STABILITY ANALY SIS OF F -4C PHANTOM AIRCRAFT: …

i

ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

STABILITY ANALYSIS OF F-4C PHANTOM AIRCRAFT: MODELLING AND

SIMULATING

GRADUATION PROJECT

Yusuf Kadri KAYAR

Department of Aeronautical Engineering

Thesis Advisor: Prof. Dr. Elbrus CAFEROV

JUNE, 2021

ii

ISTANBUL TECHNICAL UNIVERSITY FACULTY OF AERONAUTICS AND ASTRONAUTICS

STABILITY ANALYSIS OF F-4C PHANTOM AIRCRAFT: MODELLING AND

SIMULATING

GRADUATION PROJECT

Yusuf Kadri KAYAR

110160542

Department of Aeronautical Engineering


JUNE, 2021

iii

Yusuf Kadri Kayar, student of ITU Faculty of Aeronautics and Astronautics

student ID 110160542, successfully defended the graduation entitled “STABILITY

ANALYSIS OF F-4C PHANTOM AIRCRAFT: MODELLING AND

SIMULATING”, which he prepared after fulfilling the requirements specified in the

associated legislations, before the jury whose signatures are below.


Istanbul Technical University

Jury Members: Asst. Prof. Dr. Hayri ACAR


Dr. Cemil KUTCEBE


Date of Submission : 14 June 2021

Date of Defense : 28 June 2021

iv

To my family,

To my thesis advisor Prof. Dr. Elbrus CAFEROV

v

FOREWORD

Initially, i must state that i am so grateful and honored to be taking graduation project

from my dear advisor Prof. Dr. Elbrus CAFEROV, who enlightened my path and

guided me with his trust and help. I also want to thank you to my parents Selim

KAYAR and Sevinç KAYAR who have been always supporting me throughout my

education life with a great love and sacrifice.

June 2021 Yusuf Kadri KAYAR

vi

vii

TABLE OF CONTENTS

Page

FOREWORD ......................................................................................................................... v

TABLE OF CONTENTS ................................................................................................... vii

ABBREVIATIONS ........................................................................................................... viii

LIST OF SYMBOLS ........................................................................................................... ix

LIST OF TABLES ............................................................................................................. xii

LIST OF FIGURES .......................................................................................................... xiii

SUMMARY .......................................................................................................................... xv

1. INTRODUCTION ..............................................................................................................1

2. F-4C PHANTOM AIRCRAFT ..........................................................................................2

2.1 Design .............................................................................................................................3

2.2 Development ...................................................................................................................4

2.3 Specifications ..................................................................................................................4

3. INTRODUCTION TO STABILITY .................................................................................6

3.1 Static Stability .................................................................................................................6

3.2 Dynamic Stability ...........................................................................................................8

4. LONGITUDINAL MOTION ..........................................................................................10

4.1 Longitudinal Approximation ........................................................................................11

4.2 Short-Period Approximation ........................................................................................12

5. STABILITY OF F-4C PHANTOM AIRCRAFT ..........................................................14

5.1 Stability Derivatives .....................................................................................................17

5.2 Longitudinal Motion ....................................................................................................20

5.2.1 Short-Period Dynamics ....................................................................................20

5.2.2 Long-Period Dynamics ....................................................................................22

5.3 Lateral Motion ..............................................................................................................23

5.3.1 Roll Dynamics ..................................................................................................23

5.3.2 Dutch-Roll Approximation ..............................................................................24

5.4 Longitudinal Motion Modes ........................................................................................27

5.5 Lateral Motion Modes ..................................................................................................28

6. TIME RESPONSES .........................................................................................................30

7. ROOT LOCUS ANALYSIS METHOD ..........................................................................32

8. AUTOPILOT MODELLING .........................................................................................33

9. CONCLUSION ................................................................................................................38

REFERENCES .....................................................................................................................39

APPENDICES ......................................................................................................................40

APPENDIX A .....................................................................................................................41

APPENDIX B .....................................................................................................................42

viii

ABBREVIATIONS

BLC : Boundary Layer Control

GE : General Electric

MATLAB : Matrix Labaratory

NASA : National Aeronautics and Space Administration

SAS : Stability Augmentation System

TURAF : Turkish Air Forces

USAF : United States Air Fores

USAN : Unites States Navy

ix

LIST OF SYMBOLS

Symbol Desription Units

Aspect ratio -

Wing span Feet

Chord Feet

Inner chord Feet

Outer chord Feet

Drag coefficient -

Parasite drag coefficient -

Induced drag coefficient -

Skin friction coefficient -

Lift coefficient -

Roll moment coefficient -

Pitch moment coefficient -

Zero-angle of attack pitch

moment coefficient -

Yaw moment coefficient -

Pressure coefficient -

Side force coefficient -

Drag force Lbf.ft.sec2

Oswald efficiency number -

Slenderness(ratio between

length and diameter) -

Force Ft/sec2

Aerodynamic Forces -

Speed of sound in air Ft/sec

Altitude Feet

Moments of inertia referred to

body axis Slug.ft

2

Product of inertia referred to

body axis Slug.ft

2

Mass Slugs

Mach number -

Roll rate, angular velocity

about x axis Rad/sec

Pitch rate, angular velocity

about y axis Rad/sec

Dynamic pressure Lbs/ft2

x

Symbol Desription Units

Pitch moment Pound-force-feet

Mean aerodynamic chord Feet

Yaw moment Pound-force-feet

Reynolds number -

Planform area Feet2

Zero-lift drag area Feet2

Reference area Feet2

Wetted area Feet2

Wing area Feet2

Thickness Feet

Time Sec

Time to damp half the

amplitude Sec

True air speed Ft/sec

Air speed Ft/sec

Velocity in the x body axis Ft/sec

Velocity in the y body axis Ft/sec

True air speed Ft/sec

Velocity in the z body axis Ft/sec

xi

Symbol Description Units

α Angle of attack Rad

β Sideslip angle Rad

Γ Dihedral angle Rad

δ Deflection angle of a control

surface Rad

η Real part of the Dutch roll

eigenvalue -

θ Pitch angle Rad

λ Eigenvalue -

λ Taper ratio (Ratio between tip

chord and root card) -

˄ Sweep angle Rad

μ Imaginary part of the Dutch

roll eigenvalue -

ξ Damping ratio -

ρ Air density -

σ Real part of the phugoid and

short period eigenvalues -

Φ Roll angle Rad

ψ Yaw angle Rad

ω Oscillation frequency 1/sec

ω Damped frequency 1/sec

ω Natural undamped frequency 1/sec

xii

LIST OF TABLES

Page

Table 1.1: F-4 Phantom II General Characteristics ................................................ 5

Table 1.2: F-4 Phantom II Performance Characteristics ......................................... 5

Table 5.1: Stability and Inertia Characteristics of F-4C ....................................... 16

xiii

LIST OF FIGURES

Page

Figure 1.1: F-4C Phantom Aircraft ........................................................................... 1

Figure 2.1: A U.S. Navy F-4B bombing over Vieatnam, 25 November 1971 .......... 2

Figure 2.2: Retired TURAF F-4E Phantom II, housed at the Istanbul

Aviation Museum ................................................................................... 2

Figure 2.3: The McDonnell F3H-G/H model, 1954 ................................................. 3

Figure 2.4: An F4H-1F aboard an aircraft carrier, April 1960 ................................. 3

Figure 2.5: Iran Air Force F-4 Phantom II refueling through a boom in

Iran-Iraq War, 1982 ............................................................................... 4

Figure 2.6: Three side views of the F-4E/F .............................................................. 4

Figure 3.1: Positive Static Stability ............................................................................ 7

Figure 3.2: Neutral Static Stability ............................................................................ 7

Figure 3.3: Negative Static Stability .......................................................................... 7

Figure 3.4: Positive Dynamic Stability ...................................................................... 8

Figure 3.5: Neutral Dynamic Stability ....................................................................... 9

Figure 3.6: Negative Dynamic Stability .................................................................... 9

Figure 4.1: The long-period (phugoid) mode ........................................................... 10

Figure 4.2: The short-period mode .......................................................................... 10

Figure 5.1: F-4C General Arrangement ................................................................... 14

Figure 5.2: Flight Conditions of F-4C ..................................................................... 15

Figure 5.3: Roll motion ............................................................................................ 23

Figure 5.4: Dutch Roll Motion ................................................................................. 24

Figure 6.1: Time response of the transfer function for the change in angle of attack

to the change in elevator angle .............................................................. 30

Figure 6.2: Time response of the transfer function for the change in pitch rate


Figure 6.3: Time response of function for the change in velocity

to the change in the transfer elevator angle ........................................... 31

Figure 6.4: Time response of the transfer function for the change in pitch angle


Figure 7.1: A root locus analysis done in MATLAB ............................................... 32

Figure 8.1: An autopilot control panel used in World War II .................................. 33

Figure 8.2: The modern autopilot control system of Airbus A340 .......................... 33

Figure 8.3: Pitch displacement autopilot system of F-4C ........................................ 34

Figure 8.4: Root locus of inner loop ........................................................................ 34

Figure 8.5: The block diagram with the value of .................................... 35

Figure 8.6: The root locus of outer loop .................................................................. 35

Figure 8.7: Zoomed version of Figure 8.6 ................................................................ 36

Figure 8.8: The block diagram with the value of and ...... 36

Figure 8.9: The time response of .......................................................................... 37

xiv

STABILITY ANALYSIS OF F-4C PHANTOM AIRCRAFT: MODELLING

AND SIMULATING

SUMMARY

In this graduation project it is aimed to make the stabiltiy analysis of a fighter jet.

After pre-research about fighter jets, F-4C phantom aircraft was chosen. By using the

Newton’s second law, stability equations were obtained through derivatives of

motion equation. Stability analysis covers both longitudinal emotion and lateral

emotion in dynamic stability. The analysis was done for a special case. In this case, a

F-4C phantom aircraft flies at a speed of 230 ft/s at sea level.

Initially, stability derivatives were obtained with the parameters which were found in

pre-research. After, transfer functions were obtained with the help of MATLAB. And

then time responses were found and graphics were plotted in MATLAB. Stability

analysis was carried out with longitudinal motion. After completing these analyzes,

autopilot modeling was done for time response of the transfer function of the change

in pitch angle to the change in elevator angle. A point was selected with the root

locus method and gain constant was obtained. And then this value was used in outer

loop. Since all these calculations done in matrix systems and results should be shown

by plotting graphics, MATLAB (matrix labaratory) was used as the appropriate

programming language.

The required data for this stability analysis for the F-4C phantom aircraft was taken

from “AIRCRAFT HANDLING QUALITIES DATA” by Robert K. Heffley and

Wayne F. Jewell prepared for NASA (National Aeronautics and Space

Administration). F-4C phantom aircraft was chosen for this thesis because it was

used in Vietnam War by USAF (United States Air Forces) in bombing roles and it

has been still available in inventory of Turkish Air Forces. It is so interesting that

despite being an old aircraft, it has been still used by TAF with the modernizations.

Hence it worths to make research about it.

xv

ÖZET

Bu mezuniyet çalışmasında bir savaş uçağının kararlılık analizinin yapılması

hedeflenmektedir. Bunun için F-4C hayalet savaş uçağı seçilmiştir. Newton’un ikinci

yasasını kullanarak hareket denkleminin türevlerinden kararlılık denklemleri elde

edilmiştir. Kararlılık analizi dinamik kararlılıkta hem uzunlamasına hareketi hem de

yanal hareketi kapsamaktadır. Bu analiz belirli bir durum için yapılmıştır. Bu

durumda F-4C uçağı deniz seviyesinde 230 ft/s hızında uçmaktadır.

Öncelikle, kararlılık türevleri yapılan ön araştırmada bulunan parametrelerle elde

edilmiştir. Sonra transfer fonksiyonları MATLAB yardımıyla elde edilmiştir ve

zaman cevapları bulunmuştur ve grafikler MATLAB’da çizdirilmiştir. Kararlılık

analizi için uzunlamasına hareket seçilmiştir. Bu analizlerden sonra, otopilot

modellemesi için eğim açısındaki değişimin elevator açısındaki değişime aktarım

fonksiyonunun zaman cevabı kullanılmıştır. Root-locus yöntemi kullanılarak bir

nokta seçilmiş ve kazanç sabiti elde edilmiştir. Sonra bu değer dış halka için

kullanılmıştır. Bütün bu hesaplamalar matris yöntemiyle yapıldığı için ve sonuçların

grafiklerde çizdirilip gösterileceği için uygun bir programlama dili olan MATLAB

kullanılmıştır.

F-4C hayalet savaş uçağı analizi için gerekli tüm bilgi Robert K. Heffley ve Wayne

F. Jewell tarafından NASA (Ulusal Havacılık ve Uzay Dairesi) için hazırlanmış

“AIRCRAFT HANDLING QUALITIES DATA” makalesinden alınmıştır. Bu tez

çalışması için F-4C hayalet uçağı seçilmiştir çünkü bu uçak Vietnam Savaşı’ nda

Birleşik Milletler Hava Kuvvetleri tarafından bombardıman uçağı ularak

kullanılmıştır ve günümüzde ise hala Türk Hava Kuvvetleri envanterinde

bulunmaktadır. Çok eski bir uçak olmasına rağmen yapılan modernizasyonlarla THK

tarafından kullanılması ilgi çekici ve araştırmaya değer bi konudur.

xvi

1

1. INTRODUCTION

Figure 1.1 : F-4C Phantom Aircraft

In this thesis, stability analysis of F-4C phantom aircraft is performed by using

longitudinal equations. Longitudinal and lateral stability are analyzed and autopilot

modelling done. Before starting stability analysis, F-4C is introduced to the readers

with briefed information in terms of history, design and development process and

specifications. It is important to know in which date this aircraft was created, in

which wars it was used, which developments and modernizations have been applied

on it. After having enough general information about F-4C aircraft, it is necessary to

give information about two kinds of stability which are static and dynamics

stabilities. And then dynamic stability is explained in more details because this

concept should be examined more to understand stability analysis. Longitudinal and

lateral motions are explained before starting to obtain stability derivatives. In order to

perform stability analysis stability derivates should be obtained from the data which

was taken from pre-research sources. Stability analysis will be made only for one

condition as it was stated before. MATLAB programming language will be used as

the most appropriate analysis tool. It is important to state once again that the required

data for stability analysis was taken from a reliable source that named “AIRCRAFT

HANDLING QUALITIES DATA” by Robert K. Heffley and Wayne F. Jewell who

prepared for NASA. Algorithm codes are used to obtain stability derivatives. The

obtained derivatives are placed in state space models and results were obtained for

different modes which are called as short and long (phugoid) methods for

longitudinal dynamic stability while are called as Spiral, Roll, and Dutch Roll modes

for lateral dynamic stability. Finally, in order to perform autopilot modelling, time

response of the transfer function for the change in pitch angle to the change in

elevator angle is used.

2

2. F-4C PHANTOM AIRCRAFT

F-4 Phantom was produced by McDonnell company for the Unites States Navy

(USN) in 1961. It is a long-range supersonic jet figther-bomber which has two seats

and two engines. This aircraft is a large fighter and can reach a speed of over Mach

2.2. It can carry about 20,000 pounds of weapons including air-to-air missiles air-to-

ground missiles, and variety of bombs. It has so many records of high speed and

high altitude in the wars.

The F-4 Phantom was used in Vietnam War by the Unites States Air Forces, Navy

and Marine Corps. It became an important weapon in the ground-attack missions.

This aircraft was also operated by 11 countries and used in other wars. Israel used

these aircrafts in Arab-Israel War and Iran used them in Iran-Iraq War. Since F-4

Phantom was created many years ago, this aircraft still serves in active service to

some countries which are Iran, South Korea, Greece and Turkey. F-4 Phantom was

also used against Islamic State militans in Middle East.

Figıre 2.1: A U.S. Navy F-4B bombing over Vieatnam, 25 November 1971

Figure 2.2 : Retired TURAF F-4E Phantom II, housed at the Istanbul Aviation

Museum

3

2.1 Development

In 1952, McDonnell company started internal studies with United States Navy. Navy

was in a great need of new aircraft that can both attack to enemies and defend the

Navy. So Mcdonnel company devoloped an aircraft model which is shown figure

below.

Figure 2.3: The McDonnell F3H-G/H model, 1954

This figure shows the origins of F-4 aircraft. The Unites States Navy would order

these aircrafts but then they changed their minds after the upcoming Grumman

XF9F-9 and Vought XF8U-1 aircrafts. Because these aircrafts satisfied the needs and

fulfilled the expectations for supersonic fighter aircraft.

Figure 2.4: An F4H-1F aboard an aircraft carrier, April 1960

McDonnell company did not stop development process and designed another

prototype which is called XF4H-1. This prototype was designed to have a capacity of

carrying four radar-guided missiles named AAM-N-6 Sparrow III. It has two J79-

GE-8 engines which sat low section in the fuselage in order to satisfy more capacity

of internal fuel. The wings were designed in a way to improve control at high angle

of attacks and had a leading edge sweep of 45° and had blown flaps which satisfy an

advantage of handling in low speeds. 23° of anhedral was given the tailplane in order

to improve control at high angle of attack. In addition, air intakes has different kinds

of geometry rampts for regulating aitflow to the engines at supersonic speeds. To

4

perform carrier missions, the landing gear was designed to withstand high loads and

AN/APQ-50 radar helped to maintain flight in all weather conditions.

Production process brought many upgrades for the radar, nose and canopy. After

upgrades F-4 has a larger radar antenna, sharper nose, and new canopy which was

rebuilt to improve pilot’s eyeshot for better visibility. And rear cockpit, second seat,

is made larger to prevent claustrophobia due to its narrow and small room. In many

years, more types of F-4 Phantom was produced and began to operated in variety of

different operations.

2.2 Design

The F-4 Aircraft was designed as tandom-seat for two people. It is a fighter-bomber

as a interceptor which was operated in the United States Navy. Its main role was to

defend the Unites States Navy fleet against enemy aircrafts. It has maximum takeoff

weight over 60,000 lb, has a top speed of 2.2 Mach, climb rate over 41,000 ft/min. It

has 9 external hardpoints that can carry heavy weapons about 20,000 pounds. These

weapons are air-to-air and air-to-surface missiles. F-4 was designed without an

internal cannon as other interceptor aircrafts.

Figure 2.5: Iran Air Force F-4 Phantom II refueling through a boom in Iran-Iraq

War, 1982

2.3 Specifications

Figure 2.6: Three side views of the F-4E/F

5

F-4 Phantom II specifications were shown in two tables below which are general

characteristics and perfomance data of this aircraft.

Crew 2

Length 63 feet 0 inch

Height 16 feet 5 inch

Wing area 530 square feet

Wingspan 38 feet 5 inch

Airfoil NACA 0006.4–64 root, NACA 0003-64

tip

Aspect Ratio 2.77

Maximum takeoff weight 61,795 lb

Maximum landing weight 36,831 lb

Empty weight 30,328 lb

Gross weight 41,500 lb

Fuel capacity

1,994 US gal internal, 3,335 US gal with

2x 370 US gal external tanks on the outer

wing hardpoints and either a 600 or 610

US gal tank for the center-line station.

Powerplant

2 × General Electric J79-GE-17A after-

burning turbojet engines, 11,905 lbf thrust

each dry, 17,845 lbf with afterburner

Table 1.1 F-4 Phantom II General Characteristics

Cruise speed 510 knots

Maximum speed Mach 2.23, 1,280 knots at 40,000 feet

Combat range 370 nmi

Ferry range 1,457 nmi

Service ceiling 60,000 feet

Rate of climb 41,300 ft/min

Wing loading 78 lb/sq feet

Lift-to-drag ratio 8.58

Thrus-to-weight ratio 0.86 at loaded weight, 0.58 at maximum

takeoff weight

Takeoff roll 4,490 feet at 53,814 lb

Landing roll 3,680 feet at 36,831 lb

Table 1.2 F-4 Phantom II Performance Characteristics

6

3. INTRODUCTION TO STABILITY

Initially, it is important to know and understand what stability means in flight

mechanics. Stability is the natural tendency of an aircraft to return to its equilibrium

position after having any disturbance. Atmospheric events or pilot’s actions may

cause disturbance. For example, turbulent air or wind gradients are atmospheric

events that are seen and experienced by pilots frequently. Hence, it is so critical to

maintain stability of aircraft after any disturbances.

Stability can be satisfied with manually by pilot’s commands but this is not enough

and safe method. Because it will be difficult and fatiguing for pilot to keep aircraft

stable after each time aircraft gets disturbed. So there must be an automatic control in

aircraft that will make it easier to keep aircraft in stable position. This automatic

control is satisfied by flight control systems that are placed to aircrafts. Stability

Augmentation System (SAS) is an example for a flight control system. This system

actuates the aircraft flight controls in order to dampen out the disturbance because

artificial damping is needed.

Stability control is important for aircraft’s perfomance. Aircraft with poor stability

control causes poor handling qualities and poor handling qualities will result poor

and weak flight performance which can be dangerous for pilot’s life and aircraft.

Regardless how aircraft is designed perfecly, flight performance will be achieved by

good stability control. Hence, stability and control studies on good handling in every

condition that aircraft is going through and how control systems are designed in

order to achieve best flight perfomance for aircraft. In the next chapters, static

stability and dynamic stability will be explained in detail.

3.1 Static Stability

In earlier chapter, stability is defined as the natural tendency of an aircract to return

its equilibrium position after having a disturbance. Static stability means when a

system perturbed it gives an instantaneous response which is a statically stable

system will firstly move back towards its equilibrium position. There are three types

of static stability which are positive, neutral and negative stabilities.

An aircraft that has positive static stability tends to return its equilibrium altitude

when it has perturbation. Let’s assume that an aircraft is flying in an specific altitude

and it hit some turbulance. As a result of turbulence the nose pitches up. And then

the nose lowers and move back towards to its original altitude with the help of

automatic control systems in aircraft.

7

Figure 3.1: Positive Static Stability

On the other hand, an aircraft that has neutral static stabiliy tends to stay in its new

altitude after having perturbation. So it means there is no tendency to return its

equilibrium point. For example, if aircraft hits turbulence and the nose pitches up at

some degrees, aircraft will stay at that degrees.

Figure 3.2: Neutral Static Stability

Finally, an aircraft that has negative static stability tends to maintain moving away

from its original altitude after having perturbation. For example, if aircraft hits

turbulence and the nose pitches up, after aircraft will maintain the moving away and

continue to pitch up. In most of aircrafts, this situation is undesirable.

Figure 3.3: Negative Static Stability

8

Static stability is very important matter in aircrafts. Some aircrafts are designed to be

very stable. For example commercial aircrafts. Because in these type of aircrafs

maneuver is not desired. The purpose of aircraft performance determines the required

properties. If stability is increased, aircraft will flight safer and stable. But this is not

a desired property for fighter jets. These aircrafts need high manuevering skills.

Hence, their designs tend to be very unstable.

3.2 Dynamic Stability

Dynamic stability means how an aircraft responds over time after it is disturbed from

its equilibrium state. In dynamic stability not only initial tendancy, but also the

amplitudes of the response due to disturbance decay in finite time to return its

equilibrium state. But it is important to point out that if an aircraft is statically stable,

it does not mean that it will also automatically satisfy dynamic stability. However, a

system must be statically stable in order to be dynamically stable. There are three

types of dynamic stabilities which are positive, neutral and negative dynamic

stabilities.

Aircraft that has positive dynamic stability has oscillations that dampen out over

time. For example, if an aircraft is trimmed for level flight and pilot pull back on the

yoke and leave it free, the nose will instantenously begin pitching down. How much

pilot pitched up the nose initially will effect how much the nose will pitch down and

then pitch will nose up again over time but this will be less than pilot’s intial control

input. The pitching will end over time and aircraft will return back to its original

state.

Figure 3.4: Positive Dynamic Stability

Aircraft that has neutral dynamic stability has oscillations that never dampen out. For

example, if a neutrally dynamic stable aircraft is pitched up, aircraft will nose down

and then nose up again. And the oscillations will contunie in this status if pilot does

not change this situation. In theory, this situation can last forever.

9

Figure 3.5: Neutral Dynamic Stability

Finally, aircraft that has negative dynamic stability has oscillations that goes unstable

over time if pilot does not control it. For example if an aircraft has a negative

dynamic stability, it has pitch oscillations that will be amplified more over time.

Figure 3.6: Negative Dynamic Stability

So far, the types of static stability and dynamic stability are explained in detail with

the help of figures. As a result, the type of aircraft and its purpose determines which

stability type is used for that aircraft. For example, stable aircrafts such as Cessna

training aircraft and commercial aircrafts are designed to be statically and

dynamically stable. Because in these type of aircrafts, it is important to get trim point

easily for safe flight. However, fighter jets such as F-4 and F-16, they are designed to

be unstable. Because in these type of aircrafts high maneuverability is desired.

10

4. LONGITUDINAL MOTION

The motion of an aircraft in flight process has tree translation motions which are

vertical, horizontal and transverse motions and three rotational motions which are

pitch, yaw, and roll motions. To analyze aircraft motion in flight, rigid-body

equations of motion are used in order to simplify the complexity of the flight process.

There will be two assumptions to be done. In first assumption, aircraft will make

small deviations from its equilibrium flight state. In second assumption, equations of

motion will be divided into two groups in order to analyze motion of the aircraft.

Longitudinal equations consist of the X-force, Z-force, and pitching moment

equations while lateral equations consist of the Y-force, rolling and yawing moment

equations.

The parameters of oscillation motions are the period of time which is needed for one

complete oscillation and the time which is needed to damp to half-amplitude or the

time to double the amplitude (for a dynamically unstable motion). The longitudinal

motion consists of two different types of oscillations. These are short-period

oscillation which is called as the short-period mode and long-period oscillation

which is called as the phugoid (long) mode. In phugoid mode, air-speed, pitch angle,

and altitude vary in a large amplitude. However, angle-of-attack doesn’t vary in a

large-amplitude like other parameters, on the contrary there is almost no variation.

When aircraft is perturbed it re-establishes itself to go back towards its equilibrium

flight condition. This process happens slowly in phugoid mode so inertia forces and

damping forces are very low. In this case, the oscillation usually is controlled by

pilot and this period lasts from 20 seconds to 60 seconds.

Figure 4.1: The long-period (phugoid) mode

In short-period, oscillations are usually heavily damped with a period of a few

seconds. In this motion, aircraft is pitching rapidly about the center of gravity and

angle of attack variation is seen. The time to damp the amplitude to one-half of its

value only takes a second.

Figure 4.2: The short-period mode

11

4.1 Longitudinal approximations

Longitudinal approximations will be done according to assumptions that are

mentioned before. Before starting approximations, it will be helpful to consider that

long-period can be referred as an interchange of potential and kinetical energy in

terms of equilibrium altitude and airspeed. Long-period is designated by an angle of

attack that does not change. In this case, analysis will start with neglection. As it is

seen in the equation below, momentum equation is neglected and due to no variation

in angle of attack, AoA will be zero.

Homogenous longitudinal state equation takes this state according to assumptions

made above:

* + [

] * θ+

When equality is solved, eigenvalues of the long-period approach are found:

| |

or

|

|

Expanding this determinat gives

or

[ √

]

The frequency and damping ratios expressions are obtained as

√

√

12

The forward speed and the lift to drag ratio are inversely proportional to the

frequency of oscillation and the damping ratio, respectively. The phugoid damping

decreases as the aerodynamic efficiency (L/D) increases, as we can see from this

approximation. When pilots fly an airplane under visual flight rules, the phugoid

damping and frequency will differ a lot, but they will always find the plane to be safe

to fly. Low phugoid damping, on the other hand, can become very objectionable if

they are flying the plane under instrument flight rules. The lift to drag ratio of the

airplane will have to be reduced to increase the damping of the phugoid motion.

Since such a choice would decrease the airplane's output, the designer would reject it

and look for another option, such as an automatic stabilization system to provide the

proper damping characteristics.

4.2 Short-Period Approximation

In short-period approximation, will be assumed in order to obtain short

period mode of motion. In this case, the longitudinal state space equation yields

[ ] [

] [ ]

Using the relationship, this equation can be written in terms of the angle of attack.

By using following formula, derivatives can be replaced due to and with

derivatives due to and . The derivative is defined as is

( )

It also can be shown as

The state equations for the short period approximations can be rewritten as by using

these expressions

[ ]

[

]

[ ]

When equality is solved, eigenvalues of the short-period approach are found:

| |

which yields

13

[

( )]

Characteristic equation is obtained from the determinant obtained above

( )

The roots are obtained from the characteristic equation,

(

)

[(

)

( )]

Damping and frequency can be expressed as:

[( )]

(

)

14

5. STABILITY OF F-4C PHANTOM AIRCRAFT

Before starting stability analysis of F-4C, it will be useful to give more information

about this aircraft. The F-4C is a tactical fighter of the United States Air Forces

whose primary role is to engage in all-weather air-to-air missile combat. Ailerons in

conjunction with spoilers on a swept wing provide lateral control. Longitudinal

stability and control are provided by a swept stabilator. A conventional fin-rudder

configuration is used to provide directional stability and control. Complete span

leading edge flaps and inboard plain trailing edge flaps, combined with blowing-type

boundary layer control (BLC), minimize landing speed. When a complete flap

deflection occurs, boundary layer control is automatically induced.

Figure 5.1: F-4C General Arrangement

15

Figure 5.2: Flight Conditions of F-4C

16

Aircraft F-4C

W

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

Table 5.1: Stability and Inertia Characteristics of F-4C

17

5.1 Stability Derivatives

By using data given of Stability and Inertia Characteristics of F-4C in table 1.1,

dimensional longitudinal stabillity derivatives are determined in the following

calculations. The dynamic pressure Q and the following equations are

(

) ( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

[ ( )( )]( )

( )( )

( )( )

[ ( )( )]( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

( )( )

18

( )

( )( )

( )( ) (

( )( ))

( )( ) (

( ))

( ) (

( ))

( )( )

( )( )

( )( )

19

( )( ) (

)

( )( ) (

)

( )( ) (

)

( )( ) (

)

20

5.2 Longitudinal Motion

In this section, short-period dynamics and long-period dynamics are determined.

5.2.1 Short-Period Dynamics

The equation with control input from the elevator in state space form is written as

[ ]

[

]

[ ]

[

]

[ ]

Propulsion system control is neglected and Laplace transform of this equation yields

( ) ( ) ( )

( )

( ) ( ) [ ( )] ( ) (

)

Dividing equations by ( ) algebraic equations are obtained

( ) ( )

( ) ( )

( )

( ) ( )

( ) [ ( )]

( )

( )

Solving these equations with Cramer’s rule yields

21

( )

( ) ( )

( )

|

( )

|

|

( ) ( )

|

From the shorthand notation to express function polynomials

( )

( ) ( )

( )

The transfer function for the change in pitch rate to the change in elevator angle is

shown as

( )

( ) ( )

( )

|

( )

|

|

( ) ( )

|

or

( )

( ) ( )

( )

The transfer function for the change in angle of attack to the change in elevator

angle:

( )

( )

( )

( ) ( )( )

(

)

( )( )

( )

The transfer function for the change in pitch rate to the change in elevator angle:

22

( )

( ) (

)

( )

( ( )

) ( )

( )

(

)

( )( )

( )

5.2.2 Long-Period Dynamics

The state space equation is given for the long-period approximations

* + [

] * + [

] [

]

Laplace transformation of this equation is

( ) ( ) ( ) ( ) ( )

( ) ( )

( )

( )

By setting ( ) to 0 and solving transfer functions

( ) ( )

( )

( )

( )

( )

( )

( )

( )

then it yields

( )

( )

|

|

|

|

similarly

( )

( )

|

|

|

|

23

equation is obtained. The transfer function for the change in velocity to the change in

elevator angle:

( )

( )

( )( )

( )( )

The transfer function for the change in pitch angle to the change in elevator angle:

( )

( ) (

)

(( )( )

)

( )( )

5.3 Lateral Motion

In this section roll dynamics and dutch roll approximation are determined.

5.3.1 Roll Dynamics

Figure 5.3: Roll motion

24

The transfer function ( ) ( ) and ( ) ( ) can be obtained by taking

the Laplace transform of the roll equation:

( ) ( ) ( )

The transfer function for the change in roll rate to the change in aileron angle:

( )

( )

( )

The transfer function for the change in roll angle to the change in aileron angle:

( )

( )

( )

[ ( )]

5.3.2 Dutch Roll Approximation

Figure 5.4: Dutch Roll Motion

The approximate equations are given for the final simplified transfer function of

Dutch roll motion

[ ] [

(

)

] * + [

] [

]

Taking the Laplace transform yields

(

) ( ) (

) ( )

( )

( ) ( ) ( ) ( ) ( )

25

By setting ( ) and ( ) to 0 and solving equations, transfer functions are

obtained as follows:

(

) ( )

( ) (

) ( )

( )

( )

( ) ( )

( )

( )

Solving for transfer function yields

( )

( )

|

|

|

|

( )

( )

|

|

|

|

( )

( )

|

|

|

|

( )

( )

|

|

|

|

The transfer function for the change in sideslip angle to the change in rudder angle:

( )

( )

( )( ) ( )( )

( ) ( )( )

( )( )

26

The transfer function for the change in yaw rate to the change in rudder angle:

( )

( )

( )( ) ( )( )

( ) ( )( )

( )( ) ( )( )

The transfer function for the change in sideslip angle to the change in aileron angle:

( )

( )

( )( )

( ) ( )( )

( )( ) ( )( )

The transfer function for the change in yaw rate to the change in aileron angle:

( )

( )

( )( )

( ) ( )( )

( )( ) ( )( )

27

5.4 Longitudinal Motion Modes

[

]

[

( )( ) ( )( ) ( )( )

]

[

]

The eigenvalues can be determined by finding eigenvalues of the matrix A:

| |

The resulting characteristic equation is obtained by using MATLAB

The solution of the characteristic equation yields the eigenvalues:

( ) ( )

( ) ( )

When comparing the responses of the short and phugoid modes, it is clear that the

short period mode is indeed heavily damped, but the phugoid mode is just lightly

damped. Despite the phugoid mode's light damping, it rarely causes problems for the

pilot since its time scale is long enough that minor control inputs may compensate

for disturbance-induced excitation of this mode. Short mode and phugoid mode

approximations were also used to produce transfer functions.

28

5.5 Lateral Motion Modes

[

(

)

]

[

]

[

]

The eigenvalues can be determined by finding eigenvalues of the matrix A:

| |

The resulting characteristic equation is obtained by using MATLAB.

The solution of the characteristic equation yields the eigenvalues:

( ) ( )

( )

( )

The Dutch Roll mode is represented by the first pair of roots, whereas the rolling and

spiral modes are represented by the real roots. To put it in perspective, the period of

the Dutch Roll mode is on par with the longitudinal short period mode, but it is much

more lightly damped.

A typical aircraft's response to lateral and directional disturbances is composed of

two exponential modes and one oscillatory mode, as seen below:

29

A heavily damped rolling mode whose time to damp to half amplitude is mostly

determined by the roll damping ;

A spiral mode that is usually only lightly damped, if at all, and is potentially

unstable. For this mode, the dihedral effect has a significant stabilizing factor,

whereas stability is destabilizing;

The Dutch Roll mode, which consists of a coordinated yawing, rolling, and side

sliding motion, is a lightly damped oscillatory, intermediate frequency mode. The

dihedral impact is often destabilizing in this mode, whereas stability is stabilizing..

30

6.TIME RESPONSES

( )

( )

Figure 6.1: Time response of the transfer function for the change in angle of attack

to the change in elevator angle

( )

( )

Figure 6.2: Time response of the transfer function for the change in pitch rate to the

change in elevator angle

31

( )

( )

Figure 6.3: Time response of function for the change in velocity to the change in the

transfer elevator angle

( )

( )

Figure 6.4: Time response of the transfer function for the change in pitch angle to

the change in elevator angle

32

7. ROOT LOCUS ANALYSIS METHOD

Before modelling autopilot, it is necessary to explain the importance of the root locus

method. Root locus analysis is a graphical tool for studying how the roots of a

system vary with adjustment of a system parameter, most frequently a gain in a

feedback system, in control theory and stability theory. This is a technique developed

by Walter R. Evans and used as a stability criterion in the field of classical control

theory to determine the system's stability.

The root locus of a feedback system is the graphical representation of the possible

positions of its closed-loop poles for various values of a system parameter in the

complex s-plane. The angle condition, which is a constraint in mathematics that the

locus of points in the s-plane on which a system's closed-loop poles reside satisfies

this condition, is satisfied by the points that make up the root locus. And then the

magnitude condition, which is a constraint satisfied by the locus of points in the s-

plane where a system's closed-loop poles are located, can be used to get the value of

the parameter for a certain point on the root locus. If to summarize, root locus gives

the answers of the questions; what happens when the gain of the controller changes,

will the system be stable and will the response change? Questions of these answers

will be answered in next chapter when autopilot modelling is done.

Figure 7.1: A root locus analysis done in MATLAB

33

8. AUTOPILOT MODELLING

An autopilot system controls aircraft without requiring pilot’s constant manual

control but it does not mean that autopilot replace the pilots. Autopilot system assists

the pilot in operating control of aircraft. In this way, pilot can focus on other

operations such as weather and on-board systems.

Figure 8.1: An autopilot control panel used in World War II

In the figure above an old type of autopilot model is seen which is used in World

War II. This autopilot system helped pilots of United States Arm Air Corps aircrafts

on a true heading and altitude for many hours. The autopilot system was further

developed and control algorithms and hydraulic servomechanicsms were improved.

Radio-navigation was added to this system in order to have capability of flying in the

night and in bad weather conditions. And developments were maintained to gain an

ability of takeoff and landing completely under the control of autopilot system.

Modern autopilots include computer software systems in order to control the aircraft.

Aircraft’s current position is read by the software and then aircraft is guided by a

flight control system. Here you see in the figure below an modern autopilot control

system.

Figure 8.2: The modern autopilot control system of Airbus A340

34

The pitch displacement autopilot system of F-4C was designed and shown in the

figure below. Autopilot with and inner and outer loops is

Figure 8.3: Pitch displacement autopilot system of F-4C

The root locus transfer function for the inner loop is given below

(

) (

)

The inner-loop root locus is

Figure 8.4: Root locus of inner loop

was selected for the inner loop. The closed-loop system for the inner loop

is

(

)

35

The block diagram for the system with the value of can be shown as

Figure 8.5: The block diagram with the value of

After getting the block diagram, a value will be determined. Hence, required

root locus transfer function for the outer loop is given as

Then the outer-loop root locus is

Figure 8.6: The root locus of outer loop

36

Figure 8.7: Zoomed version of Figure 8.6

was selected for the closed loop transfer function so autopilot can be

written as

( )

( )( )( )

And then the pitch displacement autopilot with the value of and

can be shown as

Figure 8.8: The block diagram with the value of and

37

, the unit step, was put in order to get the response of

Figure 8.9: The time response of

In the figure above time response of is shown. This graphic can be changed with a

lead-compensator if time response is desired to be improved.

38

9. CONCLUSION

In this graduation project, stability analysis and autopilot modelling of F-4C aircraft

was aimed and done with the help of MATLAB. F-4C was chosen for this project

due to its importance in history. This aircraft was used in important wars by many

countries. More importantly, although this aircraft is old and out of service in many

countries, Turkey has been still using this aircraft with the new developments and

modernizations. It has been serving well for the Turkish Air Force in the operations

has been done in Middle East against terrorist groups and organizations.

The information which includes the required data for the stability analysis was taken

from a reliable source which is “AIRCRAFT HANDLING QUALITIES DATA” by

Robert K. Heffley and Wayne F. Jewell prepared for NASA (National Aeronautics

and Space Administration). After the data required for stability analysis was taken

from this source, stability derivatives were obtained for both, longitudinal and lateral

motions. In longitudinal motion of F-4C aircraft, the transfer function for the change

in angle of attack to the change in elevator angle and the transfer function for the

change in pitch rate to the change in elevator angle were found. On the other hand, in

the long-period dynamics, the transfer function for the change in velocity to the

change in elevator angle and the transfer function for the change in pitch angle to the

change in elevator angle were found. As a result, four different time responses were

found. This process was also done for lateral motion. For roll mode, the transfer

function for change in roll angle to change in aileron angle was found, as well as the

transfer function for change in roll angle to change in aileron angle. In addition, four

separate transfer functions for Dutch-Roll dynamics were obtained. And then the time

responses for both motions, characteristic equations and eigenvalues for various modes

were calculated. With the help of MATLAB, graphics of time responses for lateral

motion were plotted by using the transfer function found.

Finally, modelling autopilot system was done. The time response of the transfer function

for pitch angle change to elevator angle change was used. Gain value was selected for

the inner root locus and then was put in the block diagram. An outer-loop was obtained

in this manner, and then the aircraft's time response graph for this transfer function was

obtained. According to analysis, stability of F-4C was observed to be very low because

F-4C is fighter jet plane and in this aircraft low stability was desired in order to have

high maneuvrability and controllability.

39

REFERENCES

[1] Jafarov E., Lecture Notes, Istanbul Technical University, 2006.

[2] Nelson,R. Flight Stability and Control, 2nd Ed.

[3] Yechout T.R., Morris S.L., Bossert D.E., and Hallgren W.F. Introduction to

Aircraft Flight Mechanics: Performance, Static Stability, Dynamic Stability, And

Classical Feedback, AIAA, 2003.

[4] Heffley, R. K.; and W. F. Jewell. Aircraft Handling Qualities Data. NASA CR-

2144,

December 1972.

[5] Hull D.G., Fundamentals of Airplane Flight Mechanics, Springer, 2007.

[6] Roskam J., Airplane Flight Dynamics and Automatic Controls, Roskam Aviation

and Engineering Corporation, 1979

[7] Hull D.G., Fundamentals of Airplane Flight Mechanics, Springer, 2007.

[8] Kosar, Durmaz, Jafarov, Longitudinal Dynamics Analysis of Boeing 747-400

(WSEAS proceedings).

40

APPENDICES

APPENDIX A Codes of determining eigenvalues

APPENDIX B Codes of determining time responses

41

APPENDIX A

%Longitudinal characteristic equation and eigenvalues

A = [-0.058032239 0.043164476 0 -32.2; -0.219419417 -

0.306707579 230 0; 0.000138297 -0.001671324 -0.450157794

0; 0 0 1 0];

charpoly(A) %Characteristic equation

e = eig(A) %Eigenvalues

%Lateral characteristic equation and eigenvalues

A = [-0.078535365 0 -1 0.14; -8.043680031 -1.179004413

0.888587885 0; 1.835527350 -0.010080161 -0.248127043 0; 0

1 0 0];

charpoly(A) %Characteristic equation

e = eig(A) %Eigenvalues

42

APPENDIX B

% Time response of the transfer function for the change

in angle of attack to the change in elevator angle

title('Step Response')


in pitch rate to the change in elevator angle


% Time response of function for the change in velocity to

the change in the transfer elevator angle


num=[-0.028776317 -1.417915941]

den=[1 0.756865373 0.522471373]

sys=tf(num,den)

step(-0.1*sys)

ylabel('Angle of Attack')

num=[-1.404962057 -0.419850763]

den=[1 0.756865373 0.522471373]

sys=tf(num,den)

step(-0.1*sys)

ylabel('Pitch Rate')

num=[-0.926597409]

den=[1 0.058032239 0.030718718]

sys=tf(num,den)

step(-0.1*sys)

ylabel('Velocity’)

43


in pitch angle to the change in elevator angle


% Root locus of inner loop for autopilot modelling

h = tf([0.28776317 0.01669954],[1 10.058032239

0.611041108 0.30718718]);

num=[0.028776317 0.001669954]

den=[1 0.058032239 0.030718718]

sys=tf(num,den)

step(-0.1*sys)

ylabel('Pitch Angle')

STABILITY ANALY SIS OF F -4C PHANTOM AIRCRAFT: …

Documents