Page 46 - Mechatronics with Experiments

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propulsion force for the airplane is provided by jet engines (or propeller engines in smaller
planes). There are four major groups of forces acting on an airplane (Figure 1.26):
1. Gravity force (F ).
g
2. Thrust force generated by the engines (F ).
t
3. Lift force generated by the movable wings (flight control surfaces) (F ).
l
4. Drag force due to friction between the air and aircraft body (F ).
d
When the thrust force is in the horizontal direction, the vertical acceleration of the
airplane is determined by the difference between lift force and gravitational force, whereas
the horizontal acceleration is determined by the difference between the thrust force and
drag force. Gravitational force is always in the z-direction of the global coordinate system.
Thrust force direction is a function of the attitude (pitch angle) of the airplane. The lift
and drag forces are aerodynamic forces and are strongly dependent on the relative speed
between the airplane body and air flow, as well as the aerodynamic shape of the airplane,
that is the configuration of the movable wing sections.
There are six coordinate variables to be controlled in an airplane motion: x , y , z G
G
G
position coordinates of the center of mass of the airplane relative to a reference on earth,
the orientation orientation , , (yaw, roll, pitch) of the airplane relative to a reference
frame.
By convention, airplane motion can be described in two groups:

1. longitudinal motion that includes the motion along axial direction x and vertical
direction z as well as pitch rotation motion about the y axis,
2. lateral motion that includes the rotational motions of roll and yaw.

The longitudinal and lateral motion variables are lightly coupled to each other, that is,
motions in the lateral plane result in small motions in the longitudinal plane, and vice versa.
The position coordinates are obtained by the integral of the speed vector of the airplane
along its flight path. The linear speed vector of the airplane is determined by the integral of
the acceleration vector which is determined by the net force vector acting on the center of
mass of the airplane.
For simplicity, if we assume the thrust force and the drag force are in the x-direction,
and lift and gravitational forces are in the z-direction, the dynamic relationship for motion
of the airplane in the x and z directions can be expressed as (Figure 1.26),
m ⋅ ̈ x (t) = F (t) − F (t) (1.13)
G
d
t
m ⋅ ̈ z (t) = F (t) − F (t) (1.14)
g
G
l
where x (t), y (t), z (t) are the coordinates of the center of mass of the airplane with
G G G
respect to a fixed reference frame, F , F , F , F are thrust, drag, lift and gravity forces. For
l
g
d
t
simplicity we show them as if they act as center of mass of the airplane as single vectors.
In reality, they are distributed over the whole plane and depend on the current position of
the control surface. The motion in y-direction is negligable since the accelerations in the
y-direction are due to the small aerodynamic drag forces resulting from the relative speed
of the airplane and air in the y-direction.
m ⋅ ̈ y (t) =−F (t) (1.15)
G dy
where F (t) is the drag force in the y-direction. However, there are finite drag forces in the
dy
y-direction, and hence the resulting motion in y-direction. The airplane path and orientation
is modified to correct for the unintended motion in the y-direction. Reference [4] gives a