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Proceedings of the ASME 2012 International Design Engineering Technical Conferences &Computers and Information in Engineering Conference
IDETC/CIE 2012August 12-15, 2012, Chicago, IL, USA
DETC2012-70990
DETERMINATION OF STATE SPACE MATRICES FOR ACTIVE VIBRATIONCONTROL USING ANSYS FINITE ELEMENT PACKAGE
A.H. DarajiSchool of Mechanical and Systems Engineering
Newcastle UniversityNewcastle Upon Tyne, United Kingdom
J.M. HaleSchool of Mechanical and Systems Engineering
Newcastle UniversityNewcastle Upon Tyne, United Kingdom
Email:[email protected] Email:[email protected]
ABSTRACTThis paper concerns optimal placement of discrete
piezoelectric sensors and actuators for active vibration control,
using a genetic algorithm based on minimization of linear
quadratic index as an objective function. A new method is
developed to get state space matrices for simple and complex
structures with bonded sensors and actuators, using the ANSYS
finite element package taking into account piezoelectric mass,
stiffness and electromechanical coupling effects.
The state space matrices for smart structures are highly
important in active vibration control for the optimisation of
sensor and actuator locations and investigation of open andclosed loop system control response, both using simulation and
experimentally.
As an example, a flexible flat plate with bonded
sensor/actuator pairs is represented in ANSYS using three
dimensional SOLID45 elements for the passive structure and
SOLID5 for the piezoelectric elements, from which the
necessary state space matrices are obtained.
To test the results, the plate is mounted as a cantilever and
two sensor/actuator pairs are located at the optimal locations.
These are used to attenuate the first six modes of vibration
using active vibration reduction based on a classical and
optimal linear quadratic control scheme. The plate is subject to
forced vibration at the first, second and third naturalfrequencies and represented in ANSYS using a proportional
derivative controller and compared with a Matlab model based
on ANSYS state space matrices using linear quadratic control.
It is shown that the ANSYS state space matrices describe the
system efficiently and correctly.
Keywords. Vibration control, piezoelectric sensor/actuator
pair, genetic algorithm, optimal placement, electric charge,
ANSYS state space matrices.
1. INTRODUCTIONActive vibration control is often considered superior to
passive control, being a high response, smarter and lighter
solution to the problem of structural vibration. In this area
researchers have reported the importance of discrete
piezoelectric sensors and actuators and their locations, rather
than a distributed piezoelectric sensor or actuator covering a
whole surface, which causes low sensing and actuating effect
Kumar and Narayanan showed that misplaced sensors and
actuators cause problems due to lack of observability and
controllability [1]. Kapuria and Yasin demonstrated that indirect feedback control, multiple segmentation of electrodes
leads to faster attenuation of the closed loop response for the
same gain with the same optimal control output weighting
parameters [2].
Several published works have investigated plates and shells
with distributed piezoelectric sensor/actuator pairs for active
vibration control. Tzou and Tseng modelled such a mechanica
structure (plate/shell) using the finite element method and
Hamiltons principle. They proposed a new piezoelectric finite
element model including an internal electric degree of freedom
[3]. Detwiler et al modelled a laminated composite plate
containing distributed piezoelectric sensor/actuator pairs using
finite element and variational principles based on first ordershear deformation theory [4].
Optimal placement for sensors and actuators has been
investigated for beams, plates and shells to achieve controller
optimality using the genetic algorithm. Wang et al studied
optimal location and size (length) of a single piezoelectric
sensor/actuator pair bonded on a beam, based on controllability
index maximization as an objective function[5]. Devasia et a
proposed minimization of quadratic index as an objective
function using a simple search algorithm for placement and
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sizing of a single piezoelectric sensor/actuator bonded to a
uniform beam. They reported that minimization of quadratic
index gives better results than controllability for placement and
sizing [6].
Location optimization of a single piezoelectric actuator was
investigated by Sadri et al for a simply supported plate based on
modal controllability maximisation as an objective function [7].
Two piezoelectric actuators and piezofilm sensors were
optimised using controllability, observability and spillover as
an objective function to suppress the first three modes of
vibration by Han et al [8]. Quek, et al optimised two
piezoelectric sensor/actuator pairs bonded on a cantilever plate
based on modal controllability to suppress the first two modes
of vibration [9]. Peng, et al studied optimal placement of four
sensor/actuator pairs to control the first five modes of vibration
based on grammian controllability index maximization [10].
Optimal placement of ten sensor/actuator pairs was researched
using minimization of linear quadratic index as an objective
function to suppress the first six modes of vibration. It was
reported that a LQR controller required lower peak actuator
voltage than classical methods[1]. Bruant et al investigatedoptimal placement of two and three actuators to suppress the
first five modes and considered the sixth, seventh and eighth
mode as residual modes. Maximization of observability or
controllability index is used as an objective function [11].
Optimization of the number of piezoelectric sensor/actuator
pairs is investigated by Roy and Chakraborty for composite
beams and shells using a modified genetic algorithm
mqximising controllability index as an objective function [12].
A new placement strategy including a conditional filter is
proposed by Daraji and Hale to reduce the genetic algorithm
search space and explore the global optimal configuration of
ten and four piezoelectric sensor/actutor pairs, respectively, to
attenuate the first six modes of vibration[13]. Effect of structuresymmetry on optimal placement of sensors and actuators has
also been studied by Daraji and Hale using minimization of
linear quadratic index as an objective function. They have
found symmetrical configurations of actuators for symmetrical
structures and asymmetrical actuators configurations for
asymmetrical structures and the symmetrical piezoelectric
configuration gave higher vibration attenuation than published
asymmetrical configurations[14]. A half and quarter
chromosome technique has been developed by Daraji and Hale
to reduce genetic algorithm search space by more than 99%
when locating ten and eight sensor/actuators pairs on a flat
plate with linear quadratic index minimization as an objective
function [15].A new method is proposed in this work to determine state
space matrices from the ANSYS finite element package taking
into account piezoelectric mass, stiffness and electromechanical
coupling effects. This is a highly reliable and flexible method
for describing the response of simple and complex structures by
implementing the state space matrices both in simulation and
experimentally. In this work, a flexible plate with bonded
piezoelectric sensor/actuator pairs is investigated, based on
finite element and Hamiltons principle. Optimal placement of
two sensor/actuator pairs is investigated to suppress the first six
modes of vibration for an isotropic cantilever plate using a
genetic algorithm. Minimization of linear quadratic index is
taken as an objective function. The plate with two
sensor/actuator pairs in optimal locations is represented in
ANSYS using three dimensional SOLID45 elements for the
passive structure and SOLID5 for the active piezoelectric
elements to determine the controller state space matrices taking
piezoelectric mass, stiffness and electromechanical coupling
effects. Vibration reduction for a cantilever plate is
investigated, using ANSYS and Matlb simulation based on
ANSYS state space matrices to test the correctness of this
work.
2. NOMENCLATURE State matrix Sensor surface area Actuator input matrix
Differential matrix relating strain to
nodal displacement
Output sensor matrix Electric charge applied to an actuatorq Modal Electric charge induced in s/a Actuator, plate and sensor thickness Linear quadratic index Jacobin determinant Feedback gain matrix Piezoelectric electromechanical
coupling and capacitance matrix Number of modes Optimal LQR weighted matrices Number of actuators
State vector
Modal displacement, velocity andacceleration Natural frequency Mode shape mass normalised Actuator and sensor voltage Damping ratio Mode shape spectrum normalised Piezoelectric permittivity3. MODELLING
3.1 Finite Element State Space Matrices Determination
The plate with piezoelectric patches bonded to its surface to
form sensors and actuators in pairs is modelled using finite
element and Hamiltons principle based on Mindlin-Reissne
plate theory. An isoparametric two dimensional element is
chosen for modelling with four nodes and three degrees o
freedom per node. The full derivation and parameters are
explained by Daraji and Hale, the dynamic equation in moda
coordinates and state space matrices are[14];
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A finite element program has been modified based on finite
element codes in reference [16] to determine sensor/actuator
electormechanical coupling, capacitance matrices (equations 9
and 10), natural frequencies and mode shape mass normalised
to get state space matrices (equations 6 and 9).
3.2 ANSYS State Space MatricesThe plate with bonded piezoelectric sensor/actuator pairs is
represented in the ANSYS finite element package using threedimensional SOLID45 elements for the passive structure and
SOLID5 for active sensor and actuator elements. The sensor
and actuator electrode surface is connected by a single terminal
either to collect the induced charge as a result of mechanical
strain in a sensor/actuator or to apply feedback voltage to an
actuator, both in real time experiment and in ANSYS, for
vibration suppression.
The plate with two sensor/actuator pairs in optimal locations
is analysed as a free vibration problem in ANSYS to get the
first six natural frequencies, mode shape mass and spectrum
normalised, and modal charge induced in the sensors and
actuators. It is assumed that the charge induced on the sensor
and actuator surface is distributed equally and related to theelement nodal displacement in x and y directions by factors
depended on the piezoelectric element node location.
The electric modal charge induced on a singlepiezoelectric sensor or actuator surface is equal to
electromechanical coupling matrix multiplied by modeshape spectrum normalized mode shape as follows;
For single actuator and mode number , equation (12
becomes:
Where is a modal charge accumulated on a sensor or
actuator electrode at mode number. are modeshape spectrum normalised for sensor/actuator element surface
nodes 1, 2, .
. The sensor/actuator nodes factors
equal 1
for any node does not share other single actuator or sensor
elements nodes, 2 for any node shareing two elements nodes
and 4 for node shareing four elements nodes.
Where and are refer to piezoelectric senso
permittivity, area and thickness respectively. The first six
natural frequencies and mass normalised modes shape can be
determined from ANSYS package, and substituting equations
(14) and (17) in state space matrices equations (6) and (7) to get
ANSYS state space matrices.
The matrices and are individual modal stateinput actuator and output sensor matrices where subscript (i)refers to the mode number. The state matrices for number ofmodes and number of actuators are:
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4. CONTROL LAW AND OBJECTIVE FUNCTIONLinear quadratic optimal controller design is based on
minimization of performance index . Values of a positive-definite weighted matrix dimension and
dimension are controlled by the value of theperformance index, where
represent the number of
modes and actuators, respectively. These matrices areestablished by the relative importance of error and controller
energy, high value of giveing high vibration suppression.Optimal control system design for a given linear system is
realised by minimization of performance index .
Ogata has shown it is possible to follow this derivation to
design a linear quadratic controller [17], which leads to the
following Riccati equation:
Solution of the Reduced Riccati equation (24) gives the
value of matrix ; if matrix is positive definite then thesystem is stable or the closed loop matrix isstable. Feedback control gain can be obtained after substitution
of in equation (25).Minimization of linear quadratic cost function Jis taken as
an objective function to optimise gain and piezoelectric
actuator locations[18]. It can be seen from the Riccati equation
(24) that the Riccati solution matrix is a function of actuatorlocation matrix [B] while the matrices and areconstant for a particular control system. The linear quadratic
cost function J is equal to the trace .The minimum value of gives optimal piezoelectric actuator location andminimum feedback gain . So;
Fitness= Where
plate dimension
5. GENETIC ALGORITHMIn 1975, Holland invented the genetic algorithm, a heuristicmethod based on survival of the fittest or the principle of
natural evolution. It has been continuously improved and is
now a powerful method for searching optimal solutions [19].
The working mechanism of the genetic algorithm is represented
by two stages: firstly selection of the breeding population from
the current whole population, and secondly reproduction. The
process is started by defining a population of individuals a
random from the search space, the chromosome of each being
made up of two random numbers in the range 01-100
representing the locations of the two sensor/actuator pairs on
the plate. This is the population of the first generation. In the
selection process, the fitness function value for each individua
is calculated using these genetic values as data, and the
breeding population defined as those with the highest value o
fitness function. The reproduction process is closely based on
sexual reproduction. Pairs of individuals from the breeding
population share their genetic material to produce offspring
containing a combination of their parents genes.
Many strategies have been developed for the reproduction
process, but all involve crossover and mutation. In crossover
the chromosome of each parent is broken and two new
chromosomes formed from the pieces. In mutation, one o
more genes in a childs chromosome are changed randomly. In
this way crossover explores the known regions of the search
space by testing different combinations of genes that have been
shown to promote high fitness, while mutation helps tomaintain diversity in the population and so explore new regions
of the search space. The process then continues for many
generations until the population converges on a single optima
solution, which is to say that the chromosomes of all members
of the breeding population are almost identical.
The plate was divided into 100 elements encoded by
sequential numbers 01, 02, , , 100; each of them representing a
possible location of a sensor/actuator pair as shown in Figure 1
As implemented in this work, a chromosome contains three
genes, which is the number of piezoelectric actuators to be
optimised plus one to store the fitness value.
Placement strategy for discrete sensors and actuators using a
genetic algorithm based on a conditional filter is proposed by
Daraji and Hale is used in this work [13]. Its main features are:
1. Suitable values of and are set by the user.2. The state matrix of dimension is
prepared for the first six modes of vibration according to
the equation (18).
3. One hundred chromosomes were chosen randomly fromthe search space to form the initial population.
4. The input (actuators) matrix is calculated for each
chromosome and for the first six modes of vibration
according to equation (20).
5. A fitness value is calculated for each member of the
population based on the fitness function, according to
equation (26), and stored in the chromosome string to save
future recalculation.
6. The chromosomes are sorted according to their fitness
value and the 50 chromosomes with the lowest fitness
values (i.e. the most fit) are selected to form the breeding
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population, called parents. The remaining, less fit,
chromosomes are discarded.
7. The members of the breeding population are paired up in
order of fitness and crossover applied to each pair, the
crossover point being selected randomly and is different
for each parent. This gives two new offspring (child)
chromosomes with new properties.
8. A mutation rate of 5% is used on the child chromosomes.
9. The new chromosomes are filtered for repeated genes. It
is a physical requirement of this work that there be two
sensor/actuator pairs, so more than one gene for a
particular location is meaningless. The filter tests for
repeated genes, and if detected replaces one with a gene
from the search space.
10.The input (actuators) matrix is calculated for each child
chromosome and thereafter the process is repeated from 5
for a preset number of generations.
6. RESULTS AND DISCUSSION
6.1 Research ProblemA flat cantilever plate dimensioned mm is
mounted rigidly from the left hand edge as shown in Figure 1.
The plate is descritised to one hundred elements sequentially from left to right and down to up as shown in the
Figure. Optimal placement of two piezoelectric sensor/actuator
pairs is investigated to suppress the first six modes of vibration.
Active vibration reduction is investigated by matlab simulation
based on the state space matrices taken for ANSYS finite
element package. The plate and piezoelectric properties are
listed in Table1.
6.2 Natural FrequenciesThe first six natural frequencies and mode shapes for the
cantilever plate were investigated using ANSYS. The plate is
represented using two dimensional SHELL63 elements, three
dimensional SOLID45 and the results are
Table 1. PLATE AND PIEZOELECTRIC PROPERTIES
Properties Plate Piezoelectric PIC225
Modulus, GPa 210 -------
Density, Kg/m 7810 7810
Poissons ratio 0.3 -------Thickness, mm 1.9 0.5Length, width, mm 500, 500 50, 50 , C/m2 --------- -7.15 , GPa -------- 123,76.7,97.11 (F/m) ---------
compared with experimental. The results converged with mesh
refining to constant values and it was shown that the mesh of SHELL63 elements gave good accuracy for the firs
six natural frequencies compared with a finer mesh, with the
three dimensional element SOLID45 and with experiment as
shown in Table2.Sensor and actuator placement complexity is limited by the
number of finite elements and number of actuators to be
optimised and it is important to find the smallest number of
finite elements in order to minimise the computatioal cost. The mesh of SHELL63 elements was found to be ideal.Half-power bandwidth was used to determine damping ratio
for each mode using the experimental frequency response
graphs. The frequency difference between the half powe
(-3dB) points on each modal peak n was measured and the
damping ratio calculated as /2n and the results shown in
Table2.
Table 2.NATURAL FREQUENCIES
Mode (Hz)
Element Type Shell63 6.71 16.85 55.61 78.71 80.39 106.96Shell63 6.62 16.27 44.10 54.59 62.96 110.03Shell63 6.59 16.17 41.32 52.37 59.79 104.70Shell63 6.59 16.15 40.62 51.80 59.00 103.31Solid45) 6.59 16.15 40.44 51.68 58.86 103.18Experimental 5.90 16.90 37.30 51.60 58.20 101.00
Exper. 19.7 10.6 5.19 4.52 1.09 2.6996.3 Piezoelectric Location OptimizationThe genetic algorithm described in section 5 was used to
find optimal locations for two actuators bonded on a 0.5m
square cantilever plate, fixed rigidly on the left hand edge. The
optimal solution obtained by progressive convergence of the
population is shown in Figures 2, 3 and 4, in which the
population is distributed around the circle with radius R
representing its fitness value (smaller radius means higher
fitness).
0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0 .35 0. 4 0. 45 0. 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
100
01
Figure1. CANTILIEVER PLATE MOUNTED RIGIDLY FROMTHE LEFT HAND EDGE DESCRITISED TO ONE HUNDREDELEMENTS NUMBERED SEQUENTIALLY FROM LEFT TO
RIGHT AND DOWN TO UP
10
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At the first generation (Figure 2a), the random population
is very diverse with representatives of high and low fitness and
the full range in between. This first generation population is
shown in another form in Figure 2b, where each point
represents an actuator location for one of the individuals in the
population. In the first generation these locations are widely
distributed, having been selected at random.
After ten generations (Figure 3a) the population has almost
converged to a high fitness value close to the centre of the
circle. Figure 3b shows that the genes have begun to cluster in
three locations.
After twenty generations (Figure 4) the population has
converged to a level of higher fitness for all individuals. This is
shown most clearly in Figure 4b, with all chromosomes coding
for actuators at the most effective two sites. It can be seen that
the optimal piezoelectric actuator locations are symmetrically
distributed about the axis of symmetry. This optimal location
of two piezoelectric actuators shown in Figure 4b is in
agreement with reference [9].
6.4 Optimal Location ValidationThe genetic algorithm program was run multiple times for
the cantilever plate to test the reliability of the optimized
actuator locations. The results are shown in Figure 5, which
gives an indication of the progress of each run by plotting the
fitness value for the fittest member of each generation. It can
be seen that the final fitness value is the same in each run
though the path by which it is reached is different in each case.
0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0. 05 0. 1 0. 15 0. 2 0. 25 0. 3 0. 35 0. 4 0. 45 0. 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Figure 2. ONE HUNDRED CHROMOSOMES FORTHE FIRST RANDOM POPULATION FOR R=1 ANDQ=10
11, (a)FITNESS VALUE DISTRIBUTION, (b)
GENES (ACTUATORS) DISTRIBUTION ON THECANTILEVER PLATE
Figure 3.CHROMOSOME FITNESS PROGRESSION AFTERTEN GENERATIONS, (a)FITNESS VALUES DISTRIBUTION,
(b) GENE DISTRIBUTION ON THE CANTILEVER PLATE
Figure 4. CHROMOSOMES FITNESS
PROGRESSION AFTER TWENTY GENERATIONS,(a)FITNESS VALUES DISTRIBUTION, (b)OPTIMALGENES DISTRIBUTION FOR THE CANTILEVER
PLATE MOUNTED RIGIDLY FROM THE LEFTHAND EDGE
(a)
(a) (b)
(b)
(b)
Figure 5. OPTIMAL FITNESS VALUE FOR THE BESTMEMBER AT EACH GENERATION IN EACH OF
TWELVE RUNS OF THE COMPUTER PRGRAM, EACHRUN IS SHOWN IN A DIFFERENT COLOUR
(a)
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6.5 State Space Matrices DeterminationThe state space matrices are determined for the cantilever
plate with two piezoelectric sensor/actuator pairs in optimal
location for the first six modes of vibration using ANSYS. The
plate is represented using three dimensional SOLID45 elements
and the piezoelectric sensor/actuator pairs by SOLID5.
Part of ANSYS APDL program to calculate state space
matrices is shown below.*SET,DIS
*SET,QS
*DIM,DIS,ARRAY,108,6
*DIM,QS,ARRAY,6,2
*SET,SNN,0
*DO,IVOLU,4,5
VSEL,S,VOLU,,IVOL,,,1
*SET,SNN,SNN+1
NSEL,R,LOC,Z,-0.0005
*GET,MINUMN,NODE,0,NUM,MIN
*GET,MAXUMN,NODE,0,NUM,MAX
*DO,JM,1,6,1*set,IJJ1,0
*set,IJJ2,0
*set,IJJ3,0
*DO,IJ,MINUMN,MAXUMN,1
*SET,IJJ1,IJJ1+1
*SET,IJJ2,IJJ1+1
*SET,IJJ3,IJJ2+1
*GET,DIS(IJJ1,JM),NODE,IJ,UX
*GET,DIS(IJJ2,JM),NODE,IJ,UY
*GET,DIS(IJJ3,JM),NODE,IJ,UZ
*SET,IJJ1,IJJ3
*ENDDO*GET,QS(JM,SNN),NODE,ANTOP(SNN),RF,AMPS
*ENDDO*ENDDO
The first six natural frequencies determined by ANSYS
including the effects of piezoelectric mass and stiffness are as
follows;
The modal damping ratiosare determined experimentally
and given in Table 2.
6.6 ANSYS State Space Matrices ValidationProportional-differential and optimal linear quadratic
control schemes are implemented to attenuate vibration for the
cantilever plate with two sensor/actuator pairs located in the
optimal locations and simulated using ANSYS and the Matlab
based on the ANSYS state space matrices respectively.
Firstly, the plate is represented in ANSYS using three
dimensional SOLID45 elements for the plate and SOLID5 for
the sensors and actuators. The plate was driven at the first
second and third resonant frequencies for six seconds until it
reached nearly steady amplitude, and then the controller was
activated to show the effect of active vibration reduction. A
proportional differential (PD) control scheme is realised in the
ANSYS test by taking each sensor output voltage and feeding i
to the actuators after modifying it by the PD controller. The
output voltage of the two sensors is shown in Figures 6(a1)(a2) and (a3), two actuator feedback voltages in Figures 7(a1)
(a2) and (a3), and plate free end displacements at coordinates
( 0,0.5) and (0.5,0.5m) in Figures 8(a1), (a2) and (a3).
Secondly, the same scenario was applied to the Matlab
program implementing the ANSYS state space matrices
(section 6.5) using optimal linear quadratic control scheme and
the equivalent results are shown in Figures 6, 7 and 8(b1, b2
and b3).
Figure 6 shows the correctness of the ANSYS state space
matrices, in which sensors output voltage response before
controller activation (open loop) for the first three natura
frequencies ANSYS test in Figures 6(a1), (a2) and (a3) is quite
similar to the Matlab test in Figures 6(b1), (b2) and (b3) andreached the same steady state voltage amplitude value before
activation of the controller in all cases.
The ANSYS state space matrices are also validated by the
open loop response prior to activation of the controller shown
in Figure 8(a) and (b) for the two case studies. In this figure, the
displacements of the two corners at the free end of the plate
were measured directly using ANSYS, which is a trustworthy
result. In the equivalent Matlab simulation, the displacement
were measured by modal estimation ( ) based on the
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information of two actuators at locations 01 and 91 (just 2% of
the total plate). The response in the Matlab simulation reached
60% of the ANSYS value. Considering the coarseness of the
modal estimation, this gives confidence in the ANSYS state
space matrices.
In real time experiment design, the controller depends on
estimator state space matrices and sensor output voltage. In this
work (Figure 7), Matlab test was given sensor voltage >95%
with respect to ANSYS test for the first three modes and this
result approve work correctness.
It can be observed form Figure 6 after controller activation
that the PD controller in Figure 6(a) gives a higher overshot and
lower sensor suppression voltage than the LQR controller in
Figure 6(b). Similarly, Figure 7 shows that the feedback
voltage to the actuators by the LQR controller gives highe
response, lower steady feedback voltage and shorter peakvoltage time than the PD controller .
These results further validate the state space mode
developed here, and also show the effectiveness of active
vibration control, even when using a very limited number of
sensors and actuators provided they are well located. It is
evident that the modal dynamic equations describing the system
by state space matrices do indeed model the system accurately
since the results shown in Figures 6(b), 7(b) and 8(b), obtained
a1
b1
b2
b3
a3
a2
Figure 6. OPEN AND CLOSED LOOP SENSORS
OUTPUT VOLTAGE RESPONSE FOR CANTILIEVER
PLATE SUBJECTED TO SINSOUDAL DISTURBANCE
VOLTAGE AT THE FIRST, SECOND ANDTHIRD NATURAL FREQUENCIES , AT ACTUATOR
LOCATION 01, (a1,a2,a3)ANSYS PACKAGE
RESULTS USING PD COTROLLER P=20,D=10,
(b1,b2,b3)MATLAB RESULTS USING ANSYS STATE
SPACE MATRICES USING LQR CONTROLLER (R=1,Q=10
9First , 10
8SECOND AND THIRD MODE)
First mode
First mode
Matlab
Third mode
Second mode
Second mode
Matlab
Third mode
Matlab
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using them, give such good agreement with the ANSYS finite
element results in Figures 6(a), 7(a) and 8(a).
CONCLUSIONA new method has been developed to determine state space
matrices for a flexible structure with bonded piezoelectric
sensors and actuators using the ANSYS finite element package
taking into account piezoelectric mass, stiffness and
electromechanical coupling. This makes use of mode shapes
natural frequencies and modal electric charge induced on the
piezoelectric surface obtained using ANSYS to determine statespace matrices.
Optimal locations of two sensor/actuator patches and
controller gains have been investigated for a cantilever plate
using the genetic algorithm based on minimization of linear
quadratic index as an objective function. The optimal location
is validated by running the computer program multiple times
and obtaining the same optimal by different routes in each case.
b2
a1
a2
b1
b3
a3
Figure 7. OPEN AND CLOSED LOOP ACTUATORS
FEEDBACK VOLTAGE FOR CANTILIVER PLATE
SUBJECTED TO SINSOUDAL DISTURBANCE
VOLTAGE AT THE FIRST SECOND ANDTHIRD NATURAL FREQUENCIES ,AT ACTUATOR
LOCATION 01, (a1,a2,a3)ANSYS PACKAGE RESULTS
USING PD COTROLLER P=20,D=10, (b1,b2,b3)MATLAB RESULTS BASED ON ANSYS STATE SPACE
MATRICES USING LQR CONTROLLER (R=1, Q=109
First , 108SECOND AND THIRD MODE)
First modeThird mode
First mode
Matlab Third mode
Matlab
Second mode
Second mode
Matlab
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8/10/2019 Determination of State Space Matrices for Active Vibration
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The state space matrices have been validated by comparing
the open loop transient response of a square cantilever plate
tested using ANSYS simulation based on physical displacemen
coordinates with that from a Matlab program based on modal
state space matrices obtained from ANSYS for the first threeresonance frequencies.
Vibration reduction has been studied for the two cases
using proportional and optimal linear quadric control schemes
respectively. It is shown that a reduction of 75% (-13dB) can
be obtained with just two sensor/actuator pairs in optima
locations.
Figure 8. OPEN AND CLOSED LOOP FREE END PLATE
DISPLACEMENT RESPONSE AT PLATE COORDINATES
(0.5,0),(0.5,0.5), SUBJECTED TO SINSOUDAL
DISTURBANCE VOLTAGE AT THE FIRST,SECOND AND THIRD NATURAL FREQUENCIES ,AT
ACTUATOR LOCATION 01, (a1,a2,a3)ANSYS PACKAGERESULTS USING PD COTROLLER P=20,D=10.(b1,b2,b3)
MATLAB RESULTS BASED ON ANSYS STATE SPACE
MATRICES USING LQR CONTROLLER
(R=1, Q=109
First , 108SECOND AND THIRD MODE)
Third modeFirst mode
Second mode
a1
a2
a3
First mode
MatlabThird mode
Matlab
Second mode
Matlab
b3
b2
b2
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8/10/2019 Determination of State Space Matrices for Active Vibration
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