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for more information on path performance problems.
2.2 Equations of Motion for Flight in Vertical Plane
Let us consider an airplane whose fiight path is contained in a vertical plane as
show in Fig. 2.2. Let V be the flight velocity that is directed along the tangent to
the flight path: Let the tangent to the flight path'at a given instant of time make an
angle y with respect to the local horizontal. The angle y is usually called the flight
path angle. Let 0 be the inclination of the airplane reference line or the zero-lift
line with respect to the local horizontal so thatPthe angle of attack a - 0 - y. Let
R be the radius of curvature of the fiight path in the vertical plane.
The external forces acting on the airplane are lift L acting normal to the flight
velocity, the drag D parallel to the flightYpath and in opposite direction to the flight
velocity V, the thrust T in the forward direction mak~:g an angle e with respect to
the flight path, and the weight W in the direction of gravity directed towards the
center of the Earth.
Resolving the forces along and normal to the flight path, we have
W (1V
T cos e - D - W sin y = -7. (2.5)
g dt
W I/2
T sin e +L - W cos y - -. (2.6)
gR
Equations (2.5) and (2.6) describe the accelerated motion of the airplane in a
vertical plane. Generally, the thrust inclination e is ver}t small. Therefore, it is
tal
Fig. 2.2 Forces achng on an airplane in fiightin a verticalplane.
AIRCRAFT PERFORMANCE
71
usual in performance analyses to assume that the thrust is aligned with the flight
path (e = 0).
For the special case of an airplane whose flight path is a straight line in the verti-
cal plane and whose fiight'velocit)r is constant, acceleration terms on the right-hand
sides of Eqs. (2.5) and (2.6) vanish. The performance of an airplane based on this
assumption is called static performance. Examples of static performance are steady
level fiight, steady climb, range, and endurance in constant- velocity cruise. Per-
formance problems that involve acceleration terms are accelerated climbs, takeoff,
and landing.
Assuming that the thrust vector is aligned with the fiight path, equations for
static performance are given by
T-D- Wsiny -0
L- Wcosy -O
(2.7)
(2.8)
In addition to these equations of motion, we also need the following kinematic
relations to calculate distances with respect to the ground:
x -. V cos y
h = V siny
(2.9)
(2.10)
where x and h are the horizontal and 'vertical distances measured with respect to the
origin of a suitable coordinate system fixed on the ground. The "." over a symbol
denotes differentiation with respect to time. :fhus, x -. dx/dt is the horizontal
velocity with respect to the ground, and h : dhldt is the rate of increase of
altitude or the rate of climb. In calm air, the velocity with respect to the ground
and that with respect to air are equal. In the presence of the wind, x = V + VLV,
where vw is the wind velocity and "-" applies to the headwind and "+" for the
tailwind.
Let us introduce some nondimensional parameters so that we can express the
above equations in a more compact form. Let
E = g-
L
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