Page 47 - Mechatronics with Experiments

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JWST499-Cetinkunt
JWST499-c01
INTRODUCTION 33
more accurate dynamic relationship between aerodynamic forces generated by the control
surfaces, thrust force, and aircraft motion.
Orientation speeds are similarly obtained by integral of the angular accelerations
which are determined by the net moments about the respective axes by all of the above
referenced forces.
d
( ⃗ H (t)) = ⃗ T (t) (1.16)
G
G
dt
⃗
⃗
I 3x3 ⋅ (t) + (t) × ⃗ H (t) = ⃗ T (t) (1.17)
G
G
where
⃗ H (t) = H (t) ⋅⃗ i + H (t) ⋅ ⃗ j + H (t) ⋅ k ⃗ (1.18)
G Gx Gy Gz
= I ⋅ ⃗ w(t) (1.19)
3x3
⃗ T (t) = T (t) ⋅⃗ i + T (t) ⋅ ⃗ j + T (t) ⋅ k ⃗ (1.20)
G Gx Gy Gz
⃗ w(t) = w (t) ⋅⃗ i + w (t) ⋅ ⃗ j + w (t) ⋅ k ⃗ (1.21)
x y z
dw (t) dw (t) dw (t)
y
x
z
⃗ (t) = ⋅⃗ i + ⋅ ⃗ j + ⋅ k ⃗ (1.22)
dt dt dt
⃗
= (t) ⋅⃗ i + (t) ⋅ ⃗ j + (t) ⋅ k (1.23)
k
x
y
⎡ I | −I xy −I xz ⎤
|
xx |
|
I 3x3 = ⎢ −I yx | | I yy | | −I yz ⎥ (1.24)
⎢ | | ⎥
⎣ −I | −I zy | I zz ⎦
zx
H (t) = I xx ⋅ w (t) − I xy ⋅ w (t) − I ⋅ w (t) (1.25)
y
z
xz
x
Gx
H (t) =−I yx ⋅ w (t) + I yy ⋅ w (t) − I ⋅ w (t) (1.26)
Gy
y
x
z
yz
H (t) =−I ⋅ w (t) − I ⋅ w (t) + I ⋅ w (t) (1.27)
z
y
Gz
zz
zy
zx
x
where ⃗ H (t) is the angular momentum vector of the airplane with respect to a coordinate
G
frame fixed to the center of mass of the airplane.
⃗ T (t) is the net moment vector about the axes of the coordinate frame due to all
G
the forces acting on the airplane (gravity, thrust, lift (forces generated by flight control
surfaces), drag forces).
I 3x3 is the moment of inertia matrix which is symmetric (I xy = I , I xz = I , I yz = I ),
zy
yx
zx
and ⃗ w(t) is the angular velocity vector, ⃗ (t) is the angular acceleration vector, of the airplane,
with respect to the same coordinate frame. The coordinate frame xyz is fixed to the airplane
center of mass and moves with it. Hence if the weight distribution of the airplane does
not change, the I 3x3 matrix is constant. However, due to consumed fuel and movement of
passengers during the flight, the inertia matrix (I 3x3 ) changes slowly. In addition, the center
of mass coordinates of the airplane also change during flight due to the same reasons. Fuel
stored in the wings and other parts of the airplane body is pumped to the engines in such a
way that the change in the center of mass location is smooth and slow as a function of time.
The movable surfaces on the two wings and on the tail of the airplane are used to
effect the aerodynamic forces, hence the lift and the orientation of the airplane (Figures 1.25
and 1.26). Aerodynamic forces (lift and drag forces) acting on the airplane are functions of
1. relative speed between the airplane and the air (hence the speed of the airplane, as
well as the air speed (direction and magnitude) and turbulance conditions),
2. shape of the airplane, where some of the shapes are adjustable during flight, such as
the control surfaces on the wings and the tail.