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AUTOMATION OF CONTROL DURING CLIMBING
During climbing it is necessary to find the optimal trajectory of motion. The change in aircraft
energy depending on the flight parameters and engine power setting must be taken into account. Here we
use such term as power altitude Нp :
Hp 
E
,
mg
where the total mechanical energy of aircraft is
mV 2
 mgH .
2
The first component is kinematical energy, and the second term is potential energy of aircraft.
The rate of change in power altitude (so called power rate of climb Vуp)
E
V yp 
dH p
dt

V dV dH V dV


 Vy ,
g dt
dt
g dt
defines the power properties of aircraft.
From the equation of forces that act on aircraft during climbing:
mV  P cos   X a  G sin  ,
and taking into account that sin   V y V 1 , G = mg we will obtain:
P cos   X a
V 
V  Vy 
V,
g
mg
that is
V yе 
P cos   X a
V.
mg
(6.11)
Since the thrust P and drag Xa forces depend on
the altitude and speed of flight, then the power arte of
Vyp
H=11 km
climb Vyp also depends on these parameters.
H=5 km
Approximated form of this dependence (dependence of
H=11 km
power rate of climb on altitude and M-number) is
H=13 km
shown in fig. 6.8. For supersonic aircraft the value Vyp
H=13 km
H=15 km has two maximums in the range of subsonic and
supersonic speeds.
The power rate of climb is the differential
characteristic of aircraft capabilities. It characterizes
the power of aircraft on each altitude. But in practice
1
M
the integral characteristics of aircraft are important:
Fig. 6.8
time of climb, time to reach the given altitude with
simultaneous getting of cruising speed, distance of
climbing, fuel consumption and others. As a rule, to find the optimal program of climbing we can use the
following functional:
H=0
J  kt  (1  k )Gf ,
(6.12)
where k is weight coefficient (0  k 1); G f is the relative fuel consumption.
The automation problem is to select such program of climbing H = f (M) or M = f (H), and such
program of engine power setting in the form of function Gf from one of parameters t , L, H , M , H p ,
which provide the minimum of functional (6.12).
With k = 1 the time of climbing to power altitude is minimized. With k = 0 the fuel consumption is
minimized. For intermediate value of k we can receive either minimal fuel consumption under required
time of climbing, or minimal time of climbing under the limited fuel consumption. In this case we can
find the required value of k, that minimizes the functional (6.12), by using the relation between
parameters в Gf and  t .
The typical relation between parameters Gf and  t for
supersonic aircraft is shown in fig. 6.9.
But in many cases it is not recommended to minimize only
t or Gf because the other parameter becomes too high. Using
the dependences Gf (t) or t ( Gf ), we can get the minimal
value of one of parameter with limitation on the other. But it is
very convenient to use the integral characteristics.
In particular, the problem of finding the optimal program
of climbing with minimum time comes to the finding the extreme
of functional Jt on time of climbing:
t2


J t   k  1  k Gf dt ,
(6.13)
t1
where Gf is the fuel consumption per second.
Taking into account the relations
dt 
dH p
dt
dH p 
,
dH p
V yp
we will get from (6.13) the functional Jp, which is used to find the optimal program of climbing:
Jp 
H p2

H p1
k  1  k  Gf
dH p .
V yp
(6.14)
From (6.14)
min
dJ p
dH p
and taking into account (6.11):
max
dH p
dJ p
 max
 max
V yp
k  1  k Gf
,
P cos   X a V .
mg k  1  k  Gf 
(6.15)
The expression (6.15) is used in algorithms of finding the optimal program of climbing. The
optimal programs of climbing are calculated as dependences of true airspeed Vtrue or M-number on the
altitude H. Let’s consider the technique of finding of such programs.
First of all, we will consider the simplest case of minimization of climbing time. With k  1 it is
necessary to find the program
M = f (H),
which provides in any point the following condition (under assumption that cos  1)
P  X a  V .
max
mg
In this case for given engine power settings, aircraft weight and drag coefficient we can find the
dependences:
P = f (H, M);
Xa = f (H, M).
Then in the coordinate system ( H, M ) we plot the grid of curves
Hp  H 
V2
= const.
2g
On this grid in the coordinate system ( H, M ) we plot the dependences
V yp 
P  Xa
V = const.
mg
For the supersonic aircraft these curves have the following forms shown in fig. 6.10.
Fig. 6. 10
Points of maximal values of V yp for definite values of Нp (tangency points of graphs V yp to one
of lines of grid Нp) are connected by straight lines in order to have the graphic analogy of optimal
program of climbing. You can see that the program has two sections: subsonic and supersonic with
discontinuity between them. It is connected with additional extreme of power arte of climb dependence
in the supersonic speed range. The points of discontinuity are connected by line with constant energy,
and this section of program is called the acceleration stage.
The range of possible rate of climb is limited (fig. 6.10) by the minimal and maximal available
speed of flight. They are defined by the dynamic pressure on the average altitudes, and by the maximal
available M-number on altitudes greater 11 km. If the sections of calculated trajectory are beyond these
limitations, then they are replaced by sections along the boundaries.
Let’s consider now the case when k  1 . It is necessary to find not only the program М=f(H), but
also the program of power engine settings. Here we must find the parameters H , M , Gf for every value
of power altitude Нp in order to fulfill the following condition
max
P  X a  V
.
mg[k  1  k Gf ]
This problem can be solved as previous one graphically (fig. 6.11).
H
Hp = const
Gf1  Gf2  Gf3
Gf2  Gf opt
with G f2
V ye
with G f1
dJ e
with G f3
 const
M
Рис. 6.11
For this we build the surface
dH p
dJ p

P  X a  V
 const
mg [k  1  k  Gf ]
in the space of parameters H , M , Gf and find the tangency point with cylindrical surface Нp = const.
Then we will repeat this operation for several values of Нp, then connect received points by lines and
finally we will get the optimal program of climbing
M = f (H); Gf = f (H).
Fig. 6.13 shows the typical programs of climbing for subsonic civil aircraft as function Vtrue =
f(H).
H
Vmin
1
2
3
4
(Vmax)M
11 km
(Vmax)q
Vtrue
Fig. 6.13
The optimal program of energy increasing (curve 4 in fig. 6.13) corresponds to minimal time to
reach the required altitude with given (limited) speed. It is realized with greater speeds in comparison
with the program of climbing with minimal time (curve 2). The last program nevertheless is in the range
of higher speeds than the program of climbing to minimal en-route altitude (curve 1). The limitations on
fuel consumption are taken into account by the nominal power engine settings.
The optimal program of climbing with minimal fuel consumption (curve 3) shows us that it is
advantageous to reach the cruising altitude first and only then to accelerate and reach the cruising speed.
This program is base one for transport civil aircraft.