A FEEDBACK LINEARIZATION APPROACH FOR PANEL FLUTTER
SUPPRESSION WITH PIEZOELECTRIC ACTUATION
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
_____________________________________
Oluseyi Olasupo Onawola
Certificate of Approval:
______________________________ ______________________________
Subhash C. Sinha, Co-Chair Winfred A. Foster, Jr., Co-Chair
Professor Professor
Mechanical Engineering Aerospace Engineering
______________________________ ______________________________
Robert S. Gross John Y. Hung
Associate Professor Professor
Aerospace Engineering Electrical and Computer Engineering
______________________________ ______________________________
Gilbert L. Crouse, Jr. George T. Flowers
Associate Professor Dean
Aerospace Engineering Graduate School
A FEEDBACK LINEARIZATION APPROACH FOR PANEL FLUTTER
SUPPRESSION WITH PIEZOELECTRIC ACTUATION
Oluseyi Olasupo Onawola
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
December 19, 2008
iii
A FEEDBACK LINEARIZATION APPROACH FOR PANEL FLUTTER
SUPPRESSION WITH PIEZOELECTRIC ACTUATION
Oluseyi Olasupo Onawola
Permission is granted to Auburn University to make copies of this dissertation at its
direction, upon the request of individuals or institutions and at their expense. The author
reserves all publication rights.
_________________________________
Signature of Author
_________________________________
Date of Graduation
iv
DISSERTATION ABSTRACT
A FEEDBACK LINEARIZATION APPROACH FOR PANEL FLUTTER
SUPPRESSION WITH PIEZOELECTRIC ACTUATION
Oluseyi Olasupo Onawola
Doctor of Philosophy, December 19, 2008
(M.S., University of Delaware, 2004)
(M.Eng., University of Ilorin, 1998)
(B.Sc., Obafemi Awolowo, 1990)
122 Typed Pages
Directed by Winfred A. Foster and Subhash C. Sinha
A panel is subject to dynamic instability when induced aerodynamic loads under the
supersonic/hypersonic environment result in a self-excited oscillation called panel flutter.
The panel of an aircraft that flies at supersonic speed or a structural panel that is in fluid
flow at such regime may experience panel flutter. A plate with highly distributed
piezoelectric actuators and sensors connected to processing networks, referred to as
intelligent plate can actively control its vibrations. The objective of this research is to
analytically demonstrate panel flutter suppression using piezoelectric actuation based on
feedback linearization controllers.
A nonlinear control system is formulated using the nonlinear dynamic equations for
a simply supported rectangular panel with piezoelectric layers based on Galerkin?s
method with modal expansions of nonlinear partial differential equation obtained from
v
von K?rman large-deflection plate theory, which accounts for the structure nonlinearity.
The nonlinear equations also account for loads such as externally applied in-plane loads,
aerodynamic loads, and electrical displacements. The aerodynamic loads are given by the
first-order piston theory or the quasi-steady supersonic theory. The control inputs are
given by the electric fields required to drive the actuators based on piezoelectric
actuation, which is modeled by linear piezoelectric constitutive relations. Outputs of the
nonlinear system are feedback and used to transform it into an equivalent controllable
linear system in new coordinates by formulating nonlinear feedback control laws, which
cancel the nonlinear dynamics resulting in a linear system. The pole placement technique
is then employed to make the states of the feedback linearized models locally
asymptotically stable at a given equilibrium.
Numerical simulations are carried out for the closed-loop systems at dynamic
pressures higher than the critical dynamic pressures for the onset of panel flutter, where
limit-cycle motions are generated. The simulated systems show that the closed-loop
systems based on the controllers effectively suppress panel flutter limit-cycle motions
with the generated piezoelectric bending actuations as control inputs. Therefore, with the
feedback linearization controllers developed, the limit-cycle motion of panel flutter can
be completely suppressed or the panel can be made flutter free if the controller gains are
carefully selected.
vi
ACKNOWLEDGMENTS
The author gives thanks to God in whom resides all knowledge, all wisdom, and all
power; for it is He, who has made it so that order masquerades as randomness.
The author expresses his thanks to Dr. W. A. Foster and Dr. S. C. Sinha, my co-
advisors, for creating the opportunity for me to work on this research. He is also
gratefully indebted to them for their support, advice, and guidance. He would also like to
thank Dr. J. Y. Hung, Dr. R. S. Gross, and Dr. G. L. Crouse for serving as members of his
committee. He has found their advice, assistance, and contributions highly invaluable.
He acknowledges the roles played by the chair of the department, Dr. J. E. Cochran,
and the office staff, Ms. Ginger Ware and Ms. Evia Vickerstaff, for creating conducive
environment for this study. He also would like to thank Bob and Frances Stevenson,
James and Cathey Donald, Dr. Henry Y. Fadamiro, Dr. Charlotte Ward, Amit Gabale,
and Samuel Taiwo Adedokun for making the period of this study a memorable one for
him and his family. Cherished is the memory of Virgil Stark who was an encourager.
The author thanks his lovely wife, Alice, for her endurance, patience, and love.
What an amazing experience it has been to be blessed with his two daughters, Joy and
Felicia. Without their constant support this research may not have been possible.
The author thanks the Lord for giving him inspiration through the life of his late
mother, Mrs. Felicia Atinuke Onawola (nee Morohunfola), from whom he learnt the
lessons of perseverance, sacrifices, and inner strength.
vii
Style manual or journal used American Institute of Aeronautics and Astronautics Journal
Computer software used Microsoft Word
viii
TABLE OF CONTENTS
LIST OF TABLES x
LIST OF FIGURES xi
NOMENCLATURE xiii
1. INTRODUCTION 1
1.1 Panel Flutter ??????????????????????????. 4
1.2 Intelligent Structures ??????????????????????? 8
1.3 Panel Flutter Suppression ????????????????????? 13
1.4 Objectives and Scope ??????????????????????.. 15
2. PIEZOELECTRICITY 17
2.1 Characteristics of Materials ???????????????????? 17
2.2 Electrical Enthalpy and Electric Fields ???????????????.. 20
2.3 Linear Piezoelectric Constitutive Relations ?????????????... 21
3. FORMULATIONS 23
3.1 Aerodynamic forces ??????????????????????? 24
3.2 Displacement Field Theory ???????????????????? 24
3.3 Nonlinear Strain-Displacement Relations ??????????????.. 25
3.4 Constitutive Equations ?????????????????????? 27
3.5 Dynamic Version of the Principle of Virtual Work ??????????... 28
3.6 Stress Resultants and Bending Couples ???????????????. 29
3.7 Piezoelectric In-plane Forces and Moments ?????????????.. 31
3.8 Nonlinear Equations of Motion ??????????????????. 34
3.9 Nonlinear Modal Equations ???????????????????... 37
4. FEEDBACK LINEARIZATION 40
4.1 Mathematical Background ????????????????????. 42
4.1.1 Lie derivatives ??????????????????????. 43
4.1.2 Lie brackets ???????????????????????. 44
4.1.3 Frobenius theorem ????????????????????.. 44
4.1.4 Diffeomorphism and state transformations ???????????. 44
4.1.5 Controllability ??????????????????????. 45
4.2 Single-Input Single-Output (SISO) System ?????????????... 45
4.2.1 Relative degrees ?????????????????????.. 45
ix
4.2.2 Exact linearization ????????????????????... 46
4.2.3 Partial linearization ????????????????????. 48
4.3 Multi-Input Multi-Output (MIMO) System ?????????????... 50
4.4 Application to Panel Flutter ???????????????????... 53
4.4.1 Control of the first mode ??????????????????. 53
4.4.2 Control of the second mode ????????????????? 58
4.4.3 Control of both first and second mode ????????????? 62
5. NUMERICAL SIMULATIONS 66
5.1 Limit-Cycle Motions of the Panel flutter ??????????????.. 72
5.2 Suppression of Panel Flutter due to Aerodynamic Loads only ??????. 77
5.3 Suppression of Panel Flutter due to Combined Aerodynamic and Externally
Applied Inplane Loads ?????????????????????.. 87
6. CONCLUSIONS AND RECOMMENDATIONS 96
6.1 Conclusions ?????????????????????????? 96
6.2 Recommendations ???????????????????????.. 98
REFERENCES 100
APPENDIX 107
x
LIST OF TABLES
Table 1.1 Panel flutter theories ?????????????????????.. 4
Table 5.2 The geometrical and material properties of the intelligent panel ????.. 67
xi
LIST OF FIGURES
Fig. 1.1 Nonlinear oscillations of a simply-supported plate ???????...??.. 3
Fig. 1.2 Comparison of experimental results and first-order piston theory solutions? 6
Fig. 2.1 Geometrical orientation of an active material showing the poling direction? 18
Fig. 3.1 Geometrical properties of a panel with bonded piezoceramics patches??... 32
Fig. 5.1 A simply-supported plate showing actuators for the first mode?????... 69
Fig. 5.2 A simply-supported plate showing actuators for the second mode????.. 70
Fig. 5.3 A simply-supported plate showing actuators for first and second modes?? 71
Fig. 5.4 Panel deflection of a simply-supported plate at the mid-span in the flow
direction ??????????????????????????... 74
Fig. 5.5 Time history of uncontrolled panel deflection, at 1500=? and 0=mxR ??.. 75
Fig. 5.6 Phase plot of uncontrolled panel deflection, at 1500=? and 0=mxR ???.. 76
Fig. 5.7 Phase plot of the zero dynamics for the panel at 1500=? and ,0=mxR
using first mode as the output ??????????????????.. 79
Fig. 5.8 Time history of panel deflection and control effort with feedback
linearization controller, at 1500=? and ,0=mxR using first mode as the
output ???????????????????????????... 80
Fig. 5.9 Phase plot of the panel with feedback linearization controller, at 1500=?
and 0=mxR , using first mode as the output ?????????????. 81
Fig. 5.10 Phase plot of the zero dynamics for the panel at 1500=? and 0=mxR shows
a new equilibrium, when second mode is the output ??????... 82
xii
Fig. 5.11 Time history of panel deflection and control effort with feedback
linearization, at 1500=? and 0=mxR , using second mode as the output??. 83
Fig. 5.12 Phase plot of the panel with feedback linearization controller, at 1500=?
and 0=mxR , using second mode as the output ???????????? 84
Fig. 5.13 Time history of panel deflection and control effort with feedback
linearization, at 1500=? and 0=mxR , using first and second modes as the
output ???????????????????????????.. 85
Fig. 5.14 Phase plot of the panel with feedback linearization controller, at 1500=?
and 0=mxR , using first and second mode as the output ????????.. 86
Fig. 5.15 Phase plot of the zero dynamics for the panel at 380=? and ,2pi?=mxR
using first mode as the output ??????????????????.. 88
Fig. 5.16 Time history of panel deflection and control effort with feedback
linearization controller, at 380=? and ,0=mxR using first mode as the output 89
Fig. 5.17 Phase plot of the panel with feedback linearization controller, at 380=?
and ,2pi?=mxR using first mode as the output ???????????? 90
Fig. 5.18 Phase plot of the zero dynamics for the panel at 380=? and 2pi?=mxR
shows a new equilibrium, when second mode is the output ??????? 91
Fig. 5.19 Time history of panel deflection and control effort with feedback
linearization, at 380=? and 2pi?=mxR , using second mode as the output ? 92
Fig. 5.20 Phase plot of the panel with feedback linearization controller, at 380=?
and 2pi?=mxR , using second mode as the output ??????????? 93
Fig. 5.21 Time history of panel deflection and control effort with feedback
linearization, at 380=? and 2pi?=mxR , using first and second modes as the
outputs ???????????????????????????.. 94
Fig. 5.22 Phase plot of the panel with feedback linearization controller, at 380=?
and 2pi?=mxR , using first and second mode as the outputs ??????? 95
xiii
NOMENCLATURE
,
1
a ,
2
a ,
n
a
nm
A ,
n
A = modal amplitudes
a, b = plate length, plate span
,A ,B D = extensional, coupling, bending stiffnesses
3
b ,
4
b = electro-elastic coupling coefficients
E
ijkl
C ,
E
ijkl
Q , []
E
Q = elastic constant matrix
1
C ,
2
C ,
3
C ,
4
C ,
5
C ,
6
C = nonlinear modal amplitude terms
d
c = aerodynamic damping term
fv
c = flow speed coupling term
3k
c ,
4k
c = linear stiffness terms
312
c ,
330
c ,
403
c = nonlinear stiffness terms
ijk
d , []d = piezoelectric strain coefficients
i
D = electric displacement
e = piezoelectric stress coefficient
E , ,
i
E {}E = electric field
E ,
s
E ,
p
E = elastic constant
s
E ,
p
E = aluminum elastic constant, piezo ceramic elastic constant
f = dielectric permittivity
h ,
p
h ,
s
h = panel, piezo ceramic, substructure thicknesses
H = electrical enthalpy
()?H = Heaviside function
K = kinetic energy
?
M = Mach number
{}M = bending stress couple
{}N = stress resultants
i
P = Polarization vector
q ,
a
q = dynamic pressure
31
R ,
32
R = moment-electric field coefficients
t = time
T = transpose
u , v , w = displacement field along ?x , ?y , ?z directions
xiv
0
u ,
0
v = midplane displacement fields
U = internal strain energy
V = workdone due to external force
w , W = deflection or transverse displacement
max
w ,
max
W = maximum deflection
W
&
= rate of change of deflection
x , y , z = displacement field components
? , ? = delta
()?? = dirac
ijk
e = piezoelectric stress coefficients
? ,
ij
? , []? , {}? = strains
{ }
p
? , { }
0
? = piezoelectric strain, midplane strain
? ,
cr
? = dynamic pressure, critical dynamic pressure
{}? = piezoelectric induced strain vectors
{}? ,
x
? ,
y
? ,
xy
? = curvature
?
?
kl
, []
?
? , []
?
? = dielectric permittivity
xy
? = shear strain
xy
? = shear stress
? = nondimensional time
a
? = air density
? , ? = potential, airy stress potential
? = gradient
ij
? , []? , {}? = elastic stress components
o
m = mass per unit area
1
1. INTRODUCTION
Panel flutter is the self-excited oscillation of a plate or shell when exposed to
airflow along its surface [1]. This is a dynamic instability phenomenon in the
supersonic/hypersonic speed regime, and is induced by the aerodynamic loads, which act
only on one side of a panel. This differs from aeroelastic wing flutter, where the flow acts
on both sides of the wing. Generally, flutter is an oscillatory aeroelastic instability
characterized by the loss of system damping due to the presence of unsteady aerodynamic
loads [2].
The consequences of aeroelastically induced motion are structural failures, and they
have been observed in research aircraft, launch vehicles for spacecraft, and jet engines.
The earliest reported structural failures that can be attributed to panel flutter were the
failures of early German V-2 rockets during World War II [3, 4]. Panel flutter can be
experienced by a vehicle that flies at a supersonic speed in the air. The skin panels
experience sustained vibrations with associated limit cycle oscillations that can result in
structural failures by fatigue due to the aerodynamic pressure on the vehicle surface.
Experiments indicate that there are critical dynamic pressures (air flow speeds)
above which panel flutter exists. At dynamic pressures below these critical dynamic
pressures the panel has random oscillations with small amplitudes. These are small
compared to the panel thickness, and they die out with time. Linear structural theory
predicts the critical dynamic pressure value above which the panel motion becomes
2
unstable and grows exponentially with time, but it only predicts the flutter boundary and
the corresponding structural flutter frequency. At dynamic pressures above the critical
pressure, the amplitude of vibration becomes large, and it is on the order of the panel
thickness, the effect of in-plane stretching forces becomes significant and acts as a
restoring force, while the aerodynamic forces tend to increase the amplitude. Therefore,
the interplay of the mid-plane stretching forces, which generally restrain the motion and
cause stability, and the aerodynamic forces, which grows the amplitudes and causes
instability, results in the bounded limit cycle oscillations that are observed. This is shown
in Fig. 1.1. Therefore linear theory becomes insufficient, and nonlinear structural theory,
which is based on von K?rm?n large deflection plate theory, is suggested for further
analysis.
Flexible structures, such as satellites, atmospheric re-entry vehicles, and other
aerodynamic vehicles are generally lightly damped due to low structural damping in the
materials used and the lack of other forms of damping. In these structures, vibrations
have long decay times that can lead to fatigue, instability, or other problems associated
with the operation of the structures. One of the earliest works of actively controlling the
vibrations of these flexible structures using active materials is by Bailey and Hubbard [5]
who developed an active vibration damper for a cantilever beam using a distributed-
parameter actuator and distributed-parameter control theory. When these structures are
made with highly distributed actuators, sensors, and processing networks [6], they are
referred to as intelligent structures. The study of aeroelastic phenomena has received
serious attentions in the past few decades in two particular areas of interest, namely wing
flutter and panel flutter, and efforts have been made to develop controllers for these
3
classes of problems. Much attention has been focused in the literature on active control of
wing flutter using nonlinear control techniques, but very little has been done in the area
of panel flutter control using such techniques. Moon, S. H. et al. [7] noted that the system
to be controlled is both nonlinear and underactuated, and that it is better to control
nonlinear systems using a nonlinear control method. A nonlinear controller using a
feedback linearization control method was proposed and applied to suppress panel flutter
using a finite element method.
Fig. 1.1 Nonlinear oscillations of a simply-supported plate
4
1.1 Panel Flutter
There are voluminous works on panel flutter over several decades, with most
analyses placed in one of five categories [8] based on the structural and aerodynamics
theories employed, and they are described in [9 -14]. They are shown in Table 1.1. The
first category is the linear structural theory and quasi-steady aerodynamic theory [9, 10].
The second is the linear structural theory and full linearized (inviscid, potential)
aerodynamic theory [11, 12]. The third is the nonlinear structural theory and quasi-steady
aerodynamic theory [13-18]. The fourth is the nonlinear structural theory and the full
linearized (inviscid, potential) aerodynamic theory [19, 20], and the fifth is the nonlinear
structural theory and the nonlinear piston aerodynamic theory [21].
The aerodynamic pressure, which acts on one side of the panel surface, is developed
as a function of the panel motion. Linearized potential flow theory is recommended for
air speeds close to Mach one, quasi-steady linear (first-order) piston theory is employed
for supersonic air flow ( 2>?M ), and nonlinear (third-order) piston theory is
recommended for the hypersonic regime ( 5>?M ). Structural theory can be linear or
nonlinear depending on the order of magnitude of the transverse deflection compared to
the panel thickness.
Table 1.1 Panel flutter theories
Type Structural theory Aerodynamic theory Mach number
1 linear Quasi-steady piston 52 << ?M
2 linear Full-linearized potential 51 << ?M
3 nonlinear Quasi-steady piston 52 << ?M
4 nonlinear Full-linearized potential 51 << ?M
5 nonlinear Nonlinear piston 5>?M
5
Linear panel flutter can be solved with the Fourier method in the frequency domain.
The critical dynamic pressure and flutter boundary are found by increasing the
aerodynamic pressure until two linear frequencies coalesce. The two values of
frequencies, which are real become a complex pair. Beyond the flutter boundary, the
panel will undergo fluttering motion, and the amplitude of the panel motion diverges, but
various experiments indicate that the amplitude grows to a limiting value, which becomes
stable, nearly sinusoidal and independent of the initial conditions. This motion is called
limit cycle oscillation. This phenomenon is explained by the interplay between damping
due to the structural nonlinearities and instability due to aerodynamic pressure effect. The
transverse deflection of the panel is of the order of the panel thickness when it undergoes
limit cycle oscillation in the fluttering zone, so linear analysis is inadequate. In order to
account for the geometric nonlinearity, von K?rm?n large-plate theory is usually
employed in nonlinear panel flutter problem, and it agrees well with experimental results
[22] as shown in Fig. 1.2.
The analysis of nonlinear panel flutter involves analytical techniques such as
Galerkin or the Rayleigh-Ritz method, which is used to reduce the partial differential
equations of motion to a set of ordinary, nonlinear, integral-differential equations in time
for the modal amplitudes. The integral terms are omitted, if quasi-steady aerodynamic
theory is used instead of the linearized full (inviscid, potential) aerodynamic theory. The
linear panel flutter problem can be obtained, if the nonlinear terms are omitted [23]. The
set of ordinary differential equations obtained is further solved by a direct integration
method, harmonic balance method, or perturbation method.
6
Fig. 1.2 Comparison of experimental results and first-order piston theory solutions
The numerical time integration, when employed, produces the time-displacement
history, from which limit cycle oscillation is obtained.
The harmonic balance method has been widely and successfully applied to
nonlinear panel analysis [1, 2, 14, 15, 17, 18, 21, 25, 26]. Using this method, Fung [2,
14] and Kobayashi [17] solved 2-D plates, and Librescu [18] developed general solutions
for rectangular and cylindrical specific orthotropic plates. Eastep and McIntosh in [21]
used a Rayleigh-Ritz approximation to Hamilton?s variational principle instead of
Galerkin?s method to set up the equations of motion in the spatial domain for the solved
rectangular plates. Kuo, Morino and Dugundji [1] also solved the nonlinear panel flutter
problem for rectangular plates. Eslami [25] studied specific orthotropic panels. Yen and
7
Lau [26] studied the dynamical behaviour of a hinged-hinged 2-D plate excited by
supersonic flow.
Perturbation methods are used to solve problems with small nonlinearity due to the
assumption of small disturbance from an equilibrium position, and they have been used to
solve panel flutter for rectangular plates by Morino [1, 27], and by Eslami [25] for
specific orthotropic plates.
The nonlinear theory of supersonic panel flutter is deterministic. Ibrahim and Orono
[28] investigated stochastic nonlinear flutter of a simply-supported 2-D isotropic panel
subjected to random in-plane forces. The aerodynamic loading was modeled using a first-
order quasi-steady piston theory. A general moment equation for two- and three-mode
interactions was derived by using the Fokker-Planck equation approach. The stochastic
nonlinear flutter was studied using Gaussian and first-order non-Gaussian closure
schemes. They concluded that the nonlinear random flutter of panels in terms of four and
more modes can adequately be determined by using the Gaussian closure scheme.
The other alternative approaches to Galerkin?s method and modal expansion are
numerical methods (finite difference and finite element representations) and separation of
variables or so-called exact solutions. The former is particularly useful in solving
nonlinear panel flutter problems without simple boundary conditions or problems with
equations of motion with various terms which makes the analytical solution improbable.
Survey of various applications of finite element methods to nonlinear panel flutter can be
found in Han and Yang [29] up to 1983, Gray and Mei [8] up to 1991, Zhou et al. [30] up
to 1994.
8
1.2 Intelligent Structures
Pierre and Jacques Curie [31] discovered that some crystals produce charges on
their surfaces when compressed in particular directions, and those charges are
proportional to the applied pressure. These charges are withdrawn when the applied
pressure is removed. It was also found that these crystals become strained when they are
electrically polarized. This effect is called piezoelectricity, and it is exhibited by
crystalline materials, such as quartz and rochelle salt.
Nowadays, the most commonly used piezoelectric materials include ceramics called
lead zirconate titanate (PZT), and polymers such as poly-vinylidene fluoride (PVDF),
Macro Fiber Composites (MFC) and Active Fiber Composites (AFC).
Piezoelectric materials act as a generator by converting mechanical energy into
electrical energy when pressure is applied, and this is known as the sensor mode or the
direct effect. Conversely, it acts as motor by converting electrical energy into mechanical
energy, when electric field is applied to it, and this is known as the actuator mode or
converse effect. It also acts as a capacitor for storing electrical energy. These materials
have been used extensively in electromechanical transducers, such as ultrasonic
generators, filters, strain gages, pressure transducers, accelerometers, sensors, and
actuators because of their direct and converse effects.
Piezoelectric layers or patches are usually bonded to or embedded in the surface of a
structure. The mechanical/electrical behavior of these flexible structure members can
then be monitored or modified by the piezoelectric layers or patches used as sensors or
actuators.
9
Actuation strain is the component of strain that is due to stimuli other than
mechanical stress, and it can be produced by piezoelectric materials. This strain
physically causes induced strains to be produced. The potential applications for induced
strain actuators are their uses as highly distributed actuators in intelligent structures.
Therefore, flexible structures can be controlled by the use of smart sensors and actuators.
Intelligent structures having distributed actuators with induced strain actuations can be
used to design structures with intrinsic vibration and shape control capabilities. Some
studies have been carried out on induced strain actuation for beams [6, 32, 33, 34, 35]
and plates [36, 37]. The actuation strain is modeled into the constitutive relations as is
usually done with thermal strain. In [6], both static and dynamic models were developed
for segmented piezoelectric and substructure couplings. These were incorporated into the
Bernoulli-Euler beam equations, and these models were refined into three types [32]: the
uniform strain model with only extensional strain in the actuator for surface bonded
actuators; the Bernoulli-Euler or consistent strain model, which accounts for both
extension and bending in the actuator and is applicable to both surface bonded or
embedded actuators; and finite element models which account for extension, bending and
shear in the actuator and structure. Experimental results were used to validate the beam
actuation models presented.
The static model of the mechanical coupling of the segmented piezoelectric
actuators accounts for only pure bending of the elastic substructure, therefore Im and
Atluri [34] proposed a refined model, which includes the transverse shear forces, axial
forces and the bending moments induced by actuators.
10
Crawley and Lazarus [36] formulated a general model of the induced strain
actuation of plates with various boundary conditions and externally applied loads for both
isotropic and anisotropic plates that are entirely or partially covered with piezoelectric
actuators in various orientations, either bonded to or embedded in the substrates. This
model combines both the actuators and the substrates into one integrated structure, and it
is referred to as the ?consistent plate model.? This model considers the induced strain
actuators to be plies of a laminated plate. There is an assumption of consistent
deformations in the actuators and the substrates. The strain distribution is assumed to
result from a linear combination of in-plane extensional (constant strain through the
thickness) and bending (linearly varying through the thickness) displacements.
Hagood, Chung and von Flotow [38] modeled the effects of dynamic coupling
between a structure and an electrical network through the piezoelectric effect. Burke and
Hubbard [39, 40] applied a spatially shaped distributed actuator for the vibration control
of a simply supported beam, and this distribution facilitates the control of desired
vibrational modes.
Static and dynamic models have been derived for segmented piezoelectric actuators
that are bonded to elastic substructures or embedded in laminated structures [6]. These
models are used to predict the response of a structural member to a command voltage
applied to the actuators and give guidance as to the optimal locations for their
placements.
Dimitriadis, Fuller and Rogers [41] extended the static and dynamic models
developed in [6] for piezoelectric elements bonded to and embedded in one dimensional
beams to two dimensional plates by estimating the load induced by the actuators to the
11
supporting elastic structures. The results were used to selectively excite and suppress
particular vibrational modes leading to improved control behavior.
A conservation of strain energy model has been used to determine the equivalent
force and moment induced by finite-length spatially-distributed induced strain actuation
attached to or embedded in laminate beams and plates using the applied moment on the
cross-section of the edges of the actuators [35]. This model was extended into developing
classical laminated plate theory (CLPT) for a laminate plate with induced strain actuators
for actuator patches that are spatially distributed [37].
This ?consistent plate model? has been experimentally verified and has been shown
to be the most accurate representation of the actual behavior of both discrete surface
bonded or embedded actuators, either segmented or continuous.
The placement of actuators primarily is dependent on the mode to be controlled. The
placement of piezoelectric actuators for controlling particular free vibration modes was
considered by Crawley and de Luis [6]. Lee [42, 43] developed a piezoelectric laminate
theory based on modal sensors and actuators. These modal sensors/actuators sense and
actuate the modal coordinate of a particular mode of a beam or plate. They are also used
to excite or measure combinations of modes. Tanaka [44] placed a number of sensor
patches on a structure to measure the response of a number of modes. Results
demonstrate that modes can be selectively excited and that the geometry of the actuator
shape affects the distribution of the response among modes [41 ? 44].
A piezoelectric material can be used as an actuator or a sensor, but when it is made
to simultaneously effect deformations and sense the strain in structural members, thereby
combining both functions in a single device, then it is referred to as a ?self-sensing
12
piezoelectric actuator,? or simply ?simultaneous sensor actuator? (SSA), and Anderson
and Hagood [45], and Anderson, Hagood and Goodliffe [46] presented a coupled
electromechanical model for such a SSA. They also investigated issues relating to its
implementation in both open and closed-loop experiments performed on a cantilevered
beam. Typically, the current drawn by the piezoelectric material is ignored when it is
used as an actuator. When the current drawn is taken into account, there is the possibility
of reconstructing the actuator strain from a voltage-driven piezoelectric. Dosch, Inman
and Garcia [47] developed a technique for using a self-sensing actuator in a closed-loop
that is truly collocated and effective in vibration suppression of intelligent structures.
In the past few decades, a tremendous amount of research has been devoted to the
vibration control of structures. While passive control improves the performance
characteristics of a structure through the use of materials or devices that enhance the
damping and/or stiffness characteristics of the structure, active control achieves the
desirable performance characteristics through feedback control, whereby actuators apply
forces or moments to a structure based on the structural response measured by the
sensors.
Some of the research in the field of vibration control of flexible structures using
piezoelectric sensors and actuators include efforts by Plump and Hubbard, Sung and
Chen, Chen et. al., Joseph [48 - 51]. They studied structures that are able to sense and
control their own behaviors, so as to achieve much higher levels of operational
performance than conventional materials and structures. A technique called positive
position feedback (PPF) for vibration suppression in large space structures was also
investigated, and this technique makes use of generalized displacement measurements.
13
These works also include suppression of elastodynamic responses of high-speed flexible
linkage mechanisms by employing a state feedback optimal control scheme.
Piezoceramics are used to generate the control inputs, and they are also used as sensing
devices.
1.3 Panel Flutter Suppression
The effectiveness of using passive or active control of flexible structures has been
demonstrated by many researchers. However, in the area of panel flutter suppression
using piezoelectric materials, only a few research efforts have been reported [52 - 57].
Frampton, et. al. [57] investigated the active control of panel flutter with piezoelectric
transducers by implementing direct rate feedback control, and they demonstrated a
significant increase in the flutter boundaries.
Chuh Mei and his research group [54, 56, 58 - 60] have carried out extensive
research on the suppression of nonlinear panel flutter using piezoelectric actuators. They
used both the finite element method and Galerkin?s method with modal expansion. The
finite element models account for nonlinear stiffness matrices, thermal and aerodynamic
loads on the panel. Optimal control was used to actively suppress large-amplitude, limit
cycle flutter motions of rectangular plates at supersonic speeds using piezoelectric
actuators.
Moon. S. H. et al. [61, 62] investigated both active and passive suppression schemes
for nonlinear flutter of composite panel. Optimal controllers based on linear optimal
control theory were designed for active suppression schemes, while piezoelectric
14
actuators connected with an inductor-resistor series shunt circuits were used for the
passive suppression. An active/passive hybrid piezoelectric network was also formulated.
Since the previous studies on panel flutter suppression used optimal controllers for
linearized models, Moon, S. H. et al. [5] applied a nonlinear controller using a feedback
linearization control method to suppress panel flutter using the finite element method.
This technique was also employed in developing nonlinear control techniques for a
prototypical wing sections with torsional nonlinearies at Texas A. & M. [63]. Locally
asymptotically stable (nonlinear) feedback controllers for a range of flow speeds and
elastic axis locations were derived for this aeroelastic system using partial feedback
linearization techniques when either the pitch or plunge is chosen as the output. This
leads to a partial input-output feedback linearizing coordinate transformation with the use
of a single trailing-edge control surface. As a result, the associated zero dynamics of the
subsystem was studied, and it was found that it can also be locally asymptotically stable.
Full feedback linearization was also carried out with two trailing-edge control surfaces.
When the nonlinear partial feedback linearization is constructed so as to explicitly control
the pitch degree of freedom, the zero dynamics of the closed-loop system are linear. But,
when the nonlinear partial feedback linearization explicitly controls the plunge degree of
freedom, closed-loop stability of the zero dynamics is considerably more difficult. It is
shown that there exist locations of the elastic axis and speeds of the
subsonic/incompressible flow for which feedback strategy exhibits a wide range of
bifurcational phenomena.
15
1.4 Objectives and Scope
There is much research going on in the development of intelligent systems using
active materials, and some of these efforts include the development of intelligent plates.
Panel flutter has also posed tremendous challenges to aeroelasticians, and has generated
lots of research in the design of structural surfaces exposed to aerodynamic loads,
especially in supersonic environments. The application of these intelligent plates in
aircraft or vehicles and surfaces in a fluid medium has the potential of making these
surfaces actively respond to external stimuli. These intelligent plates have actuators and
sensors embedded or bonded to their surfaces, and they are connected to processors
which modify the signals so that these intelligent plates are able to react to stimuli that
can cause large deflections and instability resulting in the failure of the panel. With these
developments, advanced aircraft or vehicles and surfaces in a fluid medium can operate
in harsh environments.
The main objective of this research is to investigate a technique for suppressing the
fluttering of a fluid loaded flat panel or flat panel with aerodynamic loads, which is also
acted upon by in-plane forces. This problem is also widely known as panel flutter
suppression. The technique that is used is based on nonlinear control theory. The main
idea is to transform the nonlinear panel flutter problem into an equivalent controllable
linear problem that can be written in simple Brunovsky canonical form by the method
called feedback linearization. This involves developing nonlinear feedback control laws,
which cancel the nonlinear dynamics resulting in a linear system, and a pole placement
16
technique is then employed so as to make the states of the linearized feedback models
locally asymptotically stable at a given equilibrium.
The active materials used in this investigation are piezoelectric ceramics, and they
have dual effects coupling their electrical and structural properties. The
electromechanical quantities involved are presented in Chapter 2 and these lead to linear
piezoelectric constitutive relations.
Equations of motion are given for a flat panel with bonded and distributed actuators
and sensors subject to aerodynamic loads, in-plane loads and applied electric fields.
These equations are coupled nonlinear partial differential equations, which are reduced to
nonlinear ordinary differential equations in time, and presented in state-space format, in
Chapter 3.
In Chapter 4, feedback linearizing controllers are developed for a fluid loaded flat
panel with limit cycle oscillations at a dynamic pressure above the critical dynamic
pressure with or without externally applied in-plane loads. The suppression of the
oscillations after the onset of flutter is investigated. Numerical simulations are carried out
in Chapter 5 to study the feedback linearized methodologies, and to establish the stability
of the resulting states. The conclusions to this investigation are presented in Chapter 6.
17
2. PIEZOELECTRICITY
Piezoelectric materials are active materials that are either ceramic or polymeric.
Ceramic materials include lead zirconante titanates (PZT) and single crystals; while
polymeric materials include polyvinylidene fluoride (PVDF), macro fiber composite
(MFC) and Active Fibers (AFC). These materials are bonded to the surface of, or
embedded into flexible structural members, so that actuation and sensing are achieved at
the material level.
2.1 Characteristics of the Materials
A ceramic material is made active by a poling process, which is the application of a
large external electrical field, which aligns the randomly orientated unit cells in the
medium. One can consider a material with geometrical orientation as shown in Fig. (2.1).
The coupling coefficients are defined by the direction of the poling. Three mutually
perpendicular directions are shown along the axes 1, 2 and 3. There are also three other
modes, and these are 4, 5, and 6, which represent shear in the 1, 2 and 3 directions,
respectively. The material, is poled as shown, in the 3-direction, so that the polarization is
taken to be along that direction. The piezoelectric strain coefficient, piezoelectric stress
coefficient, and permittivity are represented as ,d ,e and ,? respectively. They have 2
subscripts. The first indicates the direction of the electric field, while the second indicates
the direction of strain. Therefore, the piezoelectric coefficients are ,31d ,32d ,33d ,15d ,25d
18
and 36d . The most commonly used coefficient is 31d , and this implies that electric field is
applied in the direction of polarization, 3-direction, while, the induced strain is in the 1-
direction.
Fig.2.1 Geometrical orientation of an active material showing the poling direction
1
Poling
direction
2
3
19
Piezoelectric materials have distinctive effects. They develop an electrical charge
when subjected to mechanical stress in the direct piezoelectric effect, and conversely they
develop mechanical strain when subjected to an electrical field. Therefore, they can
convert electrical energy into mechanical energy and vice-versa. The applied electric
potential produces an electrical field across the material that induces mechanical strain in
it, while in reversal; the application of stress to the same material generates electrical
charges on it.
The direct piezoelectric effect is the production of both positive and negative
electric charges on the corresponding surfaces, and it results in the deformation that takes
place under external pressure (stress). Thus, there is polarization of the medium due to
deformation in the absence of an electric field, iE , and the relationship between the
polarization vector, iP , is given as
jkijki eP ?= or jkijki dP ?= (2.1)
The converse piezoelectric effect is the mechanical deformation which results from the
application of electric field, iE , due to the polarization of a medium, and the relationship
is given as
iijkjk Ed=? (2.2)
Therefore, there is strong coupling between the deformation fields and internal electric
fields.
20
2.2 Electrical Enthalpy and Electric Field
The electrical enthalpy H describes the amount of energy stored in a material and is
defined in [64]. The electrical enthalpy can be written as
ii DEUH ?= (2.3)
whereU is the total internal energy, E and D are the electric field and displacement
vectors, respectively.
Toupin[65] formulated electric enthalpy density using a polynomial approximation
based on a power series expansion about the natural state of a piezoelectric medium. The
result is as shown
),( EHH ?= (2.4)
where, ? is strain, and E is the electric field.
lkkljkiijkklijijkl
E EEEeCH ?????
2
1
2
1 ??= (2.5)
{ } [ ] { } { } [ ]{ } { } [ ] { }EEeEQH TTET ????? 2121 ??= (2.6)
where, EijklC is used interchangeably with Q , and EijklC , ijke , and ??ij are elastic stiffness
constants, piezoelectric stress constant, and dielectric permittivity, respectively.
The electric field vector is the negative gradient of the electric potential,? , and it is
assumed to vary linearly in the thickness, kt , direction, that is,
???=E (2.7)
{ }TzE00=
with
ktz
E ??=
21
2.3 Linear Piezoelectric Constitutive Relations
Although there are many nonlinear phenomena in piezoelectric materials, linear
constitutive relations are often used to describe the behavior of piezoelectric layers. The
mechanical stress and strain vectors,? and ? , respectively, are related through the
electric enthalpy H . The linear piezoelectric constitutive equations obtained from [66]
are given in Eq. (2.8) and Eq. (2.9).
kkijklijkl
E
ij
ij EeC
H ?=
?
?= ?
?? (2.8)
kikklikl
i
i EeE
HD ??? +=
?
??= (2.9)
and, in matrix form
{ } [ ] { } [ ] { }EeQ TE ?= ?? (2.10)
[ ]{ } [ ] [ ]EeD ??? += (2.11)
Just as in the relationship between mechanical stress and mechanical strain, the
piezoelectric stress coefficient is proportional to the piezoelectric strain coefficient, with
elastic material properties as the constants of proportionality. Hence,
[ ] [ ][ ]EQde = (2.12)
substituting Eq. (2.12) into Eq. (2.8), one obatins
{ } [ ] { } [ ] { }( )EdQ TE ?= ?? (2.13)
We can write
[ ] [ ] { } [ ][ ]( ) { }EQdQ TEE ?= ??
[ ] { } [ ] { }( )EdQ TE ?= ? (2.14)
22
One can observe that the piezoelectric induced strain is the product of the applied electric
field and piezoelectric strain coefficient of the material, and it is written as
{ } [ ] { }Ed T=? (2.15)
The free permittivity matrix[ ]?? is easier to obtain than the clamped permittivity
matrix[ ] ,?? We can use the relationship below to relate the two.
[ ] [ ] [ ][ ] [ ]TE dQd?= ?? ?? (2.16)
Similarly, one can re-write the expression for the electrical displacement density using
Eq. (2.11), Eq. (2.12), and Eq. (2.16) as
{ } [ ][ ] { } [ ] [ ][ ] [ ]( ){ }EdQdQdD TEE ?+= ???
[ ][ ] { } [ ] { }( ) [ ] { }EEdQd TE ??? +?= (2.17)
In this section, both the linear piezoelectric constitutive relations and linear electrical
displacement density relations have been obtained. They are given by
( ) ??
?
?
?
?
??
?
?
?
?
??
??
?
??
??
?
?
??
??
?
??
??
?
?
?
?
?
?
?
?
?
?
?
?
=
??
??
?
??
??
?
kxy
yy
xx
ki
kxy
yy
xx
kkxy
yy
xx
d
d
d
E
v
v
v
?
?
?
?
?
?
100
01
01
2
1
(2.18)
and
{ }
( )
)(
)(
)(
)(
2
1
)()(
100
01
01
k
iiik
k
xy
yy
xx
k
i
xy
yy
xx
k
k
xyyyxx
k
i E
d
d
d
E
v
v
v
dddD ??
?
?
?
+
??
?
?
?
?
??
?
?
?
?
??
??
?
??
??
?
?
??
??
?
??
??
?
?
?
?
?
?
?
?
?
?
?
?
=
(2.19)
respectively.
23
3. FORMULATIONS
In this chapter, the generalized nonlinear dynamic equations for a simply supported
rectangular panel with piezoelectric layers are presented.
The flat plate or panel is considered to be an intelligent plate, and it is made up of
the host substructure and piezoelectric materials embedded within the host or bonded to
the surface of the host. The panel considered in this study is thin. The piezoelectric
materials are in the form of distributed patches or continuous layers, while the host
substructure is considered to be an isotropic material.
Many research efforts have been conducted in the field of vibration control of
structures using piezoelectric actuators and sensors. In the case studied here, the structure
is a simply supported rectangular intelligent plate or panel, and its generalized nonlinear
dynamic equations are derived as in [54]. The intelligent plate is considered to undergo
large transverse displacement of the order of the plate thickness, therefore, von K?rman
large-deflection plate theory, which accounts for the structure nonlinearity, is used for
modeling the plate deflection. The linear piezoelectric theory is used to derive the
equations of piezoelectric actuation and sensing, and first-order piston theory or the
quasi-steady supersonic theory is used to model the aerodynamic force due to the
supersonic fluid flow.
24
3.1 Aerodynamic Forces
In the case where the fluid flow over a panel is considered as aerodynamic loads or
forces, this problem is sometimes referred to as panel flutter, in the literature. The
aerodynamics pressures can be represented by quasi-steady first-order piston theory, full
linearized (inviscid, potential) aerodynamic theory, or nonlinear piston aerodynamic
theory. The aerodynamic theory that is applied in this study is the quasi-steady first-order
piston aerodynamic theory, and it is employed to model the aerodynamic pressure when a
flight vehicle is in the supersonic airflow regime. This theory describes the aerodynamic
loads on a skin panel as pressure on a piston in a long narrow tube with a given velocity,
and this is expressed as in [24].
?
?
?
?
?
?
?
?
?
?
+
?
?
?=?
??
?
t
w
VM
M
x
wq
p
a
1
1
22
2
2
?
(3.1)
where
2
2
1
?
= Vq
aa
? is the dynamic pressure,
a
? is the air density,
?
V the free stream
airflow speed,
?
M the Mach number, w the transverse displacement of the panel, and
1
2
?=
?
M? .
3.2 Displacement Field Theory:
The displacement field theory is based on Kirchhoff?s hypothesis, and it states that
line elements which originally are perpendicular to the middle surface of the plate remain
straight and normal to the deformed middle surface, and there is no change in length. The
displacement field, which comprises longitudinal u , and normal displacements v , in the
plane of the plate, and transverse displacement w , can be written as
25
x
zwtyxuu
,00
),,( ?=
y
zwtyxvv
,00
),,( ?=
),,(
0
tyxww= (3.2a,b,c)
3.3 Nonlinear Strain-Displacement Relations:
In panel flutter, the plate displacement can be of the order of the thickness of the
plate due to both static and dynamic instabilities and the associated limit cycle.
Therefore, the plate is considered to undergo large displacement, and one can use von
K?rm?n?s theory, which considers nonlinear strain-displacement relations.
2
0
0
2
1
?
?
?
?
?
?
?
?
+
?
?
=
x
w
x
u
xx
?
2
0
0
2
1
?
?
?
?
?
?
?
?
?
?
+
?
?
=
y
w
y
v
yy
?
y
w
x
w
x
v
y
u
xy
?
?
?
?
+
?
?
+
?
?
=
00
0
? (3.3a,b,c)
The flat panel is considered to be an isotropic material, and it is thin, so that the ratio of
the length or width over thickness of the panel is greater than 20. The rotary inertia and
transverse shear deformation effects are negligible, hence from the assumptions of
Kirchhoff?s hypothesis, the transverse strain components
zz
? ,
xz
? and
yz
? are taken to be
negligible, so that, one can write
0=
?
?
=
z
w
zz
?
26
0=
?
?
+
?
?
=
x
w
z
u
xz
?
0=
?
?
+
?
?
=
y
w
z
v
yz
? (3.4a,b,c)
and from von K?rm?n?s strain-displacement relations, one obtains
2
2
2
0
2
1
x
w
z
x
w
x
u
xx
?
?
?
?
?
?
?
?
?
?
?
+
?
?
=?
2
2
2
0
2
1
y
w
z
y
w
y
v
yy
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
=?
yx
w
z
y
w
x
w
x
v
y
u
xy
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
=
200
2
1
?
In vector form
{}
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
=
?
?
?
?
?
?
?
?
?
?
=
yx
w
z
y
w
x
w
x
v
y
u
y
w
z
y
w
y
v
x
w
z
x
w
x
u
xy
yy
xx
200
2
2
2
0
2
2
2
0
2
2
1
2
1
2
1
?
?
?
? (3.5.)
that is
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
xy
y
x
xy
yy
xx
xy
yy
xx
z
?
?
?
?
?
?
?
?
?
0
0
0
where, the middle-surface strain components are
27
{}
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
+
?
?
=
?
?
?
?
?
?
?
?
?
?
=
y
w
x
w
x
v
y
u
y
w
y
v
x
w
x
u
xy
yy
xx
00
2
0
2
0
0
0
0
0
2
1
2
1
2
1
?
?
?
? (3.6)
and the curvatures are given by
{}
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
=
yx
w
y
w
x
w
xy
y
x
2
2
2
2
2
2
?
?
?
? (3.7)
or, simply
{} { } { } {}????
?
z
m
?+=
00
{} { } {}??? z?=
0
3.4 Constitutive Equations
In the analysis carried out in this study, both the elastic and the piezoelectric properties of
the piezoelectric ceramic utilized are included. The stress-strain relations for an
active/piezoelectric layer in an intelligent structure are given by the linear piezoelectric
constitutive equations obtained in Eq. (2.18), that is,
{} []{ } {}{ }( )
p
p
zQ ???? ??=
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
36
32
31
3
2
1
2
)1(00
01
01
1
d
d
d
e
v
v
v
v
E
xy
yy
xx
p
p
p
p
p
xy
yy
xx
?
?
?
?
?
?
(3.8a)
28
It can easily be observed that when any layer is passive or the piezoelectric properties of
an active/piezoelectric layer is not activated, the stress-strain relations are given in Eq.
(3.8b). This is simply achieved by taking the electric field term to be zero in the linear
piezoelectric constitutive equations derived, and they are the same as in the literatures for
purely passive layers.
{} [ ] { } {}( )??? zQ
s
?=
0
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
xy
yy
xx
s
s
s
s
s
xy
yy
xx
v
v
v
v
E
?
?
?
?
?
?
)1(00
01
01
1
2
1
2
(3.8b)
3.5 Dynamic Version of the Principle of Virtual Work
The equations of motion are derived using the dynamic version of the principle of
virtual work or Hamilton?s principle. The derivation accounts for both the elastic work
done and piezoelectric effect.
()0
0
=??+
?
dtWKVU
t
e
???? (3.9)
where K? and U? are the virtual kinetic energy and virtual internal strain energy of the
system,
e
W? is the virtual electrical energy, and V? is the virtual work done due to
external forces and the applied surface charge only.
The virtual internal strain energy U? is given as:
()
??
?
?
++=
0
2
2
2
h
h
dzdxdyU
xyxyyyyyxxxx
?????????? (3.10a)
29
The virtual external applied load V? is given as:
wdxdyppV
as
?? )(
0
?+??=
?
?
(3.10b)
where
s
p? = static pressure differential on the surface of the plate, excluding
aerodynamic loading, and
a
p? is the aerodynamic loading over the surface of the plate.
The virtual kinetic energy K? is given as:
dxdy
t
w
t
w
mK
o
?
?
?
?
?
?
?
?
?
?
=
?
?
0
?? (3.10c)
The variational quantities obtained in Eq. (3.10) are substituted into Eq. (3.9), and after
carrying out the appropriate integration across the plate thickness, quantities such as the
stress resultants and bending couples can be obtained. After carrying out the integration
by parts and applying appropriate variational statements, these become the von K?rm?n
equations for a plate with large deflections, with the piezoelectric terms included.
Reference can be made to [13, 54, 67] where appropriate derivations were carried out.
3.6 Stress Resultants and Bending Couples
Stress resultants, ,N and bending couples, ,M are the forces and couples per unit
width, and they are defined as
{}{}(){}()
?
?
=
2
2
,1,
h
h
dzzMN
k
? (3.1)
Substituting Eq. (3.8) into the above equations result in the constitutive relations for the
laminated panel used as an intelligent plate with piezoelectric ceramics bonded to both
surfaces of the host layer. The stress components in the plate are integrated over each
30
layer thickness, and thereafter the stress resultants in each layer are summed for the
whole plate. Therefore, the stress resultants and bending couples are given by
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
p
p
M
N
DB
BA
M
N
?
?
0
(3.12)
where
p
N and
p
M are the piezoelectric induced inplane forces and bending moments.
The stiffness terms are the extensional stiffness matrix,[ ]A , bending stiffness matrix,[ ]D ,
and coupling (stretching-bending) stiffness matrix,[ ]B ; and these are represented as
[][][](){}( )
?
?
=
2
2
2
,,1,,
h
h
dzzzDBA
k
? (3.13)
The piezoelectric layers are assumed to be symmetrically bonded to the host layer,
therefore the laminate does not exhibit coupling between bending and stretching, hence
the coupling matrix, [] 0=B . Generally, we consider that the Poisson ratio for both host
and piezoelectric materials are similar, hence vvv
ps
== . The stiffness matrices are
represented as
[]
?
?
?
?
?
?
?
?
?
?
?
?
=
)1(00
01
01
1
2
1
2
v
v
v
v
Eh
A (3.14a)
[]
?
?
?
?
?
?
?
?
?
?
?
=
)1(00
01
01
2
1
v
v
v
DD (3.14b)
with the equivalent panel elastic constant,
())(
1
spss
hhEhE
h
E ?+= (3.15a)
31
and equivalent panel bending stiffness
())(
)1(12
1
333
2
spss
hhEhE
v
D ?+
?
= (3.15b)
3.7 Piezoelectric In-plane Force and Moment
The piezoelectric materials are assumed to be perfectly bonded to the entire top and
bottom of the panel surfaces; so that classical analytical approaches can be applied to the
problem of panel flutter in this research.
The piezoelectric actuators produce the actuation strain that physically causes
induced strains to be produced. The actuators are used as modal actuators [42], which
actuate the modal coordinate of a particular mode of the panel. They are also used to
excite and measure combinations of modes [44] when they are used as sensors.
The piezoelectric layers are also assumed to be segmented so that only the desired
portions of the piezoelectric layers are activated. This arrangement provides the
opportunity to consider the piezoelectric materials as patches at the activated areas only.
The mechanical/electrical behavior of the flexible panel are monitored or modified with
these piezoelectric layers or patches acting as actuators and sensors. The piezoelectric
patches are taken to be rectangular in shape; therefore, the piezoelectric layer is divided
into
x
c
N by
y
c
N elements.
t
E
3
and
b
E
3
are the electric fields on the top and bottom
piezoelectric layers, respectively. The overall thickness of the panel is ,h the length of
the panel in the air flow direction is ,a and the span is .b The thickness of each
piezoelectric patch or layer is ,
p
h and the thickness of the host layer is .
s
h The
geometrical properties of the panel are shown in Fig. (3.1).
32
Fig. 3.1 Geometrical properties of a panel with bonded piezoceramic patches.
The inplane force induced by the piezoelectric layers, or patches per unit length or
simply piezoelectric force per unit length is represented as
{} []{} []{}
??
??
?=
2
2
2
2
s
h
s
h
h
h
dzQdzQN
k
p
kk
p
k
p
?? (3.16)
()
mp
p
p
xx
Evddh
v
E
N
3231
2
1
2 +
?
=
mp
p
Edh
v
E
31
1
2
?
= (3.17a)
()
mp
p
p
yy
Edvdh
v
E
N
3231
2
1
2 +
?
=
mp
p
Edh
v
E
31
1
2
?
= (3.17b)
h
Y
b
a
X
Z
s
h
p
h
33
and
0=
p
xy
N (3.17c)
Where
3
E is the effective electric field applied on the top and bottom layers which
produces only in-plane force, and denoted by
m
E given as
()
bt
m
EEE
33
2
1
+= (3.18)
The induced bending moment actuation per unit length, or simply the piezoelectric
bending moment per unit length is
{} []{} []{}
??
??
?=
2
2
2
2
s
h
s
h
h
h
zdzQzdzQM
k
p
kk
p
k
p
?? (3.19)
where
3
E is the effective electric field applied on the top and bottom layers, which in this
case is referred to as
bij
E producing only bending moment, and it is given as
()
b
ij
t
ijbij
EEE
33
2
1
?= (3.20)
and defining
()
spp
p
hhhd
E
R +
?
=
3131
1 ?
()
spp
p
hhhd
E
R +
?
=
3132
1 ?
(3.21a,b)
The piezoelectric induced bending moment actuation due to discontinuously attached or
embedded piezoelectric patches [54] can be written as
()()[]()()[]
jj
N
i
N
j
iibij
P
x
yyHyyHxxHxxHERM
x
c
y
c
??????=
?
==
??? 1
11
131
()()[]()()[]
jj
N
i
N
j
iibij
P
y
yyHyyHxxHxxHERM
x
c
y
c
??????=
?
==
??? 1
11
132
(3.22a,b)
34
where the Heaviside function H is given by
()
?
?
?
<
?
=?
ax
ax
axH
0
1
(3.23)
() ()[]axH
x
ax ?
?
?
=??
()
?
?
??
?=?
nnn
agdxxgax )1)(()(
)(
?
3.8 Nonlinear Equations of Motion
The governing differential equations for an isotropic plate with finite length and
piezoelectric ceramic actuators bonded as patches or layers on a host plate subject to
large deflection due to flow velocity over its surface and combined equivalent inplane
loads, combined equivalent bending moments, aerodynamic load, and static pressure
differential are derived in [67, 24]. The inclusion of piezoelectric terms can also be
reviewed in [54]. The von K?rm?n?s large deflection plate equations are represented as
p
y
w
N
x
w
N
y
M
x
M
t
w
mp
yx
w
yxy
w
xx
w
y
wD
ac
y
c
x
c
y
c
x
o
s
??
?
?
+
?
?
+
?
?
?
?
?
?
?
?
???
??
?
??
??
?
?
?
?
??
+
?
?
?
??
=?
2
2
2
2
2
2
2
2
2
222
2
2
2
2
2
2
2
2
4
2
(3.24)
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
?
=??
2
2
2
2
2
2
4
1
y
w
x
w
yx
w
Eh
(3.25)
The plate deflection is w,
ppsso
hhm ?? 2+= , ? is the Airy stress potential function,
c
x
N ,
c
y
N are combined equivalent inplane loads, and
c
x
M ,
c
y
M are combined equivalent
bending moments.
The aerodynamic pressure loading is assumed to be
35
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
=?=?
?
t
w
UM
M
x
wq
ppp
s
1
1
22
2
2
?
(3.26)
Membrane inplane loads are given by
?
?
?
?
?
?
?
?
?
=
a
m
x
dx
x
w
a
Eh
N
0
2
2
? ,
? ?
?
?
?
?
?
?
?
?
?
=
b
m
y
dy
y
w
a
Eh
N
0
2
2
?
Induced piezoelectric inplane loads are given by
mp
p
p
x
Edh
v
E
N
31
1
2
?
= ,
mp
p
p
y
Edh
v
E
N
31
1
2
?
=
Induced piezoelectric bending moments are given by
()()[]()()[]
jj
N
i
N
j
iibij
P
x
yyHyyHxxHxxHERM
x
c
y
c
??????=
?
==
??? 1
11
131
()()()()
jj
N
i
N
j
iibij
P
y
yyHyyHxxHxxHERM
x
c
y
c
??????=
?
==
??? 1
11
132
The inplane stress resultants are
2
2
y
N
x
?
??
=
2
2
x
N
y
?
??
=
yx
N
xy
??
??
?=
2
(3.27)
The inplane equations of equilibrium:
0=
?
?
+
?
?
y
N
x
N
xy
x
36
0=
?
?
+
?
?
x
N
y
N
xyy
(3.28)
are satisfied by ,? the Airy stress potential function.
The boundary conditions for a plate that is simply-supported on the four edges are
0
),(),0(
),(),0(),(),0(
2
2
2
2
=
?
?
=
?
?
====
x
yaw
x
yw
yawywyauyu (3.29)
0
),()0,(
),()0,(),()0,(
2
2
2
2
=
?
?
=
?
?
====
y
bxw
y
xw
bxwxwbxvxv (3.30)
The solution to the nonlinear equation given in Eq. (3.24), which is the displacement, can
be represented as combination of linearly independent mode shapes. The assumed
solutions must satisfy the given boundary conditions given in Eq. (3.29) and Eq. (3.30).
Therefore, for a rectangular plate, simply-supported on all edges, one can assume that the
transverse deflection can be written as
??
?
?
?
?
?
?
?
?
?
?
?
?
=
nm
nm
b
ym
a
xn
Aw
??
sinsin (3.1)
with the longitudinal axis of the plate in the flow direction. One can simply retain only
the first spanwise mode for panel flutter limit cycle analyses, hence 1=m , and the
transverse deflection can be simplified as
?
?
?
?
?
?
?
?
?
?
?
?
?
=
n
n
b
y
a
xn
Aw
??
sinsin (3.2)
Therefore, substituting Eq. (3.32) into Eq. (3.24) above [24], one obtains the nonlinear
differential equations in time, and Lai et al. [54] obtained the additional term for
piezoelectric bending moment:
37
[]
r
r
rn
nyxn
n
a
n
a
rn
nr
a
b
a
RnRa
b
a
n
d
da
c
d
ad
?
+
??
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+++ )1(1
2
22
2
22
2
2
24
2
2
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?++
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
2424
65
43
4
2
2
1
2
3
4
CC
CC
b
a
C
b
a
Cn
a
D
Eh
n
?
j
j
i
i
x
c
y
c
y
y
x
x
N
i
N
j
bij
b
y
a
xn
E
b
a
nb
a
a
nb
R
Dhb
a
11
coscos
4
11
2
31
3
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+=
??
==
??
(3.33)
using the below non-dimensional quantities
h
A
a
n
n
=
,
2
1
4
0
?
?
?
?
?
?
?
?
=
am
D
t?
,
c
xx
N
D
a
R
2
?= ,
c
yy
N
D
a
R
2
?=
DM
qa
?
=
3
2
? ,
D
qa
?
?
3
2
=
0
m
a
?
=
?
? ,
?
=
M
c
a
?
,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
2
2
2
1
2
M
M
c
a
3.9 Nonlinear Modal Equations
Approximate deflections of the given system can be obtained using a linear
combination of two modes. Hence, the modal nonlinear equations lead to a set of two
coupled nonlinear differential equations, and they are given as
For :1=n
21
2
22
1
2
2
241
2
1
2
3
8
aa
b
a
RnRa
b
a
n
d
da
c
d
ad
yxa
???
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?++
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
2424
65
43
4
2
2
1
21
3
4
CC
CC
b
a
C
b
a
Cn
a
D
Eh
?
38
j
j
i
i
x
c
y
c
y
y
x
x
N
i
N
j
bij
b
y
a
xn
E
b
a
nb
a
a
nb
R
Dhb
a
11
coscos
4
11
2
31
3
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+=
??
==
??
(3.34a)
For :2=n
12
2
22
2
2
2
242
2
2
2
3
8
aa
b
a
RnRa
b
a
n
d
da
c
d
ad
yxa
???
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+++
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?++
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++
2424
65
43
4
2
2
1
22
3
4
CC
CC
b
a
C
b
a
Cn
a
D
Eh
?
j
j
i
i
x
c
y
c
y
y
x
x
N
i
N
j
bij
b
y
a
xn
E
b
a
nb
a
a
nb
R
Dhb
a
11
coscos
4
11
2
31
3
??
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+=
??
==
??
(3.34b)
1
C ,
2
C ,
3
C ,
4
C ,
5
C and
6
C are nonlinear terms of the modal amplitudes, and they are
defined in the appendix.
The above equations can be rewritten by defining their coefficients with the
quantities defined in the appendix, and these equations become
ubaacacacacaca
kfvd 3
2
21312
3
133013211
=+++?+ &&&
ubaacacacacaca
kfvd 42
2
1421
3
240324122
=+++++ &&&
(3.35)
with
1
a and
2
a as the amplitudes of the first mode and second mode, respectively, and the
coupling is caused by both the nonlinear terms and the flow velocity over the flat panel.
The coefficients, ,
d
c ,
fv
c ,
3k
c ,
4k
c ,
312
c ,
330
c ,
403
c and ,
421
c of the two coupled nonlinear
ordinary differential equations in Eq. (3.35) are non-negative quantities, which are easily
obtained by expansion and collection of the coefficients of the modal amplitudes
1
a and
39
2
a and their derivatives. One writes the equations of motion in Eq. (3.35) as a set of first-
order differential equations using state space format. State variables are defined as
{}{}
TT
aaaaxxxx
21214321
,,,,,, &&=
and the system with Single Input can be written as:
uxgxfx )()( +=& (3.6)
)(xhy = (3.7)
where,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????
????
=
2
2
1421
3
24034241
2
21312
3
13303132
4
3
)(
xxcxcxcxcxc
xxcxcxcxcxc
x
x
xf
dkfv
dkfv
(3.36.a)
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
4
3
0
0
)(
b
b
xg (3.36.b)
and
n
Rx? is the state vector,
m
Ru? is the control vector, and
nn
RRf ?: is a
sufficiently smooth nonlinear function of its argument.
nn
RRg ?: , is a sufficiently
smooth nonlinear function of its argument.
40
4. FEEDBACK LINEARIZATION
The mathematical modeling of most physical systems results in nonlinear systems,
and in order to achieve the desired dynamic behavior for such systems, feedback control
systems are often designed which make the closed-loop systems achieve the specified
objectives. There are numerous ways to design a feedback control system. There are both
linear and nonlinear feedback control systems. The former are usually based on an
approximate linearized model of an actual nonlinear system about the equilibrium point,
while the later are based on the actual nonlinear system. There are various types of
nonlinear control techniques, and these include a technique called feedback linearization.
Feedback linearization is achieved by exact state transformations and feedback,
rather than by linear approximations to the system dynamics, and this implies that the
original system models are transformed into equivalent linear models of a simpler form.
Feedback linearization problems have attracted considerable attention, and have been
used successfully in practical control problems, such as control of helicopters, high
performance aircraft, industrial robots, and biomedical devices.
Panel flutter with its associated limit cycle motions, if not suppressed, can lead to
failure of the panel. Flutter suppression of a panel with distributed and embedded or
bonded active materials can be achieved using feedback control with distributed active
materials acting as sensors and actuators, or self-sensing actuators.
41
The sensors can be made to sense the outputs (motions) of the panel, and the sensed
signals are modified by the feedback controllers and then used to actuate the panel
through the actuators. This active system is used to stabilize the motion of the panel so
that the states have a locally asymptocally stable origin, which is the main control
objective.
Design of linear controllers requires that an equilibrium point be selected, usually
the system origin, and corresponding to the state of a panel without deflection. The
formulated nonlinear system is linearized about this equilibrium point with the
assumption that there are only very small displacements of the states from the origin, but
in panel fluttering these displacements can be large, therefore, the assumption of small
displacements about the origin is invalid. The design of linear feedback controllers only
extends the flutter free region of the panel; it does not effectively suppress the fluttering,
since fluttering involves large displacements from the origin.
With feedback linearization, the nonlinear panel flutter problem is transformed
using output feedback into an equivalent controllable linear system that is in simple
Brunovsky canonical form. This involves the formulation of nonlinear feedback control
laws, which cancel the nonlinear dynamics. The pole placement technique is then
employed to make the states of the feedback linearized model locally asymptotically
stable to the origin.
42
4.1 Mathematical Background
In this section, the mathematical tools required for linearization by feedback control
are developed. The nonlinear control system is first transformed into the Brunovsky form by
a change of coordinates and state feedback, and then linear controllers are designed to control
the linearized system. A thorough review of feedback linearization can be found in the
literature [68-69].
A single-input single-output (SISO) closed-loop system is given in Eq. (4.1) below:
uxgxfx )()( +=& (4.1a)
)(xhy = (4.1b)
where
n
Rx? is a vector of states,
p
Ru? is the input vector,
m
Ry? , f and g are smooth
vector fields on
n
R and h a smooth (i.e., an infinitely differentiable) nonlinear function.
If the input feedback u and coordinate transformations of the states )(x? are applied, such
that,
vxxu )()( ?? += (4.2)
)(xz ?= (4.3)
where v is the external reference input, and the coordinate transformation )(x? has the
following properties
(i) )(x? is invertible,
n
Rx??
(ii) )(x? and )(
1
z
?
? are both smooth mappings
43
then, )(x? is the ?normal form? of special interest, which provides suitable change of
coordinates in the state space. The nonlinear closed-loop system in Eq. (4.1) is transformed to
the new coordinates to become a linear closed-loop system given in Eq. (4.4).
BvAzz +=& (4.4)
4.1.1 Lie derivatives
Let RRh
n
?: be a smooth scalar function, and
nn
RRf ?: be a smooth vector
field on
n
R , then the Lie derivative )(xhL
f
is the directional derivative of a function
)(xh along the direction of the vector )(xf .
RRxhL
n
f
a:)( ,
)(
)(
)( xf
x
xh
xhL
f
?
?
?
= (4.5)
Lie derivatives may be generated recursively, and they are defined as
hhL
f
=
0
(4.6a)
()
()
f
x
hL
hLLhL
i
f
i
ff
i
f
?
?
?
==
?
?
1
1
, ni ,,2,1 K= (4.6b)
Similarly, if
nn
RRg ?: is a smooth vector field, then
RRxhL
n
g
a:)(
)(
)(
)( xg
x
xh
xhL
g
?
?
?
= (4.7)
also, the scalar function )(xhLL
fg
is defined as
( )
)(
)(
)( xg
x
xhL
xhLL
f
fg
?
?
= (4.8)
44
4.1.2 Lie Bracket
Let f and g be two vector fields on
n
R . The Lie bracket of f and g written as
[]gf , is a third vector defined as
[] )(
)(
)(
)(
, xg
x
xf
xf
x
xg
gf
?
?
?
?
?
= (4.9a)
[] gadgLfLgf
ffg
=?=, (4.9b)
Repeated Lie brackets can be defined recursively by,
ggad
f
=
0
(4.10a)
[ ]gadfgad
i
ff
10
,
?
= , ni ,,2,1 K= (4.10b)
4.1.3 Frobenius Theorem
A nonsingular distribution { }
m
fffspan ,,,
21
K=? is completely integrable if, and
only if, it is involutive. The distribution?is involutive if the Lie bracket [ ]
ji
ff , of any
pair of vector fields
i
f and
j
f belongs to the distribution?, that is,
,??
i
f ??
j
f [ ] ???
ji
ff , (4.1)
where
mi
ff ,,K are smooth vector fields locally spanning?.
4.1.4 Diffeomorphisms and State Trannsformations
A function
nn
RR ?? : , defined in a region ?, is called a diffeomorphism if it is
smooth, and if its inverse
1?
? exists and is smooth. If the region ? is the whole
space
n
R , then )(x? is called a global diffeomorphism, but if the transformations are
45
defined only in a finite neighborhood of a given point, then it is a local diffeomorphism
about the given point.
4.1.5 Controllability
A system is said to be controllable if and only if it is possible, by means of the input,
to transfer the system from any initial state ,)(
00
xtx = to any other state
ff
xtx =)( in a
finite time .0
0
??tt
f
The controllability matrix for a nonlinear system in Eq. (4.12 ) is
given by
[ ]
m
r
f
r
fmffm
gadgadgadgadggC
1
1
1
11
,,,,,,,,
??
= KKK (4.12)
with relative degree, nr ?
4.2 Single-Input Single Output (SISO) System
Consider a single-input single output (SISO) nonlinear system of the form given in
Eq. (4.1).
4.2.1 Relative degree
The system given by Eq. (4.1) is said to have a relative degree r at a particular point
0
x if
0)(
1
=
?
xhLL
i
fg
, for all x in a neighborhood of
0
x 1,,2,1 ?= ri K
0)(
1
?
?
xhLL
r
fg
Intuitively, relative degree is the number of times one has to differentiate the output
function, )(xh to obtain an expression where the input u appears explicitly.
46
4.2.2 Exact linearization
Full feedback linearization or exact linearization is carried out when the relative
degree of the nonlinear system is the same as the dimension of the system, that is nr = .
Consider a SISO (Single-Input, Single-Output) nonlinear system
uxgxfx )()( +=&
)(xhy =
repeatedly differentiating the output
x
x
h
y &&
?
?
=
[]uxgxf
x
h
)()( +
?
?
=
uxhLxhL
gf
)()( += (4.14)
where )()( xf
x
h
xhL
f
?
?
? , )()( xg
x
h
xhL
g
?
?
?
by repeated differentiations of the output ,y r times, we obtain
0)(
1
=
?
xhLL
i
fg
, 1,,2,1 ?= ri K (4.15a)
0)(
1
?
?
xhLL
r
fg
(4.15b)
[ ],)()( xhLLxhLL
i
fg
i
fg
= (4.15c)
[ ])()(
1
xhLLxhL
i
ff
i
f
?
= , 1,,2,1 ?= ri K , (4.15d)
and )()(
0
xhxhL
f
?
uxhLLxhLy
r
fg
r
f
r
)()(
1)( ?
+= (4.16)
47
for reference trajectory
vy
r
=
)(
(4.17)
where the new input v is chosen to cancel the nonlinear dynamics in Eq. (4.16), that is,
)(
)(
1
xhLL
xhLv
u
r
fg
r
f
?
?
= , (4.18)
4.2.2.1 Nonlinear coordinate transformation:
The nonlinear system is transformed to the normal form by r functions ),(xh
),(xhL
f
?, )(
1
xhL
n
f
?
when the relative degree is same as the system dimension, that is,
nr = , and these form a new set of coordinate functions around the point
0
x .
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?=
?
hL
hL
hL
h
xz
n
f
f
f
1
2
)(
M
, (4.19)
)(
1
zx
?
?= exists and is unique
n
Rx?? , so that
21
zz =&
32
zz =&
M
)()()(
1
tuxhLLxhLz
n
fg
n
fn
?
+=& or uzazbz
r
)()( +=& (4.20)
Output
1
zy = (4.21)
48
For reference trajectory
vz
r
=& (4.2)
where
()vzb
za
u +??
?
= ))((
))((
1
(4.23a)
or
()vxhL
xhLL
u
r
f
r
fg
+?=
?
)(
)(
1
1
(4.23b)
4.2.3 Partial linearization
Nonlinear system with relative degree less than the dimension of the system ( nr < )
cannot be fully feedback linearized, but can only be partially linearized. In this case, it
can be transformed into the ?normal form? of the feedback linearization.
4.2.3.1 Nonlinear coordinate transformation:
In the case where the relative degree is less than the system dimension, that is,
nr < , r functions ),(xh ),(xhL
f
?, )(
1
xhL
r
f
?
provide a partial set of new coordinate
functions around the point
0
x . It is possible to find rn? more functions
)(,),(
1
xx
nr
?? K
+
so that
0)( =xL
ig
? nir ??+? 1 and x? around
0
x
The nonlinear system is transformed into the normal form by these functions.
49
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
hL
hL
h
x
r
f
f
1
: Ma , (4.24)
),(
1
??
?
?=x exists and is unique
n
Rx??
Therefore, the new variables areh , hL
f
,K ,
1?r
f
L in ? coordinates, and )(,),(
1
xx
nr
?? K
+
in
? coordinates. The nonlinear system is transformed to the new ),( ?? coordinates, that is,
21
?? =
&
32
?? =
&
M
)(),(),( tuab
r
????? +=
&
()??? ,q=& (4.25)
Output
1
?=y (4.26)
for reference trajectory
v
r
=?
&
(4.27)
hence, choosing the new input as
)(),(),( tuabv ???? += (4.28)
where, )),((),(
1
????
?
?= hLb
r
f
, (4.29a)
)),((),(
11
????
??
?= hLLa
r
fg
(4.29b)
50
The original input can be written as
()vb
a
u +??
?
= )),((
)),((
1
??
??
(4.30a)
or,
()vxhL
xhLL
u
r
f
r
fg
+?=
?
)(
)(
1
1
(4.30b)
The system in Eq. (4.25) is partially linear. The system is decomposed into a linear
subsystem with
th
r -order dynamics and a possibly nonlinear subsystem with
th
rn )( ? -
order dynamics, which has been rendered unobservable, and this part of the dynamics
describes the internal behavior of the system, and it is referred to as the internal
dynamics. It is given by
()??? ,q=&
It is necessary to check the stability of the internal dynamics so as to determine if it
is stable, otherwise, the feedback linearized system is useless. Therefore, the internal
behavior of the system is studied when the input and the initial conditions are chosen so
as to constrain the output to remain identically zero, and this is called the zero dynamics,
and it is given by
()?? ,0q=&
4.3 Multi-Input Multi-Output (MIMO) System
Consider a MIMO (Multi-Input, Multi-Output) nonlinear system of the form given
in Eq. (4.31). In this analysis, it is assumed that the system has the same number, m , of
input and output channels.
51
?
=
+=
m
i
ii
uxgxfx
1
)()(& (4.31a)
)(
11
xhy =
M
)(xhy
mm
= (4.31b)
The outputs can be repeatedly differentiated until one of the outputs appears explicitly. If
i
r is assumed to be the smallest integer, then,
()
?
=
?
+=
m
j
ji
r
fgi
r
f
r
i
uhLLhLy
i
j
ii
1
1
(4.32)
with 0
1
?
?
ji
r
fg
uhLL
i
j
, for at least one j
If at least one of the inputs appears at
j
r differentiation in
j
r
j
y , such that, ,0
1
?
?
j
i
r
fg
LL then
one can define a matrix
mm
RxE
?
?)(, such tat
?
?
?
?
?
?
?
?
?
?
=
??
??
)()(
)()(
)(
11
1
1
1
1
1
11
1
xhLLxhLL
xhLLxhLL
xE
m
r
fgmm
r
fg
r
fg
r
fg
mm
m
L
MOM
L
(4.3)
A system is said to have vector relative degree, },,{
1 m
rr K at
0
x , if
0)( ?xhLL
i
k
fg
i
, 20 ???
i
rk
for, mi ,,1 K= , and the matrix )(
0
xE is nonsingular [sastry].
nrr
m
?++K
1
, and total scalar relative degrees is given by
m
rrr ++= K
1
4.3.1 Nonlinear coordinate transformation:
The normal form for MIMO nonlinear system is obtained based on the functions
),(xh
i
),(xhL
if
?, )(
1
xhL
i
r
f
i
?
generated by the Lie derivatives in Eq. (4.32).
52
i
i
h=
1
?
if
i
hL=
2
?
M
i
r
f
i
r
hL
i
i
1?
=? , for mi ??1
[ ])(,,)(),(,,)(,,),(,,)()(
11
1
1111
1
xxxxxxcolx
mrrmr
mi
?????? KKKK
+
=?
from the differentiations in Eq. (4.32), one can write;
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
mm
r
f
r
f
r
m
r
u
u
xE
hL
hL
y
y
m
MMM
111
)(
1
11
(4.34a)
hence the state feedback control laws are formulated so that the nonlinear dynamics can
be cancelled, and they are written as;
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
??
m
r
m
r
r
f
r
f
y
y
xE
hL
hL
xE
u
u
MMM
1
1
1
1
1
1
1
1
2
1
)()(
or
vxE
hL
hL
xEu
r
f
r
f
)()(
1
1
1
1
1
1
??
+
?
?
?
?
?
?
?
?
?
?
= M (4.34b)
for the reference trajectories
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
m
r
m
r
v
v
y
y
m
MM
11
1
(4.34c)
and these yield the linear closed loop system represented as
ii
21
?? =
&
53
M
i
r
i
r
ii
?? =
?1
&
?
+=
m
j
j
i
ji
i
r
tuab
i
)(),(),( ?????
&
(4.35a)
i
i
y
1
?= (4.35b)
4.4 Application to Panel Flutter Suppression
The technique of linearizing a nonlinear system by feedback control presented in
Section 4.1 is applied to the resulting nonlinear system from the mathematical modeling
of a flat panel with embedded or bonded distributed piezoelectric patches subjected to
both aerodynamic loads and externally applied inplane forces carried out in Chapter 3.
The dynamic analysis reveals that vibrations with large amplitudes exist, and
nonlinearities in the system give rise to limit cycle motions. The amplitudes of the
vibrational modes are sensed by piezoelectric sensors, and these are represented as the
outputs. The inputs are the actuation of the panel by the piezoelectric actuators. The
output signals from the sensors are feedback through the linearizing controllers
developed in this research to the actuators, and these are used to suppress the fluttering of
the panel by placing the poles of the linearized system so they are stable.
4.4.1 Control with first mode as the ouput:
In this section, it is considered that the state-space representation of the panel
fluttering dynamics has a single input signal fed to the actuators distributed over the
surface of the panel and single output. Therefore, the analysis for single-input single-
54
output nonlinear system represented by Eq. (4.1) in Section 4.2 can be applied to the
panel flutter nonlinear dynamics given by Eq. (3.36) and Eq. (3.37), and these presented
as Eq. (4.36) below:
uxgxfx )()( +=& (4.36a)
)(xhy = (4.36b)
where
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????
????
=
2
2
1421
3
24034241
2
21312
3
13303132
4
3
)(
xxcxcxcxcxc
xxcxcxcxcxc
x
x
xf
dkfv
dkfv
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
4
3
0
0
)(
b
b
xg
and the output function chosen is the amplitude of the first mode, and it is given as
1
)( xxhy ==
differentiating the chosen output
0
3
?=bhLL
fg
? relative degree, 2=r
3
3
1330
2
21312132
2
xcxcxxcxcxchL
dkfvf
????=
Since the relative degreer is 2, while the system is a set of four first-order differential
equations, then one can only carry out partial feedback linearization of the system with
the chosen output, therefore, the x coordinates of the original domain becomes ),( ??x in
the transformed coordinates.
55
Using the computed Lie derivative, the normal form is given as:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?
4334
2
3
1
2
1
4
3
2
1
xbxb
x
x
x
hL
h
f
?
?
?
?
?
?
(4.37)
where
3
? and
4
? are defined such that 0
1
=?
g
L and 0
2
=?
g
L .
The Jacobian matrices of the transform and inverse transform are given below, and they
are nonsingular and are well defined, since for any input, .0
3
?b
3
b
dx
d
=
?
Therefore, this system with the chosen output has a transformation that is global
diffeomorphism, and inversion of the coordination transformations can be carried out
globally. The original states are obtained in terms of the linearizing coordinates as given
in Eq. (4.18) below
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
224
3
2
1
1
4
3
2
1
1
??
?
?
?
b
b
x
x
x
x
x (4.38)
The system dynamics for panel flutter given in Eq. (4.36), is partially feedback linearized
and it is represented in the new coordinates as
56
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
????+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????
?
????
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
2
3
1330
2
113121314
3
2
3
24
1
2
14211
3
1403143
224
3
2
3
13301
2
1312131
2
2
1
2
1
1
??????
??
?????
??
??????
?
?
?
?
?
dkfvdfvk
dkfv
cccccb
bb
b
cccccb
b
b
ccccc
&
&
&
&
u
b
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
0
0
0
3
(4.39)
and for reference trajectory:
2
?
&
=v
from eq. 4.19
huLLhL
fgf
+=
2
2
?
&
therefore, the control input u is designed to cancel the nonlinear dynamics, hence
hLL
hLv
u
fg
f
2
?
= ,
( )()
2
3
13301
2
1312131
3
1
??????
dkfv
cccccv
b
u ?????= (4.0)
Output:
1
?=y (4.1)
Substituting Eq. (4.20) into Eq. (4.19), the resulting system is partially linearized with the
linearized subsystem given as;
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
v
0
00
10
2
1
2
1
?
?
?
?
&
&
(4.2)
57
They are made asymptotically stable by pole placements;
?? Kv ?=)( , or
2110
?? kkv ??=
4.4.1.1 Internal dynamics
The internal dynamics are given by the subsystem below:
()
224
3
1
1
??? ?= b
b
&
()
2
3
1330
2
113121314
3
2
3
24
1
2
14211
3
14031432
??????
??
??????
dkfvdfvk
cccccb
bb
b
cccccb ????+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
???????=&
(4.3)
4.4.1.2 Zero dynamics
The zero dynamics are given below:
set {}{}0,0,,,,,
212121
??????? == , hence,
3
2
1
b
?
? ?=&
( )
3
1403214342
???? cccbcb
dkfv
??+=&
(4.4)
The feedback linearized subsystem presented in Eq. (4.42) is a second-order
dynamic system, with the modal amplitude of the first mode and its derivative as the two
new states in the transformed coordinates. The designed controllers are proportional to
the two new states therefore the controlled subsystem simply becomes a damped mass
spring oscillator in the new coordinates in terms of the first mode. The modal amplitude
of the second mode and its derivatives constitute the zero dynamics presented in Eq.
(4.44).
58
4.4.2 Control with second mode as the output:
In this case, the output from the system dynamics is taken as the amplitude of the
second mode of the panel flutter sensed by the distributed sensors attached to the panel at
appropriate locations, and a single input signal fed to the actuators distributed over the
surface of the panel. Therefore, again, the analysis for single-input single-output
nonlinear system in Section 4.2 can be applied also to the system presented in Eq. (4.36).
uxgxfx )()( +=&
)(xhy =
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????
????
=
2
2
1421
3
24034241
2
21312
3
13303132
4
3
)(
xxcxcxcxcxc
xxcxcxcxcxc
x
x
xf
dkfv
dkfv
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
4
3
0
0
)(
b
b
xg
and the output is given as
2
)( xxhy == (4.5)
differentiating the chosen output
0
4
?=bhLL
fg
? relative degree, 2=r
4
3
24032
2
1421241
2
xcxcxxcxcxchL
dkfvf
?????=
The relative degree,r , is 2, while the system is a set of four first-order differential
equation, therefore, one can only carry out partial feedback linearization of the system
59
with the chosen output, therefore, the x coordinate of the original domain
becomes ),( ??x in the transformed coordinates
Using the computed Lie derivatives, the normal form is given as:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=?
4334
1
4
2
2
1
4
3
2
1
xbxb
x
x
x
hL
h
f
?
?
?
?
?
?
(4.6)
where
3
? and
4
? are defined such that 0
1
=?
g
L and 0
2
=?
g
L .
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
2
223
4
1
1
4
3
2
1
1
?
??
?
?
b
b
x
x
x
x
x (4.7)
The Jacobian matrices of the transform and inverse transform are given below, and they
are nonsingular and are well defined, since for any input, .0
4
?b
4
b
dx
d
?=
?
The partial feedback linearized system becomes:
()
()
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+????+??????
+
?????
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
4
2
4
233
13301
2
13121314
2
1142112
3
1403143
223
4
2
3
14031
2
1421141
2
2
1
2
1
1
bb
b
cccccbcccccb
b
b
ccccc
dkfvfvdk
dkfv
??
???????????
??
??????
?
?
?
?
?
&
&
&
&
60
u
b
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
0
0
0
4
(4.8)
reference trajectory:
2
?
&
=v
from eq. 4.48
huLLhL
fgf
+=
2
2
?
&
therefore, input u is designed so that the nonlinear dynamics are cancelled.
hLL
hLv
u
fg
f
2
?
= ,
( )()
2
3
14031
2
1421141
4
1
??????
dkfv
cccccv
b
u ??????=
Output:
1
?=y (4.9)
The linear subsystem of the feedback linearized system is given below:
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
v
0
00
10
2
1
2
1
?
?
?
?
&
&
(4.50)
and by pole placements; ?? Kv ?=)(, or
2110
?? kkv ??=
The new control input v is chosen, so that Eq. (4.50) is linear and in canonic form,
therefore, the linear control gains
0
k and
1
k are designed so that the subsystem is
asymptotically stable.
61
4.4.2.1 Internal dynamics:
The internal dynamics are given by the subsystem below:
()
223
4
1
1
??? += b
b
&
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+????+??????=
4
2
4
233
13301
2
13121314
2
1142112
3
14031432
bb
b
cccccbcccccb
dkfvfvdk
??
????????????&
(4.51)
4.4.2.2 Zero dynamics
The zero dynamics are given below in Eq. (4.52), by setting:
{}{}0,0,,,,,
212121
??????? == ,
hence,
4
2
1
b
?
? =&
( )
3
13304213432
???? cbccbcb
dkfv
???=&
(4.52)
As a reversal to the case in previous section, the feedback linearized subsystem
presented in Eq. (4.50) is second-order, with the modal amplitude of the second mode
and its derivative as the two new states in the transformed coordinates. The designed
controllers are also chosen to be proportional to the two new states. The modal amplitude
of the first mode and its derivatives constitute the zero dynamics presented in Eq. (4.52).
62
4.4.3 Control with both first and second modes as the outputs:
The distributed active sensors embedded in or bonded to the panel are located so
that the amplitudes of both the first and second modes of the fluttering panel are sensed
separately as outputs. Similarly, input signals are fed to the actuators distributed over the
surfaces of the panel so as to independently actuate both the first and second modes of the
panel. This is a case of a multi-input multi-output nonlinear system, and the analysis
presented in Section 4.3 can be applied to the multi-input multi-output active panel
undergoing fluttering given in Eq. (4.31) below:
2211
)()()( uxguxgxfx ++=& (4.53)
)(
11
xhy =
)(
22
xhy = (4.54a,b)
where
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????
????
=
2
2
1421
3
24034241
2
21312
3
13303132
4
3
)(
xxcxcxcxcxc
xxcxcxcxcxc
x
x
xf
dkfv
dkfv
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
41
31
1
0
0
)(
b
b
xg ,
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
42
32
2
0
0
)(
b
b
xg
and the outputs are:
111
)( xxhy ==
222
)( xxhy ==
63
From the computed Lie derivatives, we have
3
2
213122
3
1330131
2
xcxxcxcxcxchL
dfvkf
??+??=
4
3
24032
2
14212412
2
xcxcxxcxcxchL
dkfvf
?????=
The vector relative degree, }2,2{},{
21
=rr , and the total scalar relative degree is 4.
?
?
?
?
?
?
=
4241
3231
)(
bb
bb
xE (4.5)
The coordinate transformations of the x coordinates in terms of the new coordinates? are
given as:
1
11
?=x
2
12
?=x
1
23
?=x
2
24
?=x
The Jacobian matrices of the transform and inverse transform are given below, and they
are nonsingular and well defined.
1?=
?
dx
d
Hence, the coordinate transformation is global diffeomorphism.
The original system dynamics given in Eq. (4.31) are represented in the new coordinates
as:
2
42
32
1
41
31
2
1
21
1421
1
1
2
2
32
1403
2
14
2
2
1
2
22
1
1
1312
2
1
31
1330
1
13
1
2
2
2
2
1
1
2
1
1
0
0
0
0
)()(
)()(
u
b
b
u
b
b
ccccc
ccccc
fvdk
dfvk
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?????
??+??
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??????
?
??????
?
?
?
?
?
&
&
&
&
(4.56)
for reference trajectories:
1
1
2
v=?
&
64
2
2
2
v=?
&
(4.57a,b)
where the new control input are
)()()()()(
2
1
21
1
111
tuatuabv ??? ++=
)()()()()(
2
2
21
2
122
tuatuabv ??? ++=
The state feedback control laws are formulated so that the nonlinearities in eq. (4.56) are
cancelled, and they are given as
()
1
2
22
1
1
1312
2
1
31
1330
1
131
42
1
)()( ??????
dfvk
cccccv
b
u ??+???
?
=
2
1
21
1421
1
1
2
2
32
1403
2
142
32
)()( ?????? cccccv
b
fvdk
??????
?
+ (4.58a)
1
2
22
1
1
1312
2
1
31
1330
1
131
41
2
)()( ??????
dfvk
cccccv
b
u ??+???
?
=
()
2
1
21
1421
1
1
2
2
32
1403
2
142
31
)()( ?????? cccccv
b
fvdk
??????
?
+ (4.58b)
41324231
bbbb ?=?
The feedback linearized models, which are two fully decoupled second-order dynamic
system given in Eq. (4.59) and Eq. (4.60) become
Mode 1:
1
2
1
1
?? =
&
1
1
2
v=?
&
, (4.59a)
where
1
2
1
1
1
1
1
01
?? kkv ??= by pole placements (4.59b)
The first output
1
11
?=y
65
Mode 2:
2
2
2
1
?? =
&
2
2
2
v=?
&
, (4.60a)
where
2
2
2
1
2
1
2
02
?? kkv ??= by pole placements (4.60b)
The second output
2
12
?=y
Once again, just as in the previous two sections, the new inputs are chosen so that
each resulting mode is asymptotically stable. One can observe that there is no internal
dynamics when the original multi-input multi-output nonlinear system is employed
because the original system given in Eq. (4.53) and Eq. (4.54) is fourth-order with two
control inputs having total scalar relative degree of 4, that is, .nr =
66
5. NUMERICAL SIMULATIONS
A simply supported rectangular panel with two piezoelectric layers segmented into
rectangular patches is used for the numerical simulations. The panel is an aluminum
panel, while the piezoelectric layers are lead zirconate titanate (PZT) ceramics. It is
assumed that the PZT patches are perfectly and symmetrically bonded to the rectangular
aluminum panel to form an active panel. The geometry and the material properties of the
intelligent panel are given in Table 5.1.
The mathematical model of the active panel dynamics is presented in Chapter 3.
This model, which is represented as a set of modal nonlinear differential equations,
accounts for various forces acting on the intelligent panel including aerodynamic loads,
externally applied in-plane loads and electrical displacements. The aerodynamic loads,
which are represented by the nondimensional dynamic pressures, induce instability of the
intelligent panel resulting in panel flutter with associated limit cycle motions. The
electrical displacements produced the actuation of the piezoelectric ceramics that are used
to suppress the limit cycle motions through output feedback linearizing control developed
in this research. The feedback linearizations transform the nonlinear models to simple
Brunovsky canonical forms.
67
Table 5.1 The geometrical and the material properties of the active panel
Host layer Actuator
Material Aluminum Lead zirconium titanate
Length (in.) a : 12.0
p
x : a1.0
Width (in.) b : 12.0
p
y : b6.0
Thickness (in.)
s
h : 0.05
p
h : 0.005
Mass density (Ib-
sec
2
/in
4
)
s
? :
3
102588.0
?
?
p
? :
3
107101.0
?
?
Young?s modulus (psi)
s
E :
6
104.10 ?
p
E :
6
100.9 ?
Poisson?s ration
s
? : 0.3
p
? : 0.3
Charge constant (in./v) -
31
d :
9
10478.7
?
??
Charge constant (in./v) -
32
d :
9
10478.7
?
??
Charge constant (in./v) -
36
d : 0
Coercive Field (v/in.) -
max
e : 15243
68
A Runge-Kutta integration scheme was used to simulate the modal nonlinear
models. The integration time step was chosen to be about one tenth of the smallest period
of the normal modes, that is, .0015.0=?? Initial conditions were chosen arbitrarily, but
the same values were used for all the simulations. Generally, any chosen initial
conditions still result in limit cycle motions. Panel flutter with associated limit cycle
motions were obtained by the integrations, and the suppression of these limit cycle
motions were demonstrated by activating the controllers at specified time. Two linear
normal modes were used to model panel flutter in this research. The calculations were
conducted in time domain.
The PZT patches were used as both actuators and sensors simultaneously. These
patches were activated independently, so that the motions of the panel were sensed and
actuated at desired locations on its surfaces, therefore, the active panel was controlled by
single input and multi-input signals through these actuators. Three cases were considered:
the first case was when the first mode was sensed as output signal, and it was shown in
Fig. (5.1), the second case was when the output signal of the second mode was sensed as
shown in Fig. (5.2), and the third case was when both the first and the second modes were
sensed as shown in Fig. (5.3). The output signals were modified and fed back through the
actuators as shown above.
69
Fig. 5.1 A simply-supported plate showing the actuators for the first mode
a
b
70
Fig. 5.2 A simply-supported plate showing the actuators for the second mode
a
b
71
Fig. 5.3 A simply-supported plate showing the actuators for first and second modes
a
b
72
5.1 Limit-Cycle Motion of the Panel Flutter
The panel flutter is induced by the aerodynamic pressure on one side of the panel.
The critical dynamic pressure is calculated by eigenvalue analysis of the linear system,
and this is given by its open-loop roots.
Based on the panel dynamics, the aerodynamic pressure affects both the damping
terms and the flow coupling terms in the system. The panel exhibits free oscillations
when there is no aerodynamic pressure, and there are no other damping and nonlinear
effects in the system. Linear eigenvalue analysis shows that it has purely imaginary
eigenvalues. The application of aerodynamic pressure introduces both damping and flow
coupling terms, so the panel exhibits damped oscillations, and the system has complex
eigenvalues with negative real parts, which lead to the decay of the oscillation of the
panel. As the dynamic pressure (? ) is increased, the rate of decay increases until it
reaches a critical point, at which there exists a pair of purely imaginary eigenvalues, with
the other eigenvalues having negative real parts, and this signifies the onset of panel
flutter. At this critical point, the dynamic pressure is called the critical dynamic pressure
( 385=
cr
? ), and the system becomes critical. Beyond this critical point, the pair of
purely imaginary eigenvalues becomes eigenvalues with positive real parts, the motion of
the panel diverges, and the system becomes unstable by linear analysis and the amplitude
of the panel deflection diverges, but the structural nonlinearity due to the effect of the in-
plane stretching forces becomes significant and acts as a restoring force, and the
amplitude stays at a certain value with limit-cycle motion of the panel, and fluttering of
the panel is sustained.
73
In [54], six linear normal modes were used for numerical analysis, but the critical
dynamic pressures using two to six linear normal modes were presented. While the
critical dynamic pressure obtained is 515 for six linear normal modes, it is 385 for two
normal linear modes. Although, four or six linear modes are required for obtaining a
converged limit-cycle amplitude and frequency [13], several research works have been
presented with two normal modes [10, 16, 17, 21, 27].
The model was run with the dynamic pressure set to 1,500, which is about 3.9 times
the critical dynamic pressure. An aerodynamic damping coefficient of 01.0=
a
c was
used. Fig. (5.4) shows the deflection profile of the mid-span of the panel in the flow
direction at a specific instant of time. The position of the maximum deflection, ,
max
w of
the panel is at about 68.5% of the panel length. The time history of the deflection of the
position of maximum deflection is shown in Fig. (5.5a), and it reflects the panel flutter
that is taking place, and the existence of limit cycle motion is shown in Fig. (5.5b).
74
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
x/a
w/
h
Fig. 5.4 Panel deflection of a simply-supported plate at the mid-span in the flow
direction
75
0 1 2 3 4 5 6
-4
-3
-2
-1
0
1
2
3
4
Nondimensional Time
w/
h
Fig. 5.5 Time history of uncontrolled panel deflection, at 1500=? and 0=
m
x
R .
76
-4 -3 -2 -1 0 1 2 3 4
-250
-200
-150
-100
-50
0
50
100
150
200
250
w/h
w
d
o
t/h
Fig. 5.6 Phase plot of uncontrolled panel deflection, at 1500=? and 0=
m
x
R .
77
5.2 Suppression of Panel Flutter due to Aerodynamic Load only
At a dynamic pressure, 1500=? , panel flutter limit cycle motions are obtained first,
and then the controllers are activated to suppress them at a selected time. In order to
suppress the panel flutter limit cycle motion, a closed-loop system with feedback
linearization controllers developed in this research were used. The linearized systems in
the transformed coordinates were in canonical controllable forms; hence, the pole-
placement techniques were used to select the control gains, such that the roots of the
closed-loop systems were entirely in the left half of the complex plane, hence, the
feedback linearized system becomes asymptotically stable. The control inputs are the
electric fields, generated by the electric potentials applied on the PZT patches. There are
maximum allowable electric fields, above which depolarization of the piezoelectric
property takes place, but that is not one of the objective of this research.
The PZT patches sense the magnitude of the output of the first mode, second mode,
or both first and second modes of the limit cycle motions of the panel flutter. These are
the three cases shown in Fig. (5.1-3). In each case, the selected output is fed to the
controllers that modify the signals, and is fed back to the actuators, and this actuates the
panel so that the magnitudes of limit cycles are suppressed, until the sensor senses no
deflection of the panel from the equilibrium.
For the three cases, plots of the zero dynamics, plots of the time histories for the
panel at the position of maximum deflection, plots of the normalized control inputs, and
phase plots are shown in Fig. (5.7 ? 14). The zero dynamics for the single-input nonlinear
systems show that they are asymptotically stable. The limit cycle motions are suppressed
78
and the selected point stabilized at the undeflected point, except for the second case,
where the second mode is used as the output, the selected point stabilizes at a new
equilibrium. See Fig. (5.10) and Fig. (5.11) for the phase plot of the zero dynamics and
plot of time history, respectively. This is confirmed by placing the poles of the closed-
loop system at other various locations.
79
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
u1
u2
Fig. 5.7 Phase plot of the zero dynamics for the panel at 1500=? and 0=
m
x
R , using the
first mode as the output.
80
0 5 10 15
-4
-2
0
2
4
6
Nondimensional Time
w/
h
0 5 10 15
-0.5
0
0.5
1
Nondimensional Time
N
o
rm
al
i
z
ed C
ont
rol
I
nput
Fig. 5.8 Time history of panel deflection and control effort with feedback linearization
controller, at 1500=? and 0=
m
x
R , using the first mode as the output.
81
-4 -3 -2 -1 0 1 2 3 4 5
-300
-200
-100
0
100
200
300
w/h
w
d
o
t/h
Fig. 5.9 Phase plot of the panel with feedback linearization controller, at 1500=? and
0=
m
x
R , using the first mode as the output.
82
-4 -3 -2 -1 0 1 2 3 4
-1.5
-1
-0.5
0
0.5
1
1.5
u1
u2
Fig. 5.10 Phase plot of the zero dynamics for the panel at 1500=? and 0=
m
x
R shows a
new equilibrium, when the second mode is the output.
83
0 5 10 15
-4
-2
0
2
4
Nondimensional Time
w/
h
0 5 10 15
-1
-0.5
0
0.5
Nondimensional Time
N
o
rm
al
i
z
ed C
ont
rol
I
nput
Fig. 5.11 Time history of panel deflection and control effort with feedback linearization
controller, at 1500=? and 0=
m
x
R , using the second mode as the output.
84
-4 -3 -2 -1 0 1 2 3 4
-300
-200
-100
0
100
200
300
w/h
w
d
o
t/h
Fig. 5.12 Phase plot of the panel with feedback linearization controller, at 1500=? and
0=
m
x
R , using the second mode as the output.
85
0 5 10 15
-4
-2
0
2
4
Nondimensional Time
w/
h
0 5 10 15
-1
-0.5
0
0.5
1
Nondimensional Time
N
o
r
m
al
i
z
ed C
ont
r
o
l
I
nput
s
input 1
input 2
Fig. 5.13 Time history of panel deflection and control efforts with feedback linearization
controller, at 1500=? and 0=
m
x
R , using first and second modes as outputs.
86
-4 -3 -2 -1 0 1 2 3 4
-500
0
500
1000
1500
2000
2500
w/h
w
d
o
t/h
Fig. 5.14 Phase plot of the panel with feedback linearization controller, at 1500=? and
0=
m
x
R , using first and second modes as outputs.
87
5.3 Suppression of Panel Flutter due to Combined Aerodynamic and Externally
Applied In-plane Forces
In this section, it is considered that an externally applied in-plane load is on the
panel with the aerodynamic load. The latter is set at a dynamic pressure, ,380=? and the
former is set at a normalized in-plane load ,
2
??=
x
R that is, a compressive load. At this
dynamic pressure, without the externally applied load, the panel is stable, that is, there is
no panel flutter, but the applied in-plane load causes the panel to flutter at lower dynamic
pressure. For this condition, the critical dynamic pressure
cr
? is 325. In order to suppress
panel flutter limit cycle motion due to these conditions, the same controllers with the
same closed-loop roots pole placed as in the previous section are employed. These poles
can be placed in different places for better quality of suppression.
The panel flutter in each case is suppressed. Plots of the zero dynamics, plots of the
time histories, plots of the normalized control, and the phase plots are shown in Fig.
(5.15-22). The time histories show that the point of maximum deflection stabilizes at the
undeflected point, except for the second case again, where the second mode is used as the
output. For the second case, the selected point stabilizes at a new equilibrium, although
this is not revealed in Fig. (5.16), but it can be observed with higher value of dynamic
pressure.
88
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
u1
u2
Fig. 5.15 Phase plot of the zero dynamics for the panel at 380=? and
2
??=
m
x
R , using
the first mode as the output.
89
0 5 10 15
-1
-0.5
0
0.5
1
Nondimensional Time
w/
h
0 5 10 15
-1
-0.5
0
0.5
Nondimensional Time
N
o
rm
al
i
z
ed C
ont
rol
I
nput
Fig. 5.16 Time history of panel deflection and control effort with feedback linearization
controller, at 380=? and
2
??=
m
x
R , using the first mode as the output.
90
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
-30
-20
-10
0
10
20
30
w/h
w
d
o
t/h
Fig. 5.17 Phase plot for the panel with feedback linearization controller, at 380=? and
2
??=
m
x
R , using the first mode as the output.
91
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
-6
-4
-2
0
2
4
6
8
10
12
14
x 10
-3
u1
u2
Fig. 5.18 Phase plot of the zero dynamics for the panel at 380=? and
2
??=
m
x
R shows
a new equilibrium, when the second mode is the output.
92
0 5 10 15
-1.5
-1
-0.5
0
0.5
1
1.5
Nondimensional Time
w/
h
0 5 10 15
-1
-0.5
0
0.5
1
Nondimensional Time
N
o
rm
al
i
z
ed C
ont
rol
I
nput
Fig. 5.19 Time history of panel deflection and control effort with feedback linearization
controller, at 380=? and
2
??=
m
x
R , using the second mode as the output.
93
-1.5 -1 -0.5 0 0.5 1 1.5
-30
-20
-10
0
10
20
30
w/h
w
d
o
t/h
Fig. 5.20 Phase plot for the panel with feedback linearization controller, at 380=? and
2
??=
m
x
R , using the second mode as the output.
94
0 5 10 15
-1
0
1
2
3
4
Nondimensional Time
w/
h
0 5 10 15
-1
-0.5
0
0.5
1
Nondimensional Time
N
o
r
m
al
i
z
ed C
ont
r
o
l
I
nput
s
input 1
input 2
Fig. 5.21 Time history of panel deflection and control efforts with feedback linearization
controller, at 380=? and
2
??=
m
x
R , using first and second modes as outputs.
95
-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
-30
-20
-10
0
10
20
30
w/h
w
d
o
t/h
Fig. 5.22 Phase plot for the panel with feedback linearization controller, at 380=? and
2
??=
m
x
R , using first and second modes as outputs.
96
6. CONCLUSIONS AND RECOMMENDATIONS
6.1 Conclusions
Feedback linearization is based on nonlinear control theory, and it has been used in
this study to transform the nonlinear panel flutter problem into an equivalent controllable
linear problem that can be written in simple Brunovsky canonical form by the chosen
outputs. This takes into account the nonlinear characteristics of panel flutter dynamics in
the design of nonlinear feedback controllers. Nonlinear feedback control laws are
developed and used to cancel the nonlinear dynamics resulting in a linear problem. The
pole placement technique is then employed so as to make the states of the feedback
linearized model locally asymptotically stable at a given equilibrium.
Using this approach of feedback linearization, nonlinear dynamic equations of an
intelligent panel subject to aerodynamic loads with or without externally applied in-plane
load are transformed into linear equations in the new coordinates. This intelligent plate
has piezoelectric actuators and sensors symmetrically bonded to its surfaces. The
piezoelectric actuations of the piezoelectric layers enable the plate to actively respond to
external stimuli that cause large deflections and instability resulting in the failure of the
panel due to fatigue. With this development, advanced aircraft or vehicles and surfaces in
a fluid medium can operate in supersonic environments by the use of this intelligent
panel.
97
The nonlinear dynamic are nonlinear coupled partial differential equations obtained
from von K?rman large-deflection plate theory accounting for the structure nonlinearity,
and reduced to nonlinear modal equations using two normal modes by Galerkin?s method
with modal expansion. The nonlinear modal equations are transformed to state-space
format, using the amplitudes of the modes and their derivatives as the states, and
presented as a nonlinear control system. Linear panel flutter analyses are carried out to
determine the critical dynamic pressures
cr
? at which there are onsets of panel flutter limit
cycle motions. At dynamic pressures above the critical dynamic pressure, limit cycle
motions are considered large, therefore nonlinear panel flutter analysis is employed. The
piezoelectric actuation of the active panel drives the actuators to suppress the panel flutter
associated limit cycle motions, and it is carried out by the piezoelectric bending moment
generated by the electric field, which is considered as the control input, and it is applied
on the actuators.
In selecting the output to linearize the nonlinear control system, three outputs are
considered, and these are the first mode, second mode, and both first and second modes.
These are three cases for which numerical simulations are carried out. The closed-loop
systems for the first two cases are classified as single-input single-output nonlinear
systems, and only partial feedback linearization is carried out, therefore, there are internal
dynamics, which are established to be locally asymptotically stable. In the third case, the
closed-loop system is classified as multi-input multi-output nonlinear system, and full
feedback linearization is carried out, thus, in this case, there are no internal dynamics.
The closed-loop systems for the three cases are numerically simulated at much
higher dynamic pressures than the critical dynamic pressures so that limit-cycle motions
98
are generated. The simulated systems show that the closed-loop systems based on the
controllers effectively suppress panel flutter limit cycle motions with the generated
piezoelectric bending actuations as control inputs. Therefore, with the feedback
linearization controllers developed, the limit cycle motion of panel flutter can be
completely suppressed if the controller gains are carefully selected.
The flutter free dynamics are also achieved if the actuators are activated before the
critical dynamic pressure is reached, therefore, the dynamic pressure of the panel can be
allowed to exceed the critical dynamic pressure
cr
? without flutter. This approach is
practically more feasible than the suppression of limit-cycle motions, when aircraft wing
or air vehicle surface is loaded with aerodynamic loads.
6.2 Recommendations
Based on the studies carried out in this research, there are ample opportunities to
improve and extended the effort here, and some of these are highlighted below:
The technique used in this study can be used to linearize the nonlinear dynamics for
panel flutter reduced to nonlinear modal equations based on Galerkin?s method with
modal expansions using five or six linear modes.
In this research, feedback linearization has shown a promising opportunity to
develop a flutter free intelligent panel, and this provides tremendous opportunity for
aeroservoelasticians in terms of research and development of aircraft wings with superior
performance in a supersonic environment. Therefore, it is necessary that a physical
system be built, with the analysis in this study and other future analytical works used as
benchmarks.
99
In the present effort, the maximum allowable electric field that can lead to
depolarization of the piezoelectric ceramics is not considered as a limitation, but for
practical system, it is. Therefore, optimal control technique can used to design the
controllers for the feedback linearized system.
The mathematical model of panel flutter is idealistic, and the actual system
possesses uncertainties, therefore, there is a need to compensate for the uncertainties in
the system by designing adaptive and robust controllers.
100
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107
APPENDICES
The expressions of )6,,1( K=iC
i
of eq. 3 are given in [13] as;
()[ ]
?
?
+
?
m
b
a
ma
m
C
2
2
2
2
1
1 ?
?
()[ ]
?
?
+
?
m
b
a
ma
m
C
2
2
2
2
2
1 ?
?
[ ]),(),(),({
2
3
nrmsnrmsmsraaaC
rs
msr
m
++??+?
???
???
[ ]}),(),(),( nrmsnrmsms +????+ ???
where
()[]
2
2
4)(
)(
),(
b
a
ms
msm
ms
++
?
=?
()[]
2
2
4)(
)(
),(
b
a
ms
msm
ms
+?
+
=?
=),( ms? if 0== ms
= if 0?= ms
= ifms ?
[ ]),(),(),(){(
2
4
rnmsnrmsmsmsraaaC
rs
msr
m
?+++++?
???
???
[ ]}),(),(),()( rnmsnrmsmsms ??++??+ ???
where
=),( ns? if 0== ns
= if 0?= ns
= ifns ?
108
[ ]),(),(),(){(
22
5
nrmsnrmsmsmsraaaC
rs
msr
m
++??++?
???
???
[ ]}),(),(),()(
2
nrmsnrmsmsms +?????+ ???
[]),(),({
2
6
nrmsnrms
ms
m
raaaC
rs
msr
m
++??+
+
?
???
??
[ ]}),(),(),( nrmsnrmsms +????+ ???
where
ms
m
ms
?
=),(? if ms ?
0= if ms =