【Aerodynamic phenomena of an A/C】
(A) Forewords:
○ The A/C motions are subjected to the actions of various aerodynamic forces and moments.
Rudder
M
L
L
Elevator
Aileron
D
Y
T
N
Aerodynamic forces and
moments of an A/C
○ Complicated A/C motions result due to the interactions among various mechanisms through
which the aerodynamic forces and moments of the vehicle are varied. The aerodynamic
forces and moments of an A/C are varied through the following categories of mechanisms:
(A) Parasitic aerodynamic interactions:
● This phenomena will involve the aerodynamic forces and moments that are
induced by the A/C motions,
● Two types of these motion-related aerodynamic interactions exist:
○ Change in force (moment) that is the result of a motion occurs along the same
degree of freedom (DOF) of the force (moment) that is being changed. This
phenomenon, generally termed the damping of the motion, can occur on
motions along each and every DOF.
○ Change in forces (moments) that are the results of a motion occurs along a
degree of freedom (DOF) which is different those of the forces (moments) that
are being changed. This phenomenon, generally termed the couplings among
motions, may occur on motions along some, but not necessary all, DOF.
--- A motion can occur with no coupling to motions along other DOF
--- It is also possible that a motion along one DOF and generate coupling
forces and moments alone more than one of the other DOF.
(B) Pilot initiated aerodynamic interactions:
● This phenomena will involve the aerodynamic forces and moments that are
induced by the deflections of the control surfaces or the engine throttle.
○ Detailed definitions and quantitative expressions of these aerodynamic phenomena follow.
(B) Aerodynamic damping of the A/C motions:
● Damping of forward motion (or speed damping):
○ The phenomenon involves the change in drag, denoted as D _ u , due to change in
flight speed, denoted as U .
○ For a || U || 1 , we will have D _ u  Du  U where Du is the proportional
constant of the linear relationship. In general, Du  D _ u / U .
--- With || U || 0 , Du  D / u . Therefore, we often use Du  D / u .
--- The derivative, D / u , which represents the strength of the phenomenon is
normally termed the speed damping coefficient.
● Damping of vertical motion (or vertical damping):
○ The phenomenon involves the change in lift, denoted as L _ w , due to change in
vertical velocity, denoted as w .
○ For a || w || 1 , L _ w  Lw  w with Lw  L _ u / w , or Lw  L / w .
○ The derivative, L / w , which is essentially a proportional constant, represents the
strength of the phenomenon and is termed the vertical damping coefficient.
● Damping of the sideslip motion:
○ The phenomenon involves the change in side force, denoted as Y _ v , due to
change in sideslip velocity, denoted as v .
○ For a || v || 1 , Y _ v  Yv  v with Yv  Y _ v / v , or Yv  Y / v .
○ The derivative, Y / v , which is also a proportional constant, represents the strength
of the phenomenon and is termed the sideslip damping coefficient.
● Damping of the pitching motion:
○ The phenomenon involves the change in pitching moment, denoted as M _ q , due
to change in pitching velocity, denoted as  q . Still, it is assumed that || q || 1 ,
and therefore M _ q  Mq  q with Mq  M _ q / q , or Mq  M / q ,
○ The derivative, M / q , which is still a proportional constant, represents the
strength of the phenomenon and is termed the pitching damping coefficient.
● Damping of the rolling motion:
○ The phenomenon involves the change in rolling moment, denoted as L _ p , due to
change in rolling velocity, denoted as p . Again, it is assumed that || p || 1 , and
therefore L _ p  L p  p with L p  L _ p / p , or L p  L / p ,
○ The derivative, L / p , which is still a proportional constant, represents the strength
of the phenomenon and is termed the rolling damping coefficient.
● Damping of the yawing motion:
○ The phenomenon involves the change in yawing moment, denoted as N _ r , due to
change in yawing velocity, denoted as r . Still, it is assumed that || r || 1 , and
therefore N _ r  Nr  r with Nr  N _ r / r , or Nr  N / r ,
○ The derivative, N / r , which is still a proportional constant, represents the strength
of the phenomenon and is termed the yawing damping coefficient.
(C) Aerodynamic couplings among the A/C motions:
● Couplings generated by the change in forward speed, u :
○ Forward motion to vertical motion coupling (or flight speed to lift coupling):
--- The phenomenon involves the change in lift, denote as L _ u , due u .
--- Again, L _ u  Lu  u with Lu  L / u or Lu  L / u .
--- The derivative, L / u , a proportional constant representing the strength of the
phenomenon, is termed the speed to lift coupling coefficient.
○ Forward motion to pitching motion coupling (or Flight speed to pitch coupling):
--- The phenomenon involves the change in pitching moment, M _ u , due u .
Again, M _ u  Mu  u with Mu  M  / u or Mu  M / u .
--- The derivative, M / Mu is termed the speed to pitch coupling coefficient.
● Couplings generated by the change in vertical speed, w :
○ Vertical motion to forward motion coupling (or Vertical speed to drag coupling):
--- The phenomenon involves the change in drag, D _ w , due w . Still, we
assume that D _ w  Dw  w with Dw  D _ w / w or Dw  D / w . The
derivative, D / w , is termed the vertical speed to drag coupling coefficient.
○ Vertical motion to pitching motion coupling (or Vertical speed to pitch coupling):
--- The phenomenon involves the change in pitching moment, M _ w , due to
w . Still, M _ w  Mw  w with Mw  M _ w / w or Mw  M / w . The
derivative, M / w is termed the vertical speed to pitch coupling coefficient.
○ Vertical acceleration to pitch coupling (or AOA derivative to pitch coupling)::
 , will lead to a change in
--- It is found that changes in vertical acceleration, w
 is assumed with
pitching moment, denote as M  . Still, a M   MZ  w
 or MZ  M / w , being the proportional constant.
MZ  M  / w
  (1/ U )  M /  . The derivative,
--- Moreover w   U ; hence, M Z  M / w
M /  is then termed the AOA derivative to pitch coupling coefficient.
● Couplings generated by the change in sideslip velocity, v :
○ Sideslip to yaw coupling:
--- The phenomenon involves the change in yawing moment, N _ v , due to v .
Again, N _ v  Nv  v is assumed with Nv  N _ v / v or Nv  N / v .
--- The derivative, N / v , is termed the sideslip to yaw coupling coefficient.
○ Sideslip to roll coupling:
--- The phenomenon involves the change in rolling moment, L _ v , due to v .
Again, a L _ v  Lv  v is assumed with Lv  L _ v / v or Lv  L / v .
--- The derivative, L / v , is termed the sideslip to roll coupling coefficient.
● Roll to yaw coupling , couplings generated by the change in rolling velocity, p :
○ The phenomenon involves the change in yawing moment, denote as N _ p , due to
p . Still assuming N _ p  Np  p with Np  N _ p / p or Np  N / p .
○ The derivative, N / p , is termed the roll to yaw coupling coefficient.
● Yaw to roll coupling , couplings generated by the change in yawing velocity, r :
○ The phenomenon involves the change in rolling moment, denote as L _ r , due to
r . Again, a L _ r  Lr  r is assumed with Lr  L _ r / r or Lr  L / r .
○ The derivative, L / r , is termed the yaw to roll coupling coefficient.
(D) Effectiveness of the control surfaces:
● The elevator, the aileron, and the rudder constitute the 3 primary control surfaces of an A/C
● There are primary functions of these control surfaces, as well as their secondary effects.
● Among the primary control functions of the control surfaces, a change in the elevator
deflection, denoted as  e , will generate a change in the pitching moment, denoted here
as M _  e . Similarly, a change in the aileron deflection, denoted as  a , will generate a
change in the rolling moment, L _  a , and a change in the rudder deflection, denoted as
 a , will generate a change in the yawing moment, N _  r ..
○ For a ||  e || 1 , M _  e  M   e with M  M _  e /  e or M  M /  e .
The derivative, M /  e , which represents the strength of the elevator in changing
the pitching moment of an A/C, is termed the elevator effectiveness.
○ For a ||  a || 1 , L _  a  L   a with L  L _  a /  a or L  L /  a and
L /  a is termed the aileron effectiveness in changing the rolling moment.
○ For ||  r || 1, N _  r  N   r with N  N _  r /  r or N  N /  r and
N /  r termed the rudder effectiveness, in changing the yawing moment
● As secondary effects, a  a will also generate a change in the yawing moment, N _  a ,
and a  a will also generate a change in the rolling moment, L* _  r .
○ For a ||  a || 1 , N _  a  N   a with N  N _  a /  a or N  N /  a ,
The derivative, N /  a , is termed the aileron to yaw coupling coefficient.
○ For a ||  r || 1, L _  r  L   r with L  L _  r /  r or L  L /  r , The
derivative, L /  r , is termed the rudder to roll coupling coefficient.
● For most A/C, || L /  r |||| N /  r || and || N /  a |||| L /  a || .
(E) Additional terms of the aerodynamic phenomena:
● The effect of the stabilizer on the pitching moment Two mechanisms exists.
(A) Through the selection of its installation angle, denoted in this note as it .
○ By selecting a proper value of it , a constant pitching moment is produced in
order to balances the A/C for a designated trimmed (static) flight.
○ However, it can not be changed once the A/C is built. As a result, the stabilizer
is normally regarded as a secondary control surface.
(B) Through the change in tail relative wind in a pitch motion.
○ This aerodynamic effect will presents a damping to the pitching motion.
※ The effectiveness of the stabilizer on these functions is represented by another
derivative, M /  t , which measures the change in pitching moment due to changes
in  t , the stabilizer angle-of-attack which includes it
● Change in thrust, thereby altering the balance of force in the x-axis. Two sources exist.
(A) Change in thrust due to change in flight speed, an additional speed damping effect:
○ The formula for this aerodynamic effect is T _ u  Tu  u , Tu  T / u
○ The derivative, T / u , is termed the thrust to speed derivative.
(B) Change in thrust due to change in engine throttle, an additional control to the A/C:
○ The formula for this aerodynamic effect is T _   T  u , T  T /  T
○ The derivative, T / T , is termed the throttle control effectiveness.
(F) Overall aerodynamic effects on the A/C motions:
● In a A/C maneuver, the velocities and the angular velocities of the vehicle will vary,
thereby producing nonzero values of u , v , w , p ,  q , and r , all at the same
time. As a result, those aerodynamic phenomena of individual motions will add up and
produce the following overall aerodynamic effects on the A/C.
○ Overall aerodynamic forces, denoted as X aero , along the x-axis:
X aero  (Tu  Tu )  u  Dw  w  T  T
○ Overall aerodynamic forces, denoted as Yaero , along the y-axis:
Yaero  Yv  v
○ Overall aerodynamic forces, denoted as Z aero , along the z-axis:
Z aero  Lu  u  Lw  w
○ Overall aerodynamic moments, denoted as Laero , along the x-axis:
Laero  Lp  p  Lr  r  Lv  v  L   a  L   r p
○ Overall aerodynamic moments, denoted as M aero , along the y-axis:
M aero  Mu  u  Mw  w  Mw  w  Mq  q  M   e
○ Overall aerodynamic moments, denoted as N aero , along the z-axis:
N aero  Nv  v  Nr  r  Np  p  N   r  N   a
● Remarks:
○ These aerodynamic effects represent the external forces and moment of an A/C. A
complete equations of motions (EOM) of an A/C can be formed by combing these
aerodynamic terms with the gravitational terms and inertial terms of the vehicle.
○ The parameters of these equations, which are summarized as follows, will be
collectively termed as the coefficients of aerodynamic phenomena (CAP),
Du  D / u , Lu  L / u , Mu  M / u . Tu  T / u :

Lw  L / w , Dw  D / w ,
Mw  M / w ., MZ  M / w
M q  M / q , M  M /  e , Yv  Y / v , Lv  L / v .
Nv  N / v , L p  L / p , N p  N / p ,
Nr  N / r ,
Lr  L / r .
L  L /  a , N  N /  a ,
N  N /  r , Lr  L /  r , T  T /  T