively multiplying these factors, it can be quite small. Also,
evaluation of 2) & 3) above is quite difficult, needing reference to good
books on aerodynamics, see Refs 2 & 3.
It is recommended, however, that 1) above should always be evaluated
because it is important, and easy to do, and 2) above is well described in Ref.
3. I usually ignore 3) above due to lack of data.
Sometimes I use one of these good
books, sometimes another; this is because design is largely a matter of
judgment.
Pitch stability calculations
Since designing models requires
numeracy, here are some of the equations I use. First, pitch stability, looking
at figure 1:
The X dimensions
are measured from the AC of the leading surface(s). The AC of a surface is
taken to be at the 1/4 chord of the corresponding MAC.
X12 is the distance between the
AC’s of leading and trailing surfaces, i.e., Ls. From the leading
surface AC, XN is the distance to the NP, and X1 that to the CG.
Fraser showed that PC = X1/X12 -
1/ (AW/E*AS+1); note: “*” is “x” (multiply) in computer characters.
E is the effectiveness of the
trailing surfaces; usually E<1; AW = area of leading surface(s), and AS = area
of trailing surface(s).
AW and AS can each consist of a
group of surfaces, e.g., biplane, triplane or whatever. PC is dimensionless.
By writing PC=0 the above
equation yields:
XN = X12/ (AW/E*AS+1), then,
PC = (X1-XN)/X12 so that the
stability coefficient is obtained for a given set of dimensions; alternatively,
X1 = XN+PC*X12 so that the CG is
located for a given stability coefficient.
Figure 1 shows the format for a
conventional model; not shown is a reference line, from which the drawing
dimensions are reckoned. For example, it is preferable (essential?) to locate
the CG from a real point like a wing root LE than from a theoretical point like
the AC of a surface. For unconventional model layouts, the reference line may
be placed anywhere convenient and the signs + or – in the above equations are
suitably chosen. In the difficult case of the three-surface layout (like
Piaggio ‘Avanti’), the leading surface is de-stabilizing and is ‘subtracted’.
Three Surface
The aerodynamic center of lifting surfaces
Next, we should address the task
of locating the AC of a group of surfaces. A lifting surface may consist of a
group of like surfaces vertically disposed (biplane, triplane, etc.), each
comprising one or more multi-taper surfaces. There is no need for the tedious
calculations to find a monoplane surface equivalent to those of a biplane or
triplane, as suggested by some authors. I call each panel of a multi-taper
surface an “element”. The AC of an element lies on the quarter-chord line (by
convention), and it can be located very accurately by using simple equations;
the drawing method so beloved by authors is a waste of time.
Elements bounded by straight lines are considered to be
trapezoids, i.e. a quadrilateral with one pair of parallel sides, requiring
only simple arithmetic derived by calculus. Conics - circular, elliptical and
parabolic shapes - are also dealt with by schoolroom calculus, and I have
developed simple expressions for locating the AC, relative to a reference line,
of many of the possible plan forms.
Complex surface shapes like those
with a straight line LE and a curved TE, for example, are readily handled by
employing a program based on what was called “Simpson’s Rule” in my schooldays.
The AC of a multi-taper surface e.g., a wing comprising several different
elements is found by calculating the weighted mean of area times distance of AC
from the reference line of all the elements of a group.
It is therefore simple, using a
computer, to make changes in the dimensions of the model and get the new result
very rapidly.
At this point yet another design
choice must be highlighted. If a body, e.g. fuselage or engine nacelle,
obscures a lifting surface from above or below, should the over-laid area be counted
as lifting? There is plainly a difference of opinion in current texts; my
tedious study of several classic full-size aircraft revealed that the basis for
items like wing area is confused - some do count the over-laid area, some do
not, and they do not bother to tell the reader. My firm opinion is that the
over-lay does not count, that is, lifting surfaces are those having an airfoil,
and are clear of a fuselage or engine nacelle. I have found support for this
view from practical results at the flying field. However, by all means make
some allowance for body shape in special cases.
Tailless Tractor Delta
Calculations to locate an AC
Trapezoid:
Now, let us look at figure 2:
what we need to know is the value of M, the distance between the reference line
and the AC, and the area A for an element.
Let the taper ratio T/C = R
M=MAC*CP+L/S*B, where CP=0.25,
i.e., the quarter chord point.
And, the area A=S*(C+T)/2
Then, L=S/3*(1+2*R)/ (1+R), and,
MAC=2*C/3*(1+R+R2)/ (1+R).
Several
trapezoids:
Looking at figure 3, a string of
two or more elements, like B-747 - (2); or SAAB ‘Viggin’ - (3).
To locate
the AC of the complete surface (‘ACW’) we calculate the weighted mean of all
the values of M*A to obtain the distance MW of ACW from the reference line:-
MW=(M1*A1+M2*A2+M3*A3)/(A1+A2+A3)
Note that the usual published graphical
method is limited to one element, whereas we can handle any number of elements
by calculation. Remember, expressions like these have to be written one time
only in the computer.
It is worth noting that the MAC
of the string of elements does not necessarily lie between a leading and
trailing edge.
Ellipse:
Referring to figure 4, the “Spitfire” wing was composed of two quarter -
ellipses having the same span S and different root chords C1 and C2.
Our
object is to find the value of M, L, MAC and the area A. Here we have to use
calculus; it is interesting that the solutions for the first three do not
appear in any of my textbooks, though values for MAC and L were given in a UK
magazine article, that for L was not quite correct, and the value of M, which
is what we really need to know, was ignored.
From calculus, we have the
general solution for all ellipses:-
Note: “PI” = 3.141593, the ratio
of circle circumference to diameter.
L = S*(1-(8/(3*PI))2)0.5 = 52.9% of S (very nearly)
MAC = C*8/(3*PI) = 85% of C (very
nearly).
A=C*S*PI/4 =78.54% of C*S (very
nearly).
To cut a fairly lengthy story
short, just tell the computer that: - M = (C1+C2)*0.2122+ C1*0.1512
Math buffs might like to check this:
let C2=3*C1, then M=C1, i.e., the AC is on the quarter chord. Think about it!
Parabola:
The parabola is first cousin to
the ellipse. There were some Russian aircraft with parabolic wings, and I
recently found this shape a good solution to the problem of designing a tractor
tailless model with a very low AR of 1.5.
Calculus shows that the following convenient expressions apply, looking at
figure 5:
M=0.4*C, and,
A=C*S*2/3. Also, MAC=0.8*C, where C =
root chord.
Compound shapes:
A surface having, for example, a straight
line LE with an elliptical TE (e.g. Republic P-47) requires a math treatment
where the chord is described by the sum of the y values of an expression for a
straight line and for a curve, as shown in figure 6.
For any element, MAC =
(2/area) * integral (chord2) from 0 to ½ span, see Ref 4, p 27.
This integral is messy, however,
the problem is solved by writing ‘chord’ as a function of x then dividing the
span into a number of narrow (‘dx’) chord wise strips. The chord is determined
from the equations for the leading and trailing edges. For example, that for a
straight line LE could take the form:
c1 = C1- x/S * (C1 – T), or: c1 =
C1 – x*tan(z).
That for an elliptical TE might
be: c2 = C2 * (1 – x2/S2)0.5.
The area of each chordwise strip
= (c1 + c2)*dx.
Then, using (in BASIC, or
equivalent) a “for-next” iteration, add up the values of (c1 + c2)2 and divide by the sum of the values of (c1 + c2), which gives the MAC. Then
find the corresponding value of x, by substituting the corresponding value of either
c1 or c2 into one of the two equations (for the leading or trailing edge). Then
the AC can easily be located. A good value for ‘dx’ is 0.001*S, so that there
will be 1000 iterations.
Back in 1986-90, when this
particular study was made, computers were so slow that there was time to queue
at the coffee machine while running; 10 to 15 years later, the result comes up
instantly. I have run this type of program for a range of compound shapes. No
such solution for this particular problem has appeared in print, to my
knowledge.
Tailless:
Neither Fraser nor anyone else,
to my knowledge, has suggested a method of locating the CG of a tailless model
other than by the ‘popular’ method, i.e. a static margin ahead of the AC of the
whole wing. However, the elevons normally produce a down force at the wing TE
to achieve positive pitch stability.
My proposal is that if the
elevons are treated as a conventional stabilizer, then calculation can be
performed by the ‘proper’ method, as for tailed models. The lifting surface is
what remains with the elevon removed. The ‘wing’ is treated as consisting of
several trapezoids, as shown in figure 7, with, maybe, one or two full chord
elements beside the elevon, and an element ahead of it having reduced chord;
the ‘tail’ is the elevon trapezoid.
I have designed and built various
such models, ranging from high AR backswept types like Northrop, through Deltas
like Convair ‘Dagger’ to a low AR plane like the Handley-Page 115 of 1965. The
lower the AR, the greater the error of the popular method, which results in the
CG being so far forward as to make the landing flair difficult or impossible.
The proper method, however,
involves an unknown factor, that is, the value of E for an elevon. What I have
done is to build and fly a number of such models, and from the observed results
arrived at an empirical value of E=0.2 which is cautiously recommended. An
expert author or two has ridiculed my idea, and they may be right; however, it
seems that none can support the popular method by theory or practice.
Tailless Tractor
V-tail design
Ref. 5 featured an NACA report
(#823 11/44) which showed that one should take care. It is usually considered
that the surface areas are obtained by dividing the equivalent flat and upright
surface area(s) by the cosine and sine of the vee angle to the horizontal,
respectively. However, the report points out that these trig ratios should be squared because the rate of change of angle of attack is also changed by the same
ratio. That is to say, V-tail surfaces should be larger than is commonly
thought.
The V-tail has always seemed to
me to be a good choice, especially for slope-soarers, and my observations
suggest that the NACA were probably right. Comparing otherwise similar models,
with orthodox or V-tails, the latter appeared to need the NACA increased area
in order to achieve similar pitch and yaw stability.
It can be shown that, designed by
the NACA method, the tail surface (‘wetted’) area is the same whether in Vee or
conventional form.
Yaw stability
It can be said that there is
positive yaw stability if the centre of lateral area (CLA) is trailing the CG.
Just what constitutes lateral (side) area is open to discussion. Fin(s) and
rudder(s) can be treated in the same way as horizontal lifting surfaces, and
the calculations are similar. It is reasonable to add portions of the lateral
area of bodies like fuselage, nacelles, and even airscrew(s); guidelines for
this are difficult to find (but try Ref. 2). This is a nice test of designer judgment,
interesting but not too critical.
Then, having summed the yaw
moments of the vertical surfaces (area times distance from CG to AC), in the
same way as for “PC”, how should we normalize it so as to get a similar yaw
stability coefficient “YC”? Personally, I divide by the span of the largest
lifting surface, on the intuitive grounds that, if the main wingspan is large,
then its yaw inertia is important. Others are free to disagree; a current
author simply makes his fin and rudder “look right”, and I have no problem with
that.
However, experience suggests that
it is useful to have a coefficient for yaw stability, so that one can make
coherent changes to a design. This method has worked quite well for me, having
found an empirical value of YC as a guideline for each kind of design. That is
to say, I use values of YC as follows, for: Conventional = 0.05 - 0.07,
Tailless = 0.04 - 0.08, Canard. = 0.07 - 0.1, and Glider = 0.03 - 0.04.
It may also be useful to relate
spinning to the value of YC, for, in my experience, if it is large then a spin,
as an aerobatic manoeuvre, may be impossible.
Trim:
Trim means the relative angles of
incidence of each group of lifting surfaces as specified on the build plan;
“in-trim” means that no control input is required, and “out of trim” means that
input is continuously applied.
What is the distinction between
“unstable”, and “out of trim”? The texts have usually explained this well:
roughly speaking, if you can control the model by pushing a stick or sticks
consistently in one direction, then it is merely out of trim. If, however, you
have to keep chasing the model by waggling the sticks, because it continually
gets away from you (divergent flight path), then it is unstable, and you may
feel lucky to land it in one piece. I know the feeling!
Looking
at figure 8, a model is in trim if the sum of all forces acting vertically and
horizontally, and of all moments about the CG, equal zero. The forces are lift,
drag, mass and thrust, and the moments are due to those forces not acting
through the CG, plus the pitch couples due to cambered lifting surfaces. For
models however, it is usual to leave out thrust and drag because we have great
difficulty in assigning values. So I treat power models as gliders, expecting
that the errors of omission are not serious.
We need to know the lift
coefficients of the surfaces and their moments about the CG, and their pitch
couples (which depend upon the pitch coefficients of the airfoils). These
values are all dependent on airspeed when it is between stalling and maximum
speeds. The calculations for pitch stability gave the lift moment arms and
surface areas, and textbooks supply an airfoil pitch coefficient - which is
usually constant over the values of CL corresponding to the working
speed range.
The actual value of the pitching
couple of a lifting surface is proportional to the MAC (or the mean chord), the
area, and dynamic pressure Q, where Q=rho/2*v2; and rho=air density,
v=airspeed. Angles of incidence are based on the difference between angle of
attack ALPHA and the angle of zero lift, = ALPHA0 for the chosen airfoil.
Returning to figure 8, the task
is to evaluate L1 and L2, which in turn yields the CL and ALPHA of
each group of surfaces (leading and trailing) so that equilibrium is obtained.
The relative angle of incidence is often called the “decalage”, from the French
word for ‘angular displacement’. This is calculated for several speeds from
minimum to (a notional) maximum.
Having chosen the trim speed,
which corresponds to neutral elevator, the decalage of the lifting surfaces is
now known. They are then arranged relative to the fuselage (if any) so that the
flying attitude of the model is as you want it. Think “Whitley”, the WW2 RAF
bomber! For different layouts such as three-surface, the diagram is altered to
suit, same as for stability.
In flight, pilot input to
elevators or elevons obtains trim at all other airspeeds. In the case of the
“Flying Flea”, control is uniquely done by rotating the leading wing, which
changes the decalage; the range is surprisingly small, perhaps 5 degrees from Vmin to Vmax.
Similarly, the ‘all-flying tail’,
popular in modern gliders, controls by varying the decalage.
The “Flying Flea”, tandem
wing.
Pitch controlled by
rotating the leading wing a few degrees.
Trim calculations:
Written for BASIC, we would have:
L1+L2=W where W = weight; forces resolved vertically.
L1*X1+M1+M2=L2*X2 i.e., the sum of couples and moments about
CG = zero.
The general solution of these
simultaneous equations is as follows:
L1=(W*X2-M1-M2)/X12
And, L2=W-L1
So, for a group of lifting
surfaces, lift L=CL*A*Q, and pitch couple M=CM*A*MAC*Q, where CM=airfoil pitch
coefficient,
and, CL=lift coefficient = LIFT
SLOPE*ALPHA, where ALPHA = angle of attack (radians).
To solve for the difficult
three-surface case, one way to go is to combine two surfaces pragmatically, so
that the simultaneous equation is valid, then alter the combination and try again.
It can be done.
In general, the angle of
incidence =ALPHA-ALPHA0 where ALPHA0 = the zero lift angle of the airfoil.
Trailing surfaces may be affected
by downwash from leading surfaces; this is a matter of judgment in the context
of trim, similarly to the effect of the rate of change of downwash in the
stability calculations. Simple airfoil theory states that: downwash (degrees) =
18.24*CL/AR, at the TE of the surface.
But, if the trailing surface is distant from the wake centre by
more than half the chord of the leading surface - as for a T-tail - the effect
of downwash is greatly reduced. On the other hand, in the case of a very close
up trailing surface - such as the “Flea” - experience has shown that the above
equation serves as a reasonable guideline. There are other expressions for
downwash to be found in texts, especially ref 2.
Since we want to know the
decalage (DEC) over the speed range, repetitive calculation is required, ideal
for a PC, too laborious “by hand”. Also, since most textbooks are pre-PC, it is
necessary to alter nearly all the symbols that have been used for decades in
textbook equations. The PC may not recognize suffixes, subscripts, lower case
or fancy Greek letters, so each of us has to invent our own. My practice is to
use mnemonic strings, these may look lengthy but the PC memory is hardly taxed.
Since we are dealing with force, length and time for trim calculations, we have
to get the units right, which is up to each user - imperial or metric, for
example. I work in inches and ounces
for designing, and convert to feet and pounds for dynamics.
To find Vmin, and
decalage DEC vs. airspeed V from Vmin to Vmax:-
First, re-write dynamic pressure
Q as a coefficient for V2: Q =
0.002378/(2*144)
The sub-routine is (ignore my line
numbers):-
1540
FWPM=Q*V^2*CMWF*AWF*MCWF/12 FWPM =
Leading surface pitch moment, CMWF =
Leading surface pitch coefficient (1/4
chord), AWF = Leading surface area,
MCWF = Leading surface mean chord or MAC.
1550 RWPM = Q*V^2*CMWR*AWR*MCWR/12 As 1540 but trailing surface pitch moment.
1560
CLWF=(W/16*X2/12-FWPM-RWPM)/(Q*V^2*AWF*X12/12) CLWF = Leading surface CL.
1570 CLWR=
(W/16-Q*V^2*AWF)/(Q*V^2*AWR)
CLWR = Trailing surface CL.
1580 AAWF=CLWF/SLWF Leading surface angle of attack. SLWF = lift slope, a, = ðCL/ða
1585 INCWF=AAWF+AA0WF Angle of incidence. AA0WF = zero lift angle.
1590 AAWR=CLWR/SLWR Trailing surface, angle of
attack.
1600 DW=18.24*CLWF/ARWF Downwash, leading surface. The value 18.24
may be varied depending on the distance downstream and height h of the trailing
surfaces.
1610 INCWR=AAWR+AA0WR+DW Trailing surface, angle of incidence.
1620 DEC=INCWF-INCWR Decalage
1630 RNWF=V/12*MCWF/.000156 Leading surface Reynolds number.
1640 RETURN Return to GOSUB command.
To find Vmin, which corresponds
to the condition CLWF 
CLMWF (leading surface, CL, CLmax) we
guess the lowest likely stalling speed for a start value for V; then to the sub-routine
and back, then comparing CLWF to the value of CLMWF 
0.05, say. The “gate”
of 0.1 is notional, found by trial and error/experience; recommended as a
starter. If CLWF fails to pass through the “gate” (yes, it is quite like penning
sheep), then we change the value of V by adding or subtracting a small notional
quantity, the “step”. Somewhere earlier in the program, then, will be a line
like this:-
20 STEP=.05, Increment of airspeed, ft/s.
So, to find Vmin, we
can write:-
900 V=30 A guess at the value of Vmin
910 GOSUB 1540
920 IF CLWF>(CLMWF+.05) THEN
V=V+STEP:GOTO 910 V too low,
ie., V<Vmin
930 IF CLWF<(CLMWF-.05) THEN
V=V-STEP:GOTO 910 V too
high, ie., V>Vmin
940 VMIN=V Stalling speed (CLWF 
CLMWF)
Next, to construct a Table
showing decalage vs airspeed, plus anything else available such as incidence, CL,
downwash and RN, we can write:-
950 IF VMIN<30 THEN V=30 ELSE
IF VMIN<40 THEN V=40:GOSUB 1540 The
values 30 and 40 are given here from an actual example; this is trial and error
time. Next, we obtain numbers for the Table:-
960
V1=V:INCWF1=INCWF:INCWR1=INCWR:DEC1=DEC:CLWF1=CLWF:DW1=DW
970 V=V+10:GOSUB 1540 And so on.
The value 10 is a notional “step” in V; often a step of 5 is
adequate. This iteration ceases at some preset high value of V, the data is
then sorted into tabular form for printing; a typical result might look like
this:
V ft/sec 40 50 60
70 80
Airspeed
INCWF +9.3
+2.7 +0.4 -0.9 -1.8 Leading
surface incidence, deg.
INCWR +0.2
-2.7 -3.7 -4.3 -4.7 Trailing
surface incidence, deg.
Decalage +9.0 +5.4
+4.2 +3.4 +2.9 Difference, deg.
In order to draw the model plans,
it is necessary to choose the airspeed for trim setting; some may choose a
fairly low speed corresponding to take off or approach, others may prefer high
cruise or maximum speed. I generally go for the former.
At all other airspeeds, some
pilot input to the elevator (or, strictly speaking, the pitch control) is
required, though it may be so small that we do not notice it.
It is entirely feasible to mix
throttle and elevator in a programmable Tx to try for horizontal flight
whatever the throttle opening; be warned, this a trial and error operation -
adjustments should be quite small. Very gratifying when more-or-less right!
Decalage is usually positive,
that is, the leading surfaces are at greater incidence than the trailing
surfaces.
Some current authors state that decalage must be positive for pitch stability. This is not so, because in
configurations like the “Flying Flea” the downwash effect evidently results in
the model being trimmed with negative decalage, yet there is positive
stability. That is my experience, based on two “Fleas”, which has told me not
to believe everything I read.
Tandem wing, twin tractor
Stall Behaviour:
Following a discussion of pitch
stability, a note is required on the ultimate loss of pitch stability that is,
when the model stalls. Fraser’s statements do not apply since we are not now
looking at the effects of a micro-gust. At the stall, either the leading or the
trailing surfaces reach CLMAX, with reduction in CL to follow, as ALPHA
increases. It is clear that the design, which now includes the choice of
airfoil sections, should ensure that the leading surface(s) stall first while
the trailing surface(s) are still increasing CL as ALPHA increases.
For unorthodox models like the
tandem wing and canard, it is suggested that airfoil camber and thickness
should be less in the leading surface(s) than in the trailing surface(s), so
that the leading surface(s) should stall first.
Canard, Pusher
In this connection, I have lately
used airfoils as follows: Flea, leading & trailing = NACA 23012. Swept
canard, leading = SD6060; trailing = NACA 2415, or MB253515. Tandems, leading =
Gottingen 796; trailing = Gottingen 797.
For the two latter types, the
leading surface airfoil is thinner and less cambered than that of the trailing
surface(s). Some of my earlier experimental swept canards and tandem wings
employed thicker and more cambered airfoils up front, on the grounds that the
leading surfaces are more heavily loaded than the trailing surfaces. They were
stable, but stall behaviour was poor - the trailing surfaces lost lift first,
very untidy.
Burnelli ‘lifting fuselage’ – the one built in Britain, which served
thru WW2
My design procedure summarized
1. Sketch the plan
and side elevations of the model using a CAD program, like DesignCAD,
reflecting the object(s) of the design, e.g. performance, maneuverability, load
carrying, or just “unorthodox, how well will it fly?” This is where a design
“vision” begins to become practical.
2. Calculate the
CG location for a preferred value of PC; my particular choice of a minimum
value is: PC = -0.08. I call this CG location the “aerodynamic CG”.
3. Calculate the CG location of the model when
constructed, which I call the “structural CG”, using estimated values for the
weight and cg location of every item, from motor(s) thru wings, tails, and
fuselage, down to servos, battery, Rx and switch. Find the CG locations
corresponding to fuel tank(s) empty and full. For this operation, almost any
guess is better than nothing. With a little experience, it is surprising how
well the location of the structural CG can be forecast.
4. Run and re-run
steps 1, 2 & 3, changing dimensions and shifting components until the
structural and aerodynamic CG agree reasonably well. This should be at the
worst case of fuel empty or full, and can include ballast.
5. Concurrently
with positioning the CG to ensure reasonable pitch stability, obtain
estimations of yaw stability, and, having chosen the airfoils, the decalage. It
is not difficult to estimate stalling speed as well. These things are done by
simple extensions to the pitch stability program.
6. Make working
drawings using CAD, while quickly checking the effects of dimensional or other
changes using the calculation programs.
To sum up, if you wish to design
a conventional model then by all means use the simple magazine article
‘popular’ method for locating the CG; however, be aware that the method is
incorrect and should never be used for an unconventional design; instead the
‘proper’ method is recommended both on the basis of sound mathematics and
practical results.
Tandem Wing
Book References:
1)
‘Equilibrium, stability & the load on your tail’ by David Fraser,
pp101-119, Soartech #6, 1985
2)
“The Design of the Aeroplane” by Darrol Stinton, pub. Granada, 1983
3)
“Model Aircraft Aerodynamics”, by Martin Simons, pub. Argus, 1987
4)
“Theory of wing sections”, by Abbott & von Doenhof, pub. Dover, NY, 1949.
5)
Soartech #2, 1983
