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Lateral Directional Approximations to Aircraft

A Dissertation

Submitted in Partial Fulfilment of the

Requirements for the Degree of

Master of Engineering

in Aerospace Engineering

By

Joel George

Department of Aerospace Engineering

Indian Institute of Science

Bangalore - 560 012

India

July 2005

Abstract

The generalized approximate equations governing the lateral-directional modes of an aircraft with con-

ventional configuration are discussed in this report. Such approximations existing in the open literature

are reviewed, evaluated and many of them are shown to be inaccurate. The derivation of new simple,

accurate and consistent approximations is presented. Some work in the area of high angle of attack flight

dynamics is also reported.

The most frequently used governing equations of an aircraft are a set of linearized, first order ordinary

differential equations representing the dynamics of a six degree of freedom rigid aircraft. These equations

do not provide an insight into the participation of and the role played by different parameters (stability

derivatives) in dynamics. Such an insight is important for efficient classroom teaching, aircraft design

and control law algorithm formulation. A quick tour of the existing literature reveals that good approx-

imations exist for longitudinal modes. However, not all of the lateral-directional modes have simple, yet

accurate approximate representation in spite of the presence of umpteen number of these approximations

in literature. The existing approximations are tested for their accuracy over a wide spectrum database with

data for different types of airplanes in various flight conditions. It is found that accurate approximations

exist only for the spiral mode. New approximations which are accurate and simple are developed for roll

root and dutch roll frequency. Toward this a simulation package was developed in MATLAB to study the

physics of modes by visualization. Extensive simulation studies were done using the simulation package.

These simulations revealed that the roll mode of at least some airplanes involve a considerable amount of

participation of yaw and sideslip, apart from roll. This finding disproves the traditional notion that the

roll mode consist purely of rolling motion. Inspired by this finding, new approximation to roll mode root

is developed which is shown to be accurate and simple. An accurate dutch roll frequency approximation

is shown to be heavily depended on a good roll approximation. The new roll root approximation is used

to obtain an excellent approximation to dutch roll frequency. Finally, an attempt was made to derive

handling quality criteria at high angles of attack. The result obtained is reported. Lack of time precluded

further pursuit of this interesting and sparsely known realm of dynamics.

In a nutshell, this report is an exhaustive review of the lateral-directional approximations and provides

new exemplary approximations.

i

Contents

Abstract i

List of Tables vii

List of Figures ix

Nomenclature x

1 Introduction 1

1.1 Mathematical Modelling of Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Linearization of Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Decoupling of Aircraft Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.1 Solution of the Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.2 Why an Analytical Solution? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.3 Why an Approximate Solution? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4.4 Desired Characteristics of Approximations . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Flight Dynamics at High Angles of Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.6 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.7 An Overview of the Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Governing Equations of the Lateral Directional Dynamics 5

2.1 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

ii

Contents iii

2.2 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Simplified Form of Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Factorization of the Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Approximations to the Spiral Mode 9

3.1 A Brief Description of the Spiral Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Existing Spiral Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 Traditional Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.2 Quasi-steady Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2.3 Small Root Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.4 The IITB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.5 The Bu Aer Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.6 Kolk’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2.7 Livneh’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Evaluating the Accuracies of Spiral Approximations . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Selection of the Airplane Database . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.2 Database of Roskam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.3 The Measure of Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.4 Calculation of Dimensional Stability Derivatives . . . . . . . . . . . . . . . . . . . 16

3.4 Comments on Existing Spiral Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Roll Mode Approximations 19

4.1 An Introduction to the Roll Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 Roll Mode Approximations in the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.1 One Degree of Freedom Approximation . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.2 Two Degree of Freedom Approximation . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.3 Three Degree of Freedom Approximation . . . . . . . . . . . . . . . . . . . . . . . 20

Contents iv



4.2.6 Approximate Factorization Approximation . . . . . . . . . . . . . . . . . . . . . . 21

4.2.7 Mengali and Giulietti’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Analysis of Accuracies of Roll Approximations . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Development of A New Roll Mode Approximation 25

5.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5.2 Detailed Analysis of the Roll Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.2.1 The Simulation Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.2 Observations and Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Modifying the New Roll Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.4 Remarks on the New Roll Mode Approximation . . . . . . . . . . . . . . . . . . . . . . . . 27

6 Dutch Roll Approximations 29

6.1 The Dutch Roll Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2 Existing Dutch Roll Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6.2.1 One Degree of Freedom Approximation . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2.2 Two Degree of Freedom Approximation . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2.3 Three Degree of Freedom Approximation . . . . . . . . . . . . . . . . . . . . . . . 32

6.2.4 Lanchester’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2.5 Seckel’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.2.6 Russel’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.2.7 Phillips’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35


6.2.9 The IITB Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36


6.2.11 Hancock’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Contents v

6.2.12 Etkin and Reid’s Approximation for Dutch Roll Damping . . . . . . . . . . . . . . 36

6.3 Evaluation of Accuracies of the Dutch Roll Approximations . . . . . . . . . . . . . . . . . 37

6.3.1 Evaluation of Dutch Roll Frequency Approximations . . . . . . . . . . . . . . . . . 37

6.3.2 Evaluation of the Dutch Roll Damping Approximations . . . . . . . . . . . . . . . 39

6.4 Concluding Comments on Dutch Roll Approximations . . . . . . . . . . . . . . . . . . . . 39

7 A New Accurate Approximation for the Dutch Roll Frequency 41

7.1 Derivation of the Novel Dutch Roll Frequency Approximation . . . . . . . . . . . . . . . . 41

7.2 Evaluation of the Accuracy of the New Approximation . . . . . . . . . . . . . . . . . . . . 42

8 High Angle of Attack Flight Dynamics 43

8.1 Salient Characteristics at High Angles of Attack . . . . . . . . . . . . . . . . . . . . . . . 43

8.2 Lateral-Directional Departure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.2.1 Modes of Departure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.2.2 Departure Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.3 Lateral-Directional Mode Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.4 Development of Lateral-Directional Handling Quality Criteria . . . . . . . . . . . . . . . . 45

8.5 Generation of High Angle of Attack Data for F16 . . . . . . . . . . . . . . . . . . . . . . . 46

8.5.1 Calculation of Non-dimensional Derivatives . . . . . . . . . . . . . . . . . . . . . . 46

8.5.2 Calculation of Dimensional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 48

8.6 Evaluation of Accuracy of Proposed Approximation . . . . . . . . . . . . . . . . . . . . . 51

8.7 Remarks on the New Handling Quality Criteria . . . . . . . . . . . . . . . . . . . . . . . . 51

9 Conclusions 53

9.1 Highlights and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A The Airplane Database 55

A.1 Cessna 172 (Airplane A) @ low altitude cruise . . . . . . . . . . . . . . . . . . . . . . . . . 56

Contents vi

A.2 Beechcraft 99 (Airplane B) @ power approach, low and high altitude cruise . . . . . . . . 57

A.3 Marchetti S211 (Airplane C) @ approach, normal and high altitude cruise . . . . . . . . . 58

A.4 Learjet 24 (Airplane D) @ approach, maximum and low weight cruise . . . . . . . . . . . 59

A.5 F4C (Airplane E) @ power approach, subsonic and supersonic cruise . . . . . . . . . . . . 60

A.6 Boeing 747 (Airplane F) @ power approach, low and high altitude cruise . . . . . . . . . . 61

B The Simulation Package 62

B.1 Salient Features of the Simulation Package Developed . . . . . . . . . . . . . . . . . . . . 62

B.2 Choice of Eigenvector as Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

C Flying qualities and Airworthiness Criteria 68

C.1 Definition of Airplane Class, Flight Phase and Level of Handling Qualities . . . . . . . . 68

C.1.1 Definition of Airplane Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

C.1.2 Definition of Flight Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

C.1.3 Definition of Levels of Flying Qualities . . . . . . . . . . . . . . . . . . . . . . . . 71

C.2 Control Forces Required of the Pilot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

C.3 Requirements for Dynamic Longitudinal Stability . . . . . . . . . . . . . . . . . . . . . . 72

C.4 Requirements for Dynamic Lateral-Directional Stability and Roll Response . . . . . . . . 72

C.4.1 Requirements for Dynamic Lateral-Directional Stability . . . . . . . . . . . . . . . 72

C.4.2 Requirements for Roll Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

D Data for F16 76

D.1 Geometry and Inertias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

D.2 Nondimensional Force and Moment Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 77

D.3 Body Axes Six Degree of Freedom Equations for Aircraft . . . . . . . . . . . . . . . . . . 78

References 80

List of Tables

3.1 Airplanes in Roskam’s database . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Relations to convert non-dimensional stability derivaties to dimensional derivatives . . . . 16

3.3 Accuracies of existing spiral approximations . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Accuracies of existing spiral approximations (contd.) . . . . . . . . . . . . . . . . . . . . . 18

4.1 Accuracy of existing roll mode approximation . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Accuracy of existing roll mode approximation (contd.) . . . . . . . . . . . . . . . . . . . . 24

5.1 Accuracy of the new roll mode approximation . . . . . . . . . . . . . . . . . . . . . . . . . 28

6.1 Accuracy of dutch roll frequency approximations . . . . . . . . . . . . . . . . . . . . . . . 37

6.2 Accuracy of dutch roll frequency approximations (contd.) . . . . . . . . . . . . . . . . . . 38

6.3 Accuracy of dutch roll frequency approximations (contd.) . . . . . . . . . . . . . . . . . . 38

6.4 Accuracies of dutch roll damping approximations . . . . . . . . . . . . . . . . . . . . . . . 39

6.5 Accuracies of dutch roll damping approximations (contd.) . . . . . . . . . . . . . . . . . . 40

6.6 Accuracies of dutch roll damping approximations (contd.) . . . . . . . . . . . . . . . . . . 40

7.1 Accuracy of the new dutch roll frequency approximation . . . . . . . . . . . . . . . . . . . 42

C.1 Short-Period Damping Ratio Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

C.2 Spiral Stability - Minimum Time to Double Amplitude, T2S. . . . . . . . . . . . . . . . . 73

C.3 Minimum Dutch Roll Frequency And Damping . . . . . . . . . . . . . . . . . . . . . . . . 73

C.4 Maximum Roll-Mode Time Constant (Seconds) . . . . . . . . . . . . . . . . . . . . . . . . 74

vii

List of Tables viii

C.5 Roll Performance Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

C.6 Roll Performance for Air-to-Air Combat . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

C.7 Roll Performance for Ground Attack with External Stores . . . . . . . . . . . . . . . . . . 75

D.1 Nondimensional damping derivatives with angle of attack (α) . . . . . . . . . . . . . . . . 77

D.2 Rolling moment coefficients for F16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

D.3 Yawing moment coefficients for F16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

D.4 Cx and Cm variation with α and elevator deflection . . . . . . . . . . . . . . . . . . . . . . 78

D.5 Cz - The Z force with α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

List of Figures

3.1 Stable and unstable spiral modes following a disturbance, illustrating the participation of

yaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Participation of roll and loss of altitude in an unstable spiral mode . . . . . . . . . . . . . 10

3.3 Definition of Stability axis (Xs-Zs) and Body axis (Xb-Zb) . . . . . . . . . . . . . . . . . . 17

5.1 A typical session with simulation/animation package developed in MATLAB with a GUI . 26

6.1 A flowchart illustration of onset and propagation of the dutch roll . . . . . . . . . . . . . 30

8.1 Clβ variation with sideslip (β) at an angle of attack of 10◦ . . . . . . . . . . . . . . . . . . 47

8.2 Computed lateral-directional stability derivatives for F16 for varying angles of attack . . . 48

8.3 The Root Locus of lateral-directional roots - the change in roots with angle of attack . . . 50

8.4 Comparison of exact damping of combined spiral-roll mode and the damping given by

proposed approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

B.1 The GUI window to input data to simulation program . . . . . . . . . . . . . . . . . . . . 64

B.2 The SIMULINK model of the aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.3 Control panel GUI for the simulation package . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.4 Model of an aircraft used for animation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

B.5 A turboprop airplane model developed in MATLAB . . . . . . . . . . . . . . . . . . . . . 67

B.6 An F16 model used for visualization of aircraft motions . . . . . . . . . . . . . . . . . . . 67

ix

Nomenclature

English Alphebet

A coefficient of characteristic equation, system matrix

A1 ratio of inertias (Ixz/Ixx)

a0 coefficient of characteristic equation




B coefficient of characteristic equation

B1 ratio of inertias (Ixz/Izz)

b wing span

C coefficient of characteristic equation

Cl coefficient of rolling moment

Clp nondimensional variation of rolling moment coefficient with roll rate

Clr nondimensional variation of rolling moment coefficient with yaw rate

Clβ nondimensional variation of rolling moment coefficient with sideslip

Clδanondimensional variation of rolling moment coefficient with aileron deflection

Clδrnondimensional variation of rolling moment coefficient with rudder deflection

Cm coefficient of pitching moment

Cn coefficient of yawing moment

Cnpnondimensional variation of yawing moment coefficient with roll rate

Cnrnondimensional variation of yawing moment coefficient with yaw rate

Cnβnondimensional variation of yawing moment coefficient with sideslip

Cnδanondimensional variation of yawing moment coefficient with aileron deflection

Cnδrnondimensional variation of yawing moment coefficient with rudder deflection

Cy side force coefficient

c mean aerodynamic chord

c1 inertial coefficient


x

Nomenclature xi








D coefficient of characteristic equation

E coefficient of characteristic equation

g acceleration due to gravity

I identity matrix

Ixx moment of inertia; subscript b: body axis, s: stability axis

Ixz product of inertia; subscript b: body axis, s: stability axis

Iyy moment of inertia; subscript b: body axis, s: stability axis

Izz moment of inertia; subscript b: body axis, s: stability axis

L rolling moment, matix of left eigen vectors

l left eigen vector

Lp dimensional variation of rolling moment with roll rate

L′p primed dimensional variation of rolling moment with roll rate

Lr dimensional variation of rolling moment with yaw rate

L′r primed dimensional variation of rolling moment with yaw rate

Lβ dimensional variation of rolling moment with sideslip

L′β primed dimensional variation of rolling moment with sideslip

M pitching moment

m mass of aircraft

N yawing moment

Np dimensional variation of yawing moment with roll rate

N ′p primed dimensional variation of yawing moment with roll rate

Nr dimensional variation of yawing moment with yaw rate

N ′r primed dimensional variation of yawing moment with yaw rate

Nβ dimensional variation of yawing moment with sideslip

N ′β primed dimensional variation of yawing moment with sideslip

P roll rate

p perturbed roll rate

Q pitch rate

q perturbed pitch rate

q1 steady state dynamic pressure

R yaw rate, matrix of right eigen vectors

Nomenclature xii

r right eigen vector

r perturbed yaw rate

s Laplace transform variable

T2ptime to double amplitude for phugoid

T2stime to double amplitude for spiral mode

U velocity component in X direction

U1 steady state velocity

V velocity component along Y direction

W velocity component in Z direction

X force along X axis, X axis

x state vector

Y force along Y axis, Y axis

Yp dimensional variation of side force with roll rate

Yr dimensional variation of side force with yaw rate

Yβ dimensional variation of side force with sideslip

Z force along Z axis, Z axis

Greek Alphebet

α total or perturbed angle of attack

β total or perturbed sideslip angle

Γ flight path angle, inertial coefficient

δa aileron defelection

δe elevator defelection

δr rudder defelection

ζD dutch roll damping

ζp phigoid damping

ζsp short period mode damping

ζsr combined spiral-roll damping

Θ angle of attack

Θ1 steady state angle of attack

Λ diagonal matrix of eigen values

λ eigen value

λr roll mode root

λs spiral root

Φ roll angle

φ perturbed roll angle

Ψ yaw angle

Nomenclature xiii

ψ perturbed yaw angle

ωnDdutch roll frequency

ωnsrcombined spiral-roll mode frequency

Chapter 1

Introduction

This introductory chapter is a general discussion on aircraft dynamics, equations of motion and

approximations to the modes of aircraft.

1.1 Mathematical Modelling of Aircraft

The simplest way to model an aircraft is to assume it as a point mass. Although such an approach,

sometimes called the energy approach, gives interestingly amazing solutions like the the dive-zoom path

as the minimum time to climb trajectory for a supersonic aircraft [1], it does not capture all the dynamics

of the aircraft which is of interest to an analyst or a designer. Real aircraft have flexible structures.

However, it is seen that equations of motion derived under rigid body assumption provide an acceptable

model of the actual airplane.

The equations of motion of a rigid aircraft are nonlinear. It is a set of twelve nonlinear first order ordinary

differential equations. Out of the twelve equations, six are kinematic and six are dynamic. The kinematic

equations can be separated into three navigation equations and three equations for Euler angles. The

dynamic equations consist of three force and three moment equations. The navigation equations are

uncoupled with the remaining nine. This simplifies the analysis; the remaining nine equations can be

considered as a set and the navigation equations may be considered separately.

1.2 Linearization of Equations of Motion

The resulting nine nonlinear equations can be linearized about a steady state, which is an equilibrium

point. Such linearization is reasonable with no apologies because be it a climb, a level cruise, a coordinated

turn, a power approach and for that matter, even in a pull up, the aircraft is in a steady state, or it

operates about an equilibrium point most of the time. The interest of a designer or an analyst is in the

1

Chapter 1. Introduction 2

behaviour of airplane about one of these equilibrium points.

1.3 Decoupling of Aircraft Dynamics

The linearized equations of motion of a rigid aircraft can be decoupled into longitudinal dynamics which

involve the motions in the plane of symmetry of the aircraft and lateral-directional dynamics which consist

of the out of plane of symmetry motions. The longitudinal dynamics is characterized by two oscillatory

modes - the phugoid mode which is a long period lightly damped mode and the short period mode which

is a high frequency highly damped mode. The lateral-directional dynamics consist of the spiral mode,

the roll mode and the dutch roll mode.

1.4 Characteristic Equation

The longitudinal dynamics is represented by a characteristic equation which is a fourth order polynomial

equation called the quartic. The roots of this quartic characteristic equation give the modes. The

characteristic equation of the lateral-directional dynamics is a fifth order polynomial equation called the

quintic, the roots of which gives the modes of lateral-directional dynamics. One of the roots of this quintic

equation is zero which shows a neutrally stable mode. The fact that the dynamics of the aircraft is not

affected by the direction in which the aircraft heads give rise to this neutrally stable mode. Thus with

one zero root, the characteristic equation of the lateral-directional dynamics can be reduced to a quartic.

1.4.1 Solution of the Characteristic Equation

The solution of the characteristic equation gives the roots of non-oscillatory modes and the frequency and

damping of oscillatory modes. The quartic characteristic equation can be numerically solved to obtain

the roots which give the modes of the aircraft. However, the numerical solution does not give an insight

into how the different parameters (stability derivatives) affect the modes.

1.4.2 Why an Analytical Solution?

The need for an analytical equation are

• A knowledge of how the frequency and damping of each mode is affected by various stability

derivatives is very important in class room education as far as understanding and appreciating the

behaviour of aircraft is concerned.

• The aviation regulation authority requirements for an aircraft dictates the bounds for the frequency

and the damping of various modes. Any aircraft design should comply to these specifications. Thus,


it is of at most importance for a designer to know which parameter to tune to attain the desired

requirements. A numerical solution does not give a clue, thus necessitating an analytical solution.

• Moreover, an analytical solution gives information about which of the stability derivatives are

dominant as far as a particular mode is concerned. Such an information is important for a control

law algorithm designer in that he can tailor his algorithm so as not to spend computational time

and effort in real time estimation of the non-dominant stability derivatives.

1.4.3 Why an Approximate Solution?

The quartic equation has an analytical solution [2]. The analytical technique used to find the roots of

a quartic polynomial equation is known as Ferrari’s method. However, the obtained analytical solutions

are too lengthy to make any reasonable sense out of it. Symbolic softwares like MATHEMATICA [3]

gives solutions which runs into pages. This does not serve the purpose. Approximate expressions for the

roots of characteristic equation which are simple, yet give reasonable accuracies are needed.

The mammoth expression obtained as a solution to the quartic characteristic equation of the aircraft

motion can be reduced to a simpler yet accurate form. This is because, most of the terms in this huge

expression are not significant when compared to the few dominant terms. Thus, with the omission of

insignificant terms, approximate and at the same time accurate enough expressions for the roots can be

arrived at.

However, the most popular method of obtaining accurate approximations is by making meaningful phys-

ical assumptions about the modes. For example, an exemplarily accurate approximation for short period

mode of the longitudinal dynamics can be arrived at by assuming that the forward velocity remains

constant during this mode. However, this approach may not work always and the obtained expressions,

many times, may have to be modified to attain desired accuracy for the approximation.

1.4.4 Desired Characteristics of Approximations

The approximate equations of the aircraft motion should possess certain desirable qualities in that it

should be 1) simple, 2) accurate and 3) consistent. The approximation must be simple and succinct so

as to give a physical insight. However, the accuracy of the results should not be compromised much for

the simplicity. Moreover, the approximation should be consistent so that accurate results are obtained

not only for a particular case but for a wide variety of aircraft and for different flight conditions.

1.5 Flight Dynamics at High Angles of Attack

Modern day fighter aircraft require high angle of attack manoeuvring capabilities. Thus the high angle

of attack flight dynamics is of current interest to the aerospace industry. One or more of the modes of an


aircraft can go unstable during high angle of attack flights, making the airplane go rapidly out of control.

This phenomenon is called departure and may be catastrophic. Thus the accurate prediction of onset of

departure is important. Also, at high angle of attack, some of the real modes may combine to become an

oscillatory mode or an oscillatory mode may split to form two real modes. Thus the aircraft behaviour

and response changes drastically at high angle flights. This point towards the need for good handling

quality criteria in the high angle of attack regime.

1.6 Scope

Better and better understanding of the aircraft behaviour has resulted in the formulation of a lot of ap-

proximations to aircraft motion. As far as the longitudinal mode approximations are concerned, very good

approximations for short period mode existed. The problem of finding an accurate approximation for the

phugoid was addressed, among others [5, 11–14, 21], by Pradeep [4]. In an elegant paper titled ‘A Century

of Phugoid Approximations’, he reviewed the then existing phugoid approximations, demonstrated the

inaccuracy and inconsistency of many of them and derived a new phugoid approximation.

An extensive survey of the existing literature, as put down in the subsequent chapters of this report,

reveals that no good approximations exist for at least some modes of lateral and directional dynamics of

the aircraft.

Also in the high angle of attack regime, the literature lacks the presence of accurate departure criteria

and good handling quality criteria.

1.7 An Overview of the Report

This section gives an overview of the organization of rest of the report. Having stated the importance of

the study of approximations, next chapter puts down the exact equations governing the lateral-directional

dynamics of an aircraft from which all approximations are to be derived. Chapter 3 reviews the existing

spiral approximations, and shows that accurate approximations exist for the spiral mode. The roll mode

approximations present in the literature are evaluated for their accuracy and consistency in Chapter

4. The development of a new roll approximation is discussed in Chapter 5. The following chapter, i.e.

Chapter 6, enumerates and studies the existing dutch roll approximations in the literature. Chapter 7 is

a description on the new dutch roll frequency approximation. Chapter 8 describes the works done during

this project on high angle of attack flight dynamics. The last chapter summarizes the report and gives a

few direction for future work.

Chapter 2

Governing Equations of the Lateral Directional Dynamics

The equations governing the lateral-directional dynamics of an aircraft is presented in this chap-

ter. A proper appreciation of these equations is important as, in many cases, they are the

starting point to the derivation of approximations to lateral-directional modes.

2.1 Linearized Equations of Motion

The exact equations governing the motion of an aircraft are nonlinear and the longitudinal and lateral-

directional dynamics are coupled. These nonlinear equations can be linearized about an equilibrium

point, say, the steady 1g level flight. In the linearized set of equations, the lateral-directional dynam-

ics is decoupled from the longitudinal dynamics. The linearized equations governing lateral-directional

dynamics assume the following form.

β =Yβ

U1β +

Yp

U1p− (U1 − Yr)

U1r +

g cos Θ1

U1φ

p−A1r = Lββ + Lpp+ Lrr

r −B1p = Nββ +Npp+Nrr (2.1)

φ = p

ψ = r

The notations used here are those followed by Roskam [5]. The first three equations in the above

set account for the dynamics and the remaining two are the kinematic equations. In the equations

representing dynamics, the first one is the side force equilibrium and the other two are rolling and yawing

moment balances respectively.

5

Chapter 2. Governing Equations of the Lateral Directional Dynamics 6

2.2 Characteristic Equation

The set of governing differential equations (2.1) is converted into algebraic equations by taking the Laplace

transform. The set of algebraic equations thus obtained can be simultaneously solved to obtain a fifth

degree polynomial characteristic equation given by

s(As4 +Bs3 + Cs2 +Ds+ E) = 0 (2.2)

where

A = U1(1−A1B1)

B = −Yβ(1−A1B1)− U1(Lp +Nr +A1Np +B1Lr)

C = U1(LpNr − LrNp) + Yβ(Lp +Nr +A1Np +B1Lr)− Yp(Lβ +NβA1)

+ U1(LβB1 +Nβ)− Yr(LβB1 +Nβ) (2.3)

D = −Yβ(LpNr − LrNp) + Yp(LβNr −NβLr)− g cos Θ1(Lβ +NβA1)

+ U1(LβNp −NβLp)− Yr(LβNp −NβLp)

E = g cos Θ1(LβNr −NβLr)

2.3 Simplified Form of Equations of Motion

In the equation set (2.1), the angular acceleration equations (second and third equations) are coupled

through the inertial parameters A1 and B1 which are ratios of moments of inertia. Decoupling the angular

accelerations, the equation set can be rewritten in the following form.

β =Yβ

U1β +

Yp

U1p− (U1 − Yr)

U1r +

g cos Θ1

U1φ

p =(Lβ +A1Nβ)

1−A1B1β +

(Lp +A1Np)1−A1B1

p+(Lr +A1Nr)

1−A1B1r

r =(Nβ +B1Lβ)

1−A1B1β +

(Np +B1Lp)1−A1B1

p+(Nr +B1Lr)

1−A1B1r (2.4)

φ = p

ψ = r

For sake of brevity and easy comprehension, the following notations are introduced.

L′i =Li +A1Ni

1−A1B1& N ′

i =Ni +B1Li

1−A1B1(2.5)

where the subscript i is either β, p or r.


Using the new notation, the set of equations (2.4) can be written in a compact form as given below.

β =Yβ

U1β +

Yp

U1p− (U1 − Yr)

U1r +

g cos Θ1

U1φ

p = L′ββ + L′pp+ L′rr

r = N ′ββ +N ′

pp+N ′rr (2.6)

φ = p

ψ = r

The above set of equations can be denoted in the vector form as x = Ax, where x = [β p r φ ψ]T . The

characteristic equation of such a linear dynamical system is given by det(sI−A) = 0 which in the present

case takes the form

s(s4 + a3s3 + a2s

2 + a1s+ a0) = 0 (2.7)

where

a3 =B

A= −Yβ

U1− L′p −N ′

r

a2 =C

A= (L′pN

′r − L′rN

′p) +

Yβ

U1(L′p +N ′

r)−Yp

U1L′β +N ′

β −Yr

U1N ′

β

a1 =D

A= −Yβ

U1(L′pN

′r − L′rN

′p) +

Yp

U1(L′βN

′r −N ′

βL′r)−

g

U1cos Θ1L

′β (2.8)

+ (L′βN′p −N ′

βL′p)−

Yr

U1(L′βN

′p −N ′

βL′p)

a0 =E

A=

g

U1cos Θ1(L′βN

′r −N ′

βL′r)

It is to be noted that equation (2.7) is same as equation (2.2) as both are the characteristic equations of

the same system.

2.4 Factorization of the Characteristic Equation

The modes of the lateral-directional dynamics are the Spiral and the Roll modes which are non-oscillatory

and the oscillatory Dutch Roll mode. Therefore, the roots of the characteristic equation should give

the time constants of the real modes and frequency and damping of the oscillatory mode. Using this

information, the characteristic equation (2.7) can be factorized as follows.

λ4 + a3λ3 + a2λ

2 + a1λ+ a0 = (λ− λs)(λ− λr)(λ2 + 2ζDωnDλ+ ω2

nD) (2.9)

where

• λs is the spiral root,

• λr is the roll root,

• ωnDis the dutch roll frequency and


• ζD is the dutch roll damping.

Equating the coefficients on both sides of equation (2.9) and using equation (2.8), the following set of

relations can be obtained.

a3 =B

A= 2ζDωnD

− λr − λs

a2 =C

A= ω2

nD− 2ζDωnD

(λr + λs) + λrλs

a1 =D

A= 2ζDωnD

λrλs − (λr + λs)ω2nD

(2.10)

a0 =E

A= ω2

nDλrλs

It should be observed that these relations contain λs, λr, ω2nD

and 2ζDωnD, the approximations for

which are sought for. If good approximations exist for a couple of these, accurate approximations for the

remaining could be obtained from one of the relations given in (2.10). This is one of the strategies to

be employed in this report, for the derivation of new simple, yet accurate and consistent approximations

to lateral-directional modes. Such an approach was earlier used by Pradeep [4] for the derivation of a

phugoid approximation.

Chapter 3

Approximations to the Spiral Mode

Approximate expressions for the spiral mode root, which is a real root, existing in the literature

are enumerated and evaluated for their simplicity, accuracy and consistency, in this chapter. It

is found that good approximations exist for this mode as long as the magnitude of spiral root

is small in comparison to the root of the roll mode.

3.1 A Brief Description of the Spiral Mode

The spiral mode consists mostly of roll (φ) and yaw (ψ) with very little participation of sideslip (β).

Usually, the spiral root is small in magnitude and therefore it is a slow mode. Most of the airplanes have

unstable or marginally stable spiral modes.

Figure 3.1 illustrates the participation of yaw in the spiral mode. The figure depicts the behaviour of

two aircraft, one with stable and other with unstable spiral mode, after the spiral mode is excited due to

some disturbance.

An illustration of the amount of roll involved in the spiral mode is given by Figure 3.2, which depicts an

unstable spiral. As shown in the figure, once an aircraft enters into a spiral mode, in the absence of any

control or power corrections, the aircraft will lose altitude continuously.

3.2 Existing Spiral Approximations

An almost exhaustive survey of the literature reveals that at least five different approximations exist

for the spiral mode root. The number of methods or techniques that exist is more than the number

of approximations itself as some of the methods, although different in spirit, lead to the same final

expression. This section presents the derivation of these approximations.

9

Chapter 3. Approximations to the Spiral Mode 10

Y

X

Stable Spiral

Unstable Spiral

Figure 3.1: Stable and unstable spiral modes following a disturbance, illustrating the participation ofyaw

Z

Y

Unstable Spiral

Figure 3.2: Participation of roll and loss of altitude in an unstable spiral mode


3.2.1 Traditional Approximation

Spiral mode, as seen above, is dominated by φ and ψ with very small β. However, the aerodynamic

forces in spiral mode depend on β, φ and ψ. Assuming that the φ forces are the weakest, it is possible to

approximate the spiral root by eliminating φ degree of freedom. With these premises and assumptions,

the equation of motion (2.1) becomes

0 = 0

−A1r = Lββ + Lrr

r = Nββ +Nrr (3.1)

0 = 0

Note that the kinematic equation ψ = r, present in equation (2.1), which corresponds to the neutrally

stable yaw mode, is omitted in the above set of equations and will be excluded from all further listings

for the sake of brevity and clarity.

The equation set (3.1) gives a one degree of freedom (first degree polynomial) characteristic equation,

the solution to which is

λs =LβNr −NβLr

Lβ +NβA1(3.2)

For most of the airplanes, A1 ≈ 0 and therefore NβA1 � Lβ . Thus, the traditional approximation as

given by Roskam [5] and Nelson [6] is

λs =LβNr −NβLr

Lβ(3.3)

3.2.2 Quasi-steady Approximation

Any turning motion of the aircraft can be assumed to be a quasi-steady process. In a quasi-steady

turn, the centripetal force due to yaw rate balances side force due to rotation of lift vector, assuming

aerodynamic side force to be small. With the assumption that Θ1 ≈ 0, this forms the first equation in the

equation set (3.4) given below. Also, the angular accelerations p and r are neglected as they are small.

Thus the equations (2.1) governing the lateral-directional motion of the aircraft reduces to the following

form.

0 = −r +g

U1φ

0 = Lββ + Lpp+ Lrr

0 = Nββ +Npp+Nrr (3.4)

φ = p

The characteristic equation corresponding to above set of equations is given as

λs =g

U1

(NβLr − LβNr)(LβNp −NβLp)

(3.5)


Hancock [7] and Babister [8] give such an approximation which is derived using the quasi-steady turn

assumption.

Cook [9] goes through similar arguments to arrive at an expression for the spiral root which same as that

given above except that Yr, the side force due to yaw rate, is kept instead of neglecting it. This gives the

spiral mode root as

λs =g

(U1 − Yr)(NβLr − LβNr)(LβNp −NβLp)

(3.6)

In the above derivations to obtain an expression for the spiral root, the angular accelerations, p and r,

were neglected assuming that they are small. Relaxing such an assumption and at the same time holding

on to the quasi-steady turn argument, the governing equation (2.6) becomes as given below.

0 = −r +g

U1φ


r = N ′ββ +N ′

pp+N ′rr (3.7)

φ = p

Taking the Laplace transform of the above equation, the characteristic equation can be formed as

s2 +

[−L′p +

L′βN ′

β

(N ′

p −g

U1

)]s+

g

U1

(L′βN ′

β

N ′r − L′r

)= 0 (3.8)

This is the two degree of freedom characteristic equation for the combined roll and spiral modes. Thus

the above equation is of the form λ2 + b1λ+ b0 = 0, where

λ2 + b1λ+ b0 = (λ− λr) (λ− λs)

= λ2 + (−λr − λs)λ+ λrλs (3.9)

Equating the the coefficients of the above equation and assuming that the roll root (λr) is much large

compared to the spiral root (λs), an approximation for the spiral mode can be arrived at in the following

manner.

b1 = −λr − λs ≈ −λr

b0 = λrλs (3.10)

⇒ λs = −b0b1

Substituting into above equation, the coefficients of equation (3.8), Stevens [10] obtains an approximate

expression for the spiral root as

λs =− g

U1

(L′βN

′r −N ′

βL′r

)L′β

(N ′

p −g

U1

)− L′pN

′β

(3.11)


3.2.3 Small Root Approximation

The spiral root is usually smaller in magnitude by one or two orders when compared to the other roots

of the characteristic equation. Therefore, a good approximation for the spiral root can be obtained by

retaining only the lower order terms in the characteristic equation (2.2), which then becomes Dλ+E = 0.

Thus, the spiral approximation is given as

λs = −ED

Substituting the expressions for D and E from equation (2.3) the above equation can be written as

−g cos Θ1(LβNr −NβLr)−Yβ(LpNr − LrNp) + Yp(LβNr −NβLr)− g cos Θ1(Lβ +NβA1) + (U1 − Yr)(LβNp −NβLp)

(3.12)

Such an approximation is given by McRuer et. al. [11], Russel [12], Seckel [13], Etkin and Reid [14] and

McLean [15]. The number of citations of this approximation itself indicate the popularity of and the

confidence that scientific community has on this approximation. Later in this chapter, it is shown that

this approximation almost accurately describes the spiral mode of conventional airplanes.

3.2.4 The IITB Approximation

Ananthkrishnan and Unnikrishnan [16] of IIT Bombay arrive at an expression through a set of arguments

different from any of those given above. However, the final expression obtained for the spiral root is a

simpler form of the equation (3.12) which is given below.

λs =g(LβNr −NβLr)

Yβ(LpNr − LrNp) + gLβ − U1(LβNp −NβLp)(3.13)

3.2.5 The Bu Aer Approximation

The Bu Aer report [17] claims to have done an approximate factorization of the complete characteristic

equation to obtain an expression which could actually be arrived at by neglecting many non-dominant

terms in equation (3.12). The expression given in the report is

λs =g(LβNr −NβLr)

YβLpNr + gLβ + U1NβLp(3.14)

3.2.6 Kolk’s Approximation

As seen in subsection 3.2.3, a first approximation to spiral mode root can be obtained as −E/D. A more

accurate approximation could be obtained by correcting the first approximation using Lin’s method [18].

Such an approximation that Kolk [19] provides for the spiral root is given below.

λs = − E(D − CE

D

) (3.15)


where C, D and E are as given in equation (2.3) or equation (2.8).

It is interesting to note that all the approximations which were arrived at till now, in spite of the manner

of derivation, can be obtained from −E/D by neglecting or adding certain terms or the others.

3.2.7 Livneh’s Approximation

The literal approximation for the spiral mode given by Livneh [20] is as follows.

λs =Lβg cos Θ1

(1−A1B1)U1λr ωn2D

λs (3.16)

where

λs =LβNr − LrNβ

Lβ

λr = −Lp

ωn2D = Nβ +

NrYβ −NβYr

U1

3.3 Evaluating the Accuracies of Spiral Approximations

The previous section listed out and in all feasible cases gave the derivation of spiral approximations

existing in literature. These approximations should be tested for their accuracies. In this section, these

approximations are evaluated to see how well they predict the spiral mode for different airplanes in various

flight conditions.

3.3.1 Selection of the Airplane Database

Most of the text books give an example or at the most two, to illustrate the accuracy of approximations.

However, this approach does not ensure the consistency and generality of the approximations being

evaluated. Therefore, a ‘wide spread database’ with data for all classes of airplanes and for various

different flight conditions of these airplanes is required.

Databases given by appendix C of Roskam [5] and that given by Heffley and Jewell [23] comply to the

above requirement and have been favourites of researchers. In this report, the database of Roskam will

be used, against which, the approximations are tested.

3.3.2 Database of Roskam

Roskam provides data for six modern aircraft in a total of sixteen flight conditions. The database spans

over aircraft ranging from a small piston engine airplane through regional turboprop to a wide-body jet


transport and a supersonic fighter. A variety of flight conditions like cruise, power approach, low and high

altitude cruise, maximum and low weight cruise and subsonic and supersonic cruise have been considered

and the data pertaining to the same have been given in this database.

The aircraft in database provided by Roskam, their types and flight conditions for which data are

available are listed in Table 3.1 The data for above set of aircraft in different flight conditions given

Aircraft Representative of: Flight ConditionsCessna 172

A small, single piston engine (1) Power approachgeneral aviation airplaneBeech M99 (1) Power approach

B small, twin turboprop (2) Low altitude cruiseregional commuter airplane (3) High altitude cruiseSIAI–Marchetti S211 (1) Power approach

C small, single jet engine (2) Normal cruisemilitary training airplane (3) High altitude cruiseGates Learjet M24 (1) Power approach

D twin jet engine (2) Maximum weight cruisecorporate airplane (3) Low weight cruiseMcDonnell Douglas F4C (1) Power approach

E twin jet engine (2) Subsonic cruisefighter/attack airplane (3) Supersonic cruiseBoeing 747 (1) Power approach

F large, four jet engine (2) High altitude cruisecommercial transport airplane (3) Low altitude cruise

Table 3.1: Airplanes in Roskam’s database

by Roskan include information on the geometry and inertias, the steady state flight conditions and the

non-dimensional stability derivatives. This is presented in Appendix A of the report.

3.3.3 The Measure of Accuracy

The metric chosen to measure and compare the accuracies of various approximations is the percentage

error defined as

%Error =Exact Value − Approximate Value

Exact Value× 100 (3.17)

The exact value is obtained through the numerical solution of the complete quartic characteristic equa-

tion while the approximate value is that given by an approximation. A good approximation will have

percentage errors close to zero for all the airplanes and all the flight conditions in the database. High

errors in all the cases shows that the approximation is inaccurate and that in a few cases points towards

the inconsistency of the approximation.


3.3.4 Calculation of Dimensional Stability Derivatives

The aircraft database of Roskam gives non-dimensional stability derivatives. However, the governing

equations (2.1) involves the stability derivatives in dimensional form. The dimensional stability derivatives

can be calculated from the non-dimensional derivatives, using the information of steady state flight

condition, geometry and inertias, from the expressions given in table below. The moments and products

Table 3.2: Relations to convert non-dimensional stability derivaties to dimensional derivatives

Yβ =q1SCyβ

mYp =

q1SCyp

2mU1Yr =

q1SCyr

2mU1

Lβ =q1SbClβ

IxxLp =

q1Sb2Clp

2IxxU1Lr =

q1Sb2Clr

2IxxU1

Nβ =q1SbCnβ

IzzNp =

q1Sb2Cnp

2IzzU1Nr =

q1Sb2Cnr

2IzzU1

of inertias used in these expression should be in the stability axis as the governing equations (2.1) are

valid only for stability axis. However, the inertia data given by Roskam are with respect to body axis

(The distinction between the stability and body axis is made clear in figure 3.3). Therefore, it is necessary

to transform the inertias from body axis to stability axis and this is accomplished through the following

relations. Ixxs

Izzs

Ixzs

=

cos2 α1 sin2 α1 − sin 2α1

sin2 α1 cos2 α1 sin 2α1

12 sin 2α1 − 1

2 sin 2α1 cos 2α1

Ixxb

Izzb

Ixzb

where α1 is the steady state angle of attack and the subscripts s and b denotes the quantities in stability

and body axis respectively.

3.4 Comments on Existing Spiral Approximations

The percentage errors of the various spiral approximations over the chosen database is listed in Table 3.3

and Table 3.4

As seen from the table, the traditional approximation for the spiral mode, given by Roskam and others,

is grossly inaccurate which makes Roskam claim that “the spiral simplification is at best dubious”[5].

However, Roskam’s statement loses its generality after the inspection of accuracies of some other approx-

imations as presented in Tables 3.3 and 3.4.

For example, the approximation as in equation (3.15) is exemplary in that the error is almost zero in all


Figure 3.3: Definition of Stability axis (Xs-Zs) and Body axis (Xb-Zb)

the test cases. The maximum percentage error is 0.35 for this approximation given by Kolk. However,

this expression for the spiral root which contains C, D and E, as seen from equation (3.15), after the

substitution of these quantities from equation (2.3), becomes sufficiently complex so as not to get any

insight or physical understanding into the mechanism behind the spiral mode. Thus, this approximation,

although performs well in the domains of accuracy and consistency, is not simple.

Consider the approximation given in equation (3.11) as given by Stevens. It is very accurate with the

magnitude of percentage error less than one for eleven out of sixteen test cases. It should be appreciated

that such an accuracy is being achieved in spite of being a very simple expression, unlike the approximation

given by Kolk. For one case the error percentage is greater than seven which of course is a black mark

on the consistency or generality of this approximation.

On the contrary, the spiral approximation given as −E/D, equation (3.12), is consistent over the test

cases although it is not as accurate as the above stated approximation in individual cases.

3.5 Concluding Remarks

The spiral approximations in the literature were derived, listed down and evaluated for their accuracies,

in this chapter. It is observed that simple yet accurate and consistent approximations for spiral mode

do exist. The small root approximation to the spiral mode given as −E/D was found to be a reasonable

spiral approximation.


Airplane Flight eqn 3.3 eqn 3.5 eqn 3.6 eqn 3.11 eqn 3.12Condition % Error % Error % Error % Error % Error

A 1 −3256.72 −3.83 −4.68 −0.71 0.221 −412.56 −18.55 −19.60 1.53 2.51

B 2 −1388.59 −8.92 −10.23 −1.79 1.133 −994.99 −9.55 −10.06 −0.30 0.831 −249.19 −51.92 −53.69 −7.35 −5.35

C 2 −2878.93 −4.20 −4.69 −0.88 −0.303 −1863.41 −6.02 −6.37 −0.89 −0.511 −150.91 −77.66 −78.48 −4.57 −3.81

D 2 −623.17 −15.92 −16.05 −0.04 0.243 −1287.42 −7.67 −7.84 −0.24 0.091 −212.08 −60.84 −60.84 −0.45 1.64

E 2 −1025.86 −9.46 −9.46 0.54 1.023 −2780.93 −4.04 −4.04 −0.02 0.381 −252.39 −42.63 −42.63 0.01 4.64

F 2 −1039.46 −12.56 −12.56 −2.29 −1.183 −696.12 −14.71 −14.71 −0.16 2.16

Table 3.3: Accuracies of existing spiral approximations

Airplane Flight eqn 3.13 eqn 3.14 eqn 3.15 eqn 3.16Condition % Error % Error % Error % Error

A 1 1.01 −7.33 −0.00 −11.771 5.43 4.61 −0.10 −22.74

B 2 2.65 8.16 −0.01 −0.113 2.02 7.50 −0.01 −5.381 −3.12 −8.64 −0.35 −61.20

C 2 0.32 14.89 −0.00 11.123 0.08 12.84 −0.00 7.611 −3.01 −23.30 0.07 −143.50

D 2 0.38 −0.94 0.00 −17.333 0.53 −3.10 0.00 −12.131 −6.27 −3.58 0.02 −69.37

E 2 0.63 −7.97 0.01 −19.603 −0.18 −0.18 0.00 −4.191 2.45 −36.10 −0.13 −123.88

F 2 −1.40 1.23 0.01 −8.073 1.98 −4.4 0.01 −20.12

Table 3.4: Accuracies of existing spiral approximations (contd.)

Chapter 4

Roll Mode Approximations

The amazing sight of a rolling fighter airplane brings eternal bliss for the eyes and standing

ovation to the mind. This chapter reviews the roll approximation existing in literature and

evaluates their accuracies. It is found that no simple yet accurate expression for roll mode root

exist.

4.1 An Introduction to the Roll Mode

The roll mode, for most of the airplanes, consists of purely rolling motion. However, extensive simulation

studies conducted during the course of this project over different types of aircraft for various flight

conditions revealed that the roll modes of at least some airplanes involve the participation of yaw and

side slip which if not accounted for will result in an incomplete and inaccurate representation of the

roll mode. Therefore, any good approximation developed for roll should take into account the above

observation.

4.2 Roll Mode Approximations in the Literature

Many distinct approximations for roll mode exists in literature. This section gives an almost exhaustive

list of existing approximations.

4.2.1 One Degree of Freedom Approximation

The roll mode consists largely of rolling motion. Therefore, it can be approximated as having only one

degree of freedom. Thus all the variables in equation (2.1) except p and φ are set to zero. The equation

19

Chapter 4. Roll Mode Approximations 20

(2.1) then becomes

0 = 0

p = Lpp

0 = 0 (4.1)

φ = p

The characteristic equation of above set of equations is

λ(λ− Lp) = 0 (4.2)

which gives the rolling approximation as

λr = Lp (4.3)

Such an approximation, called the one degree of freedom approximation, is given by Roskam [5], Nelson

[6], Hancock [7], Babister [8], Blakelock [21] and Ananthkrishnan [16].

Etkin and Reid [14], McLean [15], Seckel [13] and Cook [9] follow a similar approach but use equation

(2.6) instead of equation (2.1) to arrive at an approximation for roll mode as

λr = L′p (4.4)

4.2.2 Two Degree of Freedom Approximation

To obtain a two degree of freedom approximation, as given by Russel [12], the side force equation is set to

zero. It is assumed that the yawing moment generated by the derivative Np is balanced by that generated

by the side slip derivative Nβ . Also assuming the yawing velocity to be small the equation (2.1) can be

written as

0 = 0

p = Lββ + Lpp

0 = Nββ +Npp (4.5)

φ = p

This gives the rolling approximation as

λr = Lp − LβNp

Nβ(4.6)

4.2.3 Three Degree of Freedom Approximation

The roll mode involves very small sideslip (β) motions. Thus in the equation (2.6), β and β can be set

to zero in the side force equation. Further with assumptions that Yr ≈ 0,Θ1 ≈ 0, Yp ≈ 0, the three

degree of freedom equation for the combined spiral and roll mode becomes same as equation (3.7) and


the resulting characteristic equation is given in equation (3.8). Then, the first equation in the set (3.10)

gives an approximation for the roll root which can be written as follows.

λr = L′p −L′βN ′

β

(N ′

p −g

U1

)(4.7)

McRuer [11] and Stevens [10] gives such an approximation to the roll mode – the three degree of freedom

approximation.


Kolk [19] gives an approximate expression for roll root as

λr = −b32 + b0b32 + b1

(4.8)

where

b2 =B

A− E

D

[1 +

EC

D2

]b1 =

C

A− E

D

[1 +

EC

D2

]b2 (4.9)

b0 =D

A− E

D

[1 +

EC

D2

]b1

Such an expression is arrived at by using Lin’s Method [18] of approximate factorization to obtain

approximate roots of an algebraic equation.


The literal approximation for roll root as given by Livneh [20] is

λr ≈ λr + LβNp −B1λr − g cos Θ1/U1

(1−A1B1)(λ2

r − 2ζDωnDλr + ωn2D

) (4.10)

where

λr = −Lp

ωn2D = Nβ +

NrYβ −NβYr

U1(4.11)

2ζDωnD = −Nr −Yβ

U1

4.2.6 Approximate Factorization Approximation

Comparing the coefficients in equation (2.9) an approximate expression for roll mode root can be obtained

as

λr ≈ −D

C(4.12)


Simplifying the expressions for C and D in equation (2.3) by neglecting relatively small derivatives, the

roll root becomes

λr = Lp +gLβ

U1Nβ+YβLpNr

U1Nβ(4.13)

This approximation is given by Bu Aer Report [17].

4.2.7 Mengali and Giulietti’s Approximation

Yet another innovative approximation derived through the comparison of coefficients is given by Mengali

and Giulietti [22]. Assuming the spiral root to be very small, the remaining characteristic polynomial

can be written as

s3 +B

As2 +

C

As+

D

A= (s2 + 2ζDωnDs+ ωn

2D)(s− λr) (4.14)

Equating the the coefficients on both sides of the equation

B

A= −λr + 2ζDωnD (4.15)

C

A= ωn

2D − 2ζDωnDλr (4.16)

D

A= −ωn

2Dλr (4.17)

[−λr × (4.15)]− (4.16) gives

− (C/A)− (B/A)λr = (λr − ωnD)(λr + ωnD) (4.18)

Divde (4.16) by ωnD and combine (add and subtract) with (4.15) to obtain

C/A

ωnD

− (B/A) = (λr + ωnD)(1− 2ζD) (4.19)

−C/AωnD

− (B/A) = (λr − ωnD)(1 + 2ζD) (4.20)

(4.19)× (4.20) using (4.18) gives

(− (B/A)λr − (C/A))(1− 4ζ2

D

)= (B/A)2 − (C/A)2

ωn2D

(4.21)

Noting that1

ωn2D

= − λr

(D/A)and 1 � 4ζ2

D equation (4.21) becomes

− (B/A)λr − (C/A) = (B/A)2 +(C/A)2

(D/A)λr (4.22)

This gives the approximation for roll mode as

λr = −

[(B/A)2 + (C/A)

][(B/A) + (C/A)2 / (D/A)

] (4.23)


Table 4.1: Accuracy of existing roll mode approximationAircraft Flight eqn 4.3 eqn 4.4 eqn 4.6 eqn 4.7

Phase % Error % Error % Error % ErrorA 1 0.20 0.20 −8.53 −11.88

1 14.84 11.47 13.91 −50.59B 2 4.56 3.75 10.45 −29.10

3 15.36 12.55 20.59 −62.921 7.24 5.91 0.28 −84.08

C 2 1.44 1.37 16.16 −3.313 3.00 2.83 15.87 −5.461 49.61 49.34 30.23 −22.06

D 2 15.09 15.07 13.78 −1.163 11.81 11.17 8.34 −12.181 3.55 −9.91 7.16 5.85

E 2 8.04 8.10 −0.79 −2.993 −1.02 −1.58 −1.02 −0.191 13.60 15.00 −44.75 −49.68

F 2 6.19 6.13 8.84 0.563 10.24 10.30 3.40 −8.21

Using equation (2.8) and simplifying assumptions such as |YβN′r/U1| �

∣∣∣N ′β

∣∣∣, ∣∣N ′pL

′r

∣∣ � ∣∣∣N ′β

∣∣∣ and

|YβL′r/U1| �

∣∣∣L′β∣∣∣ the above equation can be reduced as

λr = −

(L′p +N ′

r +Yβ

U1

)2

+N ′β + L′p

(N ′

r +Yβ

U1

)−L′p −N ′

r −Yβ

U1+[N ′

β + L′p

(N ′

r +Yβ

U1

)]2/(N ′

βλcr

) (4.24)

where

λcr = −L′p +

L′βN ′

β

(N ′

p −g

U1

)

4.3 Analysis of Accuracies of Roll Approximations

As for the spiral approximations, the accuracy of roll approximations also should be verified for different

types of aircraft and different flight conditions to establish the extent of its generality. The database of

Roskam [5], as discussed in Chapter 3, which is representative of a wide spectrum of airplanes and flight

conditions will be used for this.

The metric for the evaluation of the accuracies of various approximation is again the percentage error in

the root computed through the approximate expression relative to the exact root.

The accuracies of various roll mode approximations as discussed above over the chosen database is given

in Table 4.3 and Table 4.2.

As seen from the tables, most of the existing roll mode approximations are inaccurate and inconsistent

except for the approximation in equation (4.24) which performs well baring one case - A1. The ap-


Table 4.2: Accuracy of existing roll mode approximation (contd.)Aircraft Flight eqn 4.8 eqn 4.10 eqn 4.13 eqn 4.24

Condition % Error % Error % Error % ErrorA 1 −1.81 −0.64 −4.99 8.58

1 −0.38 −1.93 −6.99 −0.33B 2 −3.04 −0.69 −6.20 5.08

3 −0.14 −1.79 5.95 −1.441 −1.16 −7.04 −39.45 −4.85

C 2 −1.51 0.06 −2.64 1.303 −1.00 −0.08 −2.33 0.341 −1.67 −15.88 −0.53 −0.28

D 2 −0.93 −0.50 0.99 −0.683 −0.05 −1.24 4.27 −0.601 −14.68 14.82 −44.15 −0.72

E 2 −1.36 −1.49 −2.18 −1.293 −2.43 0.18 −4.91 −1.221 −1.89 −10.57 −51.69 −1.72

F 2 −7.00 1.26 −3.88 −1.053 −5.12 −2.81 −6.10 −1.98

proximation given by equation (4.8) is good except for the power approach of the aircraft E and high

altitude cruise of aircraft F. However, this approximation is too huge an expression to receive any worthy

appreciation.

Approximation in equation (4.13) gives good accuracies except for the power approach cases. The same

is the case with approximation given by Livneh, equation (4.10), but in this case, again the expression is

large.

It has been observed that a good rolling approximation holds the key to an accurate dutch roll frequency

and damping approximations as will be seen in a subsequent chapter. Thus an accurate but simple and

consistent expression for roll mode approximation is desirable. The development of a new accurate roll

mode approximation is presented in next chapter.

Chapter 5

Development of A New Roll Mode Approximation

The existence of a simple, accurate and consistent roll approximation is necessary to arrive at a

good dutch roll frequency approximation. This chapter meets this demand by deriving such a

roll approximation.

5.1 Derivation

While deriving the three degree of freedom dutch roll approximation, Russel [12] approximately factorizes

the quartic characteristic polynomial into two quadratics as follows.[λ2 +

(−Nr −

Yβ

U1+Lβg cos Θ1/U1

L2p +Nβ

)λ+

(L2

p +Nβ +Yβ

U1Nr

)]×[

λ2 +(−Lp −

Lβg cos Θ1/U1

L2p +Nβ

)λ+

(L2

p −LβNrg cos Θ1/U1

L2p +Nβ

)](5.1)

Although Russel does not proceed to do this, an approximate expression for roll mode can be derived

from the above equation. The exact factorization of the characteristic polynomial is[λ2 + 2ζDωnD

λ+ ω2nD

][(λ− λr) (λ− λs)] (5.2)

Thus the first quadratic of equation (5.1) corresponds to the dutch roll mode and the second quadratic

corresponds to the combined spiral and roll mode. From equations (5.1) and (5.2) we get[λ2 +

(−Lp −

Lβg cos Θ1/U1

L2p +Nβ

)λ+

(L2


L2p +Nβ

)]≈[λ2 + (−λr − λs)λ+ λrλs

](5.3)

Comparing the coefficients of λ and neglecting spiral root, which is one or two orders of magnitude less

than the roll root, an approximation for roll mode root is obtained as

λr = Lp +Lβg cos Θ1/U1

L2p +Nβ

(5.4)

25

Chapter 5. Development of A New Roll Mode Approximation 26

However, this approximation does not predict the roll mode root very accurately.

5.2 Detailed Analysis of the Roll Mode

The reason for the inaccuracy of above developed approximation and all other roll mode approximations

was investigated in detail. For this purpose, a simulation package with a Graphical User Interface (GUI)

was developed in MATLAB.

Figure 5.1: A typical session with simulation/animation package developed in MATLAB with a GUI


5.2.1 The Simulation Package

A general initial condition or disturbance usually excites all the modes of a system. In the simulation

package developed, only one mode of the lateral-directional dynamics was excited at a time for detailed

scrutiny by carefully choosing the initial condition as follows. By choosing the initial condition of the

system as the eigenvector of a mode, only that particular mode can be excited. A proof for this is given

in Appendix B.

Using this the roll mode of different aircraft in various flight conditions were throughly analyzed. The

salient features of the simulation package developed is given in Appendix B. A typical session with this

simulation package is shown in Figure 5.1

5.2.2 Observations and Inference

The detailed simulation studies thus conducted led to the following important observation. The roll

mode of most of the airplanes involve almost pure roll. However, at least in some aircraft,

the contribution of yaw and sideslip to the roll mode is significantly high so as to be ignored.

This observation leads to the inference that a good roll mode approximation should respect the

participation of yaw and sideslip in the roll mode.

5.3 Modifying the New Roll Approximation

The implication of the above made inference is that the sideslip derivative Yβ and the cross-coupling

derivatives Lr and Np should find respectable positions in the expression for the roll mode approximation.

Taking this into account, the equation (5.4) can be modified (inspired by equations (4.7) and (4.13)) as

λr = L′p +L′β

(g

U1−N ′

p

)(L′2p +N ′

β)+Yβ(L′rN

′p − L′pN

′r)

(L′2p +N ′β)

(5.5)

In this approximation, the participation of yaw and side slip in the roll mode is ensured through the

presence of yaw in roll equation (Lr), roll in yaw balance (Np) and the side force term (Yβ).

5.4 Remarks on the New Roll Mode Approximation

The accuracy of above approximation over the selected database is given in Table 5.1. As seen from

the table, the new approximation is almost exact except for one case. It is to be appreciated that the

expression is really small and yet captures the whole physics of the roll mode so as to give an accurate

result for the wide database chosen. This demonstrates the simplicity and the generality of the new

approximation derived.


Table 5.1: Accuracy of the new roll mode approximationAircraft Flight eqn 5.5

Condition % ErrorA 1 −0.43

1 −0.94B 2 0.20

3 −2.431 −5.21

C 2 0.423 0.281 −10.62

D 2 0.103 −0.941 1.70

E 2 −0.323 −0.111 −1.75

F 2 2.403 0.85

Chapter 6

Dutch Roll Approximations

Dutch roll mode is the only oscillatory mode of the lateral-directional dynamics. The attempts

over years to approximately represent this oscillatory mode is consolidated and laid down in this

chapter. It is seen that although some of the dutch roll frequency approximations are reasonably

accurate, the dutch roll damping approximations in the literature are not satisfactorily accurate.

6.1 The Dutch Roll Mode

As stated above, dutch roll is the only oscillatory mode of the lateral-directional dynamics. The onset

and propagation of this periodic oscillatory mode is illustrated through a flow chart in Figure 6.1. The

periodic sinusoidal pattern seen even in the flow chart is interesting to be noted.

The aviation regulation authorities have stringent requirements on the dutch roll frequency and its damp-

ing as this directly relates to the riding quality of the airplane and thus the passenger and crew comfort.

Therefore it is essential for a designer to get an insight into what affects the frequency and damping of the

dutch roll. The dutch roll approximations do a fairly good job in providing this physical understanding.

6.2 Existing Dutch Roll Approximations

A list of dutch roll approximations which exist in the literature and their derivations wherever it is appli-

cable is given in this section. It has been found that there exists at least fourteen different approximations

for dutch roll frequency and twelve distinct approximations for dutch roll damping.

29

Chapter 6. Dutch Roll Approximations 30

Sharp gust from right side

Yaw to right

Yaw to left

Yaw to right

Yaw to left

Aircraft yaws to right

Roll to left

Roll to left

Translation to left

Roll to right

Roll to right

Translation to right

Translation to right

Aircraft translates to left

Translation to left

weather cock reaction on vertical tailincreaded drag

increaded lifton left wing

increased lift

on left wing

left wing forward

increased drag

right wing forward

tilted lift

tilted lift vector

weather cock effect

Figure 6.1: A flowchart illustration of onset and propagation of the dutch roll


6.2.1 One Degree of Freedom Approximation

The simplest of all the dutch roll approximations is given by Hancock [7] and Blakelock [21]. In this

approximation it is assumed that in a dutch roll, the yaw and the side slip are in opposite phases. This

amounts to stating that β = −ψ or β = −r. Neglecting the effect of roll in the yaw equation, the equation

(2.1) can be written as

β = −r

0 = 0

r = Nββ +Nrr (6.1)

0 = 0

Thus the simplest characteristic equation of the form s2 + 2ζDωnDs + ω2

nD= 0 representing the dutch

roll can be obtained from above set of equations as

s2 −Nrs+Nβ = 0 (6.2)

This gives the following approximations for dutch roll frequency and dutch roll damping

ω2nD

= Nβ (6.3)

2ζDωnD= −Nr (6.4)

Seckel [13] uses the primed derivative equation (2.6) instead of equation (2.1) and thus obtains the one

degree of freedom dutch roll approximation as given below.

ω2nD

= N ′β (6.5)

2ζDωnD= −N ′

r (6.6)

6.2.2 Two Degree of Freedom Approximation

For airplanes with small dihedral effect (Clβ ), the dutch roll mode mainly consists of side slipping and

yawing. Thus an approximation to dutch roll can be obtained by assuming that the sum of rolling

moments must be zero at all times and thus eliminating the roll equation and the rolling degree of

freedom. The equations of motion as given in eqn (2.1) then becomes

β =Yβ

U1β − (U1 − Yr)

U1r

0 = 0

r = Nββ +Nrr (6.7)

0 = 0

The characteristic equation which represents dutch roll as obtained from the above set of equations is

s2 −(Nr +

Yβ

U1

)s+

(YβNr

U1+Nβ −

NβYr

U1

)= 0 (6.8)


This gives the approximations for dutch roll frequency and dutch roll damping as follows.

ω2nD

=1U1

(YβNr +NβU1 −NβYr) (6.9)

2ζDωnD= −

(Nr +

Yβ

U1

)(6.10)

While Roskam [5] gives such an approximation, Babister [8] arrives at similar expressions with an addi-

tional assumption of Yr to be small. The approximation due to Babister is given below.

ω2nD

=1U1

(YβNr +NβU1) (6.11)

2ζDωnD= −

(Nr +

Yβ

U1

)(6.12)

McRuer et. al. [11], Etkin and Reid [14] and McLean [15] give the same dutch roll approximations as

above except that they use equation (2.6) instead of equation (2.1) to arrive at the following expressions

for the two degree of freedom dutch roll frequency and dutch roll damping approximations.

ω2nD

=1U1

(YβN

′r +N ′

βU1

)(6.13)

2ζDωnD= −

(N ′

r +Yβ

U1

)(6.14)

Cook [9] follows the same approach as above but retains Yr to obtain a dutch roll approximation as of

that of Roskam but the derivatives in this case are primed. The dutch roll approximations given by Cook

is

ω2nD

=1U1

(YβN

′r +N ′

βU1 −N ′βYr

)(6.15)

2ζDωnD= −

(N ′

r +Yβ

U1

)(6.16)

6.2.3 Three Degree of Freedom Approximation

McRuer et. al. [11] and McLean [15] derive a three degree of freedom approximation by neglecting

gravity terms, rolling acceleration due to yaw rate (L′rr) and yaw acceleration due to roll rate (N ′pp) in

the equation (2.6) to form following set of equations.

β =Yβ

U1β − r

p = L′ββ + L′pp

r = Nββ +Nrr (6.17)

φ = p

The characteristic equation corresponding to this set of equations is given by

s(s− L′p

) [s2 +

(−Yβ

U1−N ′

r

)s+

(N ′

β +YβN

′r

U1

)]= 0 (6.18)


In this characteristic equation, the free s corresponds to the spiral mode, (s− L′p) factor to the roll root

and the quadratic term belongs to the dutch roll. This leads to the same dutch roll approximation as

that given in equations (6.13) and (6.14). The same approach is also followed by Stevens [10] to derive

the three degree of freedom dutch roll approximation.

6.2.4 Lanchester’s Approximation

All the dutch roll approximations discussed till now considers dutch roll to be dominated by yawing

and sideslipping. Lanchester’s approximation claims and assumes the dutch roll to be a periodic motion

involving mainly roll and sideslip. Thus the yaw degree of freedom is suppressed in the equation of

motion. Such an approximation due to Lanchester is documented by Babister [8]. With the additional

assumptions that p and Yp are small, the equations of motion becomes

β =Yβ

U1β +

g

U1cos Θ1φ

0 = Lββ + Lpp

0 = 0 (6.19)

φ = p

The characteristic equation corresponding to this set of equations is

s2 − Yβ

U1s+

gLβ

U1Lpcos Θ1 = 0 (6.20)

Thus the Lanchester approximation for dutch roll is given by

ω2nD

=gLβ

U1Lpcos Θ1 (6.21)

2ζDωnD= −Yβ

U1(6.22)

6.2.5 Seckel’s Approximation

In attempting to derive a better approximation for the dutch roll, Seckel [13] considers the participation

of roll and yaw in the dutch roll but assumes that during dutch roll, the aircraft moves in a straight line

path. Thus the side force equation is omitted noting that β = −r. The equations of motion then can be

written as

β = −r


r = N ′ββ +N ′

pp+N ′rr (6.23)

φ = p

This leads to the characteristic equation for combined roll and dutch roll as

s3 −(L′p +N ′

r

)s2 +

(N ′

β + L′pN′r − L′rN

′p

)s−

(L′pN

′β −N ′

pL′β

)= 0 (6.24)


Such a characteristic equation for the combined roll and dutch roll modes is also given by Schmidt [24].

Seckel continues by assuming that the roll root is Lp and is large compared to Nr. Thus, the above

polynomial can be approximately factorized to obtain a second degree characteristic polynomial equation

representing dutch roll which is given as

s2 −

(N ′

r −N ′

pL′r

L′p+N ′

pL′β

L′p2

)s+

(N ′

β −N ′

pL′β

L′p

)(6.25)

The dutch roll approximation as given by this characteristic equation is

ω2nD

= N ′β −

N ′pL

′β

L′p(6.26)

2ζDωnD= −

(N ′

r −N ′

pL′r

L′p+N ′

pL′β

L′p2

)(6.27)

6.2.6 Russel’s Approximation

Russel [12] derives the three degree of freedom dutch roll approximation in the following manner. Ne-

glecting small and cross coupling derivatives except those due to sideslip, equation (2.1) can be written

as

β =Yβ

U1β − r +

g cos Θ1

U1φ

p = Lββ + Lpp

r = Nββ +Nrr (6.28)

φ = p

These equations lead to the characteristic equation given by

λ4 +(−Nr −

Yβ

U1− Lp

)λ3 +

(Yβ

U1Nr +Nβ + Lp

(Yβ

U1+Nr

))λ2 +(

−Lp

(Yβ

U1Nr +Nβ

)− g cos Θ1

U1Lβ

)λ+

g cos Θ1

U1LβNr = 0 (6.29)

Russel gives the approximate factorization of above characteristic equation into quadratic factors (this is

the same as equation (5.2) of Chapter 5 and was used in the derivation of a new roll mode approximation)

as [λ2 +

(−Nr −

Yβ

U1+Lβg cos Θ1/U1

L2p +Nβ

)λ+

(L2

p +Nβ +Yβ

U1Nr

)]×[

λ2 +(−Lp −

Lβg cos Θ1/U1

L2p +Nβ

)λ+

(L2


L2p +Nβ

)]≈ 0 (6.30)


In the above equation, the first quadratic corresponds to the characteristic equation for the dutch roll

mode. This gives the dutch roll approximation due to Russel as

ω2nD

=1U1

(NβU1 +NrYβ + L2

pU1

)(6.31)

2ζDωnD= −

(Nr +

Yβ

U1− g cos Θ1Lβ

U1

(L2

p +Nβ

)) (6.32)

6.2.7 Phillips’s Approximation

Phillips [25] gives the following dutch roll approximation.

ω2nD

=(

1− Yr

U1

)Nβ +

Yβ

U1Nr +RDs −

(Nr +

Yβ

U1

)2

2

≈(

1− Yr

U1

)Nβ +

Yβ

U1Nr +RDs (6.33)

2ζDωnD= −

(Nr +

Yβ

U1−RDc

+RDp

)(6.34)

where RDsis named as the dutch roll stability ratio, RDc

is called the dutch roll coupling ratio and RDp

is named as dutch roll phase divergence ratio by Phillips. The definition of these ratios are given below.

RDs=

Lβ

U1Lp[g − (U1 − Yr)Np]−

YβLrNp

U1Lp

RDc=LrNp

Lp

RDp=

g (LrNβ − LβNr)

U1Lp

(Nβ +

YβNr

U1

) − RDs

Lp


The approximate dutch roll representation as given by Livneh [20] as given below.

ω2nD≈ ω2

nD+ Lβ

B1ω2nD

+ λr(Np − g cos Θ1/U1)

(1−A1B1)(λ2


) (6.35)

2ζDωnD≈ 2ζDωnD

− LβNp −B1λr − g cos Θ1/U1

(1−A1B1)(λ2


) (6.36)

where λr, ωn2D and 2ζDωnD are defined as in equation (4.11).


6.2.9 The IITB Approximation

Ananthkrishnan and Unnikrishnan [16] of IIT Bombay obtain a dutch roll approximation which is can

be written as

ω2nD

=(Nβ +

Yβ

U1Nr

)−(Lβ

Yβ

U1Lr

)Np

Lp+

g

U1

(Lβ

Lp

)(6.37)

2ζDωnD= −

(Nr +

Yβ

U1

)(6.38)

Here the approximation given for dutch roll damping is same as that given by equation (6.10).


The dutch roll frequency and the dutch roll damping approximation given by Kolk [19] is

ω2nD

= b0

(b22 + b1b32 + b0

)(6.39)

2ζDωnD=b1 − b0

(b22 + b1b32 + b0

)(b32 + b0b22 + b1

) (6.40)

where b0, b1 and b2 are defined in equation (4.9).

6.2.11 Hancock’s Approximation

Hancock [7] derives an approximation for dutch roll frequency and damping which he claims is an extension

to the work of Thomas [26]. By approximate factorization of the quartic characteristic equation by means

of an order of magnitude analysis, he arrives at an approximation given as

ω2nD

=C

A− B

2A

(B

A− D

C

)(6.41)

2ζDωnD=

12

(B

A− D

C

)(6.42)

6.2.12 Etkin and Reid’s Approximation for Dutch Roll Damping

Etkin and Reid [14] derive an approximation for dutch roll damping using the first equation of the set

(2.10). This gives an expression for dutch roll damping as

2ζDωnD=B

A+ λr + λs (6.43)

An approximation for (λr + λs) is already available from the combined roll and spiral representation as

given in equation (3.8) and B/A is given by equation (2.8). Substituting these into the above equation


Airplane Flight eqn 6.9 eqn 6.11 eqn 6.13 eqn 6.15Condition % Error % Error % Error % Error

A 1 10.35 9.63 9.63 10.351 4.33 3.50 30.55 31.14

B 2 −4.67 −5.87 19.70 20.803 −12.12 −12.64 37.22 37.501 33.12 32.37 48.18 48.76

C 2 −14.16 −14.70 3.07 3.523 −11.58 −11.94 5.38 5.691 18.51 18.14 20.49 20.85

D 2 −0.37 −0.49 0.75 0.863 −1.13 −1.30 10.60 10.741 38.78 38.78 −9.24 −9.24

E 2 9.08 9.09 1.85 1.853 4.99 4.99 −0.20 −0.201 48.30 48.30 28.13 28.13

F 2 1.36 1.36 0.48 0.483 7.23 7.23 5.17 5.17

Table 6.1: Accuracy of dutch roll frequency approximations

results in a new approximation for dutch roll damping given as

2ζDωnD= −

[N ′

r +Yβ

U1−L′βN ′

β

(N ′

p −g

U1

)](6.44)

6.3 Evaluation of Accuracies of the Dutch Roll Approximations

In this section, the dutch roll approximations existing in the literature are evaluated for their accuracies

over the chosen database.

6.3.1 Evaluation of Dutch Roll Frequency Approximations

The accuracies of the various dutch roll frequency approximations in percentage relative error as defined

in equation (3.17) over the chosen database of airplanes and flight conditions as listed in Table 3.1 are

given in Tables 6.1–6.3.

Most of the text books state that approximation to dutch roll frequency is accurate. The depth of

confidence in these statements is evident from the inspection of the aforesaid tables listing accuracies

of various approximations. As seen from the tables, most of the approximations are good except for

power approach cases. However, the accuracies for the power approach cases are unacceptable. A couple

of good approximations worth mentioning are those given equations (6.35) and (6.39). They perform

exceptionally well except for one case each. Nevertheless, the expressions for these approximations are

so huge to make any sense out of them.



A 1 11.25 11.25 97.02 3.50 −1340.821 5.62 32.56 77.93 4.72 −232.21

B 2 −1.40 23.91 93.04 4.61 −507.303 −10.71 38.80 89.61 −4.38 −233.811 34.70 49.39 69.46 30.04 −294.46

C 2 −13.36 4.05 96.63 2.70 −286.383 −11.04 6.02 94.80 2.93 −196.421 19.31 21.62 20.94 −11.16 4.88

D 2 −0.16 1.07 83.72 −1.69 −6.903 −0.81 11.04 91.87 −4.60 −78.991 39.54 −7.99 71.45 42.24 3.77

E 2 9.50 2.28 90.37 0.49 −17.303 5.22 0.04 96.58 5.22 −5.281 52.02 32.44 67.82 16.60 −126.23

F 2 2.47 1.60 90.65 5.27 −20.433 9.69 7.65 86.00 2.72 −57.24

Table 6.2: Accuracy of dutch roll frequency approximations (contd.)


A 1 −0.43 −0.92 −1.15 1.78 385.241 −18.76 −5.52 −19.61 0.38 19.26

B 2 −5.32 0.22 −6.48 2.95 110.303 −15.66 6.25 −16.12 0.14 10.271 −2.62 1.04 −3.14 1.14 32.78

C 2 −0.36 2.74 −1.08 1.48 22.823 −1.86 2.59 −2.35 0.99 8.351 −92.15 3.84 −93.55 1.65 4.19

D 2 −18.22 −0.27 −18.36 0.92 −0.703 −13.24 0.73 −13.43 0.05 −0.021 11.92 −2.24 11.92 12.80 0.16

E 2 −9.26 −1.10 −9.29 1.34 −1.563 1.56 0.31 1.56 2.38 −0.961 −17.52 −25.38 −18.94 1.85 16.71

F 2 −5.20 −0.72 −5.10 6.54 −1.073 −13.77 −4.61 −13.95 4.87 0.20

Table 6.3: Accuracy of dutch roll frequency approximations (contd.)


Aircraft Flight eqn 6.10 eqn 6.14 eqn 6.4 eqn 6.6Condition % Error % Error % Error % Error

A 1 −2.58 −2.58 8.16 8.161 −275.02 −260.40 −194.60 −179.98

B 2 −39.59 − 28.54 −7.39 3.663 −506.98 −424.60 −373.22 −290.841 −86.60 −23.27 −33.00 30.32

C 2 −34.45 −10.40 8.23 32.293 −57.10 −25.87 −7.22 24.011 308.32 302.59 233.04 227.32

D 2 −66.80 −65.71 3.90 5.003 −521.28 −481.09 −296.80 −256.611 43.16 −14.98 57.70 29.52

E 2 −48.90 −52.53 −7.29 −10.933 5.61 3.43 33.26 31.081 −121.87 −147.30 −60.72 −86.16

F 2 −10.76 −11.72 18.37 17.403 −44.59 −45.67 −1.79 −2.86

Table 6.4: Accuracies of dutch roll damping approximations

6.3.2 Evaluation of the Dutch Roll Damping Approximations

This section evaluates the accuracy of dutch roll damping approximations. The accuracies of various

approximate dutch roll damping representations are given in Tables 6.4–6.6.

As seen from the tables, none of the approximations give even a satisfactory, if not good, representation

of the dutch roll damping. Thus the use of dutch roll damping approximations is discouraged by most of

the researchers.

6.4 Concluding Comments on Dutch Roll Approximations

This chapter showed that dutch roll frequency approximations are accurate enough except for power

approach cases whereas the dutch roll damping approximations are grossly inaccurate. An improved

approximation to the dutch roll frequency is derived in the next chapter.



A 1 89.26 7.67 −0.70 0.061 19.58 −10.71 −166.53 −54.04

B 2 67.80 19.00 −32.40 −28.533 −33.77 147.56 −422.45 −389.551 46.40 5.38 −8.63 −59.58

C 2 57.31 34.87 −26.61 −58.623 50.12 35.82 −38.70 −88.351 175.28 −517.94 25.41 −1672.20

D 2 29.29 159.43 −11.09 880.213 −124.48 442.93 −349.75 59.351 85.45 −153.65 93.74 183.06

E 2 58.39 24.31 −5.02 314.213 72.35 −109.80 14.90 101.811 38.85 −44.10 −23.72 156.16

F 2 70.87 −14.91 6.22 56.053 57.20 20.40 −14.02 72.05

Table 6.5: Accuracies of dutch roll damping approximations (contd.)


A 1 5.41 0.61 −222.58 −112.071 45.98 −1.85 −139.20 −1443.109

B 2 −5.46 −0.44 −103.06 −242.953 37.96 −1.31 −122.49 −2831.731 67.32 −2.10 −41.08 −994.42

C 2 −23.63 −3.26 −55.05 −52.883 −25.59 −4.41 −29.40 −115.191 −133.08 22.29 14.87 778.90

D 2 0.29 −43.24 45.59 −135.383 35.92 −2.23 −0.66 −1478.691 24.24 −39.21 34.02 42.63

E 2 6.75 −23.29 34.26 −116.733 2.07 −62.72 45.08 7.351 71.45 −4.28 −62.54 −655.43

F 2 0.98 −50.44 39.17 −24.583 6.78 −16.94 14.61 −115.44

Table 6.6: Accuracies of dutch roll damping approximations (contd.)

Chapter 7

A New Accurate Approximation for the Dutch Roll Frequency

The derivation of a simple, accurate and consistent approximation for the dutch roll frequency

is presented in this chapter.

7.1 Derivation of the Novel Dutch Roll Frequency Approximation

Recall that the equation set (2.10) was obtained by equating the coefficients of the characteristic equation.

The final equation of this set is

a0 =E

A= λsλrω

2nD

(7.1)

From the above equation, an expression for dutch roll frequency is obtained as

ω2nD

=(E/A)λsλr

(7.2)

Assuming that the spiral root, λs, is (−E/D), which is a good approximation to the spiral mode, an

approximate expression for dutch roll frequency can be obtained from equation (7.2) as

ω2nD

=− (D/A)

λr(7.3)

This equation shows that a good roll mode approximation is the key to a good dutch roll frequency

approximation. An excellent approximation for roll is already available in equation (5.5) which was

derived in Chapter 5. Substituting this into the above equation, the new approximation for dutch roll

frequency can be obtained as

ω2nD

=− (D/A)

L′p +L′β(

g

U1−N ′

p)

(L′2p +N ′β)

+Yβ(L′rN

′p − L′pN

′r)

(L′2p +N ′β)

(7.4)

41

Chapter 7. A New Accurate Approximation for the Dutch Roll Frequency 42

Aircraft Flight eqn 7.6Condition % Error

A 1 1.132 −0.61

B 2 1.563 2.691 11.46

C 2 0.463 0.611 13.51

D 2 −0.173 1.131 −3.42

E 2 −0.323 −0.041 0.62

F 2 −0.253 −0.79

Table 7.1: Accuracy of the new dutch roll frequency approximation

The exact expression for D/A in given equation (2.8). However, this can be simplified by neglecting

certain non-dominant terms. The simplified approximated expression for D/A is given below.

D

A≈ L′βN

′p −N ′

βL′p − L′β

g

U1(7.5)

Substituting the above expression for D/A in equation (7.4) the approximate expression for dutch roll

frequency becomes

ω2nD

=

g

U1L′β +N ′

βL′p − L′βN

′p

L′p +L′β(

g

U1−N ′

p)

(L′2p +N ′β)

+Yβ(L′rN

′p − L′pN

′r)

(L′2p +N ′β)

(7.6)

This approximation is very simple compared to all other existing dutch roll approximations.

7.2 Evaluation of the Accuracy of the New Approximation

The accuracy of the new approximation developed is given in Table 7.1. Performance of this approxima-

tion is exemplary except for two cases. In nine out of sixteen cases considered, the percentage error is

less than one. This depicts the consistency of the newly developed dutch roll frequency approximation.

Chapter 8

High Angle of Attack Flight Dynamics

High angle of attack flight dynamics has captured recent attention of the aerospace research

community owing to the requirement of high angle of attack manoeuvring capabilities for the

new generation fighter airplanes. This chapter describes the salient features of the flight

dynamics at high angles of attack, some of the research opportunities in this area and the work done

during this project on high angle of attack handling quality criteria.

8.1 Salient Characteristics at High Angles of Attack

At very high angles of attack, the aerodynamics forces and moments acting on the aircraft are entirely

different from that at low angles of attack. High angle of attack phenomena like flow separation and

vortex breaking make the flow field highly nonlinear thus making it almost impossible to model the

aerodynamics involved. The complicated nonlinear aerodynamic forces and moments leads to a totally

different flight dynamics and thus different airplane behaviour and handling qualities at high angles of

attack.

There are two important transitions that take place at high angles of attack which are of interest.

• The first is that one or more of the modes of the aircraft goes unstable which makes the airplane go

rapidly out of control. This is called departure. Thus a criteria which predict the onset of departure

become important.

• The second important transition that take place at high angles of attack is the coupling or splitting

of the modes. For example, in the lateral-directional dynamics, at high angles of attack, the spiral

and the roll modes, which are usually real modes, combine to form a single oscillatory mode. This

behaviour is important from the handling qualities point of view as the oscillatory response of the

combined spiral-roll mode takes the pilot by surprise. This points toward the necessity of good

43

Chapter 8. High Angle of Attack Flight Dynamics 44

handling quality criteria at high angles of attack which does not seem to exist as of now.

8.2 Lateral-Directional Departure

As stated earlier, a departure occurs when the airplane goes rapidly out of control.

8.2.1 Modes of Departure

In the lateral-directional dynamics the known departure manifests mainly in three ways.

Wing Rock

Wing rock which occurs at high angle of attack consists of oscillations mainly in roll. This is a nonlinear

phenomenon. Dutch roll at high angles of attack can become primarily a rolling motion. Dutch roll

damping may become negative for small amplitudes of oscillation and becomes positive for large ampli-

tudes. Then small amplitude roll oscillations are unstable which build up to a stable roll oscillation of

fixed large amplitude, thus exhibiting a limit cycle oscillation.

Nose Slice

Nose slice is an abrupt divergence in yaw. Yawing moment at high angle of attack due to the asymmetric

breaking of forebody vortices exceeds the control authority of the rudder which makes the airplane

unstable. This divergence in yaw is usually not controllable by pilot.

Wing Drop

Wing drop is the roll divergence which is usually encountered at high subsonic speeds rather than at

high angles of attack. Wing drop may possibly be a manifestation of negative stiffness of dutch roll

accompanied by a low damping. Even though the reason for this is not so clear it may be attributed

to the asymmetric stall (or the unstall) leading to asymmetric loading of the starboard and port wings

and a resulting rolling moment. Wing drop may be caused at high speeds due to shock boundary layer

interaction and resulting flow separation. At high angles of attack the associated phenomena are flow

separation and vortex breaking. These phenomena are very sensitive to Mach number and local angles

of incidence. Therefore a small yaw disturbance or a small roll rate will initiate asymmetries. The roll

rate during a typical wing drop is not so high and is controllable by a pilot.


8.2.2 Departure Criteria

The departure criteria is one which predicts the onset of departure. Although umpteen different criteria

exist for lateral-directional departure, the most commonly accepted one is

C ′nβ> 0 (8.1)

However this is found to be not so good a criterion as reported by Hancock [7]. Thus the current

literature lacks a good lateral-directional departure criteria.

8.3 Lateral-Directional Mode Coupling

It is seen that at high angles of attack the spiral and the roll modes combine to form an oscillatory mode

which is sometimes called the ‘lateral phugoid’. Such an oscillatory response is not desirable and is not

generally associated with good flying qualities. However, the aviation regulation authorities permit such

a behaviour for category B and C flight phases with a stipulation that the damping of the combined

spiral-roll mode (2ζsrωsr) should be greater than a given number. The definition of airplane class, flight

phase and level of handling qualities are given in Appendix C. The literature does not seem to have a

good criteria for handling qualities at high angles of attack.

8.4 Development of Lateral-Directional Handling Quality Criteria

The development of handling quality criteria for lateral-directional dynamics at high angles of attack is

attempted here. The combined spiral-roll dynamics can be represented as

λ2 + 2ζsrωsrλ+ ω2sr = 0 (8.2)

The handling quality requirement on this mode is

2ζsrωsr > k (8.3)

where the desired values of k for different level of handling qualities as defined in Appendix B is given by

Hodgkinson [27] and is reproduced in the table below.

Flight Phase Level Desired kCategory B & C 1 1Category B & C 2 0.6Category B & C 3 0.3

From equation(8.3) it is clear that development of a good handling quality criterion implies the need

for an accurate expression for 2ζsrωsr. This can be obtained in the following manner. The combined


spiral-roll dynamics is also given as

(λ− λs)(λ− λr) = 0

λ2 + (−λr − λs)λ+ λrλs = 0 (8.4)

From equation (8.2) and equation (8.4) comparing the coefficients of λ

−λr − λs = 2ζsrωsr (8.5)

During the derivation of a roll root in Chapter 5, it was assumed that spiral root is small in comparison to

the roll root and was neglected. From equation (8.5) it is clear that the approximate expression obtained

for roll in Chapter 5 is actually an approximation for −2ζsrωsr (= λr + λs). However, except at high

angles of attack, the spiral root is very small and thus the then obtained expression accurately represents

roll.

With a little bit of simplification of the expression obtained in Chapter 5 (equation (5.5)), an approximate

expression for damping of the combined spiral-roll mode is proposed as

2ζsrωsr ≈ −L′p +L′βN ′

β

(N ′

p −g

U1

)(8.6)

It is required to validate the accuracy of above proposed approximation. For this purpose, the data for

an aircraft with combined spiral-roll mode is needed. As the coupling of spiral and roll roots happen

at high angles of attack, the data for an airplane at some high angle of attack is required. Stevens and

Lewis [28] provide the data for F16 for angles of attack up to 45o which can be used for this validation

purpose. For completeness, this data appears as Appendix D of this report.

8.5 Generation of High Angle of Attack Data for F16

The generation of data for F16 at high angles of attack involve the calculation of dimensional stability

derivatives at those angles. The first step towards this is the estimation of non-dimensional stability

derivatives from the given data.

8.5.1 Calculation of Non-dimensional Derivatives

The p and r derivatives, Cyp, Cyr

, Clp , Clr , Cnpand Cnr

at a given angle of attack is found by direct

interpolation of the values given in the look-up table D.1 of Appendix D.

The non-dimensional derivative Cyβis evaluated as follows. The side force coefficient Cy is given as

Cy = −0.02β + 0.021δa20

+ 0.086δr30

From the above equation

Cyβ=∂Cy

∂β= −0.02(deg)−1

= −0.02(

180π

)(rad)−1 (8.7)


Figure 8.1: Clβ variation with sideslip (β) at an angle of attack of 10◦

−30 −20 −10 0 10 20 30−0.06

−0.04

−0.02

0

0.02

0.04

0.06

β

Cl β

local slope = Cl

β

The rolling moment coefficient Cl is

Cl = Cl(α, β) + Clδa(α, β)δa + Clδr

(α, β)δr

The values of Cl(α, β), Clδa(α, β) and Clδr

(α, β) are given in Tables C.2 of Appendix D in form of look-up

tables. However for a symmetric aircraft flying at steady state and zero sideslip, the aileron and rudder

deflections are zero. Thus the dihedral effect derivative (Clβ ) is given as

Clβ =∂Cl(α, β)

∂β

∣∣∣∣1

=dCl(α1, β)

dβ

For a given α1 the variation of the rolling moment coefficient (Cl) with β can be obtained by interpolation

from Table C.2 of Appendix D. For example, the Cl variation with β at α1 = 10◦ is plotted in figure 8.1.

Clβ at desired β can be obtained as the slope of the curve at that angle of sideslip as shown in figure.

For current studies, the interest is in Clβ at zero sideslip and is given by the slope at origin of Cl versus

β curve. The yawing moment for F16 is given as

Cn = Cn(α, β) + Cnδa(α, β)δa + Cnδr

(α, β)δr

where Cn(α, β), Cnδa(α, β) and Cnδr

(α, β) look-up tables are given in Table C.3 of Appendix D. The

weathercock stability derivative (Cnβ) is then obtained using the same approach as used for Clβ .


Figure 8.2: Computed lateral-directional stability derivatives for F16 for varying angles of attack

0 20 40−3

−2

−1

0

Cy β

0 20 40−0.5

0

0.5

Cl β

0 20 40−0.5

0

0.5

Cn β

0 20 40−4

−2

0

2

Cy p

0 20 40−0.5

0

0.5C

l p

0 20 400

0.1

0.2

0.3

Cn p

0 20 40−2

−1

0

1

2

Cy r

α0 20 40

−1

−0.5

0

0.5

Cl r

α0 20 40

−1.5

−1

−0.5

0

Cn r

α

This completes the process of estimation of non-dimensional derivatives. The nonterminal stability deriva-

tives computed for various angles of attack are plotted in figure 8.2

8.5.2 Calculation of Dimensional Derivatives

The dimensional derivatives can now be computed from the obtained non-dimensional derivatives using

relations (table 3.2) given in Chapter 3. However, this requires the knowledge of the steady state velocity

(U1) to calculate the steady state dynamic pressure (q1). Obtaining the steady state velocity for a given

angle of attack is a tedious and difficult task which is accomplished as follows.

• Trim the aircraft

– The equations of motion of an aircraft is of the form

x = f(x,u) (8.8)

where f(.) is a nonlinear function of the state vector x and the control input vector x. In case


of six degree of freedom equations for a rigid aircraft

x = [U V W Φ Θ Ψ P Q R]T

u = [δa δr]T

The notations used are that of Stevens and Lewis [28]. The complete set of nonlinear first order

ordinary differential equations for a six degree of freedom rigid aircraft are given in Appendix

D.

With the definition of α, β and VT as

tanα =W

U(8.9)

sinβ =V

VT(8.10)

V 2T = U2 + V 2 +W 2 (8.11)

the variables U , V and W in x can be replaced by VT , β and α. Thus the modified state

vector is

x = [VT β α Φ Θ Ψ P Q R]T

– When the aircraft is trimmed, it is in a steady state and therefore

x = 0 (8.12)

– From equation (8.8) this implies that at trim

f(x,u) = 0 (8.13)

∗ For a steady flight, some variable in x are known while others are not. For example, for a

steady level flight all the variable in the state vector x are zero except VT , α and Θ. For

level flight (zero flight path angle), Θ = α. Thus two variables α and VT remains out of

which one can be fixed while the other has to be solved for. For the purpose of evaluation

of stability derivatives at a particular angle of attack, α is fixed leaving VT to be solved.

∗ Thus the problem of trimming the aircraft involves solving of the nonlinear equation

f(x,u) = 0 for VT and the control input vector u.

∗ The problem of solving the above nonlinear equation can be formulated in an optimization

problem frame work as follows. The optimization problem is an unconstrained minimiza-

tion problem which can be stated as given below.

– Find [VT uT ]T which minimizes the cost function which is the L2 norm of f(x,u). The function

lsqnonlin of MATLAB’s optimization toolbox used for optimization. The steepest descent

optimization technique with a quadratic line search was used. The total velocity (VT ) thus

obtained through optimization is same as U1 and can be used to calculate the dimensional

derivatives.


−5 −4 −3 −2 −1 0 1−4

−3

−2

−1

0

1

2

3

4

Re(λ)

Im(λ

)

intial roll root initial spiral

inital dutch roll

combined spiral−roll

Figure 8.3: The Root Locus of lateral-directional roots - the change in roots with angle of attack


• Once the stability derivatives are known, the linearized characteristic equation can be formed the

roots of which gives the modes of the aircraft. Figure 8.3 gives the locus of lateral-directional roots

with change in angle of attack. It can be seen from the figure that as angle of attack increases the

spiral and roll root move toward each other and join to form a combined mode whose roots are

complex conjugates. The angle of attack at which spiral and roll root coupling take place, for the

aircraft under consideration, happens to be at around 20◦.

8.6 Evaluation of Accuracy of Proposed Approximation

The accuracy of the combined spiral-roll mode damping approximation as proposed in equation (8.6) is

investigated using the F16 data generated as above. It is seen that the aircraft under consideration has

a combined spiral-roll mode from an angle of attack of about 20◦ to 43◦. This gives a wide window to

test the accuracy of the proposed approximation.

Figure 8.4 gives a comparison of the exact and proposed approximate damping for the combined spiral-

roll mode of the F16. As seen from the figure, the approximate expression is accurate for angles of attack

upto 28◦. Beyond 30◦, the approximation is not so accurate. However, it predicts the trend well.

8.7 Remarks on the New Handling Quality Criteria

From equations (8.3) and (8.6) the proposed handling quality criteria is

−L′p +L′βN ′

β

(N ′

p −g

U1

)> k (8.14)

As seen in the previous section, the above expression is not so accurate. The above criteria can be at-

tempted to be given as a requirement on Clβ which may not only reduce the inaccuracy of the expression

but also will be a good contribution as the literature lacks any guidelines for a desirable dihedral effect.

In case of longitudinal dynamics, a general guideline or thumb-rule states that the static margin should

be around 10%. This means that if the center of gravity of an airplane is about 0.1 times mean aerody-

namic center ahead of the neutral point, the airplane is expected to have desirable longitudinal stability

characteristics. Similarly, for good directional stability, it is said that Cnβshould be greater than 0.0001

per degree. Such a guideline is absent as far as lateral dynamics is concerned. Therefore it will be a

substantial contribution to existing literature if a criterion for desirable dihedral effect can be provided.

In an attempt to do so, the equation (8.14) can be rewritten as a condition on Lβ as

L′β >

(k + L′p

)(N ′

p −g

U1

)N ′β (8.15)

This equation can be nondimesionalized to obtain a condition on Clβ . However, no seemingly meaningful

expression could be arrived at during the course of this project and thus this still remains an open

problem.


Figure 8.4: Comparison of exact damping of combined spiral-roll mode and the damping given by proposedapproximation

20 25 30 35 40 45

−0.2

0

0.2

0.4

0.6

α

2 ζ sr

ωsr

ExactProposed Approx.

Chapter 9

Conclusions

It is said that ‘an artist never really finishes his work, he merely abandons it’. This is because

he can always add to, modify and go on beautifying his work for ever in an attempt to attain

perfection. However, perfection can never be attained but only be approached. Therefore at

some point of time he has to trade-off perfection for time and he gives up a work for another. The same

is the case with scientific research too. Modifications and fine tunings are always possible. However, all

these are time bound and there is a limitation to one’s progress. This chapter concludes this report by

briefly recapitulating the project and provides a few recommendations for future work.

9.1 Highlights and Conclusions

This report gave an exhaustive review of the existing lateral-directional approximations, namely the

spiral, the roll and the dutch roll approximations. The approximations existing in literature were almost

exhaustively listed and their accuracies were evaluated. It was shown that good approximations exist for

spiral mode. However, no accurate yet simple approximation is available for roll mode. To gain a physical

insight in to the nature of roll mode, a simulation package with graphical user interface was developed

in MATLAB. Through extensive simulation studies (using the simulation package developed) undertaken

during the course of this project, the conventional notion that roll mode consist purely of rolling motion

was shown to be incorrect. The simulations revealed that the roll mode of at least some aircraft involved

apart from roll, yaw and sideslip which were significant enough to be ignored. Based on these observations

an inference was drawn which states that a good roll mode approximation should respect the participation

of yaw and sideslip. Taking this in to account, an accurate yet simple approximation for roll mode was

derived. Using this approximation an excellent approximation for dutch roll frequency was later arrived

at. The new approximations derived are probably the best in the literature. It was also attempted to

get an expression for handling quality criteria at high angles of attack. However, this work could not be

completed due to lack of time.

53

Chapter 9. Conclusions 54

9.2 Future Work

No worthy approximation for dutch roll damping exist in the literature. This was shown in Chapter 6.

Therefore, the development of an accurate dutch roll damping approximation will be a great contribution.

Similarly, literature lacks the presence of a good lateral-directional departure criteria. In the high angle

of attack regime, derivation of a good handling quality criteria, which was started but could not be

completed during this project, will be an excellent piece of research to be taken up.

Appendix A

The Airplane Database

The airplane data given by Roskam [5] which is required for the calculation of lateral-directional mode

roots are presented here. The database consists of the following aircraft.

• Airplane A: representative of Cessna 172. Cessna 172 known as Skyhawk is a small four seat

light aircraft with single piston engine. It is a general aviation airplane. This aircraft is one of the

favourites when it comes to flight training and because of its high wing configuration it is widely

used for sight seeing or tour flights.

• Airplane B: representative of Beechcraft 99. It is a fifteen seater airplane from Beech Aircraft

Corporation which was taken over by Raytheon Aircraft Company in early nineties. Beechcraft 99

is a regional commuter airplane powered by twin turboprop engines.

• Airplane C: representative of Marchetti S211. Supplied by Aermacchi, this Italian two seater

aircraft is military jet pilot trainer. This small aircraft uses single turbofan engine.

• Airplane D: representative of Learjet 24. Think of ‘private jets’ and Learjet is the first name that

comes to one’s mind. Learjet 24 is the successor of Learjet 23 (which was not so pilot friendly)

with improved low speed behaviour and both were huge commercial successes. The distinguishable

characteristics of this business jet with twin turbojet engine are the fuel tanks at wing tip and the

‘T-tail’.

• Airplane E: representative of F4C Phantom II. This is a fighter cum attack airplane with twin jet

engine. An aircraft with supersonic flying capabilities, F4C was extensively used in Vietnam war.

• Airplane F: representative of Boeing 747. Boeing 747 commonly known as ‘jumbo jet’ literally

reduced the distance between places. This commercial transport airplane is the first wide body jet

and is propelled by four jet engines.

55

Appendix A. The Airplane Database 56

A.1 Cessna 172 (Airplane A) @ low altitude cruise

Cessna 172

Airplane A is representative of Cessna 172, a small, single piston engine general aviation airplane. The

data for airplane A is given in table below.

Flight Condition (1) CruiseAltitude (ft) 5000Air Density (slugs/ft3) 0.00205Speed (ft/sec) 219Initial Attitude (Θ1 in rad) 0Geometry and InertiasWing Area (ft2) 174Wing Span (ft) 35.8Weight (lbs) 2645Ixxb

(slug ft2) 948Izzb

(slug ft2) 1967Ixzb

(slug ft2) 0Lateral-Directional DerivativesClβ −0.089Clp −0.47Clr 0.096Cnβ

0.065Cnp

−0.030Cnr

−0.099Cyβ

−0.310Cyp −0.037Cyr

0.210


A.2 Beechcraft 99 (Airplane B) @ power approach, low and high altitude cruise

Beech M99

The data for airplane B is given in table below. Airplane B is representative of Beech M99 aircraft which

is a small twin turboprop regional commuter airplane.

Flight Condition (1) Power (2) Cruise (3) CruiseApproach (low) (high)

Altitude (ft) Sea level 5000 20000Air Density (slugs/ft3) 0.002378 0.00205 0.001268Speed (ft/sec) 170 360 450Initial Attitude (Θ1 in rad) 0 0 0Geometry and InertiasWing Area (ft2) 280 280 280Wing Span (ft) 46 46 46Weight (lbs) 11000 7000 11000Ixxb

(slug ft2) 15189 10085 15189Izzb

(slug ft2) 34141 23046 34141Ixzb

(slug ft2) 4371 1600 4371Lateral-Directional DerivativesClβ −0.130 −0.130 −0.130Clp −0.500 −0.500 −0.500Clr 0.060 0.140 0.140Cnβ

0.120 0.080 0.080Cnp

−0.005 0.019 0.019Cnr −0.204 −0.197 −0.197Cyβ

−0.590 −0.590 −0.590Cyp

−0.210 −0.19 −0.19Cyr

0.390 0.390 0.390


A.3 Marchetti S211 (Airplane C) @ approach, normal and high altitude cruise

Marchetti S211

SIAI Marchetti S211 is a small single engine military training airplane. Airplane C is a representative of

this aircraft and the data for power approach, normal and high altitude cruise of this airplane are given

in the table below.

Flight Condition (1) Power (2) Cruise (3) CruiseApproach (normal) (high)

Altitude (ft) Sea level 25000 35000Air Density (slugs/ft3) 0.002378 0.001066 0.000739Speed (ft/sec) 124 610 584Initial Attitude (Θ1 in rad) 0 0 0Geometry and InertiasWing Area (ft2) 136 136 136Wing Span (ft) 26.3 26.3 26.3Weight (lbs) 3500 4000 4000Ixxb

(slug ft2) 750 800 800Izzb

(slug ft2) 5000 5200 5200Ixzb


0.160 0.170 0.170Cnp −0.030 0.090 0.080Cnr

−0.310 −0.260 −0.260Cyβ

−0.940 −1.000 −1.000Cyp

−0.010 −0.140 −0.120Cyr

0.590 0.610 0.620


A.4 Learjet 24 (Airplane D) @ approach, maximum and low weight cruise

Learjet M24

Airplane D represents Gates Learjet M24. The data for airplane D in power approach, maximum and

low weight cruise is given in table below. Learjet M24 is a twin jet engine corporate airplane.

Flight Condition (1) Power (2) Cruise (3) CruiseApproach (max. wht.) (low wht.)

Altitude (ft) Sea level 40000 40000Air Density (slugs/ft3) 0.002378 0.000588 0.000588Speed (ft/sec) 170 677 677Initial Attitude (Θ1 in deg) 1.8 2.7 1.5Geometry and InertiasWing Area (ft2) 230 230 230Wing Span (ft) 34 34 34Weight (lbs) 13000 13000 9000Ixxb

(slug ft2) 28000 28000 6000Izzb

(slug ft2) 47000 47000 25000Ixzb


0.150 0.127 0.124Cnp

−0.130 −0.008 −0.022Cnr

−0.260 −0.200 −0.200Cyβ

−0.730 −0.730 −0.730Cyp

0.000 0.000 0.000Cyr 0.400 0.400 0.400


A.5 F4C (Airplane E) @ power approach, subsonic and supersonic cruise

F4C

McDonnell Douglas F4C is a twin jet engine fighter/attack airplane. Airplane E of Roskam’s database is

a representative of F4C. The data for airplane E for three flight conditions is given in the table below.

Flight Condition (1) Power (2) Subsonic (3) SupersonicApproach cruise cruise

Altitude (ft) Sea level 35000 55000Air Density (slugs/ft3) 0.002378 0.000739 0.000287Speed (ft/sec) 230 876 1742Initial Attitude (Θ1 in deg) 11.7 2.6 3.3Geometry and InertiasWing Area (ft2) 530 530 530Wing Span (ft) 38.7 38.7 38.7Weight (lbs) 33200 39000 39000Ixxb

(slug ft2) 23700 25000 25000Izzb

(slug ft2) 133700 139800 139800Ixzb


0.199 0.125 0.090Cnp

0.013 −0.036 0.000Cnr

−0.320 −0.270 −0.260Cyβ

−0.655 −0.680 −0.700Cyp 0.000 0.000 0.000Cyr

0.000 0.000 0.000


A.6 Boeing 747 (Airplane F) @ power approach, low and high altitude cruise

Boeing 747

Airplane F represents Boeing 747. The data for this wide body, four jet engine commercial transport

airplane for flight conditions - power approach, high and low altitude cruise - are given in the table below.

Flight Condition (1) Power (2) Cruise (3) CruiseApproach (high) (low)

Altitude (ft) Sea level 40000 20000Air Density (slugs/ft3) 0.002378 0.000588 0.001268Speed (ft/sec) 221 871 673Initial Attitude (Θ1 in deg) 8.5 2.4 2.5Geometry and InertiasWing Area (ft2) 5500 5500 5500Wing Span (ft) 196 196 196Weight (lbs) 564000 636636 636636Ixxb

(slug ft2) 13.7× 106 18.2× 106 18.2× 106

Izzb(slug ft2) 43.1× 106 43.1× 106 43.1× 106

Ixzb(slug ft2) 0.83× 106 0.97× 106 0.97× 106

Lateral-Directional DerivativesClβ −0.281 −0.095 −0.160Clp −0.502 −0.320 −0.340Clr 0.195 0.200 0.130Cnβ

0.184 0.210 0.160Cnp

−0.222 0.020 −0.026Cnr

−0.360 −0.330 −0.280Cyβ

−1.080 −0.900 −0.900Cyp 0.000 0.000 0.000Cyr

0.000 0.000 0.000

Appendix B

The Simulation Package

B.1 Salient Features of the Simulation Package Developed

The simulation/animation package developed in MATLAB to study the lateral-directional modes of an

aircraft highly exploits the graphical user interface and animation capabilities of MATLAB.

The aircraft data is input to the program through a GUI window as shown in figure B.1. The data can

be either typed in or loaded from a file.

The core of the software is the SIMULINK model of aircraft as shown in figure B.2. The linearized

aircraft model is formed from the input data. The SIMULINK model will simulate the time evolution of

the states for a given initial condition. The initial condition is chosen as the eigenvector of a mode so as

to excite that mode alone.

A control panel as shown in figure B.3 is used to select the simulation of desired mode. As seen in the

figure the desired mode (spiral, roll or dutch roll mode) can be selected and visualized in top, side, front

and isometric views. The control panel can also be used to load new aircraft data.

For aesthetic purpose of visualization and animation, three models of aircrafts were developed entirely in

MATLAB. Model in figure B.4 is the representative of a general aircraft. Figure B.5 shows representation

of a turboprop. A fighter aircraft (F16) model used is shown in figure B.6

B.2 Choice of Eigenvector as Initial Condition

To excite only a particular lateral-directional mode of the aircraft, the simulation package developed uses

the eigenvector of that mode as initial condition and computes the time evolution of the system. Such a

choice of initial condition is justified as follows.

62

Appendix B. The Simulation Package 63

Claim . The desired mode alone of a system can be excited by choosing the initial state vector as the

eigenvector of that particular mode

Proof

Consider a system with following dynamicsx = Ax

where x is an (n × 1) vector and A an (n × n) matrix.The solution of this differential equation is

x(t) = eAtx(0) (B.1)

where x(0) is the initial state vector.Now define the right eigenvector r of the system matrix A as

Ar = λr

where λ is an eigenvalue of A.Similarly, the left eigenvector l of A is defined as

lT A = λlT

R is a matrix with columns as the right eigenvectors of A

R = [r1 r2 · · · rn]

L is defined as a matrix with rows as left eigenvectors of A

L =

26664lT1lT2...lTn

37775It can be shown that

RΛL = A

where Λ is a diagonal matrix with eigenvalues of A as its entries.Using this information, the equation (B.1) can be rewritten as

x(t) = ReΛtLx(0)

From the definition of R, L and Λ, the above equation becomes

x(t) =

nXi=1

eλitrilTi x(0) (B.2)

Now invoke the property of eigenvector matrices which is

RL = LR = I

where I is the identity matrix.From the above equations it can be inferred that

lTi rj =

(1 if i = j,

0 otherwise.

Thus in equation (B.2) if the initial state vector x(0) is chosen as rj (eigenvector of jth mode) then the using theresult as in above equation, the equation (B.2) becomes

x(t) = eλjtrj

Thus it is seen that by choosing initial condition as the eigenvector of the jth mode, only the jth mode is excited

with system response involving eigenvalue (λj) and eigenvector (rj). Thus the claim is proved.


Figure B.1: The GUI window to input data to simulation program


Figure B.2: The SIMULINK model of the aircraft


Figure B.3: Control panel GUI for the simulation package

Figure B.4: Model of an aircraft used for animation


Figure B.5: A turboprop airplane model developed in MATLAB

Figure B.6: An F16 model used for visualization of aircraft motions

Appendix C

Flying qualities and Airworthiness Criteria

The purpose of this appendix is to provide the definition of airplane classes, flight phases and levels flying

quality. It also presents some insight into military flying qualities and airworthiness criteria related to

airplane stability and control.

The advantage of the Military Specification - Flying Qualities of Piloted Airplanes, MIL-F-8785B [29] is

that it is analytical in its set-up. That is to say, the specifications are given in a numerical manner such

that it gives the designer an ‘analytical’ method by which to design toward achieving desired dynamic

handling characteristics.

The civilian Federal Airworthiness Regulations (FAR 25) on the other hand provide only sporadic guid-

ance in the area of dynamics.

Both these provide a reasonable amount of numerical design guidance in the area of static stability and

control.

C.1 Definition of Airplane Class, Flight Phase and Level of Handling Qualities

In specifying handling quality criteria it is necessary to recognize differences in types of airplanes, in types

of flying maneuvers to be expected during some phase of flight and in failure states of airplane systems.

Those differences are recognized extensively in the criteria of MIL-F-8785 B and a basic understanding

of them is essential. For that reason the following definitions are given.

68

Appendix C. Flying qualities and Airworthiness Criteria 69

C.1.1 Definition of Airplane Classes

Class I

Small, light airplanes such as:

1. light utility

2. primary trainer

3. light observation

Class II

Medium weight, low-to-medium maneuverability airplanes such as:

1. heavy utility/search and rescue

2. light or medium transport/cargo/tanker

3. early warning/electronic counter measures/ airborne command, control or communications relay

4. anti-submarine

5. assault transport

6. reconnaissance

7. tactical bomber

8. heavy attack

9. trainer for Class II

Class III

Large, heavy, low-to-medium maneuverability airplanes, such as:

1. heavy transport/cargo/tanker

2. heavy bomber

3. heavy patrol/early warning/electronic counter measures/ airborne command, control or communi-

cations relay

4. trainer for Class III


Class IV

High-maneuverabilitt airplanes, such as:

1. fighter-interceptor

2. attack

3. tactical reconnaissance

4. observation

5. trainer for Class IV

C.1.2 Definition of Flight Phases

Category A

Nonterminal flight phases generally require rapid maneuvering, precision tracking or precise flight path

control. Typical Category A flight phases are:

1. Air-to-air combat (CO)

2. Ground attack (GA)

3. Weapon delivery/launch (WD)

4. Aerial recovery (AR)

5. Reconnaissance (RC)

6. In-flight refuelling (receiver) (RR)

7. Terrain following (TF)

8. Anti-submarine search (AS)

9. Close formation flying (FF)

Category B

Nonterminal flight phases are normally accomplished using gradual maneuvers without precision tracking

although accurate flight-path control may be required. Typical Category B phases are:

1. Climb (CL)

2. Cruise (CR)


3. Loiter (LO)

4. In-flight refuelling (tanker) (RT)

5. Descent (D)

6. Emergency descent (ED)

7. Emergency deceleration (DE)

8. Aerial delivery (AD)

Category C

Terminal flight phases are normally accomplished using gradual maneuvers and usally require accurate

flight-path control. Typical Category C flight phases are :

1. Take-off (T.O.)

2. Catapult take-off (C.T.)

3. Approach (P.A.)

4. Wave-off or go-around (WO)

5. Landing (L)

C.1.3 Definition of Levels of Flying Qualities

• Level 1: Flying qualities clearly adequate for the mission flight phase.

• Level 2: Flying qualities adequate to accomplish the mission flight phase but some increase in

pilot workload or degradation in mission effectiveness exists.

• Level 3: Flying qualities such that the airplane can be controlled safely, but pilot workload is

excessive or mission effectiveness is inadequate or both. Category A flight phases can be terminated

safely and Category B and C flight phases can be completed.

The required levels of flying qualities are tied into the probability with which certain system failures can

occur. For example, it is desired to have:

• at least Level 1 for airplane normal (no failure) state

• at least Level 2 after failures that occur less than one per 100 flights.

• at least Level 3 after failures that occur less than once per 10,000 flights.

Flying quality levels below Level 3 are not allowed except in special circumstances.


C.2 Control Forces Required of the Pilot

Maximum Control Forces Allowed per FAR 23 and FAR 25

Values in pounds of force asapplied to the control wheel pitch roll yawor the rudder pedalsFor temporary applicationStick 60 30 —Wheel (applied to Rim) 75 60 —Rudder Pedal — — 150For prolonged application 10 5 20

C.3 Requirements for Dynamic Longitudinal Stability

MIL-F-8785B specifies the following requirements for dynamic longitudinal stability.

Phugoid

level 1 ζp ≥ .04level 2 ζp ≥ 0level 3 T2p ≥ 55 sec

Short Period

The minimum acceptable short period damping ratios are specified in Table C.1.

Table C.1: Short-Period Damping Ratio LimitsCategory A and C FlightPhases Category B Flight PhasesζSP ζSP ζSP ζSP

Level Minimum Maximum Minimum Maximum1 0.35 1.30 0.30 2.002 0.25 2.00 0.20 2.003 0.15* 0.15*

* May be reduced at altitudes above 20,000 feet if approved by the customer.

C.4 Requirements for Dynamic Lateral-Directional Stability and Roll Response

C.4.1 Requirements for Dynamic Lateral-Directional Stability

MIL-F-8785B specifies the following dynamic stability requirements.


Spiral Stability

The combined effect of spiral stability, flight control-system characteristics, and trim change with speed

shall be such that following a disturbance in bank of up to 20 degrees, the time for the bank angle to

double will be greater than the values in Table C.2. This requirement shall be met with the airplane

trimmed for wings-level, zero-yaw-rate flight with the cockpit controls free.

Table C.2: Spiral Stability - Minimum Time to Double Amplitude, T2S

FlightPhase

Class Category Level 1 Level 2 Level 3I &IV A 12 sec 12 sec 4 sec

B&C 20 sec 12 sec 4 secI & IV All 20 sec 12 sec 4 sec

Dutch Roll Stability

Minimum Dutch roll frequency and damping characteristics are specified in Table C.3.

Table C.3: Minimum Dutch Roll Frequency And DampingLevel Flight Class MinζD* MinζDωnD

* Min ωnD

PhaseCategory rad/sec rad/sec

A I, IV 0.19 0.35 1.0II, III 0.19 0.35 0.4**

1 B All 0.08 0.15 0.4**I, II-C, 0.08 0.15 1.0

C IVII-L, III 0.08 0.15 0.4**

2 All All 0.02 0.05 0.4**3 All All 0.02 —- 0.04**

* The governing damping requirement is that yielding the larger value of ζD.

** Class III airplanes may be excepted from the minimum ωnDrequirements, subject to approval of the

customer and provided that certain lateral directional response requirements stated in MIL-F-8785B are

met.

C.4.2 Requirements for Roll Response

MIL-F-8785B specifies the following roll response requirements.


Roll Mode

The roll-mode time constant, TR, shall be no greater than the appropriate value in Table C.4.

Table C.4: Maximum Roll-Mode Time Constant (Seconds)Flight Phase

Class Category Level 1 Level 2 Level 3I, IV A 1.0 1.4II, III 1.4 3.0All B 1.4 3.0 10

I, II-C,* IV C 1.0 1.4II-L*, III 1.4 3.0

* C = Carrier Based

* L = Land Based

Roll Performance Response

Bank angle response to lateral control is a very important handling quality parameter. Roll perfor-

mance can usually be estimated from the single-degree-of -freedom model.Roll performance is specified

as minimum bank angle reached in some time after initiation of lateral cockpit control motion. Table C.5

contains the specifications by airplane class, flight phase and handling quality level.

Additional or alternate roll performance requirements are specified for Class IV airplanes as in Tables

C.6 and C.7. These requirements take precedence over Table C.5.

Table C.5: Roll Performance RequirementsFlight

Class Phase Level 1 Level 2 Level 3Category

A φt = 60o in 1.3 sec φt = 60o in 1.7 sec φt = 60o in 2.6 secI B φt = 60o in 1.7 sec φt = 60o in 2.5 sec φt = 60o in 3.4 sec

C* φt = 30o in 1.3 sec φt = 30o in 1.8 sec φt = 30o in 2.6 secII A φt = 45o in 1.4 sec φt = 45o in 1.9 sec φt = 45o in 2.8 secII B φt = 45o in 1.9 sec φt = 45o in 2.8 sec φt = 45o in 3.8 sec

II-L0 C* φt = 30o in 1.8 sec φt = 30o in 2.5 sec φt = 30o in 3.6 secII-C C* φt = 25o in 1.0 sec φt = 25o in 1.5 sec φt = 25o in 2.0 sec

A φt = 30o in 1.5 sec φt = 30o in 2.0 sec φt = 30o in 3.0 secIII B φt = 30o in 2.0 sec φt = 30o in 3.0 sec φt = 30o in 4.0 sec

C* φt = 30o in 2.5 sec φt = 30o in 3.2 sec φt = 30o in 4.0 secA*** φt = 90o in 1.3 sec φt = 90o in 1.7 sec φt = 90o in 2.6 sec

IV B φt = 90o in 1.3 sec φt = 90o in 1.7 sec φt = 90o in 2.6 secC* φt = 90o in 1.3 sec φt = 90o in 1.7 sec φt = 90o in 2.6 sec

* For takeoff, the required bank angle can be reduced proportional to the ratio of the maximum rolling


moment of inertia for the maximum authorized landing weight to the rolling moment of inertia at takeoff,

but the Level 1 requirement shall not be reduced below the listed value for Level 3.

*** At altitudes below 20,000 feet at the high speed boundary of the Service Flight Envelope Level 3

requirements may be substituted for the Level 2 requirements with all systems functioning normally.

*** Except as specified for combat in Tables C.6 and C.7.

Air-to-air combat. For Class IV airplanes in Flight Phase CO, the roll performance requirements are:

Table C.6: Roll Performance for Air-to-Air CombatTime to roll through

90degrees 360 degreesa. Level 1 1.0 seconds 2.8 secondsb. Level 2 1.3 seconds 3.3 secondsc. Level 3 1.7 seconds 4.4 seconds

Ground attack with external stores. The roll performance requirements for Class IV airplanes in Flight

Phase GA with large complements of external stores may be relaxed from those specified in Table C.7,

subject to approval by the procuring activity. For any external loading specified in the contract however,

the roll performance shall be not less than:

Table C.7: Roll Performance for Ground Attack with External Storesa. Level 1 90 degrees in 1.7 secondsb. Level 2 90 degrees in 2.6 secondsc. Level 3 90 degrees in 3.4 seconds

Appendix D

Data for F16

This appendix gives the data for F16 given by Stevens and Lewis [28]. Weight and moments of inertia,

wing geometry and aerodynamic foces and moments as a function of angle of attack and sideslip are the

data provided.

F16

D.1 Geometry and Inertias

Mass and Interias

Weight (lbs) : W = 20500

Moments of Inertia (slug-ft2) : Ixx= 9496

Iyy= 55814

Izz = 63100

Ixz= 982

Wing Dimensions

76

Appendix D. Data for F16 77

Span = 30 ft

Area = 300 ft2

mac = 11.32 ft

Reference CG location

Xcg = 0.35 c

D.2 Nondimensional Force and Moment Coefficients

Aerodynamic Forces and Moments

The coefficients of aerodynamic forces and moments and the stability derivatives are given as a function

of angle of attack (α) and the sideslip (β). The data for angles of attack from −10◦ to 45◦ and sideslip

form 0◦ to 30◦ are provided.

The damping derivatives for F16 is given in the table D.1.

Table D.1: Nondimensional damping derivatives with angle of attack (α)α −10 −5 0 5 10 15 20 25 30 35 40 45Cxq −0.267 −0.11 0.308 10.34 20.08 20.91 20.76 20.05 10.5 10.49 10.83 10.21Cyp −0.108 −0.108 −0.188 0.11 0.258 0.226 0.344 0.362 0.611 0.529 0.298 −2.27Cyr 0.882 0.852 0.876 0.958 0.962 0.974 0.819 0.483 0.59 1.21 −0.493 −1.04Czq −28.8 −25.8 −28.9 −31.4 −31.2 −30.7 −27.7 −28.2 −29 −29.8 −38.3 −35.3Clp

−0.36 −0.359 −0.443 −0.42 −0.383 −0.375 −0.329 −0.294 −0.23 −0.21 −0.12 −0.1

Clr−0.126 −0.026 0.063 0.113 0.208 0.23 0.319 0.437 0.68 0.1 0.447 −0.33

Cmq −7.21 −5.40 −5.23 −5.26 −6.11 −6.64 −5.69 −6 −6.2 −6.4 −6.6 −6.0Cnp 0.061 0.052 0.052 -0.012 -0.013 -0.024 0.05 0.15 0.13 0.158 0.24 0.15Cnr −0.38 −0.363 −0.378 −0.386 −0.37 −0.453 −0.55 −0.582 −0.595 −0.637 −1.02 −0.84

The rolling moment coefficients of F16 are given in table D.2.

Table D.2: Rolling moment coefficients for F16Rolling moment coefficient as a function of angle of attack and sideslip

β α =−10 −5 0 5 10 15 20 25 30 35 40 450 0 0 0 0 0 0 0 0 0 0 0 05 −0.001 −0.004 −0.008 −0.012 −0.016 −0.022 −0.022 −0.021 −0.015 −0.008 −0.013 −0.015

10 −0.003 −0.009 −0.017 −0.024 −0.03 −0.041 −0.045 −0.04 −0.016 −0.002 −0.01 −0.01915 −0.001 −0.01 −0.02 −0.03 −0.039 −0.054 −0.057 −0.054 −0.023 −0.006 −0.014 −0.02720 0 −0.01 −0.022 −0.034 −0.047 −0.06 −0.069 −0.067 −0.033 −0.036 −0.035 −0.03525 0.007 −0.01 −0.023 −0.034 −0.049 −0.063 −0.081 −0.079 −0.06 −0.058 −0.062 −0.05930 0.009 −0.011 −0.023 −0.037 −0.05 −0.068 −0.089 −0.088 −0.091 −0.076 −0.077 −0.076

Clδa- change of rolling moment with aileron deflection

β α = −10 −5 0 5 10 15 20 25 30 35 40 45-30 -.041 -.052 -.053 -.056 -.05 -.056 -.082 -.059 -.042 -.038 -.027 -.017-20 -.041 -.053 -.053 -.053 -.05 -.051 -.066 -.043 -.038 -.027 -.023 -.016-10 -.042 -.053 -.052 -.051 -.049 -.049 -.043 -.035 -.026 -.016 -.018 -.014

0 -.04 -.052 -.051 -.052 -.048 -.048 -.042 -.037 -.031 -.026 -.017 -.01210 -.043 -.049 -.048 -.049 -.043 -.042 -.042 -.036 -.025 -.021 -.016 -.01120 -.044 -.048 -.048 -.047 -.042 -.041 -.02 -.028 -.013 -.014 -.011 -.0130 -.043 -.049 -.047 -.045 -.042 -.037 -.003 -.013 -.01 -.003 -.007 -.008

Clδr- change of rolling moment with rudder deflection

β α = −10 −5 0 5 10 15 20 25 30 35 40 45−30 0.005 0.017 0.014 0.01 −0.005 0.009 0.019 0.005 0 −0.005 0.011 0.008−20 0.007 0.016 0.014 0.014 0.013 0.009 0.012 0.005 0 0.004 0.009 0.007−10 0.013 0.013 0.011 0.012 0.011 0.009 0.008 0.005 −0.002 0.005 0.003 0.005

0 0.018 0.015 0.015 0.014 0.014 0.014 0.014 0.015 0.013 0.011 0.006 0.00110 0.015 0.014 0.013 0.013 0.012 0.011 0.011 0.01 0.008 0.008 0.007 0.00320 0.021 0.011 0.01 0.011 0.01 0.009 0.008 0.01 0.006 0.005 0 0.00130 0.023 0.01 0.011 0.011 0.011 0.01 0.008 0.01 0.006 0.014 0.02 0

The data required to calculate the yawing moment coefficient of F16 for various angles of attack and

sideslip are given in the table D.3.


Table D.3: Yawing moment coefficients for F16Yawing moment coefficient as a function of angle of attack and sideslip

β α = −10 −5 0 5 10 15 20 25 30 35 40 450 0 0 0 0 0 0 0 0 0 0 0 05 0.018 0.019 0.018 0.019 0.019 0.018 0.013 0.007 0.004 −0.014 −0.017 −0.03310 0.038 0.042 0.042 0.042 0.043 0.039 0.03 0.017 0.004 −0.035 −0.047 −0.05715 0.056 0.057 0.059 0.058 0.058 0.053 0.032 0.012 0.002 −0.046 −0.071 −0.07325 0.074 0.086 0.093 0.089 0.08 0.062 0.049 0.022 0.028 −0.012 −0.002 −0.01330 0.079 0.09 0.106 0.106 0.096 0.08 0.068 0.03 0.064 0.015 0.011 −0.001

Cnδa- yawing moment variation with aileron

β α = −10 −5 0 5 10 15 20 25 30 35 40 45−30 0.001 −0.027 −0.017 −0.013 −0.012 −0.016 0.001 0.017 0.011 0.017 0.008 0.016−20 0.002 −0.014 −0.016 −0.016 −0.014 −0.019 −0.021 0.002 0.012 0.016 0.015 0.011−10 −0.006 −0.008 −0.006 −0.006 −0.005 −0.008 −0.005 0.007 0.004 0.007 0.006 0.0060 −0.011 −0.011 −0.01 −0.009 −0.008 −0.006 0 0.004 0.007 0.01 0.004 0.0110 −0.015 −0.015 −0.014 −0.012 −0.011 −0.008 −0.002 0.002 0.006 0.012 0.011 0.01120 −0.024 −0.01 −0.004 −0.002 −0.001 0.003 0.014 0.006 −0.001 0.004 0.004 0.00630 −0.022 0.002 −0.003 −0.005 −0.003 −0.001 −0.009 −0.009 −0.001 0.003 −0.002 0.001

Cnδr- yawing moment variation with rudder

β α = −10 −5 0 5 10 15 20 25 30 35 40 45−30 −0.018 −0.052 −0.052 −0.052 −0.054 −0.049 −0.059 −0.051 −0.03 −0.037 −0.026 −0.013−20 −0.028 −0.051 −0.043 −0.046 −0.045 −0.049 −0.057 −0.052 −0.03 −0.033 −0.03 −0.008−10 −0.037 −0.041 −0.038 −0.04 −0.04 −0.038 −0.037 −0.03 −0.027 −0.024 −0.019 −0.0130 −0.048 −0.045 −0.045 −0.045 −0.044 −0.045 −0.047 −0.048 −0.049 −0.045 −0.033 −0.01610 −0.043 −0.044 −0.041 −0.041 −0.04 −0.038 −0.034 −0.035 −0.035 −0.029 −0.022 −0.00920 −0.052 −0.034 −0.036 −0.036 −0.035 −0.028 −0.024 −0.023 −0.02 −0.016 −0.01 −0.01430 −0.062 −0.034 −0.027 −0.028 −0.027 −0.027 −0.023 −0.023 −0.019 −0.009 −0.025 −0.01

The variation of the longitudinal derivatives Cx and Cm with angle of attack and elevator deflection are

given in table D.4.

Table D.4: Cx and Cm variation with α and elevator deflectionCx - X force variation with α and δe

δe α = −10 −5 0 5 10 15 20 25 30 35 40 45−24 −0.099 −0.081 −0.081 −0.063 −0.025 0.044 0.097 0.113 0.145 0.167 0.174 0.166−12 −0.048 −0.038 −0.04 −0.021 0.016 0.083 0.127 0.137 0.162 0.177 0.179 0.167

0 −0.022 −0.020 −0.021 −0.004 0.032 0.094 0.128 0.130 0.154 0.161 0.155 0.13812 −0.040 −0.038 −0.039 −0.025 0.006 0.062 0.087 0.085 0.100 0.110 0.104 0.09124 −0.083 −0.073 −0.076 −0.072 −0.046 0.012 0.024 0.025 0.043 0.053 0.047 0.040

Cm - pitching moment variation with α and δeδe α = −10 −5 0 5 10 15 20 25 30 35 40 45−24 0.205 0.168 0.186 0.196 0.213 0.251 0.245 0.238 0.252 0.231 0.198 0.192−12 0.081 0.077 0.107 0.11 0.11 0.141 0.127 0.119 0.133 0.108 0.081 0.093

0 −0.046 −0.02 −0.009 −0.005 −0.006 0.01 0.006 −0.001 0.014 0 −0.013 0.03212 −0.174 −0.145 −0.121 −0.127 −0.129 −0.102 −0.097 −0.113 −0.087 −0.084 −0.069 −0.00624 −0.259 −0.202 −0.184 −0.193 −0.199 −0.15 −0.16 −0.167 −0.104 −0.076 −0.041 −0.005

The Y force coefficient for F16 is calculated as follows.

Cy = −0.02β + 0.021δa20

+ 0.086δr30

The Z force coefficient variation with α, the angle of attack and the elevator deflection (δe) is calculated

using the expression given below where the value of s is given in table D.5.

Cz = s(1− β2)− 0.19δe25

Table D.5: Cz - The Z force with αα −10 −5 0 5 10 15 20 25 30 35 40 45s 0.77 0.241 −0.1 −0.416 −0.731 −1.053 −1.366 −1.646 −1.917 −2.12 −2.248 −2.229

D.3 Body Axes Six Degree of Freedom Equations for Aircraft

The complete set of nonlinear ordinary differential equations for a rigid six degree of freedom aircraft is

given as follows.


Force Equations

U = RV −QW − g sin θ +X

m

V = −RU + PW + g sinφ cos θ +Y

m

W = QU − Pv + g cosφ cos θ +Z

m

Kinematic Equations

φ = P + tan θ(Q sinφ+R cosφ)

θ = Q cosφ−R sinφ

ψ =Q sinφ+R cosφ

cos θ

Moment Equations

P = (c1R+ c2P )Q+ c3L+ c4N

Q = c5PR− c6(P 2 −R2) + c7M

R = (c8P − c2R)Q+ c4L+ c9N

Navigation Equations

˙PN =U cos θ cosψ + V (− cosφ sinψ + sinφ sin θ cosψ)

W (sinφ sinψ + cosφ sin θ cosψ)

PE =U cos θ sinψ + V (cosφ cosψ + sinφ sin θ sinψ)

W (− sinφ cosψ + cosφ sin θ sinψ)

h =U sin θ − V sinφ cos θ −W cosφ cos θ

where

Γc1 = (Iyy − Izz)Izz − I2xz Γc2 = (Ixx − Iyy + Izz)Ixz

Γc3 = Izz Γc4 = Ixz

c5 =Izz − Ixx

Iyyc6 =

Ixz

Iyy

c7 =1Iyy

Γc8 = Ixx(Ixx − Iyy) + I2xz

Γc9 = Ixx Γ = IxxIzz − I2xz

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