1、NASA CONTRACTOR REPORT STOP - A COMPUTER PROGRAM FOR SUPERSONIC TRANSPORT TRAJECTORY OPTIMIZATION by Lawrence H. Stein, Malcolm L. Mathews, and Joel W. Freak Prepared by THE BOEING COMPANY Seattle, Wash. for Langley Research Cede? / NATIONAL AERONAUTICS AND SPACE ADMINISTRATION l WASHINGTON, 0. C. l
2、 MAY 1967 l Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-TECH LIBRARY KAFB, NM NASA CR-793 - A COMPUTER PROGRAM FOR SUPE TRANSPORT TRAJECTORY OPTIMIZATION By Lawrence H. Stein, Malcolm L. Matthews, and Joel W. Frenk Distribution of this report is
3、provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it. Prepared under Contract No. NAS l-5293 by THE BOEING COMPANY Seattle, Wash. for Langley Research Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by
4、the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-ABSTRACT An IBM 7094 digital computer program using the steepest ascent technique has b
5、een developed for optimizing the flight path of a supersonic transport air- craft from the start of climb through cruise and descent. The program is suf- ficiently versatile so that other vehicles besides the SST may have their flight paths optimized. These vehicles include: space boosters, ICBMs, r
6、eentry vehicles, scramjets, and vehicles with air-augmented rocket propulsion. The program incorporates a 3-D point mass simulation of a vehicle moving in relation to a spherical, rotating earth. The inverse-square law for gravity and 1962 U. S. standard atmosphere are used. ;- The optimization is a
7、ccomplished with an automatic step-size controller and with automatic control variable weighting matrices to allow problem solu- tion in a single computer run. Automatic plotting capability is included. Mul- tistaged vehicles and problems involving variable initial conditions may be optimized. Pitch
8、 angle, bank angle, wing sweep, and throttle setting are the control variables for the program. Inequality constraints are available on all control variables as well as on parameters affecting geopolitics, passenger comfort, structural loads, and engine operation. The geopolitical constraints includ
9、e sonic boom over-pressure and maximum and minimum altitude. ii Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-FOREWORD This report was prepared by the Missile and Information Systems Division of The Boeing Company, Seattle, Washington. It presents
10、the final documentation of the analytical development and users manual for the Supersonic Transport Optimization Program (STOP). The program was developed by Boeing for the Langley Research Center under contract NAS l-5293. The contract was admin- istered by the National Aeronautics and Space Admins
11、tration under the direction of Mr. J. R. Elliott, with Mr. David F. Thomas, Jr. acting as contract mon- itor. The Supersonic Transport Optimization Program was obtained from the NASA Request for Proposal L-5347. Development of the program began in August 1965 and was completed in September 1966. Dr.
12、 L. H. Stein was responsible for program development. The closed loop guidance techniques and the control formulation were developed by Mr. M. L. Matthews. Mr. Matthews also assisted in a major part of the program development as well as in the solu- tion of the twenty check cases. Mr. J. W. Frenk wa
13、s in charge of programming and was assisted by Mr. D. A. Watson. Mr. Watson was also responsible for the plotting capability for the program. The work was performed under the direc- tion of Mr. E. G. Haugseth of the Boeing Missile and Information Systems Di- vision. This report, together with a comp
14、anion document, Supersonic Transport Trajectory Optimization - Example Solutions (NASA Contractor Report No. 66247) plus the FORTRAN source program listings, binary object deck, and symbolic object deck concludes the work prescribed under contract NAS l-5293. iii Provided by IHSNot for ResaleNo repr
15、oduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CONTENTS SUMMARY INTRODUCTION SYMBOLS ANALYTICAL DEVELOPMENT Equations of Motion Coordinate System Control Variables Kinematic Equations App
16、lied Forces. Basic Equations of Motion Auxiliary State Variables Inequality Constraints Steepest-Ascent Technique Statement of Problem Adjoint Equations Variational Equations Automatic Convergence Iterative Procedure Variable Initial Conditions Weighting Functions Matrix Nominal Trajectory Generatio
17、n Angle of Attack from Known 9 Angle of Attack for a Known V Throttling for a Known V Guidance Modes Use of Guidance Options Additional Options Gamma Tilt Circular Satellite Maximum Payload Conclusions and Recommendations PROGFXM USERS MANUAL Program Assumptions and Limitations As sum ptions Limitat
18、ions Page 1 4 5 10 10 10 10 12 13 18 19 20 25 25 26 30 34 34 37 39 42 43 44 45 45 47 48 48 48 49 49 50 50 50 51 V Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CONTENTS (Cont. ) Page Input Data Preparation 52 PR0GRA.M OPERATION 77 Nomenclature - FO
19、RTRAN COMMON 77 Output Description Printed Output Punched Card Gutput Magnetic Tape Output Sample Problem Statement of Problem Sample Input Sample Output Operating Information Program Setup Data Setup 92 92 97 100 102 102 103 103 114 General Machine Operation 114 116 116 Tape or Disk Requirements 11
20、6 End-of-Run Indication 116 Special Machine Operating Information 119 Programming Information 122 Basic Program Flow Subroutine Descriptions and Flow Diagrams Program Flexibility Program and Data Overlay Equivalence Trouble Shooting Plotting Information Limitations 122 127 279 284 287 287 297 Input
21、Controls Required for Plotting Multiple-Curve Identification 297 297 298 REFERENCES 299 APPENDIX A - Control Variable Choice for Point Mass Equations of Motion 300 vi Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CONTENTS (Cont. ) Page APPENDIX B -
22、 Program Equations, Variables, and Constants Defined Equations of Motion Partial Derivatives Auxiliary Printout Variables VAR Array CONST Array Miscellaneous COMMON Internal Program Indicators APPENDIX C -Program Control Logic 322 305 305 306 313 314 316 317 319 vii Provided by IHSNot for ResaleNo r
23、eproduction or networking permitted without license from IHS-,-,-STOP- A COMPUTER PROGRAM FOR SUPERSONIC TRANSPORT TRAJECTORY OPTIMIZATION By Lawrence H. Stein, Malcolm L. Matthews, and Joel W. Frenk Boeing Aerospace Group SUMMARY An IBM 7094 digital computer program using a steepest-ascent procedur
24、e has been developed for optimizing the flight path of supersonic transport aircraft from the start of climb through cruise and descent. This document describes the analytical development of the supersonic transport optimization program (STOP) and presents a manual for program users. Program capabil
25、ity is summarized below. 1. The program is capable of optimizing the entire flight of an SST from a given low-speed, low-altitude condition at the start of the flight to a given low-speed, low-altitude condition at the end of the flight. The program will also optimize the climb, cruise, and descent
26、phases of a mission separately. 2. The program incorporates a 3-D point-mass simulation for a vehicle moving in relation to a spherical, rotating earth. The inverse-square law for gravity is used. The 1962 U.S. standard atmosphere is included as a single subroutine. 3. The optimization is accomplish
27、ed with an automatic step-size controller so that, in general, only one pass on the computer is required for a solution. The program will generate a nominal trajectory for starting the iterative procedure. Automatic plotting capability is included. STOP is sufficiently versatile so that other vehicl
28、es besides the SST may have their flight paths optimized. These vehicles include: space boosters, ICBMs, reentry vehicles, scramjets, and vehicles with air-augmented propulsion. Even problems com- pletely divorced from flight path optimization may be solved with a minimum of reprogramming. 4. Payoff
29、 functions, terminal constraints, and stopping parameters may be selected from the list of 40 state variables (defined by equations of motion and enroute placards) given in figure 1. The program will optimize any one of the state variables while simultaneously satisfying 14 terminal constraints (one
30、 of which is considered to be the stopping condition). Inequality con- straints, imposed by the user, are considered as terminal constraints. 5. Pitch angle, bank angle, wing sweep, and throttle setting are the control vari- ables for the program. The user may select any subset of these variables fo
31、r a given problem. 1 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-L VARIABLE VARIABLE NC PLOT FORTRAN SYMBOL DESCRIPTION . UNITS TYPE INDEX INDEX INDEX NAME STATE (EQUATION : 121 1 X(Kl) W WEIGHT LB 122 2 XlK2) H ALTITUDE Fr OF MOTION) 3 I23 3 xK3
32、t Y RELATIVE FLIGHT PATH ANGLE DEG 4 124 4 X(K4) RELATIVE VELOCITY FPS 5 I25 5 XW) ; LATITUDE ANGLE DEG 6 126 6 XlK6) 7 127 7 X(K7) !t“ RELATIVE HEADlNG ANGLE DEG LONGIlUDE ANGLE DEG a 128 a X(K8) TD DUMMY TIME SEC 9 9 iii 10 X(K9) RNG PATH RANGE ALONG EARTHS SURFACE N MI 10 XiKlD) AHI AERODYNAMIC H
33、EATING INTEGRAL F-T-LB/FT2 11 11 X(Kl1) A V IDEAL RELATIVE DELTA VELOCITY FPS 12 z: E X(K12) GL GRAVllY LOSS FPS I3 14 ;i 14 X(Kl3) DL DRAG LOSS FPS X(K14) TVL THRUST VECTORING LOSS FPS ii 135 15 X(K15) ER RELATIVE SPECIFIC ENERGY Fl 136 16 X(K161 :i 137 17 X(K17) 138 la X(K18) AVAILABLE FOR EXPANSI
34、ON 19 139 19 X(K19) 1 NOT DEFINED I 20 140 20 X(K2D) STATE ii 141 21 X(K21) 0“ PITCH ANGLE PLACARD = Fit) DEG2SEC (ENROUTE 142 22 X(KiZ) q+ BANKANGLEPLACARD =F(t) DEGSEC PLACARD) 23 143 23 X(K23) ?* THROTTLE PLACARD = F(H,M) SEC 24 144 24 X(K24) 11 WING SWEEP PLACARD = F(H,M) DEGSEC iii 145 25 X(K25
35、) (Y. ANGLE OFAllACK PLACARD - F1H.M) DEG2SEC 146 26 X(K26) NOT DEFINED 27 147 27 X(K27) HD ALTITUDE RATE PLACARD = F(t) FPS 28 148 28 X(K28) Q DYNAMIC PRESSUREPLACARD = F(t) ,PSF12 SEC :i 149 29 XlK291 DYNAMIC PRESSURE PLACARD * F(M) (PSFISEC 150 30 X(K30) L Q“PlACARD = F(M) (PSF-DEGj2SEC 31 31 : 3
36、2 X(K31) TEMT TOTAL TEMPERATURE PLACARD - F(t) I“R12SEC 32 X(K32) N NORMAL LOAD FACTOR PLACARD = F(H,M) ;4 33 z 34 X(K33) RPA“ RESULTANTPHYSIOLOGICALACCEL. PLACARD = Fit1 ZSECISEC X(K34) H ALTITUDE PLACARD = F(M) FT2SEC ;6 155 35 (35) AP SONIC BOOMOVERPRESSURE PLACARD nF(A,B) (PSFI2SEC 156 36 X(K36t
37、 M“ MACHNUMBER PLACARD =F(H) SEC ;B 157 37 xK37l NOT DEFINED si ; Xw38l Y“ GAMMA PLACARD =F(H,M) DEG2 SEC 39 X(K39) NOTDEFINED * 40 160 40 X(K40) NOTDEFINED -. - - -_-. .- FIGURE I: STATEVARIABLES -.- . -. .-.- _I. ._ Provided by IHS Not for ResaleNo reproduction or networking permitted without lice
38、nse from IHS-,-,-6. Aerodynamic and engine options are available for receiving the data in the forms commonly used for most types of vehicles. 7. Multistaged vehicles and those where external stores are jettisoned as a function of time may be optimized. 8. Inequality constraints may be imposed on pa
39、rameters affecting geopolitics, passenger comfort, control limitations, structural loads, and engine opera- tion. The geopolitical constraints include sonic boom overpressure and maximum and minimum altitude. 9. The initial conditions can be varied by the program to obtain increased per- fo rmance .
40、 This option eliminates the need for a preliminary search to deter- mine the neighborhood of initial conditions for an optimal flight path. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-INTRODUCTION The need to optimize transport aircraft flight pa
41、ths is becoming apparent because (IL) the next generation will include supersonic transports for which the climb and acceleration phase consumes an appreciable part of the vehicle fuel, and (2) a large part of the flight is constrained by enroute placards. The steepest-ascent method has been success
42、fully used to optimize rocket- boost trajectories, reentry-vehicle trajectories, and orbital transfers giving substantial performance gains. There have been numerous attempts to optimize flight paths of airbreathing interceptor-type aircraft with a relatively high thrust- to-weight ratio using the s
43、teepest-ascent technique (ref. 1). These methods have been successful to a degree, but, when applied to low-thrust-to-weight-ratio air- craft, have resulted in flight path instability problems that have made convergence difficult. Recognizing that, in the past, optimization techniques have produced
44、significant performance gains for many classes of vehicles, NASA initiated the present study under RFP L-5347 (ref. 2). The purpose of the study was development of a digital computer program that would optimize the flight path of a supersonic transport with realistic operational constraints. A suffi
45、cient number of cases would be run to substantiate the program and define the most appropriate flight paths for selected SST configurations. A computer program general enough to optimize the SST would have the capability to optimize trajectories for many classes of vehicles including rockets, air-au
46、gmented rot kets, airbreather vehicles, and gliders. The technique used for optimization is the steepest-ascent method given by Bryson (ref. l), which follows the direct approach of determining a maximum. The optimum flight path for the present study is the solution to the nonlinear dif- ferential e
47、quations of motion that satisfy the imposed constraints and maximizes or minimizes one of the state variables. The development of a computer program to satisfy the SST requirements and overcome problems associated with the flight path instabilities and complex engine characteristics required some ch
48、anges in the usual methods used in optimization program s. Significant contributions to the program were made by NASA/LRC personnel. The use of pitch angle as a control variable was suggested by J. R. Elliott as a technique to overcome the flight path instabilities. The formulation of jet engine data into a form compatible with the optimization procedure was suggested by W. E. Foss, Jr. This form of data uses thrust input as T/P and weight flow as W/P, where P is the ambient atmospheric pressure. Contribu- tion to the form of the SST aerodyn
