Vibration Suppression of Large Space Structures using an
Optimized Distribution of Control Moment Gyros
by
Stephen Alexander Chee
A thesis submitted in conformity with the requirementsfor the degree of Masters of Applied ScienceGraduate Department of Aerospace Studies
University of Toronto
Copyright c© 2011 by Stephen Alexander Chee
Abstract
Vibration Suppression of Large Space Structures using an Optimized Distribution of
Control Moment Gyros
Stephen Alexander Chee
Masters of Applied Science
Graduate Department of Aerospace Studies
University of Toronto
2011
Many space vehicles have been launched with large flexible components such as booms and
solar panels. These large space structures (LSSs) have the potential to make attitude control
unstable due to their lightly damped vibration. These vibrations can be controlled using a
collection of control moment gyros (CMGs). CMGs consist of a spinning wheel in gimbals
and produce a torque when the orientation of the wheel is changed. This study investigates
the optimal distribution of these CMGs on LSSs for vibration suppression. The investigation
considers a beam and a plate structure with evenly placed CMGs. The optimization allocates
the amount of stored angular momentum possessed by these CMGs according to a cost
function dependent on how quickly vibration motions are damped and how much control
effort is exerted. The optimization results are presented and their effect on the motions of
the beam and plate are investigated.
ii
Acknowledgements
I would like to thank my supervisor Dr. Christopher Damaren for his guidance in this project
and throughout my studies at UTIAS. I would like to also extend my thanks to my family,
friends, and peers for their support and encouragement.
iii
Contents
Abstract ii
Acknowledgements iii
List of Tables vi
List of Figures vii
1 Introduction 1
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 System Dynamics and Control 4
2.1 Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Energy and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.3 Motion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Plate Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.2 Energy and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 Motion Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.1 Elastic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.2 Gyroelastic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Stabilization using State Feedback . . . . . . . . . . . . . . . . . . . . 31
2.4.2 Linear Quadratic Regulator . . . . . . . . . . . . . . . . . . . . . . . 32
iv
3 The Optimization Problem 35
3.1 Optimization Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Gradient-Based Optimization with Equality Constraints . . . . . . . . . . . 36
3.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.1 Beam Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3.2 Plate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Optimization Results 42
4.1 Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.1 Placement of a Single CMG . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Distribution of Multiple CMGs . . . . . . . . . . . . . . . . . . . . . 44
4.1.3 Cantilevered Boundary Conditions . . . . . . . . . . . . . . . . . . . 45
4.2 Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Placement of a Single CMG . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 Distribution of Multiple CMGs . . . . . . . . . . . . . . . . . . . . . 61
5 System Response 71
5.1 Initial Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 Beam Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.2 Plate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Conclusions 82
Bibliography 84
Appendices 87
A Optimized Distributions 87
B Distribution Costs 90
v
List of Tables
3.1 Beam Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Plate Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1 Initial Deformation Beam Performance for c = 1.00 . . . . . . . . . . . . . . 73
5.2 Initial Deformation Beam Performance for c = 0.01 . . . . . . . . . . . . . . 74
5.3 Initial Deformation Plate Performance for c = 1.00 . . . . . . . . . . . . . . 78
5.4 Initial Deformation Plate Performance for c = 0.01 . . . . . . . . . . . . . . 78
A.1 Scaled Optimized Distribution for Free Beam . . . . . . . . . . . . . . . . . . 88
A.2 Scaled Optimized Distribution for Cantilevered Beam . . . . . . . . . . . . . 88
A.3 Scaled Optimized Distribution for Free Plate . . . . . . . . . . . . . . . . . . 89
B.1 Distribution Objective Function Values for Free Beam (Scaled) . . . . . . . . 90
B.2 Distribution Objective Function Values for Cantilevered Beam (Scaled) . . . 91
B.3 Distribution Objective Function Values for Free Plate . . . . . . . . . . . . . 91
vi
List of Figures
2.1 Free Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Cantilevered Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Beam CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Free Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Plate FEM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Plate CMG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 CMG Distribution for the Beam . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 CMG Distribution for the Plate . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.1 The Single CMG Problem for a Free Beam . . . . . . . . . . . . . . . . . . . 42
4.2 Cost of the Placement of a Single Gyro for a Free Beam . . . . . . . . . . . . 47
4.3 Free Beam Elastic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Free Beam Single CMG Gyroelastic Modes (h = 0.01) . . . . . . . . . . . . . 49
4.5 Free Beam Single CMG Gyroelastic Modes (h = 1.00) . . . . . . . . . . . . . 50
4.6 Free Beam Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.7 Free Beam CMG Distribution Gyroelastic Modes (c = 0.01) . . . . . . . . . 52
4.8 Free Beam CMG Distribution Gyroelastic Modes (c = 1.00) . . . . . . . . . 53
4.9 Cost of the Placement of a Single Gyro for a Cantilevered Beam . . . . . . . 54
4.10 Cantilevered Beam Elastic Modes . . . . . . . . . . . . . . . . . . . . . . . . 55
4.11 Cantilevered Beam Single CMG Gyroelastic Modes (h = 0.01) . . . . . . . . 56
4.12 Cantilevered Beam Single CMG Gyroelastic Modes (h = 1.00) . . . . . . . . 57
4.13 Cantilevered Beam Optimization Results . . . . . . . . . . . . . . . . . . . . 58
4.14 Cantilevered Beam CMG Distribution Gyroelastic Modes (c = 1.00) . . . . . 59
4.15 The Single CMG Problem for a Free Plate . . . . . . . . . . . . . . . . . . . 60
4.16 Cost of the Placement of a Single Gyro for a Plate . . . . . . . . . . . . . . . 63
4.17 Free Plate Elastic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
vii
4.18 Free Plate Single CMG Gyroelastic Modes (h = 0.01) . . . . . . . . . . . . . 65
4.19 Free Plate Single CMG Gyroelastic Modes (h = 1.00) . . . . . . . . . . . . . 66
4.20 Plate Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.21 Plate Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.22 Free Plate CMG Distribution Gyroelastic Modes (c = 0.01) . . . . . . . . . . 69
4.23 Free Plate CMG Distribution Gyroelastic Modes (c = 1.00) . . . . . . . . . . 70
5.1 Initial Deformation Beam Response for Optimal Distribution . . . . . . . . . 74
5.2 Initial Deformation Beam Response for Uniform Distribution . . . . . . . . . 75
5.3 Initial Deformation Beam Response for Ends Distribution . . . . . . . . . . . 76
5.4 Initial Deformation Plate Response for Optimal Distribution . . . . . . . . . 79
5.5 Initial Deformation Plate Response for Uniform Distribution . . . . . . . . . 80
5.6 Initial Deformation Plate Response for Corner Distribution . . . . . . . . . . 81
viii
Chapter 1
Introduction
Over the past decades, many large space vehicles have been launched. One prominent
example of such a vehicle is the International Space Station (ISS). The ISS is composed of
many large components with low stiffness due to the design restrictions on mass including
its main integrated truss which spans 110 m and its solar arrays which are 34 m by 12 m.
With the deployment of large space structures, control is complicated since the body can
no longer be considered rigid. These large spacecraft are lightly damped and vibrate at low
frequencies. These flexible motions have little stability margin and can be made unstable
by attitude control. There exists an extensive body of work investigating the dynamics and
control of these structures, and more information into these topics can be found in [1] and [2].
Control moment gyros (CMGs) have a long legacy as actuators used for attitude control
of spacecraft. A CMG consists of a wheel that is rotating at constant speed with gimbals
facilitating the ability to change the wheel’s axis of rotation. When attached to a space
system, imposing a rate change of the gimbal angles results in a change in the angular
momentum of the rest of the system. A comprehensive description of the dynamics of
CMGs can be found in [3] and [4]. CMGs are typically single gimballed or double gimballed.
Single gimballed control moment gyros (SGCMGs) have the advantage of having less mass
and occupying less volume than double gimballed control moment gyros (DGCMGs), which
are desirable characteristics for any component used in space systems. However, having only
a single gimbal restricts the direction of the torques SGCMGs can produce. When used in a
multiple set for attitude control, there exist singularities in which the set of SGCMGs are not
able to produce a net torque. Research has been done to develop control methods to avoid
these singularities and to escape them when they are encountered [5]. Sometimes CMGs
are used in a couple called V-gimballed Gyroscopes or Scissored-pair CMGs. Scissored-pair
1
Chapter 1. Introduction 2
CMGs consist of two CMGs placed such that their gimbal axes are parallel and during
operation their gimbal angles are equal in magnitude and opposite in direction. The appeal
of this configuration is that it produces torques in a single axis and avoids the otherwise
occurring reaction torques. Investigations have been made into the application of Scissored-
pair CMGs to robotics as discussed in [6]. The ability to produce a torque in a single axis
is well suited for the application because the Scissored-pair CMGs could be used to produce
torques in the joint axis.
CMGs are typically used for the fine-pointing of spacecraft. However, work has been
done on the suppression of vibrations in flexible bodies using CMGs and other spinning
wheel actuators. Aubrun and Margulies, in [7], present a preliminary study into the use of
a V-gimballed CMG to damp vibrations in large flexible structures. One of the findings of
this study was that having many small CMGs may be advantageous to having a single large
CMG for damping purposes. Hablani and Skelton consider a distribution of rotors for the
control of a flat plate spacecraft model in [8]. In [9], Shi and Damaren develop a control law
for vibration suppression of a cantilevered beam using a SGCMG at the free tip and present
experimental results. Study [10] investigates the use of a CMG for the manoeuvring and
vibration suppression of a flexible truss arm undergoing constant slewing motion. This study
is primarily concerned with the performance of the feedforward/feedback control scheme used
by the CMGs to accomplish both manoeuvring and vibration suppression. During this study,
an optimization is also performed for the placement of the CMG to minimize vibration during
slewing manoeuvres.
There also exists a body of work pertaining to the consideration of gyroelastic bodies.
Following the notion that many CMGs are advantageous for vibration suppression, these
studies treat gyroscopic influences, infinitesimal amounts of stored angular momentum dis-
tributed about an elastic body, as an inherent property of the body as mass and stiffness.
D’Eleuterio, in [11], derives the partial differential equations of motions of a gyroelastic con-
tinuum and gyroelastic vibration modes are found that can be used as basis functions to
describe the general motions of the continuum. In [12], Damaren derives the controllability
and observability conditions for vibration suppression and shape control of a gyroelastic body
in terms of gyroelastic modes. The elastic bodies considered in these studies are continua
such as beams and plates. In space applications, structures are often composed of truss-like
structures whose overall behaviour can closely mimic those of a beam or a plate. Studies
have been done into methods to model the properties of these truss structures so that they
can be treated as continuous body beams and plates. Some of these techniques are reviewed
Chapter 1. Introduction 3
by Noor in [13].
1.1 Objectives
This study is concerned with finding the optimal distribution of DGCMGs used for vibration
suppression of elastic bodies. This problem is approached through considering a series of
DGCMGs placed uniformly across a free beam, a cantilevered beam, and a free plate, and
using numerical optimization to allocate the amount of angular momentum stored by the
CMGs’ spinning wheels. CMGs with more stored angular momentum will have more control
authority since they can produce larger torques using smaller changes in their gimbal angles.
Thus this optimization will effectively place more stored angular momentum where it will
benefit the control objective.
1.2 Outline of the Thesis
To begin an investigation into this problem, a model for the system is required. Chapter 2
derives the motion equations for the free beam and plate and describes the control law used
for vibration suppression. The beam model is derived using a Ritz discretization and the
plate model is derived using a finite element discretization. Using these models, the kinetic
and potential energies are derived, and the motion equations are found by using Lagrange’s
equations. These equations are described in modal coordinates, and since the model for the
system is linear, a modal analysis can provide insight into the response of these systems. A
discussion of the elastic and gyroelastic modes of these models are discussed in Section 2.3.
Finally, the Linear Quadratic Regulator scheme used to control the vibration of the system
is described in Section 2.4.
Chapter 3 describes the formulation of the optimization objective used in this analysis.
This chapter also gives an brief overview of gradient-based optimization, the class of the
method used to reach an optimized result according to the optimization objective. This is
followed by a discussion of the optimization results in Chapter 4.
Chapter 5 simulates the response of the optimized systems to an initial elastic deforma-
tion. The results of the simulations are used for a comparison of the response of beams and
plates with other distributions of stored angular momentum for sets of DGCMGs.
Chapter 2
System Dynamics and Control
Flexible bodies such as beams and plates are considered to be continuous systems whose mo-
tions can be described using differential equations and a set of boundary conditions. With
the exception of very simple problems, it very difficult to solve these equations analytically
so numerical methods are typically used to approximate the continuous system as a discrete
system. In the current analysis, the Ritz Method and the Finite Element Method are em-
ployed to discretize the gyroelastic systems and Lagrange’s equations of motion are employed
to describe the dynamics of the system. Descriptions of the kinematics of the systems employ
the use of vectrix notation; for a description of this notation refer to [14].
The resulting system of motion equations is linear, and thus a modal decomposition
can be performed to describe the motions of the system in modal coordinates. The elastic
modes of the beam and plate are used to develop a state feedback controller for vibration
suppression. The state feedback controller takes the form of a Linear Quadratic Regulator
which yields an optimal controller in terms of the energy associated with the systems’ elastic
modes and the control effort of the CMGs.
2.1 Beam Model
2.1.1 Kinematics
Consider a flexible beam undergoing bending with free boundary conditions and cantilevered
boundary conditions. These systems are illustrated in figures 2.1 and 2.2.
4
Chapter 2. System Dynamics and Control 5
F−→0
F−→def = CdefF−→und
1,x
2,y
3,z
1
2
3
F−→und
1
2
3 ρ−→
Figure 2.1: Free Beam
F−→0
F−→def = CdefF−→und
1,x
2,y
3,z
1
2
3
F−→und
1
2
3ρ−→
Figure 2.2: Cantilevered Beam
Chapter 2. System Dynamics and Control 6
The beam is assumed to have uniform cross-section, uniform density per spanwise length
ρ, and has spanwise length l. The stiffness properties of the beam are given by B2 for
the stiffness to bending in the 2-axis, B3 for the stiffness to bending in the 3-axis, and the
parameter B is defined as B =√B2B3.
To describe the bending motions of the beam, three frames are introduced: the inertial
frame, F−→0, the local frame, F−→und, and the deformed frame, F−→def. F−→0 is a right-handed
orthogonal frame in inertial space which will serve as a reference frame for the full body
motions of beam. Positions in the 1 direction are denoted by the variable x, positions in
the 2 direction are denoted by y, and positions in the 3 direction are denoted by z. For
free boundary conditions, it is assumed that the beam is nominally placed in this reference
frame such that its center is at the origin of F−→0 and its span is in the 1-axis. It is also
assumed that the displacements of the beam are small and are nominally about the positions
x ∈ [−l/2, l/2], y = 0, and z = 0. For this reason, the frame F−→und is defined with the same
orientation as the inertial frame and attached at a position along the inertial 1 direction with
x ∈ [−l/2, l/2]. F−→def is a right-handed reference frame attached to the deformed beam with
the 1 direction pointed tangent to the line drawn through the center of the cross-sections of
the beam and the 2 and 3 directions are orthogonal as depicted in figure 2.1. As consistent
with classical beam theory, the beam is essentially treated as a one-dimensional body in
which displacements of the line through the center of the cross-section along the span of the
beam are considered. For cantilevered boundary conditions, it is assumed that the fixed end
of the beam is at the origin of the inertial frame and the other two frames are defined in the
same matter as the free beam case except that the span of the beam is over x ∈ [0, l].
For small bending deflections, displacements in the inertial 1 direction are considered
negligible. Thus, motions are treated as being planar in the 2-3 plane. As depicted in figures
2.1 and 2.2, deformation of the beam from F−→und to F−→def is given by a translation described
by ρ−→ and a rotation that can be described by the rotation matrix Cdef = F−→def · F−→Tund [14].
Since the motions are considered planar, ρ−→ is given by
ρ−→ = F−→Tund
0
v(x, t)
w(x, t)
,
where the component of the deflection in the 2 direction is denoted by v(x, t) and the
component of the deflection in the 3 direction is denoted by w(x, t). This represents a
Chapter 2. System Dynamics and Control 7
continuous displacement field along the span of the beam. To discretize the continuous
system, a Ritz discretization is used in which displacements are described as an expansion
in terms of basis functions
v(x, t) =N∑
α=1
ψv,α(x)qv,α(t)
= [ψv,1(x) · · ·ψv,N(x)]︸ ︷︷ ︸
ΨTv (x)
qv,1(t)...
qv,N(t)
︸ ︷︷ ︸
qv(t)
= ΨTv (x)qv(t),
w(x, t) =N∑
α=1
ψw,α(x)qw,α(t)
= [ψw,1(x) · · ·ψw,N(x)]︸ ︷︷ ︸
ΨTw(x)
qw,1(t)...
qw,N(t)
︸ ︷︷ ︸
qw(t)
= ΨTw(x)qw(t),
where qv,α(t) and qw,α(t) are undetermined coefficients dependent on time, and ψv,α(x) and
ψw,α(x) are specified basis functions parameterized by α. With this separation of variables,
the translational velocities of the deformation field are given by
v(x, t) = ΨTv (x)qv(t),
w(x, t) = ΨTw(x)qw(t).
In the current analysis, the basis functions are chosen to be the exact mode shapes of an
elastic free beam as derived from classical beam theory for beams with uniform cross-section
and density. For the free beam, the basis functions for the elastic modes are given by
ψv,α(x) = ψw,α(x) =cosh(λαx) + cos(λαx) + Cα[sinh(λαx) + sin(λαx)]√
ρl
Chapter 2. System Dynamics and Control 8
[15], where x = x/l + 1/2,
Cα =cos(λα)− cosh(λα)
sinh(λα)− sin(λα),
and λα satisfies the frequency equation
cosλα coshλα = 1.
The λα terms are parameterized such that λ1 is the solution to the frequency equation with
the smallest magnitude, and λα increases in magnitude for increasing α. It is apparent that
λα = 0 satisfies the frequency equation. The solutions corresponding to λα = 0 are the free
beam’s rigid body modes. The rigid rotation modes for the free beam, corresponding to
α = 1, are given by
ψv,1 = ψw,1 =x
√112ρl3
, (2.1)
and the translational rigid modes are omitted because they are uncontrollable with CMGs.
For control of rigid body translation, another actuation system, such as an assortment of
thrusters, is required. For the cantilevered beam, the basis functions for the elastic modes
are give by
ψv,α(x) = ψw,α(x) =cosh(λαx)− cos(λαx)− Cα[sinh(λαx)− sin(λαx)]√
ρl
[15], where x = x/l,
Cα =coshλα + cosλαsinhλα + sinλα
,
and λα satisfies the frequency equation
cosλα coshλα = −1.
Recall that the local rotations of the deformed beam from the local frame can be described
by
Cdef = F−→def · F−→Tund.
Let ω−→ represent the angular velocity of F−→def with respect to F−→und, where the magnitude
of ω−→ is the rate of rotation and the direction of ω−→ is the instantaneous axis of rotation.
Expressed in F−→def,
ω−→ = F−→Tdefωd.
Chapter 2. System Dynamics and Control 9
To find a representation of ωd in terms of the rotation matrix, recall that the rotation
from F−→und to F−→def is given by
F−→Tund = F−→
TdefCdef
Taking the time derivative from of both sides as seen in F−→und yields
0−→ = F−→T
defCdef + F−→
TdefCdef. (2.2)
From vector calculus it can be shown that the time derivative of F−→def as seen in F−→und is
given by
F−→def= ω−→× F−→def.
Substituting this result in equation 2.2 yields
0−→ = ω−→× F−→TdefCdef + F−→
TdefCdef
= ωTd F−→def × F−→
TdefCdef + F−→
TdefCdef
= F−→Tdef(ω
×d Cdef + Cdef).
So the time derivative of Cdef is given by
Cdef = −ω×d Cdef
Thus in the deformed frame ω−→ can be expressed in terms of the rotation matrix as
ω×d = − CdefC
−1def
= − CdefCTdef.
The rotations of F−→def from F−→und is a direct result of the displacement field due to bending.
Furthermore, these rotations are also continuous along the beam and can be approximated
using the same discretization as for the displacement field when small deflections are con-
sidered. As a first step toward this discretized representation, consider Euler’s Theorem.
Euler’s Theorem states that the general displacement of a rigid body with one fixed point is
a rotation about an axis through that point. It follows from this theorem that the rotation
from F−→und to F−→def can be expressed by a rotation ϕ about an axis described by the unit
Chapter 2. System Dynamics and Control 10
vector a. The resulting rotation matrix is of the form
Cdef = cosϕ1+ (1− cosϕ)aaT − sinϕa×.
It can be shown that this parameterization of the rotation allows the angular velocity ω to
be described by
ω = ϕa− (1− cosϕ)a×a+ sinϕa.
Recognize that the second order approximation of the Taylor expansion of cosϕ and sinϕ
are give by
cosϕ.= 1− 1
2ϕ2, sinϕ
.= ϕ.
Substituting this expansion in ω and defining ϕ = ϕa yields
ω = ϕa− 12ϕ2a×a+ ϕa
= ϕa+ ϕ2a︸ ︷︷ ︸
ϕ
−12ϕa×a− 1
2ϕ a×a︸︷︷︸
0
= ϕ− 12ϕa×(ϕa+ ϕa)
= ϕ− 12ϕ×ϕ.
For small rotations of the deformed beam about the local 2- and 3- axes due to bending, ϕ
can be approximated as
ϕ(x, t) =
0
ϕ2(x, t)
ϕ3(x, t)
=
0
−∂w∂x(x, t)
∂v∂x(x, t)
.
These rotations can be expressed using the Ritz discretization used for describing the dis-
placement field. If the derivatives of the basis functions used to describe the displacments
are denoted by
φv,α =∂ψv,α
∂xφw,α =
∂ψw,α
∂x,
Chapter 2. System Dynamics and Control 11
then the derivative of the displacements can be expressed as
∂v
∂x(x, t) =
N∑
α=1
φv,α(x)qv,α(t)
= [φv,1(x) · · ·φv,N(x)]︸ ︷︷ ︸
ΦTv (x)
qv,1(t)...
qv,N(t)
︸ ︷︷ ︸
qv(t)
= ΦTv (x)qv(t),
∂w
∂x(x, t) =
N∑
α=1
φw,α(x)qw,α(t)
= [φw,1(x) · · ·φw,N(x)]︸ ︷︷ ︸
ΦTw(x)
qw,1(t)...
qw,N(t)
︸ ︷︷ ︸
qw(t)
= ΦTw(x)qw(t).
Thus, the rotations are given by
ϕ(x, t) =
0
−ΦTw(x)qw(t)
ΦTv (x)qv(t)
,
and the angular velocity of the beam in the local deformed frame, as a second order approx-
imation, is given by
ω.= ϕ− 1
2ϕ×ϕ.
Since both the translations and rotations of the beam can be expressed in terms of the
Ritz expansion, the global vector of degrees of freedom used to express the motion equation
is defined according to the weighting of the elastic modes of the system:
q =
[
qv
qw
]
.
Double gimballed control moment gyros (DGCMGs) are used to control the vibrations
Chapter 2. System Dynamics and Control 12
2
β2
1
3
β3
ωs
F−→
CMG = F−→
def
(a) β2 = β3 = 0
F−→CMG
F−→def
1
1
2
2
3
3
γ
(b) β2, β3 6= 0
Figure 2.3: Beam CMG
of the beam. The DGCMGs rotors are rotating at a constant angular velocity ωs and are
gimballed so that they are free to rotate through β2 and β3 as illustrated in figure 2.3(a).
Consider a single CMG. Let F−→CMG be a frame positioned at the center of the rotor and
rotated from the local deformed frame F−→def through the gimbal angles. When the gimbal
angles are zero, F−→CMG coincides with F−→def and the spin axis of the rotor is pointed parallel
to the spanwise direction of the beam. When the gimbal angles are non-zero, the rotation
of the rotor in the deformed frame is given by angle γ and the axis a as illustrated in figure
2.3(b). The rotation matrix from F−→def to F−→CMG is
CCMG = cos γ1+ (1− cos γ)aaT − sin γa×
and the angular velocity of the rotation from F−→def to F−→CMG is given by
ω = γa− (1− cos γ)a×a+ sin γa.
Let γ = γa. Then for small β2 and β3,
γ =
0
β2
β3
.
Chapter 2. System Dynamics and Control 13
The first order approximation for the rotation matrix is
CCMG.= 1− γ×,
and the first order approximation for the angular velocity of F−→CMG in the deformed frame
is
ωCMG.= γ =
0
β2
β3
.
2.1.2 Energy and Work
The dynamical equations for this system can be found using Lagrange’s equations
d
dt
(∂L
∂q
)
− ∂L
∂q= f,
where L is the Lagrangian, q are the generalized coordinates, and f is the total generalized
non-conservative force on the system. These equations can be arrived at through Hamilton’s
Principle and the Calculus of Variations.
To use these equations, the Lagrangian must first be found. The Lagrangian is defined
as L = T − V , where T is the kinetic energy of the system and V is the stored potential
energy of the system.
Kinetic Energy
The kinetic energy of the system is given by the kinetic energy of the beam and the kinetic
energy of the gyros
T = Tbeam +∑
i
Tgyro,i.
Assuming that the beam has a uniform cross section and uniform density per spanwise
length, ρ, then for small displacements v and w the kinetic energy of the beam is given by
Tbeam = 12ρ
∫ l
0
v2dx+ 12ρ
∫ l
0
w2dx
= 12
[
qTv qT
w
][
M 0
0 M
][
qv
qw
]
= 12qTMq,
Chapter 2. System Dynamics and Control 14
where M is the mass matrix which can be thought of as accounting for the inertial influences
of the beam. It has the form
Mαβ = ρ
∫ l
0
ψαψβdx,
and is symmetric and positive definite.
In the present configuration, the rotor of each CMG is nominally spinning about the
1-axis of F−→CMG with constant angular velocity ωs and is gimballed so that it can rotate
through β2 and β3 as illustrated in figure 2.3(a). If the moment of inertia of the rotor about
the spin axis is denoted by Is, then the nominal angular momentum for the i-th CMG is
hs,i = Is,iωs,i.
In the CMG frame, the angular momentum of the rotor is given by
hi,CMG =
hs,i
0
0
In the deformed frame, the angular momentum of the rotor is given by
hi,def = CCMGhi,CMG
Using the first order approximation for the rotation matrix, this yields
hi = (1− γ×)hi,CMG =
1 −β3 β2
β3 1 0
−β2 0 1
hs,i
0
0
=
hs,i
hs,iβ3,i
−hs,iβ2,i
.
The kinetic energy of a single gyro is given by
Tgyro,i = hTi ωi
= hTi (αi − 1
2α×
i αi)
= (hs,iβ3,i ˙α2,i − hs,iβ2,i ˙α3,i)− 12hs,i(α2,iα3,i − α3,iα2,i)
= − hx,iβ3,iΦTw,iqw − hs,iβ2,iΦ
Tv,iqv
− 12hs,iq
TvΦv,iΦ
Tw,iqw + 1
2hs,iq
TwΦw,iΦ
Tv,iqv
Chapter 2. System Dynamics and Control 15
= −[
qTv qT
w
][
hs,iΦv,i 0
0 hs,iΦw,i
]
︸ ︷︷ ︸
Bi
[
β2,i
β3,i
]
︸ ︷︷ ︸
βi
+ 12
[
qTv qT
w
][
0 −hs,iΦv,iΦTw,i
hs,iΦw,iΦTv,i 0
]
︸ ︷︷ ︸
Gi
[
qv
qw
]
= − qTBiβi +12qTGiq (2.3)
For multiple gyros the kinetic energy can be expressed as
Tgyros =N∑
i=1
Tgyro,i
= − qT[
B1 B2 . . . BN
]
︸ ︷︷ ︸
B
β1
β2
...
βN
︸ ︷︷ ︸
β
+12qT (G1 +G2 + . . .+GN)
︸ ︷︷ ︸
G
q
= − qTBβ + 12qTGq,
where B is the input matrix and G is the gyric matrix. The input matrix accounts for the
forces resulting from the control inputs and the gyric matrix are the influences due to the
stored angular momentum of the CMGs. The gyric matrix is skew-symmetric.
Thus, the kinetic energy for the system can be expressed as
T = 12qTMq+ 1
2qTGq− qTBβ.
Strain Energy
The strain energy of the system is due to the bending of the beam. Classical beam theory
yields
V = 12B2
∫ l
0
v2xx(x, t)dx+12B3
∫ l
0
w2xx(x, t)dx.
If the Ritz expansion for v and w is substituted into the strain energy, that yields
V = 12qTvKvqv +
12qTwKwqw,
Chapter 2. System Dynamics and Control 16
where
Kv = B2
∫ l
0
ψα,xxψβ,xxdx,
Kw = B3
∫ l
0
ψα,xxψβ,xxdx.
Thus the strain energy can be expressed as
V = 12qTKq
with
K =
[
Kv 0
0 Kw
]
,
where K is stiffness matrix. It is symmetric and positive definite for the cantilevered beam
and positive semi-definite for the free beam.
Non-Conservative Work
For this model, structural damping cannot be predicted and thus it is common practice to
assume proportional damping [2]. The non-conservative work due to proportional damping
is given as
δWnc = δqTDq,
where D is the damping matrix. It has the form D = diag{2ζαωα}, where the ζα terms are
the damping ratios and the ωα terms are the elastic beam frequencies associated with the
mode shapes chosen as our basis functions. For large flexible space structures, the ζα’s are
normally taken to be between 0.001 and 0.01 [2]. The elastic beam frequencies can be found
by solving the following eigenvalue problem
ω2iMei = Kei
where ωi is the natural frequency associated with the i-th mode, and ei is the corresponding
eigenvector. The generalized force vector associated with damping is given as
f = −Dq.
Chapter 2. System Dynamics and Control 17
The damping matrix D is positive definite for the cantilevered beam and positive semi-
definite for the free beam.
2.1.3 Motion Equations
From the kinetic and strain energies T and V , the Lagrangian, L = T − V , is formed. Fi-
nally, substituting the Lagrangian and the generalized non-conservative force into Lagrange’s
equationsd
dt
(∂L
∂q
)
− ∂L
∂q= f,
yields
d
dt
(∂L
∂q
)
=d
dt
(∂
∂q(12qTMq+ 1
2qTGq− qTBβ − 1
2qTKq)
)
=d
dt
(Mq+ 1
2Gq−Bβ
)
= Mq+ 12Gq−Bβ,
∂L
∂q=
∂
∂q(12qTMq+ 1
2qTGq− qTBβ − 1
2qTKq)
= −12Gq−Kq,
(
Mq+ 12Gq−Bβ
)
+ (12Gq+Kq) = −Dq,
and rearranging the terms results in
Mq+ (G+D)q+Kq = Bβ. (2.4)
In this formulation, M is the positive-definite mass matrix, G is the skew-symmetric gyric
matrix due to the stored angular momentum of the CMGs, D is the viscous damping matrix,
K is the stiffness matrix, β is a vector of the gimbal angles of the CMGs, and B is the input
matrix. With cantilevered boundary conditions D and K are positive-definite matrices, and
with free boundary conditions D and K are positive-semidefinite matrices.
Chapter 2. System Dynamics and Control 18
2.2 Plate Model
2.2.1 Kinematics
In the current analysis, a flexible plate undergoing bending is considered with free boundary
conditions. This system is illustrated in figure 2.4. The plate has a length a in the 1 direction,
F−→
0
F−→
und
F−→
def = CdefF−→
und
ρ−→
1, x
2, y
3, z
Figure 2.4: Free Plate
length b in the 2 direction, and thickness t in the 3 direction.
As with the beam case, consider the following three frames: the inertial frame, F−→0, the
local frame, F−→und, and the deformed frame, F−→def. The inertial frame F−→0 is a right-handed
reference frame attached to the center of the nominal position of the undeformed plate as
depicted in figure 2.4. The 1- and 2- axes are parallel to the edges of the undeformed plate.
Positions in the 1 direction are denoted by the variable x, positions in the 2 direction are
denoted by y, and positions in the 3 direction are denoted by z. It is assumed that the
displacements of the plate are small and are nominally about the positions x ∈ [−a/2, a/2],y ∈ [−b/2, b/2], and z = 0. For this reason, the frame F−→und is defined with the same
orientation as the inertial frame and attached at a position in the 1-2 plane of the inertial
frame with x ∈ [−a/2, a/2] and y ∈ [−b/2, b/2]. F−→def is a right-handed reference frame
attached to the deformed plate with the 3 direction pointed perpendicular to the plate as
shown in figure 2.4.
For small bending deflections, displacements in the inertial 1 and 2 directions are con-
sidered negligible. Thus, motions are treated as being solely in the 3 direction. As depicted
in figures 2.4, deformation of the beam from F−→und to F−→def is given by translation described
by ρ−→ and a rotation that can be described by the rotation matrix Cdef = F−→def · F−→Tund.
Chapter 2. System Dynamics and Control 19
Since the motions are considered only in the 3 direction, ρ−→ is given by
ρ−→ = F−→Tund
0
0
w(x, y, t)
.
This represents a continuous displacement field throughout the plate. To discretize the
continuous system, the Finite Element Method is used. For this case rectangular plates
composed of rectangular finite elements are considered. A 16 degrees of freedom model is
used as developed in Zienkiewicz [16].
1,x
2,y
3,z,w
1,xe
3
2,ye
1 2 3
4 i
7
l
kj
Figure 2.5: Plate FEM model
Consider a rectangular element with corners denoted i, j, k, and l, as illustrated in figure
2.5. The degrees of freedom at node n is defined by an, and the degrees of freedom for the
element is defined by ae for nodes n = i, j, k, l:
an =[
wn
(∂w∂x
)
n
(∂w∂y
)
n
(∂2w∂x∂y
)
n
]T
ae =[
aTi aT
j aTk aT
l
]T
where wn is the out of plane displacement at node n. In this element model, the displacements
Chapter 2. System Dynamics and Control 20
are approximated using the following Hermitian polynomial functions
H101(x) = 1− 3
x2
L2+ 2
x3
L3
H111(x) = x− 2
x2
L+x3
L2
H102(x) = 3
x2
L2+ 2
x3
L3
H112(x) = −x
2
L+x3
L2.
This allows the displacement field to be expressed as
w(xe, ye) = Nae,
where N is a vector containing the shape functions with respect to local element coordinates
xe and ye [16]. This vector has the form
N = [ Ni Nj Nk Nl ],
where
Ni = [ H101(x)H
101(y) H1
11(x)H101(y) H1
01(x)H111(y) H1
11(x)H111(y) ]
Nj = [ H102(x)H
101(y) H1
12(x)H101(y) H1
02(x)H111(y) H1
12(x)H111(y) ]
Nk = [ H102(x)H
102(y) H1
12(x)H102(y) H1
02(x)H112(y) H1
12(x)H112(y) ]
Nl = [ H101(x)H
102(y) H1
11(x)H102(y) H1
01(x)H112(y) H1
11(x)H112(y) ].
These functions relate the nodal degrees of freedom to the displacement field. The system’s
degrees of freedom can then be expressed as
q =
a1
...
aN
.
Since a simple plate structure is considered, the local coordinates do not have to be rotated
to be expressed in the inertial frame.
Recall that the rotation of the deformed frame from the local frame is given by the
Chapter 2. System Dynamics and Control 21
rotation matrix
Cdef = F−→def · F−→Tund.
As with the beam case, this rotation can be expressed in terms of an angle ϕ about an axis
a. By letting ϕ = ϕa, the second-order approximation of the angular velocity of F−→def with
respect to F−→und as seen in F−→def is given by
ω.= ϕ− 1
2ϕ×ϕ.
Since the displacements are assumed small, the rotations are also small and the components
of ϕ can be approximated using the slopes of the displacement field,
ϕ =
ϕ1
ϕ2
0
=
(∂w∂y
)
−(∂w∂x
)
0
.
As with the beam case, it is desired that the system equations be described in terms
of the elastic modal coordinates. To achieve this end, the eigenvalue problem of the elastic
system must be considered. Thus it is necessary to develop the motion equations in terms
of the nodal degrees of freedom before performing the coordinate transformation.
As for the beam case, DGCMGs are used to control the vibrations of the plate. The
DGCMGs rotors are rotating at a constant angular velocity ωs and are gimballed so that
they are free to rotate through β1 and β2 as illustrated in figure 2.6(a). Consider a single
CMG. Let F−→CMG be a frame positioned at the center of the rotor and rotated from the
local deformed frame F−→def through the gimbal angles. When the gimbal angles are zero,
F−→CMG coincides with F−→def and the spin axis of the rotor is perpendicular to the surface of
the deformed plate. When the gimbal angles are non-zero, the rotation of the rotor in the
deformed frame is given by angle γ and the axis a as illustrated in figure 2.6(b). The rotation
matrix from F−→def to F−→CMG is
CCMG = cos γ1+ (1− cos γ)aaT − sin γa×
and the angular velocity of the rotation from F−→def to F−→CMG is given by
ω = γa− (1− cos γ)a×a+ sin γa.
Chapter 2. System Dynamics and Control 22
1
β1
3
2
β2
ωs
F−→
CMG,
F−→
def
(a) β1 = β2 = 0
F−→CMG
F−→def
3
3
1
1
2
2
γ
(b) β1, β2 6= 0
Figure 2.6: Plate CMG
Let γ = γa. Then for small β1 and β2,
γ =
β1
β2
0
.
The first order approximation for the rotation matrix is
CCMG.= 1− γ×,
and the first order approximation for the angular velocity of F−→CMG in the deformed frame
is
ωCMG.= γ =
β1
β2
0
.
2.2.2 Energy and Work
Kinetic Energy
As with the beam model, Lagrange’s method will be used to formulate the motion equa-
Chapter 2. System Dynamics and Control 23
tions. The kinetic energy of the plate is given by
Tplate =12
∫∫
S
ρw2 dxdy
where ρ is the density per area of the plate and S is the area of the plate. As with the beam
case, the density of the plate is assumed to be uniform. The plate can be divided according
to the separate elements, whose kinetic energies are given by
Te =12
∫∫
Se
ρw2e dxdy.
The kinetic energy can be expressed in terms of the element’s nodal degrees of freedom,
Te =12
∫∫
Se
ρaeTNTNae dxdy
= 12aeT ρ
∫∫
Se
NTN dxdy
︸ ︷︷ ︸
Me
ae.
Thus the kinetic energy of the whole plate is given by
Tplate =∑
i
Te,i
=∑
i
12aeTi Me,ia
ei
= 12˙qTM ˙q.
In the present configuration, the rotor of each CMG is spinning about the 3-axis of F−→CMG
with constant angular velocity ωs and is gimballed to rotate through β1 and β2 as illustrated
in figure 2.6(a). If the moment of inertia of the rotor about the spin axis is denoted by Is,
then the nominal angular momentum for the i-th CMG is hs,i = Is,iωs,i. In the deformed
frame, the angular momentum of the rotor is given by
hi,def = CCMGhi,CMG.
Chapter 2. System Dynamics and Control 24
Using the first order approximation for the rotation matrix yields
hi = (1− γ×)hi,CMG =
1 0 β2
0 1 −β1−β2 β1 1
0
0
hs,i
=
hs,iβ2
−hs,iβ1hs,i
.
The kinetic energy of CMG’s rotor is given by
Tgyro,i = hTi ωi
= hT (ϕi − 12ϕ×
i ϕi)
= hs,iβ2,iϕ1,i − hs,iβ1,iϕ2,i +12(hs,iϕ2,iϕ1,i − hs,iϕ1,iϕ2,i)
=[
ϕ1,i ϕ2,i
][
0 hs,i
−hs,i 0
][
β1,i
β2,i
]
+ 12
[
ϕ1 ϕ2
][
0 hs,i
−hs,i 0
][
ϕ1,i
ϕ2,i
]
.
Recall that for small rotations
(∂w
∂y
)
i
= ϕ1,i
(∂w
∂x
)
i
= −ϕ2,i.
This allows the kinetic energy of the ith gyro to be expressed as
Tgyro,i =
[˙(
∂w∂x
)
i
˙(∂w∂y
)
i
] [
hs,i 0
0 hs,i
][
β1,i
β2,i
]
+ 12
[˙(∂w∂x
)
i
˙(∂w∂y
)
i
] [
0 hs,i
−hs,i 0
]
(∂w∂x
)
i(∂w∂y
)
i
.
Let ji denote the element number on which the gyro is positioned, then the displacement
field w can be expressed in terms of the element’s nodal degrees of freedom, aeji, and shape
functions, N,
Tgyro,i = aeji
T[
∂NT
∂x∂NT
∂y
][
hs,i 0
0 hs,i
][
β1,i
β2,i
]
+ 12aeji
T[
∂NT
∂x∂NT
∂y
][
0 hs,i
−hs,i 0
][∂N∂x
∂N∂y
]
aeji
Chapter 2. System Dynamics and Control 25
= aeji
T[
hs,i∂NT
∂xhs,i
∂NT
∂y
]
︸ ︷︷ ︸
Bi
[
β1,i
β2,i
]
+ 12aeji
T
(
hs,i∂NT
∂x
∂N
∂y− hs,i
∂NT
∂y
∂N
∂x
)
︸ ︷︷ ︸
Gi
aeji
= aeji
TBiβi +12aeji
TGiaeji.
The total kinetic energy due to the gyros are given by
Tgyros = Tgyro,1 + Tgyro,2 + . . .+ Tgyro,N
= aej1
TB1β1 +12aej1
TG1aej1+ ae
j2
TB2β2 +12aej2
TG2aej2+ . . .
+ aejN
TBNβN + 12aejN
TGNaejN
= ˙qT Bβ + 12˙qT Gq.
Thus the total kinetic energy of the system is given by
T = Tplate + Tgyros
= 12˙qTM ˙q+ ˙qT Bβ + 1
2˙qT Gq.
Strain Energy
The strain energy of the system is due to the bending of the plate. The strain-displacement
relationship for a plate element in bending is given by
ǫ = Cae,
where
C = L∇N,
and the differential operator L∇ is defined as
L∇ =[
∂2
∂x2
∂2
∂y22 ∂2
∂x∂y
]
.
The stress-strain relationship is given by
σ = Dǫ = DCae.
Chapter 2. System Dynamics and Control 26
For a thin plate, the matrix D is given by
D =Et3
12(1− ν2)
1 ν 0
ν 1 0
0 0 (1− ν)/2
,
where ν is Poisson’s ratio, E is the elastic modulus, and t is the plate thickness. The strain
energy density is given by
U0 =
∫ ǫ
0
σdǫ
=
∫ ǫ
0
Dǫdǫ
= 12ǫT Dǫ
= 12aeT C
TDCae
Thus the strain energy of an element is given by
Ve,i =
∫
U0dxdy
= 12aeTi
∫
CTDCdxdy
︸ ︷︷ ︸
Ke,i
aei .
The strain energy of the entire plate is given by the summation of the strain energy of each
element
V = Ve,1 + Ve,2 + . . .+ Ve,N
= 12aeT1 Ke,1a
e1 +
12aeT2 Ke,2a
e2 + . . .+ 1
2aeTN Ke,Na
eN
= 12qT Kq.
Non-Conservative Work
As for the beam case, proportional damping is assumed. Thus the non-conservative work
due is given as
δWnc = δqT D ˙q,
Chapter 2. System Dynamics and Control 27
where D is the damping matrix. It has the form D = E−Tdiag{2ζαωα}E−1, where the ζα
terms are the damping ratios and the ωα terms are the elastic beam frequencies associated
with the mode shapes chosen as the basis functions. The matrix E contains the eigenvectors
associated with the undamped elastic system and is used to transform the diagonal matrix
in modal coordinates to the physical coordinates consistent with q. The eigenvalue problem
corresponding to the elastic system is given by
ω2i Mei = Kei
where ωi is natural frequency corresponding to the i-th mode and ei is the eigenvector
corresponding to the i-th mode. E is given by
E =[e1 e2 . . . eN
].
The generalized force vector associated with damping is given as
f = −D ˙q.
2.2.3 Motion Equations
From the kinetic and strain energies T and V , the Lagrangian, L = T − V , is formed. Fi-
nally substituting the Lagrangian and the generalized non-conservative force into Lagrange’s
equationsd
dt
(∂L
∂ ˙q
)
− ∂L
∂q= f,
yields
M¨q+ (G+ D) ˙q+ Kq = Bβ.
In this formulation, M is the positive-definite mass matrix, G is the skew-symmetric gyric
matrix due to the stored angular momentum of the CMGs, D is the positive-semidefinite
damping matrix, K is the positive-semidefinite stiffness matrix, β is a vector of the gimbal
angles of the CMGs, and B is the input matrix.
As with the beam case, it is desired for the degrees of freedom of the motion equations
to be expressed in terms of the systems elastic modes rather than the nodal displacements.
This can be achieved by solving the eigenvalue problem of the elastic system and using the
resulting eigenvectors to perform a change of coordinates. Recall that matrix E, was a row
Chapter 2. System Dynamics and Control 28
of the eigenvectors of the elastic system. Define a modal vector q such that
q = E−1q
Thus the equation of motion becomes
MEq+ (G+ D)Eq+ KEq = Bβ.
Furthermore, pre-multiplying by ET , yields
ETMEq+ ET (G+ D)Eq+ ET KEq = ET Bβ.
Thus the modal equations of motion can be expressed as
Mq+ (G+D)q+Kq = Bβ. (2.5)
where M = ETME, G = ET GE, D = ET DE, K = ET KE, and B = ET B.
2.3 Modal Decomposition
2.3.1 Elastic Modes
When analyzing flexible systems, it is often helpful to consider the undamped free vibration of
the system. For instance, the modal coordinates used as the degrees of freedom of the systems
and the damping model employed are both obtained using the modes of the undamped free
vibration of the beam and plate. For the linear elastic case, such systems are governed by
the equation
Mz+Kz = 0,
where M ∈ Rn×n is the positive definite symmetric mass matrix, K ∈ R
n×n is the positive
semidefinite symmetric stiffness matrix, and x ∈ Rn×1 is the vector of degrees of freedom.
Since M is positive definite, this can be rewritten as
z = −Υz (2.6)
Chapter 2. System Dynamics and Control 29
whereΥ = M−1K. Υ is symmetric and thus diagonalizable. This property can be used to de-
couple the system of equations represented by equation 2.6. Let the eigenvalues ofΥ be given
by λi and the corresponding eigenvectors by ei. Furthermore let Λ = diag{λ1, λ2, . . . , λn}and E = [e1 . . . en]. Then the matrix Υ can be expressed as
Υ = EΛE−1.
Substituting this back into equation 2.6 yields
z = −EΛE−1z. (2.7)
Let η = E−1z, and pre-multiply equation 2.7 by E−1. This yields
η = −Λη
which is a series of decoupled equations
η1 = −λ1η1...
ηn = −λnηn.
These second order linear equations have the known solution
ηi = ci cos(ωit+ ϕi),
where ci is the modal amplitude, ωi =√λi is the modal frequency and ϕi is mode’s phase.
The eigenvectors create a basis that spans Rn. The result is that all of the solutions to
equation 2.6 can be expressed in terms of a linear combination of these decoupled motions
z(t) = e1η1(t) + . . .+ enηn(t).
Thus ηi is called a modal coordinate and ei represents the corresponding the mode shape.
Chapter 2. System Dynamics and Control 30
2.3.2 Gyroelastic Modes
A similar notion of modes can be had for gyroelastic systems as developed in [17]. Con-
sider the same systems considered in Section 2.1 and Section 2.2, except with no structural
damping (D = 0) and with the gimbal angles fixed in their nominal positions (β = 0). The
resulting linear first order differential equation describing the system dynamics would be
[
q
q
]
︸ ︷︷ ︸
x
=
[
−M−1G −M−1K
1 0
]
︸ ︷︷ ︸
A
[
q
q
]
︸ ︷︷ ︸
x
(2.8)
x = Ax.
The eigendecomposition of the matrix A can form a basis through which the system of
equations x = Ax can be decoupled
η1 = λ1η
...
ηn = λnη.
where λi are the eigenvalues of A with the corresponding eigenvector ei. These equations
have the known solution
ηi(t) = ηi(0) exp(λit).
Since this system is developed using the first order representation of a second order differen-
tial equation, the eigenvalues are either equal to zero or appear in complex conjugate pairs
λα = ±jωα. The corresponding eigenvectors are in general complex, and may be written as
ei = ui + jvi.
In describing these uncoupled motions, it is only the real part that is considered,
xi = Re{ei exp(jωit)}= ui cos(ωit)− vi sin(ωit).
Chapter 2. System Dynamics and Control 31
As a result, all of the solutions to equation 2.8 can be expressed in terms of a linear combi-
nation of the real part of these decoupled motions
x(t) = Re{e1η1(t)}+ . . .+ Re{enηn(t)}.
Thus ηi is the modal coordinate and ei represents the corresponding mode shape of a gy-
roelastic mode. It should be noted that the Re{ei exp(jωit)} are not unique since they will
be identical for the modes corresponding to complex conjugate pairs. In [18], Meirovitch
presents a way to divide the real and imaginary parts into separate eigenvalue problems
from which it can be shown that for λi = jωα and λk = −jωα, the substitution ei = uα and
ek = vα can be made without loss of generality.
2.4 Optimal Control
2.4.1 Stabilization using State Feedback
The dynamics of the beam and the plate both result in dynamical equations of the form
Mq+ (G+D)q+Kq = Bβ.
This equation is a second order differential equation in time. It can be converted into a first
order equation
[
q
q
]
︸ ︷︷ ︸
x
=
[
−M−1(G+D) −M−1K
1 0
]
︸ ︷︷ ︸
A
[
q
q
]
︸ ︷︷ ︸
x
+
[
B
0
]
︸ ︷︷ ︸
B
β︸︷︷︸
u
x = Ax+ Bu. (2.9)
where x ∈ Rn is the state vector, u ∈ R
m is the control input vector, A ∈ Rn×n is the
open-loop system matrix, and B ∈ Rn×m is the input matrix. This system is linear and time
invariant.
The control objective is to suppress the vibrations of the system. This requires the design
of a controller u that asymptotically stabilizes the system about the equilibrium x = 0. In
order to find u, it is required that the system be controllable.
Assuming that the full state x is available for measurement and that the system is
Chapter 2. System Dynamics and Control 32
controllable, a linear state feedback control law of the form
u = Fx
can be used. The resulting closed-loop system is given by
x = (A+ BF)x
= Ax.
The solution to this equation is
x(t) = exp(At)x(0).
Let the eigenvalues of A be λi with corresponding eigenvectors ei. If the eigenvalues of A
are distinct then the eigendecomposition of A is given by
A = EΛE−1,
where Λ = diag{λi} and E = row{ei}. It can be shown that
exp(At) = E exp(Λt)E−1,
exp(Λt) = diag{exp(λit)}.
If F is chosen such that the eigenvalues of A have negative real parts, then exp(λit) → 0 for
t → ∞. Consequently, the system is asymptotically stable about the equilibrium x = 0 if
the eigenvalues of A have negative real parts, since
x(t) → 0 as t→ ∞.
2.4.2 Linear Quadratic Regulator
In addition to stabilizing the system, it is desired that the controller yield desirable per-
formance. To this end, the controller design for these systems will be of the form of a
Linear Quadratic Regulator (LQR). The LQR controller is optimal in that it yields the state
Chapter 2. System Dynamics and Control 33
feedback controller that minimizes the cost function
J =
∫ ∞
0
(xTQx+ uTRu)dt,
where Q is a matrix which penalizes the system’s states, and R is a matrix which penalizes
the control inputs. Through the use of dynamic programming, it can be shown that the
unique controller that minimizes J is of the form
u = −R−1BTPx,
where P is the solution of the algebraic Riccati equation:
ATP+PA−PBR−1BTP+Q = 0.
Furthermore, in [19] it is shown that for u = −R−1BTPx,
J = xT0Px0.
where x0 is the state of the system at time t = 0.
Since the control objective is to suppress the vibrations of the system, it would seem
appropriate to penalize the states x according to the mechanical energy of the vibrating
beam. To this end, consider
Q = diag{M,K}.
This yields
xTQx =[
qT qT
][
M 0
0 K
][
q
q
]
= qTMq+ qTKq
= 2(Telastic + Velastic).
It should be noted that in the case of an unconstrained body, this choice of penalization
does not penalize the rigid rotation of the beam from F−→und. It penalizes only the rates of
the rigid modes since K is only positive semidefinite. To penalize the rigid rotation of the
beam, Q is modified to be
Q = diag{M,K}+Qrigid. (2.10)
Chapter 2. System Dynamics and Control 34
where
Qrigid =
{
q for diagonal positions corresponding to the rigid rotation states,
0 else where,,
q is an arbitrary constant chosen to place the eigenvalues of the closed loop system matrix
associated with the rigid modes.
To penalize the control effort, the matrix R is chosen to have the form
R = r1,
where r is an arbitrary constant chosen to place the eigenvalues of the closed system matrix.
This effectively penalizes the rate changes of the gimbal angles of the gyros uniformly.
Chapter 3
The Optimization Problem
The objective of this study is to find the optimal distribution of CMGs on elastic bodies
for vibration suppression. The optimization problem is defined according to a cost function
that measures how well the elastic modes of the system are damped and the control effort
used to damp the motions. The problem is constrained according to the amount of stored
angular momentum carried by the CMGs. A solution to this problem is approached using
gradient-based numerical optimization techniques. Gradient-based optimizers require initial
conditions and only converge on local minima, so sets of different initial conditions are
considered for both the beam and plate cases to improve the likelihood of arriving at the
best result.
3.1 Optimization Objective
The objective of this study is to find the optimal distribution of CMGs for vibration sup-
pression of beams and plates. For the models developed in Chapter 2, the distribution of
the CMGs are dependent on the locations and amount of stored angular momenta for each
CMG.
To find the optimal distribution of control moment gyros, the same objective function as
the optimal LQR problem is considered:
J =
∫ ∞
0
(xTQx+ uTRu
)dt = xT
0Px0,
where x and u are defined in equation 2.9. Since the initial state x0 is unknown, it can be
considered as a random variable with a second-order moment E{x0xT0 } = X0. From [20],
35
Chapter 3. The Optimization Problem 36
it can be seen that minimizing J = xT0Px0 is equivalent to minimizing J = tr(PX0); by
assuming X0 = 1, J = tr(P) is taken as the objective function. This makes the optimization
objective a weighting between how quickly the vibrations are damped and the amount of
control effort used.
Since both the location and amount of stored angular momentum influence the control
of the structure in a coupled manner, the amount of stored angular momentum is varied
while keeping the locations of the control moment gyros fixed. This allows a fixed number
of CMGs to be distributed evenly about the body, and the optimization will be concerned
with the allocation of the stored angular momentum. Let h = [hs,1, hs,2, . . . , hs,N ]T be an
array of the amount of nominal stored angular momentum for N CMGs. This distribution
of stored angular momentum is subject to the constraint:
N∑
k=1
h2s,k = hTh = c2,
or c(h) = hTh− c2 = 0.
Thus the optimization problem considered here can be stated as
minsJ(h), s = { h | h ∈ R
N , c(h) = 0}.
3.2 Gradient-Based Optimization with Equality Con-
straints
To approach a solution to the optimization problem posed, a gradient-based optimization
technique is employed. This section gives an overview of some features of these techniques.
Consider a smooth function f(x) where x is a vector x = [x1, x2, . . . , xn]T . The necessary
conditions for optimality of the unconstrained problem
minxf(x)
are given by
‖∇f(x)‖ = 0, ∇2f(x) ≥ 0,
Chapter 3. The Optimization Problem 37
where ∇f(x) and ∇2f(x) are the gradient and hessian of f(x) [21]. The problem considered
in Section 3.1 is a constrained optimization problem with one equality constraint,
minsJ(x), s = { x | x ∈ R
n, c(x) = 0}.
At a stationary point, the total differential of the objective function has to be equal to
zero. For a feasible point, the total differential of the constraints has to be equal to zero.
Solving for these conditions can be difficult if the constraint equations are incorporated into
the problem directly. However, this constrained optimization problem can be solved by
transforming it into a unconstrained optimization problem through the use of the method
of Lagrange multipliers [21]. Consider the Lagrangian
L(x, λ) = J(x)− λc(x),
where λ is a scalar called a Lagrange multiplier. Since the differentials of the objective
function and constraint are zero at a feasible stationary point,
dL(x, λ) = dJ − λdc
=n∑
i=1
(∂J
∂xi− λ
∂c
∂xi
)
dxi
= 0.
Furthermore, as the dxi are independent,
∂J
∂xi− λ
∂c
∂xi= 0, (i = 1, 2, . . . , n).
Therefore necessary conditions for a feasible stationary point can be expressed in terms of
the Lagrangian as
∂L∂xi
=∂J
∂xi− λ
∂c
∂xi= 0, (i = 1, 2, . . . , n),
∂L∂λ
= c = 0.
These are known as the Karush-Kuhn-Tucker (KKT) conditions and serve as the first order
necessary conditions for the optimum of a constrained problem [21].
The problem in Section 3.1 is difficult to solve analytically because of its dependency
Chapter 3. The Optimization Problem 38
on the solution to the Riccati equation. For this reason nonlinear numerical techniques are
employed. Currently, an interior-point algorithm has been applied to the problem using the
fmincon function in the MATLAB environment.
The algorithm uses one of two techniques at each step: a Newton step or a conjugate
gradient step. Both of theses techniques are gradient-based methods. Algorithms for uncon-
strained gradient-based optimization follow a common iterative procedure [21]:
0. Initial conditions x0 and a stopping tolerance ε are established. The iteration counter
is given by k.
1. The stopping criterion is given by |∇f(xk)| ≤ ε. If the condition for convergence is
satisfied, then the algorithm stops and the current point is the solution.
2. The vector dk, defining the direction in n-space along which to search, is computed.
3. The positive scalar αk, that will define the step size, is found. This scalar must satisfy
the condition f(xk + αkdk) < f(xk).
4. Set xk+1 = xk + αkdk, k = k + 1 and repeat from 1.
These methods can only ensure that a local minimum is reached and are thus dependent on
the choice of initial conditions. A detailed explanation of the optimization algorithm used
by MATLAB can be found in [22].
3.3 Implementation
3.3.1 Beam Case
Both the free and cantilevered beams were modelled using 20 basis functions in the Ritz
expansion discussed in Section 2.1.1. For the free beam case, the basis functions include the
two rigid rotations, the first nine elastic modes in the 2-axis and the first nine elastic modes
in the 3-axis. For the cantilevered case, the functions include the first ten elastic modes in
the 2-axis and the first ten elastic modes in the 3-axis. The properties of the beam are taken
to be the same for both boundary conditions. These values are included in Table 3.1.
Chapter 3. The Optimization Problem 39
F−→0
1
2
3
h2−→
h1−→
h20−→
· · ·
· · ·
(a) Free Beam
F−→0
1
2
3
h1−→
h2−→
h20−→
· · ·
(b) Cantilevered Beam
Figure 3.1: CMG Distribution for the Beam
Property Symbol Value
Beam length l 100 m
Mass per length ρ 6.200 kg/m
Stiffness to bending in 2-axis B2 1.5765× 109 N-m2
Stiffness to bending in 3-axis B3 1.5×B2
Proportional damping ratios ζα 0.01
Rigid rotation penalization (Eq. 2.10) q 100
Table 3.1: Beam Properties
The beam is outfitted with 20 CMGs. For the free beam case, these CMGs are distributed
evenly from one end to the other end with a separation distance of l/19. For the cantilevered
beam, the CMGs are spaced starting at l/20 from the fixed end and spaced evenly to the
free end. As mentioned in Section 2.1.1, the stored angular momentum of the CMGs are
nominally parallel to the spanwise direction. These configurations are illustrated in Figure
3.1.
Since gradient-based optimization techniques are used, initial conditions are required.
These techniques only ensure a local minimum is reached, so the optimization is done with
seven different initial conditions and the best results are presented in Chapter 4. The fol-
lowing initial conditions were considered:
Chapter 3. The Optimization Problem 40
• a uniform distribution with
hs,i =1
c√N, (3.1)
• and distributions based on sinusoidal functions with
hs,i =1
C
∫ bi
ai
sin(mπx
l
)
dx, (3.2)
or hs,i =1
C
∫ bi
ai
cos(mπx
l
)
dx, (3.3)
where ai = xi +l2− l
2Nand bi = xi +
l2+ l
2Nfor the free beam, ai = xi − l
2Nand
bi = xi +l
2Nfor the cantilevered beam, C is chosen to satisfy the constraint hTh = c2,
and m = 1, 2, 3.
3.3.2 Plate Case
The plate case modelled using the finite element method as discussed in Section 2.2.1. For
the optimization, a uniform 16×16 element model is used and the first 49 modes are retained.
The properties of the plate are taken from the Purdue model presented in [23]. The Purdue
model is a plate structure with a rigid hub at the center. For this analysis, the rigid mass
at the center of the plate in the Purdue model is omitted. The plate properties are given in
Table 3.2.
Property Symbol Value
Plate length a 12.5 km
Plate width b 5 km
Mass per area σ 0.2662 kg/m2
Modulus of rigidity D 20× 108 N
Poisson’s ratio ν 0.3
Proportional damping ratios ζα 0.01
Rigid rotation penalization (Eq. 2.10) q 1× 10−4
Table 3.2: Plate Properties
The plate is outfitted with 49 CMGs. These CMGs are distributed in a 7 × 7 grid. As
mentioned in Section 2.1.1, the spin axes of the CMGs’ flywheels are nominally perpendicular
to the plate. This configuration is illustrated in figure 3.2.
The optimization is done with ten different initial conditions and the best results are
presented in Chapter 4. The following initial conditions were considered:
Chapter 3. The Optimization Problem 41
• a uniform distribution with
hs,i =1
c√N., (3.4)
• a corner distribution with
{
hs,i =1
c√4, i = {1, 7, 43, 49}
hs,i = 0, otherwise, (3.5)
• and distributions based on sinusoidal functions with
hs,i =1
C
∫ di
ci
∫ bi
ai
sin(mπx
a
)
sin(nπy
b
)
dxdy, (3.6)
or hs,i =1
C
∫ di
ci
∫ bi
ai
cos(mπx
l
)
cos(nπy
b
)
dxdy, (3.7)
where ai = xi +a2− a
2Nx, bi = xi +
a2+ a
2Nx, ci = yi +
b2− b
2Ny, di = yi +
b2+ b
2Ny, C is
chosen to satisfy the constraint hTh = c2, m = 1, 2, and n = 1, 2.
F−→
0
1
2
3
h1−→
h7−→
h2−→
h43−→
h49−→
h8−→
· · · · · ·
···
···
h9−→
h14−→
· · ·
h44−→
· · · · · ·
Figure 3.2: CMG Distribution for the Plate
Chapter 4
Optimization Results
The results of the optimization problem discussed in Chapter 3 are presented. Comparisons
are made to the case in which only a single CMG is used for vibration suppression. The
effect that the optimized distribution has on the motions of the body are investigated through
consideration of the systems’ gyroelastic modes.
4.1 Beam
4.1.1 Placement of a Single CMG
Before approaching the optimization problem as defined in Section 3.3, it may be instructive
to consider the simpler problem of the cost of the placement of a single gyro. This problem
is illustrated in Figure 4.1 for the free beam case. Recall from Section 2.1.1 that the CMG
is nominally pointing in the spanwise direction.
F−→0
1
2
3
h−→
x
Figure 4.1: The Single CMG Problem for a Free Beam
42
Chapter 4. Optimization Results 43
Figure 4.2 shows the cost J associated with a single gyro at different positions along the
span of the beam. Plots are included for different values of h. For each h, a value of r
was chosen to ensure the closed loop eigenvalues of the system were reasonably placed. It is
necessary to change r because the control torque resulting from a given change in the gimbal
angles for a CMG is proportional to its stored angular momentum. In these figures, scaled
values are given for J , h, and r according to J = J/√
ρBl2, h = h/√
ρBl2 and r = r/(ρl2).
Figure 4.2(a) shows a case where the amount of stored angular momentum is relatively
low. It can be seen that there are many local maxima and minima. The locations of the
maxima correspond closely to the positions of the nodes in φv,α(x) or φw,α(x), the spatial
derivative of the elastic mode shapes in x. It is intuitive that locating the CMG at these
locations would be more costly since a mode is not controllable at its nodes. This can be
illustrated by considering the form of the control matrix in equation 2.3. Recall that the
control torque from the gyro is given by
Biβi =
[
hs,iΦv,i 0
0 hs,iΦw,i
][
β2,i
β3,i
]
,
where
Φv,i =
φv,1(xi)...
φv,N(xi)
, Φw,i =
φw,1(xi)...
φw,N(xi)
,
and xi is the spanwise position of the gyro. Thus if at xi, φv,α(xi) = 0 or φw,α(xi) = 0, that
mode will not be controllable though the CMG. Furthermore, it can be seen in Figure 4.2(a)
that the least costly location for the CMG is at the ends of the beam. This is also intuitive
since the elastic modes’ shapes do not have a node in φv,α(x) and φw,α(x) at x = l/2 and
x = −l/2. Thus, at the beam ends, all of the modes are controllable and thus benefit from
the control given by the CMG.
For low amounts of stored angular momentum, the gyroelastic modes correspond closely
with the elastic modes. This can be seen in figure 4.3, which shows the elastic modes of the
free beam, and figure 4.4, which shows the gyroelastic modes of the beam with one gyro at
x = l/2 and stored angular momentum h = 0.01. Recall from Section 2.3.2 that the motions
Chapter 4. Optimization Results 44
of a gyroelastic mode α are described by
xα = uα cos(ωαt)− vα sin(ωαt).
In Figure 4.4, the mode shape for uα is visualized by the thick black line with many thin
lines connecting it to the 1-axis and vα is visualized by the thick grey line. The circles and
ellipses planar to the 1-axis show the evolution of the gyroelastic mode through a full period
according to the gyroelastic frequency ωα. The frequencies presented in the figure are scaled
according to ωα = ωα
√
ρl4/B.
Figure 4.4 shows some interesting effects that gyricity has on the system. Particularly,
there appears to be a coupling between the elastic modes. The momentum bias of the single
gyro couples the two rigid rotations in a precessional mode illustrated in Figure 4.4(a). Even
for the vibrational modes the coupling effects become apparent with this level of gyricity
introduced to the system.
For increasing amounts of stored angular momentum it becomes apparent that the beam’s
ends are no longer the least costly position. This trend can possibly be explained by the
effect that larger amounts of stored angular momentum have on the gyroelastic modes of
the system. Figure 4.5 shows the gyroelastic modes of the beam with one gyro at x = l/2
and stored angular momentum h = 1.00. It is apparent that there is no longer so close a
correspondence between the elastic modes and the gyroelastic modes. Furthermore, it would
appear that having the CMG at the end of the beam flattens out the gyroelastic mode shapes
at the end for either uα or vα. This makes the gyroelastic modes less controllable from that
position. It is possible that this is due to the resistance of bodies with large amounts of
stored angular momentum to change its axis of rotation. It also explains why having the
CMG at the end is more costly with large amounts of stored angular momentum.
4.1.2 Distribution of Multiple CMGs
Now consider the optimization problem outlined in Section 3.1. As with the case of the single
gyro, different cases are considered according to the amount of stored angular momentum
in the system. The penalty r is taken to be the same as for the single gyro case with
c = h√
ρBl2. These constraint values can be scaled according to c = c/√
ρBl2. A total of
20 CMGs were used in the optimization.
Gradient-based optimization requires initial conditions to be given to the optimizer. For
Chapter 4. Optimization Results 45
each case, the optimizer was run using seven different initial conditions. These initial condi-
tions are discussed in Section 3.3. This was done because gradient-based optimization can
only guarantee convergence on a local minimum, and the minimum arrived at is dependent
on the initial conditions provided. Figure 4.6 shows the distribution obtained with the lowest
cost for each case. The amount of stored angular momentum for the gyros in the optimized
distributions are included in Appendix A and the costs of the optimized distributions and
the initial conditions are included in Appendix B.
From these plots it can be seen that as the amount of stored angular momentum is
increased, more stored angular momentum is distributed away from the ends of the beam.
This is consistent with the ends of the beam no longer being the optimal position for the
single gyro placement case. Curiously, the minima that the optimizer arrives at for c > 0.50
has the direction of the nominal stored angular momentum alternate between positive and
negative. This effectively lowers the total angular momentum of the system, which in turn
introduces a rigid precessional mode with low frequency. Figures 4.7 and 4.8, show the
gyroelastic modes of the optimal distributions for c = 0.01 and c = 1.00. From figure 4.7,
it can be seen that for c = 0.01, there is little difference between the modes and frequencies
for lower amounts of stored angular momentum as when there is a gyro at one end or both.
On the other hand for the case with c = 1.00, the optimal distribution of stored angular
momentum results in the gyroelastic modes being closer to the elastic modes in both shape
and frequency than for a single gyro at the beam end.
4.1.3 Cantilevered Boundary Conditions
The optimization was also carried out for the beam with cantilevered boundary conditions.
The results show the same trends as with the free beam case. For low amounts of stored
angular momentum, the optimal location of a single gyro is at the tip and the gyroelastic
modes for the single gyro at the tip and for the optimized distribution show little change from
the elastic modes in shape and frequency. For greater amounts of stored angular momentum
the optimal location of a single gyro moves away from the tip and the gyroelastic mode
shapes and frequencies diverge from the elastic modes for the gyro at the tip. The gyroelastic
modes for larger amounts of stored angular momentum are closer to the elastic modes for
the optimized distribution.
Figure 4.9 shows the values of the objective function with respect to the location of a
single gyro along the span of the beam. Curiously, in Figure 4.9(e) to Figure 4.9(f), it can
Chapter 4. Optimization Results 46
be seen that the optimal location of the gyro is at the fixed end of the beam. The CMG
would have no controllability at this location since all of the modes have a node for φv,α(0)
and φw,α(0). This suggests that it is less costly to let the beam rely on its internal damping,
which is assumed to be proportional in this analysis, than to try and control it with a single
CMG.
Figures 4.10 show the first six elastic modes of the cantilevered beam. Figures 4.11 and
4.12 show the gyroelastic modes for a single CMG at the tip of the cantilevered beam with
h = 0.01 and h = 1.00 respectively. The flattening of the gyroelastic modes at the tip for h =
1.00 can be seen. Figure 4.13 shows the optimized distributions for the cantilevered beams for
different values of c. The amount of stored angular momentum for the CMGs in the optimized
distributions are included in Appendix A and the costs of the optimized distributions and
the initial conditions are included in Appendix B. The tendency for the optimizer to find
a minimum which alternates the direction of the stored angular momentum for adjacent
CMGs can be consistent with the free beam case. Figure 4.14 shows the gyroelastic modes
for the optimized distribution for c = 1.00. It can be seen that the gyroelastic mode shapes
and frequencies for this distribution are closer to the elastic modes than for a single CMG
at the tip of the beam with h = 1.00.
Chapter 4. Optimization Results 47
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.022
0.024
0.026
0.028
0.03
0.032
0.034
0.036
0.038
0.04
0.042
x/l
JObjective Function versus CMG Location
(a) h = 0.01, r = 0.025
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.028
0.03
0.032
0.034
0.036
0.038
0.04
0.042
0.044
x/l
J
Objective Function versus CMG Location
(b) h = 0.05, r = 0.625
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.032
0.034
0.036
0.038
0.04
0.042
0.044
x/l
J
Objective Function versus CMG Location
(c) h = 0.1, r = 2.5
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.05
0.055
0.06
0.065
0.07
0.075
0.08
x/l
J
Objective Function versus CMG Location
(d) h = 0.5, r = 50
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.105
x/l
J
Objective Function versus CMG Location
(e) h = 1.0, r = 200
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
x/l
J
Objective Function versus CMG Location
(f) h = 5.0, r = 5000
Figure 4.2: Cost of the Placement of a Single Gyro for a Free Beam
Chapter 4. Optimization Results 48
uα3,vα3
uα2,vα2
x
(a) ω = 20.22
uα3,vα3
uα2,vα2
x
(b) ω = 24.76
uα3,vα3
uα2,vα2
x
(c) ω = 55.73
x
uα3,vα3
uα2,vα2
(d) ω = 68.25
x
uα2,vα2
uα3,vα3
(e) ω = 109.2
uα3,vα3
uα2,vα2
x
(f) ω = 133.8
Figure 4.3: Free Beam Elastic Modes
Chapter 4. Optimization Results 49
x
uα3,vα3
uα2,vα2
(a) ω = 0.1200
uα3,vα3
uα2,vα2
x
(b) ω = 20.17
x
uα2,vα2
uα3,vα3
(c) ω = 24.78
x
uα2,vα2
uα3,vα3
x
(d) ω = 55.59
x
uα2,vα2
uα3,vα3
(e) ω = 68.28
x
uα2,vα2
uα3,vα3
(f) ω = 109.0
Figure 4.4: Free Beam Single CMG Gyroelastic Modes (h = 0.01)
Chapter 4. Optimization Results 50
uα3,vα3
uα2,vα2
x
(a) ω = 4.561
uα3,vα3
uα2,vα2
x
(b) ω = 7.561
uα3,vα3
uα2,vα2
x
(c) ω = 28.47
x
uα3,vα3
uα2,vα2
(d) ω = 35.33
uα2,vα2
uα3,vα3
x
(e) ω = 70.82
x
uα3,vα3
uα2,vα2
(f) ω = 86.94
Figure 4.5: Free Beam Single CMG Gyroelastic Modes (h = 1.00)
Chapter 4. Optimization Results 51
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Stored Angular Momentum Distributionh/c
x/l
(a) c = 0.01, r = 0.025
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Stored Angular Momentum Distribution
h/c
x/l
(b) c = 0.05, r = 0.625
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.6
−0.4
−0.2
0
0.2
0.4
Stored Angular Momentum Distribution
h/c
x/l
(c) c = 0.10, r = 2.5
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.4
−0.2
0
0.2
0.4
0.6
Stored Angular Momentum Distribution
h/c
x/l
(d) c = 0.50, r = 50
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Stored Angular Momentum Distribution
h/c
x/l
(e) c = 1.00, r = 200
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Stored Angular Momentum Distribution
h/c
x/l
(f) c = 5.00, r = 5000
Figure 4.6: Free Beam Optimization Results
Chapter 4. Optimization Results 52
uα2,vα2
uα3,vα3
x
(a) ω = 0.1681
x
uα3,vα3
uα2,vα2
(b) ω = 20.14
uα2,vα2
uα3,vα3
x
(c) ω = 24.83
x
uα2,vα2
uα3,vα3
(d) ω = 55.48
x
uα2,vα2
uα3,vα3
(e) ω = 68.44
x
uα2,vα2
uα3,vα3
(f) ω = 108.8
Figure 4.7: Free Beam CMG Distribution Gyroelastic Modes (c = 0.01)
Chapter 4. Optimization Results 53
uα2,vα2
uα3,vα3
x
(a) ω = 0.6733
x
uα2,vα2
uα3,vα3
(b) ω = 19.75
uα3,vα3
uα2,vα2
x
(c) ω = 24.95
x
uα2,vα2
uα3,vα3
(d) ω = 55.22
uα3,vα3
uα2,vα2
x
(e) ω = 68.34
x
uα2,vα2
uα3,vα3
(f) ω = 109.4
Figure 4.8: Free Beam CMG Distribution Gyroelastic Modes (c = 1.00)
Chapter 4. Optimization Results 54
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.03
0.035
0.04
0.045
0.05
0.055
x/l
JObjective Function versus CMG Location
(a) h = 0.01, r = 0.125
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.04
0.042
0.044
0.046
0.048
0.05
0.052
0.054
x/l
J
Objective Function versus CMG Location
(b) h = 0.05, r = 3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.046
0.048
0.05
0.052
0.054
0.056
0.058
x/l
J
Objective Function versus CMG Location
(c) h = 0.1, r = 12.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
x/l
J
Objective Function versus CMG Location
(d) h = 0.5, r = 250
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
x/l
J
Objective Function versus CMG Location
(e) h = 1.0, r = 1× 103
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
x/l
J
Objective Function versus CMG Location
(f) h = 5.0, r = 1× 104
Figure 4.9: Cost of the Placement of a Single Gyro for a Cantilevered Beam
Chapter 4. Optimization Results 55
uα2,vα2
uα3,vα3
x
(a) ω = 3.177
x
uα2,vα2
uα3,vα3
(b) ω = 3.891
uα3,vα3
uα2,vα2
x
(c) ω = 19.91
uα3,vα3
uα2,vα2
x
(d) ω = 24.39
uα3,vα3
uα2,vα2
x
(e) ω = 55.75
uα3,vα3
uα2,vα2
x
(f) ω = 68.28
Figure 4.10: Cantilevered Beam Elastic Modes
Chapter 4. Optimization Results 56
uα3,vα3
uα2,vα2
x
(a) ω = 3.175
uα3,vα3
uα2,vα2
x
(b) ω = 3.893
x
uα2,vα2
uα3,vα3
(c) ω = 19.86
uα3,vα3
uα2,vα2
x
(d) ω = 24.41
uα3,vα3
uα2,vα2
x
(e) ω = 55.61
x
uα3,vα3
uα2,vα2
(f) ω = 68.31
Figure 4.11: Cantilevered Beam Single CMG Gyroelastic Modes (h = 0.01)
Chapter 4. Optimization Results 57
x
uα2,vα2
uα3,vα3
(a) ω = 0.9866
uα3,vα3
uα2,vα2
x
(b) ω = 4.966
x
uα2,vα2
uα3,vα3
(c) ω = 7.028
x
uα2,vα2
uα3,vα3
(d) ω = 28.48
x
uα2,vα2
uα3,vα3
(e) ω = 35.37
x
uα2,vα2
uα3,vα3
(f) ω = 70.82
Figure 4.12: Cantilevered Beam Single CMG Gyroelastic Modes (h = 1.00)
Chapter 4. Optimization Results 58
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Stored Angular Momentum Distributionh/c
x/l
(a) c = 0.01, r = 0.125
0 0.2 0.4 0.6 0.8 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Stored Angular Momentum Distribution
h/c
x/l
(b) c = 0.05, r = 3
0 0.2 0.4 0.6 0.8 1
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Stored Angular Momentum Distribution
h/c
x/l
(c) c = 0.10, r = 12.5
0 0.2 0.4 0.6 0.8 1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Stored Angular Momentum Distribution
h/c
x/l
(d) c = 0.50, r = 250
0 0.2 0.4 0.6 0.8 1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Stored Angular Momentum Distribution
h/c
x/l
(e) c = 1.00, r = 1× 103
0 0.2 0.4 0.6 0.8 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Stored Angular Momentum Distribution
h/c
x/l
(f) c = 5.00, r = 1× 104
Figure 4.13: Cantilevered Beam Optimization Results
Chapter 4. Optimization Results 59
x
uα2,vα2
uα3,vα3
(a) ω = 3.133
uα3,vα3
uα2,vα2
x
(b) ω = 3.939
uα3,vα3
uα2,vα2
x
(c) ω = 19.44
x
uα2,vα2
uα3,vα3
(d) ω = 24.91
x
uα2,vα2
uα3,vα3
(e) ω = 54.81
x
uα2,vα2
uα3,vα3
(f) ω = 69.36
Figure 4.14: Cantilevered Beam CMG Distribution Gyroelastic Modes (c = 1.00)
Chapter 4. Optimization Results 60
4.2 Plate
4.2.1 Placement of a Single CMG
As with the beam case, the cost of the placement of a single gyro is also considered for
the plate. Recall from Section 2.1.1 that the CMG is nominally pointing in the direction
perpendicular to the face of the plate as illustrated in figure 4.15.
F−→
0
1
2
3
x
yh−→
Figure 4.15: The Single CMG Problem for a Free Plate
Figure 4.16 shows the cost, J , associated with a single gyro at different positions about
the plate. In this figure, the values for h and r are scaled according to h = h/√Dσa4 and
r = r/(σa2). It can be seen that there are some similar trends as with the beam case. For
instance, for low amounts of stored angular momentum the least costly position of the gyro
is at the corners of the plate. As the amount of stored angular momentum is increased, the
least costly position moves toward the center of the plate. The reason behind this can be
found by considering the modes of the system. Consider the first eight elastic modes for the
plate as depicted in figure 4.17. The frequencies presented in the figure are scaled according
to ωα = ωα
√
σa4/D. At the corners, the mode shapes all have non-zero slope. With small
amounts of stored angular momentum, the gyroelastic modes closely mimic the shapes of
the elastic modes, as can be seen in figure 4.18 which depicts the gyroelastic modes for a
single gyro at the lower right corner of the plate (x = a/2, y = b/2) with stored angular
momentum h = 0.01. Since these modes have non-zero slope at corner with the CMG, they
are controllable through a CMG. This can be illustrated by considering the input matrix for
the CMG on the plate.
Biβi =
[
hs,i∂NT
∂xhs,i
∂NT
∂y
] [
β1,i
β2,i
]
.
∂N/∂x and ∂N/∂y are the slopes of the shape functions for the plate element at the location
Chapter 4. Optimization Results 61
of the CMG. Thus for a given mode shape, if slopes at the location of the CMG are non-zero,
the inputs β1,i and β2,i will affect the motions of that mode.
The higher cost for greater amounts of stored angular momentum at the plate corners
can again be attributed to the flattening effect that the large amount of stored angular
momentum has on the modes of the system. This can be seen by comparing the elastic
modes to the modes with a single gyro at the plate corner (x = a/2, y = b/2) for h = 1.00
shown in figure 4.19. Since the modes have small slopes at the position of the CMG relative
to the mode shape, the inputs β1,i and β2,i will have little effect on the motions of the modes.
4.2.2 Distribution of Multiple CMGs
For the optimization problem outlined in Section 3.1, a grid of 7 by 7 CMGs was used. As
with the case of the single gyro, different cases are considered according to the amount of
stored angular momentum in the system. The penalty r is taken to be the same as for
the single gyro case with c = h√Dσa4. These constraint values can be scaled according to
c = c/√Dσa4.
For each case, the optimizer was run using ten different initial conditions as discussed
in Section 3.3. Figure 4.20 and Figure 4.21 show the distributions obtained with the lowest
cost for each case. The amount of stored angular momentum for the CMGs in the optimized
distributions are included in Appendix A and the costs of the optimized distributions and
the initial conditions are included in Appendix B.
The trends in the optimized stored angular momentum distribution for the plate does
not seem as strongly apparent as for the beam. But there are some interesting similarities
that can be seen in the beam and the plate results. For instance, for low amounts of stored
angular momentum, the optimum distribution places the stored angular momentum at the
corners of the plate. This follows the optimum placement of the single gyro being at the
corners of the plate as the optimum placement of the single gyro for the beam was at the
beam’s ends. Furthermore, there is a distinct regime in the distributions for c ≥ 0.5 where
the stored angular momentum of CMGs along the edges x = −a/2 and x = a/2 alternate
between positive and negative directions. This is also similar to the beam, for which the
distributions for c ≥ 0.5 had alternating directions along the span of the beam.
The alternating nature of the optimized distributions for c ≥ 0.5 in the plate case result
in the gyroelastic mode shapes that do not have near zero slope at the edges of the plate
where most of the stored angular momentum is located. This can be seen in figure 4.23
Chapter 4. Optimization Results 62
which show the first four gyroelastic modes for c = 1.0. This effect on the mode shapes may
also explain why the optimized results do not place most of the stored angular momentum
toward the center of the plate, which is the more favourable position in the single gyro case.
−0.5
0
0.5 −0.2
0
0.2
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
x 104
y/ax/a
J
(a) h = 0.001, r = 5× 10−5
−0.5
0
0.5 −0.2
0
0.2
1.9
2
2.1
2.2
2.3
2.4
2.5
x 104
y/ax/a
J
(b) h = 0.005, r = 0.001
−0.5
0
0.5 −0.2
0
0.2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
x 104
y/ax/a
J
(c) h = 0.01, r = 0.005
−0.5
0
0.5 −0.2
0
0.2
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75
2.8
x 104
y/ax/a
J
(d) h = 0.05, r = 0.15
Chapter 4. Optimization Results 63
−0.5
0
0.5 −0.2
0
0.2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
x 104
y/ax/a
J
(e) h = 0.1, r = 0.4
−0.5
0
0.5 −0.2
0
0.2
2.5
3
3.5
4
x 104
y/ax/a
J
(f) h = 0.5, r = 5
−0.5
0
0.5 −0.2
0
0.2
2.5
3
3.5
4
4.5
5
x 104
y/ax/a
J
(g) h = 1.0, r = 10
−0.5
0
0.5 −0.2
0
0.2
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
x 105
y/ax/a
J
(h) h = 5.0, r = 700
Figure 4.16: Cost of the Placement of a Single Gyro for a Plate
Chapter 4. Optimization Results 64
(a) ω = 21.45 (b) ω = 32.99
(c) ω = 59.63 (d) ω = 70.80
(e) ω = 116.4 (f) ω = 118.3
(g) ω = 140.3 (h) ω = 152.8
Figure 4.17: Free Plate Elastic Modes
Chapter 4. Optimization Results 65
uα vα
(a) ω = 0.7389
(b) ω = 21.21
(c) ω = 32.59
(d) ω = 57.79
Figure 4.18: Free Plate Single CMG Gyroelastic Modes (h = 0.01)
Chapter 4. Optimization Results 66
uα vα
(a) ω = 4.511
(b) ω = 12.66
(c) ω = 26.48
(d) ω = 40.07
Figure 4.19: Free Plate Single CMG Gyroelastic Modes (h = 1.00)
Chapter 4. Optimization Results 67
−0.5
0
0.5 −0.2
0
0.2
−0.4
−0.2
0
0.2
0.4
y/ax/a
h/c
(a) c = 0.001, r = 5× 10−5
−0.5
0
0.5 −0.2
0
0.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
y/ax/a
h/c
(b) c = 0.005, r = 0.001
−0.5
0
0.5 −0.2
0
0.2
0.1
0.15
0.2
0.25
y/ax/a
h/c
(c) c = 0.01, r = 0.005
−0.5
0
0.5 −0.2
0
0.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
y/ax/a
h/c
(d) c = 0.05, r = 0.15
Figure 4.20: Plate Optimization Results
Chapter 4. Optimization Results 68
−0.5
0
0.5 −0.2
0
0.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
y/ax/a
h/c
(a) c = 0.10, r = 0.4
−0.5
0
0.5 −0.2
0
0.2
−0.3
−0.2
−0.1
0
0.1
0.2
y/ax/a
h/c
(b) c = 0.50, r = 5
−0.5
0
0.5 −0.2
0
0.2
−0.2
−0.1
0
0.1
0.2
0.3
y/ax/a
h/c
(c) c = 1.00, r = 10
−0.5
0
0.5 −0.2
0
0.2
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
y/ax/a
h/c
(d) c = 5.00, r = 700
Figure 4.21: Plate Optimization Results
Chapter 4. Optimization Results 69
uα vα
(a) ω = 4.698
(b) ω = 19.21
(c) ω = 35.75
(d) ω = 53.55
Figure 4.22: Free Plate CMG Distribution Gyroelastic Modes (c = 0.01)
Chapter 4. Optimization Results 70
uα vα
(a) ω = 3.111
(b) ω = 11.76
(c) ω = 20.66
(d) ω = 24.51
Figure 4.23: Free Plate CMG Distribution Gyroelastic Modes (c = 1.00)
Chapter 5
System Response
To compare the optimized distribution of CMGs to other distributions in a more tangible
manner than considering the cost given by the objective function, the response of these
systems are considered for an elastically deformed initial condition imposed on the free
beam or free plate. For the free beam case, the optimized distribution is compared to a
distribution with uniform stored angular momentum at each gyro and for a distribution
with equal amounts of stored angular momentum at each end. For the free plate case, the
optimized distribution is compared to a distribution with uniform stored angular momentum
at each gyro and for a distribution with equal amounts of stored angular momentum at each
corner.
5.1 Initial Deformation
5.1.1 Beam Case
Consider the closed loop response of the beam gyroelastic system when it is initially deformed
and at rest in the shape of a parabola in the 3-axis and with the optimization constraint
c = 1.00. The parabola is described by the function
z =1
25lx2.
Thus, the ends of the beam are deformed one hundredth of the span of the beam with
respect to the beam’s centre. Figure 5.1 shows the response of the beam with the optimum
distribution shown in Figure 4.6(e). Figure 5.2 shows the response of the beam with a
71
Chapter 5. System Response 72
distribution of stored-angular momentum that is uniform across all of the CMGs. Figure 5.3
shows the response of the beam with the stored-angular momentum divided equally at the
ends of the beam. The motions shown in these figures are the elastic modes of the beam.
Modes 1 and 2 are rigid rotations described by equation 2.1, where mode 1 is the rotation
about the 3-axis, ψv,1, and mode 2 is the rotation about the 2-axis, ψw,1. The mode shapes
for modes 3, 4, 5, 12, 13, and 14 are depicted in figure 4.3. They can be identified according
to their scaled frequencies.
Since the initial conditions involve a symmetric deformation in the 3-axis, modes corre-
sponding to mode shapes in the 2-axis and asymmetric mode shapes in the 3-axis have zero
valued initial conditions as can be seen in these plots. However, the coupling introduced by
the gyricity of the system causes motions in all of the modes. This is apparent, in the opti-
mized case, through the superposition of the oscillations seen in mode 3 in the responses of
modes 1, 4, and 5, as well as the oscillations in mode 12 seen superimposed on the motions of
2, 13, and 14. Stronger still is the relationship between oscillations corresponding to similar
mode shapes in the different axes. These pairs are modes 3 and 12, 4 and 13, and 5 and 14.
The coupling of these modes is expected in light of the undamped gyroelastic modes shown
in figure 4.5.
The motions in the cases of the uniform distribution and the end distribution does not
appear as coupled. The superposition of oscillations seem restricted according to the sym-
metry of the mode shape. This coupling can be seen in modes 1, 2, 4, and 13 separate from
the coupling seen in modes 3, 12, 5, and 14.
In comparing these different distributions according to the settling time of the modes,
these plots show that the optimized distribution performs best, followed by the distribution
with the stored angular momentum at the ends, and the uniform distribution performs
worst. To make a numerical comparison, consider the settling times (ts) for modes 12 and
14, the time it takes for these modes to reach and stay within 1% of the target values as
compared to their initial conditions. These modes are considered since they have nonzero
initial conditions, and their settling times are given in Table 5.1. The settling time of the
optimized distribution for mode 12 is 11% of the settling time for the uniform distribution
and 23% of the settling time for the ends distribution. The settling time of the optimized
distribution for mode 14 is 9.2% of the settling time for the uniform distribution and 16% of
the settling time for the ends distribution. Thus it can be seen that the optimization yielded
significant reduction in settling time in suppressing the vibrations of this system.
The objective function chosen for the optimization performed considered both the penal-
Chapter 5. System Response 73
ization of states as well as control effort. Recall from Section 3.1 that the objective function
of the LQR problem is given by
J =
∫ ∞
0
(xTQx+ uTRu
)dt = xT
0Px0.
For the optimization, x0 was treated as a random variable which lead to taking
J = trace{P}.
However, now that an initial condition is considered, the norm xT0Px0 gives a direct measure
of how well each case performs according to the weighting of state and control effort used
for the optimization of the stored angular momentum distribution. These values and J are
included in Table 5.1. This norm yields little insight however because the weighting between
the states compared to the control effort was arbitrarily determined to place the closed-loop
eigenvalues. To estimate the portion of this norm allocated to the energy of the states and
the control effort used to suppress the vibrations of the system, the norms
Jx =
∫ τ
0
xTQx dt and Ju =
∫ τ
0
uTRu dt
are calculated for τ equal to the settling time for mode 12. The results are also included in
Table 5.1. From these norms, it can be seen that the optimized distribution out performs
the uniform distribution and the end distribution in both reducing the energy of the states
and exerting less control effort.
Results are also presented for the case with c = 0.01 in Table 5.2. In this case Jx and
Ju are taken for τ = 3.5 s. For this constraint, the optimized and end distributions are very
close, which is expected since the optimized distribution places most of the stored angular
momentum at the ends. However, it can still be seen that the optimized distribution out
performs the other two cases in both suppressing the energy of the modes and using less
control effort.
Distribution ts (Mode 12) ts (Mode 14) J xT
0Px0 Jx Ju
Optimum 0.57 s 0.40 s 2.69× 105 1.49× 104 8.31× 103 6.55× 103
Uniform 5.27 s 4.34 s 1.77× 106 1.48× 105 7.43× 104 7.41× 104
End 2.45 s 2.45 s 1.36× 106 7.55× 104 3.76× 104 3.78× 104
Table 5.1: Initial Deformation Beam Performance for c = 1.00
Chapter 5. System Response 74
Distribution J xT
0Px0 Jx Ju
Optimum 2.38543× 105 1.53692× 104 8.7333× 103 6.6358× 103
Uniform 3.31567× 105 1.93647× 104 1.09194× 104 8.4453× 103
End 2.38552× 106 1.53700× 104 8.7336× 104 6.6364× 103
Table 5.2: Initial Deformation Beam Performance for c = 0.01
0 0.5 1 1.5 2−0.5
0
0.5
1
1.5
2
q 1
t [s]
(a) Mode 1
0 0.5 1 1.5 2−0.6
−0.4
−0.2
0
0.2
q 2t [s]
(b) Mode 2
0 0.5 1 1.5 2−2
−1
0
1
2
q 3
t [s]
(c) Mode 3, ω = 20.22
0 0.5 1 1.5 2−5
0
5
10
q 12
t [s]
(d) Mode 12, ω = 24.76
0 0.5 1 1.5 2−0.4
−0.2
0
0.2
0.4
q 4
t [s]
(e) Mode 4, ω = 55.73
0 0.5 1 1.5 2−0.1
−0.05
0
0.05
0.1
q 13
t [s]
(f) Mode 13, ω = 68.25
0 0.5 1 1.5 2−0.2
−0.1
0
0.1
0.2
q 5
t [s]
(g) Mode 5, ω = 109.2
0 0.5 1 1.5 2−0.5
0
0.5
1
q 14
t [s]
(h) Mode 14, ω = 133.8
Figure 5.1: Initial Deformation Beam Response for Optimal Distribution
Chapter 5. System Response 75
0 1 2 3 4 5 6 7 8 9−3
−2
−1
0
1
2x 10
−5
t [s]
q 1
(a) Mode 1
0 1 2 3 4 5 6 7 8 9−5
−4
−3
−2
−1
0x 10
−5
t [s]
q 2
(b) Mode 2
0 1 2 3 4 5 6 7 8 9−10
−5
0
5
t [s]
q 3
(c) Mode 3, ω = 20.22
0 1 2 3 4 5 6 7 8 9−5
0
5
10
t [s]
q 12
(d) Mode 12, ω = 24.76
0 1 2 3 4 5 6 7 8 9−3
−2
−1
0
1
2x 10
−5
t [s]
q 4
(e) Mode 4, ω = 55.73
0 1 2 3 4 5 6 7 8 9−2
−1
0
1
2x 10
−5
t [s]
q 13
(f) Mode 13, ω = 68.25
0 1 2 3 4 5 6 7 8 9−0.6
−0.4
−0.2
0
0.2
0.4
t [s]
q 5
(g) Mode 5, ω = 109.2
0 1 2 3 4 5 6 7 8 9−0.5
0
0.5
1
t [s]
q 14
(h) Mode 14, ω = 133.8
Figure 5.2: Initial Deformation Beam Response for Uniform Distribution
Chapter 5. System Response 76
0 1 2 3 4 5−2
−1
0
1
2x 10
−4
t [s]
q 1
(a) Mode 1
0 1 2 3 4 5−4
−3
−2
−1
0x 10
−4
t [s]
q 2
(b) Mode 2
0 1 2 3 4 5−6
−4
−2
0
2
t [s]
q 3
(c) Mode 3, ω = 20.22
0 1 2 3 4 5−5
0
5
10
t [s]
q 12
(d) Mode 12, ω = 24.76
0 1 2 3 4 5−4
−2
0
2
4x 10
−5
t [s]
q 4
(e) Mode 4, ω = 55.73
0 1 2 3 4 5−5
0
5
10x 10
−5
t [s]
q 13
(f) Mode 13, ω = 68.25
0 1 2 3 4 5−0.6
−0.4
−0.2
0
0.2
t [s]
q 5
(g) Mode 5, ω = 109.2
0 1 2 3 4 5−0.4
−0.2
0
0.2
0.4
0.6
t [s]
q 14
(h) Mode 14, ω = 133.8
Figure 5.3: Initial Deformation Beam Response for Ends Distribution
Chapter 5. System Response 77
5.1.2 Plate Case
The closed-loop response of the plate with a parabolic elastic initial deformation while at
rest is also considered. The deformation is described by the equation
z =1
12ax2.
The deformations of the plate at x = a/2 and x = −a/2 are a/100. The distributions
considered in this comparison include the optimized distribution, the uniform distribution,
and the distribution with the stored angular momentum divided equally at the corners of the
plate with c = 1.00. The responses of the system for the elastic modes 1 to 8 are illustrated
in figures 5.4 to 5.6. Mode 1 is the rigid rotation of the plate about the 1-axis, mode 2 is
the rigid rotation of the beam plate about the 2-axis, and the mode shapes corresponding
to modes 3 to 8 are illustrated in figure 4.17.
As with the beam case, it can be seen that the optimized distribution damps out the
vibrations of the plate in less time than the other distributions. The parabolic initial condi-
tion is closest in shape to mode 3, so consider the settling time for mode 3 as shown in Table
5.3. The optimized distribution has a settling time for mode 3 that is 16% of the settling
time of the uniform distribution and 42% of the settling time of the corner distribution.
To compare the performance of the distributions with respect to dissipating the energy
of the states and the control effort exerted, the norms xT0Px0, Jx, and Ju are considered as
defined in the beam case. The time over which the integration is taken is the settling time for
mode 3. The values of these norms are included in Table 5.3 for c = 1.00. For this constraint,
it can be seen that the optimized distribution out performs the uniform distribution and the
corner distribution in both damping out the energy of the states and reducing the control
effort exerted.
These norms are also considered for the case with c = 0.01 in Table 5.4. In this case
the integration for Jx and Ju were taken for τ equal to the settling times of the mode 3 as
included in the table. The optimized distribution out performs the other distributions in the
weighted norm and in reducing the states. However, the uniform and corner distributions
out performs the optimized distribution in their use of control effort. It would seem that the
optimization yielded a result that was more aggressive in reducing states and conservative in
its use of control effort. Whereas in the previous beam and plate cases, the weightings Jx and
Ju are closer in magnitude, in this case Jx makes more of a contribution to the cost xT0Px0
than Ju. It can also be seen that the corner distribution has a lower settling time for mode
Chapter 5. System Response 78
3 than the optimized distribution. This result illuminates an important issue with respect
to the formulation of any optimization problem: the result is very sensitive to the way the
objective function is formulated. In this case, it is likely that for a greater r, the result
may yield a result in which the optimized result out performs the other two distributions in
both Jx and Ju or just Ju alone. It may also be the case that for different weightings the
other distributions out perform the optimized results for these initial conditions since the
objective function was taken to be J = trace{P} rather than J = xT0Px0. It is important to
ensure that the constraints and objective function properly reflect the design requirements
and objectives in order obtain a useful result when optimization is used in a design process.
Distribution ts (Mode 3) J xT
0Px0 Jx Ju
Optimum 9048 s 5303 6.43× 109 3.45× 109 2.98× 109
Uniform 55720 s 12099 5.06× 1010 2.56× 1010 2.50× 1010
Corner 21680 s 32282 2.41× 1010 1.25× 1010 1.16× 1010
Table 5.3: Initial Deformation Plate Performance for c = 1.00
Distribution ts (Mode 3) J xT
0Px0 Jx Ju
Optimum 3874 s 1.68185× 104 1.30757× 1010 9.3354× 109 3.7398× 109
Uniform 4409 s 1.73168× 104 1.37044× 1010 9.9735× 109 3.7304× 109
Corner 3309 s 1.79223× 104 1.35009× 1010 1.06205× 1010 2.8800× 109
Table 5.4: Initial Deformation Plate Performance for c = 0.01
Chapter 5. System Response 79
0 5000 10000 15000−5000
0
5000
10000
t [s]
q 1
(a) Mode 1
0 5000 10000 15000−1
0
1
2
3x 10
4
t [s]
q 2
(b) Mode 2
0 5000 10000 15000−2
−1
0
1x 10
5
t [s]
q 3
(c) Mode 3
0 5000 10000 15000−6
−4
−2
0
2
4x 10
4
t [s]
q 4
(d) Mode 4
0 5000 10000 15000−10000
−5000
0
5000
t [s]
q 5
(e) Mode 5
0 5000 10000 15000−6000
−4000
−2000
0
2000
t [s]
q 6
(f) Mode 6
0 5000 10000 15000−15000
−10000
−5000
0
5000
t [s]
q 7
(g) Mode 7
0 5000 10000 15000−5000
0
5000
10000
t [s]
q 8
(h) Mode 8
Figure 5.4: Initial Deformation Plate Response for Optimal Distribution
Chapter 5. System Response 80
0 2 4 6 8 10
x 104
−2
−1
0
1x 10
−5
t [s]
q 1
(a) Mode 1
0 2 4 6 8 10
x 104
−2
−1
0
1
2x 10
−5
t [s]
q 2
(b) Mode 2
0 2 4 6 8 10
x 104
−2
−1
0
1x 10
5
t [s]
q 3
(c) Mode 3
0 2 4 6 8 10
x 104
−10
−5
0
5x 10
4
t [s]
q 4
(d) Mode 4
0 2 4 6 8 10
x 104
−2
0
2
4x 10
−6
t [s]
q 5
(e) Mode 5
0 2 4 6 8 10
x 104
−6
−4
−2
0
2x 10
−6
t [s]
q 6
(f) Mode 6
0 2 4 6 8 10
x 104
−15000
−10000
−5000
0
5000
t [s]
q 7
(g) Mode 7
0 2 4 6 8 10
x 104
−6000
−4000
−2000
0
2000
t [s]
q 8
(h) Mode 8
Figure 5.5: Initial Deformation Plate Response for Uniform Distribution
Chapter 5. System Response 81
0 1 2 3 4 5 6 7 8
x 104
−3
−2
−1
0
1
2x 10
−5
t [s]
q 1
(a) Mode 1
0 1 2 3 4 5 6 7 8
x 104
−10
−5
0
5x 10
−5
t [s]
q 2
(b) Mode 2
0 1 2 3 4 5 6 7 8
x 104
−20
−15
−10
−5
0
5x 10
4
t [s]
q 3
(c) Mode 3
0 1 2 3 4 5 6 7 8
x 104
−6
−4
−2
0
2x 10
4
t [s]
q 4
(d) Mode 4
0 1 2 3 4 5 6 7 8
x 104
−1
−0.5
0
0.5
1x 10
−5
t [s]
q 5
(e) Mode 5
0 1 2 3 4 5 6 7 8
x 104
−4
−3
−2
−1
0
1x 10
−6
t [s]
q 6
(f) Mode 6
0 1 2 3 4 5 6 7 8
x 104
−15000
−10000
−5000
0
5000
t [s]
q 7
(g) Mode 7
0 1 2 3 4 5 6 7 8
x 104
−6000
−4000
−2000
0
2000
t [s]
q 8
(h) Mode 8
Figure 5.6: Initial Deformation Plate Response for Corner Distribution
Chapter 6
Conclusions
This study was concerned with finding the optimal distribution of DGCMGs for vibration
suppression of elastic bodies. The model used for this investigation was obtained by defining
the motions of the beam through a Ritz discretization and the motions of the plate using
finite elements. The problem was approached through considering a series of DGCMGs
placed uniformly about the elastic bodies and employing numerical optimization techniques
to allocate the amount of angular momentum stored by the CMGs’ spinning wheels. The
index used by the optimizer weighted both how quickly the vibrations were damped and
how much control effort was used. The optimization was carried out for different constraints
on the amount of angular momentum stored by the set of CMGs. The performance of the
optimized distributions were compared to other distributions by consider the response of the
elastic bodies to an imposed deformation.
It was found that the optimized distribution for lower amounts of stored angular mo-
mentum allocated more angular momentum to the CMGs toward the free tips of the beam
and corners of the plate. For greater amounts of stored angular momentum, it was found
that concentrating most of the stored angular momentum in a specific location affected the
effectiveness of the CMGs to damp out vibrations in the system. For the beam case, the
optimizer would then spread the amount of stored angular momentum about the span of
the beam while alternating the directionality of the stored angular momentum for adjacent
CMGs. For the plate case the optimizer would spread more of the stored angular momentum
about the shorter edges of the plate while also alternating the directionality of the angular
momentum of adjacent CMGs at the edge. It was found that optimized distributions would
not always out perform other distributions in both how quickly the vibrations were damped
and how much control effort was exerted in response to the imposed deformations. However
82
Chapter 6. Conclusions 83
they did out perform the other distributions in weighting both factors according to the index
used in the optimization for the cases considered.
As demonstrated in this study, optimization tools can only ensure that a series of variables
minimizes or maximizes an objective function. The results from the use of these tools are
optimal according to the metric by which they were optimized, and only provide a means to
manage design objectives. This study presented the optimization results for a distribution of
CMGs according to a specific objective function and explored the effect that the distribution
had on the motions of the system. The control strategy used was an LQR using full state
feedback of modal information of the system. The consequences of sensor placement and
state estimation were not included in this analysis. It is however an important aspect to
developing any control strategy. Another avenue that was unexplored was using a collocated
control scheme where sensor and actuators have the same locations and the control law for
any one actuator in dependent only on the information obtained from the sensor with which
it is placed. The advantage of such schemes is that the closed-loop system benefits from the
resulting passive stability characteristics. Collocated control can also avoid issues of control
spillover from unmodelled modes. This study provides a first step toward investigating the
problem of optimal placement of CMGs for vibration suppression of elastic bodies, but there
is still much work to be done before these results can be implementable on any real system.
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Appendix A
Optimized Distributions
The tables included in this appendix list the stored angular momentum for each CMG in the
optimal distributions presented in Chapter 4. The CMG numbers are illustrated in figures
3.1 and 3.2 for the different elastic bodies considered.
87
Appendix A. Optimized Distributions 88
CMGh/c
c = 0.01 c = 0.05 c = 0.10 c = 0.50 c = 1.00 c = 5.00
1 0.70703 0.59530 0.45205 −0.36444 −0.34895 0.329322 −0.01085 −0.37866 −0.53668 0.55638 0.52099 −0.483913 0.00087 0.00821 0.00869 −0.17877 −0.23613 0.258934 0.00020 −0.03867 −0.06580 0.17001 0.17093 −0.164915 0.00064 0.00555 −0.00139 −0.08891 −0.11815 0.131726 0.00023 −0.02361 −0.04688 0.10128 0.10914 −0.108767 0.00049 0.00115 −0.00568 −0.06032 −0.08466 0.096898 0.00066 −0.00840 −0.02517 0.06045 0.07934 −0.086759 0.00066 0.00077 −0.00832 −0.03755 −0.06574 0.0824010 0.00061 −0.00413 −0.01714 0.04694 0.06967 −0.0799811 0.00061 −0.00413 −0.01715 −0.03485 −0.06339 0.0807212 0.00066 0.00077 −0.00831 0.04220 0.06860 −0.0817813 0.00066 −0.00840 −0.02518 −0.04227 −0.07020 0.0876614 0.00049 0.00115 −0.00567 0.06063 0.08489 −0.0964615 0.00023 −0.02361 −0.04689 −0.08081 −0.09830 0.1098616 0.00064 0.00555 −0.00139 0.09289 0.11879 −0.1312017 0.00020 −0.03867 −0.06581 −0.13882 −0.15424 0.1670118 0.00087 0.00821 0.00869 0.18171 0.23109 −0.2598519 −0.01085 −0.37866 −0.53669 −0.51591 −0.48977 0.4887120 0.70702 0.59530 0.45204 0.35444 0.33411 −0.33186
Table A.1: Scaled Optimized Distribution for Free Beam
CMGh/c
c = 0.01 c = 0.05 c = 0.10 c = 0.50 c = 1.00 c = 5.00
1 0.00065 −0.00044 0.00584 −0.01255 −0.01106 −0.006092 0.00060 −0.00127 0.00103 −0.01163 −0.00527 0.009273 0.00157 −0.00274 0.00773 −0.01227 −0.01146 −0.013874 0.00036 0.00102 0.00641 0.00759 0.01234 0.017695 0.00174 −0.00476 0.00268 −0.01796 −0.01968 −0.022836 0.00007 0.00307 0.00913 0.01127 0.01959 0.027487 0.00203 −0.00673 −0.00150 −0.02489 −0.02989 −0.033628 −0.00029 0.00502 0.01267 0.01540 0.02929 0.039169 0.00226 −0.00745 −0.00341 −0.02925 −0.03992 −0.0466510 −0.00066 0.00663 0.01330 0.02153 0.04201 0.0542711 0.00229 −0.00623 0.00086 −0.04095 −0.05588 −0.0639212 −0.00094 0.00752 0.01006 0.03528 0.06096 0.0750413 0.00181 0.00000 0.01376 −0.06451 −0.08087 −0.0897014 −0.00097 0.00582 0.00298 0.06203 0.09242 0.1080515 −0.00001 0.01904 0.04470 −0.11364 −0.12660 −0.1335916 −0.00001 −0.00332 −0.01160 0.11909 0.15398 0.1720717 −0.00386 0.05356 0.09865 −0.21138 −0.22187 −0.2273818 0.00156 −0.01684 −0.04451 0.28252 0.33961 0.3652419 −0.06671 0.71352 0.79961 −0.75421 −0.72332 −0.7044620 0.99775 −0.69787 −0.58813 0.51528 0.49425 0.48171
Table A.2: Scaled Optimized Distribution for Cantilevered Beam
Appendix A. Optimized Distributions 89
CMGh/c
c = 0.001 c = 0.005 c = 0.01 c = 0.05 c = 0.10 c = 0.50 c = 1.00 c = 5.00
1 0.49990 0.45405 0.27167 0.14673 0.20820 0.19133 −0.17577 −0.000272 −0.00100 −0.11118 0.16293 0.10146 −0.26569 −0.28440 0.29163 −0.003253 0.00150 0.05631 0.14765 0.12788 0.16252 0.20542 −0.26953 −0.007474 0.00009 0.06100 0.14079 0.12933 0.06246 −0.19124 0.24835 0.001385 −0.00147 0.03122 0.14695 0.13221 0.11916 0.18881 −0.27045 −0.008146 0.00129 0.20072 0.16173 0.11087 0.08108 −0.26054 0.34481 0.001217 −0.49988 −0.44148 0.28019 0.14403 0.08616 0.18293 −0.21710 −0.004368 0.00778 0.07513 0.16704 0.17853 0.15036 0.01764 0.00524 0.002639 0.00224 0.02517 0.09175 0.11171 0.10595 0.10389 0.09355 −0.0143510 0.00081 0.01767 0.07616 0.10032 0.10092 0.11872 0.05650 −0.0262511 0.00000 0.01544 0.07305 0.09372 0.09108 0.10319 0.06409 −0.0074712 −0.00083 0.01354 0.07523 0.10366 0.09095 0.08812 0.05238 −0.0201513 −0.00223 0.00636 0.08958 0.12192 0.10754 0.10894 −0.01128 −0.0120114 −0.00778 −0.00461 0.16508 0.17911 0.14780 −0.01193 0.04327 0.0024215 0.00617 0.11081 0.16987 0.15870 0.12983 −0.00606 −0.02519 −0.0046016 0.00131 0.02577 0.09191 0.11708 0.10331 0.06772 0.13599 0.0017017 0.00071 0.01988 0.07663 0.10947 0.11008 0.11933 0.11971 −0.0186218 0.00002 0.01694 0.07314 0.09593 0.10472 0.15638 0.05297 −0.0317419 −0.00071 0.01560 0.07632 0.09897 0.12134 0.07647 0.06298 −0.0199820 −0.00134 0.01790 0.09167 0.11071 0.11292 0.05147 0.02888 −0.0057221 −0.00612 0.06027 0.17156 0.16402 0.11896 0.00424 0.01793 −0.0024522 −0.00005 0.17058 0.18099 0.16694 0.14370 0.00264 −0.00900 −0.0037123 −0.00002 0.02632 0.09565 0.12145 0.11572 0.06938 0.07660 −0.0297024 0.00001 0.02087 0.07951 0.11077 0.12643 0.13182 0.11389 −0.0289325 0.00001 0.02024 0.07692 0.09964 0.12122 0.09992 0.06591 −0.0415226 0.00002 0.01759 0.07952 0.11054 0.11613 0.08855 0.07403 −0.0174027 0.00001 0.02837 0.09567 0.12591 0.11907 0.12463 0.08418 −0.0228628 0.00002 0.14798 0.18098 0.18871 0.14907 0.09002 −0.01788 −0.0007929 −0.00613 0.10702 0.17155 0.17366 0.15145 0.00668 −0.02259 0.0029230 −0.00131 0.02394 0.09164 0.11408 0.10253 0.08367 0.09088 −0.0334031 −0.00070 0.01803 0.07630 0.10377 0.11566 0.11928 0.05923 −0.0442032 0.00001 0.01717 0.07314 0.08821 0.10098 0.08365 0.09392 −0.0080433 0.00070 0.02260 0.07662 0.09256 0.11161 0.03540 0.09773 −0.0119734 0.00132 0.02711 0.09192 0.10269 0.10185 0.01714 0.07241 −0.0093535 0.00619 0.09414 0.16990 0.15210 0.12183 0.04545 −0.00819 −0.0060636 −0.00774 0.06458 0.16511 0.18355 0.13214 0.06338 −0.01058 0.0130537 −0.00223 0.01979 0.08956 0.10693 0.10382 0.05457 0.05366 −0.0734138 −0.00080 0.01767 0.07522 0.08306 0.09162 0.02099 0.05283 0.0735439 0.00001 0.01812 0.07306 0.07679 0.05175 0.13429 0.09300 −0.0664140 0.00083 0.02214 0.07613 0.07855 0.04238 0.11356 0.03298 0.0924341 0.00224 0.03068 0.09172 0.09655 0.06941 0.12555 0.11675 −0.0590942 0.00776 0.09083 0.16706 0.13417 0.06596 −0.00147 0.02130 0.0118443 −0.49990 0.45072 0.28024 0.15513 −0.18502 0.20739 0.18053 0.1884244 0.00103 0.00968 0.16167 0.07145 0.25046 −0.30688 −0.27931 −0.3650345 −0.00149 0.09405 0.14696 0.10183 0.08422 0.20615 0.19975 0.3933146 0.00000 0.06058 0.14077 0.00269 −0.02018 −0.20930 −0.18772 −0.4124247 0.00151 0.08469 0.14767 0.09159 0.09506 0.22341 0.20268 0.4458848 −0.00070 −0.02371 0.16291 −0.42551 −0.41933 −0.30962 −0.28656 −0.4583349 0.49989 0.44581 0.27167 0.31901 0.30397 0.20088 0.19417 0.24263
Table A.3: Scaled Optimized Distribution for Free Plate
Appendix B
Distribution Costs
The tables listed in this appendix present the values of the cost functions for each initial
condition used in the optimization as well as the cost for the optimized distributions pre-
sented in Chapter 4. For the beam cases, Distribution 1 is the the uniform distribution
given by equation 3.1, Distribution 2 to Distribution 4 are the sine distributions given by
equation 3.2 for m = 1, 2, 3, and Distribution 5 to Distribution 7 are the cosine distributions
given by equation 3.3 for m = 1, 2, 3. For the plate case, Distribution 1 is the uniform dis-
tribution given by equation 3.4, Distribution 2 is the corner distribution given by equation
3.5, Distributions 3 to Distribution 6 are the sine distributions given by equation 3.6 for
(m,n) = {(1, 1), (1, 2), (2, 1), (2, 2)}, and Distribution 7 to Distribution 10 are the cosine
distributions given by equation 3.7 for (m,n) = {(1, 1), (1, 2), (2, 1), (2, 2)}.
Dist.J
c = 0.01 c = 0.05 c = 0.10 c = 0.50 c = 1.00 c = 5.00
1 0.03030 0.03142 0.03522 0.08247 0.16168 0.861782 0.03322 0.03359 0.03512 0.06744 0.13094 0.738723 0.03313 0.03355 0.03482 0.06365 0.12059 0.680944 0.03296 0.03333 0.03450 0.06219 0.11802 0.585025 0.02844 0.03039 0.03572 0.08207 0.15294 0.810136 0.02843 0.02996 0.03507 0.07912 0.14567 0.708397 0.02852 0.03015 0.03481 0.07848 0.13880 0.69322opt 0.02180 0.02355 0.02461 0.02417 0.02455 0.02485
Table B.1: Distribution Objective Function Values for Free Beam (Scaled)
90
Appendix B. Distribution Costs 91
Dist.J
c = 0.01 c = 0.05 c = 0.10 c = 0.50 c = 1.00 c = 5.00
1 0.044053 0.045414 0.050529 0.12927 0.28046 1.119352 0.045066 0.045626 0.048536 0.10840 0.22959 1.009473 0.045083 0.045607 0.048161 0.10293 0.21276 0.927844 0.045078 0.045527 0.047900 0.10097 0.20319 0.763995 0.043180 0.045212 0.051520 0.11696 0.23110 0.966696 0.043161 0.045047 0.051066 0.11083 0.21521 0.865287 0.043162 0.044909 0.050742 0.11028 0.20391 0.79290opt 0.035610 0.037109 0.037836 0.037166 0.037473 0.030349
Table B.2: Distribution Objective Function Values for Cantilevered Beam (Scaled)
Dist.J
c = 0.001 c = 0.005 c = 0.01 c = 0.05
1 1.898× 104 1.648× 104 1.732× 104 1.351× 104
2 1.750× 104 1.560× 104 1.792× 104 2.073× 104
3 2.141× 104 1.909× 104 2.053× 104 1.648× 104
4 2.095× 104 1.880× 104 2.094× 104 2.255× 104
5 2.097× 104 1.885× 104 2.088× 104 2.134× 104
6 2.051× 104 1.839× 104 2.049× 104 2.211× 104
7 1.816× 104 1.606× 104 1.819× 104 2.009× 104
8 1.832× 104 1.622× 104 1.834× 104 2.059× 104
9 1.823× 104 1.612× 104 1.828× 104 2.057× 104
10 1.839× 104 1.626× 104 1.842× 104 2.079× 104
opt 1.750× 104 1.548× 104 1.682× 104 1.300× 104
Dist.J
c = 0.10 c = 0.50 c = 1.00 c = 5.00
1 1.119× 104 1.141× 104 1.210× 104 6.814× 104
2 1.933× 104 2.754× 104 3.228× 104 1.197× 105
3 1.343× 104 1.208× 104 1.231× 104 4.116× 104
4 1.809× 104 1.351× 104 1.261× 104 3.983× 104
5 1.660× 104 1.328× 104 1.292× 104 4.105× 104
6 1.783× 104 1.391× 104 1.386× 104 4.154× 104
7 1.646× 104 1.570× 104 1.564× 104 7.543× 104
8 1.690× 104 1.491× 104 1.463× 104 7.384× 104
9 1.708× 104 1.571× 104 1.573× 104 8.046× 104
10 1.728× 104 1.507× 104 1.459× 104 7.551× 104
opt 9.956× 103 6.773× 103 5.303× 103 1.079× 104
Table B.3: Distribution Objective Function Values for Free Plate