Supersonic transport trajectory optimization example solutions Final report

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NASA CONTRACT REPORT-_0 66247

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SUPE _''_ "'IC TRANSPORT

IND O I.N

TRAJECTORY OPTIMIZATION

EXAMPLE SOLUTIONS

f

Frenk

Matthews, and Joel W.

Stein, Malcolm L.

Lawrence H.

_N67 16774

{'i'i'i R U) (ACCESSION NIJ I_I_ F- R)

FINAL REPORT JANUARY 1961

o j,_/

(P,_.G ES) _TEGORY_ (NASA CR OR TMX OR AD NUI,I_._R)

GPO PRICE $

PREPARED UNDER CONTRACT NO. NASI-5293

:FSTI PRICE(S) $

Hard copy (HC)

Microfiche (MF)

WASH.

AEROSPACE GROUP SEATTLE,

I July 65

PREPARED F OR

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

SUPERSONIC TRANSPORT TRAJECTORY OPTIMIZATION

EXAMPLE SOLUTIONS

By Lawrence H. Stein, Malcolm L. Matthews,

and Joel W. Frenk

NASA Contractor Report No. 66247

Distribution of this report is provided in the interest of

information exchange. Responsibility for the contents

resides in the author or organization that prepared it.

Prepared under Contract No. NAS1-5293 by

THE BOEING COMPANY

Aerospace Group

Seattle, Wash.

for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

FOREWO RD

This report was prepared by the Missile andInformation System Division of

The Boeing Company, Seattle, Washington. It presents the final docmnentation

of the results obtainedfrom hvents_ sample check cases that were solved by the

SupersonicTransport Optimization Program (STOP). The program was developed

by Boeing for the Langley Research Center under contract NASI-5293. The hventy

cases included onemulti-stage boost-to-orbit problem, one variable-sweep

reentry problem, andeighteen SSTproblems. The contract was achninistered

by the National Aeronautics and SpaceAdn_inistration under the direction of

Mr. J. R. Elliott, with Mr. David F. Thomas, Jr. acting as contract monitor.

The SupersonicTransport 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. 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 solution of the

twenty check cases. Mr. J. W. Frenk was in charge of programming and was

assisted by Mr. D. A. Watson. Mr. Watson was also responsible for the

plotting eapability for the program. The work was perfomned under the diree-

tion of Mr. E. G. Haugseth of the Boeing Missile and Information System

Division.

This report, together with a companion docmnent, STOP_ A Computer

Program for Supersonic Transport Trajectory Optimization (Reference 1) plus

the FORTRAN source program listings, binary object deck, and ssnllbolic

object deck concludes the work prescribed under contract NAS1-5293.

ii

ABSTRAC T

STOP--A Computer Program for Supersonic Transport Trajectory

Optimization (Reference 1) was used to solve twenty opthnization problems. The

problems included 1 four-stage space booster mission, one optimal flight path

for a variable-sweep reentry vehicle, and eighteen SST missions involving both

fixed-wing and variable-sweep configurations. The SST missions involved

various combination_ of enreute constrah-_t_ _u account for engine limits, struc-

tural and cruise limits, and planar and area s0nic-boom over-pressure restric-

tions.

A discussion of each of the 20 problem solutions is presented in this report.

The problem statement, results, and a brief analysis are provided along with

a computer listing of each optimal trajectory.

°°.

III

CONTENTS

SUMMARY

-)

INTRODUCTION

TWENTY SAMPLE PROBLEMS

Boosters

Scout

Reentry Vehicles

Variable Sweep

Supersonic Transport

Calibration Run

Minimum Time

Maximum Final Weight

Maxinmm Range

C ONC LUSIONS

REC OMME NDAT IONS

REFERENCES

iv

SUPERSONIC TRANSPORT TRAJECTORY OPTIMIZATION

EXAMPLE SOLUTIONS

By Lawrence H. Stein, Malcolm L. Matthews, and Joel W. Frenk

Boeing Aerospace Group

SUMMARY

This volume is a summary of solutions to 20 optimm_ flight-path problems

formulated by NASA and required by contract NAS1-5293 utilizing the super-

sonic transport optimization program (STOP). The problems were selected

to demonstrate the capability of STOP and to provide optimum flight paths

for configurations and missions consistent with current research programs

at Langley Research Center.

Vehicle characteristics used for the 20 check cases were supplied by NASA/

LRC for the purpose of program checkout, and should not be construed as officially

representing any commercial or government SST design.

The 20 problems are divided into the following general categories: 1) A four-stage solid propellant space booster; 2) A variable-sweep reentry vehicle;

3) Eighteen SST missions involving bothifixed-wing and variable-sweep

configurations.

The SST missions incorporated various combinations of enroute constraints

involving engine limits, structural and cruise limits, and planar and area sonic-

boom overpressure restrictions.

The most significant results obtained from the solution to the 20 problems

are

summarized below:

1)

The variable-sweep reentry vehicle showed little or no tendency to sweep

the wings forward from the full aft position. Also, wing sweep appeared to

2) The variable-geometry SST problems all indicated that wing sweep was

a weak control variable for minimum-time-type problem s.

3) All variable-sweep SST cases showed a loop in the altitude-Mach plane

during descent. This can be attributed to the problem statement, which

required that more fuel be consumed than was required for the specified

range.

4) In general, enginebehavior was the same for all SSTcases that incorporated

at least a cruise Maeh number restriction. The schedulewas maximum

augmentationthrough the entire climb and acceleration phase, appro_mately

military thrust during cruise ( _ = 0.8 to 1.2), followed by idle thrust during

descent. One exception was noted in the area of M = 0.8 to 1. In this region,

all of the SST cases showed a thrust level considerably lower than the aug-

mentor limit. No explanation is presently available for this anomaly.

5) For the area boom problems, the aircraft preferred to fly around the cities, making the required heading changes as early in the mission as possible.

The banking maneuvers were gentle, in all cases resulting in bank angies of

about 1 degree or less.

Bccausc of the complexi_" of vehicle aerodynamics, engine characteristics,

and enroute placards, SST missions are quite susceptible to obtaining local opti-

mum flight paths. This problem can be corrected in many eases by starting from

a new nominal path. This was accomplished for many runs, but in some cases

computer time lin_itations prohibited carrying out the iterations to the final opti-

mum. In all cases a sufficient number of iterations were performed to establish

the convergence pattern and demonstrate the trends of the optimal flight path.

IN TRODU C TION

The supersonic transport optimization program (STOP) described in Volume I

of this report is a steepest-ascent digital computer program that will optimize the

flight path of a supersonic transport from a given low-speed, low-altitude condition

at the start of a flight to a given low-speed, low-altitude condition at the end of a

flight. The purpose of this volume is to present the results of a study to demon-

strate the capability of STOP to optimize the flight path of several types of vehicles

and to perform mission studies on fixed-wing and variable-sweep SST aircraft with

operational enroute constraints.

Classes of vehicles and the missions considered in this demonstration are:

I) Four-stage solid-propellant booster (Scout) flying a boost-to-orbit mission

with a maximum payload. A q (xenroute placard was applied

2) A variable-sweep-wing reentry vehicle returning from space to perform a

maximum lateral range maneuver.

3)

A fixed-wing SST performing missions of minimum time, maximum payload,

and maximum range. Enroute placards on sonic boom overpressure, struc-

tural and cruise Mach number, and engine limits are applied.

4)

A variable-sweep SST performing minimum-time missions. Enroutc placards

are imposed on sonic boom overpressure, structural Mach number limit, and

engine limits.

The aerodynamic and propulsion characteristics for the Scout are presented

in the discussion, but because the trends of all other aerodynamic and engine data

may represent the current state of the art, they are restricted and are not pre-

sented herein.

Persons with a "need to know" may contact David F. Thomas, Jr., of the

NASA/Langley Research Center--Space Mechanics Division for further informa-

tion.

Types of enroute placards applied in the 20 problems are:

I) Maximum and minimum throttle settings as functions of altitude and Mach

numb er; 2) Structural and cruise Mach number limits as functions of altitude; 3) Planar sonic boom overpressure limits as functions of longitude.

4) Area sonic boom overpressure limits as a function of latitude and longitude.

There are many additional placards available for use in STOP but these were

not required for the 20 problems specified.

There are several special-purpose options discussed in Reference i, all of

which were used in solving the specified problems: (I) gamma tilt, (2) circular

satellite, (3) maximum payload, (4) free initial condition, and (5) nominal-

trajectory generation. The first four options were used on Scout. The nominal-

trajectory generation option was used for the calibration flight with the fixed-wing

SST. The maximum-payload option with the free initial weight was used for one

of the SST problem s.

A discussion of each of the 20 problem solutions is presented in this volume.

The problem statement, results, and a brief analysis are provided. A computer

listing for each optimal trajectory is included.

TWENTY SAMPLE PROBLEMS

Boosters

Scout.--Case i: Scout is a four-stage solid-propellant VTO booster with

fixed times specified between each stage. This problem has a twofold purpose:

first, to demonstrate the versatility of STOP; and second, to provide a calibra-

tion case that can be verified by other optimization programs.

Problem statement and results: The specified mission placed a maximum

payload (_ = Wmax) into a 300-nautical-mile circular orbit. An enroute placard

was imposed on Q_ , namely

-1200 <Q_ _ 1200 psf-degrees

(i)

Table 1 presents a summary of the problem statement and results, and

figures 1 through 3 show the optimal trajectory.

Table i: Case l--Problem Statement and Results

Parameter Desired Value Optimum Value

= weight maximum 374.51 lb

t2 = velocity circular satellite 23,587 fps

• 1 = altitude 1,822,S31 ft 1,822,902 ft

_2 = flight path angle 0 degrees 0 degrees

Enroute Placard Figure Number

_3- Q cr PF 0 degrees 2

Engine characteristics, aerodynamic data, and weight statement are shown

in tables 2, 3, and 4.

Table 2: Scout En_ne Data

Time Tvac

Stage Ae Isp

1 813.6 251. 391

0 94,600

34 108,500

44 116,000

52 32,377

66 0

2 273.307 1166.4

0 48,24O

24 67,100

27 69,200

36 68,344 I

39 0

Table 2 (Continued)

Time Tvac

stage

Isp

0 21, i00 0 277.209

15 24,600

27 22,580

30 20,532 1

33 0

0 6550 278.337

3 4700

6 5600

12 6174

24 6174

Table 3:

= 5.25 ft 2)

Scout Aerodynamic Data (SRe f

C D

CL{_

M o

_CD/a CL 2

0 0.482 0 06943

O. 24435

0.5 0.533 0 06586 O. 26499

0.75 0. 657 0 06242

O. 27960

i. 0 1. 267 0 05139 O. 33961

1.1 1. 463 0 05312 0.32859

1.5 1.227 0 07159

O. 24381

2.0 1. 032 0 08992 O. 19410

2.5 O. 890 0 10327 O. 16900

3.5 O. 737 0 11898

O. 14669

5.0 O. 640 0 12995 O. 13430

2.0 1. 366 0

Coast

2.5 1. 137

3.0 0.981

3.5 0. 870

4.0 0. 794

4.5 0.741

Vacuum

Vac,Jum

Vac uu m

Table 4: ScoutWeight Statement

Time Weight

0 Liftoff stage 1 39,606.6

66 End-burn stage 1 18,144.6

66 Jettison stage 1 case 14,889.6

88 End coast 14,889.6

88 Start burn stage 2 14,889.6

127 End burn stage 2 6437.6

127 Jettison stage 2 case 4241.6

133 End coast 4241.6

133 Start burn stage 3 4241.6

166 End burn stage 3 1639.6

166 Jettison stage 3 case 883.6

598 Start burn stage 4 883.6

622 End burn stage 4 374.5

Earth model: A spherical rotating earth and the complete 1962 ARDC

Standard Atmosphere were used.

Initial conditions of the state variables are: Initial conditions:

weight, Wo = free initial condition

tilt angle, 7 T = free initial condition

flight path angle, v R = 90 degrees

altitude, h = 0 ft

velocity, V R = 0 fps

latitude, B - 37. 849°N

longitude, )t = 75.473%V

heading, _I, R = 0 degree (East)

Control variables: Pitch angle was used as the control variable.

Tilt maneuver: A tilt maneuver was specified for Scout as shown in

table 5. The value 71. 106 was determined by STOP.

Table 5: Tilt Schedule

Time ___7

0 9O

5 90

6 7 i. 106

Free initial conditions: Free initial conditions were allowed on the tilt

angle (TT) and the launch weight (Wo). Nominal and optimal values for the

free initial conditions are shown in table 6.

Free Initial Conditions

Table 6:

Optimum Value

Variable Nominal Value

71.106 degrees

YT 80 degrees

39,606.6 lb

Wo 39,512 Ib

Maximum payload option: The maximum payload option was exercised for

this problem so that the maximum allowable last-stage propellant of 509 pounds

would not be exceeded.

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Reentry Vehicles

Variable sweep, p Case 2: Flight path optimization for a variable-sweep

reentry vehicle further illustrates STOP's capability to obtain optimal solutions

for many kinds of vehicles. The vehicle optimized is unpowered and with fixed

weight.

lateral range for a variable-sweep reentry vehicle. Table 7 presents the desired

and optimum terminal values of the performance, constraint, and stopping para-

meters.

Table 7: Variable-Sweep Reentry Vehicle Results

Parameter Desired Value

Optimum Value

a_ = latitude maximum 34.1 degrees

= velocity 425 fps 425 fps

kI'l=altitude 15,000 ft 14,721 ft

_P = flightpath angle 0 degrees -0.272 degree

•4 = longitude 30 degrees 29.94 degrees

Significant results are presented in figures 4 through 8. The optimal trajec-

tory listing is given. The reentry flight path shows the typical "skip" character-

istics common to this class of vehicles. It is interesting to note that, during the

reentry path, the longitude (figure 5) reaches a maximum of 34 degrees before

returning to the terminal value. This is due to the energy management required

to meet all terminal conditions.

Earth model: A spherical rotating earth and the complete 1962 ARDC Stan-

dard Atmosphere were used.

Initial conditions: Initial conditions of the state variables are:

altitude, h = 300,000 ft

flight path angle, _R = 0 degrees

velocity, V R = 22,983.4 fps

latitude, _ = 0 degrees

heading angle, _R = 0 degrees (East)

longitude, k = 0 degrees

Also, the weight was constant at 12,100 pounds and SRE F = 252 sq ft.

Control variables: Pitch angle and bank angle were the control variables.

Sweepback angle, although originally designated as a control variable, was found

to be ineffective because of the nature of the aerodynamic data. An indication that

sweepback (A) was not a significant control variable came from the automatic

weighting matrix, which showed the effect of sweepback to be two or three orders

of magnitude less than either pitch or bank angle. The aerodynamic data was such

that the steepest-ascent procedure could not see any advantage to sweeping the

wings forward. Maximum range occurs near (L/D)max, and since this occurs at

A = 90 degrees hypersonically, the wings are maintained fully swept. At lower

Mach numbers, the lower sweep angles provide a higher L/Dmax; but, by the time

these numbers are attained, there is no opportunity to gain significant perform-

ante prior to reaching the stopping condition. In fact, it is suspected that the in-

creased profile drag associated with the lower wing sweeps is undesirable.

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Supersonic Transport

The SST check cases are designed to show the effects of realistic enroute

placards on the optimal flight path design for typical supersonic transport air-

craft. Both fixed-wing and variable-sweep aircraft are considered. Table 8

shows the initial conditions, enroute placards, control variables, terminal con-

straints, and the performance parameter for the 18 SST check cases.

The maximum and minimum throttling placard are given in figures 9 and 10.

The planar sonic-boom placard is used in problems involving only inplane flight

paths. Table 9 shows the maximum allowable overpressure on the ground in the

plane of the flight path.

Table 9: Planar Sonic-Boom Limit

Range, n. mi. Overpressure, psf

0 2

300 2

330 1.5

270O 1.5

2730 1.6

3O0O 1.6

The area sonic boom placards are used to represent overpressure restric-

tions near cities. Figure 11 shows the overpressure limit schedule [or the area

boom problems. This figure is a cross-section through thc center of the city.

The cities were defined to be circular in the present study. Case 13 is a single

area boom problem with the center of the city positioned at a range of 1500 nau-

tical miles and on the equator. Cases 14, 15, 16, and 17 are double area boom

problems for which city centers are located at 1000 and 2000 nautical miles down

range. The center of the first city is 60 nautical miles north of the equator and

the second is 60 nautical miles south of the equator. Nominal flight paths are

along the equator for the area boom problems.

Mach number placards of two types are applied: m

o

o o

o

o

o

o o

o

o

Z

1) The structural limit is given in table 10.

Table i0: Structural Placard

Maximum Mach Number

Altitude, ft

0 1.0

30,000 1.44

59,750 2.9O

90,000 2.90

and

2) A cruise Mach number limit of 2.7.

In some cases, the structural placard limiting the Mach number to 2.9 was

inadvertently initiated at 57,000-foot altitude instead of the prescribed 59,750 feet.

This error was not discovered until some of the flight paths had been optimized.

This discrepancy was not considered significant.

During most of the climb and acceleration phase for all SST cases, the throttle

was on the maximum-augmentation placard. In all cases run, however, the throttle

history experienced a dip during the ascent phase in the Mach number range of 0.8

to i. 0 for which no conclusions have been made. (The optimal throttle schedule

for all runs tended to be on the idle placard during the majority of the descent.)

A brief discussion of each of the check cases follows. The results appear as

plots of important parameters and, in each case, the optimal trajectory listing is

included. Computer time required for optimizing the SST cases was large and,

in some cases, the iterative procedure was stopped short of a true optimum. This

was considered justified because intent of the check cases was to demonstrate con-

vergence of the program and to establish trends for study purposes. Convergence

to the terminal constraints is demonstrated within the limits given in table II.

Table ii: Terminal Constraint Tolerances

Variable Tolerance

Weight i an lh

Altitude 500 ft

Velocity 50 fps

Latitude 0.01 deg

Tolerances on the penalty functions were established by examining the time

histories on the constrained variables to determine the degree of violation. Each

case was considered separately to establish the tolerance to be allowed.

Calibration run. --Case 3: The intent of this run was to fly a prescribed flight

path (fig_tre 12) to serve as a basis for verification of data handling and computing

techniques used in STOP. This case makes use of the nominal trajectory genera-

tion option. Five phases are used to generate the flight path.

The phases are as follows:

Phase 1- Fly a prescribed h-M during ascent (mode 6) to an altitude of 55,578 feet

Phase 2 --Fly _I = 0 using pitch control (mode 10) to an altitude of 65,000 feet

Phase 3--Fly IVl= 0 using throttling control (mode 12) to the end of cruise: i.e.,

until the weight equals 263,743 pounds

Phase 4-- Fly _'R :: - 0.1 deg,/sec (mode 4) decreasing to a _R of - 2 degrees

Phase 5- Fly a descent-prescribed h-M (mode 6) to a final altitude of 1500 feet

A comparison of the reference and STOP-generated trajectories may be seen

from plots of altitude versus Mach number and liftcoefficient versus time (figures

12 and 15). Figures 13 and 14 are plots of altitude and sonic boom versus range.

Figure 16 is a plot of throttling versus time. The listing of the basic trajectory

is included. The spikes observable in figure 15 are due to the technique used by

the closed-loop g_tidance option and have a negligible effect on performance.

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5O

Minimum time. -- The minimum-time problem fornmlation is related to the

economic operation of the SST because the direct operating cost (DOC) is dependent

on the block time. The flight path optimization usually performed on SST aircraft

is the maximum-range problem. Some of the flight-path trends shown in this set

of check cases may not, in general, be familiar to those more experienced in maxi-

mum-range problems. The greater difference occurs during descent because

minimum-time descents are made at a much higher average Mach number. Descents

are similar to those usually occurring under emergency conditions. These differ-

ences may be seen by comparing the resuit_ from cases 4 (mi,_in_ui-n .......... "_ '_"

LIIII_ c:tllk.l K_V (maximum range).

Fixed-wing SST--planar sonic boom: The planar sonic boom is defined for

those problems that are restricted to one plane of motion (no banking or lateral

excursions). This type of case presents the single most difficult problem encoun-

tered in the check cases, which is a result of the nonlinear nature of planar sonic

boom: sonic-boom violation appears as a "spike" during trajectory climb and

descent phases.

The far-field sonic-boom overpressure data were modified slightly from the

table suggested by NASA/LRC. The overpressure was defined to be zero at M = 1

with a linear buildup to the full value at M = 1.2. This modification allowed the

adjoints to observe the behavior of overpressure over a wider range in the area of

the spikes and resulted in a more linear behavior of the placard while maintaining

the correct simulation in the violation areas.

The baseline case for this set of runs is case 4 because it has the most realis-

tic set of placards. The placards imposed are maximum and minimum throttling

limits, planar sonic boom, and structural and cruise Mach number limits. Mini-

mum time for the optimal flight path is 8034 seconds. The performance penalty

due to the sonic-boom placard is given by the comparison with case 5 (same as 4

with sonic-boom placard removed), which has a minimum time of 7745 seconds, a penalty of 289 seconds.

The penalty due to the structural Mach number placard is only 13 seconds, as shown by comparing case 6 (same as case 4 with structural placard removed),

with a minimum time of 8021 seconds, with case 4. This difference is small since

the case 4 optimal flight was not significantly restricted by the structural placards.

Cases 4 and 6 showed somewhat different descent profiles but nearly identical per-

formance. The differences between cases 4 and 6 flight may represent a band for

which the performance is insensitive to the path. The effect of the cruise Mach

number placard may be determined by comparing cases 4 and 7. The only differ-

ence is that the cruise Mach number is limited to 2.9 for case 7 and 2.7 for ease

4. The performance penalty for the Mach 2.7 is 738 seconds. The total time for

case 7 is 7296 seconds. With no enroute constraints applied, except the engine

throttling limits, case 8 has a minimum time performance of 6426 seconds. The

maximum Mach number for this run was 3.94, which exceeds input aerodynamic

and engine characteristics. Validity of the data obtained by extrapolation beyond

Mach number 3 is questionable.

In all cases, when a sonic-boom placard was imposed, the overpressure limit

was violated slightly during both the climb and descent. The violations are small

and are not considered significant.

Case 4 problem statement and results: Table 12 depicts desired and optimum

values of the performance, stopping parameter, and constraints. Results of the

optimum trajectory are given in figures 17 through 20.

Table 12: Case 4-- Problem Statement and Results

Desired Value

Parameter Optimum Value

8034 sec.

minimum

= time

50 degrees

f_ = longitude 50 degrees

262,039 lb

262,120 lb

_2 1 = weight

1139 ft

15OO ft

_2 = altitude

376.9 fps

366.7 fps

_3 = velocity

Enroute Placards Figure Number

_4 = THROTTLE PF

_25 = BOOM PF

_6 = MACH PF

(structural and

cruise)

Pitch angle and throttling were used as control variables.

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Case 5 problem statement and results: Table 13 depicts desired and opti-

mum values for the performance, stopping parameter, and constraints. Results

of the optimum trajectory are given in figures 21 through 24.

Table 13 : Case 5-- Problem Statement and Results

Parameter Desired Value

Optimum Value

_0 = time minimum 7745 sec.

fl = longitude 50 degrees 50 degrees

41 = weight 262,120 lb 262,152 lb

42 = altitude 1500 ft 1025 ft

43 = velocity 366.7 fps 327.5 fps

Enroute Placards Figure Number

44 = THROTTLE PF 0 23

45 = MACH PF 0 21

(structural and

cruise)

Pitch angle and throttling were used as control variables.

Note that the flight path was on the structural Mach number placard during

descent (figure 21). Sonic-boom violation is greater during descent than boost.

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Case 6 problem statement and results: The problem statement and results

are presented in table 14. Included are desired and optimum values for per-

formance, stopping parameter, and constraints. Results of the optinmm tra-

jectory are given in figures 25 through 28.

Table 14: Case 6-- Problem Statement and Results

Desired Value

Parameter Optimum Value

= time minimum 8021 sec.

12 = longitude 50 degrees 50 degrees

41 = weight 262,120 lb 262,109 lb

42 = altitude 1500 ft 1199 ft

43 = velocity 366.7 fps 356.0 fps

Enroute Placards Figure Number

44 = THROTTLE PF 0

45 = BOOM PF 0

46 = MACH PF 0

(cruise)

Pitch angle and throttling were the control variables.

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Case 7 problem statement and results: The problem statement and results

are presented in table 15. Included are desired and optimum values for perform-

ance, stopping parameter, and constraints. Results of the optimum trajectory

are given in figures 29 through 32.

Table 15: Case 7-- Problem Statement and Results

Parameter Desired Value

Optimum Value

(D = time minimum 7296 see.

i'l = longitude 50 degrees 50 degrees

262,138 ib

41 = weight 262,120 lb

1411 ft

42 = altitude 1500 ft

364.8 fps

43 = velocity 366.7 fps

Enroute Placards

Figure Number

44 = THROTTLE PF 0

3O

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(structural)

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Case 8 problem statement and results: Table 16 depicts desired and opti-

Results of

nmm values of performance, stopping parameter, and constraints.

the optimum trajectory are given in figures 33 through 36.

Table 16: Case 8--Problem Statement and Results

Parameter Desired Value ()ptimum Value

co = time mininmm 6426 see.

f_ = longitude 50 degrees 50 (legrees

_1 = weight 262,120 lb 2_;2,114 lb

tI,2 = altitude 1500 ft 1545 ft

_3 = velocity 366.7 fps 366.3 fps

Enroute Placard Figure Number

_4 = THROTTLE PF 0 36

Pitch angle and throttling parameters were used as control variables.

Note that removal of Maeh number and some-boom placards permitted the

aircraft to fly out of the region of the input data. The performance gained for

this ease, due to data extrapolation, may not be meaningful.

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Variable-sweep SST--planar sonic-boom cases: The planar sonic-boom

problems in this set are formulated similar to those for the fixed-wing SST.

The variable-sweep aircraft did not experience the same difficulty in meeting

the overpressure limits as did the fixed-wing SST. The descent phase for all

the variable-sweep cases exhibited the peculiar characteristic of a dive-climb,

statement formulation, which calls for a flight path of minimum time with ter-

minal constraints on weight, velocity, altitude, and range. The problem appears

to be overconstrained, since the variable-sweep aircraft arrive at the end of

cruise with too much fuel to be used during a normal descent. The dive-climb

maneuver is therefore the manner in which the program optimally meets all

c o nst raints.

A comparison of performance of this set of cases does not show the consist-

eacy of the fixed-wing data for similar problem statements. The problem appears

to be that the variable-sweep aircraft flight paths are more susceptible to local

optimums.

The baseline run for variable-sweep aircraft with the planar boom is case 9,

which has a structural Mach number placard and the engine placards, in addition

to the sonic-boom limit. The minimum time for the flight path was 8700 seconds.

The performance penalty associated with sonic boom is 395 seconds as shown by

comparison with case 10, which had a minimum time of 8305 seconds. The

structural Mach number placard (case 11 with minimum time of 8018) penalized

the performance by 682 seconds. The penalty due to the sonic-boom placard

is a result of forcing a higher altitude during the supersonic parts of the climb

and descent phases. The penalty due to the structural Mach number limit is

essentially a result of restricting the cruise Mach number. Case 12, uncon-