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m LIBRARY
2 Michigan State
7'00“: University

 

 

This is to certify that the
dissertation entitled

VIBRATION SUPPRESSION THROUGH STIFFNESS
VARIATION AND MODAL DISPARITY

presented by

JIMMY ISSA

has been accepted towards fulfillment
of the requirements for the

Ph.D. degree in Mechanical Engineering

 

 

 

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5/08 K IProyAcc8Pres/CIRC/Date0ue mdd

VIBRATION SUPPRESSION THROUGH STIFFNESS VARIATION
AND MODAL DISPARITY

By

Jimmy Issa

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2008

ABSTRACT

VIBRATION SUPPRESSION THROUGH STIFFNESS
VARIATION AND MODAL DISPARITY

By
Jimmy Issa

Vibration suppression is the main objective of this study. A semi-active and an ac-
tive vibration control strategy based on stiffness variation are proposed for removing
energy from vibrating structures. For the semi-active vibration control strategy, the
notion of modal disparity is introduced and exploited as a new method of vibration
suppression. For a given structure, modal disparity is a measure of the difference in the
mode shapes of the structure in two stiffness states. Modal disparity is validated exper-
imentally in a beam where stiffness variation is induced by application and removal of
constraints. In dynamical systems modeled with finite degrees-of-freedom, the appli-
cation of constraints transfers energy to the unmodeled high-frequency modes, where
it is dissipated naturally and quickly. The removal of constraints does not dissipate
energy but resets the system for the constraints to be applied again for further re-
duction of energy. Thus sequential application and removal of constraints eventually
dissipates the energy of the system completely. It is shown that energy removal is
always possible, even with a random switching schedule, except in one case where the
energy is trapped in modes that span invariant subspaces with certain orthogonality
pr0perties. The optimal locations and timing of constraint application is investigated
with the goal of maximizing energy dissipation through maximal energy transfer to
the unmodeled high-frequency modes. ,For the active control strategy, cable actua-
tors are proposed for removing energy from three—dimensional framed structures The
tension in the cables has two effects on the structure; it increases the stiffness of the

structure and applies an external load on the structure. Both effects are used in the

design of the active control strategy in which the cable tension is essentially switched
between different levels to do negative work. Experimental results are presented to

validate the efficacy of the control strategy.

Copyright by
JIMMY ISSA
2008

To my dear parents Samir and Josephine.

ACKNOWLEDGMENTS

My utmost gratitude goes to my thesis advisor Dr. Ranjan Mukherjee for his con-

tinuous guidance and endless support during my years at Michigan State University.

A special thanks goes to Dr. Steven W. Shaw for providing valuable instructions and

suggestions and for his continuous help throughout my years of graduate studies.

I would like to express my deep and sincere gratitude to Dr. Alejandro R. Diaz for

his help and support.

My thanks to Dr. Alan Haddow and Dr. Hassan Khalil for serving as members of

my Ph.D committee. They provided useful suggestions to improve this work.
My thanks to Umar Farooq, Elliot Motato, Jeffrey Rhoads, Nicholas Miller and
Nandagopal Methil—Sudhakaran for providing me valuable help and for their friend—

ship.

Finally, I would like to thank the graduate secretary Aida Montalvo for her help.

vi

TABLE OF CONTENTS

1 Introduction ................................

1.1 Background and Objectives .......................
1.2 A Semi-Active Vibration Control Strategy I ...............
1.3 Active Vibration Suppression Strategy .................

Modal Disparity .............................
2.1 Stiffness Variation And Modal Disparity ................
2.2 Spatial Coordinate Description .....................
2.3 Modal Coordinates Description .....................
2.4 Numerical Example ............................

2.4.1 Modeling .............................

2.4.2 Simulations ............................
2.5 Remarks ..................................

Experimental Verification Of Modal Disparity ...........
3.1 Introduction ................................
3.2 Modeling ..................................
3.3 Stiffness Change Mechanism .......................
3.4 Modal Space ................................
3.5 Numerical Example ............................

3.6 Experimental Verification ............... - .........
3.7 Remarks ..................................

Energy Dissipation In Dynamical Systems Through Sequential Ap-

plication And Removal Of Constraints ................
4.1 Introduction ................................
4.2 Energy Loss due to Application of a Constraint ............
4.3 Relation to Prior Work ..........................
4.4 Finite DOF Linear Systems .......................
4.4.1 Spatial coordinate description ..................
4.4.2 Energetics of constraint application and removal ........
4.4.3 Modal coordinate description ..................
4.4.4 Numerical example ........................
4.5 Controllability Issues for Linear Systems ................
4.5.1 Limitation of energy dissipation .................
4.5.2 Energy entrapment ........................
4.5.3 Numerical example ........................
4.6 Application to Nonlinear Systems ....................

vii

UNI-‘I-A

oo

22
25
26
28
30
33

36
36
39
41
42
42
46
47
50
54
54
55
59
61

4.7 Remarks .................................. 63

5 Energy Dissipation Through Optimal Application And Removal of

Constraints ................................ 66
5.1 Introduction ................................ 66
5.2 Energy Loss due to Constraint Application and Removal ....... 68
5.3 Optimization Using Genetic Algorithms ................. 70
5.3.1 Discrete System Example: N -dof Mass Spring System ..... 70

5.3.2 Continuous System Example: Membrane ............ 72

5.4 Gradient Based Optimization Method .................. 77
5.4.1 Objective function ........................ 78

5.4.2 Sensitivity Analysis ........................ 80

5.4.3 Numerical Simulations ...................... 81

5.5 Remarks .................................. 83

6 Vibration Suppression Using Cable Actuators ........... 84
6.1 Introduction ................................ 84
6.2 Background ................................ 85
6.3 Modeling Of Cabled Frame Structures ................. 89
6.4 Control Scheme Design .......................... 94
6.5 Control By Stiffness Variation ...................... 95
6.5.1 Problem Definition ........................ 95

6.5.2 Cable Placement ......................... 96

6.5.3 Numerical Simulations ...................... 98

6.6 Control by Transverse Cable Force ................... 100
6.6.1 Problem Statement ........................ 100

6.6.2 Numerical Simulations ...................... 101

6.6.3 Experiments ............................ 104

6.7 Remarks .................................. 105

7 Conclusions ................................ 107
APPENDICES ................................ 110
A ....................................... 110
B ....................................... 112

BIBLIOGRAPHY .............................. 114

viii

CHAPTER 1

Introduction

1.1 Background and Objectives

Vibration control is a subject of significant importance with applications ranging
from small scale structures like micro beam resonators to large scale structures
such as large space structures, buildings and bridges. All of these structures are
prone to disturbances and excitations, for example, thermal gradients on space
structures and wind excitation on bridges. To overcome the problem that arise due
to disturbances and excitations and provide stability, controlschemes are designed
for energy removal. Typically, the control schemes fall into two categories; the first
is active control in which the control action is based on realtime sensing of states of
the structure. The second category, passive control, is implemented by embedding
passive elements in the structure with the goal of increasing its energy dissipation
properties. Some control schemes are based on a combination of the two strategies

and can be described as semi-active control.

The goal of this study has been the exploration of new methods of vibration
suppression. Our main target has been the stiffness of the structure and we have
developed energy dissipation schemes based on stiffness variation. Two methods have

been proposed. In one method, actuators are employed to locally enforce constraints

at predefined locations on a structure. This leads to a change in stiffness of the
structure and transfer of energy from low-frequency to high-frequency modes. In the
high-frequency modes, energy is dissipated quickly by conversion to heat without the
need for active control. Since we actively transfer energy to the high-frequency modes
where they are passively dissipated due to high levels of damping, we refer to this
method as semi-active control. In the second method, an active control scheme is
investigated where a cable actuator is used to transmit control forces to the structure.
This results in a change in stiffness of the structure and simultaneous application of
external forces. The tension is actively controlled and applied only when it removes

energy from the structure.

1.2 A Semi-Active Vibration Control Strategy

The controlled redistribution of energy in vibrating structures is at the heart of
many engineering problems with important practical applications. Modal control
strategies, vibration absorbers, and some forms of energy harvesting, all rely in one
form or another on the redistribution of energy in vibrating Structures from mode
to mode and, in space, from one region of the structure to another. Recently, a
new methodology for design of structures was proposed to achieve a targeted and
purposeful redistribution of vibration energy [1], [2]. This methodology relies on
modal disparity, a quantifiable property of the structure being designed, and relies

on a carefully crafted variation in the stiffness of the structure.

Stiffness variation, by itself, is not a new concept. For example, Clark [3] and
Corr and Clark [4] proposed stiffness variation of piezoelectric actuators to accom-
plish energy dissipation in vibration control. Kurdila, et. a1 [5] proved that this
state-switching strategy reduces the energy of the system and is stable, and Ramarat-

narn and Jalili [6] implemented this idea of “switched stiffness” in vibration control

           

 

 

 

 

 

 

 

B C D B C D
Sensing Sehsifigl sensing] Sensing] Sensing] Sensing]
an
Control Control Control Control Control Control
Stiffness Variation: Modal Energy Redistribution
A’ 3*
-
Sensing] Sensing]
and and
Control Control
----------------- repeat procedure ---—-~-----------
(a) Standard modal control strategy (b) Strategy based on stiffness variation

Figure 1.1. A simple analogy to illustrate a control methodology based on the concept of
modal disparity.

experiments. In contrast to these results, where the purpose of stiffness variation is
to dissipate energy, Diaz and Mukherjee [1], [2] proposed stiffness variation for modal
energy redistribution which then can be used for energy absorption, harvesting, or dis-
sipation. Modal energy redistribution should also be differentiated from earlier work
on localization [7], [8], [9], and energy pumping [10], [11], where energy redistribution

occurs spatially.

Stiffness variation and its effect on modal energy redistribution can be explained
by means of a simple analogy where the amount of vibration energy present in a
flexible structure is represented by a certain volume of fluid that needs to be drained

away. A modal view corresponds to fluid (energy) that is distributed among a set of

discrete containers, one for each mode. Figure 1.1 depicts this situation using four
modes, labeled A, B, C and D. In traditional modal control, the amount of fluid in
each container has to be sensed separately and a controller that is capable of draining
fluid from all the containers is used (Fig.1.1a). Now consider a situation where the
fluid in only two containers, say containers C and D (Fig.1.1b), are sensed and then
drained by a simple controller. Once all the fluid is removed from these containers,
the overall fluid volume decreases, but fluid remains trapped in the other containers,
A and B. Energy redistribution of the remaining fluid among all the containers,
including moving some fluid into containers C and D, can be achieved by stiffness
variation. One step of stiffness variation, followed by draining of the fluid from con-
tainers C and D, would leave fluid in the other containers, but repeating this process

back and forth between two stiffness states will drain the fluid from all the containers.

The success of a stiffness variation approach to energy redistribution is measured
by the total amount of energy that is transferred into target modes (container C
and D) at each step, and the details of how much energy is transferred out of the
other modes, the source modes (containers A and B), at each step. The rate at
which energy is redistributed depends on the source modes. For instance, if a source
mode in one stiffness state is nearly identical to a source mode in the other stiffness
state (e.g., B and B* in Fig.1.1b), then modal energy will drain very slowly from
these modes, i.e., fluid will be essentially trapped in the corresponding containers.
To quantify the amenability of a structure to energy redistribution strategies, a
measure of energy redistribution is needed. Modal disparity is such measure. It is a
property of the structure, as well as of the device introduced to effect the change in
stiffness. In this work, we generate modal disparity in structures with the objective
of transferring energy from the low-frequency modes to the high-frequency modes,

where it can be dissipated naturally and quickly.

As part of our semi-active control strategy, we consider stiffness variation in struc-
tures through the application and removal of constraints. The effect of application
and removal of constraints on the system dynamics, is studied and modal disparity
between the resulting stiffness states is quantified in chapter 2. In chapter 3 modal
disparity is experimentally verified through redistribution of modal energy between
the modes of a clamped-clamped beam in its two stiffness states In particular, the
beam has a pin joint at mid-span that can be locked using an electromagnetic brake
or allowed to rotate freely by releasing the brake; the two stiffness states of the beam
result from locking and releasing the pin joint. In chapter 4 we pr0pose a control
strategy based on the scheme in Fig.1.1. The only difference is that the energy in
containers C* and D* are not drained by active control. Instead, these containers
are chosen to correspond to the high-frequency modes of the system such that the
energy is drained due to internal damping. In chapter 4, we model the system with
finite dof1 and treat the high-frequency modes as unmodeled dynamics of the system.
The control strategy developed relies on application and release of constraints that
effectively transfer energy to the high-frequency unmodeled modes, where they are
dissipated by conversion to heat. The constraints are applied at predefined locations
on the structure and it is shown that energy reduction is always possible except in
some special cases. With the view to obtain faster rates of energy dissipation, the
location and timing of the constraints are optimized in chapter 5. Two example

problems are considered with different optimization criteria.

1.3 Active Vibration Suppression Strategy

Active vibration suppression has a long history of research. A variety of actuators

have been employed in active control and these include, piezoelectric materials,

 

1degrees-of—freedom

thrusters, momentum wheels, and cables, and their performance depends on the type
of structure (beam, truss, frame) and their placement on the structure. The idea of
active tendon control began as a way to reduce damage in cable stayed-bridges and
buildings caused by earthquakes and wind. It has been primarily deve10ped by civil
engineers, as described, for example, in Yang et al. [12, 13]. Cables are attractive
as actuators for large-scale structures since their effects can be transmitted far from
the energy source of the actuator and, in addition, their effects can be non-local, for
example, by changing the overall stiffness of a structure. Many approaches to the
control system design for cable actuators in large-scale structures have been proposed

due to the variety of structures considered, in both civil and aerospace applications.

Among the work on cable control for civil structures, Chung et al. [14] carried
out an experiment and used an optimal control scheme to reduce the response of a
single dof building type structure under base motion. Other authors [15] used active
tendon actuators to reduce the horizontal vibration of building structures subjected
to seismic and wind excitation. Warnitchai et al. [16] carried. out an experiment to
study the feasibility of active tendon control on cable-stayed bridges using velocity
feedback control. Other authors used active tendon actuator to reduce vibration in
cable stayed beams and cable stayed bridges, for example, Magana et al. [17,18]
and Fujino et al. [19]. Achkire and Preumont [20] and Bossens and Preumont
[21] used positive integral force feedback, an energy absorbing control strategy, to
suppress the vibration of large cable-stayed bridges; the cable forces were supplied

by a specially-built large scale hydraulic actuator.

Cable structures are very attractive for aerospace applications, where weight is an
important issue. In these applications the cables not only offer lightweight structural

support, they are also used as active tendons to provide stability. In recent years

the idea of active tendon control in aerospace applications was explored and many
approaches have been proposed. Murotsu et al. [22] used a newly conceived torque
actuation devise for controlling the vibrations of a beam-like space structure using
position and velocity feedback. Okubo et al. [23] proposed a tendon control system
for the shape control of flexible space structures. Preumont et al. [24, 25] used cable
tension to stiffen and control trusses by inducing active damping. Nudehi et al. [26]
used a cable-supplied end force to suppress the transverse vibrations of a cantilever
beam. The control is based on active stiffness variation in which Lyapunov stability
theory and passivity analysis were used to determine when to apply and release

tension in the cable in order to pump energy out of the system.

In chapter 6 we extend the approach of Nudehi et al. [26] to three dimensional
frame structures. Unlike the beam problem where cable tension resulted in stiffness
variation of the beam only, cable tension in structures typically result in simultaneous
stiffness variation and application of external forces. Both of these effects are used
in the design of an active control scheme to remove energy and suppress vibration
of the structure. For the development of the control scheme, a finite element model
of cable-framed structures is proposed and the effect of the cable placement on the
structure is investigated. Both numerical simulations and experimental results are

presented.

CHAPTER 2

Modal Disparity

2.1 Stiffness Variation And Modal Disparity

The main objective of this chapter is to formally introduce the notion of modal
disparity and to propose a measure to quantify it. Modal disparity arise in structures
capable of having multiple stiffness states. Typically, some mechanism is required
to switch stiffness states and hence modal disparity is a property of the structure as

well as the mechanism used to change the stiffness.

To define modal disparity, we consider a linear system vibrating freely about its
equilibrium position. The potential energy of the system is assumed to be zero
in this equilibrium configuration, which will be referred to as the zero equilibrium
henceforth. The free vibration of the system about the zero equilibrium is described
by a set of natural frequencies and mode shapes, and an energy distribution among
the modes. When the stiffness of the system is changed (by a suitable mechanism) the
equilibrium configuration changes and the system is described by a new set of natural
frequencies and mode shapes. In the general case, the new equilibrium configuration
of the system will have some stored potential energy and therefore will not be a zero
equilibrium. Due to the change of stiffness, the total energy of the system will be

redistributed. Assuming that no energy is lost due to change of stiffness, a part of

the energy will be stored as potential energy in the new equilibrium configuration;
the rest will be distributed among the new modes of the system. When the stiffness
of the system reverts back to its original value, the system recovers its original shape
and vibrates about the zero equilibrium position. All the energy of the system is
transferred back to the modes of its original configuration. If damping is ignored and
if no loss of energy occurs due to modal truncation, the total energy of the system
is conserved but the energy distribution among the modes is different from that at
the initial time. The activation and deactivation of the stiffness variation mechanism
thus allows redistribution of modal energy. The amount of energy pumped into or
out of a specific mode is dependent on the timing of the transition between stiffness

states and difference in mode shapes of the structure in the two stiffness states.

A water-bucket analogy of modal energy redistribution due to change in stiffness,
as discussed in the paragraph above, is depicted in Fig.2.1. The level of energy in the
modes is represented by the volume of water in the buckets. When the mechanism is
activated, the equilibrium configuration changes and some volume of water is stored
as potential energy in the new equilibrium configuration. The remaining volume is
distributed between the buckets that represent the modes of the system in its new
stiffness state. When the stiffness of the system reverts back to its original value, the
total amount of water remains conserved but its distribution between the buckets

changes.

The efficacy of energy redistribution can be quantified by the difference in the
volume of water in each bucket at the initial time and after one cycle of stiffness
variation. While this will depend on the time when the stiffness is changed and
reverted back to its original value, it will also depend on the difference between the

mode shapes of the structure in the two stiffness states. A measure of the change in

1st mode 2"d mode 3rd mode

Original
configuration , - ~ 3

[ "Stiffness change-«- [

 

 

mechanism activation

Potential energy ’
lSt mode 2nd mode 3rd mode

 

 

New .
configuration

 

   

m-Stif’fness change-«-
mechanism deactivation

lSt mode 2nd mode 3rd mode

 

Original
configuration

 

Figure 2.1. Modal energy redistribution after one cycle of constraint application and
removal

the mode shapes of the system due to change in stiffness is modal disparity. Modal

disparity, defined next, contributes to modal energy redistribution.

Modal disparity is the degree to which the mode shapes of the system in
its original stifiness state are different from those in its new stifiness state. The
projection of one set of mode shapes onto the other through the mass of the system is
a good way to quantify this difference. Typically, if there is no change in the stiffness,
there will be no change in the modes and if the modes are normalized with respect
to the mass matrix, this projection will lead to the identity matrix. In the case of
stiffness variation, this projection will lead to a matrix called the modal disparity
matrix. The norm of the (i, j) element of this matrix is a measure of how close the

th mode of system in one stiffness state is to the jth mode in the other stiffness

i
state. This matrix is essential for understanding the mechanics of transition between

different stiffness states of a given system.

10

The stiffness of a system can be changed in a variety of ways. Diaz and Mukherjee,
for example proposed the use of cables [2], and active joints [27] for varying the
stiffness of three dimensional structures In this present study we consider enforcing
constraints on the system as a way of changing its stiffness. In the next section we
derive expressions to relate the system displacements and velocities before and after
the change of stiffness. In section 2.3 a modal coordinate description of the transitions
is presented and modal disparity is quantified. A numerical example is presented in
section 2.4 to illustrate energy redistribution due to modal disparity. In this example,
the axial vibration of a rod is considered. The rod is fixed at one end and free at
the other and a magnetic brake is used to restrain the motion of the free end at any

desired time.

2.2 Spatial Coordinate Description

We consider the general case of continuous systems. An N dof1 reduced order model
of such systems can be derived using finite element methods. Ignoring damping, the
equation of motion describing the behavior of the system about its zero equilibrium

position can be written as follows:
M)? + KX = 0 (2.1)

where M and K are the N x N mass and stiffness matrices of the reduced order
system and X is the N dimensional vector of generalized displacements corresponding
to the dof. This unconstrained state of the system is referred to as state a. Upon
application of a constraint, the configuration of the system changes. Using the same
finite element mesh, a reduced order model with N dof is again considered to facilitate
mathematical operations. In this state the system vibrates about its new equilibrium

position. Ignoring damping and using the new equilibrium as a reference, the equation

 

1degree-of-freedom

11

of motion of the system can be written as
M? + RY = 0 (2.2)

where I? is the new N x N stiffness matrix and Y is the N dimensional vector
of displacements corresponding to the new dof. Physically, X and Y are the
displacements of the same nodes of the finite element model but have different
origins because of the system vibrating about different equilibrium configurations.

The constrained state of the system is referred to as state ,6.

After describing the behavior of the system in each state, we derive expressions to
relate the system displacements and velocities during its transition from one state
to another. We first assume that the system is vibrating in state a, and at some
time tag, the constraint is enforced to transfer the system to state ,6. This transition
actually occurs over a brief interval of time t E [t;fi, tgfl] and results in the application
of impulsive forces. The system displacements and velocities after the application of
the constraint, X (tgfi) and X (tgfi), can be related to the system displacements and

velocities before the application of the constraint, X (tgfl) and X (tag) as follows:

my)

X(t;g)
. . (2.3)
MX(t;‘fi) = MX(t;fi)+Ia_,fl

where Ian/3 is the impulse vector. The equation above can be solved without a
problem since the number of unknowns in [aafi is equal to the number of dependent
variables in X (tgfl). The initial conditions Y(t;’fl) and Y(t:fl) of state ,8 are expressed

in terms of the final conditions in state a as

thgg) = X(t;’fl)-Xo(t:g)
Y(t:g) = X0323)

12

where X0(t;'fl) is the new equilibrium position. The equilibrium position X0(t:fl) is
determined from static analysis of the structure, as will be illustrated with the help
of an example later in section 2.4. It is important tonote that X0(t;'fi) is a function
of the switching time tag Thus, the amount of potential energy stored in the system,
E0 = %Xg(t:fl)KX0(t:fi), is time dependent. When the constraint is removed, the
system reverts back to state a. This transition occurs over a brief interval of time
t E [tgwtga], during which there will be no change in the system displacements or

momentum, i. e.

Y(t2;a) = raga)
(2.5)
3’05.) = thfia)
The initial conditions of the system in state a are calculated from the displacements

and velocities in state B using the relations

Mtg...) = Y<tga>+xo<tgg>
(2.6)
Xltfia) = mg.)

In the next section we investigate the transition between states in modal coordinates.

2.3 Modal Coordinates Description

In order to describe the system in modal space, we define <15,- and 77,-(t), i = 1, 2, . . . , N,
to be the normalized mode shapes and modal amplitudes of the unconstrained system.
Similarly, we assume wz- and V,- (t), i = 1,2,... , N, to be the normalized mode shapes
and modal amplitudes of the constrained system. The system displacements X (t) in

state a and Y(t) in state 6 can now be written as follows

13

X(t) = Z9; dm,(t) in state a
(2.7)
Y(t) = 2,121 with-(t) in state 6

The change in the system displacements and velocities due to the application of a

constraint, described by Eqs.(2.3) and (2.4), can be written as

High) = X(tggl—Xoftgg)

(2.8)
MY(t;'fl) = MX(t;fi) + [any
Substitution of Eq.(2.7) into Eq.(2.8) yields
2,121 Wdtgg) = Bill ¢i77iftggl — X00553)
(2.9)

Magnum) = M2,}:1 ¢m.-(t;fl)+Ia_.fl

The modal displacements and velocities after application of the constraint can be

computed from Eq.(2.9) as follows

”1035) = Z£1¢$M¢i 772;“; )—¢fMXO(t:gl
fl (210)
Wig) = 2.1:. «film.- may.)

In the derivation of Eq.(2.10) the identity #2311104]; 2 0 was used. This follows from

the fact that constraint forces do zero work, i.e Y(t;fl)TIa_,fl = 0.

The modal displacements and velocities after the constraint has been removed are

calculated using Eqs.(2.5), (2.6) and (2.7), as shown below
wry-(ti...) = 2.121 ifMu Mtg...) + ¢}"MXo(t:g)

(2.11)
mag.) = Z£1¢$M¢iviugal

14

If we define

_ 77100 q _ V10) l
,, <I>=[¢1 <22 451v]
n(t)= ”if” , u<t>= 2.“) , .
-nNs). .vN<t>_ “H” $2 W]

 

 

 

 

the transition between states oz and [3 described in Eqs.(2.10) and (2.11) can be

rewritten in vector form as follows

(14%) = \IITM<1>n(t;fl)—\IJTMXO(t:B)

a ——+ fl
I var.) = Wm mg)
(2.12)
f ”(7:50) _ (PTA/I‘ll V050) +¢TMX0(t:¥-fl)
s —> a ]
mega) = <I>TM\111'/(tga)

It is clear that the matrix \IITM<I> and its transpose <I>TM\II are central to the transfor-
mation between states a and 6. For this reason these matrices are chosen as measures

of modal disparity.

2.4 Numerical Example

2.4.1 Modeling

We consider the axial vibration of a linear rod with a constant circular cross—section,
as shown in Fig.2.1. The rod is fixed at one end and free at the other end. An
electromagnetic brake attached to the free end of the rod enables it to be fixed at
any desired time. In stiffness state a the brake is free whereas in state 5 the brake is

locked. The differential equation describing the motion of the system is

15

Side view """" Electromagnetic
f """"" T [ d brake

x s
—> i---->
in---» 5
<¥ L :I

Figure 2.2. A flexible rod

 

 

 

 

 

\\ \k\

II
pAii — EAu = 0 (2.13)

for all x E (0, L). The boundary conditions are

u(O, t) = 0
State a u’(L, t) = 0

State )6 u(L, t) = 6

In Eq.(2.13), E is the Young’s modulus of elasticity, p is the material density and A
is the cross-section area of the rod. In the boundary conditions, 6 is the extension
of the locked end of the bar in state B. The bar is modeled with N linear standard
finite elements, see [28] for example. Each node has one displacement dof denoted
by u(x, t). If Le denotes the length of an element, the elementary mass and stiffness

matrices take the form

 

_pALe 21
Me“ 6 [12

“l1 “l

]’ [fez—L: —1 1

The modeling of both states is facilitated by the use of a linear spring as shown in

16

Fig.2.3. It is attached between the free end of the bar and the wall. In state or the
end is free, i.e. ks = 0. In state ,6, it is fixed and this is modeled by setting k3 = 00,
i.e., setting [93 to some large number. The equations of motion of the system in both

states will take the form as in Eqs.(2.1) and (2.2).

Linear spring k,

l

 

/

Electromagnetic
brake

 

Figure 2.3. The electromagnetic brake model

2.4.2 Simulations

The material and geometric properties of the rod are assumed to be those given in

Tab.2.1. The rod was modeled by 300 finite elements, i.e., N = 300. The natural

Table 2.1. Material and geometric properties of the beam in Fig.2.2

 

 

Material Rubber
Young’s Modulus 10 M Pa
Density 600 Kg / m3

Cross section Area 7r 0.052 m2

Length 4.0 m

 

 

frequencies of the first 6 modes of the rod in each state are tabulated in Tab.2.2. The
normalized mode shapes of the rod in the two stiffness states are shown in Fig.2.4. It

is important to note that the mode shapes in state 6 do not depend on the deflection

17

of the end of the rod (6) in the brake locked configuration. This fact is justified as
follows. The extension of the end of the rod will move the equilibrium of the system
to a new position X0, X0 aé 0. If the change of variable Y = (X — X0) is used, the
new system will have the same mass and stiffness matrices as given by Eq(2.2) for all
values of 6. The only change will be the level of potential energy éXg(t:fl)KX0(t:fi)

that will be stored in the system.

Table 2.2. Natural frequencies of the beam in the two stiffness states

 

mode number, i
waia “1,37; (HZ)

 

i=1 i=2 i=3 i=4 i= i=6

 

 

a 8.07 24.21 40.34 56.48 72.62 88.76
6 16.14 32.27 48.41 64.55 80.69 96.83

stiffness state

 

 

 

 

 

 

 

 

 

 

 

 

( —0.849 0.509 0.121 0.057 -0.033 0.022\
0.340 0.728 —O.566 —0.154 0.078 —0.048
—0.218 —O.283 —0.694 0.588 —0.170 0.090
0.162 0.185 0.261 0.679 0.600 —0.179
0.129 0.140 0.170 0.250 —0.670 -0.606
(—0.107 —0.113 —0.128 —O.161 0.242 —0.664 )

\DTMe = (2.14)

 

 

Using a 6—mode model of the system in each state, we consider the scenario where
the system is vibrating in state a starting from the following initial conditions:

n(0)=[0.00 0.00 0.00 0.00 0.00 0.00]T
0(0)=[0.01 0.01 0.01 0.01 0.01 0.01]T.

After 3 seconds in state a, the brake is locked and as a result the system is transferred

to state 6. Using Eq.(2.10), the initial conditions in state 6 are calculated as follows:
u(3+)=1e"5[ —5.67 —3.85 1.68 -—1.34 -—0.28 —1.24 ]T
p(3+) = 16—2] —O.56 —O.78 —1.22 —0.04 -—0.44 0.40 ]T.

18

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x (m)

Figure 2.4. Mode shapes of the rod in the two stiffness states: the solid lines represent
the di’s and the dashed lines represent the wi’s.

19

Here 6 = 6.38e‘5 m and if ET is the total energy in the system, the energy stored in
the system after application of the constraint is 0.133 ET. After 3 seconds in state
6 the system is switched back to state a, the modal displacements and velocities are
calculated using Eq.(2.11) as

n(6+)=1e—5[18.90 —4.55 0.72 0.80 —1.51 1.58
7'7(6+)=1e_2[0.38 —0.63 1.09 —0.83 0.54 —0.08]T.

]T

A scheme illustrating modal energy redistribution during one cycle of constraint appli-
cation and removal is shown in Fig.2.5. At the first transition 13.3% of the energy is
stored as potential, 81.4% is redistributed between the first 6 modes of the system in
state 6 and the remaining 5.3%, not shown here, is transferred to the higher modes.
After removing the constraint, 91.2% of the total energy ET is rearranged in the

modes of the system in state a and 3.45% is transferred to the higher modes.

State

9

State B State a

1st 16.67% 1st 10.87% 1st 17.66%
2nd 16.67% 2nd 20.41% 2nd 14.54%
3rd 16.67% 3rd 29.31% 3rd 20.33%

13.32 % .

4th 16.67 % 4th 4:11 12.82 %
5th 16.67 % 3.59 % 5111 "12.74 %
6th

5th
16.67 % 6th 12.23 % 6th [13.07 %

 

  

 

 
 
 

 
 
   

    
  

 
 
 
 

.. time (see)

Figure 2.5. Modal energy redistribution in the bar after one cycle of constraint application
and removal

to
O

2.5 Remarks

A method for modal energy redistribution was presented in this chapter. It was shown
that modal energy of a vibrating system can be redistributed by stiffness variation,
and specifically through cyclic application and removal of constraints. The amount
of energy redistributed depends on the difference in the mode shapes of the structure
in the two stiffness states. A measure of this difference in the mode shapes of the
structure is defined as modal disparity and is quantified. In the next chapter we
provide an experimental verification of modal disparity and demonstrate the potential

for energy dissipation through modal energy redistribution.

21

CHAPTER 3

Experimental Verification Of Modal

Disparity

3. 1 Introduction

In this chapter we present experimental results to illustrate modal disparity in a struc—
ture due to change in stiffness. In an earlier work [1], [2], modal disparity of structures
with variable stiffness was computed and simulation results of modal energy redistri-
bution were provided. The objective of this chapter is to experimentally demonstrate
modal energy redistribution in a clamped-clamped beam with a variable stiffness joint.
A mathematical model of the beam using finite elements is presented in Section 3.2.
The mechanics of stiffness variation is discussed in Section 3.3. Simulation results
are presented in Section 3.4; they provide a benchmark for the experimental results

presented in Section 3.5. Concluding remarks are provided in Section 3.6.

3.2 Modeling

In this section we review the finite element procedure to model the free vibration of
a beam with a mid-span hinge, as shown in Fig.3.1. We assume that the hinge has a
built in actuator enabling it to be locked or released at any desired time. Clamped

at both ends, the beam is switched from one stiffness state to another by locking or

22

 

 

 

 

 

\\\\\\\\
////////

 

 

Figure 3.1. A clamped-clamped beam with a mid span hinge

releasing the hinge. Let A be the cross-sectional area and I be the area moment of
inertia of the beam. Let p and E be the material density and modulus of elasticity
of the beam, respectively. Assuming Euler-Bernoulli theory, the equation of motion

of the beam in the x—y plane can be written as follows:

I/Il

Ely +pAy=0 ifxE (0,L/2)orx€ (L/2,L) (3.1)

The boundary conditions are
y(0,t) = y(L, t) = 0
y (OJ) = 3! (lat) = 0
y((L/2)",t) = y((L/2)+.t)

III

6 <<L/2>-,t) = y”’<(L/2>+.t>

The beam is in stiffness state a when the hinge is free and in stiffness state )6 when
it is locked. The remaining boundary conditions needed to completely describe the

behavior of the system in the two stiffness states are as follows:

stiffness state a y::((L/2)—, t) = 0
y ((L/2)+,t) = 0
y’((L/2)—,t) = y’((L/2)+,t)

s iffness S ” N
t tat/8,6 y ((L/2)—,t) = y ((L/2)+,t)

23

6': y'(L/2'.t)

 

 

 

/\/

Figure 3.2. The hinge model

We model the beam by N standard (cubic) finite elements. Each element has two
nodes with two dof1 per node (translation in the y axis and rotation about the z
axis). The modeling of the hinge in both states is facilitated by assuming that the
node at x = L/ 2 has two rotational dof, 0’ and 0", corresponding to y’(L/2-, t) and
y’(L/2+,t), respectively. When the hinge is free (stiffness state a), 6’1 and 0" are
independent. However, when it is locked (stiffness state 6) the constraint 6’ = 6" has
to be enforced. This is achieved by adding a penalty of magnitude yer (6’ — (9")2 to
the strain energy, that is, by adding a rotational stiffness between 0’ and 19" as shown

in Fig.3.2.

K. = k. [ _[ ‘] ] (3.2)

To account for the hinge model, the stiffness matrix in Eq.(3.2) is added to the global
stiffness matrix. In the stiffness state a, the parameter k7. is set to zero and in stiffness
state 6, it is set to a large positive value. The finite element model has (2N — 1) dof.
The hinge model is shown schematically in Fig.3.2, where the hinge mass is accounted

for by the addition of a lumped mass m), at the central node.

 

1degrees—of-freedom

24

3.3 Stiffness Change Mechanism

As mentioned in the previous section, the transition from stiffness state a to stiffness
state 6 is accomplished by an actuator, having the capability to lock the hinge. This
transition is assumed to occur over a brief interval of time and results in the applica-
tion of an action-reaction pair of impulsive moments to the middle node, as shown in
Fig.3.3. Furthermore, by sensing the state of the beam, the hinge is locked only when
the beam is passing through its equilibrium configuration. Let tgfi and tgfl denote
the beginning and the end of this brief transition period. The effect of the impulsive

moments can be mathematically described by the relations

Y(t;fl) = Yugfl)
(3.3)
MY(t;fi)+Ia_,fi = MY(t:fi)

where M is the mass matrix and Y is the vector of nodal dof. Ia_.fi, the impulse

vector, takes the form shown below:

Iaafi:[02”'iCa_Ca"'10]Ti C:fi

The nonzero entries of 1.1—.3 correspond to the coordinates 61 and 6’", where C is the
impulse and r is the impulsive moment. Y(t:fl) and Y(t;'fl) are the displacements and
velocities in stiffness state 6 at the end of the transition period. They are calculated
from the values of Y(t;fi) and Y(t;3) using Eq.(3.3). Although C is an unknown,
Eq.(3.3) can be solved since two elements of Y(t;3), namely 6°l and 0", are equal. The
transition from stiffness state 6 to stiffness state a is accomplished by releasing the
hinge. If tEa and tga denote the beginning and the end of this brief transition period,
the beam displacements and velocities vectors just prior to and right after the release

of the hinge are the same, i.e.

25

\ / 1'(t) dt [16) dt /
/ g Q \

middle node

 

 

Figure 3.3. Action-reaction pair of impulsive moments

205..) = mg.) (3.5)
mg...) = mg.)

The beam behavior at the transition from one state to another is described by equa-
tions (3.3) and (3.5). Eq.(3.1) describe the beam behavior at all other times as a
clamped-clamped beam with a frictionless hinge at mid span (in stifiness state a)
or a clamped-clamped beam (in stiffness state ,6). With this notation, the stiffness

parameter 16,. (introduced in section 3.2) can be defined as follows
{ 0 stiffness state a
kr =

1600 stiffness state 5

where koo is some large positive number chosen to enforce the constraint 6’ = 0'".

3.4 Modal Space

Let gt,- and «[2,- denote the i-th normalized mode shapes of the beam in stiffness state
a and stiffness state ,6, respectively, and let 11,-(t) and 11,-(t) denote the corresponding
modal displacements. In the two stiffness states, the vector of nodal dof can be

expressed as

26

{2,21% 1 ,uz-(t) 65,- stiffness state a

)=] (3.6)

l 231—1 112-(t) w,- stiffness state ,8

The transition from stiffness state a to stiffness statefi, mathematically described by

Eq.(3.3), can now be rewritten as

Zfo/tl magi.- = Egg—IVAtZgWi
(3.7)
MZ§=Nl_1fli(t;g)¢i+Ia_.g = Mimi—1V2“ 3W

Using Eq.(3.7), the modal displacements and velocities, Vj(t:fi) and 1'1]- (tzfi), can be

expressed in terms of ,uJ-(tgfl) and uj(t a—fl) as follows

22.93;.) = Effl—lij‘watgfl)
(3.8)
2.02.7.) ——- 2351—1wa4. 77.03;.)
In the derivation of Eq.(3.8) from Eq.(3.7), we used the identity 7,0311 1043 = 0. This
is true since the entries of W, j = 1, 2, - - - (2N —- 1), corresponding to the nonzero
entries of 10,—43, namely, 01 and 6" are equal.
Using the same procedure as above, the transition from stiffness state 6 to stiffness

state 04 can be obtained from Eq.(3.5) as follows

2.0;.) = 2351‘1¢}"Mw.v.(tga)
(3.9)
rig-(ti...) = Zifi‘lqswamga)

Ifwe define (I) = [(91, p2, - -- ,¢(2N_1)] and \II = [101,102, - - - , ¢(2N—1)]a it is clear from
Eqs.(3.8) and (3.9) that elements of the matrix \IITMQ define the mapping between

modal coordinates during the transition from stiffness state a to stiffness state 6. The

transposed matrix, <I>TM\II, defines the mapping between modal coordinates during

27

the transition from stiffness state 6 to stiffness state a. These matrices will be identity
matrices if the two stiffness states are the same and any deviation from the identity

structure is a measure of modal disparity between the two stiffness states [1].

3.5 Numerical Example

Consider the beam in Fig.3.1 with the material and geometric properties in Table 3.1.

The mode shapes of the first four modes of the beam in the two stiffness states are

Table 3.1. Material and geometric properties of the beam in Fig.3.1

 

 

Material Aluminum
Young’s Modulus 71 GPa
Density 2710 K g/ m3
Cross section Area 0.05x0.0023m2
Length 2.0 m

Hinge mass 0.182 Kg

 

 

shown in Fig.3.3, and their corresponding natural frequencies in Table 3.2. The even
numbered modes in the two stiffness states are identical. This is true since the hinge
is located at mid-span where the even numbered modes have zero curvature therefore

they will not be affected by the state of the hinge, i. 6. locked or released.

Table 3.2. Natural frequencies of the beam in the two stiffness states

 

mode number, i
wai, Wfii (HZ)

 

 

 

 

 

 

 

 

 

 

 

i = 1 i = 2 i = 3 i = 4
a 1.29 8.34 9.52 27.0

stiffness state
6 2.30 8.34 14.68 27.0

 

28

 

 

 

 

 

 

 

 

 

 

0 0.5 1.0 1.5 2.0 0 0.5 1.0 1.5 2.0
x (m) x (m)

 

 

      

 

 

 

 

 

 

0 0.5 l.0 1.5 2.0 0 0.5 1.0 1.5 2.0
x (m) X (m)

Figure 3.4. Mode shapes of the beam in the two stiffness states

Using modal truncation, the matrix measure of modal disparity, \IITMq), is computed

using the first four modes as follows

0.980 0.000 0.153 0.000
0.000 1.000 0.000 0.000
0.139 0.000 0.949 0.000
0.000 0.000 0.000 1.000

\IITM<I> = (3.10)

In the matrix \IITM<I>, the fact that second and fourth rows and columns maintain
the identity structure, is an indication that the even—numbered modes in the two
stiffness states are identical. However, the non-unity value of the diagonal elements
and nonzero elements in the off—diagonal entries of odd-numbered rows and columns
indicate the presence of modal disparity between the odd-numbered modes of the two

stiffness states.
To illustrate modal energy redistribution between odd-numbered modes in the two

stiffness states, we consider the scenario where the beam is initially in stiffness state

a and vibrating purely in the third mode with a maximum amplitude X0. Ignoring

29

damping, the total energy in the system is equal to Ea = 0.5 X3 112,213. It is assumed
that the beam is switched to state 6 when it passes through its neutral position.
The modal displacements right after locking the hinge are zeros u,(t:fl) = 0, since
uz-(tgfi) = 0. The modal velocities right after switching to stiffness states ,8 can be

computed from Eq.(3.8) as follows

 

 

”191033) ‘ 0.980 0.000 0.153 0.000 0

- +

122005) 2 0.000 1.000 0.000 0.000 0 (3.11)
123(tgfi) 0.139 0.000 0.949 0.000 X06203

[1240533) _ 0.000 0.000 0.000 1.000 0

Clearly, the energy of the beam is redistributed in modes 1 and 3 in stiffness state 6.

The maximum amplitudes of these modes are

(3.12)
The modal energy in modes 1 and 3 are easily calculated as follows.
Em = 2Xglwgl = $153226: 7033 = 0.1532 E, (313)

E33 = §Xg3wg3 = l09492/r3 7.233 = 0.9492 Ea
These results will be validated through experiments in the next section.

3.6 Experimental Verification

The experimental hardware is shown in Fig.3.5. The beam has a pair of piezoelectric

2 mounted on each side at a distance of 5.0 cm from one of the clamped

3

transducers
ends. These transducers are used for excitation. A single piezoelectric strain sensor

is mounted on the beam at a distance of 5.0 cm from the other clamped end. The

 

2product of Mide Technology Corporation
3product of PCB Piezotronics

30

  

 

piezoelectric strain sensor

 

electromagnatic brakes

Figure 3.5. Experimental hardware

position of the sensor and actuators are chosen to ensure high degree of controllability
and observability of the first three modes of the system. The material and geometric
properties of the beam in the experimental setup are the same as those used in sim-
ulations and provided in Table 3.1. In this table, the hinge mass includes the mass
of the electromagnetic brakes4, shown in Fig.3.5, used for locking and releasing the

hinge. In our experiments, we chose to investigate energy redistribution between the

Table 3.3. Natural frequencies of the beam in the two stiffness states, determined experi—

mentally

 

mode number, i

 

“at: ”62' (HZ) ll
[] i: 1 i: 2

 

 

 

 

 

 

i=3
stiffness state a ll 1'40 8'24 9-70
fi [I 2.37 8.20 13.90

 

 

first three modes of the beam. This was motivated by the fact that modal disparity
can be adequately demonstrated by the first three modes and estimation of the higher

modes are more prone to inaccuracies The first three natural frequencies were ex-

 

4product of Inertia Dynamics

31

piezoelectric actuator

perimentally determined for both stiffness states and are provided in Table 3.3. The
piezoelectric transducers were used to excite the beam and the strain sensor was used
to measure beam vibration. The natural frequencies were identified as the frequencies
of excitation that resulted in maximal amplitude of vibration. The experimentally de-
termined values show good agreement with the numerically computed values in Table
3.2. We first present experimental results for two cases where the beam was initially
in stiffness state a (hinge released) and switched to stiffness state [i (hinge locked).
For the first case, Case A, the beam was excited at its second natural frequency in
stiffness state a. The stiffness of the beam was switched after termination of excita-
tion and the results, shown in Fig.3.6, indicate that the beam vibrates primarily in
its second mode in stiffness state 6. This is expected since the second mode of the
two stiffness states are identical. This can be verified from the elements of the second
column vector of the modal disparity matrix \IITMCP in Eq.(3.10). All entries of this
vector are zero except for the second entry, which is unity. For the second case, Case
B, the beam was excited at its third natural frequency in stiffness state 0. Its stiff-
ness was switched after termination of excitation and the results are shown in Fig.3.7.
Since the first and third elements of the third column vector of \IJTM<I> are nonzero,
the beam vibrates in its first and third natural frequencies in stiffness state 6. The
amplitude of these modes, immediately after the switch, can be computed based on
our analysis in the last section. These values and the values obtained from experi-
ments are both presented in Table 3.4 and they show good conformity. The plots in
Fig.3.7 indicate a small presence of the second mode in both stiffness states. It is
logical to infer that excitation of the beam introduced the second mode in stiffness
state a and energy associated with this mode transferred directly to the second mode
in stiffness state 6. For the sake of completeness, we present experimental results for
one case where the beam was initially in stiffness state ,6 (hinge locked) and switched

to stiffness state a (hinge released). The results for this case, which we denote as

32

 

sensor
9

 

I
—
in-
-
b
I-

 

 

 

 

111(t)
V10)

 

 

 

 

 

 

 

1120)
V20)

 

 

 

 

 

 

1130)
V30)

 

 

 

 

 

 

4 0 i 2 3 4
time (sec)

O
—-l'
N5
1.»

Figure 3.6. Energy redistribution between modes for Case A

Case C, are shown in Fig.3.8 and summarized in Table 3.4. For this case, the beam
vibrates in its second mode in stiffness state [3 and energy associated with this mode
isfentirely transferred to the second mode in stiffness state (1, upon switching. The
results for this case are therefore quite similar to that of Case A. The amplitude of
the third mode in stiffness state a could not be measured accurately and is marked
“xxx” in Table 3.4. The difliculty of the measurement was due to its small magnitude
coupled with waxing and waning due to beating. The beating phenomenon can be
attributed to the close proximity of the second and third natural frequencies of the

beam in stiffness state a.

3.7 Remarks

The investigation in this chapter has confirmed that changes in structural stiffness

result in modal disparity, and that this disparity permits energy to be transferred

33

 

sensor
:3

 

 

 

 

1110)
O
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
V1( )
O

 

 

 

 

#20)
V20)

 

 

 

 

 

 

1130)

 

 

 

 

time (sec)

Figure 3.7. Energy redistribution between modes for Case B

between different sets of spatial modes in a given structure. Finite element based
analysis and systematic experiments have demonstrated that the phenomena can be
modeled and quantitatively predicted. One of the keys in the modeling is to properly
account for the physics of the transition between the different stiffness states, which
results in the correct mapping of the modal energies from one set of modes to another.
With these tools in hand, it should be possible to design structural systems with built-
in mechanisms for stiffness variation for favorable modal disparity, and to predict the
efficacy of various proposed switching schemes. In the next chapter we present a
semi-active control strategy where the stiffness of the structure is actively varied to
transfer energy from the low-frequency modes to the high-frequency modes where it

can be dissipated naturally and quickly.

34

Table 3.4. Modal amplitudes immediately before and after switchings

 

amplitudes before switch

amplitudes after switch

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Stiffness mode number i actual/ expected values
switch
i=1 i=2 i=3 i=1 i=2 i=3
Case A a —-> B 0.01 0.88 0.01 0.02/0.01 0.80/0.88 0.01/0.01
Case B a —> [3 0.02 0.07 0.68 036/042 005/007 044/045
Case C B ——+ a 0.00 0.84 0.01 0.01/0.00 0.79/0.84 xxx/0.00
tpa
l l f r T r
.
8
'10 3 4 '5 6 l7 8
state [3 state a
1 - - 1 .
E; 0 9: 0
> :1.
-l -1 1 m
1 1 -
E"; 0 5320'
> :1.
-l, ,1 m
1 - 1
8m 0 v~v v v‘v .Av---fi v vvv vv v E; 0 W
> :1.
I 4 - . .1 . .
0 1 2 3 4 0 1 3 4
time (sec)

Figure 3.8. Energy redistribution between modes for Case C

35

 

CHAPTER 4

Energy Dissipation In Dynamical Systems
Through Sequential Application And

Removal Of Constraints

4. 1 Introduction

Energy dissipation is the primary objective of many control problems in dynamical
systems. For such problems, we explore the feasibility of energy dissipation through
sequential application and removal of constraints. We illustrate our basic idea with

the example of the three dof1 mass-spring system, shown in Fig.4.]. In Fig.1, the

1162 163

k .
WWW/rm m2 m3

éLil

 

 

 

 

 

 

 

 

Figure 4.1. A three dof mass-spring system

displacements of the three masses are denoted by x1, x2, and 1123. A constraint x2 = 0
is applied by instantaneously pushing the pin into the slot in mass m2 and removed
by pulling the pin out of the slot. The application of the constraint requires the

instantaneous position of mass mg to satisfy x2 = 0 but removal of the constraint

 

1degrees-of-freedom

36

can occur at any time. The application of the constraint results in an instantaneous
reduction in energy of the overall system by an amount equal to the kinetic energy of
mass mg. The removal of the constraint does not alter the energy of the system but
enables the constraint to be applied again for further reduction of energy. Except in
special situations2, sequential application and removal of the constraint will ultimately

reduce the energy of the system to zero. On reading this example, two questions arise:

1. What happens to the kinetic energy of mg when the constraint is applied ?

2. Can one generalize this idea to remove energy from dynamical systems through

sequential application and removal of constraints ?

The answer to the first question is provided in Section 4.2 but it requires that we
model the pin and/ or mass as deformable bodies. This is further explained with
the help of the next example of direct central impact, which can be found in almost
any textbook in dynamics. The answer to the second question is the subject of this

chapter and is discussed in the remaining sections.

Consider the two particles A and B, of mass m A and m B, moving to the right
along the same straight line with velocities v A and 213, as shown in Fig.4.1. If we
assume that v A > 123, particle A will eventually strike particle B. Upon impact, the
two particles will deform and at the end of the period of deformation they will have
the same velocity u. A period of restitution will then take place and at the end of
this period, the two particles will have velocities 22:4 and ob. The velocities 12:4 and

u}; can be obtained by solving the two equations

 

2An example of a special situation is where the system of masses move in a manner that maintains
x2 (t) E 0; this will occur if one of the modes have a zero displacement for mass m; and the system
vibrates in that mode.

37

mAvA+mBuB = mAuh+mBu’B
(via-vii) = 80124-03)

The first equation corresponds to conservation of linear momentum. The second
equation relates the relative velocities of the particles, before and after impact,
via the coefficient of restitution, e 6 [0,1]. In regards to this standard textbook
131’: 121’: .1614. W “’4 VH4
69. rmatiostitutio Q9.
' period ’ ' period '

Figure 4.2. Direct central impact between two masses

explanation of the impact phenomenon, we wish to make the observation that
initially the two masses are assumed to be rigid bodies, and hence they are
referred to as particles, but later they are assumed to undergo deformation, in
contradiction with the initial assumption of rigidity. In reality, the masses are
deformable bodies and their deformation excites their flexible body modes if the
material is elastic. The rigid body assumption, routinely made, simply implies that
the flexible body modes correspond to high frequencies and are not relevant to

the problem involving the rigid body motion of the masses after the restitution period.

Under the implicit assumption that the masses are deformable bodies, it is
useful to discuss the two special cases corresponding to e = 0 and e = 1. When
e = 0, the two masses have zero relative velocity after impact and this is referred
to as “perfectly plastic impact”. In the absence of restitution, the masses undergo
plastic deformation. Work is done during plastic deformation and hence the
kinetic energy of the masses after impact is less than that before impact. For

e = 1, the masses have the maximal relative velocity after impact, equal to that

38

before impact. In this case of “perfectly elastic impact”, some of the kinetic
energy is stored as potential energy during deformation of the masses. The stored
potential energy is completely converted back to kinetic energy during restitution

and consequently the kinetic energy of the masses is the same before and after impact.

In this chapter we are interested in the removal of energy in “perfectly elastic”
systems and thus we eliminate the possibility of permanent or plastic deformation.
For the two-mass example, this implies no loss in kinetic energy and this can be simply
attributed to the fact that the motion of the masses A and B are unconstrained after
impact. To understand the loss of kinetic energy due to application of a constraint, as
in the case of our 3—dof mass spring system, we consider a variation of the two-mass

problem next.

4.2 Energy Loss due to Application of a Constraint

Consider the problem of direct central impact where the two masses are perfectly
elastic, but, by virtue of some mechanism, the masses remain coalesced after they
make contact. This scenario, depicted in Fig.4.3, will result from application of the
constraint (x A —— x Bl = 0, where x A and x B are the position coordinates of the
masses. In this scenario, conservation of momentum dictates that the velocity of the

two masses after contact will be

 

 

v’ = u — 6 m3
A ("M + ms) ( )
4.1
u’ = u + 6 mA
8 ("M + me)

where u is the velocity of the center of mass, an invariant of the motion, has the

expression

39

u
’ 9° ’ u

wamng —>
u ee

steady

waxing

S
5

Gigs
gl
l

‘41
—>
Figure 4.3. Scenario depicting two masses that stay connected after impact

u_ mA’UA-i-‘mB’UB
mA+mB

 

(4.2)

and 6 has the dimension of m/ sec, and assumes a positive value during waxing, a
negative value during waning, and a zero value at points of maximum deformation.

From Eqs.(4.1) and (4.2), the kinetic energy of the masses after contact can be
computed to be

1 mAmB 62
2(mA+mB)

 

1 1 1
-mAvk2 + —vai32 = —(mA + mslu2 + (4-3)

2 2 2

The second term on the right hand side of Eq.(4.3) denotes the kinetic energy that is
converted into potential energy during deformation, and converted back into kinetic
energy when the masses regain their original shape. This cyclic conversion between
kinetic and potential energy is possible due to excitation of the flexible modes of the
masses, and this process would continue perpetually if there were no modal damping.
In reality, the high-frequency flexible modes have high modal damping, and as a result
the energy is gradually dissipated through conversion into heat. From the maximum
absolute value of 6, which corresponds to e = 1, the total amount of energy dissipated

can be shown to be

40

l mAmB [6'2

_1 mAmB
2(mA+mB) "‘03— 2(

m(UA — ”B)2
When all of this energy is dissipated at steady state, the two masses move together
with the common velocity u, as shown in Fig.4.3. In contrast with the scenario
depicted in Fig.4.2, where zero relative velocity of the masses after impact implies
plastic deformation, the two masses here retain their original shape. Clearly, the
application of a constraint for a finite duration of time results in a “perfectly plastic

impact” behavior for “perfectly elastic” material property.

4.3 Relation to Prior Work

It is clear from our discussion in Section 4.2 that application of a constraint in a
perfectly elastic finite dof system results in motion identical to a perfectly plastic
impact. This dissipates mechanical energy and provides the opportunity for energy
removal through sequential application and removal of constraints. Although such a
method of energy dissipation has not been explored in the literature, there are many
papers on energy dissipation using impact dampers. The pioneering work was done
by Paget [29] for vibration reduction in turbine blades. This motivated the analytical
studies by Lieber and Jensen [30], Grubin [31], and Warburton [32] on impact damping
in single dof systems. The extension of this work includes analytical and experimental
investigations of multi-unit impact dampers [33], and the effect of impact dampers on
multi-dof systems [34], [35], [36], and continuous systems [37], [38]. There is a large
volume of literature on impact dampers, and in that work, impact is a result of a
physical collision. The approach presented in this chapter is fundamentally different
in that the physical collision is followed by coalescence, as opposed to separation. The
coalescence causes changes in natural frequencies and mode shapes of the system and

results in energy transfer from one set of modes to another. For finite-dof systems,

41

the coalesced state also has fewer modes due to fewer dof and this “modal truncation”
results in loss of kinetic energy. The modal description of impact and redistribution
of energy due to application of a constraint has been verified experimentally for a
clamped-clamped beam in chapter 3, and in this chapter we generalize the idea for

finite-dof systems.

4.4 Finite DOF Linear Systems

In this section we investigate energy dissipation in linear systems with finite dof
through sequential application and removal of constraints. From our earlier discussion
we know that application of a constraint will result in an impact and transfer of energy
into flexible body modes of the system, where it will be dissipated. Our finite-dof
assumption simply implies that the flexible modes are unmodeled; this is justified
by the fact that the energy transferred to these modes decay rapidly and their high-

frequency dynamics have negligible effect on the dynamics of the the rigid body dof.

4.4.1 Spatial coordinate description

Consider the N -dof linear system
MX + KX = 0 (4.4)

where M and K are the N -dimensional mass and stiffness matrices and X =
[x1,x2, - -- ,xN]T is the vector of independent generalized coordinates. Upon ap-

plication of a holonomic constraint, the dynamics of the system takes the form
M? + R Y = 0 (4.5)

where M and If are the (N —1)—dimensional mass and stiffness matrices and
T
Y = [y1, y2, - - - ,y(N_1)] is the vector of independent generalized coordinates of the

42

constrained system. Both the unconstrained and constrained systems are assumed
to be undamped since our objective is to investigate energy dissipation solely due to
application of the constraint.

We denote the unconstrained system as state a. and the constrained system as
state ,8. The transition from state a to state 6 occurs over the brief interval of time
when the constraint is applied. If [t;fl, tgfi] denotes this transition interval, the effect

of the transition can be mathematically described by the relations

X(tgfl) = X(tgfl) (4.6)
MX(t2:fi) = MX(t;fl)+Ia_,5 (4.7)
and
Y(t:fi) = TafiX(t;-fl) (4.8)
Y(t:fl) = TaflXugfl) (4.9)

where 10,—. g is the N -dimensional impulse of the generalized forces and Tag is a
constant (N — 1) x N transformation matrix. Equations (4.6) and (4.7) enable us
to determine the states of the unconstrained system, X and X, immediately after
application of the constraint and Eqs.(4.8) and (4.9) determine the initial conditions
for the constrained system. The number of unknowns in the vector Ia—pfl is equal
to the number of constraints and equal to the number of dependent variables in the
vector X (tgfi), which is one in the present discussion, and therefore Eq.(4.7) can be
solved without any problem. Since the set of independent variables in the vector
X (tgfl) is not unique, the transformation matrix Tag is not unique. This will be
illustrated with an example later.

The transition from state 0 to state a occurs over a brief interval of time when

the constraint is removed. If ltEa’tgal denotes this transition interval, the effect of

43

the transition can be described by the relations

Y(tga) = Y(tga) (4.10)
Y(tga) = Y(tga) (4.11)
and
xugg:=tmarmg) Min
X(tga) = TsaYUEa) (4.13)

where Tfia is a constant and unique N x (N — 1) transformation matrix. The transi-
tions from state a to state 6 and from state B to state a are summarized in Fig.4.4.

To obtain a relationship between the mass and stiffness matrices in Eqs.(4.4) and

 

Transtion from state a to state B : Application of constraint
X(e,)=xa;,,) change of “newsman
M X(tZIB) +1643 = M X(thp) variables Y(thfi) = TaB X(tfiB)

 

 

 

 

 

 

 

 

 

 

, State 6 ,
gg §‘ """" i M Y + K Y = 0 """"" ’ +
l he]; tffi Y(t) in RN-l tifga tBa time
| State or i I i [ State a
----- Mx+Kx=0~>§ §~~Mit+Kx=0~~
X(t) in RN ; ‘ X(t) in RN

 

 

 

 

 

 

 

Transtion fi'om state [3 to state a fRemoval of constraint
new raise.) chapgeof {(68347}. rat.)
Y(tba) = Y(t’éa) variables X(tEa) = T56: Y(tEa)

 

 

 

Figure 4.4. System description before and after application and removal of constraint

(4.5), we note that the kinetic energy before and after removal of the constraint is

the same. Using Eqs.(4.11) and (4.13) we can show that

44

1
2
1
2

. _ .. . _ 1 . .

YT(tfia)MY(tfia) = .Z—XTagawxa-ga) (4.14)
. .. . 1 . .
YT(tga)M1/(tga) = §YT(tga)Tg'aMTflai/(tga)

Since the above equation holds for any Y(tga), it follows that

M = TgaMTfia (4.15)

The strain energy immediately before and after removal of the constraint is also the

same. Using Eqs.(4.10) and (4.12) we can therefore show that

1 _ ~ _ 1
5YT(3&1)K1/(6fl0,) = EXT(1:;;C,)KX(1§C,) (416)
1 .. 1

=> §YT(tEa)KY(tEa) = §YT(tEa)TgaKTBaY(tia)

Once again, since the above equation holds for any Y(tga), we can claim that
I? = Tg’aKTfla (4.17)

Equations (4.15) and (4.17) provide expressions for M and If in terms of M and
K, respectively, but the reverse transformations are not possible. The kinetic energy
before and after application of the constraint is not the same and it is not possible
to start from an expression similar to Eq.(4.14). The strain energy before and after
application of the constraint is the same and it is possible to start with an expression

similar to Eq.(4.16) and use Eq.(4.8) to obtain

1 T _ _ 1 T ~
I

1 ..
=> EXT(t:fl)KX(t:fl) §XT(t;fl)T§BKTagX (7:31.)

45

The above equation is not valid for all values of X (tgfl) since the constraint can be
applied only when the configuration of the system satisfies the constraint instanta-

neously. Therefore, we cannot claim K = TgflKTafi:

4.4.2 Energetics of constraint application and removal

The change in kinetic energy over one cycle of constraint application and removal is

given by the relation

1. . 1. _ . _
AB = ExTagfi) angfi) — §XT(tafi) anafl) (4.18)

To simplify Eq.(4.18), we premultiply Eq.(4.7) by X T(t:fi) and X T(t;fl) to obtain

XT(t:fl)MX(t;fl) = XT(t:fi)MX(t;B)+XT(t:fi)Ia_,fl (4.19)

XT(t;fi)MX(t:fl) = XT(t;B)MX(t;fi)+XT(t;B)Ia_,3 (4.20)

Assuming a workless constraint, we can claim that
XT(t;fi)Ia_,fl = 0 (4.21)

Using the symmetric property of the mass matrix, we can then show from Eqs.(4.19)
and (4.20) that

= XT(t;fl)MX(t:fi)

= XT(t;fi)MX(t;fi) + XT(t;fl)Ia_,fi (4.22)

46

Substitution of Eqs.(4.21) and (4.22) into Eq.(4.18) gives
1 .T _
AE = EX (tafl)1a_.fi (4.23)
To simplify further, we rewrite Eq.(4.7) as
X(tgfl) = X(tgfi) — M—IIaxfi (4.24)
The expression for X (tgfi) in Eq.(4.24) is substituted in Eq.(4.23) to obtain

1 'T T —T
2 (X (tgflflmw — IagfiM rang)

1 T -1
: _§Ia—»6M [CI—+3

AE

(4.25)

In simplifying Eq.(4.25) we used the relation X T(t:fi)Ia_,fl = 0 and the symmetry

of the mass matrix. The mass matrix is positive definite and thus AE S 0.

4.4.3 Modal coordinate description

Let <15, and [2,-(t), i = 1,2, - - - N, denote the linearly independent mode shapes and
the corresponding modal amplitudes in state a. Similarly, let 1,0,; and ui(t), i =
1, 2, - - - (N — 1), denote the linearly independent mode shapes and the corresponding
modal amplitudes in state 6. The mode shapes in state a, 65,-, are N-dimensional
whereas the mode shapes in state {3, 1,0,, are (N — 1)-dimensional. For convenience,
we embed the (N — 1)-dimensional mode shapes of state 6, 1b,, into N dimensions

using the transformation matrix in Eq.(4.12), namely,

an

The generalized coordinates, before and after application of the constraint, can now

be expressed as follows

47

,1 2,121 #4048 = (1)1105) t5 th/B
X(t) = l (4.27)
1

21:1 11401134 = ‘i’ 1’05) t2 15%
Where (p = [$11 ¢27'” 7¢Nl E RNXN and {I} : [IAIN/32," ' 71/;(N—1)] E RNX(N-1)
are modal matrices in states a and [3, respectively, and u = [#1, p2, - -- , ply/[T and
T
1/ = [V1, 1x2, - - - ,1/( N—1)] . The transition from state a to state 6, described earlier

by Eqs.(4.6) and (4.7), can now be written as

2.1115426) 1h = 2.1:, 742-053;.) 41>.- (4.28)
442.112.1666 = M21. 61636.”... (4.29)

From the orthogonality property of the modes we have
BTW = IUH)

where 1(N—1) is the (N —1)-dimensional identity matrix. Substituting Eqs.(4.15) and

(4.26) in the above equation we can show that
VITA/11; = I(1y_1) (4.30)

Using Eq.(4.30) and the identity $3,103 = 0, which follows from Eq.(4.21), we can
obtain the modal displacements and velocities of the constrained system in terms of

those of the unconstrained system from Eqs.(4.28) and (4.29), as follows

402;.) = 2:1. 225;“ M4.- 4.0.1,.) (431)

Drugs) = 23:1 15?!” (152' 71.0.1.) (4.32)

48

In matrix form, Eqs.(4.31) and (4.32) can be written as

”(ti-g) = F #(tgg)

19(tag) 2 F [1(tafi) (4.33)

where

r a \iJTMcr (4.34)

Repeating the above procedure for the transition from state 6 to state a, we can

similarly show that

Wis) = I‘T u(tga) 4 3
. . _ PT. t_ (.5)
”(tfia) — V( Ba)

The change in kinetic energy over one cycle of constraint application and removal can

now be expressed in terms of the modal coordinates. Starting from Eq.(4.23) and
using Eqs.(4.24), (4.27), and (4.33), we obtain

AE = %XT(t;g)M [X(t;3)—X(t;5)]
%pT(t;fl)<I>TM [wag-fl) — <I>B(t;fi)l

= %pT(t;fl)<I>TM [tr — <1>] u(tgfl) (4'36)
= —%4T(t;g)AB(t;fl)
where
A 2 (IN — FTP) (4.37)

and IN is the N -dimensional identity matrix. In the next section we show that A
is positive semi-definite and this will corroborate the observation made earlier from
Eq.(4.25), namely, that AB 3 0.

We conclude this section by deriving expressions in modal coordinates that are

equivalent to Eqs.(4.15) and (4.17). Since the kinetic energy before and after removal

49

of the constraint is the same, we can use Eq.(4.35) to show that

1 . _ . _ 1 . _ 1 . _ . _
EVTUfiall/(tfia) : El‘TltEalt‘Ull-a) : '2'VT(tfia)FFTV(tfla)

Since the above equation holds good for all {/(t‘fl’a), we claim that
H" = 1(N_1) (4.38)

Let 02,-, i = 1,2, - -- ,N, and 1221-, j = 1,2, - -- ,(N — 1), denote the natural frequencies
of the unconstrained and constrained systems, respectively. The stiffness matrices in
modal coordinates can then be defined as follows:

9 = diag [1.0%,013, - -- ,w12V]

(4.39)

S) : diag [6363,” ,agN_1)]
Since the strain energy before and after removal of the constraint is the same, we can
use Eq.(4.35) to show that

I _ ~ _ l _ _
“VT(tfia)Ql/(tfia) : §#T(tga)9“(tga) = VTUAOWQPTVUIM)

Since the above equation holds for all u(tga), we claim that

(2 = 1“an (4.40)

4.4.4 Numerical example

We consider a system with three dof in state a: and two dot in state 6, i.e. N = 3.
The system, shown in Fig.4.5 consists of four bars connected by rotational springs
except for the two middle bars, which are connected by a frictionless pin joint. In

state a, the pin joint is free and the generalized coordinates are chosen to be the

50

displacements of the ends of the bars, namely, x1, x2, x3. The pin joint is locked only

when the middle two bars are aligned, and locking the joint applies the constraint
f(X) = 3x1 — 4x2 + x3 = 0 (4.41)

and transfers the system to state B. In state 6, the generalized coordinates are the
displacements y1, y2. The relationship between the generalized coordinates of states

a and 6 can be derived from Fig.4.5 as follows:

y1 = 1231 + /\1(3£L‘1 — 4:172 + 1'3) _ 1+ 3A1 —4/\1 A1

=> T —
y2 = x3 + A2(3:1:1 — 4:132 + $3) OB 3A2 —4)\2 1 + A2
$1 = 311 1 0
1132 = 0.75y1 + 0253/2 => T30 = 0.75 0.25
333 = 92 0 1

where A1 and A2 are arbitrary non-zero constants. Clearly, T fin: is unique whereas

T 05 is not. For small displacements, the system behaves linearly and Eqs.(4.4) and

Q rotational spring © free pin joint -® locked pin joint

 

 

 

 

 

 

 

 

 

 

 

 

 

 

State a a ?
x1 X2. x3;
L U2 3112 L
‘4 ------------------------ >1 7
state B g '
Y1 Y Y2:
1. U2 31/2 _ L

 

 

 

 

 

 

 

 

 

 

Figure 4.5. A linear system with three dof in state a and two dof in state 3

(4.5) describe its dynamics in states a and B, respectively. The mass and stiffness

matrices in Eqs.(4.4) and (4.5) can be derived from the system Lagrangian [39] as

51

61 0 90 —54 0
AL k
M=p1—2183 K=9—L2——54 40—10
_0310 0—10 34
~ pAL'31 ~ It 7 —3
: —— K = —
M 3 _ 3 2L2[—3 7]

 

where p is the material density and A is the cross-sectional area of the bars, I: is the
stiffness of the rotational springs, and L is a dimension that is shown in Fig.4.5.

The transformation from state a to state 6 requires the application of an action-
reaction pair of impulsive moments3, r(t), on the middle bars, as shown in Fig.4.6. If

the impulse of these moments is denoted by

t+
C=/_a‘3¢ t dt
tap?

the impulse vector corresponding to the generalized coordinates X = [x1, x2, 5133] can

be shown to be (see Appendix-A)

[any = 55- [ 2 — 8/3 2/3 ]T (4.42)

Starting from Eq.(4.25), the change in kinetic energy over one cycle of constraint

 

13(t) t(t)
1 / W© \ 1
(I
pin joint/

Figure 4.6. Action-reaction pair of impulsive moments

 

3Such impulsive moments can be generated in experiments by an electromagnetic brake, as in
[40].

52

application and removal can now be shown to be (see Appendix-B)

AE = —11T_,flM—110_,B

z ——pAL [361(tgfi) — 4mg) + i3(t;fi)]2

13

= —'1—5%PAL {le(t;fi)l}2 S 0

where the function f () is defined by Eq.(4.41).
For the purpose of simulation, we assumed the bars to be of circular cross section
and made of aluminum. The material and geometric properties of the bars and the

stiffness of the rotational springs were assumed to be
p = 2710 Kg/m3, A = 7r(0.02)2 m2, L = 0.20 m, k = 20.0 Nm (4.43)

The system was assumed to be in state or at the initial time with the following initial

conditions

(:61, 62, x3, 6:1, 3:2, 63) = (0006,0003, —0.003, 0.00, 000,000)

in SI units. The system was switched from state a to state ,8 at the earliest opportu-
nity after 0.2 seconds when the two middle bars are aligned (f (X) = 0) and switched
back to state a after 0.2 seconds in state 6. The process was continued till the energy
of the system became negligible. Figure 4.7 shows the displacements of the bars and
the energy of the system as a function of time. The energy undergoes a step change
when the system changes state from a to 6 due to application of the constraint. It

remains constant at all other times.

53

 

10 T T f

x] (mm)

 

 

 

 

 

 

 

 

 

 

-10 1 1 1 1
‘0 r r I . fii
E 0
N
K
-IO I l M L
10 r , f
E
g 0 . “Wy‘nvx‘v‘fix “-1“
>2"
-10 1 1 1
0.% I T
5
E
Q)
C
m
0 1 1 l
0 4 12 16 20

time (sec)

Figure 4.7. Plot of displacements of the bars and total energy of the system

4.5 Controllability Issues for Linear Systems

4.5.1 Limitation of energy dissipation

we investigate the properties of the matrix A, defined in Eq.(4.37), to understand
the limitations of energy dissipation through application of constraints. Consider the

matrix FTI‘, where l" is defined by Eq.(4.34). Using Eq.(4.38) it can be shown that
2
(I‘TI‘) = rTrrTr = rTr (4.44)
This implies FTP is idempotent and hence A = (I N — FTP) is idempotent. The trace
of A can be computed as

trace[A] = trace[IN] — trace[I‘TF]
= N —trace[I‘I‘T]
= N — trace[I(N_1)]
= 1 (4.45)

54

Since the eigenvalues of idempotent matrices are all zero or unity, the trace of an
idempotent matrix is equal to its rank. From Eq.(4.45) we deduce that A has one
unit eigenvalue and other eigenvalues are all zero. If u is the normalized eigenvector

of A corresponding to its unit eigenvalue, then eigen decomposition of A gives
A = ’U’UT (4.46)
Substitution of Eq.(4.46) into Eq. (4.36) gives
1 -T __ T~ _ 1 T- .. 2
AB: ‘2“ (tafi)uv Was) = —§ ['0 ”(hall 5 0 (4.47)
Clearly, the energy of the system will not be dissipated upon application of the

constraint if any of the following conditions hold:

1. Mtgfl) = 0: the constraint is applied when the system has zero kinetic energy.

2. uT [lac—w) = 0, v,- 7é 0, i = 1,2,--- ,N: the constraint is applied when the
modal velocity vector is normal to the eigenvector of A corresponding to the

unity eigenvalue.

3. u,- = 0, i 6 Sr = {k1,k2, - -- ,kr}, uj(t;fl) = 0, Vj ¢ Sr: the kinetic energy of
the system lies in specific modes that correspond to zero entries of v.

The first and second conditions can be avoided through a proper choice of the time

when the constraint is applied. It may not be possible to avoid the third condition,

which depends on modal characteristics of the unconstrained and constrained systems

and the energy distribution of the unconstrained system. This corresponds to a

necessary condition for energy entrapment.

4.5.2 Energy entrapment

In this section we show that energy can get trapped in specific modes of the system.

The third condition in Section 4.5.1 is a necessary condition for energy entrapment

55

but not a sufficient condition. Before we present the sufficient condition, we derive
some properties of the matrix I‘, defined by Eq.(4.34). We assume that F has the

form
I“=[101 P2 PN]: I‘T=[<I1'612 (KN—1)] (4-48)

where the p’s and q’s are column vectors of dimension (N — 1) and N, respectively.

Using the identity FFT = I ( N—l) from Eq.(4.38), we can show that

1 °=i ..
qg‘rqj'={ J. . i,j=1,2,---,(N-—1) (4.49)

Since 12 is the eigenvector of A corresponding to its unit eigenvalue, 1) is also the

eigenvector of FTI‘ corresponding to its zero eigenvalue. This implies that

rTr6=0 => UTFTFU=0 => r6=0 => viq, j=1,2,---,(N—1)

Since FTP has (N — 1) repeated eigenvalues, there will be no unique set of orthonor-
mal eigenvectors. The qj’s, j = 1, 2, - -- ,(N — 1) are in the space spanned by the

eigenvectors of FTP with eigenvalue of unity. This follows directly from the relation

(FTP) PT = I‘T

which follows from the identity FFT = I ( N-l) in Eq.(4.38). Now consider the or-

thonormal matrix

pf '01
Q=[CI1 <12 (1(N—1)‘U]= 5 °
pf vN

Since [QQTL j = pgpj + 22%, we can readily establish

56

pfpj =10 — 6,2- g 1.0, i,j=1,2,-~-,N (4.50)

The sufficient condition for energy entrapment is stated next with the help of the
following theorem.

Theorem 4.1 The energy of a linear system will remain trapped in r specific modes

if F contains an orthonormal sub-matrix of dimension r. [:1

Proof: We renumber the modes of both the unconstrained and constrained systems

such that F has the form

P 2 [ A11 A12 ]
A21 A22

where A22 is the orthonormal sub-matrix of dimension r. From Eqs.(4.49) and (4.50)
we know that the p’s and q’s have norm less than or equal to unity. This implies that

A12 = A21 = 0 and hence

A11 0
F: 4.1
l 0 A22l (5)

We partition the modal coordinates of state a and state 6 as f0110ws:

#=[#1#2]. V=[V1 V2], Ital/261V (45?)

Equations (4.33) and (4.35) can now be rewritten as

V2(t:B) — 4422112033) (4 53)
929:3) = A22 #2053) l
1429;...) = 2412112 V2050.) (4 54)
'(t+)=AT1'/(t_) '
l‘2 66 22 2 66

If S222 and $222 denote the lower right r x r sub-matrices of f2 and (2, respectively,

then substitution of Eq.(4.51) into Eq.(4.40) gives

57

(122 = A229224123 (455)
Using the orthonormal property of A22 it can be easily shown that
922 = A22111221422 (4-55)

In state a, the total energy associated with modes p2 prior to application of the

constraint is

__ 1 . _ . __ 1 _. _
E2(tag) = §uf(tag)u2(tafl) + §#§(tap)922fl2(tagl

Using Eqs.(4.53) and (4.55), this can be shown to be equal to

_ 1 . . 1
E2005) = §Vi(t:5)AzzAsz2(t:gl + §vf(t;'3)A22922A321/2(t:g)
1 _ _ 1 ~
: EngtzglV2ftzg) + §Vg(t:g)9221/2(t:gl
= E2(t:fl) ' (4.57)

In state 6, the energy associated with modes 112 prior to removal of the constraint is
_ 1 -T _ . _ 1 T _ ~ _
E20530) = 51/2 (tgalV2ftga) ‘I’ 51/2 (130)9221/2ftga)
Using Eqs.(4.54) and (4.56), this can be shown to be equal to
_ 1 . . 1 ~
E2(t,3(1) = §#§(tga)Af2422/t2 (132,) + §Mf(150)A22922A22#2(t§a)

1 . . 1
-: §p£(tga)p2(tga) + ~2-u5(t§a)922#2(tgal

= E2(tga) (4.58)

58

Equations (4.57) and (4.58) imply that energy is trapped between modes pg in state

a and modes V2 in state ,6, and this completes the proof.
The results in Theorem 1 can be further extended as follows:

1. From Eq.(4.55) it can be shown that the eigenvalues of (222 are the same as those
of 022. Since (222 and (222 are both diagonal matrices, they must be identical.
Equation (4.55) now implies 5222 = 422022452 => 922A22 = Aggflgz.
Since (222 and A22 commute and $222 is diagonal with distinct entries, A22
must be diagonal [41]. Since A22 is orthonormal, it simply follows that A22 = IT,

where [T is the identity matrix of dimension r. In summary, we have

5122 = 5122, A22 = Ir (4-59)

This implies that r modes in state a will be identical to r modes in state 6.
These modes form an invariant sub-space that is not affected by the constraint

and energy in these modes remain trapped.
2. From Eq.(4.51) we can show that FTP and A will have the form

FTP: Atil/411 0 A: IN—r—AflA11 0
0 Ir ’ 0 0

From the expression of A in Eq. (4.46) it follows that r elements of u will be zero.
Clearly, the third condition in Section 4.5.1 is satisfied when the condition in
Theorem 1 is satisfied.

4.5.3 Numerical example

Consider the system in Fig.4.5 with all bars having the same length L, as shown in

Fig.4.8. Assuming the system parameters to be those given by Eq.(43), the matrices

59

I‘ and A and the eigenvector u are computed as follows:

0.9599 0.0 0.2804
PT: 0.0 1.0 , U: 0.0 ,
—0.2804 0.0 . 0.9599

(4.60)
0.0786 0.0 0.2691

A = 0.0 0.0 0.0
0.2691 0.0 0.9214

Q rotational spring ©~ free pin joint -® locked pin joint

state a g 1
x1 x2 . x31

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.8. A modified version of the system in Fig.4.5 in states a and 6

From the entries of F in Eq.(4.60) it is clear that the condition in Theorem 1 is
satisfied with r = 1. Since ng = 1.0, the energy of the system is trapped in the
second mode of state a, which is also the second mode of state 3. We can also verify
that one element of 1), namely U2, is equal to zero. Energy entrapment can be verified
from the simulation results presented in Fig.4.9, the initial conditions for which were

chosen as

(3:1, 1:2, x3, 6:1, 562, :53) = (0.006, 0.016, 0.002, 0.00, 0.00, 0.00)

in SI units. As in the previous simulation, the system was switched from state a to

state [i at the earliest opportunity after 0.2 seconds when the two middle bars are

60

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time(sec)

Figure 4.9. Plot of modal amplitudes in state a and total energy of the system

aligned, i.e., when

g(X)=1‘1—2$2+:L'3=0

and switched back to state a after 0.2 seconds in state 0. The modal amplitudes
in state (1, namely, #1, [12, 113, are shown in Fig.4.9 for the intervals of time when
the system is in state a. It is clear from these plots that the amplitudes of the first
and third modes decay to zero whereas the amplitude of the second mode remains
constant. The plot of the energy confirms that some energy of the system gets trapped

in the second mode.

4.6 Application to Nonlinear Systems

For the simulations in Sections 4.4 and 4.5, the constraints were applied when the two

middle bars were aligned, i.e., the constraints were applied based on state variable

61

information. The purpose of using state variable information was to make the analysis
tractable by switching the unconstrained system to the same constrained system
every time. Furthermore, the assumption of small displacements and linear system
behavior permitted the analysis in modal coordinates. For nonlinear systems, where
a modal coordinate description is not possible, the analysis does not benefit from
switching based on state variable information. However, the information of the states
can be used to maximize the amount of energy dissipated. We do not address this
optimization problem in our work. Using a numerical example we simply show that
a random switching schedule can be quite effective in dissipating the energy of a
nonlinear system. This will demonstrate an important advantage of our approach
that has not been explicitly stated thus far: that vibration control can be achieved
without state variable estimation and therefore without the use of sensors.

The analysis for nonlinear systems can be carried out in a manner similar to that
of linear systems presented in Section 4.4.1 and is therefore not repeated. It will be
different in two respects: (a) the mass matrix will be a function of the generalized
coordinates and not a constant matrix, and (b) the transformation matrices T03 and
Tfla will vary from one cycle of constraint application and removal to the next. This
simply reflects the fact that the constraint applied and removed will be different for
each cycle since a random switching schedule will be used.

We consider a serial chain of three links connected by revolute joints, as shown
in Fig.4.10. Each joint has a rotational spring and the second joint can be locked
or released for constraint application and removal. The unconstrained system has
three dof and its configuration is described by the generalized coordinates al, a2, 0:3.
The system is constrained by locking the second joint. In this locked configuration,
which is described by the generalized coordinates Bl, ,32, the relative angle between
the second joint and first joint is denoted by ’y. The value of y will be different for

each cycle of constraint application and removal since a random switching schedule

62

will be used. The simulation results are presented in Fig.4.11. For this simulation,

-@— pin joint with a rotational spring

  
  
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Q rotational spring
@. locked pin joint

.......................

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.10. An example nonlinear system

the mass and length of each link were assumed to be 1.0 kg and 0.5 m, respectively.
The stiffness of the rotational springs were assumed to be the same and equal to 200

N m / rod. The initial conditions were assumed to be

(61, 92, 93, 91, 9’2, 6'3) = (15.0, —10.0, 5.0, 0.0, 0.000)

where the units are in deg and deg/ sec. The second joint was locked and released
with the time interval between switchings randomly varying between 0.2 and 0.3 sec
till the total energy of the system was dissipated. The simulation results indicate
that a random switching schedule is quite effective in dissipating the total energy of

the system.

4.7 Remarks

It has been shown that a sequence of application and removal of constraints can be
used as a mechanism to extract energy of vibration from finite dimensional elastic

systems. The strategy is straight forward, as it does not rely on state—dependent

63

 

 

 

 

 

 

 

 

 

 

 

O

15

time (sec)

Figure 4.11. Plot of generalized coordinates and total energy of the nonlinear system

timing of the constraint application and thus it is easy to implement. This is partic—
ularly relevant in the case of non-linear systems, where examples show that here too
application and removal of constraints leads to removal of kinetic energy. As such,
the strategy emerges as a potential mechanism for vibration control without the use
of sensors.

The results in this chapter highlight a number of interesting features of the be-
havior of linear systems under the repeated application and removal of constraints.
In general, applying and removing constraints results in energy transfer from mode
to mode and, in the case of finite-dof systems, removal from the system altogether
in a manner akin to transferring energy to unmodeled, higher-order modes. In the
process, energy is extracted from all modes except in cases where the modes and the
constraints interact in such a way that energy is entrapped and a part of the system
is indifferent to constraint application. One can view this as a sort of orthogonality

of the system, as represented by a subset of its modes, and the constraints. This sug-

64

gests that if energy removal is the principal consideration, it may always be effected
by applying and removing a different set of constraints after every switch, carefully
crafted to affect all the modes of the system. A number of interesting ftuther investi-
gations are possible. For example, since the rate at which energy transfer takes place
depends on and can be controlled by the manner in which the constraints are applied,
it may be possible to optimize both the rate and the direction of this transfer to fit a
particular purpose, simply by designing the constraint application scheme. Since the
efficiency of a number of engineering systems relies on the manipulation of vibration
energy, the work presented can have interesting applications in the design of such
systems such as devices for vibration isolation, vibration control [1], [2], and energy
harvesting. The goal of the next chapter is to use the method of energy dissipation
proposed here but apply the constraints at specific times and locations to maximize

energy removal.

65

CHAPTER 5

Energy Dissipation Through Optimal

Application And Removal of Constraints

5. 1 Introduction

The energy dissipation approach presented in the last chapter is revisited in this
chapter with the objective of investigating efficient energy removal through an Opti-
mal sequence of constraint application. A special case of the constraint application
procedure proposed in the last chapter is used here: the constraint is applied and
instantaneously removed instead of remaining active for a finite duration of time.
In terms of the nomenclature used in the last chapter, this implies ti]; = tEa and
the system always remains in state oz. The time spent in state [3 is zero. The basic
idea is illustrated with the help of Fig.5.1, which is a modified version of the three
dof1 mass-spring system of Fig.4.1. Unlike in Fig.4.1, where a pin is inserted in the
second mass to enforce the constraint x2 = 0, here we assume that the second mass
has an electromagnetic brake that enables it to stop instantaneously. Immediately
after it stops, the brake is disabled and the mass is set free to move. In this
case, the constraint x2 = 0 is enforced and instantaneously removed. The system

configuration does not change and therefore its dof remain three, namely, x1, x2 and

x3. The application and instantaneous removal of the constraint however results

 

1degrees-of—freedom

66

in an instantaneous reduction in energy of the overall system by an amount equal
to the kinetic energy of mass m2. A repetition of the process of application and
removal of the constraint will ultimately reduce the energy of the system to zero. In
this chapter, we pose and solve several optimization problems to efficiently remove
energy from the system by applying and instantaneously removing constraints at

strategic location and specific instants of time.

 

 

Figure 5.1. A three dof mass-spring system

The above idea of energy dissipation relies on application of an impulsive force on
the system. It is however different from impact dampers, commonly used for vibration
suppression, that also apply impulsive forces. For impact dampers, the motion of the
impact mass is constrained to a specific region of the physical system, the impact
mass moves in an uncontrolled fashion and the frequency of collisions depends on the
dynamics of the overall system. In this work, we do not introduce impact masses but
apply constraints that have the equivalent effect of an impact damper. This gives us
the flexibility to control the timing as well as location of the impacts.

This chapter is organized as follows. In the next section we describe the eflect of
constraint application and removal and the loss of energy associated with this process.
In section 5.3 we consider the examples of a discrete N -dof mass spring system and
a continuous rectangular membrane fixed at all sides. For both examples, three opti-
mization problems are solved using a genetic algorithm approach. A gradient-based
optimization method is pr0posed in section 5.4 for solving Optimization problems of
the continuous rectangular membrane introduced in section 5.3. Concluding remarks

are provided in section 5.5.

67

5.2 Energy Loss due to Constraint Application and Removal

In this section we investigate energy dissipation in linear systems with finite dof
through application and instantaneous removal of constraints. From our earlier dis-
cussion we know that application of a constraint will-result in an impact and transfer
energy into the high-frequency modes of the system, where it will be dissipated nat-
urally (without the use of active control). Our finite-dof assumption simply implies
that the flexible modes are unmodeled; this is justified by the fact that the energy
transferred to these modes decay rapidly and their high-frequency dynamics have
negligible effect on the dynamics of the rigid body dof.

We consider N -dof linear systems of the form

MX+KX = 0 (5.1)

where M and K are the N—dimensional mass and stiffness matrices and X =
[x1,x2, - -- ,xN]T is the vector of independent generalized coordinates. The appli-
cation and instantaneous removal of a constraint will change the momentum of the
system but will not affect the displacements. If [t_, t+] denotes the period of time
over which the constraint is applied and removed, the evolution of the system over

this interval can be mathematically described by the relations

X (1+) = X (t‘) (5.2)

MX(t+) = MX(t—)+I (5.3)

where I is the N -dimensional impulse of the generalized forces. The change in kinetic

energy over one cycle of constraint application and removal is given by the relation

68

AE = éxTaWL) MX(t+) — éXT(t—) MX(t-) (5.4)
To simplify Eq.(5.4), we premultiply Eq.(5.3) by X T(t+) and X T(t—) to obtain

XT(t+)MX(t+) = XT(t+)MX(t-) + XT(t+)1 (5.5)
XT(t-)Mx(t+) = XT(t_)MX(t_) + XT(t-)1 (5.6)

Assuming a workless constraint, we claim
XT(t+)I = 0 (5.7)

Using the symmetric property of the mass matrix, we can then show from Eqs.(5.5)

and (5.6) that
XT(t+)MX(t+) = XT(t+)MX(t-)
= XT(t-)Mx(t+)
= XT(t-)MX(t—) +XT(t-)1 (5.8)
Substitution of Eqs.(5.7) and (5.8) into Eq.(5.4) gives
AE = —XT(t-)1 (5.9)
To simplify further, we rewrite Eq.(5.3) as
X(t‘) = X(t+) — M41 (5.10)

The expression for X (t’) in Eq.(5.10) is substituted in Eq.(5.9) to obtain

69

AE 1 (XT(t+)I — ITM‘TI)

2
= —%ITM‘II (5.11)

In simplifying Eq.(5.11) we used the relation X T(t‘f)I = 0 and the symmetry of the

mass matrix. The mass matrix is positive definite and thus AE S 0.

5.3 Optimization Using Genetic Algorithms

In this section we consider two examples: an N -dof mass spring system and a continu-
ous rectangular membrane system. For each example, several optimization problems

are posed and solved using a genetic algorithm approach.

5.3.1 Discrete System Example: N-dof Mass Spring System

We consider the N -dof mass-spring system shown in Fig.5.2. We assume that we can
apply and instantaneously remove the constraint 2;,(1) = 0, for any i E {1, 2, - - - , N }
at any time t. This physically means that mass m; is stopped and instantaneously
released. The impulse required to enforce these constraints can be easily calculated
from Eq.(5.3) since the mass matrix M is diagonal. Assuming the initial and final
times to be t = 0 and t = T, respectively, and that the constraint can be applied n
times, we pose the following optimization problems with the objective of minimizing

the total energy of the system at the final time:

P1 Given the time sequence {0 3 t1 S t2 5 S tn S T}, find the number

sequence {j1,j2,--- ,jn}, jk 6 {1,2,--- ,N}, k = 1,2,--- ,n, such that

sequential application and instantaneous removal of the constraints 2,-(tk) = 0

70

for i = jk will minimize the total energy of the system at t = T.

P2 Given i 6 {1,2,--- ,N}, find the time sequence {0 S t1 S t2 S S tn S T}
such that sequential application and instantaneous removal of the constraints
x,(tk) = 0, k = 1,2, ~ -- ,n, will minimize the total energy of the system at

t=T.

P3 Find the time sequence {0 S t1 S t2 S - - - S tn S T} and the number sequence
{j11j21"‘ ,jn}, jk 6 {1,2, - -- ,N}, k = 1,2, - -- ,n, such that sequential
application and instantaneous removal of the constraints x;(tk) = 0 for i = jk

will minimize the total energy of the system at t = T.

X X2 XN
k. rt k. 4" 2—7 k...
l-MNVWV— m. 4wvvwv1— m2 Hv‘ ------------ v— mN W

Figure 5.2. An N -dof mass spring system

 

 

 

 

 

 

 

 

 

 

The above optimization problems were solved using genetic algorithms. For each of
the problems, we assumed N = 5, n = 6, and T = 103. In SI units, the five masses

and six spring constants were chosen randomly as follows:

m1 = 0.095, m2 = 0.051, m3 = 0.132, m4 = 0.094, m5 = 0.112

k1 = 14.5, kg = 7.31, kg = 11.06, k4 = 9.85, 165 = 13.91, kg = 12.62

The initial conditions used for simulations were chosen as:

71

 

 

 

 

 

151,31 t2,j2 t3..’i3 t4.j4 155,35 136,36
P1 1.00 , 3 2.00 , 2 3.00 , I 4.00 , 3 5.00 , 1 6.00 , 3
P2 1.65 , 3 1.82 , 3 2.21 , 3 4.72 , .3 4.86, 3 5.11 , 3
P3 0.26, 3 1.00, 1 2.00, 3 2.71 , 1 4.71 , 4 4.85, 4

 

 

 

 

 

 

 

 

 

 

Table 5.1. Time and number sequence for problems P1, P2 and P3

$1 = 3.95, 232 = 3.69, 173 = 2.03, 11:4 = 4.58, 225 = 4.47

6:1 = 0.92, 6:2 = 0.17, 2:3 = 0.93, 64 = 0.41, 6:5 = 0.06

where the units are mm and mm/s. The results of problems P1, P2 and P3 are
tabulated in Tab.5.1. The plots of energr decays are shown in Fig.5.3. This figure
indicate that energy of the system is reduced by 92.5% for problem P1, by 99.2% for
problem P2 and by 99.3% for P3. For problem P1, a time sequence was given and
the number sequence was obtained through optimization. For problem P2, a number
sequence was given and the time sequence was obtained through optimization. For
problem P3, both sequences were obtained through Optimization and therefore it is

not surprising that its solution has the maximum reduction in energy of the system.

5.3.2 Continuous System Example: Membrane

We consider a uniform rectangular membrane fixed on all sides, as shown in Fig.5.4.
The natural frequencies and mode shapes for this continuous system can be written

in closed form and they take the form [42]

72

x 10'3

 

 

Pl
Energy (J)

 

 

 

 

x 10'3

 

 

P2
Energy (J)

 

 

 

O
’J—
1.

 

x 10'3

b.)
.J

 

P3
Energy (J)

 

 

 

i

l l

0 2 4 6 8
time (sec)

 

Figure 5.3. Energy decay for problems P1, P2 and P3

 

(5.12)

 

 

W,- (:c, y) = sin

where 2', m and n are integers, a and b are dimensions of the membrane as shown
in Fig.5.4, p is the mass density, and 0 is the tension per unit area. In the above
equations each value of 2' represent a mode number and is associated with a specific
combination of m and n values.

We assume that a constraint of the form z'(:rp, yp, t) = O can be applied at any time
t, to any point (mp,yp) on the membrane, 0 < 23,, < a, 0 < yp < b. The application
and instantaneous removal of the constraint is achieved by the application of an

impulsive force F16(xp,yp) at point (mp,yp). Assuming an N-mode model of the

73

 

Figure 5.4. Uniform rectangular membrane fixed on all sides

membrane, the magnitude of this force can be computed using Eq.(5.3) as follows:
The impulse vector I is calculated by projecting the impulsive force F; 6(mp, yp) on
the mode shapes of the system. The 2th entry of the impulse vector can therefore be

computed as

ll

1(2) Ab A“ Wi(-"’: 1!) Fr 5(zp. yp) dz dy

FI W-i($p, y?)

Hence, the impulse vector takes the form

I = F1 [W1(mp,yp).- - - ,WN(zp.yp)]T (5.13)

Let 1),-(t) and 1'7, (t) be the modal displacement and velocity of mode 7'. Since the mass
matrix is the identity matrix in modal coordinates, the change in modal velocities

due to this impulsive force can be calculated using Eq.(5.3) as follows

74

7720+) = 7720—) + FIWi($pa 91)) (5-14)

The magnitude of F I required to enforce this constraint 2(srp, yp, t) = 0 can now be

computed as ’
FI : _Ez-Iil Wi($payp)7li(t—)

 

(5.15)
221:1 W2” (55p, yp)2
which follows directly from
N
201310.311», 0 = Z Wi($pa yp)7'h'(t+) = 0 (5-16)
i=1

Assuming the initial and final times to be t = O and t = T, respectively, and that the
constraint can be applied 71 times, we pose the following optimization problems with

the objective of minimizing the total energy of the system at the final time:

P4 Given the time sequence {0 _<_ t1 5 t2 < S tn 5 T}, find the location

sequence {($11y1)1"' ,(anyn)}, 17min < wk < Climax: ymz'n < 311: < ymax,

k = 1, 2, - -- ,n, such that sequential application and instantaneous removal of
the constraints z'(a:,-, y,, t,) = 0 for 2' = 1, 2, - - - , n will minimize the total energy

of the system at t = T.

P5 Given the location (23p,yp) 6 {1,2,--- ,N}, find the time sequence
{0 3 t1 3 t2 3 S tn 3 T} such that sequential application and in-
stantaneous removal of the constraints 2(xp,yp,t,-) = 0, 2' = 1,2, - -- ,n, will

minimize the total energy of the system at t = T.

P6 Find the time sequence {0 3 t1 3 t2 3 g tn 5 T} and the location

sequence {($1,y1).--- Jews/11)}, 36mm < 517k < xmax, ymin < 111: < ymax,

75

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

t1 t2 t3 t4 t5 t5 t7 t3 t9
P4 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00
P5 0.73 1.06 2.30 2.86 3.84 5.18 5.83 6.58 9.56
P6 0.77 1.84 2.91 4.85 4.87 5.40 6.80 8.85 9.06
Table 5.2. Time sequence for problems P4, P5 and P6
k = 1, 2, - - - ,n, such that sequential application and instantaneous removal of
the constraints z'(:c,-, 31,-, t,) = 0 for z' = 1, 2, - - - ,n will minimize the total energy

of the system at t = T.

For our numerical simulations, we assumed n = 9, T = 10 s and

xmaz = 0.85 m
ymax = 1.05 m _

37min = 0.15 m

For the membrane, we chose the following geometric and material properties: a = 1.0
m, b = 1.2 m, p = 800 kg/m3 and a = 600 N/mz. A 10-mode model of the
membranes was considered, i.e. N = 10. The initial conditions were chosen to be the
same for all modes and equal to 17,- (to) = 0.001 and 1),-(t0) = 0.001.

The results of simulations for problems P4, P5 and P6 are shown in Tab.5.2,
and Tab.5.3. The corresponding plots of energy decay are shown in Fig.5.5. They
demonstrate an energy reduction of 86.3% for problem P4, 69.3% for problem P5 and
94.0% for problem P6. For P5, we choose 22,, = 0.3 m and 3),, = 0.7 m. For problems
P4 and P6 we imposed the additional restriction that the points of application of
constraints satisfy 33min < (ck < :rmax, ymin < yk < ymax, k = 1, 2, - - - ,n. This was

done to avoid a large impulsive force needed to enforce the constraint at points that

76

 

 

 

 

P6

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.3. Location sequence for problems P4 and P6

are close to the boundary. The location sequence for P4 and P6 are shown in F ig.5.6.

 

 

 

 

 

“10‘1
.763 ‘—n
as iT—hfi—m—
:3
LL]
4x104

 

P5
Energy (J)

 

 

 

 

 

 

 

 

 

 

 

O 1 1 1 1
4x10“4
5 ———.__L
@671
543
S:
m l
0 1 1 1 1 ‘fi—
0 2 4 6 8 10

time (sec)

Figure 5.5. Energy decay for problems P4, P5 and P6

5.4 Gradient Based Optimization Method

The membrane example in section 5.3.2 is revisited in this section and solved using. a

gradient-based optimization method. Specifically, the method of moving asymptotes

77

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

m? . ts}.

t5 9' 131:.

5 .t7 1’ t1 t8tq .t/z
b .t6 t; 't3

5 .t ‘t9 '1,

i ., 4 “*9

P4 P6

Figure 5.6. Location sequence for problems P4 and P6

(MMA) [43] is used. This method can be used to solve optimization problems for
continuous systems and thus the mass-spring system of section 5.3.1 is not revisited.
We present the derivation necessary and solve optimization problem P4 in the next
section. The derivations needed for problems P5 and P6 can be carried out in a

similar manner and are not presented.

5.4.1 Objective function

We restate problem P4 as follows: Given the time sequence {0 3 t1 3 t2 3 S

tn 3 T}, find p = {($1,y1), - -- ,(xn,yn)} E szn that will

n
maximize f = g 2 IEI, (5.17)

z=l

3min < 53]: < 513mm:

(5.18)
ymz’n < yk < ymaa:

subject to

In Eq.(5.17), I,- is the impulse vector resulting from application and instantaneous

removal of the constraint 2(2:,-, y,, t,-) = 0 at time t,. It is calculated from Eqs.(5.l3)

78

and (5.15) and takes the form
= [C$U(t;)] B, (5.19)

where
Bi = [Oa‘”aOaW1($i13/i)1"‘:WN(xiayi)lT
000 = [n1<t>,--1- ,nN<t),m<t>,--- mm?”

C. = [0,"',0,W1($',y'),°°',WN(.T‘,y')]T
z ZfiIWflmaz/DZ z z z z

 

In the above expressions, U (t) denotes the vector of state variables at time t. Starting

from the initial state U (to), U (t,- ) and I, are calculated recursively as follows.

(1051—) = (P1 [1050)
UUT) = (1’1 U(t0) +1
(105;) = (1)2(P1 U(t0) + (1)211
U(t3-) = (I>2<I)1 U(t0) + (1)211 + I2
' 2' i—l: i
U(t,‘) = II <I>, U(t0) + 2 II <I>,,I,-
j=1 j=1k=j+1
i i—l (5.20)
U(t;'") = [I <5, U(t0)+ fi <I>,,I, + I,-
j=1 j=1k=j+1
. n n—l. n
U(t;) = H <I>j U(t0)+ 2 H lej
n 71—]
U(tj{) = 1] <I>, U(t0)+ f1 <I>kI,- +In
where
ch,- = eAAt and A = ON 11" (5.21)
—9 0N

In Eq.(5.21), At = (t, — t,-_1), 2' = 1,2, - -- ,n, denotes the time interval between

successive impacts and Q = diag(w,2-) is the N x N stiffness matrix of the system. 1 N

79

and 0 N are the N x N identity and zero matrices respectively. The expression for (I),

can be obtained in closed form as follows

diag [cos(w,-At)] dz'ag [Sin(ijt)/wjl

(bi = —dz'ag w-sin(w-At) diag 008(W'At)
.7 J 3

(5.22)

where diag [cos(w,-At)], diag [sin(w,-At)/w,~], diag [wj sin(w,-At)] and
diag [cos(w,-At)] are N x N diagonal matrices with [cos(w,-At)], [sin(w_,-At)/w,~],

[0), sin(w,-At)] and [cos(w,-At)] as their j-th entries, respectively, j = 1, - -- , N.

5.4.2 Sensitivity Analysis

The gradient of the objective function in Eq.(5.17) needed for the optimization algo-

rithm is computed as follows

 

 

 

af _ " T317
3p- 1:11,. a—p (5.23)
. 314
Usmg Eq.(5.19), 5; can be computed as follows
BI,- _ . __ T60,- T3U(t,-_) T _ 8B,-
3;. _ B, a U(t,) 3,. +0, 3p + [0, 00:, )] 3;» (5.24)
GB,-

 

can be computed directly because they are

3001;)
5‘10

80-
In the above equation, ——3 and

0p 019
expressed in terms of 2:,- and y,. The evaluation of is more involved and
requires recursive computations, as shown below:

3U(ti_) _ i—l i an
‘50—- ?3 H ”a;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 5.4. Location sequence for problems 81 and S2

The gradient of the objective function in Eq.(5.23) is used in the MMA algorithm to

iteratively converge to an optimal solution starting from an initial guess.

5.4.3 Numerical Simulations

The material and geometric properties of the membrane were assumed to be the same
as those used in section 5.3.2. A 10—mode model of the membrane was considered
(N = 10) and the total time of simulation was set to T = 10 sec. The constraint
was applied and instantaneously removed nine times (n = 9) and the time sequence
was chosen to be the same as that used for simulation of P4Igiven in table 5.2. The
constraints imposed on the solution, given by Eq.(5.18), were chosen as: mm," = 0.15
m, xmax = 0.85 m, ymin = 0.15 m and ymax = 1.05 m. The initial condition was
assumed to be U (to) = 0.001[1,--- ,1]T; this corresponds to initial energy of the
system equal to 31.2 x 10’5 J. We present results of two simulations, S1 and S2,
obtained from two different initial guesses for the optimal solution. Figure 5.7 shows
the energy decay and Fig.5.8 indicates the locations of constraint application on the
membrane. The exact locations are tabulated in Tab.5.4. For simulations 81 and S2
the final energies were 6.59 x 10’5 J and 9.4 x 10"5 J, a reduction of 78.9% and
69.9%, respectively. Clearly, the results of optimization depend on the initial guess

since the solutions obtained are not global optima.

81

10

 

 

 

 

 

 

 

 

 

 

 

 

 

r 1 r lfi 1 6
r 1 r 1 4
I 1 I l 2
4w m a
x x 0
4 0 4 0
5 $55 5 38am

5 Nm

time (sec)

Figure 5.7. Energy decay for simulations SI and S2

I
1+—--——-———-—-—-————————->1

 

 

 

 

 

 

 

 

82

S1

Figure 5.8. Location sequence for simulations SI and S2

82

5.5 Remarks

In the last chapter it was shown that a strategy based on sequential application of
constraints can be used to significantly eliminate vibration energy from a finite dof
system. While it may seem obvious that imposition of a constraint that freezes the
motion of a dof will remove kinetic energy, as a general strategy energy removal by
constraint application reaches its full potential only in the context of an associated
optimization problem. A number of formulations of the optimization problem are
possible and can be adjusted depending on the design goal and the amount of free-
dom available to design the system. The formulations explored in this chapter are
simple, yet they demonstrate performance when different goals are considered. Other
formulations could, for instance, account for the cost of applying the constraint using
more complex cost models. Such modifications could be incorporated easily within
the present framework and the problems presented hint to how to use these ideas to
solve more complex problems. The genetic and the gradient-based algorithms used
are simple to implement and result in reasonable energy extraction, although no ef-
fort was invested in ascertaining whether the solutions obtained are actually global

optima.

83

CHAPTER 6

Vibration Suppression Using Cable

Actuators

6. 1 Introduction

The results presented in this chapter are part of the active control strategy discussed
in the introduction. It is an extension of the work of Nudehi et a1. [26] where cable
actuators were used for vibration suppression in a cantilever beam. In this chapter
we extend the results to the general case of frame structures. The vibration modes
of the beam studied in Nudehi et al. [26] lie in a single plane but this is not the case
for general frame structures. As a result, the cable actuators have two distinguishing
effects: the first is a parametric effect where the stiffness of the structure is affected by
the cable tension, and the second is a direct effect by which external forces are applied
to the structure. In general cabled structures both effects are present but in most
cases one of them is the dominant or sole effect. In section 6.2 we summarize the work
of N udehi et al., and extend it to motivate the present study. In section 6.3 a dynamic
model, which describes the dynamics of a structure formed by frame elements and
cables, is presented. The model is based on finite elements for the frame and a linear
cable element in which sag, coupling, and other such effects, are neglected. In section

6.4 a general control scheme is presented and described. In section 6.5, we explore the

84

 

Figure 6.1. Cantilever beam with cable-supplied end force.

idea of active stiffness variation on a frame structure, placing particular emphasis on
the importance of the cable placement on the structure, and a performance function is
proposed for quantitatively assessing its effectiveness in terms of control. Numerical
simulations for several choices of cable placements are shown for a particular frame
model. In section 6.6 the direct effect of transverse cable force is examined; we
consider the special case in which the structural stiffness change due to cable effects
is small. Numerical simulations and experimental results are presented for a sample

frame. Some conclusions and directions for future work are addressed in Section 6.7.

6.2 Background

The idea of stiffness variation for vibration control of a beam using a cable—supplied
end load was investigated by Nudehi et al. [26]. In this section we briefly review that
work and use it to demonstrate the effects of cable placement. While these results are
quite straightforward in this beam application, they provide the basis for evaluating
more complex structures, as described subsequently. The results of N udehi et al. [26]
were based on a uniform cantilever beam of length L subjected to an end load P
whose line of action is directed from the free end of the beam to its base; see Fig.6.1.
Assuming Euler-Bernoulli beam theory, the beam equation of motion and boundary

conditions can be written as follows,

IIII

Ely +Py”+pAy=0. (6.1)

85

 

T
u = -x Cx
system: beam y 2 1

 

 

 

 

 

 

 

 

 

11 11(2) 1

Is+l

 

 

 

 

 

 

 

 

 

1: 5 7-
memoryless nonlinearity low pass filter

 

 

 

Figure 6.2. Control Scheme

The geometric boundary conditions are

I

y(01t) = 0, y (01 t) = O

and the natural boundary conditions are

II III I
y (L,t)=0, Ely (L,t)+P{y(L,t)—iy(L,t)}=0.

In Eq.(6.1), E and p are Young’s modulus of elasticity and density of the beam,
respectively, I is the second moment of cross-sectional area, and y, and 3;, denote
the partial derivatives of y(:1:, t) with respect to a: and t, respectively. Note that the
end load P appears in a unique manner in the boundary conditions, but it enters
the field equation the same as if it were a compressive buckling or flutter type load.
Using the first N normalized mode shapes of a cantilever beam, the beam model is
approximated by projecting the partial differential equation onto these modes and
expressing the resulting ordinary differential equations in state space form, resulting
in:

il=$2

.2
2:2 = —K:E1 — D1E2 + 03111 (6 )

In Eq.(6.2), 2:1 is the modal amplitude vector, 2:2 is the modal velocity vector,

K is the stiffness matrix, D is the damping matrix, 11 is the cable tension (11 = P),

86

Table 6.1. Material and geometric properties of the experimental beam.

 

 

Material Aluminum

Young’s Modulus 70 GPa

Density 2730 Kg /1123
Dimensions 1.25 x 0.05 x 0.003 m

 

 

and C is the stiffness change corresponding to a unit tension. Initially, it was found
using passivity analysis, that if energy was to be removed from the system, tension
should be applied in the cable only when the output function y = —$§C$1 is positive.
Then, a low pass filter was added to remove the high frequency content of the output
y, thereby providing a signal that is consistent with the actuator bandwidth. A
memoryless nonlinearity was included to account for the unilateral nature of the
control action and to proportionally reduce the control action as the system vibration
dies out. This control scheme is depicted in Fig.6.2.

An experimental setup was built to demonstrate the approach, in which a dc motor
provided the cable tension. The material and geometric properties of the beam are
provided in Tab.6.1. The cable tension was switched between 20—22N in one case and
20—24N in another case. The 20N bias tension was applied to the cable to prevent
slack. The first mode vibration of the beam was successfully suppressed using this
approach, as shown in Fig.6.3.

In order to demonstrate the effects of cable placement, the model is adapted
here to allow for cable placement anywhere along the beam. Of course, in this case
moving the cable closer to the base will reduce the control authority of the cable,
and thus reduce its effectiveness. Our aim is to capture this effect in the equations
of motion, so that it can be investigated for more complex structural systems. In
the first configuration, designated Case I, the cable is connected from the base

to the tip, as shown in Fig.6.1. In Case II the cable is assumed to be connected

87

10 f

 

 

 

 

 

 

 

 

 

T = 20-24 N ~
:2
o o 1..
>
_10 1 1 1 1 L 1 1 L 4
0 4 “me (sec) 16 20
(a)
10 . .
T = 20-22 N .
£3
'3 0
>
_10 1 1 4 1 1 1 1 1
0 4 time (sec) 16 20

(b)

Figure 6.3. Experimental results of the first mode, showing vibration suppression from
the cable forces.

from the base to a point at :1: = 3L/4, and in Case III the cable is attached at
x = L/ 2. A two-mode approximation of the beam dynamics is considered. In
Eq.(6.2), the K and the D matrices will remain unchanged for the three cases but

the C matrix will change. These matrices, including 0 for all three cases, are given by:

K: [97.33 0.0 ] D: [0.098 0.0 J

0.0 3824.39 0.0 0.618
C in Case I C in Case 11 C in Case III
1.0 —5.28 0.16 -2.10 0.01 —0.25
—5.28 44.41 —2.10 28.81 —0.25 5.58

The results of numerical simulations for the three cases are shown in Fig.6.4. The
control scheme in Fig.6.2 was used with T = 35N, 5 = 1.0 x 6‘4, and the cut-off

frequency of the low pass filter was set at 15 rad/s. The following initial conditions

(in SI units) were used,

88

$1<0>= [2133] = [832],

l= [3:8]

It is clear that a shorter cable results in lesser control authority. It is also clear that

(6.3)

__ )
”(0) _ l 512(0)

the C matrix is key to quantifying the level of control authority and thus some norm
of C should be useful for assessing the control effectiveness of cable placement. This
observation motivates the question of finding a method for determining an optimal
location of the cable. In the case of the beam the cable placement is limited to be
along the beam, making the optimization problem simple, but in a three-dimensional
structure the cable can be placed between any two points on the structure, and the

assessment and optimization become more complicated.

6.3 Modeling Of Cabled Frame Structures

In this section we present a mathematical model that describes the dynamic behavior
of a structure made of frame elements and cables. Finite element analysis is used to
derive the equation of motion of the system. Two elements are needed to completely
describe the structural dynamics, a flame element, which is well known, and a cable
element, which is proposed below. The flame element is shown in Fig.6.5; it has
two nodes, N1 and N2, and six dof1 per node, 3 displacements and 3 rotations. It
carries an axial load P, which is taken to be positive in compression. Its geometric
and material properties are defined as follows: A is the cross sectional area, I is
the second moment of cross-sectional area, J is the polar moment of cross-sectional
area, p is the density, E is the Young’s modulus, and G is the shear modulus of

elasticity. We denote u(as, t) and fix (51:, t) as its longitudinal displacement and twist

 

1degrees-of-freedom

89

 

 

 

 

E _.
3 as 0
U
-0.1 1 1 . 1
__‘ 0.17 fl
’3? :6" 0l
U
‘0.1 l g 1 1
0.1

 

Case 111
a,
°__

 

 

Case I

 

 

Case 11

 

 

 

 

Case IH

 

l l 1

0 4 8 1 2 1 6 20
time (sec)

Figure 6.4. Simulation of decay in modal amplitudes a1 and a2 due to the control shown
in Fig.6.2, for cable configuration in Case I (connection at a: = L), Case II (x = 3L/4), and
Case III (x = L/2).

angle, respectively, and v(:1:,t) and w(a:,t) as the transverse element displacements
along the y-axis and z-axis, respectively. In terms of these parameters and variables,

the frame element linear equations of motion are given by:

pAa=AEu” pA5+EIv””+Pv”=0
6.4
-- II .. rm II ( )
pJ193=GJ6$ pAw+EIw +Pw =0

where the first two equations describe the longitudinal and the torsional vibration of

90

Figure 6.5. Frame element

the element, respectively, and the latter two equations describe the transverse vibra-
tion of the element in the x-y and 23-2 planes, respectively. As per standard notation,
f I and f denote the partial derivatives of f with respect to :1: and t, respectively. The
elementary mass matrix, stiffness matrix and geometric stiffness matrix for such an
element are standard; see Reddy [28], for example.

The effects of the element axial forces P, which are induced by cable and other
loads, are only one factor contributing to the change in stiffness of the structure.
Another factor is the spring-back forces created at the cable-structure interface points.
We define a cable element to be any part of the cable that connects any two different
nodes, say N1 and N2, on the structure. Let T be the cable element tension. When
the structure deforms, each cable element acts like a spring and applies a restoring
force at each of the nodes where it is attached to the structure. Assuming small
deformations, these restoring forces are linearly dependent on the displacements of
the interface nodes. A cable element stiffness matrix can be added to the usual
stiffness matrix to account for these effects, as follows. The 3D deflections of the
cable element are projected onto the x-y and 23-2 planes, where (11,312,, 10,-) are the
displacements of the node N,- along the :c-axis, the y—axis and the z-axis, respectively,
as shown in Fig.6.6. The tension in this element is T. The cable restoring forces
F01, Fv2,Fw1, and FW are determined flom the geometry of the figure, and are given

as follows:

91

N2
4—0 -------- 4—0 --------
FVzT /T FWzT /T

V , v

y Cable/ \

Cable Cable Cable

 

 

 

static position ,t‘r‘dynamic position static position dynamic position
x A x A
_l T Z T N _i T Z T N;
y 2 er"?— 2 y “me;
Cable in space Cable projection on the xy plane Cable projection on the xz plane

Figure 6.6. Cable element and its projections on the x-y and x-z planes

Fvl 2 —T(’Ul -- Ug)/lc le C‘.’ —T(w1 — w2)/lc
(6.5)
F1,2 2 —T(v2 — v1)/lc Fw2 2 —T(w2 — w1)/lc.

In Eq.(6.5) all nonlinear effects, such as sag in the cable, are ignored. It is also
assumed that the cable tension T and cable length are constant, that is, independent
of the deformation. Using these assumptions the restoring forces in Eq.(6.5) can be
rewritten in matrix form, flom which one can obtain the cable elementary stiffness

matrix Kc, as follows,

—Fvl 1 -1 0 v1
—Fw, _Z 0 1 0 —1 ml (66)
—Fv2 lC “-1 1 U2 .
1 —1 0
T 0 1 0 —1
=> K = — .
c ,6 _1 0 1 (6 7)
0 —1 0 1

This stiffness is mapped into the global stiffness matrix of the system. After adding

the cable structure interface to the system, and ignoring the inertial effects of the

92

cable, the numerical model is developed using finite elements and will take the form
of as

M)? + (K — TKg)X = TF. (6.8)

In this equation M and K are the mass and stiffness matrices of the system without
the cable, F is the forcing vector corresponding to a unit tension in the cable and X is
the vector of nodal displacements and rotations. K g is the geometric stiffness matrix
which includes both axial loading and cable effects, calculated assuming a unit tension.
The total geometric effect flom the cable on the system is T K g, which results flom
the attendant axial loading in the flame elements (the part of P that arises from T,
calculated in the static state) and the stiffness effect due to cable elements. Both of
these effects are proportional to T. The resulting system is analyzed to determine
the system vibration modes. A coordinate transformation is carried out to express
the equations of motion in terms of modal amplitudes. This model is truncated to
n modes, resulting in the following form for the equations of motion for the retained
modal amplitudes n,

fi+Dfi+(Q—Tkg)n=Tf (6.9)

In Eq.(6.9), kg is the n x n modal geometric stiffness matrix, Q = diag(w,-2) is the
diagonal matrix of zero-tension natural frequencies, D = diag(2£iw,-) is the diagonal
modal equivalent viscous damping matrix in which 5,- are the modal damping ratios,
and f is the modal forcing vector corresponding to a unit tension in the cable. It is
clear from Eqs.(6.8) and (6.9) that the cable has two distinct effects on the system.
The first is a parametric effect in which it alters the system stiffness, and the second is
a direct effect where it provides directly applied forces. Both effects will be considered

in the control design described in the next sections.

93

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r r
11 Frame structure y‘ -x2kgx1-x2f
l h
T (z)[ 1 1
g E V TS'l'l V

 

 

 

 

 

 

 

 

memoryless nonlinearity low pass filter

Figure 6.7. Feedback

6.4 Control Scheme Design

In this section we propose a control design for the system described in Eq.(6.9) by
generalizing the results of N udehi et al. [26]. We first write the state space form of

the closed loop in Fig.6.7 as follows:

1131 = $2

:52 = -Q$1-D$2+(kg$1+f)h(z) (6.10)
. __ 1 T T 1

z — T(32kg$1+x2f) T2

where, 221 6 RN is the vector of modal displacements, 2:2 6 RN is the vector of
modal velocities and z E R is the state of the low-pass filter. The cable tension is the
control input, that is, u = h(z) = T. Similar to the control design in Sec.6.2, passivity
analysis is used to prove that tension in the cable should be applied to maintain the
inequality uTy = uT (—x§kgxl — 2:; f) 2 0, which removes energy flom the system.
The high flequency components of y are attenuated by the use of a low-pass filter
in order to avoid spillover. To reduce chattering, the output of the filter is passed
through a memoryless nonlinearity with a saturation function.

The origin of the closed-loop system in Fig.6.7 is shown to be globally asymptoti-

cally stable using the following Lyapunov function candidate:

94

1 2
V (231,2:2,z) = 2 (x’irflxl + $5232) + T/O h(a)da. (6.11)

The derivative of the lyapunov function is found to be

V = :ifflxl + 232211232 + Th(z)2

= —$§Dx2 -- zh(z) (612)

In the above equation, V S 0 since D is positive definite and h(z) is passive. Also
V = 0 implies 2:2 = 0 and z = 0. Furthermore, flom Eq.(6.10) $2 = 0 and z = 0
implies 2:1 = 0. Since V(:z:1, 2:2, 2) is radially unbounded, we can use LaSalle’s theorem
[44] to claim global asymptotic stability of the origin (2:1, .102, z) = (0, 0,0).

In implementation of the above approach, one needs to measure or estimate vibra-
tion modal amplitudes. It is important to note that when 2 < 0 the cable tension will
be negative. To avoid this situation, a bias tension T0 is applied where To is chosen to
be more than the maximum control effort T and less than the critical load. Although
a bias tension will alter the equilibrium configuration of the system in the general case,
proper cable placement can ensure that the change in the equilibrium configuration
is insignificant. In the next two sections we conduct two separate studies. In the first
study, the parametric effect is considered, i.e., f = 0 in Eq.(6.9), and the effect of

cable placement is explored. In the second study we address the case of direct cable

control assuming the geometric stiffness kg is negligible.

6.5 Control By Stiffness Variation

6.5.1 Problem Definition

The parametric cable control of the closed loop system in Fig.6.7 is studied. It is

assumed that the cable is wrapped around the structure in a way that renders the

95

modal forcing vector to be zero, 2.6., f = 0. This corresponds to a physical situation
in which the cable tension does not affect the static configuration of the system,
at least for the truncated modal description of the system. Such arrangements are
possible, and are not difficult to achieve in structures that are essentially one or
two dimensional. An important consideration that will be investigated is how the
placement of the cable will influence the control. In this section, we will identify the
parameters that determine the control authority and present numerical simulations

for a specific example.

6.5.2 Cable Placement

An important consideration in this class of problems is to develop a systematic method
for assessing and comparing various cable placements, which will be useful for deter-
mining an optimal location that maximizes the control authority of the cable on the
structure. Actuator and sensor locations have been widely studied, especially for
linear systems; see Hamdan and Nayfeh [45] and Sadri et a1. [46], for example. Un-
fortunately, these criteria can not be applied in the present case, due to the nonlinear
nature of the system. Also, unlike other actuators, the cable tendon is unique in the
fact that the overall effect of the force transmitted to the structure can be significantly
increased by rerouting the cable through pulleys at different points on the structure.
A single actuator is needed to apply the tension in the cable which can then be con-
nected to multiple locations on the structure. Here we investigate cable placement
for an undamped system, since we are interested in energy dissipation solely due to
parametric cable control. The equation of motion for this Class of systems is a special
case of Eq.(6.9):

7') + (n — Tkg)ll = 0. (6.13)

The cable placement will affect only the geometric stiffness matrix kg. This stiffness

96

matrix is directly related to the negative work done by the cable tension that is
described by the term, Tkgn. It is clear that some norm of the geometric stiffness
matrix kg can be a measure of the control authority of the cable. The cable placement
problem will be approached by first applying a change of variables that decouples the
stiffness matrix of the system, and the desired criteria will be based on the new form
of the geometric stiffness. Let 05 be the n x n transformation matrix that diagonalize
the two symmetric matrices fl and kg. The statement of this generalized eigenvalue

problem and its consequences for the two matrices are given by:
905 = Akgqb (6.14)

with

¢Tkg¢ = diGQU/Tila ¢TQ¢ = Inxn

where A = diag(T,-) are the eigenvalues, 05 is normalized with respect to the 9 matrix,
i.e., 457’qu = Inxn, and T,’s are the buckling loads for the modes, as confirmed below.
For convenience, the eignvectors that compose 05 are ordered so that [7",] < |T,-+1|.
Performing the change of variables 17 = 050, where a is an (n x 1) vector of the new
set of displacement variables, and multiplying Eq.(6.13) by ¢T, we get the equation

of motion in terms of the new modal coordinates:

Ar} + [Inxn—sz'ag(1/T,-)] 0 = 0 (6.15)

where A = 45le is the new mass matrix. Since we have taken [T 1] < [T2] < [Tn],

the system stiffness matrix is given by:

97

 

 

This form of the stiffness matrix makes it easy to quantify the stiffness change induced
by the cable tension T. To prevent structural instability, the condition 1 — £ > 0
must hold good for all 2', 2 = 1, . . . , 72. Note that the T,’s can be positive or negative,
and stability requires that the tension in the cable to be less then the first buckling
load. The absolute values of T, indicate the extent to which the diagonal entries of
the stiffness matrix are changed flom their original values of unity (when T = 0).
The parametric effect of the cable is described by the diagonal geometric stiff-
ness matrix in Eq.(6.16) and optimal cable placement can be defined as the problem
of maximizing the deviation of this matrix flom the identity stiffness matrix, inde-
pendent of the cable tension T since it is a common factor in Eq. (6.16). This is in
conformity with the observation in section 6.2 where optimal cable placement resulted
in the geometric stiffness matrix with the “largest” norm. To maximize the norm of

kg independent of T, we propose the following performance function:

72.
F, = 21/39 (6.17)
2'=1

In the next section we consider a numerical example and determine the optimal cable

placement based on this performance function.

6.5.3 Numerical Simulations

In this section we present simulation results of vibration suppression based on the
control scheme in Fig.6.7. Consider the flame structure in Fig.6.8 and assume that
the cable can run in the plane of the structure between any three nodes. It is assumed

that the cable is rerouted through a pulley at the middle node and that the actuator

98

is fixed at one of the two end nodes, which are restricted to be different nodes of
the structure. The flame is taken to be made of aluminum pipes, the geometric and
material properties of which are shown in Tab.6.2. The structure is assumed to be
1.2 m in length and 1.0 m in height. The natural. flequencies and mode shapes are
calculated flom the mass and stiffness matrices of the system, obtained using the
FEM method described in Section 6.3. A two-mode model is then obtained through
model reduction. The natural flequencies of the tension-flee system are as follows:
f1 = 2.94 Hz and f2 = 8.13 Hz. In this example, damping is ignored. Since the
cable lies entirely in the plane of the structure, the forcing vector is zero, 2.6., f = 0.

The initial conditions, in SI units, used in this simulation are as follows:
(21 (0) 0.1
0 = = ,
m ) [ 222(0) ] [ 0.01

[2:85]

The performance function is chosen based on the first two buckling loads and is

(6.18)

932(0)

ll
[_fi
.c>.o
OD
l———I

selected to be: Fp = 1/T12 + 1/T22. The tension in the cable is switched between 0
and 50N. The low pass filter cut-off frequency is 15.9 H z and the boundary-layer
parameter is taken to be e = 10—7. The value of PP was calculated for all possible
cable configurations: three sample cable locations are shown in Fig.6.8. The cable
configuration for Case I corresponds to the maximum value of PP and this makes it
the best configuration with respect to the proposed criterion. The remaining cases,
Case II and III are shown for comparison. Numerical simulations of the response,
demonstrating and comparing vibration suppression for all three cases are shown in
Fig.6.9. The results indicate that the value of F1, is a useful measure of effective cable

configuration.

99

Table 6.2. Material and geometric properties of the flame structure

 

 

Material Aluminum
Modulus of rigidity 26.2 GPa
Pipe diameter 9.53 mm
Young’s modulus 71.0 GPa
Pipe Thickness 1.00 mm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Density 2710 leg/m3
Fp=2.06'05 F p=2.8e4"’ ’-. Fp=6.28'08
5 2 f
/ / / '--.
g l- ; g g ....
f ......... 5 2
Case I ”1." Case II ’ Case 111

 

 

 

 

 

 

 

 

 

Figure 6.8. Different cable placements

6.6 Control by Transverse Cable Force

6.6. 1 Problem Statement

In this section we consider direct cable control of the closed loop system in Fig.6.7 by
setting kg = 0 in Eq.(6.9). The tension in the cable is considered to be sufficiently
small, or placed in such a manner that it will not cause a significant change in the
stiffness of the system. Therefore, the effects of Tkg can be ignored. The control
scheme in Fig.6.7 is used by ignoring the kg term and the low pass filter, and by
switching the tension in the cable between 0 and T instead of —T and +T. For this
case, the issue of cable placement is not investigated since this problem is similar to
the actuator placement problem in linear systems, which is well know in the literature;

see, Hamdan and Nayfeh [45] and Sadri et al. [46], for example. Numerical simulations

100

 

 

Case I

 

 

 

 

:ré as" Ol

U
_0.1 1 1 1 1
0.1 ' T

 

Case 111
a
O

 

0.01 ' r

 

Case I
a2
0
1

—0.01 . . 1 .
0.01

«1‘0

'0.01 ’ l l 1 1
0.01
:6“ 0 '

~0.01 ‘ ‘ ‘
0 2 4 6 8 10
time (sec)

 

Case 11

 

Case 111

 

 

Figure 6.9. Simulation of decay in modal amplitude a1 and a2 due to the control in Fig.6.2,
for cable configuration in Case 1, Case II and Case III

are presented in the next section, followed by experimental results.

6.6.2 Numerical Simulations

Numerical simulations of vibration control of the flame structure depicted in F ig.6.10
are shown in this section to prove the feasibility of the control scheme. The structure
in Fig.6.10 is a 2D frame made of elements with circular cross-sectional areas. The
geometric and material properties of the elements are the same as those in Tab.6.2,

with the difference that the flame elements are solid rods and not hollow pipes. The

101

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. . T
....... Side view ,
cable ..........

g ......... 8
/ A
2
/
fl 0 2

y ' m y

] E side view ]
c- <——
z X z x

 

 

 

 

 

Figure 6.10. Front and side view of the flame structure

frame is assumed to be fixed at one end and have a length of 1.2 m and a height of 1.0
m. The cable is assumed to be attached as shown with 6 = 12.7 mm. After calculating
the elementary mass and stiffness matrices, the global mass and stiffness matrices are
evaluated. The mesh is refined until the natural flequencies of the first two significant
modes converge. A two mode model of the structure is used with modal damping
ratios of £1 = 52 = 0.001. The first two natural flequencies of the system are found
to be f1 = 2.19 Hz and f2 = 6.06 Hz. The first mode is a pure beam bending—like
mode and the second is a torsional mode. Both mode shapes are generated using
ABAQUS / VIEWER and shown in Fig.6.11. For the control parameters, we choose

5 = 2.0 x 10‘3 and T = 10N. The initial conditions were chosen in SI units as follows:
_ a1(0) _ 0.01
31(0) ‘ ] 222(0) ] _ [ 0.005 ’
_ 621(0) _ 0.0
352(0) _ l 42(0) l _ [‘10 °

The results of numerical simulations are shown in Fig.6.12. The cable placement is

(6.19)

based on high degree of controllability of the first two modes [45] and [46].

102

 

 

 

l”t mode - bending

 

 

 

 

 

 

2"d mode - rotational

 

Figure 6.11. Shapes of the first two modes

 

 

 

 

 

 

 

 

 

 

 

10‘3
10 x e . . . ,
c6“ 0
-10 1 1 1 1 1
x10'3
5 I I I I l
:6“ 0 w
_5 l l l l l
10
:1
07,, 1 AAAAAAAAAAA
O 2 4 6 8 10 12
time (sec)

Figure 6.12. Plot of the modal amplitudes of the 1“ and 2"" mode, and the control action
(cable tension)

103

kevlar cable ]
r.

A
l

l

]
,,.

l

/ \
/'_.__,__i

 

Figure 6.13. Experimental setup

6.6.3 Experiments

Experiments were carried out with the 2—D flame shown in Fig.6.13 which is identical
to that of the flame used in the numerical model in the last section. The aluminum
flame was built and controlled using a Kevlar cable, where the tension was provided
by a dc motor working in current-control mode. The flame is formed by aluminum
rods welded together to imitate the shape of a space structure. It is designed to
have two flequencies below 10 Hz. The cable position is the same as that used for
simulations in the previous section. An accelerometer (manufactured by PCB, with
0.5 — 3000 H z bandwidth) was placed on the structure (using a small plate welded
to the flame) at a location where the significant (i.e., modeled) modes are strongly
observable. An observer was programmed in MATLAB/SIMULINK to estimate the
amplitudes of the first two modes flom the accelerometer readings. Normalized mode
shapes with respect to the mass matrix were found numerically using finite element

methods. An impulse test was used to estimate the natural frequencies of the system;

104

 

 

 

 

 

 

 

M

 

time (sec)

Figure 6.14. Plot of the modal amplitudes of the 1“ and 2’“ mode, and the control action
(cable tension)

an impulse hammer was used for this purpose. The first two natural flequencies of
the system were found to be f1 = 2.14 H z and f2 = 5.95 H z. The damping ratios
were computed as 51 = 0.003 and £2 = 0.002 using the method of log decrement
on decaying oscillations produced after exciting each of the first two modes. The
sensed signal flom the accelerometer and the feedback signal flom the computer were
interfaced using a dSPACE DSP board with a 20 kHz sampling flequency. The tension
in the cable was switched between 6 N and 14 N; the 6 N bias tension was used to
overcome the problem of potential slack in the cable. We chose 5 = 5 x 10—4. It is
important to note that the natural frequency of the isolated cable under bias tension
is much higher than the first two natural flequencies of the structure. This precludes
the possibility of inadvertently exciting the cable modes. The experimental results
are shown in Fig.6.14. In this figure, the first two modal amplitudes are plotted they

are estimated flom the voltage output of the accelerometer.

6.7 Remarks

In this chapter we illustrate the use of cable actuators for vibration suppression in

flame structures. We model the cabled structure using finite element methods and

105

develop a general control scheme for vibration suppression. The control force is
transmitted to the structure through a cable actuator and cable tension is applied
only when it removes energy flom the system. The cable tension has two distinct
effects on the structure. The first is a parametric effect that alters the stiffness
of the structure and the second is direct effect that stems flom externally applied
forces. Each effect is considered separately. For the parametric effect, optimal cable
placement on the structure is studied and numerical simulations are presented. For
the direct effect, numerical and experimental results are presented to demonstrate

the effectiveness of the control strategy.

106

CHAPTER 7

Conclusions

The aim of this work was to explore new methods for vibration suppression in struc-
tures. Two strategies were proposed, the first one is a semi-active control strategy
and the second is an active control strategy. In the first method, the semi-active
strategy, modal energy is transferred to high-flequency modes of structures where it
is dissipated naturally and quickly due to high levels of damping. For modal energy
dissipation through redistribution, we proposed sequential application and removal
of constraints. It was found that the amount of energy pumped into or out of a mode
is dependent on the timing of the constraint application and removal and on “modal
disparity”, which is a property of a structure and the constraint. The phenomenon of
modal energy redistribution was validated in a clamped-clamped beam with a hinge
in its mid span. The beam has the capability of changing configuration on the fly
by activation and deactivation of an electromagnetic brake built in the hinge. After
experimental verification of modal disparity and modal energy redistribution, we
pr0posed a semi-active control strategy for finite-dof1 linear systems. It was shown
that energy in these systems can be pumped into the high-flequency flexible modes
through sequential application and removal of constraints. This energy is dissipated
naturally due to high levels of damping in these modes. In special cases some energy

can get trapped in particular modes that are not affected by the constraints. For

 

1finite-degrees of freedom

107

linear systems, feedback was used and constraints where applied only when the
system passed through a specific configuration. This was done only to facilitate
investigation of the system behavior in modal coordinates. The use of feedback is
not a requirement of the semi-active control strategy and this is illustrated with the
example of a nonlinear system where a random sequence of application and removal
of constraints is used for energy dissipation. The potential for vibration control
without feedback is an advantage of this approach. The full potential of the approach
can be achieved through optimal application and removal of constraints. To this end,
several Optimization problems were investigated with the goal of maximizing energy
removal for a specified number of cycles of constraint application and removal. In
the second method, the active control strategy, cable actuators where used and cable
tension was varied to remove energy flom the system. It was found that the cable
has two effects on the structure. It applies external forces and alters the stiffness
of structures. The cable placement on the structure was investigated with the goal
of increasing its control authority. Numerical simulations and experimental results

were presented to show the efficacy of vibration control with cable actuators.

Our future work in semiactive control will include applications of the control
strategy to real world problems. An important application to be explored is space
structures, radar array panels, for example. For easy deployment in space, radar
array panels are typically formed by small elements connected by joints that enable
them to fold into compact shapes that can fit into the space shuttle bay. Upon
arrival in space, these radars will be unfurled and deployed. These structures are
prone to vibration problems and one approach to mitigation of such problems is to
embed electromagnetic brakes in some of the joints; the activation and deactivation
of the brakes will result in application and removal of constraints and lead to the

dissipation of vibration energy. Another target application is control of vibration in

108

membrane-like structures, typically used in space based optical systems. The chal-

lenge here is to design non—contact actuators that can apply and release constraints.

Another important consideration for future work is to extend the optimization
problem addressed in this research as part of the semi-active control strategy.
Even though it was assumed that a single actuator is used to apply and remove
constraints, physical constraints associated with the design variables were not taken
into consideration. The actuators cannot apply and release constraints at two
different locations on a structure arbitrarily quickly and have limits on the maximum
impulsive forces that they can generate. In our future work we will address the

optimization problem by taking into consideration such constraints.

An important observation made in the course of our research on cable control of
structures has been the change in damping characteristics of the structure with the
cable in tension. In particular, it has been observed that damping of the structure
with the cable in tension is significantly higher than that without the cable. Although
this is beneficial in terms of vibration control, the phenomenon of increase in damping

remains unexplained and needs to be investigated.

109

APPENDIX A

An alternate set of generalized coordinates for state a in F ig.4.5 are the angular

displacements of the bars, 6,, 2' = 1, 2, 3, 4, described by the relations

91 = $1/L
92 = 2($2-$1)/L
63 = 2(:l:3—:I:2)/3L
64 = -:l:3/L

If the generalized forces corresponding to 2:,- be F,, 2' = 1, 2, 3, and those corresponding

to 6,- be M,, j = 1, 2, 3, 4, then, the principle of virtual work
3 4

4 3 09.
= jZM 3:18—16:32

532'
can be used to establish the relationship between F, and M ,, namely

86,
F,-= )2 M,——— 311:],-

For the numerical example in section 4.4, we have

110

M1 0 1 0 0

M2 __ —T(t) (@)_1 —2 2 0
M3 — 7(t) 8:1: ’L 0 —2/3 2/3
M4 0 0 0 —1
It follows that
F1 1 2
F2 =5 —8/3 r(t) (A.1)
F3 2/3

And flom the definition of the impulsive moment C

t+
C = _“fi 7(t)dt (A.2)
l,
it simply follows
2
C
2/ 3

111

APPENDIX B

From Eq. (4.24), the impulse vector 10H); can be expressed as

M21635) = MX(t;B) + 10,15

Substituting the expression for I O,_, 5 from Eq.(4.42) and the mass matrix flom Section

4.4.4, we get
ML 6 1 0 a123,) ML 6 1 0 4103/3) C 2
—12 1 8 3 5:21:33) —-—12 1 8 3 32033) +-,j -8/3
0 3 10 a3 3,) 0 3 10 2303,) 2/3

Using the constraint 2:2 (15:5) = [3561(t:fi) + i3(t:fi)]/4, the above equation can be
rewritten as a set of three linear equations in terms of the unknowns, 231(t21'fl), 163(tgfl),

and C, as B U = f, where

9pA/16 pA/48 —2/L 910:3)
B: 7pA/12 5pA/12 8/(3L) , U = i3(t;'g)
3pA/16 43pA/48 —2/(3L) c
alum/2 + saga/12
f = pA d31(t;fl)/12 + 2i2(t;B)/3 + i3(t;fi)/4
al<t;,,)/2+a22<t;,,>/12

112

Solving the above equation yields

13

C = —-sfipAL [321(tgfl) — 44:2(tgfi) + 23(t;,)] (3.1)

Substitution of Eq.(B.1) in the expression for AE in Eq.(4.25) leads us to the final

form

13 . _ '. .— - — 2
AE = —fi§6pAL [3.2100(3) - 4$2(tap) + 333(tapl]

113

[ll

[2]

[3]

[4]

[5]

[6]

[7]

[8]

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