Vibration Suppression of Large Space Structures …...by attitude control. There exists an extensive body of work investigating the dynamics and control of these structures, and more

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


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


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


[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.


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


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,


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


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

.


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,


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


= −[

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,


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.


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.


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.


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


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


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.


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-


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.


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


= 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.


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,


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


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)


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.


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).


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


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


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)


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,


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


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.


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:


• 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:


• 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


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


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


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


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.


−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


(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


(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


(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


(f) h = 5.0, r = 5000

Figure 4.2: Cost of the Placement of a Single Gyro for a Free Beam


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


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)


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)


−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


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


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


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


h/c

x/l

(f) c = 5.00, r = 5000

Figure 4.6: Free Beam Optimization Results


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)


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)


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


(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


(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


(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


(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


(f) h = 5.0, r = 1× 104

Figure 4.9: Cost of the Placement of a Single Gyro for a Cantilevered Beam


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


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)


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)


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


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


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


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


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


h/c

x/l

(f) c = 5.00, r = 1× 104

Figure 4.13: Cantilevered Beam Optimization Results


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)


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


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


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


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


−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


(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


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)


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)


−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


−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


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)


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-


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


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


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


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


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


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


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


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


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

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