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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 wellknown 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 NonStazionario." L'Aerotecnica, n. 5, vol. XXXII, 1952, pp. 155142. LThe author wishes to thank Professor Placido Cicala, of the Polytechnic of Turin and of the University of Cordoba, for his helpful suggestions. NACA TM 1388 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 highperformance 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 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 abovementioned 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. 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, airbreathing 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 NACA TM 1388 5 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). NACA TM 1588 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. 4a. 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 speedhtight relationship V = V(Z) which minimizes integral (10). The investigation is simplified if the properties of the function NACA TM 1388 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. NACA TM 1588 The minimal nature of the aforementioned speedheight 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 abovementioned 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. 4b. 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). NACA TM 1588 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) hc. 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) NACA TM 1588 where VQ 02 = V =  g(T R) 12 = V = Q (T R) (25) (26) The speedheight 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". 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. 5a. 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 speedheight function for the central pattern MN flown at variable descending altitude is given by (Rv) = v_ ) () ov g 6z NACA TM 1388 13 5b. 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. 6a. 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 socalled aerodynamic lag is disregarded. In other word the air forces are calculated as in a steady flight. NACA TM 1588 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. NACA TM 1588 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. 6b. The Case = 0 3V If power plant is of such a type as to justify the abovementioned 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. 6c. The Case q = Constant The "instationary solutions for minimum time and for minimum fuel consumption become identical (rocketpowered aircraft). 6d. Discontinuity of the Solutions at the Tropopause The optimum speedheight 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 speedheight relation ship can be detected at the critical altitude (Zc) of an aircraft powered by a reciprocating enginepropeller 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 speedheight 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. 6e. 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 wingandpower 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). NACA TM 1588 (5) Gliding flight: airplanes having high wing loading and good aerodynamic efficiency. of. 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 speedheight 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 airbreathing engines: The speedheight 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 abovementioned distributions of speeds. (c) Calculate the instantaneous weights of the aircraft at any altitude. (d) Determine the new optimum speedheight functions by introducing into equations (18), (25), and (28) the instantaneous weights. Calculate also the new values of integrals (10), (19), and (24). (2) Rocketpowered 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 socalled "induced" drag, which is small at low altitudes because of the low angles of attack used by rocketpowered aircraft in the climbing flight. 6g. 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 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 (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. 6h. 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 speedheight 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 speedheight 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 speedheight 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 speedheight 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 jetpropelled 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 speedheight 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 6g. 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. 172174. 2. Otten, G.: Rate of Climb Calculations, Journal of the Aeronautical Sciences, vol. 10, no. 2, February 1945, pp. 4850. 5. Hayes, W. D.: Performance Methods for HighSpeed Aircraft, Journal of the Aeronautical Sciences, vol. 12, no. 5, July 1945, pp. 345548. 4. Perkins, C. D., and. Hage, R. E.; Airplane Performance, Stability and Control. John Wiley & Sons, Inc., New York, 1949, pp. 200201. 5. Miele, A.: Problemi di Minimo Tempo nel Volo NonStazionario degli Aeroplani, Atti della Accademia delle Scienze di Torino, vol. 85, 19501951, pp. 4152. 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 JetPropelled Aircraft, Part I. Speed Along the Path in Optimum Climb, RTP/TIB Translation GDC/15/148 T, April 1944. NACA TM 1588 TABLE I SPEEDHEIGHT RELATIONSHIPS OPTIMUM FOR CLIMBING WITH MINIMUM TIME Case of Point 1 Point 2 Speedheight 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. Speedheight 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. Speedheight 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. Speedheight 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. Speedheight 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 UNIVERSITY OF FLORIDA 3 III1262 08106 U5 IIII 3 1262 08106 546 7 
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