Determination of State Space Matrices for Active Vibration

8/10/2019 Determination of State Space Matrices for Active Vibration

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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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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




11/11

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Determination of State Space Matrices for Active Vibration

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