General solutions of optimum problems in nonstationary flight

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General solutions of optimum problems in nonstationary flight
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NACA TM
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25 p. : ill. ; 28 cm.
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English
Creator:
Miele, Angelo
Langley Aeronautical Laboratory
United States -- National Advisory Committee for Aeronautics
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NACA
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Washington, D.C
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Aerodynamics -- Research   ( lcsh )
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technical report   ( marcgt )
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Abstract:
Abstract: A general method concerning optimum problems in nonstationary flight is developed and discussed. Various conditions of flight in a vertical plane (climb with minimum time, climb with minimum fuel consumption, steepest climb, descending and gliding flight with maximum time of space) are studied; the corresponding best techniques of flight, that is, the optimum speed-height relationships, are determined.
Bibliography:
Includes bibliographic references (p. 22).
Statement of Responsibility:
by Angelo Miele.
General Note:
"Translation of "Soluzioni Generali di Problemi di Oppimo in Volo Non-Stazionario." from L"Aerotecnica, n. 3, vol. XXXII, 1952."
General Note:
"Report date October 1955."

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University of Florida
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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1588


GENERAL SOLUTIONS OF OPTIMUM PROBLEMS IN

NONSTATIONARY FLIGHT*1

By Angelo Miele


SUMMARY


A general method concerning optimum problems in nonstationary
flight is developed and discussed.

Best flight techniques are determined for the following conditions:
climb with minimum time, climb with minimum fuel consumption, steepest
climb, descending and gliding flight with maximum time or with maximum
distance.

Optimum distributions of speed with altitude are derived assuming
constant airplane weight and neglecting curvatures and squares of path
inclination in the projection of the equation of motion on the normal
to the flight path.

The results of this paper differ from the well-known results
obtained by neglecting accelerations with one exception, namely, the
case of gliding with maximum range.

The paper is concluded with criticisms and remarks concerning the
physical nature of the solutions and their usefulness for practical
applications.


SYMBOLS


C weight of fuel consumed

Fa = Vz/Vzu sin 9/sin 9 acceleration factor: ratio of effective
rate of climb to the one computed neg-
lecting accelerations

*"Soluzioni Generali di Problemi di Ottimo in Volo Non-Stazionario."
L'Aerotecnica, n. 5, vol. XXXII, 1952, pp. 155-142.
LThe author wishes to thank Professor Placido Cicala, of the
Polytechnic of Turin and of the University of Cordoba, for his helpful
suggestions.






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acceleration of gravity

lift

weight of fuel consumed per unit time

airplane weight

aerodynamic drag

distance flown


sO horizontal projection of the distance flown


time


thrust

speed


Vz = V sin a

Vzu = V sin 6u


effective rate of climb

rate of climb computed from the equations of uniform
flight


altitude


Z, altitude of tropopause

6 effective path inclination (positive for climbing flight)

8u path inclination computed from the equation of uniform
flight (positive for climbing flight)


1. INTRODUCTION


In the past airplane performances have usually been determined by
neglecting accelerations, which results in great simplifications in the
equations of motion.

The small curvatures associated with the usual paths of an aircraft
in a vertical plane justify the assumption of zero centrifugal forces.
On the other hand, the inertia tangential forces (often logically dis-
regarded in the study of performances of many conventional aircraft)
must be taken into account in the analysis of high-performance jet
airplanes, because of the large values of both the velocity and its
variation with the altitude. *






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For instance, it is well known that the speed for best climb of any
type of aircraft increases with altitude incompressiblee flow). This
means that the total energy developed by the power plant is not only
used to work against the aerodynamic drag and the earth's gravitational
field, but also to increase the kinetic energy of the aircraft.

When the terms due to acceleration are neglected, then the esti-
mated climbing performances are too optimistic; the values of time and
fuel consumption calculated in this way are less than the actual ones.

On the other hand it appears that compressibility effects can some-
times decrease the speed for best climb as altitude increases, especially
in the stratosphere and for aircraft with high wing loading; in this case
the rate of climb computed from the equation of uniform flight is less
than the actual one.

The above-mentioned reasons emphasize the need for an analysis of
optimum flight conditions based on the consideration of the nonuniform
character of the motion.

The first attack to the problem was performed by F. C. Phillips
(ref. i), who proposed a kinetic energy correction to the results pre-
dicted with the equations of a uniform flight. The calculation of such
a correction was carried out by assuming a distribution of speed with
altitude identical with the one which maximizes the rate of climb com-
puted without accelerations. (See ref. 4, also.)

Other studies, due to Otten (ref. 2) and Hayes (ref. 3), extended
Phillips' results by taking into consideration the true path inclination
and the effects of compressibility.

But only in recent years the problem has been considered and
analyzed in its entirety. In particular, the author (ref. 5) has solved
the problem of climb with minimum time by using a transformation based
on Green's theorem, while Lush (ref. 6), following a study due to
Kaiser (ref. 7), has treated the same problem by an elegant graphic-
analytic method based on the concept of energy height (Ze = Z + V2ftg).

The present paper generalizes a method described in a preceding
study (ref. 5) and extends its results to the following types of flight:
climb with minimum time; climb with minimum fuel consumption; steepest
climb; descending or gliding flight with maximum time or with maximum
horizontal distance.

For each case the optimum technique of flight is determined; that
is to say the function V = V(Z) which, from given initial conditions
(VI, Z1) to fixed final conditions (V2, 22), will maximize or minimize
the time, the fuel consumption or the distance.






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Results are discussed and criticized. Their limitations are pointed
out; their field of applicability is indicated for the different types
of today's engine groups (reciprocating engine, air-breathing jet engines,
rockets).


2. BASIC HYPOTHESES


The following hypotheses are basic for all the work:

(a) Airplane weight is assumed constant.

(b) Curvatures and squares of path inclination are assumed negli-
gible with regard to their effects on that part of the drag depending on
the angle of attack. '*

(c) Power plant is of an unspecified type; but its thrust and rate
of fuel consumption are assumed to be functions of the following nature:


T = T(V, Z) (1)


q = q(V, Z) (2)


(d) Angle between the vectors T, V is not taken into consideration.

(e) Only flight paths restricted to a vertical plane are considered.

(f) The aerodynamic lag is disregarded; the air forces are calculated
as in steady flight.


5. FUNDAMENTAL EQUATIONS


The following scalar expressions can be derived projecting the
fundamental equation of the motion on the tangent and on the normal to
the flight path:


T R Q 1 + sin 8 = 0 (3)

L2 de
P Q os + -- sin = 0 (4)
g dZ






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According to the preceding hypothesis (b), equation (4) can be
substituted by


P- Q= O (5)


This approximation is important. As a matter of fact, the rate of
climb given by


Vz = V sin = (T R)V/Q (6)
V dV
1 + _
g dZ


becomes a function of V, Z, and dV/dZ only, as can be seen from
equations (1), (5), and (6), the expressions for the aerodynamic forces
and the polar.

Equation (6) shows that the effective rate of climb and the sine of
the effective path inclination can be expressed as the product of the
corresponding values obtained for uniform flight


Vzu = (T R)V/ (7)


sin de = (T R)/Q (8)


by the correction factor


Fa = 1 (9)
1 dV2
1 +-----
S2g dZ


which expresses the effects associated with the nonuniform character of
the motion.

It should be noted that the preceding equations are general; there-
fore, they contain those corresponding to gliding flight as a special
case (T = 0).






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4. CLIMBING FLIGHT


The following cases of flight are discussed:

(a) Climb with minimum time.

(b) Climb with minimum fuel consumption.

(c) Steepest climb.

It is assumed that the thrust at all times is greater than the
aerodynamic drac.


4-a. Climb With Minimum Time

The time necessary to fly from given initial conditions (VI, Z1)

to fixed final conditions (V2, Z2) is



t = dZ 2 ( dV + dZ) (10)
i Vz J


where



4 = = -(V, Z) (11)
g(T R)


'= V= (V, ) (12)



Here it is desired to aet. rmine the best flight technique; that is
to say, the particular speed-htight relationship V = V(Z) which
minimizes integral (10).

The investigation is simplified if the properties of the function





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W(V, z) 6 6D (13)



are used instead of the application of variational methods-.

The curve wu = 0 divides that zone of the (V, Z) plane which is of
practical interest for flight operations (fig. 1) into two regions:
A, where w < 0; B where w > 0.

Four cases of flight (i.e. four types of boundary conditions) are
possible according to the relative positions of points 1 and 2 with
respect to the curve u = 0:

Case I: Point 1 in zone A; point 2 in zone B.

Case II: Point 1 in zone A; point 2 in zone A.

Case III: Point 1 in zone B; point 2 in zone A.

Case IV: Point 1 in zone B; point 2 in zone B.

Here, only the first case is analyzed with the restriction that
Zl< z < Z2,

The optimum flight technique is the following:

(1) Acceleration at constant altitude Z1 from V1 to the speed
(VM) defined by z(V, Zl) = 0.

(2) Climb from Z1 to Z2 using the distribution of velocities
defined by w(V, Z) = 0.

(3) Acceleration at constant altitude Z2 from the speed (VN)
defined by c(V, Z2) = 0 to V2.




2The exact study of the problem, made using equation (4) instead of
equation (5) enables one to take into account four boundary conditions,
e.g. values of V, d corresponding to initial and final altitudes.
However, the approximations involved in the present analysis permit
one to impose only two boundary conditions, e.g., values of V at the
initial and final altitudes. In fact, values of e are a consequence,
because of equations (5) and (5), of the same solution which is being
investigated in this paper.





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The minimal nature of the aforementioned speed-height relationship
will be proved by showing that the following inequality is satisfied:


At = t T = 0 (0 dV + dZ) > 0 (14)
IK2NKM1


where T is the time necessary to pass from 1 to 2 using the optimum
path 1MN2 and t is the time necessary to fly along the arbitrary
path 1K2 (which, however, shall be physically possible under the
imposed condition T > R).

The line integral (14) can be separated into two integrals associated
with the closed circuits K2NK and KMIK



At = ( dV + dZ) + (4 dV + $ dZ) (19)
K2NK KMIK



By Green's theorem the line integrals contained in equation (15)
can be transformed into surface integrals connected with the areas
SA and SB encompassed by the above-mentioned boundaries



At ff ( 6dV dZ -
SB

SdZ (16)
JJ \a 3z)





NACA TM 1388 9


Because w is positive in SB and negative in SA, it follows
that At > 0. Thus the theorem is proved5. It must be emphasized that
the optimum path includes a line MKN along which the distribution of
speed is defined by


o (17)
6V oZ


or according to equations (7), (ll), and (12) by


6(T R)V V 0(T R)V (18)
av g az



The same problem can be studied with the help of the Calculus of
Variations. But, probably because of the basic hypotheses, only the
central part MKN of the optimum path can be determined in that waye.


4-b. Climb With Minimum Fuel Consumption

The weight of fuel necessary to fly from (Zl, V1) to (Z2, V2) is

2 ,2
C = q dZ = / (0 dV + (1 dZ (19)
1 Vz ~I


3For case II which seems to have some practical interest a quasi-
optimum solution (when V2 is not much less than VN) could be the
following:
(1) Acceleration at constant altitude Zi from V1 to VM.
(2) Climb using the speed distribution defined by w = 0 until
the altitude Z3 corresponding to V2 is reached.
(3) Climb at constant velocity V2 from Z3 to Z2.
The study of the problem of absolute minimum without the restrictive
condition ZI < Z < Z2 leads to an optimum trajectory composed of:
(1) A central pattern along which the distribution of velocities is
defined by equation (17).
(2) Two initial and final branches that must be flown in vertical
flight (ascending or descending) according to the boundary conditions of
the problem (see appendix).





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where


g(T R)


S= q qQ
-i = q V(T R)


(20)


(21)


The problem of finding the special function V = V(Z) which mini-
mizes integral (19) is analogous to the preceding one; hence, the solution
is of the same type. With reference to case I, and again with the
restriction that Z1 Z < Z2 the best flight technique comprises two
accelerated motions at the initial and final altitudes Zl, Z2 and a
central climbing path along which the distribution of speeds is defined
by


i l 3
wy(v, z) o
av az


(22)


or, according to equations (20) and (21), by


T R)V V (T R)V1
v Vg 6N [i- _


(25)


h-c. Steepest Climb


The total distance traveled by the aircraft
to (Z2, V2) is



sin 2 dV +
sin 8


flying from (Z1, V1)


(24)


2 dZ)






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where


VQ
02 = V = -
g(T R)


12 = V = -Q
(T R)


(25)



(26)


The speed-height function V = V(Z) which minimizes integral (24)
is analogous to those minimizing integrals ,(10) and (19).


With reference to case I and to the central pattern
altitude the best distribution of speeds is defined by


flown at varying


,2(V, z) )2 2 0
av az


(27)


or, according to equations (25) and (26), by


3(T R) V 6(T R)
Ov g 6Z


(28)


The horizontal projection of the distance traveled is


St dZ
=s 1 tan E


(29)


If the path inclination is sufficiently small, so that it is
justified to assume sin 8 = tan 9 equation (29) becomes identical with
equation (24). It follows that the distribution of speeds defined by
equation (28) minimizes the horizontal projection of the distance
traveled and is therefore the best from the point of view of the so-
called "steepest climb".






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5. DESCENDING FLIGHT


The following conditions of flight are examined:

(a) Descending flight with maximum time.

(b) Descending flight wih maximum horizontal distance.

Thrust is assumed at all times to be less than the aerodynamic drag.


5-a. Descending Flight With Maximum Time

Now, case III is examined (the same nomenclature of the preceding
paragraph is used); namely, deceleration from high altitude and high
speed to low altitude and low speed. This condition of flight is of
more practical interest than cases I, II, and IV.

Under the restrictive condition Z1 > Z Z2, the best flight
technique is the following (see fig. 2):

(1) Deceleration at constant altitude Z1 from V1 to VM defined
by o(V, Z1) = 0 (the function w(V, Zl) is defined by equation (15)).

(2) Descending flight from Z1 to Z2 using the distribution of
velocities defined by cm(V, Z) = 0.

(3) Deceleration at constant altitude Z2 from the speed VN
defined by o(V, Z2) = 0 to V2.

This statement can be easily proved using Green's theorem as in the
preceding paragraph. It should be noted that the distribution of speeds
necessary to climb with minimum time is of the same form as the speed-
height relationship required to descend with maximum time. However both
distributions are not numerically identical since different thrusts are
required.

The results valid for gliding flight may be derived from the above
as a limiting process by letting T -- 0.

The equation for the optimum speed-height function for the central
pattern MN flown at variable descending altitude is given by

(Rv) = v_ ) ()
ov g 6z






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5-b. Descending Flight With Maximum Horizontal Distance

The best technique for this type of flight is analogous to that
described in the preceding paragraph.

It consists of two decelerations at constant altitude Z1, Z2 and
of a central pattern defined by w2(V, Z) = 0. (Case III).

For gliding flight the best distribution of velocities can be
obtained as a particular case of equation (28) for T --- 0

6R V 3R
V(51)
8V g dZ


It should be noted that the solution defined by equation (51) is
identical with the one that can be obtained from an analysis based on
the equations of uniform flight if the variation of the drag with both
the Reynolds and Mach numbers is neglected.

As a matter of fact the aerodynamic drag depends, in this latter
case, on the dynamic pressure only. Thus the practical solution of
equation (51) is given by the equivalent expressions


bR R
R o (52)
&V dZ


6. REMARKS AND CRITICISMS ON THE ACHIEVED SOLUTIONS


A short review of the obtained results will clarify their physical
nature and will be helpful from the point of view of practical applications.


6-a. Comparison Between Stationary and Instationary Solutions5

Solutions commonly used in the practical applications of the
Mechanics of Flight are those derived from a "stationary" analysis5.


5Within the limits of the present investigation the term "stationary
(instationary) solution" means a solution obtained neglecting (taking into,
account) inertia tangential forces. This terminology is used for the sake
of brevity. The so-called aerodynamic lag is disregarded. In other word
the air forces are calculated as in a steady flight.






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They are the following:

For climb with minimum time


For climb with minimum fuel


For steepest climb


For maximum endurance in gl


O(TV RV) 0
dv


consumption


S[TV RV] o
v L q J





O(T R) _
6V
d v


hiding


6(Rv) = 0
6v


(55)







(34)







(55)






(56)


For maximum range in gliding


dR = 0
oV
av


(37)


These solutions could also be achieved as a particular case of those
given in this paper if one supposes that the motion takes place in an
ideal ambient of constant air density. In fact in this latter case the
derivatives of T, q, and R with respect to Z vanish and equations (18),
(25), (28), (50), and (51) are reduced to equations (55), (54), (55), (56),
and (57), respectively.






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It is evident that the second members of equations (18), (25), (28),
(30), and (51) express synthetically the contribution given by the accel-
eration to the equation defining the optimum speeds.


6-b. The Case = 0
3V

If power plant is of such a type as to justify the above-mentioned
assumption, the "stationary" solutions for minimum time and for minimum
fuel consumption become identical.

Therefore, it is logical to suppose that in this case the
instationary" solution for minimum time will not differ too much from
the one optimum for minimum fuel consumption. Some numerical calcula-
tions have confirmed this last concept.


6-c. The Case q = Constant

The "instationary solutions for minimum time and for minimum fuel
consumption become identical (rocket-powered aircraft).


6-d. Discontinuity of the Solutions at the Tropopause

The optimum speed-height relationships given by equations (18),
(25), (28), (50), and (51) have a discontinuity at the tropopause. This
fact is related to our manner of conceiving the standard atmosphere in
which the derivatives with respect to h of the density, temperature
and pressure have two values at Z = Z*.

As a consequence, for any case of flight there are two optimum
speeds at the tropopause, the one being deduced by introducing into
equations (18), (23), (28), (50), and (31) the properties of the standard
troposphere (Vt) and the other by introducing into the same equations the
properties of the standard stratosphere (Vs).

The mechanical meaning of the aforementioned discontinuity may be
understood using Green's theorem as in the previous sections. For
instance, in the case of a turbojet aircraft this indicates the necessity
of accelerating the aircraft at Z = Z, from Vt(Vs) to Vs(Vt) if the

Analogously a discontinuity in the optimum speed-height relation-
ship can be detected at the critical altitude (Zc) of an aircraft powered
by a reciprocating engine-propeller combination. This fact depends on
the existence of two values of &P/bZ at Z = Zc (P = shaft horsepower
of a conventional engine).





NACA TM 1388


achievement of the best climbing performances is desired (see fig. 5).
As a consequence the optimum flight technique for climb with minimum time
(case I) from (VI, Z1 < Z*) to (V2, Z2 > Z.) under the limiting conditions
Z1 Z- Z2, consists of:

(1) Acceleration at constant altitude Z1 from V1 to VM defined
by ca(V, Z1) = 0.

(2) Climb from (VM, Z1) to (Vt, Z.) using the speed-height relation-
ship defined by w(V, Z) = 0.

(5) Acceleration at constant altitude Zx from Vt to Vs.

..(4) Climb from (Vs, Z4) to (VN, Z2) using again the distribution of
speeds defined by w(V, Z) = 0.

(5) Acceleration at constant altitude Z2 from VN to V2.


6-e. Hypothesis Concerning Curvature and Path Inclination

The practical consequence of the hypothesis (b) of section (2) is
an approximate calculation of that part of the drag which depends on the
lift. The errors involved have small importance for many of the cases
of flight here considered.

In any case the following concept should be emphasized: the use of
the solutions here achieved is logical only if the errors associated with
the neglect of curvatures and squares of the path inclination are small
with respect to those avoided taking into account the tangential
accelerations.

A systematic investigation of the exact limits of applicability of
the present theory to the various types of modern aircraft is beyond the
scope of this report. However, it seems possible to anticipate that the
hypotheses concerning the.curvatures and the path inclinations art justi-
fied in the following cases:

(1) Climb with minimum time and with minimum fuel consumption: jet-
propelled aircraft and conventional aircraft with high wing-and-power
loadings.

(2) Steepest climb: turbojet aircraft with low specific thrust (T/Q)
and good aerodynamic efficiency (results concerning the steepest climb are
not only influenced by the approximations made in the projection of the
equation of motion on the normal to the flight path but also by the sub-
stitution tan 0 in lieu of sin 8 in the projection of the equation of
motion on the tangent to the flight path).






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(5) Gliding flight: airplanes having high wing loading and good
aerodynamic efficiency.


o-f. Hypothesis Concerning the Weight of the Aircraft

The weight of the aircraft changes during the flight because of the
fuel consumption. Consequently, the true optimum speed-height relation-
ships are somewhat different from the ones previously derived. The true
rates of climb are greater than those calculated by assuming Q = Constant.
According to the practical values of dQ/dt the following remarks are
formulated:

(1) Aircraft powered by air-breathing engines: The speed-height
functions previously derived are substantially correct. If more preci-
sion is desired the weight changes can approximately be taken into account
by iterating the calculations as follows:

(a) Calculate the optimum distributions of speeds with equations (18),
(25), and (28) according to the case of flight and supposing Q = Constant.

(b) Determine the approximate values of the fuel consumption on the
basis of the above-mentioned distributions of speeds.

(c) Calculate the instantaneous weights of the aircraft at any
altitude.

(d) Determine the new optimum speed-height functions by introducing
into equations (18), (25), and (28) the instantaneous weights. Calculate
also the new values of integrals (10), (19), and (24).

(2) Rocket-powered aircraft: From a purely theoretical standpoint
the results here derived cannot be considered valid for this kind of air-
craft because of the important dynamical effects associated with the
changes of the airplane weight.

Notwithstanding, the author believes that the obtained solutions
are very close to the true solutions for tropospheric flight above all
if iterative procedures, like the one outlined above, are applied.

That depends on the fact that the only term of equation (18)
depending on the weight is that part of the drag which is a function of
the lift; that is to say the so-called "induced" drag, which is small at
low altitudes because of the low angles of attack used by rocket-powered
aircraft in the climbing flight.


6-g. Additional RemarKs Concerning Centripetal Accelerations

The neglect of curvatures (and therefore of the deviation times
necessary to pass from one branch to another of the optimum path) has a






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qualitative influence on the results leading to discontinuous solutions;
while it is logical to think that the exact study of the problem made
with variational methods and using equation (4) instead of equation (5)
would bring continuous speed distributions.

Consequently, the results contained in this paper are to be con-
sidered as limiting results whose degree of agreement with experiments
increase as the ratio of sum of deviation times to the total time
decrease.


6-h. Considerations Related to the Method Used in This Paper

The main effect of the hypotheses concerning weight, curvatures
and path inclinations has been the possibility of expressing the aero-
dynamic drag as a function of the type


R = R(V, Z) (38)


It follows that if the basic hypotheses are changed formulas (18),
(25), (28), (30), and (51) may retain their validity provided the drag
remain still a function of only the speed and the altitude.


7. CONCLUSIONS


A general method concerning optimum problems in nonstationary
flight is developed and discussed. Various conditions of flight in a
vertical plane (climb with minimum time, climb with minimum fuel con-
sumption, steepest climb, descending and gliding flight with maximum
time or space) are studied; the corresponding best techniques of flight,
i.e. the optimum speed-height relationships, are determined.

Each optimum path consists of an initial and a final branch which
depend on the boundary conditions of the problem and a central portion
which is flown at variable altitude and speed.

Along this central pattern the speed-height relationship obeys the
following rule:

If X= X(V, Z) is the function whose maximum or minimum with
respect to the speed


= 0 (59)
a~v





NACA TM 1588


defines the best speed-height
is studied with the equations
tion of the same problem when
is defined by


relationship when a given optimum problem
of uniform flight, then the modified solu-
the effects of acceleration are considered


S' v g z
dV g aZ


(40)


The optimum speed-height relationships have a discontinuity at the
tropopause and differ in general from the solutions based on the assump-
tion of uniform flight with one exception, namely, gliding with maximum
range. This latter result is valid if the variation of the drag with
both the Mach and Reynolds numbers is neglected.



The practical application of the method given in this paper to the
particular case of turbojet aircraft will be published soon.






NACA TM 1588


APPENDIX


PROBLEMS OF ABSOLUTE OPTIMUM


A number of minimal problems have been treated in the preceding
paragraphs with the help of some restrictive conditions. For example,
the restrictive condition Z1 < Z ` Z2 has been used to study the climb
with minimum time (case I). In other words the problem of accelerating
and climbing from low speed and low altitude to high speed and high alti-
tude has been analyzed by considering only paths internal to the region
of space limited by the horizontal planes corresponding to the initial
and final heights.

However, it may be noted that the method given in this paper may
be easily extended to the study of problems of absolute optimum.

If no restrictive condition is imposed to the altitude, the speed-
height function optimum to climb with minimum time consists of a central
branch whose equation is still w = 0 and of two initial and final por-
tions which must be flown in vertical flight (ascending or descending)
according to the boundary conditions of the problem (see table I and
fig. 4).

This statement may be easily proved by applying Green's theorem as
shown in section 4.

The main comments concerning the paths indicated in figure 4 are
the following:

(a) For jet-propelled aircraft the hypothesis P = Q leads to
errors which are small along the line whose equation is w = 0 and
also for the vertical branches 1M and N2 provided they are flown at
high speed.

On the other hand the errors may probably be of some importance
for the vertical branches if a part of them is to be flown at relatively
low speed (that depends evidently on the boundary conditions of the
problem).

(b) The optimum speed-height relationships shown in figure 4 have
a discontinuous character. On the other hand it is logical to presume
that, should the problem be studied with the use of the exact equations
of the motion, results would have a continuous character.

Consequently, the results contained in this paper must be considered
as limiting results as shown in section 6-g.






NACA TM 1588


In addition, it might be well to bear in mind that, even if an
exact study of the problem were possible from a mathematical standpoint,
the conclusions would still have to be submitted to other limitations,
namely, those imposed by the physiological strength of the pilot and
those imposed by the structural strength of the aircraft.


Translated by A. Miele?


Translated by the author, who wishes to express his thanks to
Dr. Nathan Ness and to Mr. Lawrence S. Galowin for their kind corrections
of the English manuscript.





NACA TM 1588


REFERENCES


1. Phillips, F. C.: A Kinetic Energy Correction to Predicted Rate of
Climb, Journal of the Aeronautical Sciences, vol. 9, no. 5,
March 1942, pp. 172-174.

2. Otten, G.: Rate of Climb Calculations, Journal of the Aeronautical
Sciences, vol. 10, no. 2, February 1945, pp. 48-50.

5. Hayes, W. D.: Performance Methods for High-Speed Aircraft, Journal
of the Aeronautical Sciences, vol. 12, no. 5, July 1945,
pp. 345-548.

4. Perkins, C. D., and. Hage, R. E.; Airplane Performance, Stability and
Control. John Wiley & Sons, Inc., New York, 1949, pp. 200-201.

5. Miele, A.: Problemi di Minimo Tempo nel Volo Non-Stazionario degli
Aeroplani, Atti della Accademia delle Scienze di Torino, vol. 85,
1950-1951, pp. 41-52.

6. Lush, K. J.: A Review of the Problem of Choosing a Climb Technique
With Proposals for a New Climb Technique for High Performance
Aircraft, Aeronautical Research Council, R.M. 2557, 1951.

7. Kaiser: The Climb of Jet-Propelled Aircraft, Part I. Speed Along
the Path in Optimum Climb, RTP/TIB Translation GDC/15/148 T,
April 1944.







NACA TM 1588


TABLE I


SPEED-HEIGHT RELATIONSHIPS OPTIMUM FOR CLIMBING WITH MINIMUM TIME


Case of Point 1 Point 2 Speed-height function
Flight is in zone is in zone Brah
Branch 1M Branch MN Branch N2


II


III


IV


A


A


B


B


Vertical
dive

Vertical
dive

Vertical
climb

Vertical
climb


= 0


u= 0






w=O


Vertical
dive

Vertical
climb

Vertical
climb

Vertical
dive





NACA TM 1588


-V


Figure 1.- Speed-height relationship 1MHI,12 for climb with minimum
time; case I; Z, < Z < Z,; initial and final altitudes are either
both tropospheric or both stratospheric.


------ V


Figure 2.- Speed-height relationship IIII12 for descending flight with
maximum time; case III; Z1 > Z > Z2; initial and final altitudes are
assumed either both tropospheric or both stratospheric.


w


/
/


S> O


J=O


w>O


wO=0


2 N


I


w



NACA TM 1388


-V
Figure 3.- Speed-height relationship 1.11.tSI for climb with minimum
time; case I; Z1 < Z < Z2; initial -titude is assumed to be
tropospheric; final altitude is assumed to be stratospheric.


-V V

S 2


---w- V


--w- V


Figure 4.- Speed-height relationship 1Ll1J2 for climb with minimum
time; initial and final altitudes are assumed either both tropospheric
or both stratospheric.
NACA Langley Field, Va.


'- O 0


I


Tropopause (Z*)
- ( = O


1 M


I















~4i

V


- 4


- 0


i




*t (D











.) Sw.
Cfo 0
s "c0.
M'g 0
5 g U


9 n
> .4



4 4 )mc
g (; zd0
444 CO t
Sd o d^


U-~



(420
oPr:
Sa .0'


C4



- i :.: e
C -


- ni t~ E3

* **5 00


0 0
^Jr ar
-:-


w t



0. 0
s
, ac -i CM ii


I







I




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