Aircraft Flight Mechanics

Andrew Benson

California State University Northridge

Culminating ExperienceAE 480

Abstract

Flight Charictoristics are complex and ever changing with time. Determiniates for thesecharacteristics are things such as atmospheric effects, Thrust effectiveness, Aerodynamics and

Aircraft sizing. Once determined a value is then known for such parameters as flight length andaircraft agility helping better understand the capabilities and usefulness of the aircraft at hand.

By knowing these flight parameters aircrafts can be better fit to their application, accounting forevery aspect of what its investor would like it to accomplish.

December 16, 2012

Contents

1 Introduction to Flight 5

1.1 Cayley . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Wright Brothers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Evolution of flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Aircraft Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Flight Environment 9

2.1 Atmospheric Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Altitude Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Thrust Modeling 12

3.1 Effects of Thrust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Thrust to Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Thrust vs Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4 Thrust at Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 Thrust at Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Flight Charictoristics 16

4.1 Air Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Atmosphere Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.3 True and Indicated Air Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4 Stall Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.5 True Air Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6 Air speed Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Aerodynamic Modeling 20

5.1 Lift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2 Effects of Angle of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.3 Effects of Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.4 Effects on Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Flight Enviorment 24

6.1 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1

CONTENTS 2

7 Takeoff 267.1 Runway Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Runway Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

8 Flight Range 288.1 Improving Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288.2 Range Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

9 Elevation Change 309.1 Climb Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309.2 Elevation Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

10 Holding Pattern 3310.1 Holding Pattern Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2 Maneuverability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

11 Landing 3511.1 Runway Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

12 Aircraft Modeling 3612.1 Aircraft Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.3 Air Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.4 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612.5 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.6 Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.7 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3712.8 Ground Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

13 Appendix 3813.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.2 Homework Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.2.1 Atmospheric model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.2.2 Thrust Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2.4 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2.5 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.2.6 Rate of Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.2.7 Take off & Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

13.3 Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4113.3.1 Atmospheric Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4213.3.2 Thrust Effectsl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4313.3.3 Aerodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4413.3.4 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4613.3.5 Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4713.3.6 Rate Of Climb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.3.7 Take Off and Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

List of Figures

1.1 Caley’s Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Caley’s Whirling Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Caley’s Glider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Wright Brothers Glider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Wright Flyer 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Evolution of flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Boeing 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Temperature with respect to Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Temperature with respect to Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Pressure with respect to Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Density with respect to Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1 Thrust to Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Thrust at speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1 Air Speed Dial [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Fuel Weight Consumption [5] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3 True Air Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.1 Lift Coefficient at Altitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Lift coefficient by wing parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3 Lift coefficient with respect to drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Lift coefficient to Wing Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.1 Flight Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.1 T.O Key (Figure 2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267.2 Take off Runway Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

8.1 Aircraft Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9.1 Rate of climb characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

10.1 Turn Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

11.1 Landing Runway Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3

Nomenclature

Symbol Definition Value Units

NTrop Air Constant Troposphere 1.235 K/MNStrat Air Constant Stratosphere 0.974 K/MTSea Temperature Sea Level 273.15 KelvinG Gravity 9.81 M/s2

R Gas Constant 287.1 J/kg kPSea Pressure Sea Level 101325 N/M2

γ Gamma 1.4 N/M2

π Pi 3.14 -e Oswalls Efficiency 0.85 -ro Mean Effective Radius 6357000 MT Thrust - lbfρ Density - Kg/M3

TS Thrust Sea Level - lbfTAS True Air Speed - M/Sα Angle of Attack - DegW Weight - lbfh Sweepback Angle - degs Wetted Area - M2

V Velocity - M/S2

VStall Stall Velocity - M/SQMax Max Lift Area - m2

tv Tank Vollume - GallonsC Thrust Specific Fuel Consumption - lb/hrφ Bank angle - DegXDot Max Turn Velocity - M/Sx Runway Length - MCL Lift Coefficient - -CD Drag Coefficient - -E Wing Efficiency - -D/W Drag to Weight - -T/W Thrust to Weight - -AR Aspect Ratio - -n Loading Factor - -

4

Chapter 1

Introduction to Flight

1.1 Cayley

Flight mechanics is the study of the forces thatact on an aircraft in flight, and the way the air-craft reacts. Sir George Cayley was one of thefirst to understand this concept by turning ourfocus from lighter than air travel to heaver thanair crafts equalized with a force balance.Cayley was born in 1773 and is regarded as thefather of aviation. As the first to have a ma-jor breakthrough in heavier than air flight byfirst determining the four things needed for sus-tained flight. These were the forces of weight,lift, drag and thrust and the relationships theyhad on one another. Yet another breakthroughwas his understanding of separating these forcesas he was the first man to build an aircraft thatgenerated thrust and propulsion using two sepa-rate elements. Many before him were simulatingbirds to the effect of generating lift and thrustwith one motion to no avail.In 1779 Cayley had put together a configurationthat mimics modern airplanes. His design in-cluded a fuselage, wings and a tail section withroom for one passenger. He used this model toexpress how forces acted upon the aircraft andhow the wings would provide lift.Cayley had es-tablished the basic principles and configurationof modern aircraft 100 years before the wrightbrothers flew from Kittyhawlk. In 1804 Caleywould invent yet another device useful for deter-mining aerodynamic effects. A wing structurewas built using a whirling arm by witch he couldrotate various rudimentary airfoil shapes at rel-

Figure 1.1: Caley’s Aircraft Model

atively high speed and measure the lifting force.This was the first scientific test of airfoils butwas later improved upon because of its rotat-ing effects turbulent air was interacting the air-foil giving a higher lift result that in practice offlight.

Figure 1.2: Caley’s Whirling Arm

5

CHAPTER 1. INTRODUCTION TO FLIGHT 6

Caley put his testing to practice by buildingfull size gliders in 1849. He marked the first per-son to take flight by placing the then year oldson of one of his servants in the gliders carriageand began it down a large grass hill. This gliderachieved a short flight as it gained speed downthe hill followed by its inevitable return to earth.This accomplishment renowned Caley as the fa-ther of aircraft design.

Figure 1.3: Caley’s Glider

Four years later in 1853 Cayley built a tri-plane glider that carried a coahman 900 ft acrossBrompton Dale in the north of England beforecrashing back to earth. It was the first recordedflight by an adult in an aircraft.

1.2 Wright Brothers

After Cayley the next major step in avionics wasthe wilbur and orville wright. The wright broth-ers brought many new ideas to the production ofan aircraft such as wing wrapping and the twist-ing of the wings to provide control in the air.These features while rudimentary were the big-ging of light weight aircraft structures and whatare now called alerons.On October 1900 the wright brothers proved theconcept of wing wrapping at kittyhawlk by be-ginning down a sand hill and achieving flightwith the 17ft wingspan glider.

Figure 1.4: Wright Brothers Glider

From this success the brothers went on tobuild the wight flyer 1 with a gas fueled engineon board for propulsion. This aircraft took offwith a ramp lauch similar to the glider but ex-perienced a stall when the pilot began to climbto quickly resulting in the first ever pilot er-ror. With minor repairs made Orville pilotedthe plane under similar circumstances into theair covering 852 ft in 59 seconds.

Figure 1.5: Wright Flyer 1

CHAPTER 1. INTRODUCTION TO FLIGHT 7

1.3 Evolution of flight

There have been many advancements since kitty-hawlk in 1900 allowing aircrafts to become morereliable and fly faster and higher than ever. Overthe next one hundred years flight has greatly im-proved from a cloth wrapped wooden plane to jetpowered performance aircrafts.

Figure 1.6: Evolution of flight

This rapid improvement over the years hasbeen motivated for faster and higher flight leav-ing off with where we are today in the perfor-mance era of aircraft, where aircrafts can nowhaul heavy loads or perform complex maneu-verer. This per suite was aided by the develop-ment of aircraft modeling, improving upon cay-leys whirly arm among many other aspects find-ing ourselves in the modern age with compositesand jet engines.

1

To successfully push the envelope of flightperformance though the years an accurate flightmodel had to be constructed to properly designan aircraft for its desired task. Parameterssuch as altitude density, temperature, liftingeffects and drag effects have been expandedover the years resulting in a definitive set ofparameters to model an aircraft after. Theeffects of this modeling will effect such thingsas lifting capacity, turning radius at elevationand thrust needed. These parameters will havedirect correlation to the distance an aircraft canfly along with payload capacity and fuel needed.

CHAPTER 1. INTRODUCTION TO FLIGHT 8

1.4 Aircraft Design

To follow in the footsteps of Cayley and theWrights before and design an aircraft, param-eters must be layed out. Unlike Cayley who’sgoal was to lift a single man off the ground fora short amount of time, the task that will bediscussed further will be a jet transport aircraftwith the goal of transporting 70 passengers andtheir baggage over vast distances.

By understanding the effects of this jet trans-port class aircraft all other types of flight can bemolded by simply swapping the following param-eters.

Symbol Definition Value Units

W Weight 317515 Kgs Wetted Area 28886 ft2

QMax Max Lift Area 650 m2

tv Tank Vollume 50000 GallonsC Thrust Specific Fuel Consumption 1.5 lb/hrCL Lift Coefficient 1.2 -CD0 Parasitic Drag 0.016 -W/S Wing Loading 110 -T/WSL Thrust to Weight .32 -AR Aspect Ratio 6 -

The given parameters will remain constantthough the rest of the text. These inputs de-fine an aircraft of the stature of a Boeing 747 inrelationship to size and weight.

To properly construct an aircrafts parametershowever you must first understand the environ-ment in witch it flies. The effects of flight suchas lift, thrust and speed are ever changing as youtraverse though the atmosphere.

Figure 1.7: Boeing 747

Chapter 2

Flight Environment

2.1 Atmospheric Layers

Each layer of the earths atmosphere is measuredfor purposes of simplicity using a geopotentialaltitude. This altitude is calculated with the as-sumption that the force of gravity is held con-stant as you ascend though the atmosphere. Al-though this is not the case explained by Newtonslaw of gravitation, this assumption greatly re-duces unnecessary complexity only being off lessthan one hundredth of a percent at 35 KM. Allthe altitude calculations thusly will be measuredin geopotential height and can be converted us-ing equation 1.

Zgeometric =r0

r0 − ZgeopotentialZgeopotential (2.1)

The Troposphere is the lowest region going upto about 11 miles above sea level at latitude 45north. This layer holds most of the clouds andweather denoted by its decreasing temperatureas you gain altitude. This relationship dictatesit as polytripic, being dictated by equations 2though 5.

T2

T1= [1− g0

RT1(n− 1

n)(Z2 − Z1)] (2.2)

P2

P1= [

T2

T1]nn−1 (2.3)

ρ =P1

RT1(2.4)

a =√γRT (2.5)

The Tropopause region is an intermediarylayer from the Troposphere and the Stratosphereand is denoted by its unchanging temperatureas altitude is gained.Unlike the Troposphere thislayer is Isothermal for this reason being governedby equations 2.5 though 2.7. At the upper endof this layer resides the Jet stream and this iswhere most aircrafts will reside.

P2

P1= e

−g0RT

(Z2−Z1) (2.6)

ρ2

ρ1=P2

P1(2.7)

The Stratosphere occurs from 11-31 mileswhere only the highest clouds reach such ascirrus, cirrostratus, and cirrocumulus. This iswhere the ozone layer resides absorbing U.V lightfrom the sun. This layer is denoted by its in-creasing temperature with altitude making itpolytropic, similarly being dictated by equations2.2 though 2.5.

The Mesosphere ranges from 31 -50 milesabove the earth and is also a range that decreasesin temperature as altitude is gained. This at-mospheric layer is described using equations 2.2though 2.5.

The Mesospause separates the layer of theionosphere and the mesosphere and like theTropopause is denoted by its unchanging tem-perature with altitude. Also like the Tropopause

9

CHAPTER 2. FLIGHT ENVIRONMENT 10

this layer is controlled by equations 2.5 though2.7.

The Ionosphere contains many ions and freeelectrons created by excited photons from thesun. The ionosphere is where the Aurora Bore-alis occurs from between 50 and 400 miles. Thislayer is a part of the Thermosphere witch encom-passes the Ionosphere and Exosphere in charac-terizing them as increasing in temperature as al-titude is gained. This polytropic region followsthe rules of equations 2.2 though 2.5.

The Exosphere is the outer most layer ex-tending roughly to 800 miles. There is no hardboundary but a melding with the extent of spaceas the density of molecules becomes small enoughto not be considered fluid flow anymore butquantized molecules. This layer is as mentionedbefore a part of the Thermosphere causing tem-perature to increase as altitude is gained make-ing it similarly follow equations 2.2 though 2.5.

n=1.235 (Polytropic)n=0.974(Isothermal)

2.2 Altitude Characteristics

The characteristics of temperature, pressure anddensity were plotted as a function of altitudein geopotential space to show the importanceof designing an aircraft with these layers in mind.

Each layer can clearly be seen in the plottedarea up to 35 km were aircrafts typically flyunder normal conditions relieving their uniquecharacteristics and allowing us to design for itseffects.

When temperature is plotted as a functionof altitude an isothermal region is very clearlyshown from 11000 to 15000 m separating the tro-posphere and the stratosphere were the temper-ature with respect to altitude inverts and beginsto heat instead of cool as shown in figure 2.1.

Figure 2.1: Temperature with respect to Alti-tude

Figure 2.2: Temperature with respect to Alti-tude

In conclusion these thermal boundary layersdictate the areal performance of objects such asplanes passing though them. Having a stan-dard and universal measure of these character-istics allows us to properly design for the appli-cation at hand. Understanding the characteris-

CHAPTER 2. FLIGHT ENVIRONMENT 11

tics of each of these levels is crucial in decidingwhich layer would be best for a particular air-craft to fly in such as the knowledge of flyingin the tropopause, outside of most weather andunder the jet stream were density, pressure andtemperature are all much more predicable thanother layers.

Figure 2.3: Pressure with respect to Altitude

Figure 2.4: Density with respect to Altitude

Chapter 3

Thrust Modeling

3.1 Effects of Thrust

The relationship of an aircrafts flight character-istics is vital to be understood by aircraft de-velopers to calculate the range of altitude andvelocities of their aircraft. The effect of thrustand drag effect the ceiling altitude an aircraftcan fly and also the velocity an aircraft can flyat at a given throttle.Investigating separately the velocity characteris-tics at altitude and the velocity range as a func-tion of altitude showed the limitations of an air-craft. A Boeing 747 as follows is analyzed todetermine these very characteristics.

3.2 Thrust to Drag

In a pursuit to find a way to find the performancevelocity of an aircraft at altitude the drag andthrust must be analyzed. To plot this relation-ship the two curves of thrust and weight wereplotted along a velocity scale. To equalize theeffectiveness of these terms with respect to anyaircraft the thrust and drag can be divided by theweight of the aircraft to create a non-dimensionalterm proportioning any size aircraft. This rela-tionship for Drag can be seen in equation 1.

D

W=ρelV

2CDows 2

+2ws

πAREρelV 2(3.1)

Similarly an equation for Thrust can be de-rived as follows in equation 2.

T

W=

T

W sl(ρtpρsl

).7(ρelρtp

)1(%T ) (3.2)

To find the velocity capable the crossing pointbetween the two curves can be referenced to twovelocities, a high and low speed.By observing figure 1 it can be proven that flightvelocity is dependent on percent thrust applied.By increasing thrust percent the total thrust perweight increases moving the intersection furtheralong the velocity scale. This relationship pointcan be described as the point when the thrustproduced from the aircraft equals the drag ap-plied to the aircraft reaching a state of equilib-rium.The characteristics of the drag to weight curve isshown to be parabolic like because of the over-powering skin friction drag at low speeds transi-tioning into heavy aero drag at high speeds witha null zone in-between. The significance of thecurve not reaching the X axis shows that if yourthrottle percent falls to a min value above zerothe aircraft will not be able to fly. The pointjust before you leave this curve is where the highand low velocities meet, were the max thrust ef-ficiency occurs at the lowest drag.

12

CHAPTER 3. THRUST MODELING 13

Figure 3.1: Thrust to Drag

1

CHAPTER 3. THRUST MODELING 14

3.3 Thrust vs Elevation

Thrust performance can be simply related to thealtitude you are flying. At higher altitudes equa-tion 1 will indicate a decrease in density causingthe air to be ”thinner”. This thinning effect willallow much less mas transfer as the fuel beingjettisoned from the rear of the plane will haveless molecules to press against creating a forwardpropulsion. This relationship can be categorizedas

TaltTsl

= [ρaltρsl

]na (3.3)

Similarly an expression for TSFC can be foundby observing the relationship between Thrustand Thrust specific fuel consumption and mea-suring the difference in the imperial constants.TSFC can be characterized by

TsfcaltTsfcsl

= [ρaltρsl

]nb (3.4)

Coefficients na and nb can are found imperiallyand are as follows

Coefficient Troposphere Stratosphere

na 0.7 1.0nb 0.2 .01

Table 3.1: imperial coefficients [1]

A relationship can be made not only to thrustas a function of elevation but as velocity at aconstant elevation as well. The fundamental the-ory here being the faster you are moving the lessthrust you will be producing due to aerodynamiceffects of the air being moved out of the waywere the high momentum fuel has a lesser effect.The relationship between thrust at altitude andthrust at sea level can be shown as a function ofvelocity in the following manner.

TaltTsl

= 1− 10−3V + 10−6V 2 (3.5)

3.4 Thrust at Altitude

The value of thrust/thrustsl demonstrates a di-mensionless value of percent thrust as com-pared with that available at sea level. SimilarlyTSFC/TSFCsl is a percent value as well. Theplot of this with respect to altitude is as follows

As shown the thrust available decreases expo-nentially as a function of altitude signifying at65,000 ft an aircraft has 15% of the thrust pre-viously available at sea level. Alike the thrustTSFC also decreases exponentially, however it isa very low slope curve. It can be seen that at65,000 ft an aircraft will not only have 15% ofits initial thrust but will use 40% more fuel massto accomplish his maneuver.

These two lines were plotted on the same chartto explain the relationship between loss of thrustand increase in fuel consumption denoted byTSFC. It is very obvious to see from the chartwhy compensating the aircraft with extra fuelfor a said mission and also to expect acrobaticperformance with respect to thrust to be limitedat altitude accounting for possible problems.

CHAPTER 3. THRUST MODELING 15

3.5 Thrust at Speed

Thrust is not only a function of altitude but afuntion of speed. By plotting the aircrafts ve-locity vs thrust/thrustsl we can determine thepercent loss of thrust on top of what was foundin the previous section.

Figure 3.3: Thrust at speed

Notice how the Y axis is broken to show thedifference more evidently. The plot shows a sim-ilar trend as thrust at elevation where as youtravel faster your available thrust falls to themagnitude of 25% loss at mach 1. The moreinteresting fact about this plot is the effect ofchanging velocity as the aircrafts speed increases.It is clear to see that you loose less thrust at 5,000ft at about 5% more. The interesting fact is howthe trend line then recedes back down at 10,000ft. This effect is because of Trust’s dependenceon mach velocity which is a function of speed ofsound which is a function of temperature whichbegins to increase at 10,000 ft causing the trendlines downward back toward the sea level curve.

Chapter 4

Flight Charictoristics

4.1 Air Speed

The characteristics of the atmosphere are everchanging as you ascend from the earth. Thesechanging characteristics will have a direct effectto aircraft performance similarly and must beaccounted for by the pilot, and design team tounderstand what his interments are truly tellingthem. As an example figure 1 a pilot can adjustthe indicated air speed with reference to the trueair speed.

In this study we will look into the effects ofhow the thrust and thrust specific fuel consump-tion (TSFC) change as you ascend up to 6500ft. Similarly the effect of thrust will be studiesas you accelerate an aircraft at a constant ele-vation to pinpoint the performance capabilitiesof the aircraft under various conditions. Lastlytrue air speed will be measured as a function ofelevation along with stall speed of the aircraft.

True air speed is defined by the the speed ofthe aircraft relative to the airmass in which it isflying [4]. While indicated air speed is defined asthe airspeed read directly from the airspeed in-dicator on an aircraft, driven by the pitot-staticsystem. IAS is directly related to calibrated air-speed (CAS), which is the IAS corrected for in-strument and installation errors [4]. The differ-ence between these two can be directly related tothe density of the atmosphere which will effectthe measurement of the pitot-static tube mea-surement for IAS.

Thrust specific fuel consumption is directly re-lated to the thrust available on the aircraft and

Figure 4.1: Air Speed Dial [3]

the amount of fuel needed to carry out its mis-sion. This component can be thought of in termsof pounds of fuel per pound of thrust produced.This not only effects the amount of fuel neededbut the total weight of the aircraft with thisweight added effecting all other characteristicsof the aircraft. This effect can be seen in figure2 as a function of both speed an elevation.

16

CHAPTER 4. FLIGHT CHARICTORISTICS 17

Figure 4.2: Fuel Weight Consumption [5]

Categorizing these performance characteris-tics are vital in understanding the capabilitiesof an aircraft. This change in performance isdriven by the change in density, temperature,and other components as you ascend though theatmosphere. If not properly accounted for a pi-lot can find himself flying faster or slower thandesired with respect to ground speed or to nothave the desired thrust he finds necessary underthe circumstances.

4.2 Atmosphere Constants

To understand the characteristics of flight thecharacteristics of the standard atmosphere mustbe well understood. Up to the elevation of 65,000feet we can relate the density of the air as

ρ =P1

RT1(4.1)

where T can be related as

T2

T1= [1− g0

RT1(n− 1

n)(Z2 − Z1)] (4.2)

where n = 1.235. Constant values for the stan-dard atmosphere used are as follows.

Parameter Symbol Value Units

Gravity go 9.807 m/s2

Specific heat γ 1.4 -Reference Temp To 273.15 KSea level Temp Po 101325 Pa

Seal level Density ρ 1.225 kg/m3

Table 4.1: imperial coefficients [1]

With this understanding we can further di-vulge into the performance of flight in the stan-dard atmosphere. Further analysis of the stan-dard atmosphere and detailed graphics can befound in AE 480 HW1.

4.3 True and Indicated AirSpeed

True and indicated air speed by the way they aremeasured. IAS is measured from a pitot-statictube mounted on the aircraft and is seceptableto alterations as a function of density. Similarlyenough the atmospheric density changes as youascend in an aircraft causing the IAS to differfrom TAS as you ascend though the atmosphere.according to Bernoulli’s equation for incompress-ible flow, true air speed can be calculated by

TAS =

√2δPoρalt

(4.3)

By substituting in the value a we can solve forthe TAS as a function of pressure wich is whatis being measure on board an aircraft by

a =

√γP

ρ(4.4)

TAS =

√2a2

γ(PtPs− 1) (4.5)

As Bernoulli’s equation assumed incompress-ible flow of air this equation can only be used upto a mach number of 0.3. To allow for the expan-sion past mach 1 a ”tweak” is made to the data

CHAPTER 4. FLIGHT CHARICTORISTICS 18

to imperially match the recorded data. The fol-lowing equation is as follows and encompass TASfor all IAS speeds including past mach 1.

TAS =

√2a2

γ − 1((PtPs

)γ−1γ − 1) (4.6)

4.4 Stall Velocity

Stall velocity is similarly related to the differ-ence between IAS and TAS. As you travel in aheaver than air aircraft you must produce a suffi-cient amount of lift to compensate the weight ofthe aircraft. The lift needed is set as a functionof altitude as when the air gets less dense thereis less molecules to provide uplift to the surfacearea of the wing. We can determine this speedby starting with the lift force of an aircraft.

Flift = Waircraft =1

2ρV 2ClS (4.7)

By solving for the velocity at a specified maxC value we can calculate the stall speed.

Vstall =

√2

(ws )

ρclmax(4.8)

4.5 True Air Speed

The following plot is a relationship between thetrue air speed and indicated air speed at 250 mphand at 0.85 mach number. The third curve willbe stall speed with respect to IAS to show itseffect in relationship to aircraft velocity.

As seen from the plot IAS and TAS verygreatly as you accent into the atmosphere. IAScan be shown to be a larger value than its TAScounterpart increasing logarithmically. The blueline indicates a constant IAS of 250 MPH whilethe indicated air speed will climb upwards of1000 MPH at 65,000 ft. This difference of 750MPH or 300% is extremely significant if planinga time specific mission or distance by time cal-culation. A further fact of this plot reveals thatthe effect of altitude to TAS is greater at fasterspeeds. For instance the curve representing 0.85Mach number differs from its IAS value by 400%.

CHAPTER 4. FLIGHT CHARICTORISTICS 19

Figure 4.3: True Air Speed

Speed of sound is plotted on the same chartto express the difference between IAS and TAS.For instance a aircraft with a wing loading of 90lb/ft3 and clmax of 1.2 when traveling at an IASof 250 mph a pilot will stay above the stall speedas he approaches 65,000 ft however if he is flyingwith TAS interments at 250 mph he will quicklystall around 20,000 ft because of the change indensity at that altitude.

4.6 Air speed Effects

In conclusion we have found a relationship be-tween thrust and TAS as a function of elevation.Observing thrust as a function of elevation is cru-cial to relating the needed power of an aircraftdurring design while the diminishing TSFC val-ues indicated more fuel would need to be broughton board for a flight higher in elevation because itwould not be as effective. Thrust was also eval-uated at different speeds indicating that a air-craft traveling faster will have less thrust avail-able causing effects in top speed or performancecharacteristics. All together the effect of thrustis a very dynamic characteristic and since thrustis a vital component to an aircraft it needs to beunderstood and tabulated thusly.

Similarly important is the relationship be-tween TAS and IAS. It was noted that while IASremains constant as a aircraft ascends thoughthe atmosphere the TAS will rise nearly 200%depending on the altitude reached and the IASvalue. This effect is crucial to understand interms of stall speed as to not accidentally fallbelow when you believe you are staying at thesame TAS as you gain altitude. It is also impor-tant to time flights accordingly and not to timeyour rout at 200 mph when in reality the aircraftis traveling in excess of 700 mph. This conceptis vital for pilots to understand and be able toutilize both values to adjust their flight accord-ingly.

The performance of an aircraft is very com-plex as many parts are dependence on each othermaking tabular data and plotted curves essentialto understanding the characteristics of a aircraftas it accelerates and ascends though the atmo-sphere.

Chapter 5

Aerodynamic Modeling

5.1 Lift

The relationship between lift and flight charac-teristics are sensitively effective on many vari-ables including altitude, wing size and shape andangle of attack among others. These effects dueto lift can be broken down into three categoriesof importance for a pilot or designer being theeffect on lift compared to angle of attack, dragcoefficient and efficiency.

5.2 Effects of Angle of Attack

An aircraft with the max lifting coefficient of aBoeing 747 is charted for its relationship betweenthe angle of attach of its wings and the lift co-efficient created. This overall performance willbe effected by the size and shape of the wings aswell as the elevation and speed of the aircraft.All presumptions can be found in the appendix[1] to fill equation 1.

CL =2L

ρV 2S+

πAR

1 + [1 + ( AR2cosλ)2]

12

+ α (5.1)

By sweeping the Aspect ratio of the aircraftswings from 3 to 6 it is possible to see the rela-tionship between wing sizing and flight charac-teristics as shown in figure 2 and 3.

Figure 5.1: Lift Coefficient at Altitude

As shown in figure 2 the simple effect of alti-tude can change the dynamics between CL andangle of attack. By ascending though the atmo-sphere the Boeings wings begin to be less effec-tive as when 10,000 ft is reached the angle ofattack must be increased to 1 degree to createzero lifting force where at sea level a 0 angle wasneeded. Similarly by staying at a cruising alti-tude and varying the sizing of the wings you willsee the trend in figure 3.

Figure 3 demonstrates what common scenehints at which is how the larger the aspect ratioof a wing their effectiveness at creating lift willbe greater with a lesser angle of attack due tothe added frontal wind loaded surface.By combining both of these results on CL vs αit can be observed how the linear relationshipcan be shifted vertically via changing the cruis-ing altitude of your aircraft, or to rotate each lineabout the origin by changing the aspect ratio of

20

CHAPTER 5. AERODYNAMIC MODELING 21

Figure 5.2: Lift coefficient by wing parameters

the wings. By altering these flight characteristicsa family of curves can be observed to encompassany flight performance the designer would like.

5.3 Effects of Drag

The relationship between lift and drag is syn-onymous with aircraft design. The extent of thiscan be shown in equation 2 where an aircraftsdrag coefficient is not only dependent on the co-efficient of lift but by its square. This makesit doubly important to understand what extentdrag will inhibit aircraft performance.

CD =1

2ρV 2[CDo + (KC2

L)] (5.2)

The effect of lift and drag for a Boeing 747with constants found in the appendix [2] arecharted in figure 3 by measuring the drag perlift generated as the wings travel from 90 degreesdownwards vertical to its inverse upwards shownin figure 4.

1

It can be seen that CD increases exponentiallyas CL increases by result of increasing the an-gle of attack of the wings. This performancehindrance is due to the increase in frontal areashown by the wing section. Similar to section 1.1the ”potency” of drag to the lift of the aircraftis increased as the aspect ratio is increased dueto a larger frontal drag area of the wings.The effect of drag is important in deciding on anaspect ratio in a aircrafts design stages. As seenearlier the aspect ratio can greatly improve liftcharacteristics but will increase drag faster thanbenefiting performance creating a situation fordesign optimization depending on flight charac-teristics desired. In this case of a 747 a more”sluggish” aircraft with a lower aspect ratio isdesired for its benefits in lower drag intern lowerfuel costs to transport goods.

5.4 Effects on Efficiency

Out of the given effects lift force has on an air-craft the effect on efficiency is most important toa designer. Efficiency is defined by the lift overthe drag force causing an efficient lifting condi-

CHAPTER 5. AERODYNAMIC MODELING 22

Figure 5.3: Lift coefficient with respect to drag

tion to induce less drag than that of an alternatecondition and equated as in equation 3.

E =CL

CDo +KC2L

(5.3)

A maximum theoretical efficiency can also becalculated using equations 4 and 5. The load-ing conditions of the same Boeing 747 have beenmodeled to simulate a wing span of three sep-arate aspect ratios as discussed previously. Asshown in figure 5 the efficiency of the wing is rel-atively high at low lift values such as cruisingconditions, however decrease exponential withlift on the aircraft until reaching an asymptoteat one describing when the lift force is equal tothe drag force. This situation is understood byobserving the angle at a 90 degree pitch makinga flat plate into the wind causing all of the ”lift”force by the orientation of the origin followingthe aircraft being created only by drag pushinghorizontal to the horizon.

1

Emax =

√1

4KCDo(5.4)

CLEmax =

√CDoK

(5.5)

Figure 5 shows clearly the effect of the as-pect ratio of the wings as when increased thewings become less efficient for their given liftforce. This can be directly correlated to section1.2 were it was found how larger aspect radioedwings will induce greater drag per uplift forcebecause of their enlarged frontal area. By thesame reasoning a larger aspect ratio wing willnot have the required surface area to provide liftto negate the drag effects of the frontal area.

CHAPTER 5. AERODYNAMIC MODELING 23

Figure 5.4: Lift coefficient to Wing Efficiency

The graphical comparison of lift coef. and ef-ficiency can be compared to the theoretical maxvalue of drag and lift were the max wing effi-cacy can be found not at zero lift coefficient butat a small number ranging around 0.02. Thisshows how the efficacy curve does not asymp-toticly approach infinite efficacy but has a maxvalue maxing out shortly after falling below a liftcoefficient of 2.

The lifting characteristics of a Boeing 747 haveallot more effect on the aircraft than simple liftbut also effect the drag, performance and evenfuel consumption. By observing the effects onlift with variables such as angle of attach anddrag the importance of a properly designed wingsection becomes evident. The trade off betweenthe flight performance benifit of a high AR wingand the drawback of the low efficiency of thatsame wing are key factors to consider when de-signing for a particular application.

Chapter 6

Flight Enviorment

6.1 Flight Envelope

A flight envelope graphic equates the aerobaticregion in which an aircraft can operate. Thefollowing flight envelope is a representation of aBoeing 747 ascending to its max altitude at 250knots (TAS) also traveling at 0.8 mach. The ef-fects that will define the flight envelope the air-craft can fly in are Stall speed, max structuralvelocity and a combination of high and low ve-locities that describe the possible velocities as afunction of engine thrust.The importance of stall speed to entrap an air-crafts performance is to remain above and pro-duce enough lift to remain in flight shown inequation 3.

Vstall =

√2ws

ρCLmax(6.1)

The purpose of Max structural velocity is toboundary the velocity were the structural com-ponents of the aircraft will fail denoted by equa-tion 4.

Vqmax =

√2qmaxρ

(6.2)

Furthermore the effects of min and max veloci-ties place a boundary on the performance capa-bilities of the engine at a given thrust percentshown in equations 5 and 6.

V1 =

√√√√ TS

ρCDo[1 +

√1− 4KCDo

( TW )2] (6.3)

V2 =

√√√√ TS

ρCDo[1−

√1− 4KCDo

( TW )2] (6.4)

As shown in figure 2 the flight envelope is en-compassed by the stall speed on the low velocityend, the thrust limitations on the upper altitudesnearing the tropopause finally only limited at thehigh velocity end by the thrust limitations andthe structural integrity of the fuselage and wings.This regions highlighted in orange demonstratesthe limitations of a Boeing 747 in flight.The characteristics of the stall speed can be seento curve to the right as altitude increases indi-cating a higher velocity stall. This is the causeof a thinner atmosphere producing less lift onthe wings therefore requiring a higher velocityto stay aloft. Similarly qmax or structural limitis dependent on the density of the air impactingthe aircrafts structure causing the effect to lessenas you increase in altitude.The significance of the intersection points be-tween the flight speed of the aircraft and theflight envelope boundary in understanding themax elevation at which an aircraft can fly at agiven IAS. This crossing point also references amax TAS possible at the max altitude of the air-craft.

24

CHAPTER 6. FLIGHT ENVIORMENT 25

Figure 6.1: Flight Envelope

To determine the absolute ceiling of the air-craft a horizontal line can be drawn tangent tothe curve of V2 simulating a signal IAS at whichthe aircraft can achieve its max altitude. Thispoint is significant in understanding the theo-retical performance when designing an aircraft.Flight ceiling can be import to meet FAA guide-lines, Vision range or even aerobatic needs.

The given characteristics of a Boeing 747 havebeen determined to perform well within desiredcursing speeds and altitudes and remain withincurrent FAA regulations. The aircraft can sus-tain flight comfortably at a velocity quick enoughto provide its customers with quick transporta-tion while allowing a comfortable margin be-tween cursing speed and Vqmax.

Chapter 7

Takeoff

7.1 Runway Length

Each airport is built for a specific class of air-crafts. The size of these aircraft dictate the re-quired runway length needed to become aloft andis directly proportional to the weight of the air-craft as well as the thrust available. Equation1 expresses the the need for runway length as afunction of the dimensionless weight of the air-craft.

XT.O =1.44(WS )

g( TW )CLmaxρrunway(7.1)

7.2 Runway Modeling

By using equation 1, runway distances canbe calculated for an array of aircraft types.Distances were taken for aircraft of increasingweight to size as well as thrust available relatedto the weight of the aircraft. This gave a range ofaircraft that cover most transport jet categories.To account for the variation in altitude betweenairports a sample was taken for both sea leveland 5,000 ft MSL.

Figure 7.1: T.O Key (Figure 2)

The plot shown in figure 2 demonstrates the

length or runway needed to take off a wide arrayof jet transport aircraft from two different eleva-tions. The key demonstrates first the elevationof the airport followed by the T/W ratio.

What the plot demonstrates is the increase ofrunway needed the more weight is applied to thelifting surface of an aircraft. The second thingto see is the increased distance needed as thethrust is decreased in the aircraft. Lastly it isshown that more runway is needed at higher el-evation airports for a similar aircraft.

The final result shows a range of runwaylengths from 700ft to 5,500 ft ranging from thelightweight large thrust to heavy underpoweredaircraft.

26

CHAPTER 7. TAKEOFF 27

Figure 7.2: Take off Runway Characteristics

In conclusion the effects of wing loading as wellas thust ratio have been discussed in their effecton take off and landing distance. It was shownthat the higher the wing loading the larger thedistance required to both take off and land. Sim-ilarly the less thrust available caused the need fora longer take off runway. Lastly It was shownthat elevation change had a key roll in runwaylength as the higher above sea level an airportlies, the longer the runway must be to take off orland on.

Chapter 8

Flight Range

8.1 Improving Range

Determining the Range of a Boeing 747 is shownin two ways. The first as theorized by Breguetand the second a modern day calculation de-rived from strict F.A.A. guidelines.These equa-tions root from equation 1 however differ subtlyfrom one another but result in a very differentoutcome.

R =

∫−V EC

dW

W(8.1)

The Breguet equation makes the assumptionof a constant wing efficiency of the aircraftto simplify calculation inatvertantly modeling asystem of a cruise climb condition. Equation 2shows the relationship between range and air-craft conditions, also expressed graphically infigure 1. The Breguet equation is still utilizedtoday not for its flight path but because it rep-resents the path of maximum fuel efficacy. Byflying in a cruise climb condition an aircraft un-der the same conditions and fuel load as a con-stant elevation flight is able to go further.

R =−V E1

CLn(1−

∆Wf

W1) (8.2)

Although the Breguet equation demonstratesthe max range path the F.A.A for safety has lim-ited planes to flying in 2,000 ft planes. Thischanges the wing efficiency as you travel andburn fuel causing the effect of range to be de-termined by equation 2 and similarly plotted infigure 1.

To abide by safety regulations and to maximizethe range of an aircraft flights are currently flownunder stepped flight. This flight path is the thirdand final plot of range in figure 1 demonstratinga elevation change of 2,000 ft at every intervalof flight to maintain safety of cross traffic whileallowing for a cruise climb like condition.

R =2EmaxV

CTan−1(

∆Wf

W1E1

2Emax(1−KClE1∆Wf

W1

)

(8.3)

8.2 Range Modeling

To determine the range of a 747 parameters mustbe known of the aircraft. These parameters in-clude Wing loading, Aspect ratio, weight amongother characteristics to achieve a proper model.Each of these three range equations can be plot-ted to show the 2D flight path of the aircraftunder three conditions. A graphic representa-tion rather than a numerical output of range willshow the benefit in range vs the elevation changeburden.

Figure 1 demonstrates the effect of the cruiseclimb mentioned earlier in that it has extendedthe range of the aircraft. The first deduction forthis plot is to notice the magnitude difference inthat a full fuel tank can travel nearly 800 milesfurther if the pilot chooses to step his flight pathas opposed to flying at a steady altitude. Also tobe noticed is how the stepped flight path differsslightly from that of the Breguet equation flight.

28

CHAPTER 8. FLIGHT RANGE 29

Figure 8.1: Aircraft Range

This is found when you pass the tropopause werethe atmosphere changes characteristics via tem-perature and density. Since the stepped altitudeflight is recalculated for atmospheric conditionsat every step this equation accounts for thesechanges in aircraft performance where the stepedflight assumes the starting elevation of 31,000ft for its final calculation causing the parame-ters not to reflect the change in atmosphere asit passes the natural barriers of the tropopause.The compinsation for these characteristics alsoshow to give a stepped flight a 800 mile rangeadvantage over that of its steady flight counter-part.

The effects of these three modes of flight wereshown to differ much in terms of flight range withsteped flight shown to result in the longest rangeand a steady altitude flight for the shortest withrespect to a 747 full of fuel under the same con-ditions.

1

The climb flight path is in between the twopreviously mentioned but is mostly modeled forits likeness to stepped flight with a much simplercalculation. The error to many is negligible andincorporated into their factor of safety.

Chapter 9

Elevation Change

9.1 Climb Rate

The climb rate of an aircraft can be calculatedas a function of many things including altitude,angle of attack and velocity. The two options ofclimb being observed here are what rate of climbwill result from the steepest climb and from thefastest climb between elevations.The equation for the rate of climb of an air-craft traveling at various velocities is shown be-low. equation 1 creates an inverse parabola curveshown in figure 1 displaying the climb rate char-acteristics as a function of velocity and elevation.From this curve the characteristics for steepestand fastest climb can be determined.

R

C= [

T

W− (

QCD0

WS

+KWS cos

2(θc)

Q)]V (9.1)

To determine at which angle the aircraft canfly to accomplish the steepest climb θ steepestcan be calculated from equation 2.

θc = sin−1((T

W)max −

Cos(θc)

Emax) (9.2)

To move from the generic rate of climb curveto the steepest climb equation 3 shows the re-lationship between the angle of attack and thenon dimensional weight to surface area param-eter. Other characteristics such as atmospheredensity attribute to the amount of lift needed tokeep the aircraft aloft. This function can also berepresented graphically by drawing a line from

the origin of figure 1 and passing though the tan-gent of the respected curve to result in the largestangle between that line and the x axis. The placewere this line intersects the rate of climb curveis the velocity and rate of climb of the steepestattack angle at a given elevation.

Vθc =

√2ws cos(θc)

ρ4

√k

cD0

(9.3)

To look at the opposite end of the spectruma point on the generic rate of climb graph canbe found for the quickest climb between two al-titudes. Equation 4 shows how to calculate heangle of this flight path which will result in an an-gle less than the steepest flight but greater thanzero.

θfastestclimb = sin−1(T

W(1− τ

6)−3cos(θc)

2τE2 TW

(9.4)

Finally in determining the velocity of a flightpath though the quickst climb rate is shown inequation 5. Graphically this equation representsthe upper crest of the climb curve. This point inthe curve represents were the rate of climb is atits most before it begins to fall again resulting inthe quickest vertical climb rate.

Vfastestclimb =

√Ts τ

3ρCD0

(9.5)

30

CHAPTER 9. ELEVATION CHANGE 31

9.2 Elevation Change

Rate of climb curves were calculated for ele-vations of 10,000 , 20,000 and 30,000 ft. Thefastest climb rate and the steepest climb rate arethen extracted and tabulated to gain knolegeabout the performance of the aircraft at thegiven elevations

From the previous equation and the parame-ters of the aircraft data for the Velocity and angleof the steepest climb can be calculated thoughall three elevations as shown in table 2. Alsoincluded in table 2 is the velocity and angle ofthe fastest climb rate. There are two theta’s ineach section of the table for the purpose of it-eration as the equation for finding theta is nothomogeneous.

Table 2 : Climb Parameters

To add meaning to the values calculated forvelocity and attack angle a plot can be madeto depict the characteristics of a jet transportplane in flight. Each elevation will have a rate ofclimb characteristic vs velocity as defined in theprevious section leaving a plot shown in figure 1.

The significance of this plot is to show the dis-tinction between the fastest rate of climb and the

steepest rate of climb. at 30,000 ft the plots forboth steepest and quickest climb rates are shownwere the angle between the line and the x axis isthe angle of attack. As it shows the vertical dis-tance were the steepest climb line crosses is wellbelow the point at which the quickest interceptsat the top of the parabola. This distance onlybecomes more dramatic as you decrease in ele-vation for instance a 250 ft/s difference betweenthe steepest climb and the fastest climb at 10,000ft.

Also a key feature to this plot is the increasein angle needed to accomplish quickest flight asyou decent though the atmosphere. The mainpurpose for this is the increase density givingmore lift to the wings at a higher attack angle.

CHAPTER 9. ELEVATION CHANGE 32

Figure 9.1: Rate of climb characteristics

Chapter 10

Holding Pattern

10.1 Holding Pattern Charac-teristics

Holding pattern dynamics are important to re-main fuel efficacy while holding which means theaircraft must travel slowly and smoothly. In or-der not to fall below the stall speed holding pat-tern characteristics need to be made to charac-terize the flight speed, load factor, turn angleand radius of turn.The stall speed of a aircraft in a turn goes upas a function of the roll of the aircraft. This isdue to the loss of lift developed by the wings notbing normal to the lifting plane. The resultantof this is a velocity required defined in equation6.

Vturn =

√Ts

ρCD0

(10.1)

Also a very important factor when making aturning maneuver in an aircraft is the stress fac-tor of the wings attached. ”n” represents theload factor of the aircraft which is not to goabove 2.5 for jet transport class aircrafts.

n = Emax(T

W)max (10.2)

By remaining within the tolerances of stallspeed and loading factor a radius of turn can becalculated by basic dynamics and is as follows inequation 8.

r =V 2

g√n2 − 1

(10.3)

Lastly a pilot must remain in a comfortableroll angle as to not distribute the payload onboard. A roll angle can be calculated given theprevious parameters shown in equation 9.

φ = Tan−1V Xdot

W(10.4)

10.2 Maneuverability

Turning charismatics defined in the previous sec-tion were calculated from 1000 to 18000 ft. Thiswill express the turning characteristics of a planein a holding pattern as it transitions from itscursing altitude to its landing altitude.

33

CHAPTER 10. HOLDING PATTERN 34

Figure 10.1: Turn Characteristics

Some key features to notice about this plot isthe drastic range of radius from the two extremeelevations. at 1000 ft an aircraft is essentially tentimes as adjile and only needing to roll a fractionof the amount to gain the needed maneuver.

The performance characteristics of an aircraftoutside of its cruising conditions are found to behighly dependent on elevation. The climb rate ofan aircraft was found to be drastically differentbetween the steepest angle an aircraft can climbto the fastest angle an aircraft can climb whichis not completely intuitive.Also noted was the difference in turn perfor-mance from high to low elevations. Specificallythe amount of roll needed to complete a maneu-ver drastically decreased as you decreased in el-evation.

Chapter 11

Landing

11.1 Runway Length

runway length for a landing must be calculatedusing similar characteristics shown in equation2.

XLand = 118(W

S

1

CLmax

ρSLρEL

) + 400 (11.1)

With many similarities to equation 1, equa-tion 2 is not effected by the thrust available onthe aircraft as the engines will be in idle duringthe landing. There is also a 400 ft rolling dis-tance to come to a complete stop once toucheddown. This equation is also dependent on ele-vation meaning a runway in san fransisco wouldnot be adequate in Detroit.

Parameters for the take off and landing of theaircraft are shown in table 1.

Table 1: Input Parameters

These parameters indicate an aircraft with asimilar lifting weight but with a varying enginesize. It also expresses the importance of flaps ontakeoff and landing helping increase Cl max.

Landing, similar to take off is related via air-craft characteristics shown in equation 2. Land-ing lengths are plotted for a variation of wingloading values as well as the same two elevationspreviously mentioned in Takeoff analysis.

The plot in figure 3 shows the effects of wingloading on an aircraft while landing. The higherthe wing loading value the more runway distanceis required to land a jet transport plane. Alsoshown is the effect of altitude in that the higherthe airport above sea level the longer the runwayneeds to be. A landing range between 1,700 and5,800 ft are similar but larger than the length re-quired to take off making landing the governinglength needed for an airport.

Figure 11.1: Landing Runway Characteristics

35

Chapter 12

Aircraft Modeling

12.1 Aircraft Modeling

The design of the aircraft modeled above demon-strates the take-off to landing analysis of a jettransport aircraft. The elements that were fo-cused upon were the atmospheric effects, Airspeed, Aerodynamics, Flight envelope, Rangeand ground maneuvers. By beginning with a setof parameters including size and shape of theaircraft it was determined the limitations of themodeled aircraft in each of these aspects.

12.2 Atmosphere

Because of the atmospheres altering propertiesit was found that the thrust characteristics of anaircraft will differ with elevation. With the jettransport aircraft given it was determined fromchapter 3 that the available thrust of an aircraftwill reduce with altitude. A complement to thisfinding is the effect of a model as you travel fasteryour avalible thrust drops on the order or 25%.

12.3 Air Speed

It was found that a relationship between thrustand TAS as a function of elevation. Observ-ing thrust as a function of elevation is crucialto relating the needed power of an aircraft dur-ing design while the diminishing TSFC valuesindicated more fuel would need to be brought onboard for a flight higher in elevation because itwould not be as effective. Thrust was also eval-

uated at different speeds indicating that a air-craft traveling faster will have less thrust avail-able causing effects in top speed or performancecharacteristics. All together the effect of thrustis a very dynamic characteristic and since thrustis a vital component to an aircraft it needs to beunderstood and tabulated thusly.

Similarly important is the relationship be-tween TAS and IAS. It was noted that while IASremains constant as a aircraft ascends thoughthe atmosphere the TAS will rise nearly 200%depending on the altitude reached and the IASvalue. This effect is crucial to understand interms of stall speed as to not accidentally fallbelow when you believe you are staying at thesame TAS as you gain altitude. It is also impor-tant to time flights accordingly and not to timeyour rout at 200 mph when in reality the aircraftis traveling in excess of 700 mph. This conceptis vital for pilots to understand and be able toutilize both values to adjust their flight accord-ingly.

The performance of an aircraft is very com-plex as many parts are dependence on each othermaking tabular data and plotted curves essentialto understanding the characteristics of a aircraftas it accelerates and ascends though the atmo-sphere.

12.4 Aerodynamics

The lifting characteristics of a Boeing 747 haveallot more effect on the aircraft than simple liftbut also effect the drag, performance and even

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CHAPTER 12. AIRCRAFT MODELING 37

fuel consumption. By observing the effects onlift with variables such as angle of attach anddrag the importance of a properly designed wingsection becomes evident. The trade off betweenthe flight performance benifit of a high AR wingand the drawback of the low efficiency of thatsame wing are key factors to consider when de-signing for a particular application.

12.5 Flight Envelope

The given characteristics of a Boeing 747 havebeen determined to perform well within desiredcursing speeds and altitudes and remain withincurrent FAA regulations. The aircraft can sus-tain flight comfortably at a velocity quick enoughto provide its customers with quick transporta-tion while allowing a comfortable margin be-tween cursing speed and Vqmax.

12.6 Climb

The performance characteristics of an aircraftoutside of its cruising conditions are found to behighly dependent on elevation. The climb rate ofan aircraft was found to be drastically differentbetween the steepest angle an aircraft can climbto the fastest angle an aircraft can climb whichis not completely intuitive.Also noted was the difference in turn perfor-mance from high to low elevations. Specificallythe amount of roll needed to complete a maneu-ver drastically decreased as you decreased in el-evation.

12.7 Range

The effects of these three modes of flight wereshown to differ much in terms of flight range withsteped flight shown to result in the longest rangeand a steady altitude flight for the shortest withrespect to a 747 full of fuel under the same con-ditions. The climb flight path is in between thetwo previously mentioned but is mostly modeled

for its likeness to stepped flight with a much sim-pler calculation. The error to many is negligibleand incorporated into their factor of safety.

12.8 Ground Maneuvers

In conclusion the effects of wing loading as wellas thust ratio have been discussed in their effecton take off and landing distance. It was shownthat the higher the wing loading the larger thedistance required to both take off and land. Sim-ilarly the less thrust available caused the need fora longer take off runway. Lastly It was shownthat elevation change had a key roll in runwaylength as the higher above sea level an airportlies, the longer the runway must be to take off orland on.

Chapter 13

Appendix

13.1 References

[1] Dr.Fox, Lecture Notes(Aeronautical Engineering 480”Introduction to Aeronautics”)

[2] Anderson, John David.Introduction to Flight (seventh Edition).

[3] ”Raymer’s Aircraft Design & RDS Site.”Raymer’s Aircraft Design & RDS Site.N.p., n.d. Web. 17 Dec. 2012.

[4]”Aviation History, History of Flight,Century of Flight.” Aviation History, History of Flight, Century ofFlight”.N.p., n.d. Web. 17 Dec. 2012.

[5] Study Partner : None

13.2 Homework Questions

13.2.1 Atmospheric model

AE 480 HW#1

Explore the earths atmospheric structure(s) onthe web and write a brief description of atmo-spheric modeling; include relevant graphic im-ages from the web. Include websites explored ina bibliography at the end of your HW#1 report(see Note below).Create an Excel spreadsheet to determine thevalues of temperature, density, pressure andacoustic speed up through the atmosphere to32,000 meters ( 100,000 feet), starting at sea leveland assuming a US Standard (geopotential) at-mosphere model. Let n = 1.235 for a polytropicmodel in the troposphere, use the isothermalmodel for the lower stratosphere (-56.5C), andlet n = 0.974 for the third atmospheric layer.Output a table of values with elevations given inboth meters and feet; work in SI units (meters)and convert to feet for the second column of yourtable. 1000 meter increments will be adequate.Plot T, D and P vs. elevation from sea level to 32km ( 100,000 ft - geopotential assumption), in-clude the tropopause elevation of 11 km (36089feet) as a point in your calculations, using datafrom the spreadsheet you created above. In yourplot, let elevation be the y-axis for each of these3 plots. Add appropriate labels for key pointsand regions on your plot.

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13.2.2 Thrust Effects

AE 480 HW#2 (due September 25, 2012)

#2-1 Plot Thrust/Thrustsl and Tsfc/Tsfcsl to65,000 feet (ASL) for a US Standard Atmo-sphere.

#2-2 Plot Thrust/Thruststatic vs. Mach num-ber for 0 ¡ M ¡ 1.4 at sea level, 5000 feet (ASL),and 10,000 feet (ASL) on one chart.

#2-3 Plot the following from Sea Level to theTropopause (elevation on the vertical axis):

(a) True Airspeed for an indicated air speed(IAS) of 250 MPH

(b) True airspeed for a constant Mach number= 0.85

(c) Stall speed for a wing loading of 90 lbs/ft2and a CLmax of 1.2

13.2.3 Aerodynamics

AE 480 HW #3 (due October 02, 2012)

#3-1 Plot CL vs. α for values of aspect ratio 3,6, and 10 with a sweepback angle of 35 usingthe simpler lift curve slope expression presentedin the lecture. Presume CLmax is 1.2 and asymmetric airfoil.

#3-2 Plot the parabolic drag polar, CL vs. CD,for AR of 3, 6 and 10 with sweepback angle =35. Presume an Oswald efficiency factor of 0.85and a parasitic drag coefficient (CDo) of 0.016.

#3-3 Plot aerodynamic efficiency (E) vs. CLusing the same parameters of problem

#3-2. Compare the results with theoreticalEmax and CL@Emax.

13.2.4 Flight Envelope

AE 480 HW #4 (due October 9, 2012)Note: For transport class jet aircraft, presumethat 60 ¡ W/S ¡ 120 and that T/W ¿ 1/Emaxwith a presumed T/W)sea level rating of 0.35+/- .10. And Presume appropriate values forany missing parameters.Let W/S and T/W be input parameters for thisstudy.#4.1 Plot D/W and T/W vs. true airspeed at10,000 ft ASL for a transport class jet aircraft.For T/W, plot results for three (3) throttlesettings 60, 75& 90%, as discussed in class;ignore the impact of flight speed on thrust.#4.2 Plot the flight envelope for a jet transportclass aircraft that includes curves for:V1 = hi speed equilibrium flightV2 = lo speed equilibrium flightVstallVcruise for M inf = .85Vqmax for qmax = 650 lbf/ft2V 250 Knots IAS (below 10,000 ft)Ignore the impact of flight speed on T/W in yourcalculations; use T/W at 85% throttle adjustedfor elevation impact (note this approximatesnominal cruise T/W.)Presume CLmax 1.2 for stall calculations, i.e.without flaps.Dont overlook the fact that the atmosphericproperty model and the thrust model variationwith elevation change at the tropopause.What is your absolute ceiling?

13.2.5 Range

AE 480 - HW #5 (due October 16, 2012)#5.1 Compare the range(s) of a turbofan pow-ered jet transport aircraft having a cruise fuelweight fraction of 0.38, a sea level Tsfc of 0.65lb fuel/hour/lb thrust, a sea level rated T/W of0.33, a cruise initial W/S of 110 lbf/ft2, and aCDo = 0.0155, predicted for the three flight pro-files indicated below. Choose appropriate valuesfor any missing parameters (e.g. AR, e).

CHAPTER 13. APPENDIX 40

Presume a cruise Mach of 0.85 with cruise start-ing at 31,000 feet MSL. When you set up yourspreadsheet, create an input field for all vari-ables/parameters, and all calculated parametersplus a key results field for final answers (e.g.range, flight time) required for each scenario.Range flight scenarios:Constant elevation using the arctan expressionfor range (FAA rules).Breguet range (wishful thinking ops)Stepped altitude flight using the arctan expres-sion for each level flight step; presume 2000 footsteps (currently allowed by FAA rules).Plot elevation vs. range (in statute miles) for allthree flight profiles on the same chart.Plot W/S and % throttle required vs range, ateach altitude step, for the stepped altitude rangemodel.How long (hours) will each flight profile require?

13.2.6 Rate of Climb

AE 480 - HW #6For the problem below, select a W/S and T/Wfrom the ranges given below and choose appro-priate values for all other missing required air-craft parameters; choose values similar to thosefrom HW #5:40 ¡ W/S ¡ 120 lbf/ft20.25 ¡ T/Wsea level ¡ 0.45#6.1 Plot rate of climb as a function of flightspeed at sea level, 10,000, 20,000 and 30,000ft above MSL, applying 100% rated thrust (ad-justed for elevation; ignore the flight speed effecta 2-point line from the origin to the point oneach R/C curve corresponding to steepest climb.Overlay a curve depicting fastest climb as yougo from sea level to 30,000 feet. Express R/C infeet per minute.

13.2.7 Take off & Landing

AE 480 - HW #7#7.1 Estimate Take-off Runway Length(s)Working with the Take-off Parameter (TOP)model introduced for aircraft conceptual design,

estimate the required take-off runway lengthfor a turbofan powered jet transport aircraft.Let sea level rated T/W be a parameter withvalues of 0.25, 0.35 and 0.45. Let W/S varyfrom 40 to 120 lbf/ft2. Assume a CLmax inthe neighborhood of 1.2 without flaps. Chooseappropriate values for any missing parametersrequired to complete the assignment.Plot estimated take-off runway length in feet(sTO) vs. W/S with T/W as a parameter forboth a sea level runway and a runway at 5000ft MSL (e.g. Denver, CO). Presume a CLmaxfor take-off (with flaps) at 1.5 x CLmax with noflaps.

#7.2 Estimate Landing Runway Length(s)

Working with the Landing Parameter (LP)model introduced for aircraft conceptual design,estimate the required landing runway lengthfor the turbofan powered jet transport in #7.1above, for both a sea level runway and a runwayat 5000 ft MSL. Note that T/W does not play arole in the landing analysis. Presume a CLmaxfor landing equal to 2.5 x CLmax without flaps.Let W/S vary from 40 to 90 lbf/ft2 for landing.

Plot estimated landing runway length in feet(sL) vs. W/S with remaining fuel load as a pa-rameter for both elevations.

13.3 Raw Data

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