NsftcA ii^-mi 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1388 
 
 GENERAL SOLUTIONS OF OPTIMUM PROBLEMS IN 
 
 NONSTATIONARY FLIGHT 
 
 By Angelo Miele 
 
 Translation of "Soluzioni General! di Problemi di Ottimo in Volo 
 Non-Stazionario." L*Aerotecnica,n. 3, vol. XXXII, 1952. 
 
 LIBRARY 
 
 17011 
 
 LE, FL ■6Z<:>\i-i^ 
 
 Washington 
 October 1955 
 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1388 
 
 GENERAL SOLUTIONS OF OPTIMUM PROBLEMS IN 
 NONSTATIONARY FLIGHT "'' 
 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 eind 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 
 
 -^a ^ ^z/^zu ~- ^^"^ o/sin 9^ acceleration factor: ratio of effective 
 
 rate of climb to the one computed neg- 
 lecting accelerations 
 
 * Soluzioni Generali di Problem! di Ottimo in Volo Non-Stazionario. 
 L'Aerotecnica, n. 5, vol. XXXII, 1952, pp. 135-1^2. 
 
 ■'-The author wishes to thank Professor Placido Cicala, of the 
 Polytechnic of Turin and of the University of Cordoba, for his helpful 
 suggestions . 
 
2 NACA TM 1588 
 
 g acceleration of gravity 
 
 P lift 
 
 q weight of fuel consiomed per unit time 
 
 Q airplane weight 
 
 R aerodynamic drag 
 
 s distance flown 
 
 Sq horizontal projection of the distance flown 
 
 t time 
 
 T thrust 
 
 V speed 
 
 V2; = V sin e effective rate of climb 
 
 Vzu = V sin 9u rate of climb computed from the equations of uniform 
 
 flight 
 
 Z altitude 
 
 Z^ altitude of tropopause 
 
 9 effective path inclination (positive for climbing flight) 
 
 9 path inclination computed from the equation of \iniform 
 flight (positive for climbing flight) 
 
 1. INTROmCTION 
 
 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 assiimption 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. 
 
NACA TM 1388 
 
 For instance^ it is well known that the speed for best climb of any 
 type of aircraft increases with altitude (incompressible 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 consiimption 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. 1), 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. h, 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 trans fonnation 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 (Zg = Z + v2/2g). 
 
 The present paper generalizes a method described in a preceding 
 study (ref. 5) an<i 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 
 (V]^, Zi) to fixed final conditions (V2, Z2), will maximize or minimize 
 
 the time, the fuel consumption or the distance. 
 
NACA TM 1588 
 
 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) 
 
 1 = ci(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. FUUDAMKNTAL 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: 
 
 R - Q 
 
 g dZ 
 
 sin 9=0 
 
 P - Q 
 
 cos 
 
 V2 d9 . 
 
 + sm 
 
 g dZ 
 
 = 
 
 (3) 
 
NACA TM 1588 
 
 According to the preceding hypothesis (b), equation (k) can be 
 substituted by 
 
 P - Q = (5) 
 
 This approximation is important. As a matter of fact, the rate of 
 cLimb given by 
 
 V, = V Sin e = H-i-^Ma (6) 
 
 g dZ 
 
 becomes a function of V, Z, and dV/dZ only, as can be seen from 
 equations (l), (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/Q (7) 
 
 sin 9^ = (T - R)/q (8) 
 
 by the correction factor 
 
 1 + 
 
 , 2g dZ 
 
 vhich expresses the effects associated with the noniiniform 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). 
 
6 NACA TM 1588 
 
 k. CLIMBING FLIGHT 
 
 The following cases of flight are discussed: 
 
 (a) Climb with minimum time. 
 
 (h) Climb with minimum fuel consiimption. 
 
 (c) Steepest climb. 
 
 It is assumed that the thrust at all times is greater than the 
 aerodynamic drag. 
 
 k-a. Climb With Minimum Time 
 Tlie time necessary to fly from given initial conditions (V^, Z]_) 
 to fixed final conditions (^2' ^2) ^^ 
 
 2 2 
 t = r ^ = r ((D dV + $ dZ) (10) 
 
 Jl \ Jl 
 
 where 
 
 1> = ,^ ^, = <I'(V, Z) (11) 
 
 g(T - R) 
 
 ^ = ^(V, Z) (12) 
 
 ^ V(T - R) 
 
 Here it is desired to act. rmine the best flight technique; that is 
 to say, the particular speed- height relationship V = V(z) which 
 minimizes integral (lO). 
 
 The investigation is simplified if the properties of the function 
 
NACA TM 1388 
 
 cd(V, Z) = -^ - — 
 
 (13) 
 
 2 
 are used insteaxi of the application of variational methods . 
 
 The curve cu = divides that zone of the (V, Z) plane which is of 
 practical interest for flight operations (fig. l) into two regions: 
 A, where cd < 0; B where cjo > 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 cd = 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 
 Z-L < Z < Zg. 
 
 The optimum flight technique is the following: 
 
 (1) Acceleration at constant altitude Z]_ from Vi to the speed 
 (Vm) defined by m(V, Zi) = 0. 
 
 (2) Climb from Z-]_ to Z2 using the distribution of velocities 
 defined by a>(V, Z) = 0. 
 
 (3) Acceleration at constant altitude Z2 from the speed (V]\j) 
 defined by £jo(V, Z2) = to V2. 
 
 ^The exact study of the problem, made using equation (k) instead of 
 equation (5) enables one to take into account four boundary conditions, 
 e.g. values of V, 9 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 9 are a consequence, 
 because of equations (3) and (5), of the same solution which is being 
 investigated in this paper. 
 
8 NACA TM 1588 
 
 The minimal nature of the aforementioned speed- height relationship 
 will be proved by showing that the following inequality is satisfied: 
 
 At = t - T = (/; (<D dV + f dZ) > {Ik) 
 
 1K2NKM1 
 
 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 (ik) can be separated into two integrals associated 
 with the closed circuits K2NK and KMIK 
 
 (<D dV + ^ d2,) + (0 dV + f dZ) (15) 
 
 At 
 
 ^ K2M ^ KMIX 
 
 By Green s theorem the line integrals contained in equation (15) 
 can be transformed into surface integrals connected with the areas 
 S;^ and Sg encompassed by the above-mentioned boundaries 
 
 Sb 
 
 h . ^av 6Z (16) 
 
 \av az/ 
 
NACA TM 1388 
 
 Because cu is positive in S3 and negative in Sp^, it follows 
 
 that At > 0. Thus the theorem is proved^. It raust be emphasized that 
 the optimum path includes a line MKN along which the distribution of 
 speed is defined by 
 
 M _ ^ = (17) 
 
 or according to equations (7), (ll), and (l2) by 
 
 ^(T - R)V ^ V a(T - R)V (-^Q^ 
 
 dV S Sz 
 
 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 way^. 
 
 U-b. Climb With Minimum Fuel Consumption 
 The weight of fuel necessary to fly from {Z±, Vi) to (Z2, V2) is 
 
 ^2 r2 
 
 q_ 
 
 1 Vz 
 
 C = / ^ 6Z = ($^ dV + fi dZ (19) 
 
 --'For case II which seems to have some practicaJL interest a quasi- 
 optimum solution (when V2 is not much less than Y-^) could be the 
 following: 
 
 (1) Acceleration at constant altitude Zj_ from Vj_ to Vj^. 
 
 (2) Climb using the speed distribution defined by cu = until 
 the altitude Z5 corresponding to V2 is reached. 
 
 (5) Climb at constant velocity V2 from Zj to Z2. 
 
 The study of the problem of absolute minimum without the restrictive 
 condition Z]_ < 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). 
 
10 
 
 NACA TM 1588 
 
 where 
 
 qQ 
 
 0-, = (^ = 
 
 g(T - R) 
 
 T qQ 
 
 ^1 = ^^ = 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 Z^^ < Z < Zo the best flight technique comprises two 
 
 accelerated motions at the initial and final altitudes Z-^, Z2 and a 
 central climbing path along which the distribution of speeds is defined 
 lay 
 
 ^^2. ^^1 
 
 Sv Sz 
 
 (22) 
 
 or, according to eqioations (20) and (21), by 
 
 
 (t - r)v 
 
 V _a_ 
 
 g hz 
 
 (T - R)V 
 
 (25) 
 
 h-c. Steepest Climb 
 
 The total distance traveled by the aircraft flying from (Zj_, V]_) 
 to (Z2;, V2) is 
 
 J ■[_ sm 9 J -|_ "^ '^ 
 
 (2li) 
 
NACA TM 1388 11 
 
 where 
 
 $ = v<j = ^ ^^ ^ (25) 
 
 ^ g(T - R) 
 
 *2 = vf = -^^TTiy (2^^ 
 
 The speed-height function V = V(z) which minimizes integral (2^+) 
 is analogous to those minimizing integrals .(lO) and (l9)- 
 
 With reference to case I and to the central pattern flown at varying 
 altitude the best distribution of speeds is defined by 
 
 a^(V, Z) = ^ - ^ = (27) 
 
 or, according to equations (25) and (26), by 
 
 S(T - R) ^ V a(T - R) , nx 
 
 ^V g SZ ^ ' 
 
 The horizontal projection of the distance traveled is 
 
 dZ 
 
 (29) 
 
 tan 9 
 
 If the path inclination is sufficiently small, so that it is 
 justified to assume sin 9 = tan 9 equation (29) becomes identical with 
 equation (2^+). 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 . 
 
12 NACA TM 1588 
 
 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 Zi > Z > Z2> the best flight 
 technique is the following (see fig. 2): 
 
 (1) Deceleration at constant altitude Zi from V^ to V^^ defined 
 by a)[V, Z^) = (the fimction cjd(v, Z^] is defined by equation (15))- 
 
 (2) Descending flight from Z^ to Z2 using the distribution of 
 velocities defined by a)(V, Z) = 0. 
 
 (5) Deceleration at constant altitude Z2 from the speed Vjj 
 defined by a)(v, Zg) = 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 reqiiired 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 
 
 Sv g Sz ■ ^^"^. 
 
NACA TM 1588 13 
 
 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 Z-,, Z2 and 
 of a central pattern defined by oogCv, Z) = 0. (Case III). 
 
 For gliding flight the best distribution of velocities can be 
 obtained as a particular case of equation (28) for T — > 
 
 ^ = - ^ (51) 
 
 ^ g dZ 
 
 It should be noted that the solution defined by equation (31 ) 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 (5l) is given by the equivalent expressions 
 
 ^ = ^ = (52) 
 
 ^v az 
 
 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 Solutions^ 
 
 Solutions commonly used in the practical applications of the 
 Ifechanics of Flight are those derived from a "stationary" analysis^. 
 
 '=) II 
 
 Within 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. 
 
11+ 
 
 NACA TM 1588 
 
 They axe the following: 
 For climb with minimum time 
 
 a(TV - RV) ^ Q 
 
 (35) 
 
 For climb with minimum fuel consumption 
 
 
 TV - RV 
 
 = 
 
 (3U) 
 
 For steepest climb 
 
 5(T - R) 
 
 av 
 
 (55) 
 
 For maximum endurance in gliding 
 
 av 
 
 (36) 
 
 For maximum range in gliding 
 
 M= 
 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), 
 (23), (28), (30), and (31) are reduced to equations (33), (3^+), (35), {%) , 
 and (37 ), respectively. 
 
NACA TM 1588 15 
 
 It is evident that the second members of equations (18), (23), (28), 
 (30), and (51) express synthetically the contribution given by the accel- 
 eration to the equation defining the optimum speeds. 
 
 6-b. The Case -3^ = 
 
 Sv 
 
 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), 
 (23), (28), (30), and (31) 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 optim\im 
 speeds at the tropopause, the one being deduced by introducing into 
 equations (18), (25), (28), (30), and (31) the properties of the standard 
 troposphere (V-t) and the other by introducing into the same equations the 
 properties of the standard stratosphere (Vs ) . 
 
 The mechanical meaning of the aforementioned discontinxiity" 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- 
 sh,ip can be detected at the critical altitude (Z^) of an aircraft powered 
 by a reciprocating engine-propeller combination. This fact depends on 
 the existence of two values of 6P/6Z at Z = Zq (P = shaft horsepower 
 of a conventional engine). 
 
l6 NACA TM 1388 
 
 achievement of the best climbing performances is desired (see fig. 3)- 
 As a consequence the optimum flight technique for climb with minimum time 
 (case I) from (V^^, '^1< ^) ^° (^2' ^2 ^ ^*) under the limiting conditions 
 Zi < Z < Zg, consists of: 
 
 (1) Acceleration at constant altitude Z^ from Vi to Vm defined 
 by cu(V, Zi) = 0. 
 
 (2) Climb from (Vj^j, Z^) to (Vtj, Z^) using the speed-height relation- 
 ship defined by a)(V, Z) = 0. 
 
 (5) Acceleration at constant altitude Z^. from V^ to Vs . 
 
 ..(^) Climb from (Vs, Z*) to (Vn, Z2) using again the distribution of 
 speeds defined by a)(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 modem aircraft is beyond the 
 scope of this report. However, it seems possible to anticipate that the 
 hypotheses concerning the, curvatures and the path inclinations ar^ 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 e in lieu of sin 9 in the projection of the equation of' 
 motion on the tangent to the flight path) . 
 
MCA TM 1388 17 
 
 (5) Gliding flight: airplanes having high wing loading and good 
 aerodynamic efficiency. 
 
 6-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 optlm\am distributions of speeds with equations (18), 
 (23) J 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 instaintaneous weights of the aircraft at any 
 altitude . 
 
 (d) Determine the new optimum speed-height functions by introducing 
 into equations (18), (23), and (28) the instantaneous weights. Calculate 
 also the new values of integrals (lO), (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 
 
l8 NACA TM 1588 
 
 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 (h) 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), 
 (23), (28), (30), and (31) 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 minimimi 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 
 
NACA TM 1388 19 
 
 defines the best speed-height relationship when a given optimum problem 
 is studied with the equations of uniform flight, then the modified solu- 
 tion of the same problem when the effects of acceleration are considered 
 is defined by 
 
 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. 
 
20 NACA TM I388 
 
 APPEND DC 
 
 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 Zi < Z < Z2 has been used to study the climb 
 
 with minimum time (case l). 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 m = and of two initial and final por- 
 tions which must be flown in vertical flight (ascending or descending) 
 according to the boiindary conditions of the problem (see table I and 
 fig. h). 
 
 This statement may be easily proved by applying Green's theorem as 
 shown in section k. 
 
 The main comments concerning the paths indicated in figure h are 
 the following: 
 
 (a) For jet-propelled aircraft the hypothesis P = Q leads to 
 errors which are small along the line whose equation is co = and 
 also for the vertical branches IM 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 h 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 21 
 
 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 ana 
 those imposed by the structuraJ. strength of the aircraft. 
 
 Translated by A. Miele 
 
 7 
 
 7 
 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. 
 
22 NACA TM I388 
 
 REFERENCES 
 
 1. Phillips, F. C; A Kinetic Energy Correction to Predicted Rate of 
 
 Climb, Journal of the Aeronautical Sciences, vol. 9? no. 5^ 
 March 1914-2, pp. I72-I7U. 
 
 2. Otten, G.: Rate of Climb Calculations, Journal of the Aeronautical 
 
 Sciences, vol. 10, no. 2, February 19^3, pp. U8-5O. 
 
 3. Hayes, W. D.: Performance Methods for High-Speed Aircraft, Journal 
 
 of the Aeronautical Sciences, vol. 12, no. 3; July 1914-5 j 
 pp. 3^45-3^^8. 
 
 k. Perkins, C. D., and Hage, R. E.: Airplane Performance, Stability and 
 Control. John Wiley & Sons, Inc., New York, I9U9, pp. 200-201. 
 
 5. Miele, A.: Prohlemi di Minimo Tempo nel Volo Non-Stazionario degli 
 
 Aeroplani, Atti della Accademia delle Scienze di Torino, vol. 85, 
 1950-1951, PP- ^4-1-52. 
 
 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/l5/l48 T, 
 April igkk. 
 
NACA TM 1588 
 
 25 
 
 TABLE I 
 
 SPEED-HEIGHT RELATIONSHIPS OPTIMUM FOR CLIMBING WITH MINIMJM TIME 
 
 Case of 
 Flight 
 
 Point 1 
 is in zone 
 
 Point 2 
 is in zone 
 
 Speed- 
 
 ■height function 
 
 Branch IM 
 
 Branch MN 
 
 Branch N2 
 
 I 
 
 A 
 
 B 
 
 Vertical 
 dive 
 
 m = 
 
 Vertical 
 dive 
 
 II 
 
 A 
 
 A 
 
 Vertical 
 dive 
 
 m = 
 
 Vertical 
 climb 
 
 III 
 
 B 
 
 A 
 
 Vertical 
 climb 
 
 en = 
 
 Vertical 
 
 climb 
 
 TV 
 
 B 
 
 B 
 
 Vertical 
 
 OD = 
 
 Vertical 
 
 
 
 
 climb 
 
 
 dive 
 
2k 
 
 NACA TM 1388 
 
 Z 
 
 
 / 
 
 / 
 
 
 N/ 2 
 
 "^ ^ 
 
 
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 1 ^ 
 
 ® 
 
 y ® 
 
 P k 
 
 ^<0 / / 
 
 a; > 
 
 / 'a / 
 
 ^--6lI = 
 
 / / 
 
 
 1 ?M 
 
 
 -V 
 
 Figure 1.- Speed-height relationship 1MN2 for climb with minimum 
 time; case I; Z]^ < Z < Z2; initial and final altitudes are either 
 both tropospheric or both stratospheric. 
 
 Z 
 
 A 
 
 -► V 
 
 Figure 2.- Speed -height relationship IMN2 for descending flight with 
 maximum time; case III; Z^ > Z > Zg; initial and final altitudes are 
 assumed either both tropospheric or both stratospheric. 
 
NACA TM 1588 
 
 25 
 
 z 
 A 
 
 .V 
 
 Figure 3.- Speed-height relationship lMtSN2 for climb with minimum 
 time; case I; Z| < Z < Zg; initial altitude is assumed to be 
 tropospheric; final altitude is assumed to be stratospheric. 
 
 .V 
 
 .V 
 
 z 
 
 — ^\i — »- V 
 
 Figure 4.- Speed -height relationship 1MN2 for climb with minimum 
 time; initial and final altitudes are assumed either both tropospheric 
 or both stratospheric. 
 
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 GENERAL SOLUTIONS OF OPTIMU 
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UNIVERStTV OF FLORIDA 
 
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