C.3 Lateral Aerodynamic Derivatives
C.3.1 Aerodynamics of the coupling derivatives
○We will give methods to estimate the following coupling derivatives
------
(1) : Side slip damping
○Side force comes mostly from fuselage and vertical tail (VT).
· Side force from the VT:
---: VT lift curve slope
· Therefore,
--- of the VT is always low, so estimate carefully.
--- Side wash will be strong; need actual wind tunnel data..
(2) : Rolling moment derivative
○ is mostly the result of the redistribution of lift due to dihedral angle G.
l Change in local AOA due to side slip:
l The resulting rolling moment:
.
l For a rectangular wing with constant airfoil
;
then,
l But
and we often express . As a result,
○ For general wing shapes with taper ratio l, the sweep angle L and at high Mach number:
.
○ Or, when partial span flaps are down, add the increment
--- The subscript F indicates the corresponding parameters for the portion of the span where the flaps are down.
○The fuselage may also contributes:
--- < 0 for high wing and > 0 for low wing (: width of the fuselage.)
○Vertical tail also may contribute to , left as an exercise!
(3) : Yawing moment derivative
○ comes mostly from VT though fuselage & propeller may reduce it.
○ For the contribution from VT:
a) Yawing moment from VT due to sideslip:
--- : Distance between VT a.c. and CG
b) Therefore, .
--- We have let
○For the effects from the fuselage:
a) Yawing moment from the fuselage due to sideslip:
--- : Parameter that depends on the fineness ratio
(). Typically, : 0.8 - 0.9.
--- : volume of the fuselage
b) Then, .
○ The total will be .
(4) : Yaw to roll coupling
○ is mostly due to different velocity along wing as a result of yawing.
lRolling moment due to the wing area :
l; hence,
lBy neglecting the high order term, we thus have
lFor a rectangular wing with constant airfoil: .
Then, , ---
lTo include the effect of taper ratio l, sweep angle L and at high Mach No:
.
(5) : Roll damping
○ is mostly due to wing though the vertical tail may also contributes.
○We shall estimate the wing contribution:
l Local A.O.A change due to rolling:
l Differential roll moment element due to :
.
l For the entire wing:
--- For a rectangular wing with constant airfoil:
Then,
--- And for a tapered wing:
Notes: (a) Sweep should have no effect on this,
(b) Mach no. effect should show up in
(6) : Yaw damping
○Yaw damping is mostly due to vertical tail
lEffective vertical tail angle-of-attack due to yaw:
.
lSide force from the vertical tail:
lYawing moment due to :
l Then,
(7) : Roll to yaw coupling
○ Contributions of come from two sources: wing and the vertical tail.
○ Wing contribution is mostly due to rotation of the lift vector.
l For the shaded areas in the above figure, because , we will have
.
lHowever, and are each rotated by an angle , and the following yawing moment results:
lOverall yawing moment due to rolling:
--- For rectangular wings with constant airfoil:
Therefore,
.
--- And for a wing with taper ratio l:
.
○ Contribution from the vertical tail can be derived the same way.
C.3.2 Summarize and comments of the section
○ The following lists partial results of the lateral directional derivatives:
;
;
;
○ It is seen that
● All lateral derivatives reduce in magnitude as the air speed is increased, indicating a reduction in coupling effect at higher speed.
● and have no dimension in length. As a result, these aerodynamic coupling are independent of the aircraft size.
● On the contrary, larger values of and , which implies stronger sideslip-induced roll and yaw motions, will result for smaller A/C.
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