FLIGHT VEHICLE DESIGN THESIS REPORT

Engineering School
Flight Vehicle Dynamics and Stability
Course Project

Muhammed Ahnuf FLIHT VEHICLE STABILITY AND CONTROL
2
Table of Contents
1. INTRODUCTION..................................................................................................................... 9
2. ABSTRACT.............................................................................................................................. 9
PHASE A: STABILITY COEFFICIENT CALCULATIONS ............................................................ 9
A.1. Background on Stability and Control...................................................................................... 10
A.1.1 Longitudinal Stability..................................................................................................... 10
A.1.2 Longitudinal Control........................................................................................................ 12
A.1.3 Directional Stability......................................................................................................... 12
A.1.4 Directional Control .......................................................................................................... 13
A.1.5 Roll Control..................................................................................................................... 13
A.1.6 Longitudinal Stability and Lateral Stability Coefficient .................................................... 14
A.1.7 Directional Stability and Control Coefficient.................................................................... 14
A.2 DETAILED CALCULATIONS.............................................................................................. 17
A.2.1. Wing Contribution .......................................................................................................... 18
A.2.2. Fuselage Contribution ..................................................................................................... 18
A.2.3 Tail Contribution.............................................................................................................. 20
A.2.4 MATLAB Results.............................................................................................................. 20
PHASE B: DIGITAL DATCOM..................................................................................................... 22
B.1. DATCOM Input File.......................................................................................................... 22
B.2. DATCOM Output File ....................................................................................................... 22
B.3. Aircraft Intuitive Design (AID).............................................................................................. 22
B.4. DATCOM input Parameters................................................................................................... 23
B.5. Digital DATCOM Output Results .......................................................................................... 25
B.5. Results Comparison ............................................................................................................... 26
PHASE C: THE LONGITUDINAL AND LATERAL STABILITY CHARACTERISTICS OF
JETSTAR ....................................................................................................................................... 27
C.1.1 Estimation of Longitudinal modes of Motion........................................................................ 30
C.1.2 Longitudinal Modes of Motion......................................................................................... 31
C.1.3 Long Period Approximation............................................................................................. 32
C.1.4 Short Period Approximation............................................................................................. 32
C.1.5 Comparison of Results .................................................................................................... 33
C.2.1 Estimations of Lateral modes of Motion............................................................................... 33
C.2.2. Spiral Approximation...................................................................................................... 35
C.2.3. Roll Approximation......................................................................................................... 35
C.2.4. Dutch Roll Approximation .............................................................................................. 35
C.2.5. Comparison of Exact and Approximate Method Results .................................................. 36

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3. GRAPHICAL RESULTS ........................................................................................................ 37
4. CONCLUSION....................................................................................................................... 40
5. REFERENCES........................................................................................................................ 41
APPENDEX A : LOCKHEED DIMENTIONS........................................................................................... 42
APPENDIX B: MATLAB CODING........................................................................................................ 43
APPENDIX C: Lockheed JetStar Created by AID ....................................................................... 51
List of Figures
Figure 1; Lockheed Jetstar................................................................................................................. 9
Figure 2; Proceedure for calculating Cma due to fuselage (C.Nelson,R, 2019.)................................ 11
Figure 3; correction factor vs. finess ratio (C.Nelson,R, 2019.) ........................................................ 11
Figure 4; Variation of local flow angle along the fuselage (C.Nelson,R, 2019.)................................. 11
Figure 5; Flap effectiveness parameter (C.Nelson,R, 2019.)............................................................. 12
Figure 6; Reynolds number correction factor (C.Nelson,R, 2019.).................................................... 13
Figure 7; Wing body interference factor (C.Nelson,R, 2019.)........................................................... 13
Figure 8; JetStar Longitudinal and Lateral Coefficients (C.Nelson,R, 2019.).................................... 17
Figure 9: CG of JetStar (C.Nelson,R, 2019.).................................................................................... 17
Figure 10: Aircraft sectioning.......................................................................................................... 19
Figure 11: DATCOM Digital Input Coding ......................................................................................... 24
Figure 12: Digital DATCOM Output Coding Results....................................................................... 25
Figure 13; Phugoid mode oscillation................................................................................................ 28
Figure 14; Short-period mode oscillations ....................................................................................... 28
Figure 15; predefined conditions and geometric parameters........................................................... 30
Figure 16; comparision of matlab and datcom results...................................................................... 40
Figure 17: Top View of Lockheed Jetstar.......................................................................................... 51
Figure 18: Side View of Lockheed Jetstar ......................................................................................... 51
Figure 19: Front View of Lockheed Jetstar ....................................................................................... 52
Figure 20: Isometric View of Lockheed Jetstar ................................................................................. 52
List of Tables
Table 1: Lockheed Jetstar Wing Assumptions.................................................................................. 18
Table 2: Wing Contribution Results................................................................................................. 18
Table 3: Fuselage parameters........................................................................................................... 18
Table 4: Calculations of all sections................................................................................................. 19
Table 5: Lockheed JetStar Horizontal Tail Parameters (C.Nelson,R, 2019.) ..................................... 20
Table 6: Table 5: Lockheed JetStar Vertical Tail Parameters............................................................ 20
Table 7: Tail Contribution Results Comparison ............................................................................... 20
Table 8: Yawing Moment Derivatives ............................................................................................. 21
Table 9: Rolling Moment Derivation............................................................................................... 21
Table 10: Flight Conditions............................................................................................................. 23
Table 11: Horizontal and Vertical Distances from Reference point (Aircraft nose)........................... 23
Table 12: Selected Airfoils.............................................................................................................. 23

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Table 13: Geometric characteristics of Wing and Tails .................................................................... 24
Table 14: pitching moment Results comparison............................................................................... 26
Table 15: yawing moment Results comparison ................................................................................ 26
Table 16: yawing moment Results comparison ................................................................................ 26
Table 17: Longitudinal stability coefficients .................................................................................... 30
Table 18: Longitudinal derivatives results........................................................................................ 31
Table 19: Long and short period results using SS model .................................................................. 32
Table 20: Long period approximations ............................................................................................ 32
Table 21: Short period approximations ............................................................................................ 32
Table 22: Results comparison.......................................................................................................... 33
Table 23: Lateral stability coefficients............................................................................................. 33
Table 24: Lateral stability derivatives.............................................................................................. 34
Table 25: Dutch roll, spiral mode and roll mode .............................................................................. 35
Table 26: Spiral mode approximation.............................................................................................. 35
Table 27: Roll mode approximation................................................................................................. 35
Table 28: Dutch roll approximation................................................................................................. 35
Table 29: Results comparison between the three modes................................................................... 36

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NOMENCLATURE

6

7

8

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1. INTRODUCTION
Aircraft Stability is defined as the ability of an aircraft to maintain or return to its equilibrium
state after being perturbed. The study of aircraft stability and control mainly concerns with the
degree of ease in controlling the aircraft, its behavior towards disturbances and its flying
qualities, For such flying qualities and features to be determined, a necessary analysis is
conducted that involves the determination of a set of stability coefficients through a wide set
of complex calculation. Further details regarding these stability coefficients are outlined in this
paper. Moreover, the analysis of the stability of an aircraft in the process of designing is carried
out by the means of three main phases (A, B & C) upon which the objectives of this project is
divided on
2. ABSTRACT
The objective of this project is to estimate the stability and control coefficients of the Lockheed
Jetstar at the predefined flight condition. Using MATLAB and Digital DATCOM the
coefficients are approximated and compared t against those provided in figure 4 of “flight
stability and automatic control”. In addition, using appropriate approaches, the longitudinal
and lateral-directional stability are also analyzed as well as the open-loop characteristics of the
aircraft.
PHASE A: STABILITY COEFFICIENT CALCULATIONS
In this phase of the project a Business Jet, Lockheed Jetstar was selected to estimate the stability
and control coefficients by calculation using hand calculations and MATLAB, the values
obtained from MATLAB coding will be compared with the values given in figure 4.
Geometric Characteristics:
• Wing Area (S) = 542.5 ft4
• Wing Span (b) = 52.75 ft
• Mean Aerodynamic Chord = 10.93 ft
Aerodynamic Characteristics:
• Mach Number at Sea Level = 0.20
• Mach Number at 40,000 ft = 0.80
• Weight = 38,200 lb Figure 1; Lockheed Jetstar

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A.1. Background on Stability and Control
A.1.1 Longitudinal Stability
The longitudinal stability coefficients determine the pitching stability of the aircraft. There are
3 major contributors to Longitudinal stability are; the Wings, Horizontal Tail and Fuselage.
Wing Contribution:
The wing provides lift hence has a destabilizing effect on the aircraft overall. To determine the
effect of the wing on longitudinal stability the following equations are used:
The Wing contribution at CG to pitching moment coefficient
From this equation we obtain pitching moment at zero angle of attack
We also obtain wing’s longitudinal static stability
Tail Contribution
The tail contribution to pitching moment contribution
In this equation:
Fuselage Contribution
Multopp’s method is used to achieve fuselage contribution to pitching moment.
The equation of Multopp’s method is given as;
(1)
(2)
(3)
(4)
(5)

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In this equation:
K4 − KI = Correction factor for body finesses ratio
c = Mean aerodynamic chord
S = Wing Area
wL = Average width of fuselage
iL = Incidence angle of fuselage
∆x = Length of fuselage increments
This graph is used to find K4 − KI:
The below graph estimates variation of local flow angle of the fuselage:
Figure 2; Proceedure for calculating Cma due to fuselage
(C.Nelson,R, 2019.)
Figure 3; correction factor vs. finess ratio (C.Nelson,R, 2019.)
Figure 4; Variation of local flow angle along the fuselage
(C.Nelson,R, 2019.)

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A.1.2 Longitudinal Control
The main longitudinal control surface is the elevator which is the main contributor for the
longitudinal motion of the aircraft. It has 2 main factors and equations:
Elevator Effectiveness Equation:
Elevator Control Equation:
Where,
The graph below illustrates Flap Effectiveness Parameter:
A.1.3 Directional Stability
Several factors including wing, vertical tail and fuselage are considered to achieve Directional
stability. Yawing moment coefficient is mainly affected by the wing and fuselage, equation
below is used to calculate it:
Where,
𝑘U = 𝑊𝑖𝑛𝑔 𝑏𝑜𝑑𝑦 𝑖𝑛𝑡𝑒𝑟𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑓𝑎𝑐𝑡𝑜𝑟
𝑘de
= 𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑓𝑎𝑐𝑡𝑜𝑟
𝑆hi = 𝑆𝑖𝑑𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑓𝑢𝑠𝑒𝑙𝑎𝑔𝑒
𝑙h = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑓𝑢𝑠𝑒𝑙𝑎𝑔𝑒
(6)
(7)
(8)
Figure 5; Flap effectiveness parameter (C.Nelson,R, 2019.)
(9)

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The Vertical tail contribution to yawing moment is given below using the equation below:
A.1.4 Directional Control
The directional control is mainly accounted by rudder movement. Rudder effectiveness and
control determines the directional control; hence its equations are given below:
Rudder Effectiveness:
Rudder Control:
A.1.5 Roll Control
The ailerons and spoilers are the major contributors to roll control of the aircraft. Its equation
is given below:
Figure 7; Wing body interference factor
(C.Nelson,R, 2019.)
Figure 6; Reynolds number correction factor
(C.Nelson,R, 2019.)
(10
)
(11)
(12)
(13)

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A.1.6 Longitudinal Stability and Lateral Stability Coefficient
X- Force Coefficients:
• Due to forward speed
• Due to AOA
Z – Force Coefficients:
• Due to AOA
• Due to rate of AOA
• Due to pitch rate
• Due to elevator deflection
A.1.7 Directional Stability and Control Coefficient
Y – Force Coefficients:
• Side force due to side slip condition
• Side force due to roll rate
(13)
(14)
(15)
(16)
(17)
(18)
(19)

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• Side force due to yaw rate
• Side force due to rudder deflection
Yawing Moment Coefficients:
• Directional static stability coefficient
• Yawing moment due to roll rate
• Yaw damping coefficient
• Cross control coefficient
• Rudder control coefficient
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(26)

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Rolling Moment Coefficient
• Lateral static stability coefficient
• Roll damping coefficient
• Rolling moment due to yaw rate
• Aileron Control Power
• Cross Control Coefficient
(27)
(28)
(29)
(30)
(31)

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A.2 DETAILED CALCULATIONS
A working model of Jetstar Lockheed is used to calculate values, these values where adapted
from Appendix B of “Flight Stability And Automatic Control”, we will compare these values
with our results obtained from MATLAB.
Figure 9: CG of JetStar (C.Nelson,R, 2019.)
Figure 8; JetStar Longitudinal and Lateral Coefficients (C.Nelson,R, 2019.)

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A.2.1. Wing Contribution
In order to perform MATLAB coding to achieve stability coefficients. The table below
illustrates all possible assumptions of the necessary parameters:
Wing Area 𝑺 𝒘 542.5 ft^2
Center of aerodynamic center 𝑥qr 0.25𝑐
Center of gravity 𝑥rs 0.3𝑐
Coefficient of moment at aerodynamic
center
𝐶tqru
0
Wing span 𝑏v 53.67ft
MAC 𝑐 10.93ft
Wing taper ratio 𝜆v 0.33
Wing Aspect ratio 𝐴𝑅v 5.461
Dihedral angle ⊺ 4 ∘
Quarter chord sweep angle ∧r/~ 30∘
Wing AOA at zero lift 𝑎•v 5.70∘
Wing incidence angle 𝑖v 5∘
Vertical distance from cg to ac of VT 𝑧v 2ft
2d Wing lift slope 𝐶•qv 0.097/deg
Side body area 𝑆hi 471.725 ft^2
Distance from Cr to inboard of aileron 𝑌I 20.1ft
Distance from Cr to outboard of aileron 𝑌4 25ft
Table 1: Lockheed Jetstar Wing Assumptions
Stability Coefficients MATLAB results
𝑪 𝑳 𝟎𝒘
0.4762
𝑪 𝑳 𝒂𝒘
4.1981
𝑪 𝒎 𝟎𝒘
0.0476
𝑪 𝒎 𝒂𝒘
0.4198
Table 2: Wing Contribution Results
A.2.2. Fuselage Contribution
Length of fuselage 𝒍 𝒇 60.5 ft
Max diameter of fuselage 𝑑tqŠ 7 ft
Distance from tip to CG 𝑋t 35 ft
Angle of incidence of fuselage 𝑖h 0 ∘
Table 3: Fuselage parameters
The Correction factor for body fineness ratio is obtained from figure 3 as:
𝑙h
𝑑tqŠ
= 8
𝑲 𝟐 − 𝑲 𝟏 = 𝟎.

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Calculating Fuselage pitching moments 𝐶t•h & 𝐶tqh using fuselage segments and Multopp’s
method:
Firstly, the fuselage is distributed into eight segments to obtain precise distance between
segments, the length of the increments is set to be ∆𝑥 = 5 𝑓𝑡.
Figure 10: Aircraft sectioning
Table 4: Calculations of all sections
Thus using Multopp’s equations :
𝐶t•h = 0.02022
Thereby the Fuselage pitching moment:
𝐶tqh = 0.2963
Station ∆𝒙 𝒇𝒕. 𝑾 𝒇 𝒇𝒕 𝒂 𝟎𝒘 + 𝒊 𝒇 𝒘 𝒇
𝟐
[𝒂 𝟎𝒘 + 𝒊 𝒇]∆𝒙
1 5 2.5 5.7 ∘ 156.25
2 5 4.3 5.7 ∘ 462.25
3 5 6.5 5.7 ∘ 1056.25
4 5 6.5 5.7 ∘ 1056.25
5 5 6 5.7 ∘ 900
6 5 5.6 5.7 ∘ 784
7 5 4 5.7 ∘ 400
8 5 1.4 5.7 ∘ 49
Total 4864

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A.2.3 Tail Contribution
The following values presented in table 5 and 6 are used to develop the MATLAB code
(APPENDIX B)
Horizontal tail area 𝑺𝒕 149 ft²
Horizontal tail span 𝑏™ 24.75 ft
Horizontal tail aspect ratio 𝐴𝑅™ 4.03
MAC for Horizontal tail 𝑐™ 6.6 ft
Horizontal tail root chord 𝐶𝑟™ 9.16 ft
Horizontal tail tip chord 𝐶𝑡™ 3 ft
Tail taper ratio 𝜆™ 0.32 ft
Sweep angle at 25 percent chord Λr/~ 30°
Length from cg of wing to ac of HT 𝑙™ 23.87 ft
2D lift slope for tail 𝐶𝑙q™ 0.1/deg
Tail efficiency factor 𝑛 1
Table 5: Lockheed JetStar Horizontal Tail Parameters (C.Nelson,R, 2019.)
Vertical tail area 𝑺 𝒗𝒕 110.2 ft²
Vertical tail span 𝑏œ™ 12.4 ft
Vertical tail aspect ratio 𝐴𝑅œ™ 1.4
MAC for vertical tail 𝑐œ™ 9.5 ft
Chord root for vertical tail 𝐶𝑟œ™ 12.9 ft
Chord tip for vertical tail 𝐶𝑡œ™ 4.83 ft
Vertical tail quarter chord sweep angle Λr/~ 45°
Horizontal distance from cg to ac of the vertical
tail
𝑙œ™ 17 ft
Vertical distance from cg to ac of the vertical tail 𝑧œ 5 ft
Angle of incidence for the tail 𝑖™ -1
Table 6: Table 5: Lockheed JetStar Vertical Tail Parameters
A.2.4 MATLAB Results
The following tables (7,8 and 9) displays the recorded values obtained from the developed
Matlab code (APPENDIX B) and compared against the value provided in appendix B of “Flight
Stability and Automatic Control” (figure 8)
Pitching Moment
coefficients
MATLAB results JetStar Appendix B
𝑪 𝑳 𝒂
4.7 5
𝑪 𝒎 𝒂
-0.5 -0.8
𝑪 𝒎 𝒂
-5 -3
𝑪 𝒎 𝒒
-10.31 -8
𝑪 𝑳 𝜹𝒆
0.43 0.4
𝑪 𝒎 𝜹𝒆
-0.944 -0.81
Table 7: Tail Contribution Results Comparison

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Yawing Moment
Derivatives
𝑪 𝒏 𝑩
0.121 0.137
𝑪 𝒏 𝒑
-0.148 -0.14
𝑪 𝒏 𝒓
-0.101 -0.16
𝑪 𝒏 𝜹𝒂
0.0055 0.0075
𝑪 𝒏 𝜹𝒓
-0.064 -0.063
Table 8: Yawing Moment Derivatives
Rolling Moment
Derivatives
𝑪𝒍 𝑩
-0.103 -0.103
𝑪𝒍 𝜹𝒂
0.025 0.054
𝑪𝒍 𝜹𝒓
0.029 0.029
𝑪𝒍 𝒑
-0.523 -0.37
𝑪 𝒚 𝑩
--0.6372 -0.72
𝑪𝒍 𝒓
0.3292 0.11
𝑪 𝒚 𝜹𝒓
0.175 0.175
Table 9: Rolling Moment Derivation

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PHASE B: DIGITAL DATCOM
The US Airforce Stability (USAF) and Control Digital DATCOM is a program which runs on
a computer. It is a technique for estimating the longitudinal, control stability and dynamic
derivatives of a fixed wing aircraft. Moreover, it’s widely used in the aerospace industry.
Digital DATCOM is a program which requires an input file which contains certain geometric
description of an aircraft, the output of this are the corresponding stability coefficients
according to the specified flight conditions. Furthermore, the output file of DATCOM can be
imported into MATLAB through specified function.
B.1. DATCOM Input File
Firstly, building the input file is the most significant and time-consuming task of the Digital
DATCOM process. The input file contains two basic sets of data, namely:
• Configuration of the aircraft in terms of Fuselage, Wing, Tails, Propulsion, Flaps, CG
and Moment of Inertia.
•
• Flight conditions in terms of altitude, Reynolds Number, Mach Number and Angle of
Attack.
B.2. DATCOM Output File
The output file is produced after the input file is processed at chosen flight conditions and
geometric values of the aircraft. The output consists of the following data:
• Input file reprint with error checks.
• Aerodynamic calculations and Stability Coefficients.
B.3. Aircraft Intuitive Design (AID)
The Aircraft Intuitive Design, AID is an application invented by a student at Embry-Riddle
Aeronautical University. AID is integrated into MATLAB which aids in the process of
understanding the process of Aircraft Design. Moreover, this application illustrates how
various design characteristics impact the way an aircraft flies by interfacing user-friendly
modelling through quick aerodynamic study. AID interfaces with the Digital DATCOM, due
to this AID has been used below to design a conceptual Jetstar 3D model using dimensions of
Lockheed Jetstar. The AID 3D model can be seen from appendix c, from which it was used to
generate an input file for DATCOM.

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B.4. DATCOM input Parameters
Using the parameters of the aircraft in the AID 3D Model tool, the flight condition is
provided as shown in table 10;
NMACH 1
MACH 0.20
NALPHA 10
ALSCHD 0°,2°,4°,6°,8°,10°,12°,14°,16°,18°
WT 38,200 lbs.
Table 10: Flight Conditions
Table 11 Illustrates the input parameters of horizontal and vertical distances measured from a
reference point which is defined to be the aircraft nose.
XCG 44.9762 ft.
ZCG 0 ft.
XW 27 ft.
ZW 0.46 ft.
ALIW 0°
XH 52 ft.
ZH 2.62 ft.
ALIH -1°
XV 50 ft.
ZV 0.79 ft.
Table 11: Horizontal and Vertical Distances from Reference point (Aircraft nose)
Table 12 Illustrates the selected airfoils for the type of Wing, Horizontal tail and Vertical tail.
Wing NACA-W-4-2412
Horizontal Tail NACA-H-4-0012
Vertical Tail NACA-V-4-0012
Table 12: Selected Airfoils
Table 13 Illustrates the Geometric characteristics of the wing, horizontal tail and vertical tail.

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Wing Horizontal Tail Vertical Tail
CHRDR 12 ft. 9 ft. 11.5 ft.
CHRDBP 8.6 ft. 7.5 ft. 3 ft.
CHRDTP 5 ft. 3 ft. 4 ft.
SSPN 26 ft. 12 ft. 6 ft.
SSPNE 24 ft. 10 ft. 4 ft.
SSPNOP 17.4 ft. 9.6 ft. 0 ft.
SAVSI 30° 25° 20°
SAVSO 30° 25° 0°
CHSTAT 0.25 0.25 0.25
Table 13: Geometric characteristics of Wing and Tails
Figure 16 below Illustrates the DATCOM Digital Input coding.
Figure 11: DATCOM Digital Input Coding

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B.5. Digital DATCOM Output Results
Figure 17 displays the Digital DATCOM output file displaying the stability coefficient results
obtained from the input file.
Figure 12: Digital DATCOM Output Coding Results

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B.5. Results Comparison
Table 14, 15 and 16 illustrates the comparison of longitudinal and control stability
coefficients results achieved by MATLAB, Digital DATCOM against Appendix B (figure 8)
values.
Pitching Moment
coefficients
MATLAB results JetStar Appendix B DATCOM
𝑪 𝑳 𝒂
4.7 5 2.045
𝑪 𝒎 𝒂
-0.5 -0.8 -0.5518
𝑪 𝒎 𝒂
-5 -3 -
𝑪 𝒎 𝒒
-10.31 -8 -6.338
𝑪 𝑳 𝜹𝒆
0.43 0.4 -
𝑪 𝒎 𝜹𝒆
-0.944 -0.81 -
Table 14: pitching moment Results comparison
Yawing Moment
coefficients
𝑪 𝒏 𝑩
0.121 0.137 -0.05725
𝑪 𝒏 𝒑
-0.148 -0.14 -0.05180
𝑪 𝒏 𝒓
-0.101 -0.16 -0.02380
𝑪 𝒏 𝜹𝒂
0.0055 0.0075 -
𝑪 𝒏 𝜹𝒓
-0.064 -0.063 -
Table 15: yawing moment Results comparison
Rolling Moment
coefficients
𝑪𝒍 𝑩
-0.103 -0.103 -0.1684
𝑪𝒍 𝜹𝒂
0.025 0.054 -
𝑪𝒍 𝜹𝒓
0.029 0.029 -
𝑪𝒍 𝒑
-0.523 -0.37 -0.1241
𝑪 𝒚 𝑩
--0.6372 -0.72 0.2268
𝑪𝒍 𝒓
0.3292 0.11 0.3343
𝑪 𝒚 𝜹𝒓
0.175 0.175 -
Table 16: yawing moment Results comparison

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PHASE C: THE LONGITUDINAL AND LATERAL STABILITY
CHARACTERISTICS OF JETSTAR
The motion of the aircraft is demonstrated using 3 translational and rotational equations these
equations can be analyzed through decoupling the equations of motions into longitudinal and
lateral equations. Where the longitudinal equations embody z-force, x-force and pitching
moment equations and the lateral embody Y-force, rolling and yawing equations. For the ease
of analyzing these equations the following assumptions are considered
• The motion of the airplane can be analyzed by separating the equations into two groups
• the aircraft in motion consists of small disturbance from its equilibrium flight
conditions
• all aerodynamic forces and moments are linear functions of the flight variables
disturbances
Moreover, these equations are represented in a second order differential form to mimic a
second order system.
C.1 Stick fixed longitudinal motion
the pure pitching motion equation is expressed as
From which the equation can be compared with the standard characteristic equation of 2nd
ordered system thereby the damping ratio and the undamped natural frequency by inspection
can be written as
The roots of the characteristic equation indicates what type of response the aircraft will have if
the roots are real the response will either be pure divergence or pure substance depending on
its sign if the roots are no complex the motion will be either Damped or undamped sinusoidal
oscillation the period of the of the oscillation is related to the imaginary part of the root is
expressed as;
The time to half amplitude when the second order system is stable or the time to double
amplitude when the system is unstable is estimated using the equation shown below:
(32)
(33)
(34)
(35)

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Moreover, the number of cycles to double or half amplitude based on estimated using:
When studying the behavior of an aircraft in free flight, it is essential to analyze the free
response of an aircraft to disturbance without any control input in the longitudinal motion the
disturbed motion is characterized by 2 distinct oscillating modes, these are.
The long-period mode in which the aircrafts experiences a gradual interchange in kinetic and
potential energy at large amplitude variation of airspeed pitch angle and altitude at almost no
change in the angle of attack. This mode this mode is referred to as the phugoid mode. This
motion occurs very slowly that the effects of inertia forces and damping forces are relatively
low.
The short-period mode in which the aircraft experiences a rapid pitching about the center of
gravity. The short period mode is a heavily damped oscillation with a short time
period occurring at nearly constant speeds
The aircraft’s longitudinal motion can be illustrated by a state space model based on a set
linearized first order differential equations that yields the following representation
(36)
(37)
Figure 13; Phugoid mode oscillation
Figure 14; Short-period mode oscillations

29
Utilizing the above state space representation, the Eigen values and can then be determined to
determine whether the aircraft longitudinal motion is stable or not. In addition, certain
assumption can also be considered to approximate the stability of both modes.
Long period approximation
The long period mode can be approximated by neglecting the pitching moment equation and
assuming that the change in angle of attack is negligible in other terms;
Hence the SS model yields the following characteristic equation
Where the frequency and the damping ratio can be expressed by as
Short mode approximation
The short Mode can be approximated by dropping the X force equation and considering
∆𝑢 equals 0 thus The SS model yields the following characteristic equation
Where the frequency and the damping ratio can be expressed as;
(37)
(38)
(40)
(39)
(41)
(42)
(43)

30
C.1.1 Estimation of Longitudinal modes of Motion
All the calculations for longitudinal derivatives are performed with respect to Standard Sea
Level conditions.
Density 0.00238 slug/ft
Mach Number 0.20
Gravitational Constant 32.2 ft/s²
Dynamic Pressure 59.228 lb/ft²
Weight 38,200 lb.
Free Stream Velocity 223.096 ft/s
Wingspan 53.75 ft
Mean Aerodynamic Chord 10.93 ft
Mass 1187.29 slugs
Moment of Inertia in x axis 118,773 slug.ft²
Moment of Inertia in y axis 135,869 slug.ft²
Moment of Inertia in z axis 243,504 slug.ft²
Figure 15; predefined conditions and geometric parameters
Table 16 illustrating the Longitudinal Stability coefficients depicted in figure 8.
𝑪 𝑳 0.737
𝑪 𝑫 0.095
𝑪 𝑫 𝟎
0.095
𝑪 𝑳 𝒂
5.0
𝑪 𝑫 𝒂
0.75
𝑪 𝒎 𝒂
-0.80
𝑪 𝑳 𝒂̇
0.0
𝑪 𝒎 𝒂̇
-3.0
𝑪𝒍 𝒒
0.0
𝑪 𝒎 𝒒
-10.0959
𝑪 𝑳 𝑴
0.0
𝑪 𝑫 𝑴
0.0
𝑪 𝒎 𝑴
-.05
𝑪 𝑳 𝜹𝒆
0.4
𝑪 𝒎 𝜹𝒆
-0.81
𝑪 𝒛 𝒂̇
-2.2624
𝑪 𝒛 𝒒
-4.6229
𝑪 𝒛 𝜹𝒆
-0.5821
𝑿 𝜹 𝒆
15.7358
𝑪 𝒎 𝒖
1.0
𝑪 𝑳 𝟎
0.737
Table 17: Longitudinal stability coefficients
The table below illustrates the longitudinal derivatives estimated with the use of MATLAB.

31
𝑿 𝒖 -0.0230
𝑿 𝒘 -0.0016
𝑿 𝒒 0
𝒁 𝒖 -0.1788
𝒁 𝒘 -0.6180
𝒁 𝒘̇ -0.0067
𝒁 𝒂 -137.8841
𝒁 𝒂̇ -1.4998
𝒁 𝒒 -3.065
𝒁 𝜹𝒆 -15.7538
𝑴 𝒖 0.0116
𝑴 𝒘 -0.0093
𝑴 𝒘̇ -0.00085144
𝑴 𝒂 -2.0748
𝑴 𝒂̇ -0.1900
𝑴 𝒒 -0.5065
𝑴 𝜹𝒆 -2.093
Table 18: Longitudinal derivatives results
C.1.2 Longitudinal Modes of Motion
This section estimates the Long period and Short Period Exact and Approximate values using
the State space model of the Longitudinal Derivatives. It also estimates the time to half
amplitude, Number of cycles to half amplitude and Period of both modes of motion.
State Space Representation
¬
∆𝒖̇
∆𝒘̇
∆𝒒̇
∆𝜽̇
® = ¬
𝑿 𝒖 𝑿 𝒘 𝟎 −𝒈
𝒁 𝒖 𝒁 𝒘 𝒖 𝟎 𝟎
𝑴 𝒖 + 𝑴 𝒘̇ 𝒁 𝒖 𝑴 𝒘 + 𝑴 𝒘̇ 𝒁 𝒘 𝑴 𝒒 + 𝑴 𝒘̇ 𝒖 𝟎 𝟎
𝟎 𝟎 𝟏 𝟎
® ¬
∆𝒖
∆𝒘
∆𝒒
∆𝜽
®
= ¬
−. 𝟎𝟐𝟑𝟎 −𝟎. 𝟎𝟎𝟏𝟔 𝟎 −𝟑𝟐. 𝟐
−𝟎. 𝟏𝟕𝟖𝟖 −𝟎. 𝟔𝟏𝟖𝟎 𝟐𝟐𝟑. 𝟎𝟗𝟔 𝟎
𝟎. 𝟎𝟏𝟏𝟕 −𝟎. 𝟎𝟎𝟖𝟖 −𝟎. 𝟔𝟗𝟔𝟓 𝟎
𝟎 𝟎 𝟏 𝟎
®¬
∆𝒖
∆𝒘
∆𝒒
∆𝜽
®
With the aid of MATLAB, the characteristic equation and the eigenvalues of it are estimated:
| 𝝀𝑰 − 𝑨| = 𝟎
Where,
𝑨 = ¬
−. 𝟎𝟐𝟑𝟎 −𝟎. 𝟎𝟎𝟏𝟔 𝟎 −𝟑𝟐. 𝟐
−𝟎. 𝟏𝟕𝟖𝟖 −𝟎. 𝟔𝟏𝟖𝟎 𝟐𝟐𝟑. 𝟎𝟗𝟔 𝟎
𝟎. 𝟎𝟏𝟏𝟕 −𝟎. 𝟎𝟎𝟖𝟖 −𝟎. 𝟔𝟗𝟔𝟓 𝟎
𝟎 𝟎 𝟏 𝟎
®
Hence the characteristic equation is:
𝝀 𝟒
+ 𝟏. 𝟑𝟑𝟕𝟓𝝀 𝟑
+ 𝟐. 𝟒𝟏𝟕𝟖𝝀 𝟐
+ 𝟎. 𝟒𝟑𝟔𝟗𝝀 + 𝟎. 𝟐𝟖𝟏𝟒 = 𝟎

32
Thus, the Eigenvalues are:
𝝀 𝟏,𝟐 = −𝟎. 𝟔𝟎𝟒𝟎 ± 𝒊 𝟏. 𝟑𝟐𝟕𝟖
𝝀 𝟑,𝟒 = −𝟎. 𝟎𝟔𝟒𝟖 ± 𝒊 𝟎. 𝟑𝟓𝟗𝟔
The Table below illustrates the results of the time to half amplitude, Number of cycles to half
amplitude and Period of both modes of motion.
Long Period (Phugoid) Short Period
𝝀 𝟑,𝟒 = −𝟎. 𝟎𝟔𝟒𝟖 ± 𝒊 𝟎. 𝟑𝟓𝟗𝟔 𝝀 𝟏,𝟐 = −𝟎. 𝟔𝟎𝟒𝟎 ± 𝒊 𝟏. 𝟑𝟐𝟕𝟖
𝒕 𝒉𝒂𝒍𝒇 = 𝟏𝟎. 𝟔𝟗 𝒔 𝑡¾q•h = 1.4181 𝑠
𝑵 𝒉𝒂𝒍𝒇 = 𝟎. 𝟔𝟏𝟐𝟒 𝒄𝒚𝒄𝒍𝒆𝒔 𝑁¾q•h = 0.2426 𝑐𝑦𝑐𝑙𝑒𝑠
𝑷𝒆𝒓𝒊𝒐𝒅 = 𝟏𝟕. 𝟒𝟕𝟐𝟕 𝒔 𝑃𝑒𝑟𝑖𝑜𝑑 = 4.7320 𝑠
Table 19: Long and short period results using SS model
C.1.3 Long Period Approximation
Table 4.7 below illustrates the Long period approximation using appropriate approximation
methods and equations 38,29 and 40.
𝝎 𝒏 𝒑
0.1606
𝜻 𝒑 0.0717
𝝀 𝒑 −0.0115 ± 𝑖 0.1602
𝑷𝒆𝒓𝒊𝒐𝒅 39.2130 s
𝒕 𝒉𝒂𝒍𝒇 60.1483 s
𝑵 𝒉𝒂𝒍𝒇 1.5339 cycles
Table 20: Long period approximations
C.1.4 Short Period Approximation
Table 4.7 below illustrates the short period approximation using appropriate approximation
methods and equations 41,42 and 43.
𝝎 𝒏 𝒑
1.5453
𝜻 𝒑 0.4260
𝝀 𝒑 −0.6583 ± 𝑖 1.3980
𝑷𝒆𝒓𝒊𝒐𝒅 4.4943 s
𝒕 𝒉𝒂𝒍𝒇 1.0530 s
Table 21: Short period approximations

33
C.1.5 Comparison of Results
Modes Approximate Method Exact Method Difference
𝝀 𝟑,𝟒 = −0.0115 ± 𝑖 0.1602 𝝀 𝟑,𝟒 = −0.0648 ± 𝑖 0.3596
Long Period tÈÉÊL = 60.1483 s tÈÉÊL = 10.69 s 82.227 %
NÈÉÊL = 1.5339 cycles NÈÉÊL = 0.6124 cycles 60.070 %
Period = 39.2130 s Period = 17.4727 s 55.441 %
𝝀 𝟏,𝟐 = −0.6583 ± 𝑖 1.3980 𝝀 𝟏,𝟐 = −0.6040 ± 𝑖 1.3278
Short Period
tÈÉÊL = 1.0530 s tÈÉÊL = 1.1481 s 8.283 %
NÈÉÊL = 0.2343 cyc NÈÉÊL = 0.2426 cyc 3.421 %
Period = 4.4943 s Period = 4.7320 s 5.023 %
Table 22: Results comparison
From table 22, it can be observed that the short period approximation provides more accurate
approximations than the long periods approximations method this is evident by the difference
in approximations between both methods and the exact calculation method
C.2.1 Estimations of Lateral modes of Motion
This section involves the use of MATLAB to estimate the lateral derivatives of Lockheed
Jetstar using the lateral
Derivatives provided in (figure 8). The Table below illustrates the lateral directional stability
coefficients
𝑪 𝒚 𝑩
-0.72
𝑪𝒍 𝑩
-0.103
𝑪 𝒏 𝑩
0.137
𝑪 𝒚 𝒑
0.0
𝑪𝒍 𝒑
-0.37
𝑪 𝒏 𝒑
-0.14
𝑪 𝑳 𝒓
0.11
𝑪 𝒏 𝒓
-0.16
𝑪𝒍 𝜹𝒂
0.054
𝑪 𝒏 𝜹𝒂
0.0075
𝑪 𝒚 𝜹𝒓
0.175
𝑪𝒍 𝜹𝒓
0.029
𝑪 𝒏 𝜹𝒓
-0.063
Table 23: Lateral stability coefficients

34
The table below illustrates the results of the Lateral directional derivatives estimated with the
aid of MATLAB.
𝒀 𝜷 -19.4851
𝑵 𝜷 0.9702
𝑳 𝜷 -1.4955
𝒀 𝒑 0.000
𝑵 𝒑 -0.1193
𝑳 𝒑 -0.6462
𝒀 𝒓 0.000
𝑵 𝒓 -0.1363
𝑳 𝒓 0.1921
𝒀 𝜹𝒂 0.000
𝑵 𝜹𝒂 0.0531
𝑵 𝜹𝒓 -0.4462
𝑳 𝜹𝒂 0.7840
𝑳 𝜹𝒓 0.2054
Table 24: Lateral stability derivatives
This section estimates the Dutch roll, Spiral and Roll modes of Lateral Stability. Applying
Exact and Approximate methods using the State space model of the Longitudinal Derivatives.
It also estimates the time to half amplitude, Number of cycles to half amplitude and Period of
both modes of motion.
⎣
⎢
⎢
⎡∆𝜷̇
∆𝒑̇
∆𝒓̇
∆𝝓̇ ⎦
⎥
⎥
⎤
=
⎣
⎢
⎢
⎢
⎡
𝒀 𝑩
𝒖 𝟎
𝒀 𝒑
𝒖 𝟎
−(𝟏 −
𝒀 𝒓
𝒖 𝟎
)
𝒈𝒄𝒐𝒔𝜽 𝟎
𝒖 𝟎
𝑳 𝜷 𝑳 𝒑 𝑳 𝒓 𝟎
𝑵 𝜷 𝑵 𝒑 𝑵 𝒓 𝟎
𝟎 𝟏 𝟎 𝟎 ⎦
⎥
⎥
⎥
⎤
¬
∆𝜷
∆𝒑
∆𝒓
∆𝝓
®
= ¬
−𝟎. 𝟎𝟖𝟕𝟑 𝟎 −𝟏 −𝟎. 𝟎𝟐𝟓
−𝟏. 𝟒𝟗𝟓𝟓 −𝟎. 𝟔𝟒𝟔𝟐 𝟎. 𝟏𝟗𝟐𝟏 𝟎
𝟎. 𝟗𝟕𝟎𝟐 −𝟎. 𝟏𝟏𝟗𝟑 −𝟎. 𝟏𝟑𝟔𝟑 𝟎
𝟎 𝟏 𝟎 𝟎
®¬
∆𝜷
∆𝒑
∆𝒓
∆𝝓
®
With the aid of MATLAB, the characteristic equation and the eigenvalues are found.
|𝝀𝑰 − 𝑨| = 𝟎
Where,
𝑨 = ¬
−𝟎. 𝟎𝟖𝟕𝟑 𝟎 −𝟏 −𝟎. 𝟎𝟐𝟓
−𝟏. 𝟒𝟗𝟓𝟓 −𝟎. 𝟔𝟒𝟔𝟐 𝟎. 𝟏𝟗𝟐𝟏 𝟎
𝟎. 𝟗𝟕𝟎𝟐 −𝟎. 𝟏𝟏𝟗𝟑 −𝟎. 𝟏𝟑𝟔𝟑 𝟎
𝟎 𝟏 𝟎 𝟎
®

35
Therefore, the characteristic equation is:
𝝀 𝟒
+ 𝟎. 𝟖𝟔𝟗𝟖𝝀 𝟑
+ 𝟏. 𝟏𝟒𝟗𝟓𝝀 𝟐
+ 𝟎. 𝟕𝟕𝟕𝟔𝝀 + 𝟎. 𝟎𝟎𝟎𝟒 = 𝟎
The Eigenvalues are:
𝝀 𝑫𝒖𝒕𝒄𝒉 𝑹𝒐𝒍𝒍 = −𝟎. 𝟎𝟔𝟗𝟓 ± 𝒊 𝟏. 𝟎𝟐𝟒𝟏
𝝀 𝒔𝒑𝒊𝒓𝒂𝒍 = −𝟎. 𝟎𝟎𝟎𝟔
𝝀 𝒓𝒐𝒍𝒍 = −𝟎. 𝟕𝟑𝟗𝟏
The table below illustrates the use of the eigenvalues found to estimate the time to half
amplitude, Number of cycles to half amplitude and Period of all the modes.
Dutch Roll Mode Spiral Mode Roll Mode
𝝀 𝑫𝒖𝒕𝒄𝒉 𝑹𝒐𝒍𝒍 = −𝟎. 𝟎𝟔𝟗𝟓 ± 𝒊 𝟏. 𝟎𝟐𝟒𝟏 𝝀 𝒔𝒑𝒊𝒓𝒂𝒍 = −𝟎. 𝟎𝟎𝟎𝟔 𝝀 𝒓𝒐𝒍𝒍 = −𝟎. 𝟕𝟑𝟗𝟏
𝒕 𝒉𝒂𝒍𝒇 = 𝟏𝟎. 𝟓𝟓𝟎𝟐 𝒔 𝑡×•ØÙ•Ú = 1155 𝑠 𝑡¾q•h = 0.9378 𝑠
𝑵 𝒉𝒂𝒍𝒇 = 𝟏. 𝟕𝟏𝟗 𝒄𝒚𝒄𝒍𝒆𝒔 - -
𝑷𝒆𝒓𝒊𝒐𝒅 = 𝟔. 𝟏𝟑𝟓 𝒔 - -
Table 25: Dutch roll, spiral mode and roll mode
C.2.2. Spiral Approximation
The Table below demonstrates the results of Spiral Approximation.
𝝀 𝒔𝒑𝒊𝒓𝒂𝒍 −𝟎. 𝟎𝟏𝟏𝟕
𝒕 𝒅𝒐𝒖𝒃𝒍𝒆 59.4404 𝑠
Table 26: Spiral mode approximation
C.2.3. Roll Approximation
The Table below demonstrates the results of Roll Approximation.
𝝀 𝒓𝒐𝒍𝒍 −𝟎. 𝟔𝟒𝟔𝟐
𝒕 𝒉𝒂𝒍𝒇 1.0727 𝑠
Table 27: Roll mode approximation
C.2.4. Dutch Roll Approximation
The Table below demonstrates the results of Dutch Roll Approximation.
𝝎 𝒏 0.9910
𝜻 0.1108
𝝀 −0.1098 ± i 0.9849
Period 6.3401 s
𝒕 𝒉𝒂𝒍𝒇 6.3117 s
Table 28: Dutch roll approximation

36
C.2.5. Comparison of Exact and Approximate Method Results
Modes Approximate Exact Difference
Spiral
𝜆 = −0.0117 𝜆 = −0.005 57.264 %
𝑡×•ØÙ•Ú = 59.4404 𝑠 𝑡×•ØÙ•Ú = 138.6 𝑠 57.114 %
Roll
𝜆 = −0.6462 𝜆 = −0.7268 11.089 %
𝑡¾q•h = 1.0727 𝑠 𝑡¾q•h = 0.9534 𝑠 11.121 %
Dutch
Roll
𝝀 𝑫𝒖𝒕𝒄𝒉 𝑹𝒐𝒍𝒍 = −0.1098
± i 0.9849
𝜆ÜØ™r¾ d•••
= −0.0695 ± 𝑖 1.0241
Real: 37.76%
Imaginary: 6.719%
𝑡×•ØÙ•Ú = 6.3117 𝑠 𝑡¾q•h = 10.5502 𝑠 40.174 %
𝑁¾q•h = 0.9955 𝑠 𝑁¾q•h = 1.719 𝑠 42.088 %
Period = 6.3401 s Period = 6.135 s 3.235 %
Table 29: Results comparison between the three modes

37
3. GRAPHICAL RESULTS
This graph illustrates the response of x-velocity (forward speed) and z-velocity (vertical speed)
by Elevator control input of 5 degrees. The graph shows that there is damping in both x and z
velocity component before 60 seconds and after that it vanishes.
This graph illustrates the response of x-velocity (forward speed) and z-velocity (vertical speed)
by Elevator control input of 5 degrees. The graph shows oscillations in both x and z velocity
component till 60 seconds after that the oscillation converges

38
This graph illustrates the response of change in Roll rate, Roll angle, Sideslip and Yaw rate by
Rudder deflection of 5 degrees. The aircraft isn’t experiencing oscillations in Roll rate and Roll
angle. The aircraft is experiencing small oscillations in sideslip and yaw rate until 25 seconds
after that it gradually becomes stable.
This graph illustrates the response of Roll angle and Roll rate by Aileron control input of 5
degrees. The change in Side slip and Yaw rate is negligible.

39
The above bode diagram depicts the plot of magnitude and phase angle against the frequency.
At high frequencies, the amplitude ratio is low indicating that the effect of the elevator on the
pitch altitude within the following range of frequency is negligibly small.

40
4. CONCLUSION
In order to estimate the longitudinal and lateral stability control coefficients of the Lockheed
Jetstar at sea level, codes were developed from the basis of “ASs on the for MATLAB and
DATCOM. The results that were obtained from MATLAB and DATCOM such as the
coefficients Where compared to those in appendix B in “Flight stability and Automatic
Controls”. It was observed That the results obtained is slightly different between them and
those in appendix B(figure 8), since some geometric parameters were assumed as they where
not provided in the aircraft data sheet.
The longitudinal static stability coefficients that were obtained from MATLAB and DATCOM
Confirm that the aircraft Possesses Static longitudinal stability. The weathercock coefficients
also provide that the aircraft is directionally stable from the roll static stability coefficients, in
addition. Both methods, exact and approximation, were used to estimate the modes of the
longitudinal and lateral motions. It can also be noted throughout the report that short period
approximations are more accurate than phugoid approximations and the roll approximation has
a more enhanced approximation than spiral and Dutch roll approximations.
MATLAB DATCOM
𝑪 𝒎 𝒂
- 0.5 - 0.5518
𝑪 𝒏 𝜷
0.121 - 0.05725
𝑪𝒍 𝜷
- 0.103 - 0.1684
Figure 16; comparision of matlab and datcom results

41
5. REFERENCES
• C.Nelson, R., 1998. Flight Stability and Automatic Control. 2nd ed. Singapore:
McGraw-Hill Book Co.
• Airvectors.net. (2019). The Lockheed Jetstar & Rockwell Sabreliner. [online] Available at:
https://airvectors.net/avjetst.html [Accessed 29 Oct. 2019].
• Mathworks.com. (2019). Building Graphical Aircraft Design Tools Video. [online] Available
at: https://www.mathworks.com/videos/MATLAB-and-simulink-robotics-arena-building-
interactive-design-tools-1509569729395.html [Accessed 24 Nov. 2019].
• Mathworks.com. (2019). MATLAB and Simulink Robotics Arena. [online] Available at:
https://www.mathworks.com/videos/series/MATLAB-and-simulink-robotics-arena.html
[Accessed 14 Oct. 2019].
• En.wikipedia.org. (2019). United States Air Force Stability and Control Digital DATCOM.
[online] Available at:
https://en.wikipedia.org/wiki/United_States_Air_Force_Stability_and_Control_Digital_DAT
COM [Accessed 7 Nov 2019].
• Forum.flightgear.org. (2019). FlightGear forum • View topic - create a DATCOM input file.
[online] Available at: https://forum.flightgear.org/viewtopic.php?f=36&t=12959 [Accessed
24 Dec. 2019].

42
APPENDEX A : LOCKHEED DIMENTIONS

43
APPENDIX B: MATLAB CODING

44

45

46

47

48

49

50

51
APPENDIX C: Lockheed JetStar Created by AID
Figure 17: Top View of Lockheed Jetstar
.
Figure 18: Side View of Lockheed Jetstar

52
Figure 19: Front View of Lockheed Jetstar
Figure 20: Isometric View of Lockheed Jetstar

FLIGHT VEHICLE DESIGN THESIS REPORT

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