Evaluation of Aircraft Performance Algorithms Thesis

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Evaluation of Aircraft Performance Algorithms in Federal Aviation

Administration's Integrated Noise Model

by

Wei-Nian Su

B.S. Aerospace Engineering, Iowa State University, 1996

Submitted to the Department of Aeronautics and Astronauticsin partial fulfillment of the requirement for the Degree of

Master of Science in Aeronautics and Astronauticsat the

Massachusetts Institute of Technology

February, 1999

©1999 Massachusetts Institute of Technology. All rights reserved.

Author...........

Department of Aeronautics and AstronauticsJanuary 14, 1999

Certified b y .......................... ... .............. ......... .......... V ... o....... .. .. . . ..C r

Certe b( (Professor John-Paul ClarkeDepartment of Aeronautics and Astronautics

Thesis Supervisor

A ccepted by ................................................................... ..... ....... .. ...

Professor Jaime Peraire

Chairman, epartment Graduate Committee

MASSACHUSETTSINSTITUTEOF TECHNOLOGY

MAY 1 7 1999

LIBRARIES

-oWWW

.................... ,. ....... .. ... ............................................................................



Evaluation of Aircraft Performance Algorithms in Federal Aviation

Administration's Integrated Noise Model

by

Wei-Nian Su

Submitted to the Department of Aeronautics and Astronautics Engineeringon January 14, 1999 in partial fulfillment of the requirement for the Degree of Master of Science

in Aeronautics and Astronautics Engineering

Abstract

The Integrated Noise Model (INM) has been the Federal Aviation Administration's (FAA)standard tool since 1978 for determining thepredicted noise impact in the vicinity of airports.

A review of the aircraft performance algorithmsin the INM was conducted and improvedmodels for true airspeed, takeoff/climb thrust, level-flight thrust, and climb performance weredeveloped. The true airspeed model with air compressibility correction provides an accurateprediction over a wide range of operating conditions. The quadratic takeoff/climb thrust model asa function of Mach number, altitude, and temperature and the level-flight thrust model derived fromthe minimum -thrust-flight condition providean accurate prediction within considered airspeed and

altitude range. The climb models for constant equivalent/calibrated airspeed as well as constantclimb rate climbs introduce the flight path angle correction factor as a function of altitude, airspeed,and temperature as opposed to constant correction factor usedin INM.

Comparison of flight profiles predicted by the proposed methods and INM with the flightprofiles provided by the Delta Airlines shows that the errors in overall ground distance traversedaswell as noise contour shapes are reduced by implementing the proposed models.

Thesis Supervisor: Dr. John-Paul ClarkeTitle: Charles Stark Draper Assistant Professor of Aeronautics and Astronautics



Acknowledgments

Over the past two years, I have met many people who have made my time at MITworthwhile. I would like to take this moment to express my deep gratitude to those who have madeit possible for me to achieve this accomplishment. In particular, I would like to extend myappreciation to the following individuals and organizations.

First of all, I would like to thank my advisor, Prof. John-Paul Clarke, for his encouragementand guidance throughout this research. I also like to thank Mr. Gregg Fleming from Volpe NationalTransportation Systems Center for funding my research. In addition, I also like to thank Mr. JimBrooks from Delta Airline for providing valuable data.

Finally, I would like to thank my family members: my father, Mr. Shih-Ping Su, my mother,Mrs. Yue-Ching Lin, my sister, Yua-Hwa Su, my brother, Wei-Ping Su, and my girlfriend, MissShine-Yi Wong, for their love and support throughout my study at MIT.

This research was funded by Volpe National Transportation Systems Center, U.S.Department of Transportation, and performed in the Flight TransportationLab.



Contents

A bstract........................................................................................ ........................................... 2........

A cknow ledgm ents..................................... ................................................................................... 3Contents........................................................................................ ........................................... 4.......

List of Tables..................................................................................... .............................................

List of Figures.................. .................................................................................................. 10

Nom enclature....................................................................................... ........................................ 12

Chapter 1. Introduction......................................................... ................................................. 15

1.1 Background of INM .................................................................................. 151.2 M otivation............... ......................................................................................... 15

1.3 Overview of Thesis............................................................................................................16

Chapter 2. Atmospheric Model and True Airspeed Model.....................................18

2.1 Standard A m osphere.................................................. ............................................ 18

2.1.1 INM 's Atmospheric M odel............................... . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 A rspeed Measurement................................................................................................202.2.1 Previous Work................................................ .......................................... 21

2.2.2 True Airspeed M odel................................................. .......... ........... 22

2.3 Conclusion of Chapter 2................................................ .......................................... 24

Chapter 3. Takeoff and Clim b Thrust M odel................................................ ......... ......... 25

3.1 Previous W ork............................................................. ................................................ 25

3.2 Quadratic Thrust Model..................................................... .......................................... 26

3.3 Evaluation of Coefficients.......................................................................................... 31

3.3.1 Ante-Break Equa tion...................................................... ............................. 31

3.3.2 Post-Break Equation................................................. ........... ............ 33

3.4 Validation.................................................................................. ...................................36



3.4.1 Graphical Comparison....................................................................................36

3.4.2 Error Analysis..................................................... .......................................... 43

3.5 Conclusion of Chapter 3..................................................... ..................................... 45

Chapter 4. Level Flight Thrust Model...................................................................................46

4.1 Previous Work............................................................. ................................................ 46

4.2 E quation of Motion......................................................... ............................................. 47

4.3 Drag Polar............................................................................................. . . .......... 48

4.3.1 Drag Polar Model I.. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . .49

4.3.2 Drag Polar Model II...........................................................50

4.3.3 Effects of ReynoldsNumber on Drag Polar.....................................50

4.4 Level-Flight Thrust......................................................... ............................................. 51

4.4.1 Level-FlightThrust Model I.................................................. 52

4.4.2 Level-Flight Thrust Model II.................................................54

4.5 Validation of Level-Flight Thrust Models.............................. ............... 54

4.5.1 Comparison of Proposed Models with INM Model........................................54

4.5.2 ErrorA nalysis..................................................... .......................................... 57

4.5.3 Pro and Con Between Models............................... ......... ........ 60

4.6 Conclusion of Chapter 4................................................. .......................................... 61

Chapter 5. Climb Perform ance.................................................. ............................................ 62

5.1 Previous Work............................................................. ................................................ 62

5.2 Equation of Motion and Flight Path Angle Correction Factor................................ 63

5.3 Evaluation of Flight Path Angle Correction Factor................................. ...... 65

5.3.1 Constant Equivalent Airspeed Climb Model...........................................66

5.3.2 Exact Constant Calibrated Airspeed Climb Model.....................................67

5.3.3 Simplified Constant Calibrated Airspeed Climb Model................................ 68

5.4 Graphical Comparison of Flight Path Angle Correction Factor.............................. 71

5.5 Calculation of Flight Path Angle and Ground Distance Traversed........................... 72



5.6 Error Analysis ................................................................................................................ 72

5.6.1 ConstantEquivalent Airspeed Climb M odel................................. ..... 72

5.6.2 Constant Calibrated Airspeed Climb M odel................................. ..... 74

5.6.3 Discussion................................................... ............................................. 75

5.7 Conclusion of Chapter 5................................................ .......................................... 75

Chapter 6. Accelerated Climb Performance.............................................. ........... ........... 76

6.1 Previous W ork............................................................. ................................................ 76

6.2 ConstantClimb Rate Acceleration..................................................78

6.3 Error Analysis............................................................. ................................................. 80

6.4 Conclusion of Chapter 6.................................................... .......................................... 82

Chapter 7. Comparison of Departure Profile and Noise Contour................. ... 83

7.1 Description of Ana lysis..............................................................................................83

7.2 Boeing 727-200............................................................ ................................................ 84

7.2.1 ProcedureSteps................................................... ..........................................84

7.2.2 Flight Profile and Noise Co ntou r............................................ 85

7.2.3 Error Analysis..................................................... .......................................... 93

7.3 Boeing 737-3B2............................................................ ............................................... 95

7.3.1 ProcedureSteps...................................................................................................96

7.3.2 Flight Profile and Noise Contour............................................ 96

7.3.3 Error Analysis................................. 100

7.4 Boeing 757-200................................................................................................................101

7.4.1 Procedure Steps.................................................. ......................................... 102

7.4.2 Flight Profile and Noise Contour.............................. 103

7.4.3 Error Analysis................................. 106

7 .5 Discu ssion ............................................................................... .................................. 108

7.6 Conclusion of Chapter 7.................................................................... .................... 108

6



Chapter 8. Conclusion and Future W ork....................................................109

8.1 Conclusion of Thesis................................................................................................. 109

8.2 Future W ork................................................................................................................109

Bibliography......................... ........................................................................................... 110



List of Tables

2.1 Average error in true airspeed for MIT and INM models at standard day............................24

3.1 Ante-break corrected takeoff thrust (Fn/6) versus Mach number and pressure altitude ......... 313.2 Post-break corrected takeoff thrust (F,/8) versus Mach number, pressure altitude, and

temperature............................................ .................................................................................... 33

3.3 Error in corrected net thrust during takeoff for small commercial airplane............................ 43

3.4 Error in corrected net thrust during climb for small commercial airplane............................ 44

3.5 Error in corrected net thrust during takeoff for medium commercial airplane..........................44

3.6 Error in corrected net thrust during climb for medium commercial airplane............................44

3.7 Error in corrected net thrust during takeofffor large commercial airplane............................ 45

3.8 Error in corrected net thrust during climb for large commercial airplane......................................45

4.1 Average level-flight thrust errors per engine for small commercial jet at 5000 ft.................58

4.2 Average level-flight thrust errors per engine for large commercial jet at 5000 ft..................59

4.3 Pro of level-flight thrust model I and model II...............................................60

4.4 Con of level-flight thrust model I and model II................................................60

5.1 Error in ground distance during constant equivalent airspeed climb starting from sea level........73

5.2 Error in ground distance during constant equivalent airspeed climb starting from 5000 ft......73

5.3 Error in ground distance during constant calibrated airspeed climb starting from sea level.....74

5.4 Error in ground distance during constant calibrated airspeed climb starting from 5000 ft.......74

6.1 Error in altitude gain and ground distance traversed for the small commercial airplane...........81

6.2 Error in altitude gain and ground distance traversed for the large commercial airplane.............81

7.1 Flight procedure for Case (1) and (2)............................................................... .................. 84

7.2 Flight procedure for Case (3) and (4)............................................................... .................. 85

7.3 Overall ground distance error in feet for Case (1) to (4)..................................... . . . . . 94

7.4 Error in noise impact area in square mile for Case (1) and (2)................................. .... 94

7.5 Error in noise impact area in square mile for Case (3) and (4)..................................................94

7.6 Error in closure point distance in nautical mile for Case (1) and (2)................................. 95




7.8 Flight procedure for Case (5) and (6)................................................................ ................. 96

7.9 Overall ground distance error in feet for Case (5) and (6).........................................101

7.10 Error in noise impact area in square mile for Case (5) and (6)................................. ....101


7.12 Flight procedure for Case (7).................................................. ........................................... 102

7.13 Flight procedure for C ase (8).................................................... ...........................................102

7.14 Overall ground distance error in feet for Case (7) and (8).................................. ..... 107

7.15 Error in noise impact area in square mile for Case (7) and (8)..................................... 107

7.16 Error in closure point distance in nautical mile for Case (7) and (8)................................ 107



7.4 SEL noise contour for C ase (2)................................................ ........................................... 89

7.5 Flight profile and LAMAX noise contour for Case (3)...........................................90

7.6 SEL noise contour for C ase (3).............................................................................................. 91

7.7 Flight profile and LAMAX noise contourfor Case (4)..........................................92

7.8 SEL noise contour for Case (4)...........................................................................................93

7.9 Flight profile and LAMAX noise contourfor Case (5)..........................................97

7.10 SEL noise contour for Case (5).............................................................................................. 98

7.11 Flight profile and LAMAX noise contour for Case (6)..........................................99


7.13 Flight profile and LAMAX noise contour for Case (7)................................... ..... 103


7.15 Flight profile and LAMAX noise contour for Case (8).........................................105

7.16 SEL noise contour for C ase (8)............................................................................................ 106



Nomenclature

AIR - Aviation Information Report.

DFBR - Distance from brake release.FAA - Federal Aviation Administration.

INM - Integrated Noise Model.

MCLT - Maximum climb thrust.

MGLW - Maximum gross takeoff weight.

MGTOW - Maximum gross landing weight.

MTOT - Maximum takeoff thrust.

SAE - Society of Automotive Engineers.SLD - Satellite distance.

CD - Drag coefficient.

CDRin - Drag coefficient at minimum drag-over-lift point.

CL - Lift coefficient.

CLRmin- Lift coefficient at minimum drag-over-lift point.

D - Drag.

F - Total thrust which is equal to the number of engines times the net thrust per engine.

Fn - Net thrust per engine.

Fn/8 - Corrected net thrust per engine.

g - Gravitational constant.

h - Pressure altitude above the sea level.

hi - Altitude at the beginning of climb.

h 2 - Altitude at the end of climb.

hairport - Airport elevation.

hd - Density altitude above sea level.

L - Lift.

M - Flight Mach number.



N - Number of engines.

P - Ambient air pressure.

Po - Ambient air pressure at sea level, standard day condition.

Pairpor - Ambient air pressure at the airport.

R - Drag-over-lift ratio.

Ra - Gas constant.

Re - Reynolds number.

Run - Minimum drag-over-lift ratio.

S - Reference area.

Sa - Ground distance traversed during acceleration.

Sc - Ground distance traversed during constant calibrated airspeed climb.

T - Ambient air temperature.

To - Ambient temperature at sea level, standard day condition.

Tairpor - Ambient temperature at the airport.

TISA - Standard day ambient air temperature.

Vao - Speed of sound at sea level, standard day condition.

Vc - Calibrated airspeed.

Ve - Equivalent airspeed.

Vt - True airspeed.

Vta - True airspeed at the beginning of acceleration segment.

Vtb - True airspeed at the end of the acceleration segment.

Vtz - Climb rate.

VtR, - True airspeed corresponding to minimum-level-flight-thrust condition.

V, - Headwind velocity.

W - Aircraft weight.

c - Mean cord length of the wing.

y - Flight path angle.

Ya - Ratio of specific heat at constant pressure to specific heat at constant volume for air.



6 - Ratio of the ambient air pressure at the airplane to the air pressure at mean sea level.

0 - Ratio of the ambient air temperature at the airplane to the air temperature at mean sea level.

p - Viscosity coefficient.

- Flight path angle correction factor.

p - Ambient air density.

po - Ambient air density at sea level, standard day condition.

o - Ratio of the ambient air density at the airplane to the air density at mean sea level.



equivalent airspeed was valid. This is not true, as the air compressibility effect is no longer

negligible at high airspeed and altitude.

The takeoff and climb thrust prediction methodology described in SAE AIR 1845 was

developed for operations from airports at sea-level on a standard day, and therefore only considered

the flight profile up to 3000 ft above sea-level. A linear thrust model as a function of calibrated

airspeed, altitude, and temperature was thus adopted to calculate takeoff and climb thrust. Once the

operating altitude gets beyond 3000 ft, this linear model is no longer valid and the induced error

increases dramatically.

The current method for computing level-flight thrust involves inverting the expression used

to determine the flight path angle. Implicit in the expression for the flight path angle however, is the

assumption that the drag-over-lift ratio remains approximately constant regardless of aircraft weight

and speed. This is valid during climb as the goal of achieving altitude quickly dictates that the

airplane operates at near minimum drag-over-lift ratio, and thus maximum flight path angle. In level

flight at constant speed however, the thrust is a strong function of aircraft speed.

INM uses a correction factor to account for changes in the flight path angle associated with

headwinds and the acceleration/deceleration inherent in both of constant calibrated airspeed climb

and constant climb rate climb. Currently, this correction factor assumes that the change in flight path

angle that can be attributed to accelerated climb/descent is constant. This is not true, as the change

in flight path angle attributable to accelerated climb/descent is a function of pressure altitude and

flight airspeed.

Thus, improved models which correctly account for the areas where the existing models are

deficient as described above are required.

1.3 Overview of Thesis

This thesis covers the development of a new true airspeed model, a new takeoff and climb

thrust model, a new level-flight thrust model, an improved flight path angle model, and a new

constant climb rate climb methodology. The true airspeed model along with the description of

INM's atmosphere model is presented in Chapter 2, the takeoff and climb thrust model is presented



in Chapter 3, the level-flight thrust model is presented in Chapter 4, the flight path angle model is

presented in Chapter 5, and the constant climb rate climb methodology is presented in Chapter 6.

The comparisons of flight profile and noise contour betweenproposed method, INM, and measured

data are presented in Chapter 7. Finally, the conclusion of overall analysis is presented in Chapter

8.



Chapter 2. Atmospheric Model and True Airspeed Model

An atmospheric model that provides information about the flight environment is a necessary

tool for aircraft performance analysis. However, the standard atmosphere is a reference model only,thus it must be modified to take into account nonstandard day condition.

INM takes calibrated airspeed as one of the input parameters. An airspeed model which

accurately converts calibrated airspeed to true airspeed over a wide range of operating conditions is

needed

In this chapter, discussion of INM's atmospheric model is provided and the true airspeed

model which accounts for compressibility effect is introduced.

2.1 Standard Atmosphere

In 1920, the Frenchman A. Toussaint, director of the Aerodynamic Laboratory at Saint-Cyr-

l'Ecole, France, suggested a linear relationship between temperature and height. Toussaint's formula

was formally adopted by France and Italy in March 1920 and one year later, the NACA adopted

Toussaint's formula for airplane performance testing. With the advent of aerospace technology such

that high altitude flight as well as space flight became possible in late 1959, newtables of the

standard atmosphere were created by Air Research and Development Command (ARDC) which is

now the Air Force Systems Command [3].

Several different standard atmospheres exist all using slightly different experimental data in

their models, but the difference is insignificant below 100,000 ft. A standard atmosphere model in

common use today is the 1959 ARDC Model as shown below,

dTT, + (h - h( )T dh (2.1)

T, T 1



-g

P T Ra dT

p_ PT

p1 P T9 P

(2.2)

(2.3)

where the subscript 1 stands for the atmospheric condition at base altitude and dT/dh is the

temperature lapse rate.

2.1.1 INM's Atmospheric Model

The standard atmosphere is a reference model only and certainly does not predict the actual

atmospheric properties at a given time and place, thus modifications of Eq.(2.1) and (2.2) based on

the knowledge of given airport conditions to account for nonstandard day condition are required [4].

Tairport + (h - hairport)dh (2_ dh (2.4)

T. + (h - h dTT RgSairport airport dh a d T

T0

(2.5)Pairport - Po

Po

(2.6) =



2.2 Airspeed Measurement

A Pitot-static tube which measures the difference between the total pressure and static

pressure is commonly implemented on the airplanes to measurethe airspeed. Depending on the type

of airplanes, the airspeed reading from such measurement can be equivalent airspeed or calibrated

airspeed. For lower-speed airplanes such as small, piston engine airplanes, the airspeed readings can

be considered as equivalent airspeed, by contrast, for higher-speed airplanes such as commercial jet

transports, the airspeed readings are calibrated airspeed.

The airspeed is called low or high depending on the flight Mach number,

VM = (2.7)

V

where

V= YaRaT (2.8)

If Mach number is less than 0.3, the airflows are considered as incompressible and then the

equivalent airspeed is read. If Mach number is greater than 0.3, the compressibility must be taken

into account and then the calibrated airspeed is read. The relationship between equivalent airspeed

and calibrated airspeed is given by the following equation [5],

1 Ya - 1

V 2 P Ya (k + 6) Ya - 6]2 (2.9)e Ya- 1 po

where

-1 V2Vk + Ya a (.0



The true airspeed is obtained by dividing Eq.(2.9) by the square root of density ratio.

VVt e (2.11)

2.2.1 Previous Work

In INM, the effect of air compressibility was ignored. Instead of using Eq.(2.11), INM

assumes that the calibrated airspeed is the same as the equivalent airspeed at low airspeed and low

altitude operation [2].

VV, c (2.12)

Figure 2.1 shows the comparison between exact and INM models at standard day condition.

As the figure shows, the error increases as the flight altitude and airspeed increase, thus an improved

airspeed model is desired.



0

5300

250

w 2002

0 50 100 150 200 250 300 350Calibrated Airspeed, Vc (knots)

Figure 2.1 Comparison of exact and INM models at standard day condition.

2.2.2 True Airspeed Model

The exact equation for true airspeed as a function of calibrated airspeed and pressure ratio

is very complicated, thus an approximate model which improves the accuracy while reducing the

complexity of the exact equation is desirable. If a correction factor as a function of calibrated

airspeed and altitude were introduced in Eq.(2.12), the accuracy of true airspeed prediction would

be improved.



The airspeed model is determined by the following relationship,

(1 + Eh)V cv,- (2.13)

where E s a constant coefficient with value of -1.0925E-6 1/ft determined by Least Square method.

Figure 2.2 shows the comparison of the exact model, Eq.(2.1 1 , and MIT model, Eq.(2.13),

at standard day condition. As the figure shows, the errors at high speed and altitude are reduced with

the implementation of the correction factor.

0"0 50 100 150 200

Calibrated Airspeed, Vc (knots)

250 300 350

Figure 2.2 Comparison of exact and MIT models at standard day condition.



Table 2.1 shows the average error in true airspeed for the MIT and INM models at standard

day condition. The errors were calculated for calibrated airspeed ranging from 150 knots to 350

knots at sea level, 10000 ft, and 20000 ft respectively. As the table shows, the average errors for the

MIT model are less than half of INM's errors.

Table 2.1 Average error in true airspeed for MIT and INM models at standard day.

II MIT Model (knots) INM Model (knots)

Sea Level 0.0395 0.0395

10000 ft 1.0104 2.5503

20000 ft 2.5950 7.5404

2.3 Conclusion of Chapter 2

As the analyses show, the proposed true airspeed model provides a more accurate prediction

of true airspeed than the existing INM model. In addition, the valid operating condition for the

proposed model is wider than the current model, thus, the accuracy for any subsequent aircraft

performance calculation will be improved.



Chapter 3. Takeoff and Climb Thrust Model

During takeoff and climb operations, the maximum thrust that an aircraft may use is afunction of operating altitude, temperature, and velocity. These maximum thrust values are defined

as the Maximum Takeoff Thrust (MTOT) and the Maximum Climb Thrust (MCLT) respectively.

In this chapter, a quadratic thrust model is introduced that describes the MTOT and MCLT

as a function of pressure altitude, flight Mach number, and ambient temperature. Comparison of

measured thrust data to the thrust values predicted at varying flight conditions confirms that the

quadratic model provides a good fit within the considered flight envelope.

3.1 Previous Work

In SAE AIR 1845, corrected net thrust is determined by a linearized expansion of the thrust

at sea-level standard day conditions which is a function of calibrated airspeed, pressure altitude, and

temperature [2],

F - ) = E + FV c + Gh + HT (SAE Eq. Al)

where E, F, G and H are constant coefficients to be determined by manufactures.

As opposed to SAE AIR 1845, INM expands SAE Eq. Al to a quadratic estimate for the

altitude term and uses density altitude, hd, instead of pressure altitude [4],

F = 8 hd) E + FV c GAhd GBh d + HTISA(hd)) (3.1)

hd 51867 (1 - (h) 5.256-1) (3.2)0.003566

where E, F, GA , GB , and H are constant jet coefficients.



3.2 Quadratic Thrust Model

While the relationship between corrected net thrust and the relevant flight conditions may

be linear near the reference conditions (standard day sea level), that assumption is inaccurate at high

altitude and high flight velocity. In addition, the thrust gradient with respect to calibrated airspeed

varies with altitude which provides a poor evaluation of airspeed dependent coefficients. A

quadratic thrust model as a function of pressure altitude, flight Mach number, and ambient

temperature was found to provide an improved match betweenpredicted thrust and measured thrust.

The thrust model is determined by the following relationship,

(Fn )A = ko + kM + k2M2

+ k3 h + k4h2

(3.3)

(Fn PB = k5 + k 6 M + k 7 M 2 + k8 T (3.4)

Fn/6 = Min[ FnI)F , (Fn,/)NF (3.5)

where the subscript AB and PB stand for ante-break and post-break respectively and k s are constant

coefficients.

Eq.(3.3) calculates the ante-break thrust value while as Eq.(3.4) calculates the post-break

thrust value and the minimum of these two values is the correct thrust at the corresponding flight

condition.



Figure 3.1 shows a typical surface plot for Eq.(3.3) and Eq.(3.4). As the figure shows,

Eq.(3.3) constructs the horizontal surface while holding altitude to be constant and Eq.(3.4)

constructs the tilt surface.

x 104

5..

4,

S3.5I-

2.50.8

0-6100

Mach Number o -100Temperature (F)

Figure 3.1 Typical plot for Eq.(3.3) and (3.4) at an arbitrary altitude.



Figure 3.2 shows the advantage of using flight Machnumber as the independent variable

instead of calibrated airspeed. As the figure shows, the gradientof corrected net thrust with respect

to Mach number is constant over the entire range of altitude while thegradient of corrected net thrust

with respect to calibrated airspeed is not.

x 104

X 104

3-

2 2.5

zU,

O 2o

o

0 0.1 0.2 0.3Mach Number

0.4 0.5 0 50 100 150 200 250 300Calibrated Airspeed, VC (knots)

Figure 3.2 Effect of flight Mach number and calibrated airspeed on corrected net thrust value.



Figure 3.3 shows the quadratic relationship of thrust in pressure altitude.

shows, the linear approximation in SAE is limited to altitudes below 4000 ft.

Pressure Altitude(ft)10000

As the figure

15000

Figure 3.3 Corrected net thrust vs. altitude at Mach 0, 0.2, and 0.4.

x 104a,

3.2

2.8

2.6-I-

2.4z

S2.2o

1.8

1.6

Mach 0.2

Mach 0.4

5000I

F



Figure 3.4 shows the typical plot of thrust versus temperature. As the figure shows, the

curves collapse together regardless altitude after the engine break temperature which justifies the

independence of altitude for computation of the post-break thrust value in Eq.(3.4). The engine

break temperature at a specific altitude and Mach number is the point where the thrust value

decreases as temperature increases.

x 104

0.5-100 -50 0 50 100

Temperature CF)

Figure 3.4 Corrected net thrust vs. temperature at Mach 0 and various altitudes.



3.3 Evaluation of Coefficients

The coefficients for Eq.(3.3) and Eq.(3.4) are determined by the method of Least squares overthe desired ranges of flight Mach number and pressure altitude.

3.3.1 Ante-Break Equation

Table 3.1 shows an example of the measured data required to compute the coefficients for

Eq.(3.3). The first columns and first row define the pressure altitude and flight Mach number

respectively under which the data corresponding to ante-break corrected net thrust value was

obtained.

Table 3.1 Ante-break corrected takeoff thrust Fn/6) versus Mach number and pressure altitude.

Altitude M1 = 0 M 2 = 0.1 M 3 = 0.2 M 4 = 0.3 M 5 = 0.4 M 6 = 0.5

h (ft)

h1 =0 32382 30442 28773 27372 26242 25380

h2 = 1000 32719 30780 29110 27710 26579 25718

h3 = 2000 33041 31102 29432 28032 26901 26040

h4 = 3000 33348 31408 29739 28338 27208 26346

h5= 4000 33639 31700 30030 28630 27499 26638

h6= 5000 33915 31976 30306 28906 27775 26913

h 7 = 6000 34176 32236 30567 29166 28035 27174

h 8= 7000 34421 32482 30812 29411 28281 27419

h 9 = 8000 34651 32711 31042 29641 28511 27649

hi = 9000 34865 32926 31256 29856 28725 27864

hi = 10000 35064 33125 31455 30055 28924 28063



The coefficients for Eq.(3.3), ko, kl, k 2, k3, and k4, are computed by formulating the matrices

A1 and B1 as follows,

2

1 M2 M

1 M6 M6

1 M M,2

1 M6 M

6 6,M,

h1 h2

h, h 2

h i h 1

h 2 h 2

h 2 h 2

2 21 M, M2 h11 hl

1 M 6 M2 h l

(Fn/ 6 )(Mh) F /8 )(M )(Fnl)(M2,hl)

(Fn/ 5 )(M6,hl)

(Fn )(M,h2)

(Fn/8)(M6,h2)

(Fn/6)(M6,hl )

(Fn8)(6,h,j)

(3.6)

and then solving Eq.(3.7) below for ko, k1,, k 2 , k 3 , and k 4 .

A T )- (A T B ) (3.7)



3.3.2 Post-Break Equation

Table 3.2 shows an example of the measured data required to compute the coefficients for

Eq.(3.4). The first two columns of the table define the conditions under which the data was obtained

(altitude and the corresponding temperature). The following columns give the corrected net thrust

at different flight Mach numbers.

Table 3.2 Post-break corrected takeoff thrust (Fn/8) versus Mach number, pressure altitude, and

temperature.

Altitude Temperature M = 0 M 2= 0.1 M 3 = 0.2 M 4 = 0.3 M 5 = 0.4 M 6 = 0.5

h (ft) T (oF)

1=0 T 6 29227 26894 24747 22787 21013 19427

,=0 T 2 = 9 28314 25981 23834 21874 20101 18514

S= 0 T 3 =111 27471 25138 22991 21031 19258 17671

h =0 T = 122 26699 24366 22219 20259 18486 16899

2=1000 T 5 = 2 29508 27174 25028 23068 21294 19707

2=1000 T 6 = 5 28595 26262 24115 22155 20381 18795

2=1000 T 7 = 107 27752 25419 23272 21312 19539 17952

h2=1000 T 8 = 118 26980 24647 22500 20540 18766 17180

3=2000 T9 = 79 29718 27385 25238 23278 21505 19918

3=2000 To = 92 28806 26472 24326 22366 20592 19005

3=2000 T 11 = 103 28033 25700 23553 21593 19820 18233

h3=2000 T 12 114 27261 24927 22781 20821 19047 17461

4=3000 T 13 = 5 29999 27666 25519 23559 21786 20199

4=3000 T 14= 88 29086 26753 24606 22646 20873 19286

4=3000 T5 = 100 28244 25911 23764 21804 20030 18444

h4=3000 T16 = 111 27471 25138 22991 21031 19258 17671

h5=4000 T17 = 72 30210 27877 25730 23770 21996 20410

h=4000 T 18 = 84 29367 27034 24887 22927 21154 19567

5=4000 T 19 = 96 28525 26191 24045 22085 20311 18724

h4=4000 -T, = 107 ?77S2 25419 23272 21312 19539 1 795



Table 3.2 Continued.

6=5000 T21 = 68 30491 28157 26011 24051 22277 20690

6=5000 T 22 = 81 29578 27245 25098 23138 21365 19778

6=5000 T 23 = 93 28735 26402 24255 22295 20522 18935

h6=5000 T 24 = 104 27963 25630 23483 21523 19750 18163

7=6000 T25 = 64 30772 28438 26292 24332 22558 20971

7=6000 T 26 = 77 29859 27526 25379 23419 21645 20059

7=6000 T 27 = 89 29016 26683 24536 22576 20803 19216

h7=6000 T 28 = 100 28244 25911 23764 21804 20030 18444

8=7000 T 29 = 61 30982 28649 26502 24542 22769 21182

8=7000 T 30 = 74 30069 27736 25590 23629 21856 20269

8=7000 T31 = 86 29227 26894 24747 22787 21013 19427

8=7000 T 32 = 97 28454 26121 23975 22014 20241 18654

9=8000 T33 = 57 31263 28930 26783 24823 23050 21463

9=8000 T34 = 70 30350 28017 25870 23910 22137 20550

9=8000 T 35 = 82 29508 27174 25028 23068 21294 19707

9=8000 T 36 = 93 28735 26402 24255 22295 20522 189351o=9000 T 37 = 54 31474 29141 26994 25034 23260 21674

1O=9000 T 38 = 67 30561 28228 26081 24121 22348 20761

ho=9000 T 39 = 79 29718 27385 25238 23278 21505 19918

1o=9000 T 40 = 90 28946 26613 24466 22506 20733 19146

11=10000 T 41 50 31755 29421 27275 25315 23541 21954

h1=10000 T 42 = 63 30842 28509 26362 24402 22628 21042

h1=10000 T 43 = 75 29999 27666 25519 23559 21786 20199

h1=10000 T4= 86 29227 26894 24747 22787 21013 19427



The coefficients for Eq.(3.4), k5, k6, k7, and k8, are computed by formulating the matrices A2

and B2 as follows,

21 M 1 M

1 M2 M

1 M 6 M6

1 M1 M

1 M6 M

1 M M 2 T2

1 M 6 M6

(Fn ) M 1,hl,T 1 )

F /I )(M 2 ,hl,T1 )

(Fn/ )(M6 ,hl,T 1)

(Fn/ )(MI,hl,T2)

(Fn6)(M6,h,,T2)

F /8)(Mh T(n (M1h1,T4)

(Fn16 (M6,hl P T 4 4 )

and then solvingEq.(3.9) below for k 5, k6, k7 , and k8 .

( A 2TA 2 ) - 1 A2 TB 2 )

(3.8)

(3.9)



These constant coefficients can be derived for climb operation following the same procedures

described above, but with climb thrust data. Because all columns in A s matrices are linearly

independent, ATA is strictly positive definite and a solution will always exist.

3.4 Validation

The thrust model was validated for three aircraft models, a small commercial airplane, a

medium commercial airplane, and a large commercial airplane. This section provides the details of

such evaluation.

3.4.1 Graphical Comparison

Figure 3.5 and 3.6 show the comparison between measured data,MIT model, and INM model

for the small comercial aircraft at takeoff and climb thrust setting respectively. Figure 3.7 and 3.8

show the comparison between measured data, MIT model, and INM model for the medium

commercial aircraft at takeoff and climb thrust setting respectively. Figure 3.9 and 3.10 show the

comparison between measured data,MIT model, and INM model for the large comercial aircraft at

takeoff and climb thrust setting respectively.

As shown in figures, the current INM strategy of using the density altitude to account for the

effect of temperature leads to dramatic increases in error when temperature is lower than the standard

day condition. As a result, INM can only accurately model the thrust setting near the standard day

condition. At airport, a temperature of 70 oF during summer time and 20 OF during winter time are

quite common, thus the deficiencies in the INM model can greatly affect the accuracy of the results.



Mach 0.21.05

1

c 0.95

0.9

0.85 h=650.J

h=9500i0.8 h=5- 0.75 h=5QSt

0.7

r 0.65

0.6

0.55-100 -50

0.95

0.9

r 0.85

0.8

. 0.75 h=9h=6500i

0.7h=250f

0.65 h=50f

z0.6

0.55

0.5

0.45-100 -50

0 50 100 ITemperature (F)

Mach 0.4

0 50Temperature (F)

100 150

1

0.95o

cr 0.9

- 0.85

0.8 h=95DJ-T 0.8O h=6500J

0.75h=25QgJ

0.7 h=5 H

70.65

0.60

0.55

0.5-100 -50

0.9

0.850o

r 0.

. 0.75

0.7 h=95

0.65 h=2 h=2500f

6o. h=SnL-.

z' 0.55

0.5

0.45

0.4-100 -50

0 50 100 15Temperature (F)

- Measured Data- - MIT.... INM

Mach 0.6

0 50

Temperature (F)

100 150

Figure 3.6 Climb thrust comparison for small commercial airplane at various conditions.

t.

0

| I • | I

Mach 0.3

R .~



Mach 0.1

0 50Temperature CF

Mach 0.2. _

0.61-100 -50 0 50

Temperature (F)

1 Lo

S1.2

1.1

; 0.9

0.8

(3--

100 150

0.7

0.6-100 -50 0 50 100 150

Temperature CF)

- Measured Dat- - MIT.... INM

Mach 0.3

0 50Temperature (F)

Figure 3.7 Takeoff thrust comparison for medium commercial airplane at various conditions.

I

r

U 1.22IE

W0.9z

0.800

I-

0.7 [

, i l i l

I

Mach 0

~



Mach 0.2

0.85 I

0.75

0.7

0.65

0.6

0.55

0 50 100 150

Temperature CF

0.5 L-50

Mach 0.4

0 50 100 150

Temperature CF)

0 50 100 150

Temperature CF)

- Measured Dat- -MIT

S INM

Mach 0.5

0 50 100 150

Temperature CF)

Figure 3.8 Climb thrust comparison for medium commercial airplane at various conditions.

h=10000 ff-

h=50Q.Ot

\ ..

0.55 L-50

a 0.8

~ 0.75

0.7

52 0.65

z 0.6

0.558

0 45 L-50

Mach 0.3



Mach 0.2

0 50

Temperature (F)

Mach 0.3

0 50

Temperature (F)

100 150

0 50 100 15Temperature (F)

- Measured Data-- MIT.... INM

Mach 0.4

0 50

Temperature (F)

Figure 3.9 Takeoff thrust comparison for large commercial airplane at various conditions.

0.4-100 -50

Mach 0



Mach 0.12 . . . 1.1

.P 1ig

20.9i-

50.80

0.7

E0.6

8 5

1 0.40 50 100 150 -100

Temperature (F)

Mach 0.41

g 0.9

20.8

0.72

I-

z 0.6

8 0.5

1 0.0 50 100 150 -

Temperature (F)

-50 0 50 100 150

Temperature (F)

- Measured Dat- - MIT.... INM

Mach 0.5

100 -50 0 50 100 150

Temperature (F)

Figure 3.10 Climb thrust comparison for large commercial airplane at various conditions.

o1.2

U 1.12

0.9

0.8

0.7

0.6

n-100

F.0.9

2

z 0.8z 0.6

8 5

0.4-100 -50

r;l

Mach 0.3



3.4.2 Error Analysis

The least squared erroris calculated using the following equation,

Ave Error =(3.10)(measured value), - (computed value) ] 2

where n is the numbers of data points.

The average errors are presented in Table 3.3 to 3.8. As the tables show, the quadratic thrust

model is more accurate than the existing thrust model in INM.

Table 3.3 Error in corrected net thrust during takeoff for small commercial airplane.

Least Squared Error in Corrected Net Thrust (lb)

Mach 0 Mac 0.1 Mach 0.2 Mach 0.3

0 ft 76.7736 52.2994 62.1422 68.7801

MIT Model 3000 ft 62.0462 35.1147 22.2196 29.0389

6000 ft 69 2990 28 7856 27 7817 41.5644

9000 ft 31 8990 43 8860 24 9302 27.3985

0 t 634 7971 656 0399 648 0785 596 6737

INM Model 3000 ft 631 2608 653 7065 666 7642 636 0905

6000 ft 650 9106 645 7621 659 7622 636 3030

9000 ft 665.3742 661.6280 691.0583 684_7760



Table 3.4 Error in corrected net thrust during climb for small commercial airplane.


Mach 0.2 Mach 0.3 Mach 0.4 Mach 0.5

500 ft 20.7569 27.3014 23.8526 48.0301

MIT Model 2500 ft 22.7564 24.9581 23.2439 31.14756500 ft 28 9597 24 8293 29 1363 22 4292

9500 ft 30.3556 24 1102 30 0177 26 7810

500 ft 703 9612 669 5469 627 2996 360 7722

INM Model 2500 ft 693.5819 674.8251 650.9640 404.25556500 ft 665 3673 676 1206 674 8790 482 0347

9500 ft t 639 3356 670.3949 682.4352 527.7964

Table 3.5 Error in corrected net thrust during takeoff for medium commercial airplane.


Mach 0 Mac 0.1 Mach 0.2 Mach 0.3

0 t 306 7909 392 9152 344 6927 165 1217

MIT Model 5000 ft 342 2574 173 6205 104 1779 101 2559

7920 ft 396 7150 161 8416 72 1704 100 0924

0 t 1224 11 1209 47 924 19 534 55

INM Model 5000 ft 894.78 946.14 805.73 573.207920 ft 74900 812.39 728. 1 519.66

Table 3.6 Error in corrected net thrust during climb for medium commercial airplane.


Mach 0.2 Mach 0.3 Mach 0.4 Mach 0.5

MIT Model 500 ft 20.9499 118.9694 82.6470 82.8706

9500 ft 52 6467 131 6875 44 9727 85 9502

INM Model 500 ft 809.3244 968.9295 856.5244 836.87069500 ft 7653625 961.642 7 932 9404 959 2255



Table 3.7 Error in corrected net thrust during takeoff for large commercial airplane.

Least Squared Errorin Corrected Net Thrust (lb)Mach 0 Mach 0.2 Mach 0.3 Mach 0.4

0 t 391.3044 273.5680 256.5289 303 8417

MIT Model 4000 ft 268 9983 192 9940 186 5013 200 0846

10000 ft 206 6931 157 2474 150 8482 134 6785

0 t 1766 11 1630 45 1626 52 1768 62

INM Model 4000 ft 1434.02 1350.46 1370.69 1532.7310000 ft 1331.86 1172.15 1183.47 1294.79

Table 3.8 Error in corrected net thrust during climb for largecommercial airplane.

Least Squared Error in Corrected NetThrust (lb)Mach 0.1 Mach 0.3 Mach 0.4 Mach 0.5

0 t 326.5073 100.7679 150.0685 184.0319

MIT Model 4000 ft 170.5130 92.2014 85.7651 199.47028000 ft 154 4417 212 2085 135 9459 182 5523

12000 ft 209.5427 311.1647 207.1063 157.5070

0 t 1889 59 1886 91 1881 04 1610 29

INM Model 4000 ft 1722 34 1898 43 1999 64 1914 09

8000 ft 1570 81 1891 55 2034 95 2129 19

12000 ft L 1484 26 1891 40 2102 42 2251 60


As the analyses show, the proposed quadratic thrust model is more accurate than the existing

INM model, particularly at high altitude and nonstandard day temperature condition, and would thus

provide improved prediction of aircraft thrust over a wider range of operating conditions.



Chapter 4. Level Flight Thrust Model

Level flight segments occur either between two climbing segments or between twodescending segments, and are treated as steady flight situations, namely, the balancing of

aerodynamic forces can be applied to obtain the required thrust. In order to apply the balancing of

aerodynamic forces , a complete set of drag polars for each aircraft configuration is required which

might not be very practical in computation or desirable for manufactory intent on maintaining control

of proprietary information.

In this chapter, two drag polar models are introduced. The first is an approximation of the

drag polar based on the minimum drag-over-lift point, while the second is a constrained least squarefit of the drag-over-lift ratio as a function of the lift coefficient. Two level-flight thrust models are

then developed based on these two drag polar models.

4.1 Previous Work

As suggested in SAE AIR 1845, the level-flight thrust in INM is now computed by reversing

the flight path angle equation to get the following expression [2]

1 siny(F /8)avg (W/8)avg [R + 1.03in (SAE Eq.A15)narg N 1.03

where y is zero for level flight. This expression however, does not include any velocity dependence

despite the fact that the required thrust is known to be a function of velocity [3].



4.2 Equation of Motion

The forces acting on an aircraft in steady, straight, and level flight are shown in Fig. 4.1.

Lift, L

Drag,D Thrust, F True Airspeed,VtFlight path

Weight, W

Figure 4.1 Forces on an aircraft in level flight.

The two aerodynamic forces, lift and drag, act at the center of pressure, and the g ravitationalforce,

weight of aircraft, acts at the center of gravity of the aircraft. The lines of action of the thrust and

drag forces lie very close to each other and the center of pressure can be regarded coincident with

the center of gravity of the aircraft, so that the coupling moment is negligible. Summing forces

parallel and perpendicular to the flight path yields the equation of motion of the aircraft in steady-

level flight,

F= D - p V 2 CD S2

W = L 1 p Vt2 CL S2

(4.1)

(4.2)

Combining Eq.(4.1) and (4.2) leads to the level-flight thrust as a function of weight and drag-

over-lift ratio,

F = WR

P

(4.3)



If the aircraft weight is treated as a constant, the minimum thrust occurs at the point where the drag-

over-lift ratio, R, is at minimum. This characteristic will be utilized in the development of level-

flight thrust.

4.3 Drag Polar

The aerodynamic characteristics, CL and CD , for a conventional aircraft exhibit a quadratic

relationship of the form,

CD = ko + k, CL + k2 C 2 (4.4)

where ko , k, , and k 2 are constant coefficients which can be obtained from flight test data by the

method of least squares. Dividing Eq.(4.4) by CL , the expression of drag-over-lift ratio in terms of

lift coefficient is obtained.

R =- ko + k + k 2 C L (4.5)CL

In this section, two drag polar models, Model I and Model II, will be introduced. Both

models are approximations of the drag polar near the minimum drag-over-lift point. Since the

derived drag polar models are based on the minimum drag-over-lift point, it is necessary to define

this point before deriving the models.

The minimum drag-over-lift ratio and the corresponding lift and drag coefficients are found

by taking the derivative of Eq.(4.5) with respect to CL and setting the derivative to zero to obtain

CLRin as shown below.

C k o (4.6)k2d



Substituting Eq.(4.6) into Eq.(4.4) and Eq.(4.5) to obtain the corresponding drag coefficient and

minimum drag-over-lift ratio respectively.

C = k o + k, + k2 2LRDRm CLRmm R

R komin C L + k1 + k2 CLR -m

(4.7)

(4.8)

4.3.1 Drag Polar Model I

Near the point of minimum drag-over-lift

simplified drag polar model of the form

point, the drag polar may be represented by a

CD = CDo + k CL (4.9)

where CDo, the zero-lift drag coefficient, and k are both constant coefficients. In reality, CDo and k

are functions of flight Mach number and Reynolds number, but since the operations considered here

are departures and approaches, i.e. the flight Mach number is under 0.7 and the effect of Reynolds

number only has small impact on skin friction drag, the assumption of constant CDoand k are valid

[6].

Dividing Eq.(4.9) by CL yields the expression for drag-over-lift ratio, R.

CR C + k CL

C,(4.10)

Taking the derivative of Eq.(4.10) with respect to CL,, setting the derivative to zero, and making the

necessary substitutions, yield a modified expression for CD and R in terms of CL, CLRnn, and CDRmin



CD CDRCD 2

1R22

C D R

CL

C DR i CL 22

CLRnunC CU

1 CDrm CL

2 CL CLR)

4.3.2 Drag Polar Model II

Although the drag-over-lift ratio is described by the relatively complex expression (Eq.(4.5)),

near the minimum drag-over-lift point, the drag polar may be described by an expression of the form

R = Ri + k CL - CL Rm)2 (4.13)

where k' is a constant that can be obtained from flight test data by the method of least squares.

Since aircraft are usually operating near the minimum drag-over-lift point during level flight in order

to minimize the thrust, this expression will closely match the behavior of an aircraft in level flight.

4.3.3 Effects of Reynolds Number on Drag Polar

Reynolds number, a dimensionless number of importance and impact on aerodynamics, is

essential to the determination of skin friction drag. Reynolds number, Re, is defined as,

Re - pVP-

(4.11)

(4.12)

(4.14)



Substituting forthe lift coefficient in terms of flight conditions derived from Eq.(4.2),

2WP, a V 2 S

gives the level-flight thrust in terms of known flight parameters.

ko Po o V2 SF = W + k2W

2k 2 W+ )

Po a Vt2S

4.4.1 Level-Flight Thrust Model I

Substituting the minimum-level-flight-thrust conditioninto the level-flight thrustequation,

Eq.(4. 1 , yields the following expression,

F = Rmin W 2 (4.19)Vt Rm CDR

where VtRm is given by

Vt =CR Wm R U

and

(4.20)

(4.21)CR 2

CR P CLr = S

(4.17)

(4.18)

\ - /



4.4.2 Level-Flight Thrust Model II

Level-flight thrust modelII is based on the drag polar model II, Eq.(4.13). After substituting

Eq.(4.13) into Eq.(4.3), level-flight thrust model II is given below.

F =W [ Rm in + k ( CL- CLR. )2 ] (4.26)

Furthermore, substituting Eq.(4.17) for CL in the above equation and replacing true airspeed with

Eq.(2.13), Model II is obtained in terms of flight conditions as follows.

F = W in + k C 2 Rmin (4.27)LR (1 + E h)2 V

4.5 Validation of Level-Flight Thrust Models

This section provides a comparison between the level-flight thrust predicted by the current

INM equation and the level-flight thrust predicted by the two models described above for a small and

a large airplane. The error analysis suggests that these two models are superior to the existing INM

equation.

4.5.1 Comparison of Proposed Models with INM Model

Figures 4.2 and 4.3 show the thrust ratio per engine vs. velocity plots for a small airplane and

a large airplane with different flap settings at 5000 ft, standard day condition respectively. For other

flight altitudes, the thrust histories have similar shapes, but different thrust and airspeed ranges.

Because INM uses only one drag-over-lift ratio for any flight velocity, the curve representing INM

equation is simply a straight line. Comparison to measured data shows that using a constant drag-

over-lift ratio to determine level flight is not adequate, and by providing flight condition and



necessary parameters, the proposed models can capture the curvature of measured data.

50 Flap0 Flap

200 250Calibrated Airspeed, V (knots)

85 MGTOW, 5000 ft

150 Flap

22 F

30018

12 0

0O Measured Data- MIT Model I- - MIT Model II

INM

140 160 180 200

Calibrated Airspeed, V (knots)

300 Flap+Gear

35.5-

35-

34.5 -

34

33.5 I

130 140 150 160Calibrated Airspeed, V, knots)

Figure 4.2 Thrust ratio per engine vs. velocity for small commercial jet with various flap settings.

21 -

20

k

d

6\0\O

\O

\OA AA\AA

,o7

16 -150 220

34 F

32-

30-

28'5

24 -

Q

)0

A AA

A \ AAAAA\

O

0/

0/0 0

/

\ 9o \ 9/

\

22120 140 160 180 200


I I ( I

1

33



50 Flap0 Flap

300 400Calibrated Airspeed, V knots)

85 MGTOW, 5000 ft

200 Flap

200 250 300Calibrated Airspeed, V knots)

30 I

500 150

O0 Measured Data- MIT Model I- - MIT Model II

INM

48-

47.5

47 -

8 46.5

I-

46_

.455

350

200 25/ A A A A A A A A200 250 300 350


300 Flap+Gear

160 180Calibrated Airspeed, V (knots)

200

Figure 4.3 Thrust ratio per engine vs. velocity for large commercial jet with various flap settings.

2615 0




Because the error is not sensitive to flightaltitude, 5000 ft was selected as a representative

altitude. Three aircraft weights, 85 of maximum gross takeoff weight, 90% of maximum gross

landing weight, and the average of those two weights were used for the analysis. In addition, the

lower bound of the airspeed was set to be 1.2 times of the stall speed and the upper bound of the

airspeed was 80 knots greater than the lower bound.

Table 4.1 shows the average least squared errors in level-flight thrust per engine for the small

airplane at different configurations. Because the derivation for thrust model I is based on the

expansion of the exact thrust equation about the minimum-thrust-flight conditions, the error

propagates as the flight velocity deviates from the minimum-thrust-flight velocity. The error is

proportional to the product of aircraft weight and the difference between actual flight velocity and

minimum-thrust-flight velocity. The observed errors of 10 lb to 70 lb per engine for the small

airplane are relatively small comparing to the actual level-flight thrust (8000 lb per engine).

Table 4.2 shows the average least squared errors in level-flight thrust per engine for the large

airplane at different configurations. The constrained curve fitting, thrust model II, guarantees

agreement near the vicinity of the minimum-thrust-flight point. Due to the different aerodynamic

characteristics between the small airplane and the large airplane, the constrained least square fit

provides a more accurate fit for the small airplane than the large airplane. The observed errors of

300 lb in thrust per engine for the large airplane is again relatively small comparing to the actual

level-flight thrust (28000 lb per engine).



Table 4.1 Average level-flight thrust errors per engine for small commercial jet at 5000 ft.

Model I Average Least Squared Errors (lb)0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW

2

O0 lap 25.23 27.10 33.35

50 Flap 65.05 70.04 75.04

150 Flap 50.64 54.47 58.30

300 Flap+Gear 50.23 40.50 43.28

IF Model II Average Least Squared Errors (lb)

0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW

2

00 Flap 28.06 30.15 26.26

50 Flap 33.31 35.92 38.54

150 Flap 42.26 45.45 48.64

300 Flap+Gear 15.69 12.98 13.76

INM Average Least Squared Errors (lb)


2

00 Flap 51.09 54.97 74.01

50 Flap 97.11 104.47 111.83

150 Flap 168.07 180.81 193.55

30o Flap+Gear 90.14 72.80 77.93



Table 4.2 Average level-flight thrust errors per engine for large commercial jet at 5000 ft.

Model I Average Least Squared Errors (lb)0.9xMGLW (0.9xMGLW+0.85xMGTOW) 0.85xMGTOW

2

0OFlap 284.52 328.69 278.10

50 Flap 43.07 49.26 55.38

200 Flap 130.58 146.46 102.22

30° Flap+Gear 27.30 30.52 33.50

Model II Average Least Squared Errors (lb)


2

00 Flap 180.27 205.51 97.51

50 Flap 114.76 132.04 149.29

200 Flap 233.15 268.10 138.49

300 Flap+Gear 113.86 130.70 147.40

INM Average Least Squared Errors (lb)


2

0OFlap 654.96 754.59 635.80

50 Flap 681.48 785.14 888.81

200 Flap 318.38 366.81 292.10

300 Flap+Gear 226.10 260.50 294.81



4.5.3 Pro and Con Between Models

Table4.3 and 4.4 show the pros and cons of Thrust Model I and Model II. These

observations are based on the error analysis and consideration of the numbers of parameters needed

from manufactures.

Table 4.3 Pro of level-flight thrust model I and model II.

Model I Requires only 2 parameters.

Small errors.

Good fit in the vicinity of minimum thrust.Simple.

Model IISmall errors.

Good fit in the vicinity of minimum thrust.

Table 4.4 Con of level-flight thrust model I and model II.

Model I More complicated.

Fit depends on the aerodynamic characteristics of aircraft.Model II

Need more parameters.

More work on the evaluation of parameters.



Chapter 5. Climb Performance

The flight environment is not free of headwinds or tailwinds, so the flight path angle

observed on the ground may not be the same as the flight path angle when there is no wind. Inaddition, the increase in true airspeed that occurs during constant equivalent or calibrated airspeed

climb has a small impact on the flight path angle.

Two flight path angle models are presented in this chapter which correctly account for the

effect on the flight path angle of both wind and the acceleration during climb. One is based on the

assumption of constant equivalent airspeed climb and the other is based on the assumption of

constant calibrated airspeed climb.

5.1 Previous Work

The flight path angle model in INM implicitly assumes that the airplane is climbing at a

constant calibrated airspeed and maximum available thrust. The fundamental aerodynamic force

balance leads to the equation for flight path angle [2],

y = sin - [N n)av - R ] (SAE Eq. A8)(W/8)ag

where the correction factor, , accounts for the increased climb gradient associated with an 8-knot

headwind and the acceleration inherent in climbing at a reference equivalent airspeed of 160 knots

((=1.01 when climb speed < 200 knots and =0.95 otherwise). Because this factor was derived for

an aircraft operating from a sea-level airport on a standard day, it does not account for variations in

airport altitude and aircraft climb speed.

The ground distance, Sc , that the airplane traverses during climb, is computed using the

following equation [2].



AhSc tan

tan y(SAE Eq. A9)

5.2 Equation of Motion and FlightPath Angle Correction Factor

Lift, L VtThrust, F

Flight Path Angle, y

Drag, D

W/g dV/dt Weight, W

Figure 5.1 Aircraft in steady climb with no wind.

Figure 5.1 shows an aircraft in steady climbing flight with no winds. As the figure shows,

the velocity is aligned with the flight path and the flight path itself is inclined to the horizontal at the

angle y. As in level flight, lift and drag are perpendicular and parallel to flight velocity respectively

and the weight is perpendicular to the horizontal. Thrust is assumed to be aligned with the flight

path (i.e. neglecting the thrust setting angle) and the inertial force, W/g dV/dt, is opposite to the

direction of thrust. Summing forces parallel and perpendicular to the flight path yields the equation

of motion for climb.

w dV,F - D - Wsiny -dV,g dt

(5.1)

L - Wcosy = 0 (5.2)



Combining Eq.(5.1) and (5.2) yields the expression for the flight path angle without headwind.

F -- RWsin y =

Vt d Vt1 +g dh

(5.3)

Eq.(5.3) only accounts for the impact of acceleration, thus an additional factor must be

included to account for the effect of wind on the flight path angle. Figure 5.2 shows the geometry

of the velocity vectors when wind is considered in flight.

Vt

VW

Figure 5.2 Geometry of airspeed vectors in wind.

As the figure shows, Y2 , the flight path angle after the effect of wind is included is related

to y, , the flight path angle without wind, by the following expression.

sin y2 =V, sin y1

V cos y, - V )2 + ( V sin y )2(5.4)



After small angle approximation, Eq.(5.4) becomes

sin y 2 = sin y,. (5.5)V - V

Combining Eq.(5.3) and (5.5) forms the equation of flight path angle,

Vt

V V Fsin y Vt - R] (5.6)

Vt d V W1 +

g dh

where the term in front of the bracket in Eq.(5.6) above is referred to as flight path angle correction

factor, .

Vt 1

Vt - W V dV, (5.7)w 1+

g dh

It is clearly shown in Eq.(5.7) that the flight path angle is altitude and airspeed dependent.

5.3 Evaluation of Flight Path Angle Correction Factor

Based on the atmospheric model presented in Chapter 2, Eq.(2.1) to (2.3), the following

derivatives, temperature ratio, pressure ratio, and density ratio with respect to altitude were derived.

Because the pressure ratio correction term, the second term in Eq.(2.2), is small and the effect is

negligible in the subsequent development of necessary derivatives, it was ignored in order to simplify

the expression.

dO 1 dT

dh T dh (5.8)o



5.3.2 Exact Constant Calibrated Airspeed Climb Model

Taking the derivative of Eq.(2.1 1) with respect to altitude yields the expression for dV/dh,

dV, 1dh J-

dVd6

d6

dh

-3-1 2 do+( )V o

2 e dh(5.13)

where dVe/d6 is obtained by taking derivative of Eq.(2.9) with respect to pressure ratio, 6, while

holding calibrated airspeed, Vc, to be constant.

dV Ya Po [( 1

d6 - 1 Po Ve

k k+ -) -

ya 6 6+ 1 )Ya -1]

After combining the relevant derivatives, the value of dV/dh during a constant calibrated

airspeed climb is given by

dTR -dV, Vt

dh 2 T Ro a

1 dT

0 dh

gO YaVt ya - 1

a dhk k

g [(1+)( ya 6 6

Substituting Eq.(5.15) into Eq.(5.7) yields the exact expression for the flight path angle correction

factor during constant calibrated airspeed climb,

V, 1Vt - Vw 1 +

(5.16)

2

-V g2 g T Ra

dTR

a dh

+ -

0dT

dh

Radh

Y 06 g [(1 + k

Ya- 1 Ya 6

(5.14)

+1) Ya (5.15)

where

-1

+ 1)Y 1] (5.17)



5.3.3 Simplified Constant Calibrated Airspeed Climb Model

As shown inEq.(5.16) and (5.17), the exact solution for theflight path angle correction factor

during constant calibrated airspeed climb is very complicated and a further simplification is

desirable.

Taking the derivative of Eq.(2.13) with respect to altitude yields the expression fordV/dh,

dV t Vc (1 + Eh) do[ - d (5.18)

dh - 20 dh

After combining the relevant derivatives, thevalue of dV/dh during a constant calibrated

airspeed climb is given by

SdT

dV t V + (1 + ch)( g +1 IdT)] (5.19)dh 2 To Ra 0 dh

Substituting Eq.(5.19) into Eq.(5.7) yields the simplified expression for the flight path angle

correction factor during constant calibrated airspeed climb,

Vt 1V, - V RdT(1 + 2 adh (5.20)(1 + Eh)V + (1 + eh) + dT

1 + [E + 6 + - -)]go 2T Ra 0 dh

Figure 5.3 and 5.4 show the comparison between the exact model and the simplified model

at standard day and nonstandard day condition with 8 knots headwind respectively. The close match

between two models verifies that the simplified model, Eq.(5.20), is adequate to represent the exact

constant calibrated airspeed climb model.



h = Oft

0 0 50 1Temperature at Sea Level Airport F)

h = 20000 ft

1,

0.95 knc=200 knots

.Vc=250 knots0.9-

-50 0 50 14

Temperature at Sea Level Airport CF

I-

0 10

S0.98oaU-

S0.96-C

0

E 0.94.9 0.94

h = 10000 ft1.05

1.04

1 U,- -,

Temperature at Sea Level Airport F

- Exact Model- - Simplified Model

Figure 5.4 Comparison of flight path angle correction factor predicted by the exact and simplifiedmodels at nonstandard day, 8-knot headwind condition.

70

- , V c = 50 knots

-Vc=200 knots

= Vc=250 knots

0

CUUL 1.02C

1.01o

, 1

< 0.99

0.98 -._LI n-

U.9

0.96

0.95-5

0.9-50 0 50 100



5.4 Graphical Comparison of Flight Path Angle Correction Factor

Figure 5.5 shows the comparison of flight path angle correction factor derived by MIT and

INM models at standard day, 8-knot headwind condition.As shown in Figure 5.5, the constant

factors in INM, 1.01 and 0.95, were approximately the average values for climbing at an airspeed

greater and less than 200 knots respectively.

1 1.5 2 2.5Altitude,h (ft) x 104

Vc=210 knots0U

U,8 0.95 ----------2

Vc=250 knots0

-c

D 0.9-

0.850 0.5 1 1.5

Altitude, h (ft)

Figure 5.5 Comparison of flight path angle correction factor between MIT and INM models.

2 2.5

x 104



Table 5.1 Error in ground distance during constant equivalent airspeed climb starting from sea level.

Small Commercial Airplane: 5 deg Flap Large Commercial Airplane: 10 deg Flap

Error in Ground Distance Error in Ground Distance

Traversed (ft) Traversed (ft)

Final Altitude Ve Ve Ve Final Altitude Ve Ve Ve

(ft) 160 180 200 (ft) 185 200 220

knots knots knots knots knots knots

1500 0.41 0.50 0.58 1500 0.69 0.76 0.86

3000 0.73 1.16 1.60 3000 11.08 12.03 13.53

5000 13.84 13.50 13.19 5000 90.51 98.23 110.23

Table 5.2 Error in ground distance during constant equivalent airspeed climb starting from 5000 ft.




Final Altitude Ve Ve Ve Final Altitude Ve Ve Ve

(ft) 160 180 200 (ft) 185 200 220


6500 2.34 2.53 2.73 6500 7.42 8.10 9.15

8000 23.34 25.37 27.59 8000 72.96 79.86 90.47

10000 141.00 154.18 168.55 10000 441.56 485.30 552.86



5.6.2 Constant Calibrated Airspeed Climb Model

Table 5.3 and 5.4 show the error in approximating ground distance traversed during constant

calibrated airspeed climb starting from sea level and 5000 ft respectively.

Table 5.3 Error in ground distance during constant calibrated airspeed climb starting from sea level.

Small Commercial Airplane: 5 deg Flap _ Large Commercial Airplane: 10 deg Flap



Final Altitude V c Vc V c Final Altitude V c V c Vc

(ft) 160 180 200 (ft) 185 200 220


1500 3.27 4.45 4.97 1500 6.54 7.07 6.70

3000 14.73 16.06 19.55 3000 16.01 18.94 26.03

5000 65.48 76.58 93.38 5000 59.20 70.88 93.64

Table 5.4 Error in ground distance during constant calibrated airspeed climb starting from 5000 ft.




Final Altitude V c Vc V c Final Altitude V c V c Vc

(ft) 160 180 200 (ft) 185 200 220


6500 74.25 86.43 102.55 6500 123.12 141.38 171.94

8000 166.12 195.46 234.03 8000 251.59 291.95 359.51

10000 276.80 334.98 411.76 10000 257.98 316.22 415.77



5.6.3 Discussion

As Table 5.1 and 5.3 show, the error in ground distance traversed is not sensitive to

climb airspeed, but to altitude increment. The errors are insignificant regardless of the

altitude increment while climbing from low altitude. The altitude increment becomes

important while climbing from high altitude, however, the errors of a couple hundred feet as

shown in Table 5.2 and 5.4 resulting from altitude increment of 5000 ft while climbing from

5000 ft are acceptable comparing to the overall ground distance traversed.


In this chapter, an analytical expression for climb equation was developed and further

simplified. It was proved that the flight path angle correction factor is altitude and airspeed

dependent. As shown from Table 5.1 to 5.4, the induced error in the distance traversed from the

two-point-average approximation increases as the increment in altitude and the climb airspeed

increase, thus the climb segment should be divided into increments of less than 5000 ft while

climbing at high altitude to ensure the accuracy of the result.



Chapter 6. Accelerated Climb Performance

In this chapter, an accelerated climbmodel is developed based on the assumption of constant

climb rate acceleration. As comparing to the INM model, the result suggests that the correctionfactor in INM depends on the flight conditions and particularly flight altitude.

6.1 Previous Work

Given the initial climb conditions, a specified climb rate, and the target calibrated airspeed,

the horizontal distance traversed and gain in height are obtained using SAE Eq.A10 and Al l

respectively [2].

(1/2g) (0.95) Vtb - V )S = (SAE Eq.A10)

a N F,/)avg/ W/)ag] - Ra vg Vtz/Vta )

Ah = Sa Vtz/Vtavg)/0.95 (SAE Eq.A 1l)

where 0.95 represents the headwind effect on the ground distance when climbing at a 160-knot

reference airspeed intoan 8-knot reference headwind, and the subscript avg refers to the average

of the quantity at the beginning and ending points of the climb segment.

However, the true airspeed andpressure ratio at the end of the acceleration segment depend

on the final altitude, which is unknown. INM suggests an iterative method to compute the gain in

height and horizontal distance as shown in the following figure [4].



Given Vca, Vcb, Vtz , hi

Estimate final altitude: (h2)est =hi+250Compute Vta

Compute VtbCompute Ah and Sa(h2)new = hi + Ah

End

Figure 6.1 Computation procedures for INM model.



h 2 - h 1

tan Yavg

sin- Vtz + sin_ Vtb(6.4)

where Yavg = 2

Because the values of the true airspeed and pressure ratio at the end of acceleration require the

knowledge of the final altitude, an iterative process is developed to generate the correct result.

Instead of arbitrarily giving a final altitude, an initial estimate of the final true airspeed and altitude

as shown below can speed up the iteration.

VVtbest = Vta + V b - Va) (6.5)

ca

dVh 2est h1 + Vtbest - Vta ) dh)(Vcahl) (6.6)

Following the procedures as illustrated in Figure 6.2, the ground distance traversed and gain in height

during the acceleratedclimb are obtained.




As the analysis in this chapter shows, the correction factor, a constant in the SAE model,

depends on the flight altitude and airspeed as well as the flight environment. In addition,the error

in the distance traversed and height gained during an acceleration segment induced by the two-point-

average process is not sensitive to flight altitude, but to speed increment. However, for a well

defined flight procedure, the speed increment can always be reduced to values less than 40 knots and

thus, the accuracy of the result can be improved.



Chapter 7. Comparison of Departure Profile and Noise Contour

Three aircraft models, Boeing 727-200, Boeing 737-3B2, and Boeing 757-200, at variousflight conditions were evaluated. The overall ground distance traversed was compared to the flight

profiles provided by the Delta Airlines (to the SAE A21 committee) and the corresponding noise

contours were computed using INM Version 5.2 with improved coefficients supplied by the Boeing

Company. The results suggest that the MIT model provides a more accurate prediction of aircraft

performance parameters as well as the corresponding shape noise contour.

7.1 Description of Analysis

The flight procedures of a complete departure profile consist of takeoff, climb, and

accelerated climb, where the climboperation is assumed to be constant calibrated airspeed climb and

the accelerated climb operation is assumed to be constant climb rate operation.

The inputs to INM for noise computation are as shown below:

Number of Flight per Day: 1

Run Type: Single Metric

Noise Metric: (I) LAMAX and (II) SEL

where LAMAX represents the peak value of A-weighted sound level and SEL represents the

integration of A-weighted sound pressure over a period of time.

Comparisons were made of the flight profile, noise impact area, and closure point distance,

where the closure point distance is measured from the break release point to the outer most point of

each individual sound level.



7.2 Boeing 727-200

Two runway altitudes were used: sea level and 4000 ft. The weight of aircraft was set to

190000 lb. For the sea-level-runway, the airport conditionswere 59 oF and 840 F with 8-knot

headwind which are referred to as Case (1) and Case (2) respectively. For the 4000-ft-runway, the

airport conditions were 44.7 OF and 69.7 OF with 8-knot headwind which are referred to as Case (3)

and Case (4) respectively.

7.2.1 Procedure Steps

Table 7.1 shows the flight procedure steps for Case (1) and (2) and Table 7.2 shows the flight

procedure steps for Case (3) and (4).

Table 7.1 Flight procedure for Case (1) and (2).

Step Thrust Flap ID Operation Type

1 Max Takeoff 15 Takeoff.

2 Max Takeoff 15 Climb to 1000 ft.

3 Max Takeoff 5 Accelerate to 180 knots at 750 ft/min climb rate.

4 Max Takeoff 2 Accelerate to 200 knots at 750 ft/min climb rate



7 Max Climb 0 Climb to 2500 ft.

8 Max Climb 0 Accelerate to 250 knots at 750 ft/min climb rate.

9 Max Climb 0 Case (1): Climb to 9000 ft.

Case (2): Climb to 7600 ft.





1 Max Takeoff 15 Takeoff.2 Max Takeoff 15 Climb to 1000 ft.

3 Max Takeoff 5 Accelerate to 180 knots at 750 ft/min climb rate.



6 Max Takeoff 0 Case (3): Climb to 1700 ft.

Case (4): Climb to 1800 ft.



9 Max Climb 0 Case (3): Climb to 7400 ft.

Case (4): Climb to 6000 ft .

7.2.2 Flight Profile and Noise Contour

Figure 7.1 to 7.8 show the flight profile [8] and noise contour for Cases (1), (2), (3), and (4)

respectively.



Ground Distance (ft) x 104

5 10Ground Distance (ft)

15x 10


-5 '-15 -10 -5 0SLD (nmi)

5 10 15

Figure 7.1 Flight profile and LAMAX noise contour for Case (1).

6000

5000

4000

3000

x 10

1.45

1.4

1 1.3

S1 25

1.2

1.15

1.1

1.05'o

0- Delta-- MIT- INM



15-

S10-

5-

0

-5

-10-30 -20 -10 0 10 20 30

SLD (nmi)

Figure 7.2 SEL noise contour for Case (1).



55 dB DeltaIl'a

30 MI

INM65.dB

25

20 85.dB

S 1 5

uslo-

0-

-5-

-10-30 -20 -10 0 10 20 30

SLD (nmi)




Ground Distance (ft) x 10 4

5 10 15Ground Distance (ft) x 10'

1.6 30

1.55 0-0 Delta d - Delta13-E MIT - MIT

1 INM 25 INM

w 1.45

1.4 20

F 1.35

1 1 3 5 85-1.3 E

1.251 85 d

1

1.2

0 5 10 15Ground Distance (ft) x 04

0-

-5 -,

-15 -10 -5 0 5 10 15SLD (nmi)


90



S15

I 10

5-

0-

-5

-10-30 -20 -10 0 10 20 30

SLD (nmi)




t 4000

Ground Distance (ft) x 10 Ground Distance (ft) x 104

MIT - - mS1. 4 F INM 25 INM

1.35

1 1.3 20

1.25

1.15 C

00 5 10 15Ground Distance (ft) x 10

-5-15 -10 -5 0 5 10 15

SLD (nmi)




30- - - -. MI IINM

5 dR.25

85 dB

20-

S15-

S10

0-

-5

-10 i i i i

-30 -20 -10 0 10 20 30SLD (nmi)



Table 7.3 shows the error in overall ground distance traversed in feet and Table 7.4 and 7.5

show the error in noise impact area in square statue mile for 55 dB, 65 dB, and 85 dB sound levels.

Table 7.6 and 7.7 show the error in closure point distance in nautical miles for Case (1) and (2) and

Case (3) and (4) respectively.

DeltaI Ir



Table 7.3 Overall ground distance error in feet for Case (1) to (4).

Case (1) Case (2) Case (3) Case (4)

M T -3276.0 -1236.4 -3808.9 -3359.5

NM -4993.9 -16925.8 -4852.6 -16017.7

Table 7.4 Error in noise impact area in square mile for Case (1) and (2).

Case (1) Case (2)

55 dB 65 dB 85 dB 55 dB 65 dB 85 dB

LAMAX MIT -1.542 -0.341 -0.149 -1.34 -0.646 -0.668

INM -2.025 -0.296 0.134 3.856 5.191 0.806

SEL MIT -6.724 -3.694 -0.765 -8.211 -1.884 -0.51

INM 0.054 -2.335 0.067 4.835 5.578 9.121

Table 7.5 Error in noise impact area in square mile forCase (3) and (4).

Case (3) Case (4)

55 dB 65 dB 85 dB 55 dB 65 dB 85 dB

LAMAX MIT -6.791 -4.271 -0.62 -3.536 -1.597 -0.427

INM -14.959 -7.387 -0.581 -2.906 1.025 1.356

SEL MIT -49.294 -18.932 -4.062 -14.215 -4.96 0.253

INM -10.163 -6.79 -2.697 -25.373 -7.111 7.512



Table 7.6 Error in closure point distance in nautical mile for Case (1) and (2).

Case (1) Case (2)

55 dB 65 dB 55 dB 65 dB

LAMAX MIT -0.5476 -0.5219 -0.1172 -0.1486

INM -0.8219 -0.7964 -2.2007 -2.3433

SEL MIT -0.6254 -0.5676 -0.0343 -0.0919

INM -0.6774 -0.9238 -1.7911 -2.1749


Case (3) Case (4)

55 dB 65 dB 55 dB 65 dB

LAMAX MIT -0.6678 -0.6713 -0.4725 -0.4913

INM -0.8329 -0.839 -2.0981 -2.2474

SEL MIT -0.7092 -0.6693 -0.394 -0.4617

INM -0.8745 -0.8558 -1.7051 -2.0645

7.3 Boeing 737-3B2

Two cases were studied for the Boeing 737-3B2: the first was a 59 'F, sea-level takeoff with

8-knot headwind, which is referred to as Case (5), and the other was a 80 'F, sea-level takeoff with

8-knot headwind, which is referred to as Case (6). The weight of aircraft was set to 120000 lb.




The procedure steps for Case (5) and (6) are the same as shown in Table 7.8 below.






4 Max Climb 5 Accelerate to 180 knots at 1300 ft/min climb rate





Figure 7.9 to 7.12 show the flight profile [9] and noise contour for Case (6) and Case (7).



8000

7000

6000

5000

1 4 0 0 0

3000

2000

1000

Ground Distance (ft) x 10 4

iiI

I~

4Ground Distance (ft)

I

E

S

0P

8

x 10


-4 -2 0SLD (nmi)


2 4



0-

5---15 -10 -5

Figure 7.10 SEL noise contour for Case 5).

0SLD (nmi)



15-

-15 -10 -5 0 5 10 1

SLD (nmi)



Table 7.9 shows the errorin overall ground distance traversed in feet and Table 7.10 and 7.11

show the error in noise impact area in square statue mile and error in closure point distance in

nautical mile respectively.

100

DeltaMIT

INM



Table 7.9 Overall ground distance error in feet for Case (5) and (6).

Case (5) Case (6)

MIT -121.65 339.5

INM -689.0 1507.4

Table 7.10 Error in noise impact area in square mile for Case (5) and (6)

Case (5) Case (6)

55 dB 65 dB 85 dB 55 dB 65 dB 85 dB

LAMAX MIT -0.871 -0.265 -0.071 -0.056 -0.104 -0.063

INM -0.804 -0.132 -0.017 -0.997 -0.196 -0.031

SEL MIT 0.837 0.475 0.051 2.38 1.463 0.094

INM 0.377 1.051 0.152 0.137 0.172 0.035

Table 7.11 Error in closure point distance in nautical mile for for Case (5) and (6).

Case (5) Case (6)

55 dB j 65 dB 55 dB 65 dB

LAMAX MIT -0.1043 -0.0746 0.0823 -0.0057

INM -0.1465 -0.0442 0.158 0.1269

SEL MIT -0.0362 -0.049 0.0705 0.064

INM -0.0863 -0.1369 0.2525 0.148

7.4 Boeing 757-200

Two aircraft weights were used for the Boeing 757-200: the first was 183000 ib, which is

referred to as Case (7), and the other was 223800 lb, which is referred to as Case (8). The airport

condition was 77 OF, sea-level with no headwind.

101




The procedure steps for Case (7) and (8) are as shown in Table 7.12 and 7.13 respectively.

Table 7.12 Flight procedure for Case (7).











Table 7.13 Flight procedure for Case (8).







6 Max Climb 0 Climb to 2500 ft.7 Max Climb 0 Accelerate to 250 knots at 1200 ft/min climb rate.


102




Figure 7.13 to 7.16 show the flight profile [9] and noise contour for Case (7) and (8).

6000

5000

4000

130002000

1000


4Ground Distance (ft)

14

-- MIT12 .... IN M

8x 10

-2 0SLD (nmi)

2 4


103



15is

10-

-51-1 0) -8 -6 -4 -2 0

SLD (nmi)2 4 6 8 10


104

DeltaMITINM

65d

85 .

1 aI I I I Ii 1



4 6 8 10 0 2 4 6 8 10Ground Distance (ft) x 104 Ground Distance (ft) x 104

0 2 4 6 8 10Ground Distance (ft) x 104

55 dB- Delta

16 - MITINM

14

12F

-4 -2 0SLD (nmi)

2 4


105

ii | |



Table 7.14 Overall ground distance error in feet for Case (7) and (8).

Case (7) Case (8)

MIT -3032.1 -3722.8

INM -7509.1 -9978.8

Table 7.15 Error in noise impact area in square mile for Case (7) and (8)

Case (7) Case (8)

55 dB 65 dB 85 dB 55 dB 65 dB 85 dB

LAMAX MIT 0.15 -0.042 -0.004 0.298 -0.099 -0.012

INM 0.354 0.076 0.026 0.283 -0.086 0.021

SEL MIT 1.357 0.76 0.034 2.119 1.081 0.074

INM 3.78 1.663 0.208 3.206 1.338 0.253


Case (7) Case (8)

55 dB 65 dB 55 dB 65 dB

LAMAX MIT -0.2863 -0.1758 -0.283 -0.2528

INM -0.8286 -0.4603 -1.0807 -0.6576

SEL MIT -0.4192 -0.2961 -0.5206 -0.3931

INM -1.0603 -0.9287 -1.4668 -1.3768

107



7.5 Discussion

As shown in the case studies, the error in aircraft performance prediction for INM results

from three resources, the thrust value, the flight path angle correction factor, and the accelerated

climb algorithm.

The INM thrust model does not accurately predict the thrust at altitudes greater than 2000

ft, particularly under nonstandardday condition. The over prediction of thrust in the INM yields an

over prediction of climb performance which is the major contribution to the ground distance error

in INM.

For the constant calibrated airspeed climb segments, the constant flight path angle correction

factor, 0.95, used in INM after flap retraction is slightly higher than the actual value which produces

a slightly higher climb angle and shorter ground distance traversed during climb.

The observation of accelerated climb segments shows that the climb rate in INM does not

hold constant. The climb rate in INM decreases as altitude increases which yields a faster

acceleration and a shorter ground distance traversed.

However, since the over prediction of thrust in the INM compensates the under prediction

of the ground distance traversed, the error in overall noise impact area is actually reduced for INM.

The noise contour is in fact ill-shaped, i.e. much wider in lateral direction and much shorter in

longitudinal direction as shown in the figures. Thus, the error in noise impact area does not provide

a true indication of model s superiority. The closure point distance, on the other hand, provides a

better correlation with contour shape. The smaller the error in closure point distance is, the better

match the contour shape is.


As the analyses show, the proposed thrust model provides accurate thrust prediction over a

wide range of operating conditions. In addition, the flight path angle model and accelerated climb

model as a function of flight parameters are more realistic. As the result, the close match of aircraft

performance predicted by the MIT model yields a more accurate ground distance traversed as well

as a better fit of noise contour.

108



Chapter 8. Conclusion and Future Work

8.1 Conclusion of Thesis

The analyses presented in this thesis illustrated that the proposed performance algorithms are

more accurate than the current methodology employed in the INM. The proposed true airspeed

model takes into account the effect of the air compressibility on airspeed and is valid over a wider

range of operating condition. The proposed quadratic takeoff/climb thrust model as a function of

Mach number, altitude, and temperature accurately predicts the takeoff and climb thrust under both

standard and nonstandard day conditions. The level-flight thrust model as a function of flight

parameters derived from the minimum-level-flight-thrust condition can accurately predict the actuallevel-flight thrust within the airspeed range considered. The analytical expression for the flight path

angle correction factor has proven to be more realistic than the constant correction factor used in

INM. The proposed accelerated climb algorithm models the constant climb rate acceleration and is

more suitable for the real airplane operation. As the result, the close match of aircraft performance

parameters predicted by MIT model provides a better fit of noise contour.

8.2 Future Work

Because this research focused on the improvement of aircraft performance algorithms, it is

necessary to review the methodology for the calculation of corresponding sound exposure level and

particularly the effect of weather condition on the noise propagation in both longitudinal and lateral

directions. In addition, due to the requirement of high precision, the radar tracking of aircraft

position is no longer accurate enough. For the validating purpose, it is the future work to actually

setup equipments at airport to capture the aircraft departureand approach profiles and to record the

corresponding sound level.

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Evaluation of Aircraft Performance Algorithms Thesis

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