ISTANBUL TECHNICAL UNIVERSITY  FACULTY OF AERONAUTICS AND ASTRONAUTICS

STABILITY ANALYSIS OF F-104 STARFIGHTER : MODELLING AND SIMULATING

GRADUATION PROJECT

Tural SHIRINZADE

Department of Aeronautical Engineering

Thesis Advisor: Prof. Dr. Elbrus Caferov

JUNE, 2021

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ISTANBUL TECHNICAL UNIVERSITY  FACULTY OF AERONAUTICS AND ASTRONAUTICS

STABILITY ANALYSIS OF F-104 STARFIGHTER AIRCRAFT: MODELLING AND SIMULATING

GRADUATION PROJECT

Tural SHIRINZADE (110090903)

Department of Aeronautıcal Engineering

Thesis Advisor: Prof. Dr. Elbrus CAFEROV Anabilim Dalı : Herhangi Mühendislik, Bilim Programı : Herhangi Program

JUNE, 2021

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Name-surname,student of ITU Faculty of Aeronautics and Astronauticsstudent ID …....., successfully defended the graduation entitled “…….GRADUATION PROJECT TITLE…….…..”, which he/she prepared after fulfilling the requirements specified in the associated legislations, before the jury whose signatures are below.

Thesis Advisor : Prof. Dr. Elbrus Caferov ...... İstanbul Technical University

Jury Members : Prof. Dr. Name SURNAME ...... İstanbul Technical University

Prof. Dr. Name SURNAME ...... İstanbul Technical University

Date of Submission : 14 June 2021 Date of Defense : June 2021

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To my family,

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FOREWORD

I would like to thank my thesis advisor Prof. Dr. Elbrus Caferov, for increasing my interest in science and guiding me, my professors and friends for everything they have taught, my mother and father for always supporting me in my education life, and my friends for their encouraging behavior and helpfulness.

June 2021 Tural SHIRINZADE

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TABLE OF CONTENTS

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FOREWORD ...... v TABLE OF CONTENTS ...... vvii ABBREVIATIONS ...... ix LIST OF TABLES ...... x LIST OF FIGURES ...... x SUMMARY ...... xxi 1. INTRODUCTION ...... 1 2. F-104 STARFIGHTER ...... Error! Bookmark not defined. 2.1 General Information ...... Error! Bookmark not defined. 2.2 Stability ...... 4 2.3 Characteristics ...... 5 3. STABILITY ...... 7 3.1 Static Stability ...... 7 3.2 Dynamic Stability ...... 8 4. LONGITUDINAL MOTION ...... 10 4.1 Longitudinal Approximations ...... 10 4.1.1 Long Period Approximation ...... 10 4.1.2 Short Period Approximation ...... 11 4.2 Longitudinal Derivatives ...... 13 5. LATERAL MOTION ...... 16 5.1 Dutch Roll Approximation ...... 16 5.2 Lateral Derivatives ...... 17 6. TRANSFER FUNCTIONS AND TIME RESPONSES...... 19 6.1 Longitudinal Transfer Functions ...... 19 6.1 Lateral Transfer Functions ...... 23 7. DESIGN ...... 27 8. CONCLUSION ...... 33 REFERENCES ...... 34 APPENDICES ...... 35 APPENDIX A.1 ...... 36

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ABBREVIATIONS

SAS : Stability Augmentation System

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LIST OF TABLES

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Table 2.1 : Specifications of F-104 ...... 3

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LIST OF FIGURES Page

Figure 2.1 : F-104 Starfighter ………………...……………………………………..2 Figure 2.2 : F-104 Scheme …………...……………………………………………..4 Figure 2.3 : F-104 Characteristics …………...………………………………….…..5 Figure 3.1 : Types of Stability…………...…………………………………………..7 Figure 3.2 : Different Cases for Dynamix Stability …………...…………………….8 Figure 3.3. : Relation Between Static and Dynamic Stabilities ……………………..9 Figure 4.1 : Frequency and Damping Ratio for Long and Short Periods ………….12 Figure 4.2 : Influence of Center of Gravity Position on Longitudinal Response .....13 Figure 6.1 : Time Response of the Pitch Angle…………………..………………..19 Figure 6.2 : Time Response of the Velocity…………...……………………...…..20 Figure 6.3 : Time Response of the Angle of Attack…………...……….…………..21 Figure 6.4 : Time Response of the Pitch Rate…………...……………………..…..22 Figure 6.5 : Time Response of the Sideslip Angle for ……….……...……..23 Figure 6.6 : Time Response of the Yaw Rate for Rudder………….……...………..24 Figure 6.7 : Time Response of the Sideslip Angle for Aileron …………..………..25 Figure 6.8 : Time Response of the Yaw Rate for Aileron………………...………..26 Figure 7.1 : Inner Loop Root Locus …………...……………………………….…..28 Figure 7.2 : Outer Loop Root Locus …………...…………………………………..30 Figure 7.3 : Time Response for Autopilot …………...……………...……………..31 Figure 7.4 : Autopilot Time Response with Details …………………...…………..32

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STABILITY ANALYSIS OF F-104 STARFIGHTER AIRCRAFT: MODELLING AND SIMULATING

SUMMARY

In this graduation project, it is aimed to analyze the dynamics of a fighter aircraft and design the autopilot. The F-104 Starfighter aircraft was selected for analysis. Stability derivatives were calculated using the characteristics of the aircraft. Then, using these stability derivatives, longitudinal and lateral motion modes were investigated and transfer functions of motions were obtained. In the last part, autopilot and SAS were designed for longitudinal movement.

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1. INTRODUCTION

In this study, stability analysis and autopilot design were made for the F-104 Starfighter, which is not a very successful design in terms of stability. First of all, a brief information about the selected aircraft was given. Then the concept of stability and its two types were mentioned. After that, the motion of the aircraft and both longitudinal and lateral movements were studied. In order to facilitate the calculation of these movements, information was given about the accepted approximations. Then, the transfer functions of both longitudinal and lateral motion modes were calculated and time responses were obtained. Finally, the autopilot was designed. The autopilot was designed to be an inner and outer loop for longitudinal movement. First of all, the SAS design was made, then the outer loop was designed in line with this design. Root locus method was used in autopilot design. MATLAB was used for both autopilot design and analysis of motion modes.

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2. F-104 STARFIGHTER

2.1 General Information

The F-104 Starfighter is a fighter-bomber designed by the American Lockheed company. It has a single vertical stabilized "T" type tail structure with one general electric J-79 engine. The general electric j-79 engine is also the engine of the F-4 Phantom and F-4 2020 Terminator aircraft. The F-104 starfighter is an aircraft that does not lose much energy at high altitude and can fly very fast thanks to its short wings. But these characteristics cause problems in terms of stability and control: Because of their short wings, they could not glide as a result of any engine failure or loss of power, and they often fell for this reason. F-104 was nicknamed the Witwenmacher ("widowmaker") and Fliegender Sarg ("flying coffin") as a result of too many accidents in German production.

Figure 2.1 : F-104 Starfighter.

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Table 2.1: Specifications of F-104

Crew 1 Length 54 ft 8 in (16.66 m

Wingspan 21 ft 9 in (6.63 m) Height 13 ft 6 in (4.11 m)

Wing area 196.1 sq ft (18.22 m2)

Airfoil Biconvex 3.36% root and tip Empty weight 14,000 lb (6,350 kg)

Max takeoff weight 29,027 lb (13,166 kg) Maximum speed 1,528 mph (2,459 km/h, 1,328 kn)

Maximum speed Mach 2

Combat range 420 mi (680 km, 360 nmi)

Ferry range 1,630 mi (2,620 km, 1,420 nmi)

Service ceiling 50,000 ft (15,000 m) Rate of climb 48,000 ft/min (240 m/s) Initially Lift-to-drag 9.2 Wing loading 105 lb/sq ft (510 kg/m2) Thrust/weight 0.54 with max. takeoff weight (0.76 loaded)

Powerplant 1 × General Electric J79 afterburning turbojet, 10,000 lbf (44 kN) thrust dry, 15,600 lbf (69 kN) with afterburner

Guns 1 × 20 mm (0.787 in) M61A1 Vulcan 6- barreled Gatling cannon, 725 rounds

Hardpoints 7 with a capacity of 4,000 lb (1,800 kg),with provisions to carry combinations of 4 × AIM-9 Sidewinder Missiles

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2.2 Stability

The wing has a full span leading edge flap. Trailing edge flaps have a blowing-type boundary layer control system. Control is provided by conventional ailerons and rudder and an all- movable stabilizer. Pitch, roll, and yaw dampers are incorporated. Pitch and roll controls are fully irreversible while the yaw control is a cable-actuated rudder without boost. A bobweight is used in the longitudinal feel system. Its position is assumed to be at the pilot's location.

Figure 2.2 : F-104 Scheme.

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2.3 Characteristics

Figure 2.3 : F-104 Characteristics [2].

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3. STABILITY

Stability is the tendency of the airplane to return to its equilibrium position after a disturbance (pilot or atmospheric effect). The stability of the aircraft should be such that it does not fatigue the pilot by constantly controlling it. Stability of aircraft with poor stability can be artificially increased with Stability Augmentation System (SAS).

3.1 Static Stability

Figure 3.1 : Types of Static Stability [2].

Figure 3.1 shows three possible states of static stability. In part (a), the ball whose equilibrium state is disturbed, is forced to return to its equilibrium position with the effect of gravity, in this case the ball is statically stable. In part (b), the same force will push the ball away from its

7 equilibrium position. This is statically unstable. In the last figure, the ball is on a flat surface and if it is moved it will not return to its equilibrium position, but it will not move away either. This is the neutral stable state, which is the boundary of the stable and unstable states. As a result, to provide static stability, a force is needed to return the aircraft to its equilibrium position.

3.2 Dynamic Stability

What is important in terms of dynamic stability is the subsequent behavior of the vehicle leaving the equilibrium position.

Figure 3.2 : Different Cases for Dynamic Stability [2].

In the case of positive dynamic stability, the motion is damped and returns to the equilibrium position over time. This is called positive damping. In the case of negative dynamic stability, on the contrary, as time passes, it moves further away from the equilibrium position – negative damping. In this case artificial damping is required and this is achieved with Stability Augmentation System. SAS is an electromechanical system that detects unwanted motion and 8 provides appropriate control to achieve damping. While static stability does not guarantee dynamic stability, the reverse is true: the vehicle must be statically stable in order to be dynamically stable. The figure below illustrates this more clearly.

Figure 3.3 : Relation Between Static and Dynamic Stabilities.

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4. LONGITUDINAL MOTION

Studying the motion of aircraft can be quite complex. The airplane has three translation motions (vertical, horizontal, and transverse), three rotational motions (pitch, yaw, and roll), and numerous elastic degrees of freedom. Assumptions are accepted for convenience in examining the motion of the aircraft. One of the most important of these is the assumption that the equations of motion can be studied by dividing them into two groups. The X-force, Z-force, and pitching moment equations embody the longitudinal equations, and the Y-force, rolling, and yawing moment equations form the lateral equations.

State-space form of linearized equations of motion for aircraft for longitudinal equations:

4.1 Longitudinal Approximations

4.1.1 Long Period Approximation

LongPeriod (Phugoid) mode can be thought of as the gradual displacement of kinetic and potential energies at equilibrium altitude and velocity. In phugoid mode approximation, there is a nearly constant angle of attack while pitch attitude, altitude, and velocity change. This approximation ignores the pitching moment and angle of attack.

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Longitudinal equations becomes:

And to find eigenvalues:

4.1.2 Short Period Approximation

For the short period approximation, velocity and X-force are ignored and the equation becomes:

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Eigenvalues:

Figure 4.1 : Frequency and Damping Ratio for Long and Short Period Approximations.

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Between these two approximations, the short period is much more important. In case of high frequency and heavy damping, the aircraft responds instantly to the elevator input, but in the case of low frequency and light damping, the aircraft becomes very difficult to control.

Phugoid mode is so slow that the pilot can control it easily. However, if the damping is too low, it can be extremely tiring.

Figure 4.2 : Influence of Center of Gravity Position on Longitudinal Response [2].

4.2. Longitudinal Derivatives

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Final result for longitudinal equations:

To find eigenvalues:

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By taking determinant, we obtain Characteristic Equation:

Roots of equation:

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5. LATERAL MOTION

State-space form of linearized equations of motion for aircraft For lateral equations:

There are three approximation in lateral motion:

1. A slowly convergent or divergent motion, called the spiral mode. 2. A highly convergent motion, called the rolling mode. 3. A lightly damped oscillatory motion having a low frequency, called the Dutch roll mode.

5.1. Dutch Roll Approximation In this approximation, rolling moment is neglected. And equation becomes:

Finding characteristic equation by using determinant:

Obtaining natural frequency and damping ratio:

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5.2. Lateral Derivatives

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Final result for lateral equations:

Eigenvalues:

Characteristic Equation:

Roots:

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6. TRANSFER FUNCTIONS AND TIME RESPONSES

6. 1. Longitudinal Transfer Functions

Figure 6.1 : Time Response of the Transfer Function for the Change in Pitch Angle to the Change in Elevator Angle.

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Figure 6.2 : Time Response of the Transfer Function for the Change in Velocity to the Change in Elevator Angle.

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Figure 6.3 : Time Response of the Transfer Function for the Change in Angle of Attack to the Change in Elevator Angle.

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Figure 6.4 : Time Response of the Transfer Function for the Change in Pitch Rate to the Change in Elevator Angle.

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6. 2. Lateral Transfer Functions

Figure 6.5 : Time Response of the Transfer Function for the Change in Sideslip Angle to the Change in Rudder Angle.

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Figure 6.6 : Time Response of the Transfer Function for the Change in Yaw Rate to the Change in Rudder Angle.

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Figure 6.7 : Time Response of the Transfer Function for the Change in Sideslip Angle to the Change in Aileron Angle.

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Figure 6.8 : Time Response of the Transfer Function for the Change in Yaw Rate to the Change in Aileron Angle.

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7. AUTOPILOT DESIGN

As obtained before, F-104’s transfer function for the change in pitch angle to the change in elevator angle is:

And the block diagram for autopliot is given below:

Design of inner loop is priority.

The root locus transfer function for inner loop is:

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Plotting Root Locus for obtaining range for :

Figure 7.1 : Inner Loop Root Locus.

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For better response, higher damping ratio is selected: . The gain for this value of damping ratio is

The block diagram with is:

And the closed loop system for the inner loop is:

The root locus transfer function of outer loop is:

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Ploting the outer loop root locus:

Figure 7.2 : Outer Loop Root Locus.

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The gain for is

Time response obtained:

Figure 7.3 : Time Response for Autopilot.

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Figure 7.4: Autopilot time response with details.

Rise Time: 7.22 s

Peak Amplitude: 1.24 at 23.7 s

Overshoot: 23.7%

Settling Time: 50.1 s

Final block diagram is:

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8. CONCLUSION

Stability and control is one of the most important areas for an aircraft. The smallest mistakes made in this regard can lead to great disasters. F-104, the aircraft under review is one of the best examples of this, with its nicknames. Stability analysis and autopilot design are intertwined with both aerodynamic and structural design, and changes in any of these areas affect the others. Since the stability derivatives in the transfer function used for autopilot design are functions of aerodynamic coefficients and structural weight values, this complex relationship can be seen when looking at the formulas. For this reason, autopilot design requires an iterative study. These repetitions are sometimes tiring and it becomes easy to make mistakes. Typing a single number wrong affects the whole result and puts the design in a dead end. Therefore, stability analysis and autopilot design require meticulous and patient work. It is possible to make the autopilot's time response more reasonable with filters such as the lead compensator, but this was not possible in terms of the values obtained in this study. For the lead compensator to make a significant change in time response, the pole and zero to be eliminated must be closer to the origin than in this design. It is beyond the scope of this study to see if this autopilot designed for the F-104 has a physical counterpart. Therefore, this project has a theoretical dimension and is only an initial step towards understanding the highly complex motion of the aircraft.

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REFERENCES

[1] Heffley, R. K., and Jewell, W. F. (1972). Aircraft Hangling Qualities Data

[2] Nelson, R. C. (1998). Flight Stability and Control

[3] Ogata, K. (2019). Modern Control Engineering

[4] Yechout, T. R., Morris, S. L., Bossert D. E. and Hallgren, W. F.(2003). Introduction to Aircraft Flight Mechanics: Performance, Static Stability, Dynamic Stability, and Classical Feedback Control.

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APPENDICES

APPENDIX A: Matlab Codes

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APPENDIX A

% longitudinal and lateral motion % characteristic equation and eigenvalues

A_long=[-0.069 -0.0154 0 -32.17 -0.224 -0.172 1742 0 0.000006 -0.0103 -0.372 0 0 0 1 0];

A_lat=[-0.101 0 -1 0.0185 -47.2783 -0.8692 0.492 0 7.531 -0.0182 -0.127 0 0 1 0 0];

disp ('longitudinal') charpoly (A_long) e=eig(A_long) disp ('lateral') charpoly (A_lat) d=eig(A_lat)

% time response of the transfer function for the % change in pitch angle to the change in elevator angle num = [0.051 0.0035]; den = [1 0.069 0.0041]; sys = tf(num,den); sys step(-0.1*sys)

%root locus for outer loop num = [0.51 0.035]; den = [1 10.07 2.1 0.1366 0 ]; sys = tf(num,den); sys rlocus (sys)

% step response of autopilot num = [0.3876 0.0266]; den = [1 10.07 2.1 0.52 0.0266]; sys = tf(num,den); sys step(sys)

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