
91


97

AERODYNAMIC COEFFICIENTS AND AIRLOADS PROGRAM
AERODYNAMIC COEFFICIENTS

Spanwise Coefficients

The program AIRLOADS.BAS calculates the basic and additive
Spanwise aerodynamic lift coefficient distributions for the wing.
It combines these for the Spanwise lift coefficient distribution
for any specific total wing lift coefficient and then calculates
the associated Spanwise drag and moment coefficients for that
wing CL. Where there is a discontinuity in lift between the
aileron and flaps, the program mathematically fairs the basic
distribution.

Stall Coefficient

The program also calculates the stall lift coefficient for
every section of the wing. Then incrementally increases the total
wing CL until one of the section c^ touches the stall c^ for that
section. This provides a calculated stall C^ and angle of attack
for the wing.

Airplane Less Tail Coefficients

The program calculates the pitching moment coefficient of
the fuselage and nacelles. It adds them to the total wing moment
coefficient to provide lift, drag and moment coefficients for the
airplane less tail for any C^. It then formulates the equations
for lift, drag and moment to be used in the program FLTLOADS.BAS
to make the balancing calculations for the V-n diagrams.

Equations

Equations used to calculate the Spanwise additive lift
distribution for a Cr=l are:

cCiai=.5(moC/Mo+4S/(3.1416B)(l-(2y/B)2))
Mo=SUM(moCdy/(S/2))
where c=chord of element

c^gj=lift coefficient of element

niQ=slope of lift curve of element

S=wing area, square inches

B=span tip to tip, inches

y=element butt line, inches

dy=width of element, inches

MQ=slope of lift curve of the wing

101

FAR 23 LOADS

Equations used to calculate the unfaired spanwise basic lift
distribution are:

A^o=SUM(moar.cdy)/SUM(moCdy)

aa=ar~\lQ

cib=(mo/2)aa

cc^=unfaired basic lift of element

C^=SUM(cc^dy)

where A^Q=angle from arbitrary reference plane to zero

lift plane of the wing

a^.=angle from arbitrary reference plane to zero

lift line of each element
C]^=basic lift coefficient of wing =0
aa=angle from wing zero lift line to section lift

line

Equations used to calculate faired spanwise basic lift
distribution are:

L=a/2 if a>b

L=b/2 if b>a

where L=distance to be faired inboard and outboard of
discontinuity between flap and aileron
a= distance from discontinuity to tip
b= distance from BL 0 to discontinuity

THETA=3.1416(yi-yinbd))/(2L)
^Ib (faired)^0^-^^ ave) (ABS(COS(THETA) )+cc^ ave

where THETA= angle used in Cosine function to fair
y,= Butt Line of an element

yinbd= ydisc^
y(ji,g-= Butt Line at discontinuity of flap & aileron

cCjib ave= ave element lift at discontinuity

Tau is a correction factor for the slope of the lift curve
of the wing to account for the deviation of the wing plan form
from an ellipse. Tau is a required input to AIRLOADS.BAS. A sepa-
rate program TAU.BAS is provided to calculate Tau per ANC1(1)
"Spanwise Air Load Distribution", Army-Navy-Commerce Committee on
Aircraft Requirements, 1938.

Sweepback

AIRLOAD4.BAS is a supplemental program on your program disk
which should be used in place of AIRLOADS.BAS to calculate aero
coefficients when sweepback (of 25 percent chord) is greater than
15 degrees.

102

AERODYNAMIC COEFFICIENTS & AIRLOADS

AIR LOADS

The airplane less tail equations, weight-cg data, structural
speeds  and surface geometry will be entered into the  next  pro-
gram, FLTLOADS.BAS, to calculate data for every condition on  the
V-n  diagrams.  For each condition, the C^ for the wing  and  the
speed  of  the airplane will be calculated by that  program.  The
critical  wing conditions will be selected by the following  pro-
gram,   SELECT.BAS.  THEN  you  will  return  to  this   program,
AIRLOADS.BAS.

With the CL and V known for a specific condition, the actual
wing  air loads can be calculated by AIRLOADS.BAS.  The  spanwise
airload  distributions  for lift, drag and  pitching  moment  are
calculated.  Then  the  shear, bending moments  and  torsion  are
calculated along the quarter chord on the wing reference plane in
the airplane coordinates.

Accelerated Rolling Condition

According  to FAR 23.249(a) the rolling  acceleration  loads
may be obtained by modifying the symmetrical condition for condi-
tion A in the figure of FAR 23.333(d) which is the condition  for
stalling  speed  at  limit load factor. For  normal  and  utility
category airplanes assume 100 percent of the airload of condition
A  on one side of the airplane and 70 percent on the  other.  For
airplanes of more than 1000 pounds design weight, the  percentage
may  be increased linearly with weight up to 75 percent at  12500
pounds.  (For  acrobatic category use 60  percent  regardless  of
weight.)

In the sample calculation to follow for accelerated roll, we
found the bending moment at the plane of symmetry for condition A
(case  262) is 324,050 inch pounds. We used case 262 because  the
critical  condition  for accelerated roll was case 280  at  18000
foot altitude and CG 4 as selected by the program SELECT.BAS  and
case 262 is condition A for CG 4 at 18000 feet altitude.

Using  the  formula  for percentage on  the  other  side  we
calculate the other side is 71.43 percent or 231,469 inch pounds.
Then  the unbalanced moment accelerating the roll is  92581  inch
pounds.

At this point we will  calculate the airloads for  condition
280.  Later  the reacting moments to unbalanced moments  will  be
calculated by the wing inertia program, WINGINER.BAS.

Steady Rolling (maximum outboard torsion) Condition

According   to   FAR  23.349(b)  the   effect   of   aileron
displacement  on wing torsion may be accounted for by adding  .01
times  the  aileron  deflection  to  the  basic  airfoil   moment

103

FAR 23 LOADS

coefficient over the aileron portion of the span in the  critical

condition in 23.333(d). The aileron deflections are specified  in
FAR 23.455.

For  conventional ailerons the deflection at maneuver  speed
is  the full deflection. For deflections at V^ the deflection  is
Vp/V^  times  the  deflection at Vp. For deflections  at  V^  the
deflection  is .5 times Vp/V^. This is the FAA interpretation  as
expressed in CAM 3.222(b)T3).

In our sample the critical steady roll condition selected by
program SELECT.BAS is case 138 which is for CG 1 at V^ at  12,000
feet   altitude.   Then  the  deflection  of   the   aileron   is
(15)V /V^=15(121.3/170)=10.703 deg.

The increment of pitching moment coefficient to be added  to
the  basic  airfoil coefficient is  -.01(10.703)=-.107.  This  is
added  to  the -.03 for the basic airfoil for c^ = -.137  in  the
area of the aileron.

Sweepback

AIRLOAD4.BAS is a supplemental program on your program  disk
which  should be used in place of AIRLOADS.BAS to calculate  air-
loads  when  sweepback (of 25 percent chord) is greater  than  15
degrees.

High Mach Numbers

AIRLOAD4.BAS is a supplemental program on your program  disk
which  should be used in place of AIRLOADS.BAS to  calculate  air
loads when any Mach number is greater than Mach .5.

(Note: Either AIRLOADS.BAS or AIRLOAD4.BAS may be used to  calcu-
late aero coefficients when sweepback is less than 15 degrees but
then  AIRLOAD4.BAS should be used to calculate airloads  if  Mach
number is greater than .5).

104


116


117

FAR 23 LOADS

AIRFOIL SECTION DATA FOR BEACH BANANA

^ero lift

-1 deg for all 23000 airfoils, flap at 0 deg  **

-16 deg for all 23000 airfoils, flap at 40 deg

^zero lift = -ib deg ?or SL1L '-->UUL> airroiis, ri
c^o = .01 for all 23000 airfoils, flap at 0 deg

-m
dc^/da --

BL

0

0

0

0

46.5
46.5
46.5
46.5

109.279
109.279
109.279
109.279

201
201

.04 for all 23000 airfoils, flap at 40 deg
.03 for all 23000 airfoils, flap at 0 deg

-.45 for all 23000 airfoils, flap at 40 deg

f* *~

Cy= -.03 for all 23000 airfoils, flap at 0 deg
c

.1075 for all 23000 airfoils, flap at any angle

t/c Percent

16.5

16.5

16.5

16.5

15.459
15.459
15.459
15.459

14.053
14.053
14.053
14.053

12
12

Flap Deflection

0

0

40

40

0
0

40
40

0
0

40
40

RN

3x10"
9x10
3x10"

9x10

6

3xl06
9x10"
3x10"
9x10"

3xl06
9x10
3x10"
9x10"

3xl06
9x10"

c! max *
1.45

1.66
2.36
2.62

1.46
1.68
2.37
2.64

1.48
1.70
2.44
2.66

1.50
1.74

AIRPLANE DATA

Fuselage width =
Fuselage length =
Fuselage frontal
Total horizontal
Wing root c/4 at
Angle from WL to

3.833 ft.
: 26.522 ft.

area = 17.231 sq. ft.

and vertical tail area = 51.785 sq. ft.
31.8 percent of fuselage
fuselage cord nose to tail cone = -.918 deg,

* Estimated and interpolated from figure 4.10
Aerodynamics" by Dommasch, Sherby and Connolly

in "Airplane

**   Estimated from data in "Theory of Wing Sections" by Abbott
and Von Doenhoff

118

LOADS AND BALANCING CALCULATIONS FOR FLIGHT ENVELOPE
GENERAL

The  FLTLOADS.BAS  program computes all the loads  for  any

combination  of  airspeed  and  load factor  on and within  the

boundaries  of the flight envelopes specified in FAR  23.333  and
FAR 23.345.

The  data  necessary  to make these  calculations  has  been
provided  by  the  weights,   geometry,   aero  coefficients  and
structural speeds programs.

The  points  or  conditions on the flight  envelope  of  FAR
23.333  must  be calculated not only for each airspeed  and  load
factor  but  for  each  altitude  up  to  the  maximum operating
altitude.   For  single naturally aspirated engine airplanes with
gross  weight of 6000 pounds or less,  dive speed of mach  .4  or
less  and max operating altitude of 15000 feet or less,  the  FAA
will  probably  accept sealevel loads only.  For other  airplanes
with  maximum operating altitudes of 20,000 feet or  less,  three
altitudes should be investigated -- sealevel,  shoulder  altitude
and  max operating altitude.  For other airplanes with  operating
altitudes   above   20,000  feet,   four  altitudes   should   be
investigated,  to  include  20,000 feet where the  gust  formulas
begin to taper.

For the flight envelope of FAR 23.345 for take-off, approach
and landing, only sealevel need be investigated.

MENU

A menu is provided to choose the activity you want as
follows:

1. Start new input

2. Read old input from disk

3. Save input on disk

4. Correct input

5. Make balancing calculations for flight conditions

6. Print input on paper

7. Save balancing calculations on disk

8. End program

START NEW INPUT

Input data required is entered by keyboard in response to
questions on the monitor. Provisions are made to verify the data
and make changes until data is correct.

INPUT FROM DISK

159

FAR 23 LOADS

Data previously saved by menu item 3 may be read into com-
puter memory without retyping.

SAVE INPUT ON DISK

Input data which has been entered by keyboard or from disk

or that has been corrected may be saved on disk under new file
name or over written on same file name.

CORRECT INPUT

This  subroutine  is not included in  the  current  version,
Version 2.02. Corrections can be made by entering data again from
the keyboard where changes can be made until data is correct.

You  can  correct the input data on your word  processor.  First,
save your current input data to a disk file using menu action  3.
Then load that file to your word processor as a non-document file
(an  ASCII file). Make the changes using your word processor  and
save the file to the same name. Then run the FLTLOADS.BAS program
using the corrected data file.

BALANCING CALCULATIONS

The  airplane  is  balanced  at each  point  on  the  flight
envelope  by  making  the  sum of moments of  all  loads  on  the
airplane about its' center of gravity equal to zero.

An iteration process is used to accomplish this. For  condi-
tions  with known speeds the aero coefficients are corrected  for
compressibility effects due to speed and altitude. Then the angle
of  attack is estimated and the total airplane load  factor,  N^,
normal  to  the  airplane reference line is  calculated.  If  the
calculated load factor N^ does not equal the required load factor
for  the condition, the angle of attack is incremented until  the
correct load factor is produced.

For conditions on the stall or C]_^^ line, the speed is  not
known.  So a speed is estimated and the angle of attack is  found
which produces the correct load factor by the process above.  The
CL  for the wing is calculated from C^ = L^/(qS).  If the  C^  of
the  wing is greater than the C^g^^LL' the ^Q^  is  incremented
and  the process repeated until the correct load factor and  lift
factor  are  attained. Each time the speed is changed,  the  aero
coefficients  are  corrected for compressibility effects  due  to
speed and altitude.

The  data for each point on the flight envelope is saved  in
computer  memory  including the name of the condition,  the  case
number, equivalent air speed, altitude, normal load factor, angle
of  attack, compressibility factor, wing lift  coefficient,  wing

160

LOADS & BALANCING CALCULATIONS FOR FLIGHT ENVELOPE

lift  normal to the airplane reference line, tail load  and  air-
plane drag load. The data is automatically printed on paper.

PRINT INPUT DATA ON PAPER

The current input data in computer memory may be printed out
on paper.

SAVE OUTPUT DATA FROM BALANCING CALCULATIONS TO DISK

After balancing calculations are complete,  the complete set
of data for the balanced flight conditions for all points on  all
the flight envelopes (V-n diagrams) must be saved on a disk  file
for input to the program SELECT.BAS which will select the  criti-
cal loads.

EQUATIONS

Equation used for balancing the moments about the center of
gravity with the horizontal tail is:

L^CMA.T+LzWCXcG-Xw^DxCZcG-^/^T-^CG)

where L^=horizontal tail load

M^_y=aerodynamic pitching moment of airplane less tail
L^ ^_rp=lift airplane less tail normal to airplane ref
Dy=drag of airplane parallel to airplane reference
XQg=fuselage station of center of gravity
X^=fuselage station of 25 percent MAC
ZQr_=waterline of center of gravity
Z^=waterline of 25 percent MAC

Equation  for aerodynamic coefficients as function of  angle
of attack:

CL=Co+ClA+C2A2+C3A3+C4A4

CD=Do+DlCL+D2CL2+D3CL3+D4CL4
C^MQ+M^A+M^+M^+M^A4

where C^=wing lift coefficient
Cp=wing drag coefficient
C^=wing pitching moment coefficient
CJ=coefficients for lift equation
DJ=coefficients for drag equation
Mj=coefficients for moment equation
A=angle of attack

161

FAR 23 LOADS

Equations used to calculate lift, drag and moment relative
to the wind are:

L^OS

D=CoQS

M=C^QSC

where Q=dynamic pressure=V-'-/295

S=wing area

C=MAC

Equations   used   to rotate coefficients into   airplane
reference are:

L^=L x COS(A) + D x SIN(A)
D^=D x COS(A) - L x SIN(A)

where Lr7=lift of wing normal to airplane reference
Dv=drag of wing parallel to airplane reference

Equations   used to adjust aerodynamic coefficients for
compressibility effects are:

G=1/(1-M2)-5
where G=Glauert Correction
M=Mach number

A=29.02436(518.69-.003566H)-5
where A=speed of sound in knots
H=altitude

s=(l-6.879xlO~6H)4258

where spratio of density of air at altitude to density at
sealevel

V^=Vp/s-5

M=V^,7A

where Vip=true speed in knots

Vg=equivalent speed in knots

CLMAX=1-19367+.32739M+10.8352M2-44.4985M3+51.8759M4-19.5434M5
^^LMAX^0"^ Macn)/CTMav(SL 1G Mach) in above equation

This equation for CTMAX as a ^"ction of Mach Number is a
least squares fit to the curve for a wing of aspect ratio of 6
with a 23016 airfoil at the root and 23009 at the tip (ref Theory
of Wing Sections by Abbott & Doenhoff). This curve is shown in
the figure on page 163. You will notice that the C^^ at Mach .7
is 50 percent of the C[^^ at Mach .1 which is about the stalling
speed of the typical light airplane.

The shape of the curve is typical for most wings. The shape
162

LOADS & BALANCING CALCULATIONS FOR FLIGHT ENVELOPE

at condition Mach number to the CJ^^ at sea level lg stall.

Then

C^^^(Cond Mach Your Plane)=R x CJ^^(SL 1G Your Plane)

CATEGORY

This version of FLTLOADS.BAS (version 2.02) accounts for
differences in regulations for normal, utility and acrobatic
categories.

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177


