B.2.2 Estimation of longitudinal aerodynamic derivatives
○ We have defined the following longitudinal aerodynamic derivatives:
Lu  1  L , L  1  L , Tu  1  T ,
m U E
m  E
m U E
Du  1  D , D  1  D ,
m U E
m  E
M   1  M , M   1  M , M   1  M ,
I y  E
I y  E
I y  E
M u  1  M ,
I y U E
M   1  M
I y  e
E
.
○ The values of these parameters are estimated from wind tunnel data, through
formulas that are derived as follows.
38
(1) Lu : The lift velocity derivative
 Lift is defined as L  Lw  12  airU 2SwCLw where CLw may also depend on
 SU 
U C 
U . As a result, Lu  1 L  air w 0  CLw  0 L  .
m U 
m
2 U 

 
--- The subscript  indicates that the derivative is evaluated at a constant  .
--- If necessary, contribution from the horizon tail can be derived the same way.
 We often express the speed U in terms of its Mach number, M  U cs
where cs is the speed of sound; hence, a better expression for Lu will be:
CL 
2g 
Lu 
 CL,trim  M
 .
UCL,trim 
2 M 
--- We have uses the relation:
 air S wU  air S wU UCL,t r i m 2m g
2g




.
m
m
UCL ,t r i m mCL ,t r iUm UCL ,t r i m
 At low subsonic region, data shows that  CL  M  0 .
Prandtl-Glauert
We can also use the Prandtl-Glauert formula:
CL
1
CL (M ) 
 C L M 0 .
M
2
M

1
1 M
In transonic region and supersonic region, estimate  CL  M from data.
39
(2) L : Vertical damping
 We have CL w  aw ; hence,
qS w
qS w C L ,trim
g

L
1
L  

 aw 

 aw 
 aw
m 
m
m C L ,trim
C L ,trim
 In general, aw , is a function of Mach No. and of the aspect ration AR :
a
--- For high AR : aw  a0 1  0 , a0  2 : the Mach No. effect.
A
2

--- For low AR : aw  2 AR
R

1 M
(Slender wing theory).
(3) Du : Drag damping
 Again, D  12  airU 2 S wCDw where CD may also depend on U ; hence,
CD 
2g

Du  1  D 
 C  M
 .
D
m u
U 0CL ,trim 
2 M 
 Estimation of  CD  M can be made from data
--- The curve of CD versus M closely resembling that of CL versus M . As a
result, we can ignore  CD  M at low Mach number.
40
(4) D : Drag-AOA derivative
 In general, CD and CL are related as follows:
2
2
aw  2
CL
C D  C DP 
 C DP 
.
 e AR
 e AR
--- The first term, CDP , is the parasitic drag and the second term is the classical
induce drag, for an elliptical wing in particular. For general designs using
non-elliptical wing, the second term is merely an approximate formula.
--- The constant, e , is called the span efficiency factor; e  0.75 for most
conventional subsonic aircraft, lower for lower AR and higher flight speed.
2
qS wC L ,trim

S
U
 aw
aw
g aw

D
air w
1
D





2
 Then, 
.
m 
m
 e AR
m
 e AR
 e AR
2
(5) Tu : Thrust-velocity derivative
 Tu is evaluated at constant throttle positions.
 The value of Tu depends on the kinds of power plant.
--- Tu  0 for propeller; Tu  0 for turbo jet and rocket; and Tu  0 for ram jet.
 Value of Tu is normally derived from experimental data.
41
(6) M u : Speed static stability
2
 We have defined M  12  air U 0 S wc CM with C M also depending on U .
 S cU 
U C 
As a result, M u  1  M  air w 0   CM  0  M 
I y U 
Iy
2 U 

 S c U  C 
 But CM   0 ; hence; M u  air w 0   M 
Iy
 U 
2
--- We often express I y in terms of I y  m  K y (See next page). Then,
 air S wc U 0 U 0CL,trim  CM 
2gc
 CM 
Mu 



  2

2

U

U
U
C
m Ky

 K y U 0CL,trim 

0 L ,trim
--- The speed could include effects of flow field and engine thrust T , as
follows:
 CM CM  M CM  T
.




U
M  U
T  U
 
--- The thrust term depends on engine type and engine location.
 A negative Mu is usually not desired. As nose-down increases M while
Mu  0 further enhance the pitch down motion.
42
【Digression: The expression of I y in terms of I y  m  K y 】
2
mass m
○ The expression, I y  m  K y , visualizes the
A/C as a ring of mass m and with a radius K y
perpendicular to the y axis. In general, K y is
termed the radius of gyration in the y axis
 K y also has an indication on the distribution
radius of gyration Ky
of A/C mass on the X_Z plane..
○ The major purpose of this expression of I y are as follows:
2
 To allow for the normalization of the A/C mass from the formula for M u ,
in order to emphasize the dependence of M u on the A/C geometry.
 To show off the fact that I y depends on the square of the A/C size.
○ Similar expressions also are used for I x and I z when dealing with the
lateral directional motions.
43
(7) M : AOA static stability
 
 We have M   1  M
I y 
U
qS w c   C M 
g c
 C 


 M 
 2
Iy
   K y C L ,trim   U
CM
 aw  static margin ; CM   0 if the A/C is statically stable
CL
--- The value of C M depends strongly on the tail size and the C.G. position.

--- CM  aw 
(8) M   1  M  : Pitching damping
I y    E
 A constant  is assumed.
--- A curved flight path results.
--- The tail feels a different angle of
attack:  t  X t U
--- The change in tail AOA results in
M   t 
 M  M Xt 


.
  U
 X t
relative
U wind
t
t
act
q Xt
Xt

  0,   0
t
t
 M  M Xt
1  Xt   M


 Then,
;
hence,
M


I y U  t
    t U
44
【Digression: The estimation of M /  t 】
○ This derivative, M /  t , which measures the change in pitching moment
due to change in the horizontal stabilizer angle, will continue to appear in
many subsequent discussions.
○ Therefore, we would like to derive a formula to estimate its value.
 From the right figure,
M
M  Lt  X t
Lt
 We also have,
t
Lt  qSt at   t
M
Xt
 Therefore,
 qSt at  X t
 t
○ As a result, the following expression for M  can be written:
2
2
2
X t qSt at X t qSt at S wCL,trim X t
S
g at
M  


 2 t 
I yU 0
I yU 0
S wCL,trim K y S w U 0CL,trim
45
 
(9) M   1  M : AOA damping
I y  E
 This derivative is evaluated at constant  .
--- It will result from a plunging motion.
Plunging flight path
--- In this case, change in AOA is due to change
  0,
in flight path angle  . Also, any change in
  0
act
 w occurs simultaneously in  t .
--- However, a   0 causes the downwash to
Xt
U
X
vary, and perturbation in pitching moment
Downwash propogation time: t  Ut
arises due to lag in the downwash (見下頁).
--- If the tail feels downwash from time required for air to move from wing to tail, i.e.
t  X t U , then the change in downwash angle due to a   0 will be:
Xt
d

d

d

  t 



 
d dt d U
--- The change in pitching moment due to  :
X
M  M    M  d  t  
 t
 t d U
 As a result,
X
M   1  M  d  t  d  M  .
I y  t d U d
46
【Downwash deficiency in a plunging motion】
In general, the change in wing AOA also
occurs on the tail, except the downwash.
A
1







.



.

Actual downwash
.
   + t) -  

0
.
  0+ t

E



U


Error in  produces a
pitching moment M.
M
 Lt
U
.
     
 d
d

But the downwash is caused by the wing
and it will need  = Xt /U to reach the tail.
Xt
U
  0+. t
  0+ t)
2) To maintain equilibrium we would need
D
   0
U
    2
With  = 0, and after some t > 0
.
1) The AOA changes to   0+ t
C
0
2
U
    1
At equilibrium
B
.
     

U

47
M 
(10) M   1 
 : Control effectiveness
I y    E
 We will have M  0 for usual sign convention. Also, it's magnitude must be
large enough to provide enough control to the aircraft.
M M
 For an all-moving tail,
; hence,


 t
qSt at  X t
X
S
g at
 t2  t 
Iy
K y S w C L ,trim
M
M
 For a trailing-edge flap type elevator,
with 0    1; hence,
 
M 

M   
 t
X t St
g at


2
K y S w C L ,trim
--- The trailing-edge flap type elevator is usually a poor design at transonic and
supersonic speeds.
48
《Longitudinal Derivatives - A Summary》
CL 
g
2g 
M
L

 aw
,
C




C L ,trim
UCL,trim  L,trim 2 M 
g aw
CD 
2g

,
Du 
 C  M
D

2


U 0CL,trim  D 2 M 
 e AR
g  c  aw
2gc
 C 
Mu  2
  M  , M   2
 static margin
K y U 0C L ,trim   U 
K y C L ,trim
Lu 
2
X
S
g at
M   M   d ,
,
M   t 2  t 
d
K y S w U 0CL,trim
X
S
g at
M     t2  t 
, 0  1
C
K y Sw
L ,trim
We have leave out T in this discussion because there exists no
formula to estimate this derivative. Normally, this derivative is
provided by the engine manufacturer.
49