falling leaf thesis 2009

8/6/2019 Falling Leaf Thesis 2009

1/99

Analysis of the Out-of-Control Falling Leaf Motion using aRotational Axis Coordinate System

Daniel C. Lluch

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State Universityin partial fulfillment of the requirements for the degree of

Master of Sciencein

Aerospace Engineering

Dr. Frederick Lutze - chairDr. Mark AndersonDr. Wayne Durham

October 1998

Blacksburg, Virginia

Keywords: nonlinear flight dynamics, coordinate system, coordinate transformationCopyright 1998, Daniel C. Lluch


2/99

Analysis of the Out-of-Control Falling Leaf Motion using aRotational Axis Coordinate System

Daniel C. Lluch

(ABSTRACT)

The realm of aircraft flight dynamics analysis reaches from local static stability to globaldynamic behavior. It includes aircraft performance issues as well as structural concerns.In the particular aspect of dynamic motions of an aircraft and how we understand them,an alternate coordinate system will be introduced that will lend insight and simplificationinto the understanding of these dynamic motions. The main contribution of this coordinatesystem is that one can easily visualize how the instantaneous velocity vector relates to theinstantaneous rotation vector, the angular rate vector of the aircraft.

The out-of-control motion known as the Falling Leaf will be considered under the light ofthis new coordinate system. This motion is not well understood and can lead to loss of theaircraft and crew. Design guidelines will be presented to predict amplitude and frequency ofthe Falling Leaf.

This work received support from NASA Langley Research Center, contract number NAG-1-1851 #1


3/99

Acknowledgments

I would like to thank all those who have helped me through the years in reference to thiswork as well as throughout my life. First and foremost, Id like to thank my parents forabsolutely always being there and caring as to what was happening in my life. I thank my

sisters for feeding me good meals and giving me a place to go during my undergraduatedays in Blacksburg. Thanks to all my friends, who have been a huge cornerstone in my life,especially friend and roommate Jin Wook Lim.

I owe where I am today in my academic career to my advisor, Dr Fred Lutze. He gave mean opportunity to prove myself many years ago when he had no obligation too, and I thankhim. The remaining members of my committee, Dr Anderson and Dr Durham have made theacademic environment extremely pleasant. I could not have asked for a better committee ormentors in this endeavor. Mr John Foster allowed this project to come my way. He made myshort stay at NASA a wonderful learning experience. Although not technically my boss, Itruly hope all my bosses will be like him. Last but certainly not least, thanks to the SimLab

crew (including Josh) for always being there and putting up with the guitar playing.

iii


4/99

Contents

1 Introduction 1

1.1 Preliminaries to Deriving Equations of Motion . . . . . . . . . . . . . . . . . 21.1.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.2 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.3 Properties of Transformation Matrices . . . . . . . . . . . . . . . . . 6

1.1.4 Some useful Relationships . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.5 Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.6 Angular Rates and Kinematics . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Deriving the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Applied External Forces . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2.2 The Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 The Equations of Motion Collected . . . . . . . . . . . . . . . . . . . . . . . 19

2 Rotational Reference Frame 22

2.1 Coordinate System Relationships . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Relating Information Among the Systems . . . . . . . . . . . . . . . . . . . . 25

2.3 Rotational Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Rotational Force and Moment Equations . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Concerns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Special Case: fixed along V . . . . . . . . . . . . . . . . . . . . . . 30

3 The Falling Leaf 32

iv


5/99

3.1 The Motion Defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Constants of the Motion . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.2 Reduced Model: Equations Collected . . . . . . . . . . . . . . . . . . 38

3.2.3 The Forces and Moments . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Model Validation and Simulation . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Analysis of the Falling Leaf 48

4.1 Predicting the Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.2 Predicting the Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.1 Applying Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Design Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A Global Aerodynamic Force and Moment Curves 76

v


6/99

List of Figures

1.1 Cartesian Polar Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Wind Body Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Thrust Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Angular Momentum for a Rigid Body . . . . . . . . . . . . . . . . . . . . . . 16

2.1 Rotational Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1 Define . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Cl vs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Real Falling Leaf Data, F-18C . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Simulation of Falling Leaf represented in {V,,,,p,r,} . . . . . . . . . . 443.5 Simulation of Falling Leaf represented in {,, , } . . . . . . . . . . . . . 45

4.1 Falling Leaf Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 sin ref vs C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 ref vs C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 vs t for simulation run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Falling Leaf Shape for various ref . . . . . . . . . . . . . . . . . . . . . . . 54

4.6 Potential Energy Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Sensitivity of final energy level to initial conditions . . . . . . . . . . . . . . 59

4.8 Approximation of final energy level . . . . . . . . . . . . . . . . . . . . . . . 61

4.9 Stable initial condition, {,, , } = {35, 65, 16/s, 62} . . . . . . . . . 65

vi


7/99

4.10 Various rolling moment curves . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.11 Simulation with rolling moment . . . . . . . . . . . . . . . . . . . . . . . 69

4.12 Simulation with - rolling moment . . . . . . . . . . . . . . . . . . . . . . . 704.13 Energy curves for various rolling moments . . . . . . . . . . . . . . . . . . . 71

4.14 Comparison of yawing moments . . . . . . . . . . . . . . . . . . . . . . . . . 74

A.1 Lift Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.2 Drag Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

A.3 Body-axis Side Force Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 82

A.4 Rolling Moment Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.5 Yawing Moment Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.6 Pitching Moment Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

vii


8/99

List of Tables

3.1 Assumptions for Reduced Model . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2 Rolling Moment Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

viii


9/99

Nomenclature

(i,

j,

k) unit vectors in the (x,y,z ) direction of the appropriate sub/superscripted coordinatesystem

(, ) 2-3 rotation angles defining the transformation from the body-fixed coordinate sys-tem to the rotational coordinate system

(,,) 3-2-1 rotation angles defining the transformation from the inertial coordinate systemto the wind axis coordinate system

(, , ) 3-2-1 rotation angles defining the transformation from the inertial coordinate sys-tem to the rotational coordinate system

(,,) Euler angles, 3-2-1 rotation angles defining the transformation from the inertialcoordinate system to the body-fixed coordinate system

(A,B,C) Defined constants in the -equation

(a,b,c,d) Coefficients of characteristic polynomial

(g1k , g2k , g3k ) Components of gravity represented in the k reference frame

(L,M,N) Body-fixed axis moments

(p,q,r) Body-fixed axis components of the relative angular rate vector between the body-

fixed coordinate system and the inertial coordinate system

(pw, qw, rw) Wind axis components of the relative angular rate vector between the wind axiscoordinate system and the inertial coordinate system

(u,v ,w) Body-fixed axis components of aircraft velocity relative to inertial space

(xk, yk, z

k) axis system of appropriate sub/superscript

Angle of attack, the angle between the xb-axis and the projection of the aircraft velocityV into the xb zb plane

p Rotation angle to rotate from general moments of inertia in some body-fixed axis systemto principal moments of inertia in the principal axis system

Sideslip angle, the angle made by the aircraft velocity V and the projection of the aircraftvelocity V into the xb zb plane

ref Defines frequency of the sine representation of the rolling moment coefficient as

ref

ij Kronecker delta operator

ix


10/99

ddt

k( F) Time rate of change of F with respect to the k reference frame

Thrust offset angle measured in the xb

zb plane

ij Alternating tensor operator

Short form of the moment of inertia constant (IxxIzz I2xz)ca/b The angular rate of frame a with respect to frame b represented in frame c

q Dynamic pressure, 12

V2

New variable used in variable transformation

Magnitude of the instantaneous rotation vector, b/h

Air density

Angle between the instantaneous velocity vector and the instantaneous rotation vector

ref Value of once steady Falling Leaf oscillation has been reached

Angle defined by A Jacobian matrix

b Wingspan

C Coefficient used in solving for ref

C1 Defined to be equivalent toCY S

2

C2 Defined to be equivalent toClmax Sb

2

ci Short form of the moments of inertia constants in a body-fixed axis system, 1 c 9Cl Rolling moment coefficient

CY Body-axis side force coefficient

Clmax Amplitude of the sine representation of the rolling moment coefficient

CY Stability derivativeCY

D Drag, aerodynamic force along the xw-axis, defined positive in the negative xw direction

DOF Degree of freedom

E Total energy

x


11/99

F Evaluated elliptic integral

g Gravity

Ixx Moment of Inertia about the xb-axis

Ixz Cross Moment of Inertia

Iyy Moment of Inertia about the yb-axis

Izz Moment of Inertia about the zb-axis

K Constant of the motion defined by an initial condition for the reduced system of equationsof motion

k Constant defining phasing between roll and yaw rates

L Lift, aerodynamic force along the zw-axis, defined positive in the negative zw direction

m Aircraft mass

Q Wind axis side force, aerodynamic force along the yw-axis, defined positive in the negativeyw direction

S Wing planform area

T Thrust, Kinetic Energy

Tji Transformation matrix from i coordinate system to j coordinate system

Tperiod Period of oscillation

Tx Fundamental transformation matrix about the x-axis through an angle

V Total aircraft velocity, Potential energy

X Aerodynamic force along the xb-axis

Y Aerodynamic force along the yb-axis

Z Aerodynamic force along the zb-axis

n Row or column vector of a transformation matrix

Ik Inertia tensor represented in the k reference frame

xi


12/99

Chapter 1

Introduction

Issues of aircraft dynamic stability have been under consideration for many years (Ref[10]).Due to high performance needs, aircraft have extended their operating region into the highangle of attack regime. Seeking departure resistant aircraft in this high angle of attackregime has led to establishing roll and yaw departure criteria. Nonlinear coupling betweenthe longitudinal and lateral-directional variables has shown to be a key driver in analyzingglobal stability (Refs [10], [9]). Stability has also been found to be a function of the dynamicstate of the aircraft (Ref [11]).

To contribute to the analysis of aircraft dynamics, focus has been placed on the relationshipbetween the instantaneous rotation vector and the instantaneous velocity vector in the effort

to understand how those vectors behave for particular maneuvers (Ref [2], [6]). The workpresented here will place all importance on the interaction between the velocity vector and theinstantaneous rotation vector to the extent of developing equations of motion that specificallyfocus on the relationship of these vectors. This representation of the equations of motionrequires developing a new coordinate system. A good choice of coordinate system allows oneto easily focus on the relationship in question.

In the analysis of vehicle dynamics, one often comes to the realization that a result can beunderstood in a simpler way if the problem is approached from a different point of view.Consider the motion of a particle shown in Figure 1.1. If the analysis was performed usingCartesian coordinates, namely x-y coordinates, it would show rapidly varying state variables.

x = f(x, y) (1.1)

y = f(x, y) (1.2)

Yet if the same analysis was done using polar coordinates, namely r- coordinates, it wouldshow a rapidly varying r and a slowly varying .

1


13/99

Daniel C. Lluch Chapter 1. Introduction 2

x

yr

Figure 1.1: Cartesian Polar Comparison

r = f(r, ) (1.3)

= f(r, ) (1.4)

An intelligent choice of the coordinate system can lend insight into a particular problem,

and can lead to assumptions, such as assuming to be a constant for this example.The idea shown above will be applied to a more complex problem. A coordinate system isproposed for the analysis of aircraft dynamics in which some of the variables will be nearlyconstant. This coordinate system will lend insight into the analysis of dynamic motions ofan aircraft. As a direct result of the use of the new coordinate system, key parameters canbe identified that establish various characteristics of selected dynamic motions. Stabilityguidelines will also be developed in terms of the new coordinate system.

1.1 Preliminaries to Deriving Equations of Motion

In order to study the dynamics of an aircraft, the laws that govern the motion must beknown. Newtons laws of motion can be applied to a rigid body to derive its equationsof motion. One must choose an axis system to represent these laws and the associatedaerodynamic forces and moments. As shown above, significant insight can be obtained froman intelligent choice of variables and coordinate system, depending on the particular motionthat is to be analyzed. In the sections that follow, the groundwork will be laid in definingthe necessary information needed to apply Newtons laws to an aircraft. The mechanics of


14/99


defining coordinate systems is shown for those coordinate systems currently in use. It willthen be shown how these coordinate systems are used in expressing Newtons laws. For anyfurther information on the content of the sections that follow, see Ref [7].

1.1.1 Coordinate Systems

The first assumption for considering aircraft dynamics is most commonly the flat-earthapproximation, which states that the ground-fixed axes are inertial. This assumption alsoimplies the gravity vector is constant in magnitude and direction. When considering stabilityanalysis, this approximation is valid since the aircraft will not be traveling far distances.Most commonly one sees the z-axis point along the direction of gravity, and the x and yaxes pointing north and east in the plane of the earths surface. One can then define the

local horizontal axes, (xh, yh, zh) as being parallel to the ground-fixed axes yet fixed at thecenter of mass of the aircraft and traveling with the aircraft. It is the local horizontal axissystem where gravity has its most simplest representation.

It is then necessary to define a coordinate system that is fixed to the aircraft and moveswith it in both translation and rotation, referred to as a body-fixed axis system, (xb, yb, zb).The origin of this system is fixed at the center of mass of the aircraft and has the x andz axes in the plane of symmetry of the aircraft with the x-axis pointing out the nose andthe z-axis pointing out the bottom. The y-axis then completes the right-hand rule with theaxis pointing in the direction of the right wing. There are various interesting choices for thebody-fixed axes. In particular, when the (xb, yb, zb) is aligned with the principal moments of

inertia, (xp, yp, zp), called the principal axis system. This particular choice of the (xb, yb, zb)significantly reduces the complexity of the inertial terms in the equations of motion yet hasthe draw back that the moments have to be expressed in the same coordinate system. Thisrepresentation of the moments may not be directly available.

Another coordinate system of interest is the wind axes system, (xw, yw, zw). The origin isfixed at the center of mass of the aircraft and has the x-axis pointing along the instantaneousvelocity vector, the z-axis is in the plane of symmetry pointing out the bottom of the aircraft,and the y-axis then completes the right-hand rule. The instantaneous velocity vector will bethe velocity of the aircraft, in both magnitude and direction, relative to inertial space. Thisparticular choice of axes significantly reduces the representation of the aerodynamic forces

and moments in the linearized problem, but does not contribute much simplification in thegeneral nonlinear problem. In addition, the terms in the inertia tensor become functions oftime.


15/99


1.1.2 Transformations

With the defined coordinate systems, information between them can be related by definingtransformations between any two of the coordinate systems. Define a rotation about thex-axis to be a 1-rotation, a rotation about the y-axis a 2-rotation, and a rotation about thez-axis a 3 rotation. Euler showed that that one can rotate to any new coordinate systemby rotating though, at most, three independent fundamental rotations. For instance, youcan get to any new coordinate system via a 1-2-3 rotation, a 3-2-1 rotation, or a 1-3-1rotation. There are twelve different combinations allowing the transformation from any twocoordinate systems. A 1-1-3 rotation is not a valid transformation between any two arbitrarycoordinate systems since the first two rotations can be considered one rotation about thex-axis. Fundamental transformation matrices can be defined about each single axis (Ref [7]):

Rotate about the x-axis:

Tx =

1 0 00 cos sin

0 sin cos

(1.5)

Rotate about the y-axis:

Ty =

cos 0 sin

0 1 0sin 0 cos

(1.6)

Rotate about the z-axis:

Tz =

cos sin 0 sin cos 0

0 0 1

(1.7)

With these fundamental rotation matrices we can now build the transformation matrixbetween any two of the coordinate systems defined in Sec 1.1.1.

Horizontal and Body-fixed Relationship: Euler Angles

In flight dynamics, the common order for rotation from the local horizontal to the body-fixedcoordinate systems is a 3-2-1 rotation. The following set of angles are defined to be the Eulerangles, [,,]:

1. Rotate through an angle about the zh-axis to an intermediate coordinate system,(x, y, z).


16/99


2. Then rotate through an angle about the y-axis to an intermediate coordinate system,(x, y, z).

3. Finally, rotate through an angle about the x-axis to reach the body-fixed axissystem, (xb, yb, zb)

Therefore the transformation matrix from the local horizontal system to the body-fixed axissystem is,

Tbh = TxTy Tz (1.8)

which after evaluation becomes,

Tbh = cos cos cos sin sin sin sin cos cos sin sin sin sin + cos cos sin cos

cos sin cos + sin sin cos sin sin sin cos cos cos

(1.9)

Horizontal and Wind Relationship

The relationship between the local horizontal coordinate system and the wind axis coordinatesystem are similar to the Euler angles. The rotations are in the same order yet three newangles are defined, [, ,]:

Twh = TxTy Tz (1.10)

resulting in:

Twh =

cos cos cos sin sin sin sin cos cos sin sin sin sin + cos cos sin cos

cos sin cos + sin sin cos sin sin sin cos cos cos

(1.11)

Body-fixed and Wind Relationship

The rotation from the body-fixed axes to the wind axes can be fully described by only twofundamental transformations, a 2-3 rotation, because the zw-axis is also in the plane ofsymmetry of the aircraft. The angles are [, ],

1. Rotate through an angle about the yb-axis to an intermediate coordinate system,(x, y, z).

2. Then rotate through an angle about the z-axis to reach the wind axes coordinatesystem, (xw, yw, zw)


17/99


Therefore,

Twb = Tz Ty (1.12)

Evaluating the above expression gives the transformation matrix from the body-fixed axissystem to the wind axis system,

Twb =

cos cos sin sin cos cos sin cos sin sin

sin 0 cos

(1.13)

With Twb defined, another relationship can be expressed between the local horizontal andthe wind axis systems which relates the eight angles [,,,,,,,] of which only five

are independent.

Twh(, ,) = Twb(, ) Tbh(,,) (1.14)

1.1.3 Properties of Transformation Matrices

The transformation matrix has a few interesting properties. The matrices that are dealt withhere are similarity transformations. Let n be any row or column vector of the transformationmatrix. The orthogonality condition states,

ni nj = 0 for i = j. (1.15)The normal condition states,

ni nj = 1 for i = j. (1.16)

Both conditions can be combined to form the orthonormal condition, which can be expressedwith the Kronecker operator,

ni nj = ij where ij = 1 i = j0 i

= j

. (1.17)

The dextral condition states that a matrix conforms to the right-handed rule. This conditioncan be expressed with the alternating tensor operator,

ni nj = ijk nk where ijk =

1 ijk = 1-2-3 and cyclic perturbations1 ijk = 3-2-1 and cyclic perturbations

0 otherwise. (1.18)


18/99


Because the purpose of the similarity transformation matrix is to only change the represen-tation of given information, it does not scale that information. A vector in space is invariantregardless of what coordinate system it is represented in. Therefore, all the above conditionsbeing satisfied leads to the following:

det Tyx = 1 (1.19)

(Interesting to note that the determinant is -1 if you are going from a right-handed to left-handed coordinate system or vice-versa and the transformation matrix is not dextral)

Consider having the transformation matrix, Tyx, yet having information expressed in the y-coordinate system, (xy , yy, zy). To express the same information in the x-coordinate system,(xx, yx, zx),

nx = Txyny = T1yx n

y = TTyxny (1.20)

The orthonormal-dextral transformation matrix has the property that its inverse is its trans-pose. Information between any two coordinate systems can be easily related once the trans-formation matrix between the two coordinate systems is defined.

1.1.4 Some useful Relationships

As mentioned in Sec 1.1.1 gravity is most naturally expressed in the local horizontal coordi-nate system, (xh, yh, zh).

gh =

00g

(1.21)Equation 1.9 can be used to express the gravity vector in the body-fixed axis system,(xb, yb, zb).

gb = Tbhgh = g

sin sin cos cos cos

(1.22)

Equations 1.9-1.13 can be used to express the gravity vector in the wind axis system,

(xw, yw, zw).gw = Twhg

h = TwbTbhgh

= g

sin sin cos cos cos

(1.23)

= g

( cos cos sin + sin sin cos + sin cos cos cos )(cos sin sin + cos sin cos sin sin cos cos )

(sin sin + cos cos cos )

(1.24)


19/99


y-body

V

u

x-body

z-body

wv

Figure 1.2: Wind Body Comparison

Similarly, the velocity vector is most naturally expressed in the wind axis coordinate system,(xw, yw, zw). By definition,

Vw = V0

0

(1.25)

Figure 1.2 shows the how the velocity vector relates to the body-fixed coordinate system,(xb, yb, zb).

Vb =

uvw

= Tbw V

w (1.26)

uvw

=

V cos cos V sin

V sin cos

(1.27)

Equation 1.27 gives the relationship between the body-fixed components of the velocity


20/99


vector and the wind axis variables. Namely,

tan =w

u(1.28)

sin =v

V(1.29)

V2 = u2 + v2 + w2 (1.30)

1.1.5 Moments of Inertia

The moments of inertia for a rigid body in a body-fixed coordinate system with an x zplane of symmetry are defined to be

Ixx =

m

(y2 + z2) dm (1.31)

Iyy =

m

(x2 + z2) dm (1.32)

Izz =

m

(x2 + y2) dm (1.33)

Ixz =

m

xz dm (1.34)

Ixy = Iyz = 0 (1.35)

It is worth noting that one may see the cross moment of inertia defined in literature asthe negative of the above thus allowing every element in the inertia tensor be positive. Thedefinition above does conform to the convention seen in aircraft dynamics. The matrix tensorfor the inertia terms which contain an x z plane of symmetry is then

I =

Ixx 0 Ixz0 Iyy 0

Ixz 0 Izz

b

(1.36)

As mentioned in Sec 1.1.1 a great deal of simplification results from choosing the body-fixedcoordinate system to line up with the principal directions of the moments of inertia. Thistransformation can be achieved through a single rotation about the yb-axis:

Tpb =

cos p 0 sin p0 1 0

sin p 0 cos p

(1.37)


21/99


with

p =

1

2 tan

1 2IxzIxx Izz . (1.38)

Therefore the principal inertia tensor is given by

I =

Ixx 0 00 Iyy 0

0 0 Izz

p

= Tpb Ib TTpb (1.39)

Tpb can also be used to represent general body-fixed forces and moments in the principalcoordinate system, (xp, yp, zp).

Fp = Tpb Fb (1.40)

1.1.6 Angular Rates and Kinematics

In deriving the equations of motion for a rigid body, information is usually expressed innon-inertial coordinate systems. It is then necessary to define the angular rate of the chosencoordinate system with respect to the inertial coordinate system. Let b/h be the angularrate of change of the body-fixed coordinate system with respect to the local horizontalcoordinate system (recall (xh, yh, zh) is considered inertial). As with any other vector, b/h,

can be represented in any coordinate system. Recall from Sec 1.1.2 the Euler rotation anglesdescribe how (xh, yh, zh) relates to (xb, yb, zb). The time rate of change between the body-fixedaxes and the local horizontal axes is most naturally expressed in a mixture of coordinatesystems and is related to the Euler angle rates as follows,

b/h = ib + j

+ k

(1.41)

with i, j, k being unit vectors along the (x, y, z) directions of the appropriate coordinatesystem. Expressing every term in the body-fixed system,

b/h =

ib +

Tb

j

+

Tbk

(1.42)

Defining the body-fixed components and expressing the transformation matrices in terms ofthere fundamental rotations,

bb/h =

pqr

=

100

+ Tx

010

+ Tx Ty

001

(1.43)


22/99


which gives:

p

qr

= 1 0

sin

0 cos cos sin 0 sin cos cos

(1.44)The inverse relationship is,

=

1 sin tan cos tan 0 cos sin

0 sin sec cos sec

pqr

(1.45)

(It is important to note that the matrix relating the body axis rates and the time rate ofchange of the Euler angles is not a transformation matrix as defined in Sec 1.1.2.)

Equation 1.45 shows that as the moments (directly) and forces (indirectly) affect the bodyaxis rates, the Euler angles have to evolve a certain way. The angular rates completely definehow the Euler angles evolve in time. This kinematic relationship is not subject to change orinterpretation.

A similar relationship exists for w/h. Written for convenience,

ww/h =

pwqwrw

=

1 0 sin 0 cos cos sin

0 sin cos cos

(1.46)

The inverse relationship is,

=


0 sin sec cos sec

pwqwrw

(1.47)

It is useful to relate the body axis rates to the wind axis rates.

w/h = w/b + b/h (1.48)

The new term in Equation 1.48 is most naturally expressed in two coordinate systems,

w/b = jb + kw (1.49)

Equation 1.48 can now be expressed in a variety of ways. In the body-fixed coordinatesystem,

p

q r

= Tbw

pwqw

rw

(1.50)


23/99


Equivalently, solving for the body axis rates and wind axis rates respectively,

p

qr

= Tbwpw + sin

qw + cos rw

(1.51)

pwqwrw

= Twb

p sin q

r + cos

(1.52)

1.2 Deriving the Equations of Motion

All the necessary information has now been presented in order to move towards derivingthe equations of motion. Because Newtons laws only hold in an inertial reference frame(non-rotating), it is necessary to introduce the time-rate of change of a rotating reference

frame. For instance, if F is represented in non-inertial reference frame A, then the time rateof change as seen by the inertial reference frame N (Newtonian) is,

d

dt

N

(F) =d

dt

A

(F) + A/N F (1.53)

Writing Newtons Law in vector form,

F = m ddt

N( V) (1.54)

If the velocity in Equation 1.54 is represented in the body-fixed coordinate system,

F = m

d

dt

b

( Vb) + b/h Vb

(1.55)

Defining the body axis force components, using the body axis velocity components, andincorporating matrix multiplication for the cross-product,

Fb =

FxbFybFzb

= muvw

+ m0

r q

r 0 pq p 0

uvw

(1.56)Evaluating this expression, the body axis force equations become,

FxbFybFzb

= m

u + qw rvv + ru pww +pv qu

(1.57)


24/99


Similarly, if the velocity in Equation 1.54 is represented in the wind axis coordinate system,

F = m

ddt

w

(V

w

) + w/h V

w(1.58)

Defining the wind axis force components, using the wind axis velocity components, andincorporating matrix multiplication for the cross-product,

Fw =

FxwFywFzw

= m

V00

+ m

0 rw qwrw 0 pw

qw pw 0

V00

(1.59)

multiplying through the wind axis force equations become,

FxwFywFzw

= m

VV rw

V qw

(1.60)

Three differential equations are not obvious from Equation 1.60. Equation 1.52 can be usedto substitute in for the appropriate wind axis rates and the wind axis force equations takethe form,

FxwFyw

Fzw

= m

V

V(p sin + r cos )V[sin (p cos + r sin ) + cos ( q)]

. (1.61)

The two sets of equations given by Equations 1.57 and 1.61 represent Newtons law, F = ma,for an aircraft. The identical information is present in both representations, it is the appli-cation that dictates which set of equations to use. In the chapters that follow, a particulardynamic motion will call for an introduction of yet another representation of the same in-formation as Equations 1.57 and 1.61. As will be shown, the choice of which axis system torepresent the equations of motion can greatly simplify the analysis and lend insight into theproblem.

1.2.1 Applied External Forces

The forces that are applied to the aircraft in Equations 1.57 and 1.61 can be broken intothree groups:

Those due to the aerodynamic flow field about the aircraft Those due to gravity (i.e. weight of the aircraft)


25/99


T

z-body

x-body

Figure 1.3: Thrust Vector

Those due to thrust produced by the aircraft

Define the aerodynamics forces as follows,

FA =

XYZ

b

FA =

DQ

L

w

(1.62)

Notice the equivalence yet also notice that the two vectors are represented in two differentcoordinate systems. They are related through the Twb transformation matrix, Equation 1.13.

Equations 1.22 and either 1.23 or 1.24 show how gravity is expressed in the body-fixed andwind axis coordinate systems respectively. Multiply the appropriate vector by the mass ofthe aircraft m, and the result is the component of weight in the desired direction. Therefore,

mg = m

g1bg2bg3

b

b

mg = m

g1wg2wg3

w

w

, (1.63)

where the components of the gravity vector are given by Equations 1.22-1.24.

The direction of thrust will be assumed to be constant in the body-fixed coordinate system.It is further assumed that the thrust vector lies in the plane of symmetry, and the anglemade with the xb-axis is defined to be as shown in Figure 1.3. Therefore the thrust vectoronly has components in the xb-axis and zb-axis.


26/99


T =

Tcos

0T sin

b

T =

Txw

TywTzw

w

(1.64)

where the wind axis components of thrust are,

TxwTywTzw

= Twb

Tcos 0

Tsin

=

cos (Tcos cos Tsin sin )sin (Tcos cos T sin sin )

Tcos sin T sin cos

(1.65)

The applied external forces in Equations 1.57 and 1.61 can now be written as,

FxbFybFzb

= Tcos + X+ mg1b

Y + mg2bTsin + Z+ mg3b

(1.66)where the gravity components are given by Equation 1.22, and

FxwFywFzw

=

Txw D + mg1wTyw Q + mg2wTzw L + mg3w

(1.67)

where the thrust components are given by Equation 1.65 and the gravity components aregiven by either Equation 1.23 or 1.24.

1.2.2 The Moments

Information about the how a rigid body rotates can be derived from using Newtons Law inanother form that defines the moments about the point that some vector r is defined from.Namely

M r F = r m ddt

N

( V) (1.68)

To understand the right side of the equation we begin with the angular momentum.

The angular momentum of the particle in Figure 1.4 is defined to be

HD = r Vm (1.69)

For the rigid body shown in Figure 1.5 with center of gravity G, Equation 1.69 can be writtenin the form

HD =

DP VP dm (1.70)


27/99


m

r

D

V

Figure 1.4: Angular Momentum

D

G

r

P

dm

Figure 1.5: Angular Momentum for a Rigid Body


28/99


Taking the time rate of change of Equation 1.70,

d

dt( HD) = d

dt( DP)

VP dm + DP

d

dt( VP)dm (1.71)

Suppose D is now set to coincide with G, the center of gravity. The first integral in Equa-tion 1.71 becomes,

d

dt( DP) VPdm =

d

dtr VPdm (1.72)

=

( r) ( VG + r)dm (1.73)

=

rdm

VG +

(r) ( r)dm (1.74)

= 0 (1.75)

The integral in the first term of Equation 1.74 is zero since the integral is being performedabout the center of mass. The second integral is zero since it is a vector crossed withitself. Likewise, when D is set to G, the second integral of Equation 1.71 is the right side ofEquation 1.68. Therefore,

MG =d

dt

N

( HG) (1.76)

Equation 1.70 can also be written in the following manner.

HD =

DP VPdm (1.77)

=

( DG + r) ( VG + r)

dm (1.78)

= DG VG

dm + DG

rdm

+

rdm VG +

r ( r)dm

(1.79)

= DG m VG + IG (1.80)In the special case when D is set to G the angular momentum can be written as,

HG = IG (1.81)Where IG represents the moment of inertia tensor.

Keeping in mind that Equation 1.76 is only valid for moments about the center of gravity,the G subscript is dropped. If H is expressed in a body-fixed coordinate system

M =d

dt

b

( H) + b/h H (1.82)


29/99


The inertia tensor, I, in a body-fixed coordinate system is constant. Therefore, using Equa-tion 1.81 the moment equations can be written for a rigid body in the form,

M = Ib ddt

b(b/h) + b/h Ib b/h (1.83)

solving for the time rate of change of b/h,

d

dt

b

(b/h) = (Ib)1

M b/h Ib b/h

(1.84)

Introducing the body components of the above vectors, assuming that thrust acts through thecenter of gravity, using matrix multiplication for the cross product, and defining = IxxIzz I2xz,

p

qr

= Izz

0 Ixz

0 1Iyy 0Ixz

0 Ixx

L

MN

0 r qr 0 p

q p 0

Ixx 0 Ixz0 Iyy 0

Ixz 0 Izz

pqr

=

Izz

{L + (Iyy Izz)rq+ Ixzpq}+ Ixz

{N + (Ixx Iyy)pq Ixzrq}

1

Iyy

{M

Ixz(p2

r2)

(Izz

Ixx)pr

}Ixz

{L + (Iyy Izz )rq+ Ixzpq}

+ Ixx

{N + (Ixx Iyy)pq Ixzrq}

(1.85)

Similarly if the angular momentum H in Equation 1.76 is expressed in the wind axis coor-dinate system,

M =d

dt

w

( H) + w/h HThis representation of the moment equation does not appear frequently since, unlike in thebody-fixed axis, the inertia tensor is not constant in the wind axis coordinate system. Themoment equations for the wind axis coordinate system can be written in the form,

M =d

dt

w

(Iw) w/h + Iw d

dt

w

(w/h) + w/h I w/h (1.86)

All the terms in the above equation are quite attainable except the time rate of change ofthe wind axis inertia tensor. Concentrating on this particular term,

Iw = Twb Ib TTwb (1.87)


30/99


By taking the time rate of change of the above equation, terms involving and areintroduced. In the general case, the time rate of change of the wind axis inertia matrix canbe quite lengthy. The (1,1)-term is printed here,

Iw(1, 1) =2sin cos Ixx cos2 + 2Ixz sin cos + Iyy Izz sin2

+ 2 cos2 Ixx sin cos Ixz cos(2) + Izz sin cos (1.88)

Every term in the time rate of change of the wind axis inertia matrix is similar in complexityto the expression above. Therefore, evaluating Equation 1.86 would be best suited for thecomputer. The moments applied to the body represented in the wind axis coordinate systemcan be written as Mw = Twb Mb. The advantage of Equation 1.86 is that one would havethe time rate of change of the wind axis rates directly available. These equations couldbe integrated in time and the kinematics could directly be calculated using Equation 1.47.Due to simplicity gained from a constant inertia tensor, the moment equations are usuallyrepresented in some body-fixed axis system.

1.3 The Equations of Motion Collected

Force Equations

Using Equations 1.57 and 1.66, the body axis force equations become,

u = 1m

(Tcos + X) + rv qw + g1bv =

Y

m+pw ru + g2b (1.89)

w =1

m(Tsin + Z) + qu pv + g3b

Using Equations 1.61 and 1.67, the wind axis force equations become,

V =1

m(cos (Tcos cos Tsin sin ) D) + g1w

= 1mV

(sin (Tcos cos Tsin sin ) Q)+p sin r cos + g2w

V(1.90)

=1

mV cos (Tcos sin Tsin cos L)

tan (p cos + r sin ) + q+ g3wV cos


31/99


The gravity components are given by Equation 1.22 and either Equation 1.23 or 1.24

g1bg2bg3b

= g sin

g sin cos g cos cos

(1.22)

g1wg2wg3w

=

g sin g sin cos g cos cos

(1.23)

=

g( cos cos sin + sin sin cos + sin cos cos cos )g(cos sin sin + cos sin cos sin sin cos cos )

g(sin sin + cos cos cos )

(1.24)

Moment Equations

Introducing the convention used in Ref [12] for the constants involving the moments ofinertias for a body-fixed coordinate system, Equation 1.85 becomes

p = (c1r + c2p)q+ c3L + c4N

q = c5pr c6(p2 r2) + c7M (1.91)r = (c8p c2r)q+ c4L + c9N

with

c1 = (Iyy Izz )Izz I

2

xz

c2 = (I

xx Iyy + Izz )Ixz

c3 =Izz

c4 =Ixz

c5 =Izz Ixx

Iyyc6 =

IxzIyy

c7 =1

Iyyc8 =

(Ixx Iyy)Ixx + I2xz

c9 =Ixx

= IxxIzz I2xz

(1.92)

Kinematic Equations

=


0 sin sec cos sec

pqr

(1.45)


32/99


Using Equations 1.47 and 1.52,

= 1 sin tan cos tan 0 cos sin 0 sin sec cos sec

Twbp

sin

q r + cos

This chapter has derived the equations of motion for an aircraft in two different representa-tions, the body-fixed axis system and the wind axis system. Both representations completelyhouse the identical dynamics and both representations are commonly used throughout thefield of aircraft dynamics. Different insight can be gained when using one representation overthe the other, and this insight is completely defined by the application in question. The nextchapter will introduce yet another and new coordinate system. The equations of motion willbe derived with the new coordinate system as the basis for derivation.


33/99

Chapter 2

Rotational Reference Frame

As mentioned in Chapter 1, a coordinate system will be introduced to lend insight to theanalysis of dynamic motions of an aircraft. The system proposed is not a good choice forthe complete flight envelope of an aircraft for reasons that will become obvious, yet it isproposed for the analytical applications that may lend understanding into the drivers andcauses of particular motions.

The rotational coordinate system, (xr , yr, zr), has its origin fixed at the center of mass ofthe aircraft. It has the x-axis pointing along the same line as the instantaneous rotationvector, b/h, but its positive direction always being in the front hemisphere of the body-fixed coordinate system (see Ref [6]). The z-axis is in the plane of symmetry of the aircraft

pointing out the bottom of the aircraft, and the y-axis then completes the right-hand rule.

2.1 Coordinate System Relationships

With the introduction of the rotational coordinate system, relationships can be derivedbetween it and the coordinate systems defined in Sec 1.1.1. Figure 2.1 shows how the thebody-fixed coordinate system relates to the rotational and wind coordinate systems.

Body-fixed and Rotational Relationship

The rotation from the body-fixed to the rotational axes coordinate system is analogous to thebody-fixed/wind relationship shown in Section 1.1.2. The relationship can be fully describedby only two fundamental transformations, a 2-3 rotation. The angles are [, ];

1. Rotate through an angle about the yb-axis to an intermediate coordinate system,(x, y, z).

22


34/99

Daniel C. Lluch Chapter 2. Rotational Reference Frame 23

V

z-body

y-body

x-body

b/h

Figure 2.1: Rotational Coordinate System


35/99


2. Then rotate through an angle about the z-axis to reach the rotational axes coordinatesystem, (xr, yr, zr)

Therefore,

Trb = TzTy (2.1)

Evaluating the above expression gives the transformation from the body-fixed axis systemto the rotational axis system.

Trb =

cos cos sin sin cos cos sin cos sin sin

sin 0 cos

(2.2)

Wind Axes and Rotational Relationship

As mentioned in Sec 1.1.2, two coordinate systems can be related by any three independentfundamental rotations. The rotation from the wind axes to the rotational coordinate systemcan be obtained by a 3-2-3 rotation. These rotations can be easily visualized in Figure 2.1and are given by the angles [,, ],

1. Rotate through an angle about the zw-axis to an intermediate coordinate system,(x

, y

, z

).2. Then rotate through an angle about the y-axis to an intermediate coordinate system,

(x, y, z).

3. Finally, rotate through an angle about the z-axis to reach the rotational axis system,(xr, yr, zr)

Therefore,

Trw = Tz TyTz (2.3)

Evaluating the above expression gives the transformation matrix from the wind axis systemto the rotational axis system,

Trw =

cos cos cos + sin sin cos cos sin + sin cos cos sin sin cos cos + cos sin sin cos sin + cos cos sin sin

sin cos sin sin cos

(2.4)


36/99


Horizontal and Rotational Relationship

The relationship between the local horizontal and the rotational coordinate systems willalso follow the convention used in Sec 1.1.2. The three new angles will be defined as thecorresponding capital Greek letter of the Euler angles, [, , ];

Trh = TxTyTz (2.5)

Evaluating the above expression gives the transformation matrix from the local horizontalaxis system to the rotational axis system.

Trh = cos cos cos sin sin

sinsincos cos sin sin sin sin + cos cos sin cos cos sin cos + sin sin cos sin sin sin cos cos cos (2.6)

Another representation for the transformation between local horizontal axes and rotationalaxes exists that relates the eight angles [,, , , ,, ,] of which only five are indepen-dent. Namely,

Trh(, , ) = Trb(, ) Tbh(,,) (2.7)

This relationship will be used later for a reduced system. Equating terms from the matrices

on either side of Equation 2.7 will give the the three independent spherical relationshipsbetween five independent angles and the remaining three dependent angles (also see Ref [6]).

2.2 Relating Information Among the Systems

The body axis rotation vector, b/h, is most naturally expressed in the rotational coordinatesystem, (xr, yr, zr). By definition,

rb/h =

00

(2.8)The inverse of the matrix in Equation 2.2 can be used to relate the body axis rates and ,

pqr

= Tbr

00

=

cos cos sin

sin cos

(2.9)


37/99


Equation 2.9 gives the relationship between the body-fixed components of the rotation vectorand the rotational axis variables. Namely,

tan = rp

(2.10)

sin =q

(2.11)

2 = p2 + q2 + r2 (2.12)

also from Figure 2.1,

= + (2.13)

It is also useful to have the body axis gravity components in terms these new variables. Inaddition to the relationship in Equation 1.22,

gb = TbrTrhgh =

g( sincos cos sincoscos sin coscossin )g(sincoscos sinsin )

g( sinsin cos sincossin sin + cos cos cos )

(2.14)

2.3 Rotational Kinematics

It is beneficial to step back and examine what is being done. The rotational coordinate

system is a coordinate system that is aligned along on the angular rate vector of anothercoordinate system, namely b/h. The rotational coordinate system has its own angular ratevector that expresses how it rotates with respect to inertial space, namely r/h. These twoangular rate vectors are different, and differ by the relative angular rate between the twosystems.

The kinematics are derived in the same way as in Sec 1.1.6.

rr/h =

prqrrr

=

1 0 sin0 cos cos sin

0 sin cos cos

(2.15)

The inverse relationship gives,

=


0 sin sec cos sec

prqrrr

(2.16)

It is useful to relate the rotational axis rates to the body-fixed axis rates.

r/h = r/b + b/h (2.17)


38/99


The new term in Equation 2.17 is most naturally expressed in two coordinate systems,

r/b =

jb + kr (2.18)

Equation 2.17 can now be expressed in a variety of ways. In the rotational coordinate system,

pr qr

rr

= Trb

00

(2.19)

Equivalently, solving for the rotational axis rates,

prqrrr

=

sin cos

(2.20)

2.4 Rotational Force and Moment Equations

A rigorous derivation will follow showing the steps to achieve a new set of force and momentequations. Taking the time rate of change of Equation 2.12, using Equation 2.9 to substitutefor the body axis rates, and dividing both sides by two,

= p cos cos + qsin + r sin cos (2.21)

Taking the time rate of change of Equations 2.10 and 2.11 respectively,

sec2 = rp p rp2=

1

p(r p tan )

=cos

cos (2.22)

cos =q

q

2

=1

(q sin )

=1

cos (q

sin ) (2.23)

The next step is to replace all the time rate of change variables on the right side of Equa-tions 2.21-2.23. Equation 2.9 is used to substitute for the (p,q,r)s on the right side ofEquation 1.91,

pqr

=

2 sin cos (c1 sin + c2 cos ) + c3L + c4N2 cos2( c5

2sin2 c6 cos2) + c7M

2 sin cos (c8 cos + c2 sin ) + c4L + c9N

(2.24)


39/99


equivalently,

p

qr

= 2

2sin2(c1 sin + c2 cos ) + c3L + c4N

2

cos2

(c52 sin2 c6 cos2) + c7M

2

2sin2(c8 cos + c2 sin ) + c4L + c9N

(2.25)Substituting for the time rates of change, Equations 2.21-2.23 become the moment equationsrepresented in the new coordinate system,

= 2 sin cos2

c1 + c5 + c8

2sin2 + (c2 c6)cos2

+ c7Msin + (c3L + c4N)cos cos + (c4L + c9N)sin cos (2.26)

= sin c8 cos2 2c2 sin cos c1 sin2

+ 1cos

(c4L + c9N)cos (c3L + c4N)sin (2.27) = cos3

c52

sin2 c6 cos2

+c7Mcos

sin2 cos

c1 + c82

sin2 + c2 cos2

cos sin

c3L + c4N

sin sin

c4L + c9N

(2.28)

Equations 2.26-2.28 are a different representation of the body-fixed moment equations givenby Equation 1.91. It is important to note that the aerodynamic moments L,M,N and the

constants involving the inertia terms ci are all still the associated values for the body-fixedaxis system. Equations 2.26-2.28 are still body-fixed moment equations that describe howb/h changes in time.

The wind axis force equations, Equation 1.90, are modified as follows. Using Equation 2.9and Equation 2.13,

p sin r cos = cos cos sin( + ) sin cos( + )= cos

cos (sin cos + cos sin ) sin (cos cos sin sin )

= cos

sin (sin2 + cos2)

= cos sin (2.29)

p cos + r sin = cos

cos cos( + ) + sin sin( + )

= cos

cos (cos cos sin sin ) + sin (sin cos + cos sin )= cos

cos (sin2 + cos2)

= cos cos (2.30)

Taking the time rate of change of Equation 2.13,

= (2.31)


40/99


The force equations in the new coordinate system can then be written as,

V =1

m

cos (Tcos cos( + ) T sin sin( + )) D

+ g1w (2.32)

=1

mV

sin (Tcos cos( + ) Tsin sin( + )) Q

+ cos sin +

g2wV

(2.33)

=1

mV cos

Tcos sin( + ) Tsin cos( + ) L

cos cos tan + sin + g3w

V cos (2.34)

where the in Equation 2.34 is given by Equation 2.27.

Using Equations 1.45 and 2.9, the kinematics are then written as

=


0 sin sec cos sec

cos cos sin

sin cos

(2.35)

The kinematics can also be represented with two other sets of angles, namely (,,) or(, , ). This representation of the kinematics is valid yet, in the general case, will intro-duce time rates of change ( , ) and (, ) respectively on the right hand side of the kinematicequations. Bringing these time rates of change on the right hand side is not desirable foranalytical purposes yet it is not an issue if running simulations using computers.

2.4.1 Concerns

Upon inspection of the equations of motion proposed, there are various problems that arise.Many of the problems are analogous to those encountered in the wind axis coordinate system.For example, if total velocity, V, or the xb-component of velocity, u, were ever zero and become undefined. These conditions are only likely for a helicopter or a tilt rotor aircraft.For the system in question,

Equations 2.27 and 2.28 become undefined when is zero is undefined when is zero, Equation 2.11 If the rotation vector is comprised of pure pitch rate, with roll and yaw rates zero, then

is 90 degrees and Equations 2.27 and 2.28 are undefined as well

Motion consisting of pure yaw rate can not be described as is undefined


41/99


As stated at the beginning of the chapter, dynamic motions are of interest. Therefore, being zero is not an issue although care must be taken as can go through zero, andgenerally does. Motion with pure a longitudinal angular rate is not an issue since a subsetof the above equations can be used to analyze that particular motion. Knowing that is90 degrees and is zero, and both angles are fixed, two degrees of freedom are removed,thus allowing the removal of the equations that cause difficulty. A similar procedure can beperformed for the analysis of a flat spin. An example follows that shows how a subset ofdynamic motions can be extracted from the equations of motion proposed above.

2.4.2 Special Case: fixed along V

Consider the case where the rotation vector lines up with the velocity vector at all times.

This condition can easily be visualized by considering todays basic rotary balance windtunnel testing techniques. From Figure 2.1, it can be seen that

= 0 ( = ) (2.36)

= (2.37)

And taking the time rate of change,

= 0 ( = ) (2.38)

= (2.39)

These conditions remove two degrees of freedom, complete dynamics can be described by a4DOF model. Namely,

V =1

m

cos (Tcos cos Tsin sin ) D

+ g1w (2.40)

=1

mV

sin (Tcos cos T sin sin ) Q

+

g2wV

(2.41)

= 2 sin cos2

c1 + c5 + c8

2sin2 + (c2 c6)cos2

+ c7Msin + (c3L + c4N)cos cos + (c4L + c9N)sin cos (2.42)

= sin

c8 cos

2

2c2 sin cos c1 sin2

+1

cos

(c4L + c9N)cos (c3L + c4N)sin

(2.43)

Two algebraic relations come from = 0 (or = ) and = . The following expressionscan be used to solve for the forces and moments necessary for the rotation vector to line up


42/99


with the velocity vector. Those expressions are given by,

sin

c8 cos2

2c2 sin cos c1 sin2

+

1

cos

(c4L + c9N)cos (c3L + c4N)sin =

1

mV cos (Tcos sin T sin cos L) sin + q+ g3w

V cos (2.44)

and

cos3c5

2sin2 c6 cos2

+

c7Mcos

sin2cos

c1 + c8

2sin2+ c2 cos2

cos sin c3L + c4N

sin sin

c4L + c9N= 1

mV(sin (Tcos cos Tsin sin ) Q) + g2w

V(2.45)

with the gravity components taking the form,

g1wg2wg3w

=

g( cos cos sin + sin sin cos + sin cos cos cos )g(cos sin sin + cos sin cos sin sin cos cos )

g(sin sin + cos cos cos )

(2.46)

Finally the kinematics can take the form,

=


0 sin sec cos sec

cos cos sin

sin cos

(2.47)

The two algebraic expressions are not very pleasing to the eye, yet they do include thenecessary conditions for the proposed motion. Further assumptions can greatly simplify theresults just presented. For instance, assuming thrust acts along the xb-axis making zero,and assuming principle moments of inertia are two assumptions that would greatly reducethe complexity of the problem.

The next chapter will introduce a new motion that will require applying assumptions andconstraints to the rotational equations of motion, Equations 2.26-2.28 and 2.32-2.34, similarto those above. Analysis will then be conducted on the reduced set of equations and will lendinsight into the motion by concentrating on how the instantaneous rotation vector interactswith the instaneous velocity vector.


43/99

Chapter 3

The Falling Leaf

In recent years, the aircraft flight envelope has developed beyond what would have beenimaginable in the time directly following the Wright brothers. Linear behaving aircraft hasnow become a subset, although still a very important one, to the global nonlinear char-acteristics exhibited in flight. Analysis techniques have been formulated to capture globaldynamics and behavior of not only aircraft Ref [4, 8], but spacecraft Ref [1], and a varietyof other vehicles .

Recently, analysis of a particular motion known as the Falling Leaf has been undertaken inRef [3]. This motion is a prime candidate for the equations of motion proposed in Chapter 2as will become evident in this chapter. Application of the methods proposed earlier will be

applied and analysis will be undertaken with a new set of variables in order to lend insightto the dynamic motion known as the Falling Leaf

3.1 The Motion Defined

The Falling Leaf is characterized by a large coupled longitudinal/lateral-directional out-of-control oscillation. Angles of attack and sideslip can easily reach values of 70 degrees. Rolland yaw rates are in-phase while pitch rate is nearly zero. The motion is sustained for aslong as 60 seconds, and recovery has not been guaranteed. Pilot input does not significantly

affect the motion (once it has started). Understanding of this motion is incomplete, anddesign criteria is non-existent until recent guidelines introduced in Ref [3]. This motion ishighly disorienting and it may by difficult for the pilot to identify that the aircraft has evenentered into the motion. Tactical effectiveness is compromised and loss of aircraft has beenattributed to this motion.

The identical assumptions from Ref [3] will be applied to the rotational equations of mo-tion, Equations 2.26-2.28 and 2.32-2.34, and a dynamicly identical model will be developed.

32


44/99

Daniel C. Lluch Chapter 3. The Falling Leaf 33

Analysis of the reduced set of equations will shed light on some issues that were unansweredin the previous work.

3.2 Model Development

The identical assumptions from Ref [3] will be applied to the rotational equations of motion.Table 3.1 lists the assumptions made by Ref [3] and translates them to the new set ofvariables.

Therefore, Equations 2.26-2.28 and Equations 2.32-2.34 take the form

=L

Ixxcos +

N

Izzsin (3.1)

0 =N

Izzcos L

Ixxsin (3.2)

0 =

2c5 sin2 +

c7M

(3.3)

V =

Y + mg2bsin

m(3.4)

= sin +

Y + mg2bcos

mV(3.5)

= cos tan (3.6)

The differential equations of interest are then

V =

Y + mg2bsin

m(3.7)

= sin +

Y + mg2bcos

mV(3.8)

= cos tan (3.9) =

L

Ixx cos (3.10)

with the following algebraic relations that must hold,

N

Izzcos =

L

Ixxsin (3.11)

M

Iyy=

2

2c5 sin2 (3.12)

The gravity term will be addressed in the next section along with the kinematics of thereduced model.


45/99


Table 3.1: Assumptions for Reduced Model

Old System New System

xT = (V , ,,p ,q ,r) xT = (V , , , , , )

zero net force in the xb and zb directions.

From Equation 1.66

FxbFybFzb

=

0Y + mg2b

0

translating to wind axis system and apply-ing these wind axis forces to the rotationalsystem

FxwFywFzw

= Twb

FxbFybFzb

=

(Y + mg2b)sin (Y + mg2b )cos

0

Pitch rate fixed at zero

q= q = 0

Equivalently the instantaneous rotationvector is constrained within the plane ofsymmetry

= = 0

In phase roll and yaw rates,

r = kp

Fixes angle of rotation vector with respectto body axis

= tan1(k)

Thrust acts along xb-axis, = 0

Principle moments of inertia, Ixz = 0 SAME


46/99


Equations 3.7-3.10 constitute the force and moment equations for a 4 degree of freedommodel identical to that of Ref [3]. The only difference will come from how the kinematicsare dealt with and how the forces and moments are represented.

3.2.1 Constants of the Motion

Three constants of the motion exist given the assumptions made in Table 3.1. These con-stants will be derived and explained. Finally, the constants will allow the problem to bereduced further, yet have the same dynamics of the higher order model. In addition to re-duction of the model, these constants will lend insight and understanding in to the conditionsand drivers of the Falling Leaf.

Force Constant

Equations 3.7-3.9 will be referred to as the force equations. The force equations can then becombined in the following manner. Rewriting Equation 3.8,

= sin cos tan

+ Vcos

V sin (3.13)

multiplying both sides by tan and moving all terms to the left hand side

tan + tan

VV = 0 (3.14)

This equation can be integrated and results in a constant of the motion.

lncos lncos ln V = ln 1K

(3.15)

ln(V cos cos ) = ln K (3.16)

taking the exponential of each side

V cos cos = K (3.17)

To get a physical understanding of what this constant of the motion means, it is useful tointroduce a new variable. Figure 3.1 shows that is the angle between the instantaneousrotation vector and the instantaneous velocity vector. The relationship between and thestate variables is as follows,

cos = cos cos (3.18)


47/99


x-body

y-body

z-body

V

b/h

Figure 3.1: Define


48/99


The above relationship is only valid when the rotation vector is constrained within the planeof symmetry of the aircraft. That is when in Figure 2.1 is zero, or equivalently, body axispitch rate q is zero.

Using the new variable , Equation 3.17 can be written as,

V cos = K (3.19)

which states that the component of velocity along the instantaneous rotation vector is aconstant. The affect of this constant on the motion of the proposed model will be analyzed.The existence of this constant can be directly linked to the first assumption in Table 3.1,which states there is zero net force in the xb and zb directions.

Kinematic Constant

As stated in Sec 1.1.6, the kinematics are not subject to change or interpretation. Givenangular rates the kinematic angles must evolve in a certain way. Yet again, a intelligentcoordinate system choice to relate to inertial space can greatly simplify the problem andlend understanding to the mechanics of the motion. Another two constants are readilyidentifiable by using the kinematics derived in Sec 2.3. The assumptions made in Table 3.1fix the rotational reference frame with respect to the body-fixed reference frame. Particularlyfor this case, = 0 and = tan1 k. With these angles being constant, Equation 2.20 reducesto

prqr

rr

=

0

0

(3.20)

Therefore, Equation 2.16 simply becomes,

=

00

(3.21)

This result is quite interesting. It states that the pitch angle and heading angle of therotational coordinate system are constant. In inertial space, the rotational coordinate systemsolely has a roll angle that changes in time. It is also important to note that this result willapply for any instance that the instantaneous rotation vector is fixed in direction (magnitudemay vary) with respect to the body-fixed axis system. Namely, when r/b in Equation 2.17is zero.

The next step is to relate the kinematic angles [, , ] with the conventional Euler angles[,,]. This relation can be shown in general by using Equation 2.7. For the reduced model


49/99


given by Table 3.1, the two sets of angles are related as follows with and being twoconstants of the motion.

tan = sin cos sin sin + cos cos cos

(3.22)

sin = cos sin sin cos cos (3.23)tan =

cos cos sin + sin (cos sin sin sin cos )cos cos cos + sin (cos sin cos + sin sin )

(3.24)

These angles come into play when describing the representation of gravity in the force equa-tions using Equation 2.14. The effect of the above constants will be analyzed. For now theseconstants will allow the dynamics of the reduced model to be completely expressed as shownin the following section.

3.2.2 Reduced Model: Equations Collected

Force Equations

= sin +

Y + mg sincoscos2cos

Km(3.25)

= cos tan (3.9)

Moment Equations

=L

Ixx cos (3.26)

Kinematics

= (3.27)

Similar to the Euler angles, the heading angle of the rotational coordinate system doesnot appear on the right hand side of the equations, thus that constant of the motion is notof interest. The other constants of the motion are given by initial condition and is givenby the relationship in Table 3.1.


50/99


3.2.3 The Forces and Moments

The equations of motion shown above still incorporate general forces and moments. Bodyaxis side force Y and rolling moment L are the only forces and moments that appear. Therepresentation of forces and moments will be addressed in general and then applied to theparticular problem posed above.

As a subset of the work presented here, aerodynamic forces and moments were consideredat high angles of attack and sideslip. Data was gathered from Ref [13] and the F-18Caerodynamic database to construct a representative curve for each of the force and momentcoefficients. An in depth review of the work will not be presented but the final product isshown in Appendix A. Static wind tunnel testing has included angle of attack test pointsup to 90. The sideslip region has not been as extreme, commonly being 15 or 20 inmost aerodynamic databases with often some of the forces and moments, such as the liftcoefficient, not having a functional dependence on sideslip because of the assumption thatthe aircraft will not experience large sideslip excursions. Ref [13] contains data for sideslipsweeps of 90 which served as a basis for construction of the curves in Appendix A inthe large sideslip regions. The data gathered showed the trends of the force and momentcoefficients that occurs throughout the large operating region consisting of 0 < < 90 and 90. For example, Cn loses stability at high angles of attack, and local Cl (not secantderivative) becomes positive at large values of sideslip possibly resulting in positive rollingmoment (secant derivative also loses stability) at extreme values of sideslip.

It is convenient to represent the aerodynamics as simply as possible yet capture the generalbehavior across the operating region. The body axis side force and rolling moment are linearin the model used by Ref [3]. Keeping in mind the results from Appendix A, it is a goodfirst approximation on a global scale to have body-axis side force as linear in . A linearrolling moment in is acceptable in the small sideslip region yet if trying to model on aglobal scale it is a poor choice. To a first approximation the rolling moment L is chosen tobe the negative of a sine function in . This representation allows for Cl to change sign as increases as well as have a positive rolling moment at large sideslip conditions.

Therefore the body-axis side force and the rolling moment coefficients are written respectivelyas,

CY = CY (3.28)

Cl = Clmax sin

ref

(3.29)

Where CY is the conventional stability derivative indicating the slope of the CY vs curve,Clmax is the maximum value of Cl as indicated by Figure 3.2, and ref = |Cl=0|=0, which isthe value of (non-zero) where rolling moment is zero, also indicated in Figure 3.2.

Conventional dimensionalization for the force and moment coefficients is used, with q being


51/99


Cl

ref

ref

lmax

lmaxC

-C

Figure 3.2: Cl vs


52/99


the dynamic pressure, 12

V2,

Y = CY qS (3.30)

L = Cl qSb (3.31)

Using the coefficients defined above and Equation 3.17 to substitute for aircraft velocity V,the side force and rolling moment can be written as,

Y = C1K2

cos2cos2(3.32)

L = C2K2 sin

ref

cos2cos2(3.33)

with,

C1 =CY S

2(3.34)

C2 =ClmaxSb

2(3.35)

Using this representation for the side force and rolling moment, Equations 3.25 and 3.26 canbe written as,

= sin +C1K

m cos +

g sincoscos2cos

K(3.25)

=C2K2

Ixx cos sin

ref

cos2cos2(3.26)

3.3 Model Validation and Simulation

The assumptions made in Table 3.1 are consistent with real observed motions exhibited byaircraft as shown in Ref [3]. Onset of the motion has been attributed to unobserved sideslipbuildup that is commonly seen in low speed, high bank angle turns. Representative initialconditions will be used and translated into the new set of variables. The constants of the

motion associated with the reduced system will be calculated. The reduced system willthen be simulated and the results presented. Real flight data can be seen in Ref [3], and isreprinted with permission in Figure 3.3.

Typical values of a fighter-type aircraft for the parameters that appear in the equations ofmotion are shown in Table 3.2. Representative initial conditions will be drawn from Fig 3.3and are summarized as follows,

{V,,,,p,r,}0 = {245 ft/s, 25, 0, 35, 15/s, 5/s, 60} (3.36)


53/99


Figure 3.3: Real Falling Leaf Data, F-18C

reprinted from Ref [3]

Table 3.2: Model Parameters

S = 400 ft2 b = 37 ft

= .001648 slugs/ft3 Ixx = 24290 slugs ft2m = 1120 slugs CY = .95

Clmax = .06 ref = 60

k = .36 = tan1(k)


54/99


The velocity time history shows the probe losing effectiveness at high angles of attack. Initialvelocity was then taken to be a low speed condition consistent with the first reading of theprobe. These initial conditions can be translated to the new set of variables in the followingmanner. Using Equation 2.13,

0 = 0 = 5 (3.37)

Using Equation 2.12,

0 = sign(p0)

p20 + r20 = 16/s (3.38)

Using Equation 3.22,

0 = tan1

sin 0 cos 0sin sin 0 + cos cos 0 cos 0

= 62 (3.39)

Using Equation 3.17, one of the constants of the motion is

K = V0 cos 0 cos 0 = 200 ft/s (3.40)

Using Equation 3.23, another constant of the motion is

= sin1 (cos sin 0 sin cos 0 cos 0) = 10 (3.41)

Therefore the initial condition for the new set of variables is

{,, , } = {35, 5, 16/s, 62} (3.42)

With the constants of the motion being K = 200 ft/s and = 10.The system consisting of Equations 3.9, 3.27, 3.25, and 3.26 can now be simulated. Theresults for these initial conditions are shown in Figures 3.4 and 3.5. In comparison to the realflight data in Figure 3.3, variables match up reasonably well. The best being the body-axisrates, reaching peaks of approximately 150/s and 50/s for p and r, respectively. Steadystate velocity for the actual data was approximately 400 ft/s compared to 350 ft/s for thesimulated data. The angle of attack and sideslip angle were estimated for the real data andwere not available for the full duration of the motion. Yet that data matches very well inmagnitude and shape of the umbrella curve ofvs that has now become a standard graphin the analysis of the Falling Leaf. The model proposed captures very well the curling overof beta that can be seen in the estimated vs plot. The curling feature can be directlyrelated to the fact that the rolling moment was represented as a sine curve as opposed tolinear for this particular case. The frequency of the motion is a close match as well, with aperiod of approximately 6.1 seconds.


55/99


0 10 20 30 40200

300

400

Vel,ft/s

0 10 20 30 40100

0

100

beta,

deg

0 10 20 30 40

0

50

100

alpha,

deg

0 10 20 30 40200

0

200

p,r

deg/s

0 10 20 30 4010

0

10

20

theta,

deg

time, s0 10 20 30 40

100

0

100

phi,deg

time, s

60 40 20 0 20 40 6010

0

10

20

30

40

50

60

70

80

beta, deg

alpha,

deg

Figure 3.4: Simulation of Falling Leaf represented in {V,,,,p,r,}


56/99


0 10 20 30 40100

50

0

50

100

beta,

deg

0 10 20 30 4040

20

0

20

40

60

80

tau,

deg

0 10 20 30 40200

100

0

100

200

omegadeg/s

time, s0 10 20 30 40

150

100

50

0

50

100

150

PHI,deg

time, s

60 40 20 0 20 40 6030

20

10

0

10

20

30

40

50

60

beta, deg

tau,

deg

Figure 3.5: Simulation of Falling Leaf represented in {,, , }


57/99


Concentrating on the vs and the vs plots, it can be seen that the shapes are identical.Furthermore, from Equation 2.13, and differ by the constant . In order to understandthe full importance of the use of in place of, focus is placed on the velocity trace. As canbe seen velocity seems to be approaching a steady-state value. Equation 3.19 then statesthat is approaching a constant value, defined to be ref. Recall from Figure 3.1, is theangle between the instantaneous velocity vector and the instantaneous rotation vector. Theright hand side of Equation 3.18 is then approaching cos ref. Equation 3.18 also shows thatat = 0, = ref and at = 0, = ref. This result is quite important and will beaddressed in the next chapter.

3.4 Summary

Due to the model structure, a near steady state limit cycle is converged upon with anyinitial disturbance in the lateral-directional variables. An understanding of the drivers andparameters that affect amplitude and frequency is needed. As shown in the previous sec-tion, the proposed model compares quite well with the known motion. Furthermore, theinitial simplification and insight that the new rotational coordinate system has provided isnoteworthy and will receive an in-depth analysis in the next chapter.

The assumptions under which the proposed model, Equations 3.9, 3.27, 3.25, and 3.26, isvalid will be reviewed.

1. Zero net force in the xb and zb directions. From Equation 1.66, this assumption statesthat the sum of the forces due to thrust, aerodynamics, and gravity offset each otherat all times. This assumption directly lends to the existence of the constant of themotion K as shown in Section 3.2.1.

2. Pitch rate is fixed at zero. Equivalently from Figure 2.1, is zero. This assumptionis the removal of a degree-of-freedom, constraining the instantaneous rotation vectorto be in the plane of symmetry of the aircraft and allows Equation 3.18 to be validat all times. An algebraic expression exists that must be satisfied in order for thisdegree-of-freedom to be removed, namely Equation 2.28 reduces to,

c7M = 2 c6 cos2

c5

2

sin2 (3.43)The pitching moment must satisfy Equation 3.43 if the pitch rate q is fixed at zero.

3. The ratio of the body-axis roll and yaw rates is a constant. This assumption is aremoval of another degree-of-freedom, fixing the angle in Figure 2.1 to be the constanttan1(k). Another algebraic equation must be satisfied in order for this degree-of-freedom to be removed, namely Equation 2.27 reduces to,

L (c3 sin c4 cos ) = N(c9 cos c4 sin ) (3.44)


58/99


Although only the in-phase case will be considered, when k is positive, this assumptionalso does allow for a strictly out-of-phase condition to be set, when k is negative. Withthis assumption and assumption #2, the rotational coordinate system is now fixedwith respect to the body-fixed coordinate system. With these two coordinate systemsfixed with respect to each other the kinematic constants of the motion exist as shownin Section 3.2.1.

4. Principal moments of inertia, Ixz = 0. This assumption is quite common in the analysisof aircraft dynamics. Inherent in this assumption is that the moments are known aboutthe principle axes. This representation of the moments is not usually the case, yet theassumption is valid when the rates experienced are small and Ixz is small relative tothe other terms in the inertia tensor. For the particular motion of the Falling Leaf,rates can reach substantial values. The q equation of Equation 1.91 was considered at

rates typical of the Falling Leaf. It was found that inclusion of a typical Ixz value canchange the sign necessary on the pitching moment M to keep q = 0 (see assumption#2). This fact must be kept in mind in validating the second assumption on this list.Using the expression shown in assumption #2, it can be easily shown that for typicalvalues c5 = .06869 and c6 = .02471, the critical value is = 17.8. (The criticalvalue for is the angle necessary to rotate the inertia tensor to principle axis, seeSec 1.1.5). This value is directly in the range for given by assumption #3. It wouldnot be possible to keep q = 0 without the inclusion of the Ixz term. For the presentwork this assumption is made from the observation of the very fact that the pitch rateq is very close to zero throughout the motion of the Falling Leaf.

5. The angle in defining the direction of thrust is zero. This assumption has no bearingon the results presented since assumption #1 removes thrust from the problem. Inher-ent in the definition of how thrust is oriented with respect to the body-fixed axis systemwas that it was constrained in the plane of symmetry of the aircraft, see Section 1.2.1.Relaxing this constraint would allow for a control to enter into the picture through theFyb-term in Equation 1.66, which in turn can possibly lead to recovery techniques byusing the thrust as a control. This possibility is more attractive for thrust vectoredaircraft its analysis is beyond the scope of this work.

6. The representation of the forces and moments are derived in Section 3.2.3. The onlyforces and moments that appear are body-axis side force and rolling moment. It

is shown to be good approximation to have side force represented as linear and therolling moment represented as the sine function shown in Figure 3.2.


59/99

Chapter 4

Analysis of the Falling Leaf

This chapter will perform an in depth analysis of the Falling Leaf motion in order to obtainan understanding of the drivers that control the characteristics of the motion. The charac-teristics of interest are amplitude, frequency, and the shape of the motion. The model thatwas developed in Chapter 3 will be used for the analysis. Recall that the state vector is nowcomprised of 4 variables, namely { , , , }. The equations are reprinted here for ease ofreference.

= sin + Y + mg sincoscos2cos

Km(4.1)

= cos tan (4.2) =

L

Ixx cos (4.3)

= (4.4)

The body-axis side force and the rolling moment in the above equations are represented asfollows,

Y = C1

K2

cos2cos2 C1 =

CYS

2 (4.5)

L = C2K2 sin

ref

cos2cos2C2 =

ClmaxSb2

(4.6)

Refer to

falling leaf thesis 2009

Documents