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Simulation of Flexible Aircraft

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Simulation of Flexible Aircraft
by
Humayoon Abbasi
A thesis submitted in conformity with the requirements
for the degree of Master of Applied Science
Graduate Department of Aerospace Science and Engineering
University of Toronto
c 2010 by Humayoon Abbasi
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Abstract
Simulation of Flexible Aircraft
Humayoon Abbasi
Master of Applied Science
Graduate Department of Aerospace Science and Engineering
University of Toronto
2010
This study aims to improve flight simulation of flexible aircraft. More specifically, this
thesis concentrates on comparing two flexible aircraft flight simulation models. Both
modeling techniques considered use the same aircraft structural and aerodynamic data
provided by the aircraft manufacturer. Simulation models were developed and tested
using a number of control inputs in both longitudinal and lateral dimensions. Time
history responses from the simulations were compared. The effect of increasing the
flexibility of the aircraft model was also studied on both models. It was found that the
two models produce very similar results for the original aircraft stiffness case. However,
the lateral response of the two models diverges as the stiffness is lowered. A number of
recommendations are made for further testing and research, based on the conclusions of
the study.
ii
Acknowledgements
First and foremost, I would like to acknowledge my thesis supervisor, Dr. Peter R. Grant,
without whose assistance none of this work would have been possible. I am very grateful
for his supervision throughout my research and study as well as for his patience and
understanding.
I would also like to express my thanks to Nestor X. Li for all his assistance in developing and troubleshooting the simulations as well as responding to my numerous requests
in a timely fashion. I would also like to thank my friends M. Umer Ahmed, Terrence
Fung, Andrew Sun, Amir Naseri, Steacie Liu, Bruce Haycock and others who made my
time spent at UTIAS pleasant and enjoyable.
Additionally, I would like to acknowledge the financial support of National Science
and Engineering Research Council with their Canada Graduate Scholarship which made
my graduate life much more comfortable than it would have been otherwise.
Last but not least, I would like to thank my family for their continuous support and
patience during my time of study at University of Toronto. As well, I am very grateful
to my wife, Isra, for her contribution in prolonging my Masters. I dedicate this to you.
Humayoon Abbasi
iii
Contents
1 Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 Background
3
2.1
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Background Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2.1
Aerodynamic Loads . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2.2
Natural Vibration Modes of the Aircraft . . . . . . . . . . . . . .
6
3 Methodology
9
4 Mean Axes Model
10
5 Fixed Axes Model
18
6 Results and Discussion
25
6.1
6.2
Comparison of Mean and Fixed Axes Models . . . . . . . . . . . . . . . .
25
6.1.1
Response to Elevator Impulse . . . . . . . . . . . . . . . . . . . .
25
6.1.2
Response to Elevator 2311 . . . . . . . . . . . . . . . . . . . . . .
30
6.1.3
Response to an Aileron Impulse . . . . . . . . . . . . . . . . . . .
34
6.1.4
Response to Aileron 2311
. . . . . . . . . . . . . . . . . . . . . .
39
. . . . . . . . . . . . . . . . . . . . .
42
6.2.1
Response to Elevator Impulse . . . . . . . . . . . . . . . . . . . .
42
6.2.2
Response to Elevator 2311 . . . . . . . . . . . . . . . . . . . . . .
53
6.2.3
Response to Aileron Impulse . . . . . . . . . . . . . . . . . . . . .
60
6.2.4
Response to Aileron 2311
70
Effect of Varying Aircraft Stiffness
. . . . . . . . . . . . . . . . . . . . . .
iv
7 Conclusions and Recommendations
78
References
80
v
List of Figures
4.1
Inertial position of a mass element
. . . . . . . . . . . . . . . . . . . . .
11
5.1
Component reference frames . . . . . . . . . . . . . . . . . . . . . . . . .
19
6.1
Pitch rate time history of aircraft response to elevator impulse . . . . . .
26
6.2
A closer look at pitch rate time history just after the input . . . . . . . .
27
6.3
Pitch angle time history of aircraft response to elevator impulse . . . . .
27
6.4
X velocity time history of aircraft response to elevator impulse . . . . . .
28
6.5
Acceleration time history of aircraft response to elevator impulse . . . . .
28
6.6
Acceleration time history focusing on the response immediately after elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.7
29
Wingtip relative deflection time history of aircraft response to elevator
impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
6.8
Pitch rate time history of aircraft response to elevator 2311 . . . . . . . .
30
6.9
Pitch angle time history of aircraft response to elevator 2311 . . . . . . .
31
6.10 X velocity time history of aircraft response to elevator 2311 . . . . . . . .
32
6.11 Acceleration time history of aircraft response to elevator 2311 . . . . . .
32
6.12 Wingtip relative deflection time history of aircraft response to elevator 2311 33
6.13 Roll rate time history of aircraft response to aileron impulse . . . . . . .
34
6.14 A closer look at the roll rate time history immediately proceeding the input 35
6.15 Roll angle time history of aircraft response to aileron impulse . . . . . . .
36
6.16 Y velocity time history of aircraft response to aileron impulse
. . . . . .
36
6.17 Acceleration time history of aircraft response to aileron impulse . . . . .
37
6.18 Focusing on acceleration time history response just after the input . . . .
37
6.19 Wingtip relative deflection time history of aircraft response to aileron impulse 38
6.20 Roll rate time history of aircraft response to aileron 2311 . . . . . . . . .
39
6.21 Roll angle time history of aircraft response to aileron 2311 . . . . . . . .
40
vi
6.22 Y velocity time history of aircraft response to aileron 2311 . . . . . . . .
40
6.23 Acceleration time history of aircraft response to aileron 2311 . . . . . . .
41
6.24 Wingtip relative deflection time history of aircraft response to aileron 2311 41
6.25 Pitch rate time history of mean axes model’s response to elevator impulse
43
6.26 Pitch rate time history of fixed axes model’s response to elevator impulse
44
6.27 A closer look at the pitch rate time history of mean axes model immediately after the elevator impulse . . . . . . . . . . . . . . . . . . . . . . .
45
6.28 A closer look at the pitch rate time history of fixed axes model immediately
proceeding the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
6.29 Pitch angle time history of mean axes model’s response to elevator impulse 46
6.30 Pitch angle time history of fixed axes model’s response to elevator impulse 46
6.31 X velocity time history of mean axes model’s response to elevator impulse
47
6.32 X velocity time history of fixed axes model’s response to elevator impulse
47
6.33 X acceleration time history of mean axes model’s response to elevator
impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
6.34 X acceleration time history of fixed axes model’s response to elevator impulse 48
6.35 X acceleration time history of mean axes model: focusing on the response
immediately proceeding the input . . . . . . . . . . . . . . . . . . . . . .
49
6.36 X acceleration time history of fixed axes model immediately proceeding
the elevator impulse input . . . . . . . . . . . . . . . . . . . . . . . . . .
49
6.37 Z acceleration time history of mean axes model’s response to elevator impulse 50
6.38 Z acceleration time history of fixed axes model’s response to elevator impulse 50
6.39 Focusing on the Z acceleration time history of mean axes model immediately after the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
6.40 Focusing on the Z acceleration time history of fixed axes model immediately after the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
6.41 Wingtip relative deflection time history of mean axes model’s response to
elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
6.42 Wingtip relative deflection time history of fixed axes model’s response to
elevator impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
6.43 Pitch rate time history of mean axes model’s response to elevator 2311 .
53
6.44 Pitch rate time history of fixed axes model’s response to elevator 2311 . .
54
6.45 Pitch angle time history of mean axes model’s response to elevator 2311 .
55
6.46 Pitch angle time history of fixed axes model’s response to elevator 2311 .
55
vii
6.47 X velocity time history of mean axes model’s response to elevator 2311 .
56
6.48 X velocity time history of fixed axes model’s response to elevator 2311 . .
56
6.49 X acceleration time history of mean axes model’s response to elevator 2311 57
6.50 X acceleration time history of fixed axes model’s response to elevator 2311 57
6.51 Z acceleration time history of mean axes model’s response to elevator 2311 58
6.52 Z acceleration time history of fixed axes model’s response to elevator 2311
58
6.53 Wingtip relative deflection time history of mean axes model’s response to
elevator 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
6.54 Wingtip relative deflection time history of fixed axes model’s response to
elevator 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
6.55 Roll rate time history of mean axes model’s response to aileron impulse .
60
6.56 Roll rate time history of fixed axes model’s response to aileron impulse .
61
6.57 Fousing on roll rate time history of mean axes model just after the aileron
impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
6.58 Focusing on roll rate time history of fixed axes model just after the aileron
impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
6.59 Roll angle time history of mean axes model’s response to aileron impulse
63
6.60 Roll angle time history of fixed axes model’s response to aileron impulse .
63
6.61 Y velocity time history of mean axes model’s response to aileron impulse
64
6.62 Y velocity time history of fixed axes model’s response to aileron impulse
64
6.63 X acceleration time history of mean axes model’s response to aileron impulse 65
6.64 X acceleration time history of fixed axes model’s response to aileron impulse 65
6.65 Focusing on X acceleration time history of mean axes model proceeding
the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6.66 Focusing on X acceleration time history of fixed axes model proceeding
the input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
6.67 Y acceleration time history of mean axes model’s response to aileron impulse 67
6.68 Y acceleration time history of fixed axes model’s response to aileron impulse 67
6.69 Focusing on Y acceleration time history of mean axes model immediately
after aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6.70 Focusing on Y acceleration time history of fixed axes model immediately
after aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
6.71 Wingtip relative deflection time history of mean axes model’s response to
aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
69
6.72 Wingtip relative deflection time history of fixed axes model’s response to
aileron impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
6.73 Roll rate time history of mean axes model’s response to aileron 2311 . . .
70
6.74 Roll rate time history of fixed axes model’s response to aileron 2311 . . .
71
6.75 Roll angle time history of mean axes model’s response to aileron 2311 . .
72
6.76 Roll angle time history of fixed axes model’s response to aileron 2311 . .
72
6.77 Y velocity time history of mean axes model’s response to aileron 2311 . .
73
6.78 Y velocity time history of fixed axes model’s response to aileron 2311 . .
73
6.79 X acceleration time history of mean axes model’s response to aileron 2311
74
6.80 X acceleration time history of fixed axes model’s response to aileron 2311
74
6.81 Y acceleration time history of mean axes model’s response to aileron 2311
75
6.82 Y acceleration time history of fixed axes model’s response to aileron 2311
75
6.83 Wingtip relative deflection time history of mean axes model’s response to
aileron 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
6.84 Wingtip relative deflection time history of fixed axes model’s response to
aileron 2311 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
76
Nomenclature
In order of appearance, grouped by section:
c
chord length
CLα
lift curve slope
q
dynamic pressure
α
angle of attack
ρ
air density
Vx
longitudinal body frame component of velocity
Vz
vertical body frame component of velocity
CD0
drag coefficient corresponding to zero angle of attack
k
Oswald efficiency factor
cs
lateral chord length
Csβ
slope of the lateral force curve
qs
dynamic pressure of the lateral force
β
angle of sideslip
Vy
lateral body frame component of velocity
x
L
the Lagrangian
T
kinetic energy of system
U
potential energy of system
W
external work done on system
(•)(e)
signifies application to an element
ψ
displacement of a finite element
n
vector of element shape functions
d
vector of the nodal displacements of the elements
ρ
density of element
f
body force vector
m
mass
k
stiffness
B
matrix consisting of shape function derivatives
D
constitutive matrix relating element stress to strain
σ
element stress
ε
element strain
M
global mass matrix
K
global stiffness matrix
F(t)
vector of external forces, possibly time varying
ωi2
eigenvalues representing the natural frequencies
φi
eigenvectors representing the natural mode shapes
xi
ρdV
mass element with volume dV
R̄
inertial position of an element
p̄
position of the element relative to a local reference system
R̄0
position of the local reference system relative to the inertial frame
ω̄
angular velocity of local reference frame relative to inertial reference frame
Ug
gravitational potential energy
ḡ
acceleration due to gravity
s̄ (x, y, z)
d¯(x, y, z, t)
undeformed position of mass element
Ue
d¯
deformed position of a mass element
strain energy
displacement
δ d¯
δt
rate of displacement
x̄cm
position of the centre of mass relative to local reference frame
M
total mass of elastic body
[I]
inertia tensor of elastic body
φi (x, y, z)
mode shapes
ηi (t)
generalized coordinates
Mi
generalized mass of the i th mode
ωi
natural vibration frequency of the i th mode
î, ĵ and k̂
unit vectors in each of the coordinate directions of the local reference
frame
φ, θ and ψ
Euler angles commonly used in rigid body aircraft equations of motion
Qi
generalized force
δW
virtual work
Qηi
generalized forces for the elastic equations
αv
angle of attack of the aircraft
φi
b
elastic displacement of a section due to bending
mac
aerodynamic moment per unit span about the line of aerodynamic centres
αs
local angle of attack of an aerodynamic section
xii
Cf
transformation matrix
Ef
time rate of change of the transformation matrix
V̄f
velocity of a point on the fuselage
Vf
translational velocity vector for reference frame xf yf zf
ωf
angular velocity vector for reference frame xf yf zf
rf
position vector associated with the mass element dmf
uf
elastic displacement vector
vf
bending velocity
αf
torsion velocity
r̃f
skew-symmetric matrix corresponding to rf
ũf
skew-symmetric matrix corresponding to uf
(•)i
component type, (•)f for fuselage, (•)w for wing and (•)e for empennage
rf i
radius vector from xf yf zf to xi yi zi
L̂i
Lagrangian density for component i, not including the strain energy
F̂ui
Rayleigh’s dissipation function density for component i
F̂ψi
Rayleigh’s dissipation function density for component i
Lui
stiffness differential operator matrix for component i
Lψi
stiffness differential operator matrix for component i
F
net force vector acting on the aircraft in terms of fuselage body axes
M
net moment vector acting on the aircraft in terms of fuselage body axes
Ûi
net force density vector for component i
Ψ̂i
net moment density vector for component i
S̃
matrix of first moments of inertia of the deformed aircraft
J
matrix of second moments of inertia of the deformed aircraft, simply
referred to as the inertia matrix
Cui and Cψi
damping matrices
p
momentum vector for the entire aircraft
Kui
bending stiffness matrix
Kψi
torsional stiffness matrix
fi
net distributed forces over the component i
δ
spatial Dirac delta function
rE
position vector of engine
FE δ (r − rE )
engine thrust
xiii
Chapter 1
Introduction
Traditionally, aircraft simulation models assume the aircraft structure to be rigid and
thus the aircraft is modeled as a rigid-body. This allows the aircraft equations of motion
to be simplified greatly. Most of the flight simulation models make use of this assumption
and the results are acceptable for a majority of aircraft present today.
1.1
Motivation
With advances in material sciences, aircraft manufacturers are increasingly making use
of lighter materials to save weight and as a result cut down on fuel costs and improve
performance. One disadvantage of using lighter materials is an increase in the flexibility
of the aircraft structures. The resulting aircraft have structures that are considerably
more deformable even when encountering forces within their design envelopes and thus
cannot be treated as rigid-bodies for flight simulation purposes.
As a result of these developments, accurate modeling tools for modeling the dynamic
response of flexible aircraft have become ever more important. Additionally, to allow
pilot-in-the-loop simulations, an added constraint is placed on these techniques that they
have to have sufficiently low computational complexity to enable real-time simulations
using readily available hardware.
1.2
Objectives
The primary goal of this effort is to identify a modeling technique for simulating dynamic
flight response of a flexible aircraft. Additionally, the following sub-objectives are also
1
Chapter 1. Introduction
2
being pursued:
1. Studying the effect of varying the number of structural modes retained in the
simulation model. For the model to be exact, an infinite number of structural
dynamics modes is required. However, as this is impossible to achieve in practice,
only a small number of modes are retained as dictated by the accuracy required,
the frequency response for the flight task being flown, and the computing resources
available.
2. Studying the effect of varying the flexibility of the aircraft structure. As the aircraft
structure becomes increasingly flexible, the structural response becomes increasingly nonlinear and the overall system response departs farther away from that of
the rigid model.
Chapter 2
Background
2.1
Literature Review
Traditionally, the fields of flight dynamics and aeroelasticity have been developed as separate disciplines. Even though the need for integrating the two fields was recognized
long ago, not much work was done to fulfill this need in the early years due to a significant increase in the complexity of the resulting problem. Also, the lack of computing
resources available at the time required the use of simplifying assumptions which allowed
for analytical solutions to be found.
One of the very first integrated analytical models for the dynamics of flexible aircraft
is found in a two-part report by Milne [5]. In the first part, a linear flexible flight dynamics
model was derived. The model consisted of linear equations about a steady state and
assumed both rigid body motions as well as elastic deformations to be small. Mean
axes (refer to Ch. 4 for a description of mean axes) constraints were specified, however,
the formulation used body axes attached to the undeformed aircraft instead of mean
axes which can change as the aircraft deforms. In the second part, longitudinal stability
analysis was performed on the aircraft model. Taylor and Woodcock [8] simplified Milne’s
[5] formulation into scalar form and used it to study selected special cases.
Buttril, Zeiler and Arbuckle [1] developed a flight dynamics model for flexible aircraft by integrating nonlinear rigid body mechanics and linear structural dynamics using
Lagrangian mechanics. The model made use of mean axes constraints and undamped
natural vibration modes. The equations included nonlinear inertial coupling terms between angular and elastic momenta. The result of these coupling terms, however, was
shown to be small. A steady-state, nonlinear aerodynamic model, augmented by an un3
Chapter 2. Background
4
steady aerodynamic model was used to calculate the aerodynamic loads. Waszak and
Schmidt [10] used Langrange’s equations to derive the equations of motion for a flexible aircraft. The equations included nonlinear rigid-body motion equations, as well as
linear equations for elastic degrees of freedom which were expressed in terms of natural vibrational modes of the aircraft. Inertial decoupling of the rigid-body motions and
the elastic deformations was achieved by applying mean axes constraints. Strip theory
was used to compute the aerodynamic loads which were then converted to generalized
forces, ultimately coupling the rigid-body and elastic degrees of freedom. The resulting
equations were very similar in appearance to the Euler equations for rigid-body motion,
except for the augmented structural equations. Consequently, these equations have been
very appealing for use in real-time flight simulation of flexible aircraft. Waszak, Buttril
and Schmidt [9] presented a number of techniques to simplify flexible aircraft models,
such as residualization and truncation of modes.
Meirovitch and Tuzcu [4] have been critical of the use of mean axes in deriving the
equations of motion for flexible aircraft. As these equations tend to be quite complex,
earlier investigators looking to derive analytical models used the mean axes to help inertially decouple the system. However, Meirovitch and Tuzcu [4] argue that in a lot of
these investigations, the additional constraints imposed by the use of mean axes were
not fully satisfied. Additionally, if one were to satisfy these constraints, the resulting
model would be more complicated than what could be derived without the use of mean
axes. Meirovitch and Tuzcu [3] developed a unified formulation of the equations of motion in terms of quasi-coordinates. The model included hybrid (ordinary and partial)
differential equations of motion which were derived without the use of mean axes. Strip
theory was used to compute the aerodynamic forces; however, the need for a more accurate aerodynamic model was alluded to. The resulting state equations were functions
of quasi-velocities, as well as generalized coordinates, velocities, and forces. A numerical
example for a model of a flexible aircraft was presented and two flight dynamics problems
were studied: the steady level cruise and the steady level turn manoeuvre. A controller
consisting of a linear quadratic regulator was designed to alleviate gust loads and ensure
flight stability.
5
Chapter 2. Background
2.2
2.2.1
Background Theory
Aerodynamic Loads
Aerodynamic forces constitute a major part of the total external forces that act on an
aircraft. Therefore, in order to simulate the flight of an aircraft, it is required that the
aerodynamic loading be computed at each time step of the simulation. There are a number of aerodynamic theories available which vary in terms of accuracy and computation
costs. Usually the deciding factor in choosing one aerodynamic theory over another is
the computation resources required for implementing that theory.
Strip theory is an aerodynamic theory that provides a reasonable estimate of aerodynamic loading on an aircraft without being too resource-intensive. This is particularly
important for simulation of flexible aircraft as the structure of a flexible aircraft can deform which can alter the aerodynamic loads exerted on the aircraft. Thus, it is important
to have a quick way of estimating the aerodynamic loads on the deformed aircraft at each
time step of the simulation.
Strip theory regards all components of the aircraft as two-dimensional surfaces. For
horizontal surfaces, the lift force per unit span is given by
l = qcCLα α
(2.1)
where c is the chord length, CLα is the lift curve slope, q is the dynamic pressure and α
is the angle of attack. The dynamic pressure is given by
2
2
q = 21 ρ(V x + V z )
(2.2)
where ρ is the air density and V x and V z are the longitudinal and vertical components
of the velocity in the component body frame, respectively.
α = tan−1 (V z V x )
(2.3)
is the expression for computing the angle of attack.
In a similar fashion, the drag force per unit span is given by
2
d = qcCD = qc CD0 + kCL2 = qc CD0 + kCLα
α2
(2.4)
where CD0 is the drag coefficient corresponding to zero angle of attack and k is a constant
dependent on efficiency. For vertical surfaces, the side force per unit span can be written
as
s = qs cs Csβ β
(2.5)
6
Chapter 2. Background
where cs is the lateral chord length, Csβ is the slope of the lateral force curve, qs is the
dynamic pressure of the lateral force and β is the angle of sideslip. The lateral dynamic
pressure can be expressed as
2
2
qs = 12 ρ(V x + V y )
(2.6)
β = tan−1 (V y V x )
(2.7)
and the sideslip angle as
where V x and V y are the longitudinal and lateral components of velocity of the aircraft.
The aerodynamic forces on the aircraft components can be written in the vector form as


l sin α − d cos α



(2.8)
0
fa = 


−l cos α − d sin α
and the lateral aerodynamic force vector as

s sin β

fs = 
 −s cos β
0


.

(2.9)
Typically, the lift, drag and lateral force per unit span are applied at the line of aerodynamic centers. Thus, the above mentioned force vectors may be integrated to obtain
the net aerodynamic forces and moments, using the line of aerodynamic centers as the
domain of integration.
2.2.2
Natural Vibration Modes of the Aircraft
Developing the mean-axes flight dynamics model of a flexible aircraft requires knowledge
of the aircraft’s structural modes. Silveira [7] performed modal analysis on the structural
model of a small business jet aircraft in ANSYS, which will be used in this project. The
following is a summary of the theory behind the modal analysis.
The Extended Hamilton’s Principle can be used to express the motion of a system
over time as
Zt2
δ
Zt2
Ldt +
t1
Zt2
(T − U + W )dt = 0
δW dt = δ
t1
(2.10)
t1
where L = T − U is the Lagrangian and T , U and W are the kinetic energy, potential
energy and external work done on the system. Representing the structure by an equivalent finite element model, Equation (2.10) can be applied to each element of the model,
7
Chapter 2. Background
as
Zt2
δ
(T (e) − U (e) + W (e) )dt = 0
(2.11)
t1
where the superscript (e) signifies application to an element. Assuming the displacement
of each finite element, ψ, to be given by
ψ = nT d
(2.12)
where n is a vector of element’s shape functions and d is a vector of the nodal displacements of the elements, then expressions can be written for the kinetic and potential
energy and work, as follows:
T
(e)
=
1
2
Z
ρψ̇ T ψ̇dV = 21 ḋT mḋ
(2.13)
V
U (e) = 12 dT kd
(2.14)
W (e) = dT f
(2.15)
where ρ is the element’s density, f is the body force vector and
Z
m = ρNT NdV
(2.16)
V
Z
k=
BT DBdV
(2.17)
V
are the mass and stiffness, respectively, wherein B is the matrix consisting of shape
function derivatives and D is the constitutive matrix relating element stress, σ, to strain,
ε, as follows:
σ = Dε.
(2.18)
Inserting Eqs. (2.13) - (2.17) in the Hamiltonian equation, Eq. (2.11), the structural
dynamic equation for an element can be derived as
md̈ + kd = f .
(2.19)
Similarly, applying the Hamilton’s Principle to the whole finite element model yields
Md̈ + Kd = F(t)
(2.20)
8
Chapter 2. Background
where M and K are the global mass and stiffness matrices, respectively, and F(t) is the
vector of external forces, which may vary with time.
The natural vibration modes of an elastic structure and their corresponding frequencies can be found by determining the homogeneous solution to Eqn. (2.20). Assuming a
solution of the form
d = φeiωt
(2.21)
−ω 2 Mφeiωt + Kφeiωt = 0
(2.22)
we get
which may be resolved by solving the generalized eigenvalue problem given below:
K − ω 2 M φ = 0.
(2.23)
This can be solved by performing an eigenvalue analysis to obtain the eigenvalues, ωi2 , and
the eigenvectors, φi , which represent the natural frequencies and mode shapes, respectively, of the vibrating structure. These mode shapes are used in the dynamic modeling
of flexible aircraft.
Chapter 3
Methodology
In order to achieve the objectives for this project as laid out in Sec. 1.2, the following
steps were performed:
1. Find data for a real aircraft, including structural, aerodynamic and geometric data.
2. Find the natural vibration modes of the aircraft.
3. Develop the rigid-body simulation model.
4. Develop the mean axes flexible simulation model of the aircraft.
5. Develop the fixed axes flexible aircraft simulation model.
6. Compare time histories of the response of the flexible models and the rigid-body
model to various inputs.
7. Compare the effect of varying the elasticity of the aircraft structure, between the
two flexible models.
8. Based on the above results, make recommendations for future work to be done in
the field of flexible aircraft modeling and simulation.
9
Chapter 4
Mean Axes Model
The simplified mean axes model for flight dynamics of flexible aircraft, described in this
chapter, is based on the formulation developed by Waszak and Schmidt [10]. It derives
nonlinear equations of motion for a deformable aircraft from first principles (Lagrange’s
equation and Principle of Virtual Work). Inertial decoupling of the rigid-body and elastic
momenta is achieved by choosing a particular form of local body reference axes, known
as mean-axes, and with the use of the aircraft’s natural vibration modes. The resulting
equations of motion are presented in scalar form, as a set of six nonlinear rigid-body
equations of motion and n linear equations governing the elastic degrees of freedom,
where n is the number of structural modes retained in the model.
Consider the mass element, ρdV , of an elastic body shown in Fig. 4.1. The inertial
position of this element is given by
R̄ = R̄0 + p̄
(4.1)
where p̄ is the position of the element relative to a local reference system oxyz and R̄0
is the position of the local reference system relative to the inertial frame OXY Z. The
kinetic energy of the whole body is
1
T =
2
Z
dR̄ dR̄
·
ρdV
dt dt
(4.2)
V
where, treating each mass element as a point mass, the derivative terms are given by
dR̄
dR̄0 δ p̄
=
+
+ ω̄ × p̄
dt
dt
δt
in which
δ
δt
(4.3)
denotes time derivative w.r.t. the local reference frame (attached to the body)
and ω̄ is the angular velocity of the local reference frame relative to the inertial reference
10
11
Chapter 4. Mean Axes Model
ρdV
x
p̄
y
o
R̄
z
R̄0
O
X
Y
Z
Figure 4.1: Inertial position of a mass element
frame. Inserting the above into the expression for the kinetic energy yields:
Z 1
dR̄0 dR̄0
dR̄0 δ p̄ δ p̄ δ p̄
δ p̄
T =
·
+2
·
+
·
+ 2 · (ω̄ × p̄) + (ω̄ × p̄) · (ω̄ × p̄)
2
dt
dt
dt δt
δt δt
δt
V
dR̄0
+2 (ω̄ × p̄) ·
ρdV . (4.4)
dt
The potential energy of an elastic body is composed of the gravitational potential energy
and the strain energy. The gravitational potential energy can be expressed as
Z
Ug = − ḡ · R̄0 + p̄ ρdV
(4.5)
V
where ḡ is the acceleration due to gravity. To derive the expression for the strain energy, lets represent the position of a mass element in an elastic body as the sum of its
undeformed position, s̄ (x, y, z), and its deformed position, d¯(x, y, z, t) (note that the
12
Chapter 4. Mean Axes Model
deformed position is a function of time), as shown:
p̄ = s̄ + d¯
(4.6)
Then the strain energy can be written as (see Ref. [10])
Z 2¯
δ d ¯
1
Ue = −
· dρdV .
2
δt2
(4.7)
V
In order to simplify the equations of motion of an elastic body, a local reference system
is used such that there is no inertial coupling between the elastic degrees of freedom and
the rigid-body degrees of freedom. Such a reference system is referred to as a “mean
axes” system. The mean axes are defined by the following condition:
Z
Z
δ p̄
δ p̄
ρdV = p̄ × ρdV = 0
δt
δt
V
(4.8)
V
which means that the linear and angular momenta due to elastic deformations are zero
at every instant. These exact mean axes constraints are often difficult to apply, however,
they can be used to derive the so-called “practical mean axes conditions” which are
much easier to apply. Recalling Eq. (4.6) and the fact that only the deformed part of
the position vector is time varying, the above equation can be reduced to
Z
Z ¯
δ d¯
δd
s̄ + d¯ × ρdV = 0.
ρdV =
δt
δt
(4.9)
V
V
¯ and
Assuming structural deformation to be small, the product of the displacement, d,
displacement rate,
δ d¯
,
δt
can be ignored. Also, assuming the mass density of each element
to be constant, Eq. (4.9) simplifies to
Z
Z
δ
δ
¯
¯
dρdV
=
s̄ × dρdV
=0
δt
δt
V
(4.10)
V
These are the practical mean axes constraints which will simplify the equations of motion
for our problem.
By applying the mean axes constraints derived above, the expression for kinetic energy
of the system (Eq. (4.4)) can be simplified as follows. Applying Eq. (4.8) to the second
term of the kinetic energy expression, we get
Z
Z
dR̄0 δ p̄
dR̄0
δ p̄
· ρdV =
·
ρdV ∼
=0
dt δt
dt
δt
V
V
(4.11)
13
Chapter 4. Mean Axes Model
Similarly, the fourth term becomes
Z
Z
δ p̄
δ p̄
· (ω̄ × p̄) ρdV = p̄ × ρdV · ω̄ ∼
= 0,
δt
δt
(4.12)
V
V
resulting in the following reduced expression for the kinetic energy of the system:
Z 1
dR̄0 dR̄0 δ p̄ δ p̄
dR̄0
T =
·
+
·
+ (ω̄ × p̄) · (ω̄ × p̄) + 2 (ω̄ × p̄) ·
ρdV . (4.13)
2
dt
dt
δt δt
dt
V
Further simplification of the model can be done by selecting the origin of the local
reference frame to lie at the instantaneous centre of mass of the elastic body, as expressed
below:
R
p̄ρdV
V
x̄cm = R
=0
ρdV
(4.14)
V
where x̄cm is the position of the centre of mass relative to the local reference frame. This
results in the last term in the kinetic energy expression, Eq. (4.4), to become zero, as
follows:
Z
dR̄0
ρdV = ω̄ ×
(ω̄ × p̄) ·
dt
Z
p̄ρdV ·
dR̄0
= 0.
dt
(4.15)
V
V
The kinetic energy expression now reduced to
Z 1
dR̄0 dR̄0 δ p̄ δ p̄
T =
·
+
·
+ (ω̄ × p̄) · (ω̄ × p̄) ρdV ,
2
dt
dt
δt δt
(4.16)
V
which can also be written as
1 dR̄0 dR̄0 1 T
1
T = M
·
+ ω̄ [I] ω̄ +
2
dt
dt
2
2
Z
δ p̄ δ p̄
· ρdV
δt δt
(4.17)
V
where M is the total mass of the elastic body and [I] is the inertia tensor. Note, in this
formulation, the inertia tensor is assumed to be constant since the elastic deformations
that would generally cause variations in the inertia tensor, are assumed to be small.
Additionally, using Eq. (4.14), the gravitational potential energy can be simplified as
Z
Z
Z
Ug = − ḡ · R̄0 + p̄ ρdV = −R̄0 · ḡ ρdV − p̄ρdV · ḡ = −R̄0 · ḡM.
(4.18)
V
V
V
Using the natural vibration modes of the elastic body (see 2.2.2), any forced motion
of the body can be described. The free vibration modes used in conjunction with the
14
Chapter 4. Mean Axes Model
practical mean axes constraints can determine the origin and the orientation of the local
reference frame which would decouple the kinetic energy expression. Once the natural
vibration modes have been found, the relative displacements of the elastic body undergoing general elastic deformations can be described in terms of the mode shapes, φi (x, y, z),
and the generalized coordinates, ηi (t), as follows:
d¯ =
∞
X
φ̄i (x, y, z) ηi (t).
(4.19)
i=1
Writing the practical mean axes constraints in terms of the above, we get
Z
Z
∞
∞
X
X
dηi
dηi
φi ρdV =
s̄ × φi ρdV = 0.
dt
dt
i=1
i=1
V
(4.20)
V
Expressing the last term in the kinetic energy expression, Eq. (4.17), in terms of the
elastic displacements defined in Eq. (4.19), results in
)
Z
Z ¯ ¯
Z (X
∞
∞
X
δ p̄ δ p̄
δd δd
dηi
dηj
φ̄i
φ̄j
· ρdV =
· ρdV =
·
ρdV .
δt δt
δt δt
dt j=1
dt
i=1
V
V
(4.21)
V
By definition, the natural vibration modes are orthogonal to each other, which can
be expressed as
Z
φ̄i · φ̄j ρdV ≡ 0,
i 6= j.
(4.22)
V
Using this property to further simplify Eq. (4.21) results in
Z
∞
X
δ p̄ δ p̄
· ρdV =
Mi η̇i2
δt δt
i=1
(4.23)
V
where Mi is the generalized mass of the i th mode, defined by
Z
Mi = φi · φi ρdV .
(4.24)
V
The kinetic energy of the elastic body can now be expressed as
∞
1X
1 dR̄0 dR̄0 1 T
Mi η̇i2 .
T = M
·
+ ω̄ [I] ω̄ +
2
dt
dt
2
2 i=1
(4.25)
Expressing the strain energy, Eq. (4.7), in terms of the natural vibration modes yields
∞
Ue =
1X 2 2
ω i η i Mi
2 i=1
(4.26)
15
Chapter 4. Mean Axes Model
where ωi is the natural vibration frequency of the i th mode.
In order to apply the Lagrange’s equation, the motion of the elastic body needs to
be described by generalized coordinates relative to an inertial reference frame. Defining
the inertial position of the origin of the local, body-fixed, mean axes reference frame (the
instantaneous centre of mass of the elastic body) as
R̄0 = xî + y ĵ + z k̂
(4.27)
where î, ĵ and k̂ are unit vectors in each of the coordinate directions of the local reference
frame. The time rate of change of the inertial coordinates is given by
dR̄0
δ R̄0
∆
=
+ ω̄ × R̄0 = U î + V ĵ + W k̂.
dt
δt
(4.28)
Using the same set of Euler angles which are commonly used in rigid body aircraft
equations of motion, φ, θ and ψ, the orientation of the local reference frame can be
defined. Then we can define the angular velocity of the local reference frame relative to
the inertial reference frame as
∆
ω̄ = pî + q ĵ + rk̂
(4.29)
p = φ̇ − ψ̇ sin θ
(4.30)
q = ψ̇ cos θ sin φ + θ̇ cos φ
(4.31)
r = ψ̇ cos θ cos φ − θ̇ sin φ
(4.32)
where
Then the inertial velocity of the elastic body in the local reference frame becomes
U = ẋ + qz − ry
(4.33)
V = ẏ + rx − pz
(4.34)
W = ż + py − qx
(4.35)
Using the above expressions for inertial position and angular and translational velocities, the kinetic energy can be expressed as

p

∞
i   1X
1h
1
2
2
2


Mi η̇i2
T = M (U + V + W ) +
p q r [I]  q  +
2
2
2 i=1
r
(4.36)
16
Chapter 4. Mean Axes Model
and the potential energy expressions become
Ug = −M g(−x sin θ + y sin φ cos θ + z cos φ cos θ)
(4.37)
∞
Ue =
1X 2 2
ωi η i Mi .
2 i=1
The above energy expressions allow the application of Lagrange’s equation
∂T
∂U
d ∂T
−
+
= Qi .
dt ∂ q̇i
∂qi ∂qi
(4.38)
(4.39)
The resulting equations are the equations of motion of flexible aircraft and can be
expressed in terms of the inertial velocities expressed in the local coordinate system (i.e.
U , V , W , p, q and r). After some algebraic manipulations, these equations are
h
i
M U̇ − rV + qW + g sin θ = QX
h
i
M V̇ − pW + rU − g sin φ cos θ = QY
h
i
M Ẇ − qU + pV − g cos φ cos θ = QZ
Ixx ṗ − (Ixy q̇ + Ixz ṙ) + (Izz − Iyy ) qr + (Ixy r − Ixz q) p + r2 − q 2 Iyz = QφB
Iyy q̇ − (Ixy ṗ + Iyz ṙ) + (Ixx − Izz ) pr + (Iyz p − Ixy r) q + p2 − r2 Ixz = QθB
Izz ṙ − (Ixz ṗ + Iyz q̇) + (Iyy − Ixx ) pq + (Ixz q − Iyz p) r + q 2 − p2 Ixy = QψB
Mi η̈i + ωi 2 ηi = Qηi i = 1, 2, . . . n.
(4.40)
(4.41)
(4.42)
(4.43)
(4.44)
(4.45)
(4.46)
Notice that the above equations have generalized force terms in them, represented
by the Qi ’s. These generalized forces are found using the Principle of Virtual Work,
expressed below:
∂
(δW )
(4.47)
∂qi
where δW is the work associated with arbitrary virtual displacements of the generalized
Qi =
coordinates. If we define the external aerodynamic and propulsive forces, relative to the
local reference frame, to be given by
∆
(4.48)
∆
(4.49)
F̄ = X î + Y ĵ + Z k̂
and the moments by
M = Lî + M ĵ + N k̂
17
Chapter 4. Mean Axes Model
then it can be shown that (Ref. [10])
QX = X,
QY = Y,
QZ = Z
(4.50)
QψB = N .
(4.51)
and
QφB = L,
QθB = M ,
Finally, the generalized forces for the elastic equations, Qηi , may be computed as
follows:
b
Z2 −l cos αv φi
Qηi =
−
b
+ (mac + le cos αs )
dφi b
dx
dy
(4.52)
b
2
where l is the aerodynamic force per unit span produced by each section of the body, αv
is the angle of attack of the aircraft, φi b is the elastic displacement of each section due to
bending, mac is the aerodynamic moment per unit span about the line of aerodynamic
centres and αs is the local angle of attack of each section. For further details see Section
2.2.1 and Ref. [10].
Chapter 5
Fixed Axes Model
The higher fidelity fixed axes flexible aircraft model is based on the integrated model
developed by Meirovitch and Tuzcu [3]. It presents an alternative approach to modeling flight dynamics of elastic aircraft without the use of mean-axes. This derivation
ultimately yields equations of motions in the form of ordinary differential equations in
vector form, in terms of quasi-coordinates and generalized coordinates, velocities and
forces. This chapter summarizes the derivation of this formulation for use in flexible
aircraft simulation.
In order to derive the equations of motion, a reference frame xf yf zf is attached to
the undeformed fuselage, as well as reference frames xw yw zw and xe ye ze to the wing and
empennage, respectively, as shown in Fig. 5.1. These local reference frames represent
the respective components body axes. Consequently, the motion of the whole system
can be described by six rigid body degrees of freedoms of the fuselage body axes and
the elastic deformation of the flexible components relative to their respective undeformed
body axes.
Using the conventional set of Euler angles used in aircraft dynamics, φ, θ and ψ, the
transformation matrix, Cf , that transforms the inertial axes XY Z into xf yf zf can be
obtained as (assuming a 321-rotation)


cψcθ
sψcθ
−sθ



Cf = 
cψsθsφ
sψcφ
sψsθsφ
cψcφ
cθsφ
−
+


cψsθcφ − sψsφ sψsθcφ − cψsφ cθcφ
(5.1)
where s = sin and c = cos. Similarly, the time rate of change of the transformation
18
19
Chapter 5. Fixed Axes Model
xf F
0f
xeV
rfeR
rfwR
Z
ywR
xwR
zwR
yeR
Rf
xeR
yf F
zeR
0
zf F
X
Y
Figure 5.1: Component reference frames
matrix, Ef , is

1
0
−sθ



.
Ef = 
0
cφ
cθsφ


0 −sφ cθcφ
(5.2)
The velocity of a point on the fuselage, V̄f , is given by
V̄f (rf , t) = Vf (t) + [r̃f + ũf (rf , t)]T [ωf (t) + αf (rf , t)] + vf (rf , t)
∼
= Vf + (r̃f + ũf )T ωf + r̃T αf + vf
(5.3)
f
where Vf and ωf are translational and angular velocity vectors for reference frame xf yf zf ,
rf is the position vector associated with the mass element dmf , uf is the elastic displacement, vf and αf are the bending and torsion velocities, respectively and r̃f and ũf are
the skew-symmetric matrices corresponding to rf and uf , respectively. The velocity of a
point on the wing or empennage can be expressed as
V̄i (ri , t) = Ci V̄f (rf i , t) + r̃iT Ci [Ωf (rf i , t) + αf (rf i , t)]
+ [r̃i + ũi (ri , t)]T [ωi (t) + αi (ri , t)] + vi (ri , t)
∼
= Ci Vf + [Ci (r̃f i + ũf i )T + (r̃i + ũi )T Ci ]ωf + r̃iT Ci (Ωf i + αf i )
+ Ci (vf i + r̃fTi αf i ) + r̃iT αi + vi
(5.4)
i = w, e
where i denotes the component type, f for fuselage, w for wing and e for empennage, Ci
is the rotation matrix from the fuselage body axes xf yf zf to the component body axes
20
Chapter 5. Fixed Axes Model
xi yi zi , rf i is the radius vector from xf yf zf to xi yi zi , Ωf i is the angular velocity of the
fuselage at rf i due to bending, expressed as
T
Ωf i = [ 0 −∂ u̇f z /∂xf ∂ u̇f y /∂xf ] (5.5)
rf i
and αf i is the elastic velocity of the fuselage at rf i due to torsion, given by
T
αf i = [ αf i 0 0 ] .
(5.6)
rf i
The hybrid equations of motion for flexible aircraft are given in Ref. [2]. These
hybrid equations, derived from Lagrange’s equation, are a mix of ordinary and partial
differential equations, in terms of quasi-coordinates, and are expressed as
d
∂L
∂L
∂L
+ ω̃f ∂V
− Cf ∂R
=F
dt ∂Vf
f
f
−1 ∂L
∂L
d
∂L
∂L
+ ω̃f ∂ω
− Ef T
=M
+ Ṽf ∂V
dt ∂ωf
∂θf
f
f
∂ L̂i
∂
− ∂∂uL̂ii + ∂∂F̂u̇uii + Lui ui = Ûi
∂t
∂vi
∂ F̂
∂ L̂i
∂
i = f, w, e
+ ∂ ψ̇ψi + Lψi ψi = Ψ̂i
∂t ∂αi
(5.7)
i
where L is the Lagrangian for the entire aircraft, Ṽf and ω̃f are skew-symmetric matrices
of Vf and ωf , respectively, Rf is the position vector of the origin Of of xf yf zf relative to
XY Z, θf is the vector of Euler angles between xf yf zf and XY Z, ψi is the elastic angular
displacement vector for component i, L̂i is the Lagrangian density for component i, not
including the strain energy, F̂ui and F̂ψi are the Rayleigh’s dissipation function densities
for component i, Lui and Lψi are stiffness differential operator matrices for component i,
F and M are the net force and moment vectors acting on the aircraft in terms of fuselage
body axes and Ûi and Ψ̂i are the net force and moment density vectors for component i,
respectively.
The hybrid equations of motion shown above include the Lagrangian L = T − U ,
in which T is the kinetic energy and U the potential energy, the Rayleigh dissipation
function densities, F̂ui and F̂ψi , which contain structural damping information and the
stiffness operators, Lui and Lψi , which are derived from the strain energy. In order to
obtain an explicit set of expressions for the flexible aircraft equations of motion, it is
necessary to derive each of these quantities in terms of the known rigid-body and elastic
coordinates of the system. The kinetic energy for the entire aircraft can be expressed as
T = Tf + Tw + Te
(5.8)
21
Chapter 5. Fixed Axes Model
where Ti are the kinetic energies of the aircraft components, given by
Z
1
Ti = 2 V̄iT V̄i dmi ,
i = f, w, e
(5.9)
Therefore, the total kinetic energy becomes
Z
Z
Z
T
T
1
1
1
T = 2 V̄f V̄f dmf + 2 V̄w V̄w dmw + 2 V̄eT V̄e dme .
(5.10)
The Rayleigh dissipation function densities can be expressed as
F̂ui = 21 cui EIi
∂ 2 u̇Ti ∂ 2 u̇i
,
∂xi 2 ∂xi 2
F̂ψi = 21 cψi GJi
∂ ψ̇iT ∂ ψ̇i
,
∂xi ∂xi
i = f, w, e
(5.11)
where cui and cψi are the bending and torsion damping functions, respectively, and EIi
and GJi are, respectively, the flexural and torsional rigidities.
As partial differential equations are difficult to work with, they are conventionally
replaced by a set of ordinary differential equations in most practical applications, with
the help of spatial discretization. In this case, the individual components of the aircraft
are discretized separately, instead of using structural modes for the entire aircraft which
can result in the loss of some geometric details of the aircraft structure. The following
gives the discretization scheme being used for this formulation:
ui (ri , t) = Φui (ri )qui (t),
ψi (ri , t) = Φψi (ri )qψi (t),
i = f, w, e
(5.12)
where Φui and Φψi are matrices of shape functions and qui and qψi are corresponding
vectors of generalized coordinates. The time derivatives of these generalized coordinates
are given by
sui (t) = q̇ui (t),
sψi (t) = q̇ψi (t),
i = f, w, e
(5.13)
The velocity vectors can now be written using the discretized form of the elastic displacements, just shown, as
V̄f (rf , t) = Vf + (r̃f + Φ̃uf quf )T ωf + Φuf suf + r̃f Φψf sψf
V̄i (ri , t) = Ci Vf + [Ci (r̃f i + Φ̃uf i quf )T + (r̃i + Φ̃ui qui )T Ci ]ωf
(5.14)
+(r̃iT Ci ∆Φuf i + Ci Φuf i )suf + Φui sui
+(r̃iT Ci Φψf i + Ci r̃fTi Φψf i )sψf + r̃iT Φψi sψi
where

0
0

∆=
 0
0
0 ∂/∂xf
0


−∂/∂xf 
,
0

0
0
i = w, e



T
,
Φuf = 
φ
0
 uf y

T
0 φuf z
Φuf i = Φuf (rf i ),
i = w, e
(5.15)
22
Chapter 5. Fixed Axes Model
Similar expressions can be written for Φψf and Φψf i . Inserting the above into the kinetic
energy expression, Eq. (5.10), we get the following compact form for the total kinetic
energy:
T = 21 VT MV
(5.16)
where V is the discrete velocity vector for the entire system, expressed as
iT
h
V = VfT ωfT sTuf sTuw sTue sTψf sTψw sTψe
(5.17)
and M = [Mmn ] is the complete system mass matrix. Refer to Ref. [4] for expressions of
the submatrices that makeup the mass matrix.
The matrix of first moments of inertia of the deformed aircraft, S̃, is given by
S̃ =
R
R
(r̃f + Φ̃uf quf )dmf + [(r̃f w + Φ̃uf w quf )CwT + CwT (r̃w + Φ̃uw quw )]Cw dmw
R
+ [(r̃f e + Φ̃uf e quf )CeT + CeT (r̃e + Φ̃ue que )]Ce dme
(5.18)
and the matrix of second moments of inertia of the deformed aircraft, J, simply referred
to as the inertia matrix can be computed as follows:
Z
J = (r̃f + Φ̃uf quf )T (r̃f + Φ̃uf quf )dmf
Z
+ [Cw (r̃f w + Φ̃uf w quf )T + (r̃w + Φ̃uw quw )T Cw ]T
× [Cw (r̃f w + Φ̃uf w quf )T + (r̃w + Φ̃uw quw )T Cw ]dmw
Z
+ [Ce (r̃f e + Φ̃uf e quf )T + (r̃e + Φ̃ue que )T Ce ]T
(5.19)
× [Ce (r̃f e + Φ̃uf e quf )T + (r̃e + Φ̃ue que )T Ce ]dme .
Similarly, substituting Eq. (5.12) into Eqs. (5.11) results in the following expressions
for the generalized Rayleigh’s dissipation functions:
T
Fui = 21 q̇ui
Cui q̇ui ,
T
Fψi = 21 q̇ψi
Cψi q̇ψi ,
i = f, w, e
(5.20)
where the damping matrices, Cui and Cψi are given by
d2 ΦTui d2 Φui
Cui =
cui EIi
dDi ,
dx2i dx2i
Di
Z
Z
Cψi =
cψi GJi
Di
d2 ΦTψi d2 Φψi
dDi ,
dx2i dx2i
i = f, w, e
(5.21)
Expressing the momentum vector for the entire aircraft by
h
iT
p = pTV f pTωf pTuf pTuw pTue pTψf pTψw pTψe ,
(5.22)
23
Chapter 5. Fixed Axes Model
we can write
p = ∂T /∂V = M V.
(5.23)
Having derived expressions for all of the required quantities, we can write the discretized
state equations which describe the dynamics of a flexible aircraft as follows (from Equation (5.7)):
Ṙf = CfT Vf ,
q̇ui = sui ,
θ̇f = Ef−1 wf
q̇ψi = sψi ,
ṗV f = −ω̃f pV f + F,
i = f, w, e
ṗωf = −Ṽf pV f − ω̃f pωf + M
ṗui = ∂T /∂qui − Kui qui − Cui sui + Qui ,
ṗψi = −Kψi qψi − Cψi sψi + Qψi ,
i = f, w, e
i = f, w, e
where, using Eqs. (5.10) and (5.14), we have
R
R
R T T
∂T
T
T T
T
T T
V̄e dme
V̄
dm
+
Φ
ω̃
C
=
Φ
ω̃
V̄
dm
+
Φ
ω̃
C
w
w
f
f
e
w
uf
e
f
uf
f
uf
w
f
∂quf
R
∂T
= ΦTuw C̃w ωf T V̄w dmw
∂quw
R
∂T
= ΦTue C̃e ωf T V̄e dme .
∂que
Additionally, from strain energy, the bending stiffness matrix can be derived as
Z
∂ 2 ΦTui
∂ 2 Φui
Kui =
[EI
]
dDi ,
i = f, w, e,
i
2
∂x2i
Di ∂xi
and the torsional stiffness matrix as
Z
∂ΦTψi ∂Φψi
dDi ,
GJi
Kψi =
∂xi ∂xi
Di
(5.24)
i = f, w, e.
(5.25)
(5.26)
(5.27)
The only remaining quantities appearing in the state equations, Eq. (5.24) are the
generalized force terms, F, M, Qui and Qψi . These forces are related to the actual loads
applied to the aircraft and can be obtained from the principle of virtual work as
R
R
R
F = Df [ff + FE δ (r − rE )] dDf + CwT Dw fw dDw + CeT De fe dDe
R M = Df r̃f + Φ̃uf quf [ff + FE δ (r − rE )] dDf
i
R h
+ Dw r̃f w + Φ̃uf w quf CwT + CwT r̃w + Φ̃uw quw fw dDw
i
R h
T
T
+ De r̃f e + Φ̃uf e quf Ce + Ce r̃e + Φ̃ue que fe dDe
T
R
R
Quf = Df ΦTuf [ff + FE δ (r − rE )] dDf + Dw r̃wT Cw ∆Φuf w + Cw Φuf w fw dDw
T
R
+ De r̃eT Ce ∆Φuf e + Ce Φuf e fe dDe
T
R
R
Qψf = Df ΦTψf r̃f [ff + FE δ (r − rE )] dDf + Dw r̃wT Cw Φψf w + Cw r̃fTw Φψf w fw dDw
T
R
+ De r̃eT Ce Φψf e + Ce r̃fTe Φψf e fe dDe
R
R
Qui = Di ΦTui fi dDi , Qψi = Di ΦTψi r̃i fi dDi ,
i = w, e
(5.28)
Chapter 5. Fixed Axes Model
24
where fi are the net distributed forces over the component i, due to gravity, aerodynamics
and control actions and FE δ (r − rE ) is the engine thrust, in which δ is the spatial Dirac
delta function and rE is the engine’s position vector. This completes the derivation of
all the quantities required by this flight dynamics model.
Chapter 6
Results and Discussion
The simulation results from the mean axes and fixed axes models were compared along
with rigid body results for two control inputs for both longitudinal and lateral controls.
6.1
6.1.1
Comparison of Mean and Fixed Axes Models
Response to Elevator Impulse
The time history plots of the aircraft response to an elevator impulse are shown in Figures
6.1 to 6.7, below.
From Figure 6.1 its clear that the two models have a very similar response in terms of
the pitch rate of the aircraft. Additionally, the rigid aircraft response is also very similar
to the flexible aircraft responses.
In Figure 6.3, the pitch angle time histories are shown for the two flexible aircraft
models as well as the rigid model response. Here, the response of the fixed axes model is
offset from the other two models. This offset is caused by the fixed axes model trimming
at a slightly different pitch angle initially, due to the fact that the structural component
of the two models (the total modal mass) is not exactly the same.
Figure 6.4 plots the longitudinal velocity response to the elevator impulse input.
Again, the two flexible models and the rigid model are in close agreement with one
another.
The acceleration time histories, (Figure 6.5), are also very similar between the three
models, except for some spikes seen in the vertical component of acceleration (Az ) of the
mean axes model. The possible cause of these spikes is discussed later (see end of Sec.
25
Chapter 6. Results and Discussion
26
Figure 6.1: Pitch rate time history of aircraft response to elevator impulse
6.2.3).
Figure 6.7 shows the vertical component of deflection of the right wingtip in response
to the elevator impulse input to the models. As can be seen, the two flexible models are
offset from one another, but have a similar trend. This offset is caused by different initial
trim values.
Overall, it can be concluded that the two models are very similar in response for the
longitudinal elevator impulse input.
Chapter 6. Results and Discussion
Figure 6.2: A closer look at pitch rate time history just after the input
Figure 6.3: Pitch angle time history of aircraft response to elevator impulse
27
Chapter 6. Results and Discussion
Figure 6.4: X velocity time history of aircraft response to elevator impulse
Figure 6.5: Acceleration time history of aircraft response to elevator impulse
28
Chapter 6. Results and Discussion
29
Figure 6.6: Acceleration time history focusing on the response immediately after elevator
impulse
Figure 6.7: Wingtip relative deflection time history of aircraft response to elevator impulse
Chapter 6. Results and Discussion
6.1.2
30
Response to Elevator 2311
2311 inputs are designed to excite a wide range of frequencies in an aircraft’s response.
2311 inputs are alternating step inputs which satisfy the time duration ratio 2:3:1:1.
Refer to [6] for details. The mean axes and fixed axes flexible aircraft models were both
simulated with an elevator 2311 input. The resulting time histories are shown in Figures
6.8 to 6.12 below. For comparison, the response of the rigid aircraft model is also included
in the plots.
Figure 6.8: Pitch rate time history of aircraft response to elevator 2311
In the pitch rate time history, it can be seen that the response of the three models is
very similar. Only slight differences in the response can be observed during the second
peak and in the region after the control input has ended. However, the magnitude of the
variation in response is negligible.
From Figure 6.9, the pitch angle time histories of the two models can be compared for
the case of an elevator 2311 input. The overall trend is the same for the two flexible models
as well as the rigid model. However, unlike the elevator impulse case, the difference in
the response of the two flexible models is more noticeable here. Overall though, it can
be said that the models are in agreement.
Once again, the longitudinal velocity time histories for the three models follow the
Chapter 6. Results and Discussion
31
Figure 6.9: Pitch angle time history of aircraft response to elevator 2311
same trend (Figure 6.10). The difference in the three models becomes the most noticeable
at the minimum, about halfway through the simulation.
As expected, the time histories for the aircraft acceleration in response to the elevator
2311 input are similar for both flexible models. The rigid aircraft model acceleration is
also in close agreement to the flexible models.
Figure 6.12 gives the time history plots for the wing tip deflection of the aircraft
relative to the wing root for both fixed and mean axes flexible aircraft models. Apart
from a slight offset caused by a difference in the initial trim values, the two models are
very similar in their response.
Chapter 6. Results and Discussion
Figure 6.10: X velocity time history of aircraft response to elevator 2311
Figure 6.11: Acceleration time history of aircraft response to elevator 2311
32
Chapter 6. Results and Discussion
33
Figure 6.12: Wingtip relative deflection time history of aircraft response to elevator 2311
Chapter 6. Results and Discussion
6.1.3
34
Response to an Aileron Impulse
To compare the lateral response of the aircraft models, the models were simulated with
an aileron impulse input same as the elevator impulse used for longitudinal response. The
time history plots of the aircraft response to the aileron impulse are shown in Figures
6.13 to 6.19, below.
Figure 6.13: Roll rate time history of aircraft response to aileron impulse
The roll rate response, Figure 6.13, for the mean axes and fixed axes models is very
similar, as well as the response of the rigid model.
The roll angle time history response to the aileron impulse input is plotted in Figure
6.15. All three of the models start off at the same value and follow the same trend
initially. However, after the control input ends, the mean axes model’s response drifts
away from that of the other two models.
Figure 6.16 shows the lateral velocity time history plots for the mean axes, fixed axes,
as well as the rigid aircraft models. It can be seen that all three models have a similar
response. The amplitude of the response is quite small though in all three cases.
The acceleration time histories for the three models’ response to the aileron impulse
are compared in Figure 6.17. The responses agree with one another for all three models,
except for the mean axes model’s lateral acceleration Ay which has noticeable spikes and
Chapter 6. Results and Discussion
35
Figure 6.14: A closer look at the roll rate time history immediately proceeding the input
starts to drift away from the other two models. The relative wingtip deflection of the
two models in response to the aileron impulse is plotted in Figure 6.19. The response
is very similar for the two models with the exception of a small offset, likely caused by
different trim points between the two models.
Chapter 6. Results and Discussion
Figure 6.15: Roll angle time history of aircraft response to aileron impulse
Figure 6.16: Y velocity time history of aircraft response to aileron impulse
36
Chapter 6. Results and Discussion
Figure 6.17: Acceleration time history of aircraft response to aileron impulse
Figure 6.18: Focusing on acceleration time history response just after the input
37
Chapter 6. Results and Discussion
38
Figure 6.19: Wingtip relative deflection time history of aircraft response to aileron impulse
Chapter 6. Results and Discussion
6.1.4
39
Response to Aileron 2311
Similar to the elevator 2311 input, an aileron 2311 input was used to simulate the response
of the models. 2311 inputs excite a wider range of frequencies in the aircraft model output.
Figures 6.20 to 6.24 show plots of the output of the flexible aircraft models along with
the rigid model’s output.
Figure 6.20: Roll rate time history of aircraft response to aileron 2311
Figure 6.20 shows a comparison of the roll rate time histories of all three models in
response to an aileron 2311. The response from all three models is very similar.
The roll angle time history of the simulation in response to an aileron 2311 is plotted
in Figure 6.21. The response from all three models is about the same during the input,
however, once the input ends the models diverge slightly.
The lateral velocity time histories for the aileron 2311 response are compared in Figure
6.22. The velocities from all three of the models match well, for the whole time interval.
The acceleration time history response to aileron 2311 is shown in Figure 6.23. All
three models follow the same trend for all three acceleration components. Only the lateral
acceleration, Ay , drifts slightly with time.
Chapter 6. Results and Discussion
Figure 6.21: Roll angle time history of aircraft response to aileron 2311
Figure 6.22: Y velocity time history of aircraft response to aileron 2311
40
Chapter 6. Results and Discussion
41
Figure 6.23: Acceleration time history of aircraft response to aileron 2311
Figure 6.24: Wingtip relative deflection time history of aircraft response to aileron 2311
Chapter 6. Results and Discussion
42
The time history for relative wingtip deflection in response to an aileron 2311 input
for the mean axes and fixed axes models can be seen in Figure 6.24. Except for a small
offset, the wingtip deflection is about the same for the two models. The offset can be
attributed to the fact that the two models trim to slightly different initial values.
By observing the response of the two flexible models given in this section, it can be
concluded that the two models produce very similar output to a variety of inputs. As
such, either of the two flexible models can be employed to simulate the flight dynamics
of this particular aircraft. However, it can be noticed from the preceding results that
the two flexible models’ responses do not differ much from the response of the rigid
aircraft model. The reason being that the aircraft used in these simulations has a fairly
rigid structure and thus it does not exhibit significant interactions between the structural
dynamics and the flight dynamics.
To study the response of the mean axes and fixed axes flexible flight simulation models,
the aircraft structural data was modified to artificially make it more flexible. The effect
of varying the stiffness of the aircraft structure on the response of the two flexible models
is presented in the following section.
6.2
Effect of Varying Aircraft Stiffness
In this section, the effect of varying the stiffness of the aircraft structure on the response
of the flexible models is analysed. Each of the models was tested with one-half and onetenth the stiffness of the original aircraft and the results were compared for the different
stiffness cases along with the rigid aircraft model. The same inputs were used as in the
previous section for consistency.
6.2.1
Response to Elevator Impulse
The time history plots of the aircraft response to an elevator impulse are shown in Figures
6.25 to 6.42, below.
Figures 6.25 and 6.26 present the pitch rate response of the mean and fixed axes
models, respectively. The original and half stiffness cases are very similar to the rigid
model. The difference becomes significant for the tenth stiffness case, for both models.
Comparing between the two models, the responses follow a similar trend with the exception of noticeable spikes in the mean axes model’s response at the beginning of the
Chapter 6. Results and Discussion
43
Figure 6.25: Pitch rate time history of mean axes model’s response to elevator impulse
input.
The pitch angle time histories in response to elevator impulse for varying stiffnesses are
given in Figures 6.29 and 6.30 for the two models. Once again, the original stiffness and
half stiffness cases are similar to rigid response while the difference becomes noticeable
in the case of one-tenth the original stiffness, for both models. The responses of the fixed
axes model are offset due to different initial trim values, especially evident in the tenth
stiffness case.
The longitudinal component of velocity, Vx , simulation time histories are shown in
Figures 6.31 and 6.32, for the mean axes and fixed axes models respectively. Again, the
original and half stiffness cases match well with the rigid case, while the tenth stiffness
cases differ noticeably.
Figures 6.33 and 6.34 show the response of the longitudinal acceleration component,
Ax . Similar to previous results, the response of the two flexible models is similar to one
another with the exception of damped oscillations that start at the onset of the input in
the mean axes model’s response.
The vertical component of acceleration, Az , is compared for various aircraft stiffnesses,
for the two flexible models in Figures 6.37 and 6.38. Similar to previous results, the two
models produce similar responses except for the oscillations in the mean axes model and
Chapter 6. Results and Discussion
44
Figure 6.26: Pitch rate time history of fixed axes model’s response to elevator impulse
the effect of varying the structural stiffness of the model only becomes noticeable when
the stiffness is decreased to one-tenth of the original stiffness.
The effect of varying the stiffness of the structure of the aircraft model on the relative
wingtip deflection in response to an elevator impulse input is demonstrated in Figures 6.41
and 6.42 for the mean axes and fixed axes models, respectively. As expected, the amount
of deformation increases with decreasing stiffness. The response of the two models is
similar for the original and half stiffness cases while there is a slightly greater offset for
the tenth stiffness case.
Chapter 6. Results and Discussion
45
Figure 6.27: A closer look at the pitch rate time history of mean axes model immediately
after the elevator impulse
Figure 6.28: A closer look at the pitch rate time history of fixed axes model immediately
proceeding the input
Chapter 6. Results and Discussion
46
Figure 6.29: Pitch angle time history of mean axes model’s response to elevator impulse
Figure 6.30: Pitch angle time history of fixed axes model’s response to elevator impulse
Chapter 6. Results and Discussion
47
Figure 6.31: X velocity time history of mean axes model’s response to elevator impulse
Figure 6.32: X velocity time history of fixed axes model’s response to elevator impulse
Chapter 6. Results and Discussion
48
Figure 6.33: X acceleration time history of mean axes model’s response to elevator impulse
Figure 6.34: X acceleration time history of fixed axes model’s response to elevator impulse
Chapter 6. Results and Discussion
49
Figure 6.35: X acceleration time history of mean axes model: focusing on the response
immediately proceeding the input
Figure 6.36: X acceleration time history of fixed axes model immediately proceeding the
elevator impulse input
Chapter 6. Results and Discussion
50
Figure 6.37: Z acceleration time history of mean axes model’s response to elevator impulse
Figure 6.38: Z acceleration time history of fixed axes model’s response to elevator impulse
Chapter 6. Results and Discussion
51
Figure 6.39: Focusing on the Z acceleration time history of mean axes model immediately
after the input
Figure 6.40: Focusing on the Z acceleration time history of fixed axes model immediately
after the input
Chapter 6. Results and Discussion
52
Figure 6.41: Wingtip relative deflection time history of mean axes model’s response to
elevator impulse
Figure 6.42: Wingtip relative deflection time history of fixed axes model’s response to
elevator impulse
Chapter 6. Results and Discussion
6.2.2
53
Response to Elevator 2311
An elevator 2311 input was used to further evaluate the effect of varying the stiffness of
the aircraft structure on the response of flexible models. The results of these simulations
are shown in Figures 6.43 to 6.54 below.
Figure 6.43: Pitch rate time history of mean axes model’s response to elevator 2311
The pitch rate response of the two models (Figures 6.43 and 6.44) is in agreement
with each other. Conforming with the previous results, both models have a very similar
response for all stiffness cases. Once again, the effect of varying the stiffness only becomes
evident when the stiffness is lowered to one-tenth of the original aircraft structure’s
stiffness.
Figures 6.43 and 6.44 show the pitch angle response to the elevator 2311 input for the
mean axes and fixed axes models, respectively. As expected the results for both models
are very similar for all stiffness cases.
The longitudinal velocity component time histories for the elevator 2311 input response are given by Figures 6.47 and 6.48. All the responses for both models match well
with each other with only slight differences in the time histories of the output. The difference in structural stiffnesses is most visible for the one-tenth stiffness cases, following
the previous results.
Chapter 6. Results and Discussion
54
Figure 6.44: Pitch rate time history of fixed axes model’s response to elevator 2311
Figures 6.49 to 6.52 show the longitudinal and vertical acceleration time histories
in response to an elevator 2311 input for both flexible models with varying structural
stiffnesses. Following the trend of the previous results there is a good match between
the results of the two models for each of the aircraft stiffnesses modeled. The only
small difference between the two models is the presence of oscillations in the mean axes
model response, especially apparent in the tenth stiffness case. The reason behind these
oscillations is discussed later in this chapter.
The wingtip deflection in response to the elevator 2311 input, for the two models,
is given in Figures 6.53 to 6.54. Apart from the results being slightly offset from one
another caused by the models trimming to different values initially, the response of the
two models is very comparable, for all three stiffness cases.
Chapter 6. Results and Discussion
55
Figure 6.45: Pitch angle time history of mean axes model’s response to elevator 2311
Figure 6.46: Pitch angle time history of fixed axes model’s response to elevator 2311
Chapter 6. Results and Discussion
56
Figure 6.47: X velocity time history of mean axes model’s response to elevator 2311
Figure 6.48: X velocity time history of fixed axes model’s response to elevator 2311
Chapter 6. Results and Discussion
57
Figure 6.49: X acceleration time history of mean axes model’s response to elevator 2311
Figure 6.50: X acceleration time history of fixed axes model’s response to elevator 2311
Chapter 6. Results and Discussion
58
Figure 6.51: Z acceleration time history of mean axes model’s response to elevator 2311
Figure 6.52: Z acceleration time history of fixed axes model’s response to elevator 2311
Chapter 6. Results and Discussion
59
Figure 6.53: Wingtip relative deflection time history of mean axes model’s response to
elevator 2311
Figure 6.54: Wingtip relative deflection time history of fixed axes model’s response to
elevator 2311
Chapter 6. Results and Discussion
6.2.3
60
Response to Aileron Impulse
To study the effect of varying the stiffness on the lateral response of the flexible aircraft
models, simulations were run with inputs to the aileron. The first input used was an
impulse, similar to the one used in Sec. 6.1.1 for the elevator. The time history plots of
the response of the models to the aileron impulse for three stiffness cases are shown in
the following Figures 6.55 to 6.72.
Figure 6.55: Roll rate time history of mean axes model’s response to aileron impulse
Figures 6.55 and 6.56 compare the time histories of the roll rate in response to an
aileron impulse, respectively, for the mean and fixed axes flexible aircraft flight simulation models with varying structural stiffness. Both models compare favorably with one
another for the original and half stiffness cases. Both models’ response shows damped
oscillations in the tenth stiffness case. However, the mean axes model’s response for the
tenth stiffness case starts to diverge from the other two responses about 15 seconds after
the input.
The roll angle simulation response to the aileron impulse is plotted in Figures 6.59 and
6.60. The response is similar between the two models for the original and half stiffness
cases. However, the roll angle of the mean axes model diverges rapidly after the control
input for the tenth stiffness case. The fixed axes model’s response does not diverge.
Chapter 6. Results and Discussion
61
Figure 6.56: Roll rate time history of fixed axes model’s response to aileron impulse
The response of the lateral component of velocity to an aileron impulse input for the
mean axes and fixed axes models is shown in Figures 6.61 and 6.62. Again, the tenth
stiffness case of the mean axes model diverges rapidly compared to the other cases which
are similar to the response of the fixed axes model.
The longitudinal and lateral acceleration response of the two flexible aircraft models
is given by Figures 6.63 to 6.68. The time histories exhibit a good match between the
two models for the original and half stiffness cases. However, the mean axes model’s
response diverges for both acceleration components.
Figures 6.71 and 6.72 give the relative wingtip deflection of the flexible aircraft models
for the three stiffness cases in response to an aileron impulse. Both models produce similar
results with a small offset caused by difference in the initial trim points of the two models.
Chapter 6. Results and Discussion
62
Figure 6.57: Fousing on roll rate time history of mean axes model just after the aileron
impulse
Figure 6.58: Focusing on roll rate time history of fixed axes model just after the aileron
impulse
Chapter 6. Results and Discussion
63
Figure 6.59: Roll angle time history of mean axes model’s response to aileron impulse
Figure 6.60: Roll angle time history of fixed axes model’s response to aileron impulse
Chapter 6. Results and Discussion
64
Figure 6.61: Y velocity time history of mean axes model’s response to aileron impulse
Figure 6.62: Y velocity time history of fixed axes model’s response to aileron impulse
Chapter 6. Results and Discussion
65
Figure 6.63: X acceleration time history of mean axes model’s response to aileron impulse
Figure 6.64: X acceleration time history of fixed axes model’s response to aileron impulse
Chapter 6. Results and Discussion
66
Figure 6.65: Focusing on X acceleration time history of mean axes model proceeding the
input
Figure 6.66: Focusing on X acceleration time history of fixed axes model proceeding the
input
Chapter 6. Results and Discussion
67
Figure 6.67: Y acceleration time history of mean axes model’s response to aileron impulse
Figure 6.68: Y acceleration time history of fixed axes model’s response to aileron impulse
Chapter 6. Results and Discussion
68
Figure 6.69: Focusing on Y acceleration time history of mean axes model immediately
after aileron impulse
Figure 6.70: Focusing on Y acceleration time history of fixed axes model immediately
after aileron impulse
Chapter 6. Results and Discussion
69
Figure 6.71: Wingtip relative deflection time history of mean axes model’s response to
aileron impulse
Figure 6.72: Wingtip relative deflection time history of fixed axes model’s response to
aileron impulse
Chapter 6. Results and Discussion
6.2.4
70
Response to Aileron 2311
Similar to the elevator input, a 2311 aileron input was used to compare the outputs of
the simulation models. The time history plots of the aircraft response to the aileron 2311
input are shown in Figures 6.73 to 6.84.
Figure 6.73: Roll rate time history of mean axes model’s response to aileron 2311
The roll rate responses of the mean axes model for the original and half stiffness cases
compare reasonably well with the fixed axes model (Figures 6.73 and 6.74). However,
for the tenth stiffness case, the mean axes model’s response is out of phase and does not
match the fixed axes model in magnitude either.
The roll angle time histories of the flexible models for an aileron 2311 input are given
in Figures 6.75 and 6.76. The two models’ response matches with one another for the
original and half stiffness cases, while the tenth stiffness cases’ response diverges for the
mean axes model.
Figures 6.77 and 6.78 present the time histories of the lateral component of velocity
of the two flexible models to the aileron 2311. The frequency of oscillations in the output
of the mean axes model varies with stiffness, as can be noticed by observing the tenth
stiffness case. However, no such trend is noticed in the response of the fixed axes model.
Furthermore, the amplitude of the oscillations increases with decreasing stiffness in case
Chapter 6. Results and Discussion
71
Figure 6.74: Roll rate time history of fixed axes model’s response to aileron 2311
of the mean axes model while the opposite is true for the fixed axes model.
Figures 6.79 and 6.80 provide the longitudinal acceleration time histories of the two
model’s response to an aileron 2311 input. The response of the mean axes model starts
to diverge after the control input, something easily noticeable in the tenth stiffness case.
Meanwhile, the response of the fixed axes model does not diverge.
The lateral component of acceleration time histories in response to an aileron 2311
are shown in Figures 6.81 and 6.82 for varying aircraft stiffness. Once again, the response
of the mean axes model diverges, while the fixed axes model’s response does not. Additionally, the mean axes model’s response exhibits small spikes which are less pronounced
in the case of the fixed axes model.
The wingtip deflection relative to the wing root in response to the aileron 2311 input
for different aircraft structural stiffness ratios is given in Figures 6.83 and 6.84. The
wingtip deflection time histories for the two models are very similar for each of the three
aircraft stiffness ratios.
Chapter 6. Results and Discussion
Figure 6.75: Roll angle time history of mean axes model’s response to aileron 2311
Figure 6.76: Roll angle time history of fixed axes model’s response to aileron 2311
72
Chapter 6. Results and Discussion
Figure 6.77: Y velocity time history of mean axes model’s response to aileron 2311
Figure 6.78: Y velocity time history of fixed axes model’s response to aileron 2311
73
Chapter 6. Results and Discussion
74
Figure 6.79: X acceleration time history of mean axes model’s response to aileron 2311
Figure 6.80: X acceleration time history of fixed axes model’s response to aileron 2311
Chapter 6. Results and Discussion
75
Figure 6.81: Y acceleration time history of mean axes model’s response to aileron 2311
Figure 6.82: Y acceleration time history of fixed axes model’s response to aileron 2311
Chapter 6. Results and Discussion
76
Figure 6.83: Wingtip relative deflection time history of mean axes model’s response to
aileron 2311
Figure 6.84: Wingtip relative deflection time history of fixed axes model’s response to
aileron 2311
Chapter 6. Results and Discussion
77
The mean axes and the fixed axes models’ response is comparable for longitudinal
inputs for the original and half aircraft stiffness ratios. Even for the one-tenth stiffness
ratio cases, the two models’ response is similar with the exception of some sharp oscillations found in the response of the mean axes model. However, these oscillations are
damped and subside fairly quickly. These oscillations are likely caused by the particular choice of the natural vibrational modes that were used in the mean axes model for
these simulations. More specifically, mode #4 which is primarily a horizontal empennage
bending mode causes these oscillations.
For the lateral cases, the response of the mean axes model does not match the fixed
axes model as well as the longitudinal cases do. The response of the mean axes model
diverges, especially for the tenth stiffness cases. The cause of this behaviour is not yet
known, however it seems to be caused by the flexible part of the model as the rigid model
does not show the same characteristics. Apart from that, the two models do correlate for
the wing deflection output. Additionally, for the original and the half stiffness cases the
diverging behaviour is not as pronounced and thus the response is comparable between
the two flexible flight models.
Chapter 7
Conclusions and Recommendations
The objective of this study was to compare two available models for the simulation of
aircraft with flexible structures. Using aircraft data available in the literature, a mean
axes model and a fixed axes model were developed. Simulations were run with a couple
of inputs to the models to obtain a comprehensive amount of data for performing the
comparative analysis. The first input used was an impulse composed of modal frequencies
obtained from Finite Element Analysis of the aircraft structure. The other input used
was a 2311 input which was used to excite a wide range of frequencies in the aircraft
dynamics. Each of these inputs was used in both longitudinal (elevator) and lateral
(aileron) control inputs.
Time history responses of the two models were compared with one another. It was
found that the two models act in a very similar manner for all inputs tested in both
lateral and longitudinal dimensions. However, comparing the response of the flexible
models with a rigid aircraft model revealed the fact that the aircraft considered in this
study is not very flexible by nature.
To overcome this problem, the stiffness of the aircraft models was scaled down and
the simulations were re-run. The effect of the increased flexibility on the response of
the two models was analyzed by comparing simulation time histories of the half-stiffness
and tenth-stiffness models. The longitudinal response of the two models was found to
resemble each other for both decreased stiffness cases. However, in the lateral case, it was
noticed that the tenth stiffness mean axes model diverges. The same behaviour was not
noticed in the tenth stiffness fixed axes model’s response. The cause for this divergent
behaviour was examined, however, no firm explanation has been found so far.
As a result of this study, a number of recommendations are made for future work
78
Chapter 7. Conclusions and Recommendations
79
in the field of flexible aircraft flight simulation. First of all, further effort is needed to
determine the cause of the divergent behaviour in the response of the mean axes model,
most evident in the tenth-stiffness cases . Second, instead of just scaling the stiffness
of the aircraft model, the whole structure of the aircraft needs to be modified to cause
more oscillatory response to control inputs in the simulations. This will allow for a more
dynamic analysis to be performed to study the performance of the two flexible flight
models. Third, to improve the fidelity of the models, a more complete aerodynamic
computation technique is needed to replace the simple aerodynamic strip theory used
in this study. Such a technique will result in capturing non-linear aerodynamic effects
which are currently ignored. Fourth, by optimizing the mode shapes included in the
two flexible models, it may be possible to obtain similar oscillations from both models.
Further effort is required in this area to help obtain a conclusive result. Last, a handling
quality analysis is recommended to study the response of the two models in a motion
flight simulator with human pilots. This will provide greater insight into the fidelity of
the model in simulating the flight dynamics of the flexible aircraft being modeled.
References
[1] C. S. Buttril, T. A. Zeiler, and P. D. Arbuckle. Nonlinear simulation of a flexible
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[2] L. Meirovitch. A unified theory for the flight dynamics and aeroelasticity of whole
aircraft. In Proceedings of the Eleventh Symposium on Structural Dynamics and
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[3] L. Meirovitch and I. Tuzcu. Integrated approach to the dynamics and control of
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[4] L. Meirovitch and I. Tuzcu. The lure of mean axes. Journal of Applied Mechanics,
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M. eng. research project, University of Toronto Institure for Aerospace Studies,
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