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dt
dx dh = v
dh dt
dx V
dh = Vy
1
=~
y
::= -E
(2.44)
(2.45)
76 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Let R = x f _ xi denote the range, which is equal to the horizontal distance covered
with respect to the ground. Then,
R = -l,( Edh (2.46)
where hi and hf are the initial and ffnal altitudes, respectively. Assuming that the
angle of attack ct is held constant so that E is constant during the glide, we have
R - EAh (2.47)
where Ah = h/.- hf is the height lost during the glide. From Eq. (2.47), we observe
that, for a given height difference, the range is maximum when E - Em, wluch
is also the condition for the fiattest glide. In other words, maximum range occurs
when the glide angle is minimum. Using Eq. (2.20), we get
Rmax = 2~~DO (2.48)
Here, we have ignored the effect of wind on the range. Actually, the range depends
on wind conditions. A headwind reduces the range, whereas a tailwind increases it.
In sailplane terminology, glide ratio is the ratio between the ground distance
traversed and the heightlost. The glide ratio is also equal to E. A high-performance
sailplane with a glide ratio of40 can cover 4 km with respect to the ground for every
100 m of height lost.
Let the rate of sink, the speed with which the glideris heading towards the Earth,
be denoted by hs. Usually, h denotes the rate of climb. Therefore, hs - -h. With
this, the rate of sink is given by
hs = -Vy
DV
-W
(2.49)
The term D V in Eq. (2.49) represents the power required PR to sustain the gliding
flight. Thus, we observe that the rate of sink is minimum when the power required
per-unit-weight is minimum, which also corresponds to the case when the param-
eter (CD] C~./2) is a minimum or (C2t2]CD) is a maximrun It can be shown that
this happens when
CDr " 3CDO (2.50)
CL :
(2.51)
AIRCRAFT PERFORMANCE
Let the value oflift coefficient when (C2/2] CD) iS maximum be denoted by CL,m.
Then,
CL,m =
=nCz
With tlus, it can be shown that
-.,), ax = -,
(2.52)
(2-53)
We observe that for glide with minimum sink rate, the induced :is three times
the zero-lift drag. The schematic variations of E and cC;.n~/)gwith angle of
attack are shown in Fig. 2.5.
The velocity Vm for glide with mirumum sink rateis given by
Vm -.
~ 0.76 'VR
(2.54)
From this relation, we observe that the velocity for the glide with minimum sink
rateis about 0*76 VR or 0.76 times the velocity for flattest glide.Whereas the flattest
glide occurs when the drag is minimum, the glide with minimum sink rate occurs
when the power required is a minimum. rfhese velocities are shown in Fig. 2.6.
CL
Jfig. 2.5 Variation of aerodynanuc parameters with lift coefficienL
78 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
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