In addition, the input and the output are related, especially for the feedback, by: In the above equation, [G] is the gain matrix of size n r, when r states are chosen for input feedback
Trang 1That is, in the shortened form, these equations can be
written as:
f_xg ¼ ½A fxg þ ½B f
Equation (9.21) is the state-space representation of
Equation (9.12), wherein the right-hand side has
deriva-tives of the forcing function
One can now obtain the transfer function of the system
from the state equation (Equation (9.21)) This can be
done if one takes the Laplace transform of Equation
(9.21), that is:
sf^xðsÞg fxð0Þg ¼ ½A f^xðsÞg þ ½B ^fðsÞ
^yðsÞ ¼ ½C f^xðsÞg þ D^fðsÞ ð9:22Þ
Here,f^xðsÞg and ^fðsÞ are the Laplace transform of the
state vectorfxðtÞg and the forcing function f ðtÞ Transfer
functions are normally derived by assuming a zero initial
condition From the first part of Equation (9.22), we
have:
f^xðsÞg ¼ ½s½I ½A 1½B ^fðsÞ ð9:23Þ
Using the above in the second part of Equation (9.22), we
can relate the output to the input, that is, the transfer
function is given by:
^yðsÞ
^fðsÞ¼ GðsÞ ¼ ½C ½s½I ½A
1½B þ D ð9:24Þ
That is, the transfer function computation involves
com-putation of ½s½I ½A 1 Hence, the determinant of
matrix½s½I ½A will give the characteristic polynomial
of the transfer function and the eigenvalue of matrix½A
will give the poles of the system
Let us now consider a simple single degree of freedom
of the spring–mass vibratory system, the governing
differential equation of which is given by:
m€xþ c_x þ kx ¼ f ðtÞ ð9:25Þwhere m is the mass of the system, c is the viscously
damped damper coefficient and k is the stiffness of the
system For state-space representation of the system, we
define the state variables x1ðtÞ ¼ xðtÞ and x2ðtÞ ¼ _xðtÞ
Using these state variables, Equation (9.25) reduces to
the following two first-order equations (state equations),written in the matrix form as:
_x1_x2
( )f
ms2þ cs þ k ð9:27ÞThis is the same as what was derived in Equation (9.7),obtained by taking a Laplace transformation on thegoverning equation
In designing controllers for multi-input multi-outputsystems, especially for structural applications, one willhave to depend extensively on the discritized mathema-tical model as that derived from FE techniques Thediscritized Finite Element governing equation of anystructure is of the form:
½M f€xg þ ½C f_xg þ ½K fxg ¼ ff g ð9:28ÞHere, [M], [C], and [K] are the mass, damping andstiffness matrices, respectively
These matrices are of size n n:fxg is the degree offreedom vector andff g is the force vector, both of whichare of size n 1 The above equation is similar to thesingle-degree-of-freedom equation (Equation (9.25)) andthe state space equation will be of the form of Equation(9.26) Hence, the state vectors for the FE equation are
fx1g ¼ fxg and fx2g ¼ f_x1g The form of Equation (9.28) and its corresponding outputvectorfyg is given by:
reduced-state-space-f_x1gf_x2g
Trang 2and [C], respectively Equation (9.29) represents a
2n 2n system That is, an n n second-order system
(Equation (9.28)), when reduced to state-space form,
becomes 2n 2n of the first-order system In addition,
the input and the output are related, especially for the
feedback, by:
In the above equation, [G] is the gain matrix of size
n r, when r states are chosen for input feedback to
reduce the response, especially for vibration control
applications Using the second part of Equation (9.29)
in the above equation, we can write the output–input
relation in terms of the state vector as:
ff g ¼ ½G ½C fxg ð9:31ÞOnce we reduce the governing equation in the state-space
form, and using Equation (9.26), one can determine the
transfer function However, normally, the system size of
the FE system is quite high, especially for dynamic
systems In order to design the control system, it is
practically impossible to consider the entire FE system
due to its large system size In most control applications
to structural problems, such as vibration or noise control,
only the first few modes are targeted for reduction based
on their energy content In such a situation, one has to
reduce the order of the system using suitable reduction
techniques The concepts of dynamic reduction are
addressed in the latter part of this chapter
9.3 STABILITY OF CONTROL SYSTEM
A control system design should adhere to some basic
concepts that ensure the stability of the system In this
section, some of the commonly used methods in
deter-mining the stability of the system are highlighted
An engineer’s definition of stability is that a system
should have enough damping to damp out all of the
transients and resumes a steady-state condition That is, a
system is said to be stable if a finite duration input causes
a finite duration response On the other hand, a system is
said to be unstable if a finite duration input causes the
response to diverge from its initial value That is, when
the output changes ‘unidirectionally’ and ‘shoots up’
with ever increasing amplitude, the system is said to be
unstable
Here, let us consider a linear system Most systems we
come across are differential equations, second-order in
time, and in most cases are equations with constantcoefficients One of the fundamental features of constantcoefficient equations is that they have exponential solu-tions of the form:
yðtÞ ¼ Aer 1 tþ Ber 2 tþ Cer 3 tþ Der 4 tþ ð9:32Þ
In the above equation, the constants A, B, C, etc aredetermined by using the initial conditions and the forcingfunctions; r1; r2, etc are the roots or eigenvalues of thecharacteristic polynomial The stability of Equation(9.32) depends on the values of r If these are negativeand real, then the output tends to zero value as t) 1.Such a system, where all of the r’s are negative and real,
is said to be a stable system If the roots of thecharacteristic polynomial are positive and real, then theoutput of Equation (9.32) grows without a bound as
t) 1 Such a system is said to be an unstable system
If all of the r’s are purely imaginary, then the systemexhibits continuous oscillations due to the presence ofsine and cosine terms in the output equation Finally, ifall of the r’s are complex, having both real and imaginaryparts, it amounts to attenuation of the response due to agrowth in time Hence, the determination of the stability
of the system amounts to determination of the roots ofthe characteristic polynomial In terms of the complexvariable s, a system is said to be stable if all of the rootsare in the left half of the s-plane and unstable if any rootsare on the imaginary axis or in the right half of thes-plane
If the system is linear, then testing of the stability ofthe system amounts to determining whether any root is inthe right half of the s-plane or on the imaginary axis Thefollowing are the different methods of testing the stability
(4) Root Locus method
(5) Using the state-space or transfer-function approach.The choice of using the above tests is ‘problem-dependent’ We will now briefly describe the abovemethods in a few sentences The reader is advised torefer to Kuo[2] for a detailed account of these methods
In the first test, a characteristic polynomial of order n
is first obtained and its roots are determined numerically.There are many elementary root-finding algorithms, such
as the Newton–Raphson technique, bisection method,
Trang 3secant method, etc For complex differential equations,
some of the more recent techniques, such as the
compa-nion matrix method or polynomial eigenvalue method,
can be used These are discussed in Chakraborty [3] In
Finite Element terminology, an n degree of freedom
model will yield a characteristic polynomial of order n
If n is very large, as in the case of the transient dynamic
problem, then solving for all poles from the characteristic
polynomial is an ‘horrendous’ task Hence, the system
size of the FE equations is reduced using proper model
order reduction
The Routh–Hurwitz criterion test gives us the number
of roots if any of these exist to the right of the s-plane It
does not give the location of these roots on the s-plane
and hence does not give any guidance for design
proce-dures It can be conveniently used for lower-order
systems and is relatively simple to implement
The Nyquist criterion [4] helps in identifying the poles
that are located on the right half of the s-plane This is a
frequency-domain technique that is based on conformal
mapping and complex variable theory The method
involves plotting the open-loop Frequency Response
Function (FRF) and looking at the frequency amplitude
at the resonant frequencies From this, one can infer on
the stability of the system The main advantage of this
criterion is that one can modify the control design by
reshaping the frequency-response plots
The root locus is again a graphical method, wherein
the curves are constructed in the s-plane that show the
response of each root of the characteristic polynomial as
a specified system parameter is varied Using this
method, it is possible to evaluate the root location for a
given value of the system parameter and also establish
the conditions for stability Again, due to the graphical
nature of the method, design procedures can be
devel-oped based on reshaping of the curves
In the state-space approach, the eigenvalues of the
state matrix [A] (see Equation 9.21) will give the poles in
the s-plane from which the stability of the system can be
assessed From the FE point of view, this method is ideal
As a part of the FE code, there are many eigenvalue/
vector extraction routines, which are used in free/forced
vibration analysis These routines can be used to extract
the pole information from the state matrix [A]
There are two other terms that are normally used in the
control theory as regards the stability of the system
These are Controllability and Observability These
terms are commonly used in the control theory as they
play an important role in the design of controllers,
particularly when using the state-space approach These
were introduced by Kalman A system is said to be ‘not
controllable’ if it does not satisfy the controllability andobservability conditions Hence, some conditions arespecified in terms of the control parameters, which asystem is made to satisfy for if it is to become con-trollable and observable These conditions can be derived
by using the following definitions A system is said to becontrollable at some time t0if it is possible to transfer thesystem from an initial state xðt0Þ to any other state in afinite interval of time by using an unconstrained controlvector A system, which is in the state xðt0Þ, is said to beobservable at some time t0, if it is possible to determinethis state from observation of the output over a finiteinterval of time
Using the above definition, we can derive the tions for both input and output controllability Here, a
condi-‘mere’ condition is stated without going into much detail.Let us consider the governing differential equation oforder n in the state-space form given in Equation (9.21).The condition of controllability of the input is that thevectors½B ; ½A ½B ; ; ½A n1½B are linearly independentand the matrix is given by:
½B ½A ½B : : : ½A n1½B
ð9:33Þwhich is of rank n or is not singular Similarly, we canstate the condition of output controllability of the stateequation given by Equation (9.21) in a similar manner.That is, we can write the output controllable matrix as:
½C ½B ½C ½A ½B ½C ½A 2½B : ½C ½A n1½B ½D
ð9:34ÞThe above matrix is of the order m ðn þ 1Þr, wherematrix [A] is n n, vector [B] is n r, [C] is m p, where m is the number
of ‘master’ dofs and p is the number of modes retained in
the transformation To and Ewins [10] discussed the
computation of a generalized inverse for a rectangular
matrix, which is given by:
As discussed by O’Callahan [9], this method allows an
arbitrary selection of the modes that are to be selected in
the ROM and the quality of the ROM does not depend
upon the location of the ‘master’ dof However, the
number of modes included in the transformation should
be more than or equal to the number of ‘master’ dofs In
addition, the frequencies and modes shapes of the ROM
are exactly the same as those of the selected frequencies
and mode shapes of the full-system model This is one of
the great advantages in the design of control systems,
wherein one has to design the same by using a limited
number-of-degrees-of-freedom model Since the reduced
mathematical model based on the SEREP can exactly
represent the dynamic characteristics of the full model,
the control theory tolerances are greatly enhanced
In addition to the above three ROM techniques, there
are three other techniques reported in the literature These
are the Condensation Modal Order Reduction Technique,
based on the Projection Operator, proposed by Dyka et
al [11] and referred to as the CMR method, the Improved
Reduced System of O’Callahan [12], referred to as the
IRS method and the Dynamic Improved Reduced System
of Friswell et al [13], referred to as the DIRS method
The above three methods are not discussed here, although
some of the results from these methods are used in the
next subsection for comparison purposes
9.5.1.4 Reduced order modeling in transient dynamics:
a comparative studyThe main objective of this section is to identify thereduction technique that results in the most accurateresponse for the given master–slave dof configuration
It was explained previously that the characteristics of thetransient dynamic problem is that the frequency content
of the forcing function is quite high In other words, thetime duration is very short, normally of the order ofmicroseconds Hence, it excites all higher-order modes.This results in very fine FE discritization and hence avery large system size Thus, when using an ROM, onehas to be very careful in choosing the master–slave dofcombination
For comparative study of different ROMs, a 2-D lever beam under plane-stress conditions and subjected toaxial impact, shown in Figure 9.1 is considered Thedimensions of the beam are 500 mm 6:0 mm 9:0 mmand the isotropic material properties are E¼ 72:0 GPa,
canti-n¼ 0:3, and r ¼ 2700 kg=m3 The time history and thefrequency spectrum of the applied load is shown inFigure 9.2 and the load is acting axially at the freeend of the cantilever beam The full system model is
40 60 80 100
Figure 9.2 Input load history and its frequency spectrum used
in the comparative study.
Trang 10descritized based on the wavelength consideration The
full system matrix is of the order 624 624 For
compar-ison of the response, three reduced-order models, namely,
the Dynamic Condensation (DC), Dynamic IRS (DIRS)
and SEREP are considered and the response is computed
for the same ‘master–slave’ dof configuration for all of the
methods Two different patterns of ‘master–slave’ dof
configurations are used in the investigation, which form
the reduced-order system matrices of order 150 150 and
50 50, respectively The configurations indicating the
spatial distribution of the ‘master–slave’ dof for the two
patterns are shown in Figure 9.3 The locations of the
‘master’ dofs at the nodes are shown by * marks
The axial velocities are plotted at the middle node of
the free end of the beam For pattern 1 with 150 dof, the
axial velocity plot is given in Figure 9.4 In this case, the
location and amplitude of the incident and first reflection
of the wave are accurately captured for all of the
reduction methods; however, the response through
SEREP is observed to be able to capture even a small
dispersion exhibited by the longitudinal wave and the
results match exactly with the full-system response Thenumber of modes included in the transformation in thecase of SEREP is equal to the number of ‘master’ dofs,that is, 150 The condensing frequency used in the DCmethod is the fundamental frequency of the system(185.09 Hz) Figure 9.5 gives a comparison of the samefor the pattern-2 (dof 50) configuration of the ‘master–slave’ dof In this case, the response by SEREP matchesaccurately with that of the full-system response, but theresponse histories by DC and DIRS have shown sometime lag in the occurrences of the reflected pulses, whilethere is no such time lag observed for the incident pulse.That is, the other two ROMs under-predict the axial wavevelocity In addition, a slight under-prediction ofresponse and perturbation is observed in the cases of
DC and DIRS The accurate matching of the response inthis case for the ROM through SEREP can be explained
by the fact that the first few eigenmodes carry maximumspectral energy, which can be observed by the FFTdiagram of the load history, as shown in Figure 9.2 andSEREP can be said to work excellently with inclusion ofthe eigenmodes that carry maximum energy
dof) for different ROMs.
Full-order response ROM (SEREP) ROM (DIRS) ROM (DC)
0.15 0.1 0.05 0 –0.05 –0.1
Trang 11Next, the transverse loading is considered For the
same cantilever beam and for two different
‘master-slave’ configurations, the free-end transverse responses
are plotted in Figures 9.6 and 9.7, respectively The ROM
simulation of wave propagation for the transverse
excita-tion is observed to be more sensitive to the order of
reduction and the ‘master–slave’ dof configuration For
the case of the ROM of order 150 (pattern 1), all of the
methods are observed to give a slight decay in the
amplitude of the transverse velocity corresponding to
the occurrence of the incident pulse
For the case of the ROM of order 50 (pattern 2),
SEREP gives the most accurate solution, which matches
exactly with the full-system response after the
occur-rence of the incident pulse peak DC and DIRS result in
an oscillatory response and a slight perturbation from the
original system response at a longer time range In all of
the simulations discussed above, for SEREP the number
of modes is always taken equal to the number of ‘master’dofs, while for DIRS and DC the condensing frequency istaken as the fundamental frequency of the system Fromthe above example, it is quite clear that the SEREPmethod is perhaps the best ROM from the computationalviewpoint That is, whatever the modes that are retained
in the transformation, those modes are accurately sented in the ROM This is particularly useful in multi-modal control, wherein the modes that requires suppres-sion are a priori assigned while designing the controllaw
repre-9.6 ACTIVE CONTROL OF VIBRATIONAND WAVES DUE TO BROADBANDEXCITATION
The FE model of a system is usually very ‘high’,especially for transient dynamics and wave-propagationproblems Designing a control system for such problems
is very difficult due to the very ‘high’ system size Forexample, in vibration-control problems, it is customary toreduce the vibration amplitudes of the first few modesusing suitable control algorithms This can be easilyaccomplished by using a reduced-order model of thecomplete system that has all of the necessary informationabout the first few relevant modes This ROM can beused in conjunction with state-space modeling or thetransfer function approach that was outlined earlier, fordesigning the control law However, transient dynamics
or wave-propagation problems are multi-modal blems That is, if we use the FE approach, then onecannot design the controller based on the first few modes,since many higher-order modes carry significant portion
pro-of kinetic energy Obtaining all pro-of the higher-ordereigenvalues/vectors is computationally very prohibitive.That leaves one with no other option but to look for analternate mathematical tool that has a smaller system sizeand yet contains all of the model information A tool thatfits into the above description is the Spectral FiniteElement Model (SFEM) which was dealt within Chapters
7 and 8 In this present chapter, a new design philosophybased on Fourier transforms is developed, wherein theexisting SFEM is modified to model the control ele-ments, namely the sensor and the actuator, and also thecontroller Since the system size of the SFEM model isvery small and also contains all modal information, nomodel-order reduction is required In addition, the con-troller can be designed for the entire eigenspectrum andhence in cases of vibration- or noise-control problems,
Figure 9.6 Comparison of transverse velocity for pattern 1
(150 dof) for different ROMs.
Full-order response ROM (SEREP) ROM (DIRS) ROM (DC)
Figure 9.7 Comparison of transverse velocity for pattern 2
(50 dof) for different ROMS.
Trang 12one can obtain the Frequency Response Function of the
structure after the feedback signal is enforced This will
quantitatively give the amount of vibration amplitude
reduction over the entire eigenspectrum This will give a
great reduction in the computational effort as opposed to
the traditional approaches One other major advantage of
the model is that one can handle arrays of sensors/
actuators and any sensor(s) can be fed to any actuator(s)
or set of actuators That is, it is quite simple to handle
both collocated and non-collocated sensor–actuator
con-figurations This aspect is extremely difficult to handle in
the traditional approaches
9.6.1 Available strategies for vibration
and wave control
Design of smart structural systems based on control of
the first few resonant modes, individually, is the most
common in practice For many vibration-control
appli-cations, this serves the control objective, since the
modal energy is distributed over the first few resonant
modes only The basic steps behind development of
such active control system models can be described as
follows:
First, an appropriate kinematics and constitutive
model is assumed For actuators or load cells mounted
on the host structure, appropriate ‘lumping’ of the
control force and actuator inertia can be considered
For surface-bonded or embedded layered sensors/
actuators, the same kinematics as the host structure
with additional constraints (for example, shear-lag to
model active/passive constrained layers,
discontinu-ous functions to represent interfacial slip while
hand-ling inclusions, air-gap, etc.) can be used
Next, one has to adopt an application-specific control
scheme For a known harmonic disturbance, a control
force can be applied in the open-loop having an
optimal phase difference with the mechanical
distur-bance An actuator force can directly be specified to
add onto the equivalent mechanical force vector For
unknown dynamic loading, and as required in most
stable controller designs, closed-loop control schemes
are to be adopted The initial configuration of the error
sensors, whose placements and numbers are to be
fixed based on optimal control performance
(observa-bility and controlla(observa-bility), can be used for feedback or
feed-forward control These error measurements are
considered as inputs to the controller under design
The controller output vector is to be used as the input
electrical signal to the actuators For an off-line
optimal control design based on a conventional mization technique, the above steps are to be repeated
opti-at every iteropti-ation while extremizing the cost tion(s) For an off-line optimal control design based
func-on soft-computing tools (e.g genetic algorithms),these solution spaces can be explored directly
Once all of the system parameters (stiffness, mass,damping, electromechanical properties of sensors andactuators, sensor locations, actuator locations, actua-tor input, etc.) for a particular configurations areavailable, one has to develop a global model for thepassive structure and senor/actuator segment usinganalytical or finite techniques Under certain cases
of electromechanical coupling, the system matricescan be decoupled into passive and active components.For the fully coupled electromechanical case, ananalytical solution can be obtained for only a fewelectromechanical boundary conditions and for this,one can use a detailed finite element model Formounted actuator or load cells, the effect of actuatorstiffness, inertia and force can be ‘lumped’
Next, one has to adopt suitable methods of systemsolution in temporal or modal space When the dis-cretized system size is large, an appropriate reduced-order modeling technique can be used Dynamic Con-densation, Proper Orthogonal Decomposition (POD)
or the System Equivalent Reduction Expansion Process(SEREP), among many reduced-order modeling tech-niques, are found useful Based on the formalism of thecontrol cost function construction, a state-space model(first-order representation) is often used instead of adirect second-order representation This is particularlysuitable for conventional designs based on the quad-ratic regulator approach, where the state-space plantmatrix, the input/output matrix, along with the requiredweighting matrices, are introduced Peak-response spe-cifications are generally found to be linear matrixfunctions of the design variables, which allow them
to be incorporated within the design framework out increasing the complexity of the optimization [14]
with-In time-marching schemes (for example, Newmarktime integration) while designing optimal control sys-tem, the control cost function is minimized, includingspecial control system features (e.g gain scheduling,feedback delay, etc.) When modal analysis is adopted,the modified dynamic stiffness matrix (including thecontributions of sensor, controller and actuator para-meters) is to be optimized so that the prescribed modesare controlled In this approach, the control efficiency isquantified in terms of reduction in the modal amplitudelevel in the frequency response
Trang 13Once the range of control system parameters and the
sensor/actuator collocation pattern is obtained,
sensi-tivity and stability studies are carried out Sensisensi-tivity
studies are important to identify the most effective
solution-space of the design parameters This also
helps in visualizing the deviation in the desired
response due to control uncertainty and measurement
of noise With the narrowed-down solution-space of
the design parameters thus obtained above, the locus
of the roots of the characteristic system, that is, poles
(resonances) and zeros (anti-resonances) of the system
transfer function for varying design parameters are
studied The range of design parameters that produces
the root locus on the right-half phase-plane are
unstable and are avoided in the final design A
secondary objective is often placed for control of
transient disturbances, which is to minimize the
tran-sient response time of the controller
For real-time automatic control systems, the off-line
design discussed above is augmented by an adaptive
filter that tunes the control gains in the presence of
measurement errors and uncertainty [15] In addition,
there are certain drawbacks of the finite dimensional
design to control a distributed parameter system, such
as control spillover This is the result of insufficient
modes considered in the MIMO state-space model
Although adaptive filters can augment the
perfor-mance of an off-line design based on a finite number
of states, better modeling techniques for distributed
parameter systems are often advantageous This is
where techniques such as SFEM score over other
methods available for the solution A modified
SFEM that includes the modeling of control elements
such as sensors, actuators and control elements is
what is called the Active Spectral Finite Element
Model (ASEM) This formulation is built on the
same lines as the FEM and removes most of the
limitations of the other off-line techniques This
method is explained in the next subsection
9.6.2 Active spectral finite element model (ASEM)
for broadband wave control
In this subsection, a generalized active spectral finite
element model (ASEM) capable of handling arbitrary
distributed sensor–actuator configurations with a PID
feedback scheme is presented The main objective is to
develop an efficient and faster computational technique
for the analysis and design of multiple sensor–actuator
configurations for active control of broadband waves in
connected composite beams The ASEM can be used to
study structure–control interactions produced by varioustypes of mounted or embedded active actuators andsensors, along with classical transducers modeled as
‘lumped’ devices Among the specific advantages arethe accurate sensors and actuators dynamics based on auniform micro electromechanical field model, considera-tion of multiple scattering of waves through structuraljoints and boundaries and near-field effects on thesensors Numerical experiments on a slender laminatedcomposite cantilever beam with a bonded piezoelectricfiber composite (PFC) (explained in Section 8.4.4, chap-ter 8) are performed Various aspects of low-authoritycontrol against parametric variations are explained Somephysical insight into the macroscopic behavior of thesePFC actuators has been reported in Bent [16] and Hagood
et al [17]
In this ASEM model, the beam network is discretizedand classified into three different classes of elements, asfollows: (1) a spectral element for finite beams withmechanical and passive properties; (2) distributed orpoint sensors; (3) distributed or point actuators A sche-matic diagram of a sensor–actuator element configuration
is shown in Figure 9.8 Here, it is assumed that that thecontroller output for a single actuator can be designedbased on a feedback signal constructed from a group ofsensors Furthermore, in Figure 9.8, the connectivitybetween the pth sensor and the qth actuator is alsoshown, where the sensor response is measured at thelocal coordinate systemðXsp; YspÞ and the actuation force
is provided at the local coordinate systemðXa ; Ya Þ
9.6.2.1 Spectral element for finite beamsThe SFEM outlined in Chapter 7 is again used here torepresent the dynamics of the beam structure For the
Sensor element
Figure 9.8 Sensor–actuator configuration for the active tral finite element model.
Trang 14spec-sake of completeness, the element-level equations
invol-ving the nodal displacement vector are repeated here
Assuming the beam to have three degrees of freedom
(two translational dofs and one rotational dof) per node
and having two nodes, the elemental displacement vector
in the frequency domain is given by:
f^ueg ¼ ^0 w^1 ^
1 ^0 w^2 ^
2
!ð9:69Þ
and the corresponding nodal force vector is given
by:
f^Fge¼ N^1 V^1 M^1 N^2 V^2 M^2!
ð9:70ÞThe use of the spectral form of solution for the governing
equation and its eventual solution in the frequency
domain results in the dynamic shape functions for the
spectral element formulation, which can be written as:
f^u0gef^wgef^yge
where ½@1 ; ½@2 ; ½@3 are the exact spectral element
shape-function matrices corresponding to the axial,
trans-verse and rotational degrees of freedom, and½ ^K e is the
exact element dynamic stiffness matrix As in the case of
the FE, the stiffness matrix in the elemental coordinate
system is transformed to the global coordinate system by
using a suitable transformation matrix
9.6.2.2 Sensor element
For illustrative purposes, a point sensor has been
con-sidered in the modeling However, it should be noted that
the formulation does allow for distributed sensors such as
piezoelectric film sensors The force-balance equations
for the sensor element are identical in form to Equation
(9.71) Based on the response measured by a
displace-ment sensorðsÞ, which is located at ðxs p; zs pÞ in the pth
sensor element (denoted by subscript sp), the actuator
input spectrum can be expressed with the help of
^I ¼Xpg^Z; H^¼ b^I; ^¼ ð^Zu;^Zw; ^ZeÞ ð9:75Þ
where ^Z is given by Equations (9.72)–(9.74) The stant g is a scalar gain and b is the actuator sensitivityparameter introduced to account for the actuator assem-bly and packaging properties (e.g the solenoid config-uration for a packaged Terfenol-D rod actuator [19],voltage-to-electric field conversion factor for plane-polarized PZT wafers, etc.) Next, after substituting for
con-^
H from the magnetomechanical (or electromechanical)force-boundary condition (Equation (8.119) in chapter 8)into Equation (9.75) and following the same procedure asused for discretizing the purely mechanical domain usingthe SFEM, the force-balance equation for the qth actuatorelement (denoted by subscript aq) in the actuator localcoordinate system can be obtained as:
f^Fgaq¼ ½ ^K aqf^uegaqþ
A33eff 0 B33eff A33eff 0 B33eff
bg^Zð9:76Þ
Trang 15½A33eff; B33eff ¼
ð
e33eff½1; z dz ð9:77Þ
defines the equivalent mechanical stiffness due to the
effective magnetomechanical (or electromechanical)
coupling coefficient eeff33 (see Equation (8.109) in chapter
8 for the PFC) for actuation in the longitudinal mode A
similar vector with non-zero second and fifth elements in
Equation (9.76) be used After substituting ^Z in terms of
the sensor element shape function matrix ½@ and the
corresponding nodal displacement vector f g from^e
Equations (9.72)–(9.74), Equation (9.76) can be rewritten
q s p is introduced to representthe Sensor–Actuator Stiffness Influence Matrix (SASIM)
This equation is transformed to the global coordinates as
in a regular FE solution This procedure leads to the final
expression for the qth actuator element with the pth
the assembled closed-loop MIMO system with a general
sensor–actuator configuration in the ASEM is obtained in
f^ug ¼ ½^T 1½ ^K þ ½ ^KðxsÞa s 1
f^Fg ð9:80Þ
In the above equation,½^T is the matrix that relates thedisplacement field to the wave vector At this stage, if atransfer-function-based concept of wave cancellation ischosen for designing the controller, Equation (9.80) pro-vides a direct way to carry out identification of appropriatecontrol gains for known sensor and actuator locations thatwill reduce certain elements off^ug to zero, and hencethe corresponding wave components can be controlled.However, the analytical approach to achieve this is limited
by the fact that one cannot obtain an explicit expressionfor the dependence of local wave components on sensorand actuator locations and other control parameters for acomplex problem, which may have more than one dis-cretized subdomain Hence, a semiautomated schemeintegrated with an ASEM is chosen to analyze thespatially rediscretized system by changing sensor loca-tions or actuator locations on an iterative basis This isfeasible because of the fast computation and small systemsize permitted by the ASEM
9.6.2.4 Numerical implementation
As the initial step, input time-dependent forces or turbances are decomposed into Fourier components byusing the forward FFT Note that all of the element-leveloperations as well as the global system-level operationsare carried out at each discrete frequency on Except forthis basic difference, the proposed program architecture
dis-is almost identical (for an open-loop configuration) to afinite element program in terms of features such as input,assemblage, solving of the system and output For aclosed-loop system, we use Equation (9.79) to implementthe explicit form of the global dynamic stiffness matrix at
a particular frequency, which is in most of the cases,
... experienceany force and the effect of associated inertia and damp-ing are not included in the transformation Upon parti-tioning the mass and stiffness matrices into submatricesand the displacement and force... (Equation(9. 49) ) and f^ug and f^fg are the frequency-domain
amplitudes’ displacement and force vectors, respectively.Again, partitioning the above in terms of matrices asso-ciated with...
reduced-state-space-f_x1gf_x2g
Trang 2and [C], respectively Equation (9. 29)