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Thus, the climb angle y is maximum when u-l, which we know is also the
condition for minimum drag in level fiight. Also, when u -.1, we have V :'VR,
CL = CZ = J~Tk-, CDO - CD/, and CD =2Jk~o.
98                 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig. 2.16    Determination of bme to climb.
Substituting u = 1 in Eq. (2.116), we get
                             Ymax =S,-.'( E  )
The rate of climb in a nondimensional form is given by
                                    u sin y = 2~. [2z. - (l3 +')]
For the rate of climb to be a maximum,
                      d(usi yrisin ) = dd (2ztl-u3_ 1) =0
            du            u
or
                                                3u4 _ 2ZU2 -1 -. O
u=
Then, the maximum rate of climb in nondimensional form is given by
(2.119)
(2.120)
(2.121)
(2.122)
(2.123)
(u sin Y)max = 2E~[2z.,  - (.1+:  )]                  (2.124)
i
3
 l
?
P
;7
v
AIRCRAFT PERFORMANCE
and the dimensional maximum rate of climb is given by
99
(RlC)nw; = ( V sin Y)max - (ri sh y)max VR                      (2.125)
As in the case of propeller aircraft, the time to climb from a given initial altitude
hi to the desired final altitude hf is grven by
t= f,,"nh
(2.126)
For minimum time to climb, the aircraft has to follow the path along which the
specific excess power is maximum at each altitude. For a propeller airr)lane, this
has to be done graphically or numerically whereas, for a jet airplane whose thrust
is independent of the flight velocity, it is possible to find an analytical solution.
We will discuss this problem in more detail when we consider e;e:gy cl:imb later
in tlus section.
2.5.2  Absoluteand Service Ce17ings
    Previously, we defined the absolute ceiling as that altitude where the two level
 flight solutions merge into one. Based on the rate of climb, we can have an alternate
definition of the absolute ceiling.
     The (R/C)max decreases with altitude as shown in Fig. 2.17 because of the drop
in thrust available or power available as the altitude increases, The altitude where
the maximum rate of climb drops to 100 ft/min (30.5 m/nun) is called service
 ceiling. Extrapolating further to (R] C)max = O, we get the absolute ceiling. For jet
 aircraft whose thrustisindependent offlightvelocity, z = 1 when the (R/C)max  = 0.
Thus, both definitions of absolute ceiling are essentially identical.
h
E
100 ftlmin
( R/C)max
Fig.2.17   Service and absolute ceilings.
 1
l;
 i/
[.::
I{i
 .:
u
x
rl
fi
E
where um.is the nondimensional flight velocity at which the rate of climb is max-
imum. The value of Um iS obtained by using the positive sign before the term
 ,ff/7~ in Eq. (2.123). The convsponding dimensional speed is equal to Um VR,
100             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
    For propeller airplanes, the service ceiling is in the range of 4-6 km, whereas
for commercialjet airplanes it is  11-17 km.
      The steady or constant-velocity climb discussed above is based on the assump-
tion that accelerations along and normal to the flight path are both equal to zero.
In this section, we will study a more general casebof climbing flight that involves
nonzero accelerations. The equations ofmotion for this class offlight are given by
Eqs. (2.5) and (2.6). With :tu0 and y = V/R, we have
           WdV
　
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