93929129 Simulation Study of Active Vibration Control of Clamped Clamped Beam by Using Two...

INTERNATIONAL JOURNAL OF COMPUTER APPLICATIONS IN

ENGINEERING, TECHNOLOGY AND SCIENCES (IJ-CA-ETS)

ISSN: 0974-3596 | October ’09 – March ’10 | Volume 2 : Issue 1 | Page:188

SIMULATION STUDY OF ACTIVE VIBRATION CONTROL

OF CLAMPED-CLAMPED BEAM BY USING TWO

PIEZOELECTRIC ACTUATORS

1S.M. KHOT,

2NITESH P. YELVE,

3PANKAJ S. PATIL

1,2

Assistant Professor, Fr. C. R. Institute of Technology, Navi Mumbai, 3 Student, Department of Mechanical Engineering, Fr. Conceicao Rodrigues

Institute of Technology

[email protected],

[email protected],

[email protected]

ABSTRACT :

Stringent behaviour requirements imposed on flexible structures have necessitated the sensing and control

of vibrations in these structures in a suitable manner. This issue is particularly important for space and

aircraft structures for which the mission requirements are crucial and the divergence from these

requirements may be considerably expensive. One of the most likely alternatives to deal with this aspect of

vibrations is the use of active vibration control, which makes the structure a Smart structure. In this paper,

active vibration control of a clamped-clamped beam using two piezoelectric actuators for controlling

multimodes is discussed. In order to design the controller, the mathematical model of the system is

required. To form such a model theoretically may be difficult or impossible for complex structures.

However, such structures may be easily modeled in finite element (FE) environment like ANSYS©. The

mathematical model required is extracted in MATLAB© from the results of modal analysis of the beam

done in ANSYS©. Since the matrices of the full model of any system are very large in general, model

reduction is attempted in MATLAB© by discarding those modes, which do not contribute to the overall

response of the system. Then by using this reduced model, design of optimal controller is achieved using

Linear Quadratic Regulator (LQR) algorithm with state feedback control law. Effect of selection of

weighting matrices of performance index of LQR on the performance of optimal controller is also reported.

Validity of using reduced model for designing optimal controller is checked by comparing its response with

that of full model. If reduced models are used for designing controllers for active vibration control of real

life complicated systems, a lot of computational time can be saved.

KEYWORDS: Smart Structures, Cantilever Beam, Piezoelectric Actuator, Optimal (LQR) Controller,

ANSYS©, MATLAB

©

1. INTRODUCTION

It is desired to design lighter mechanical systems

carrying out higher workloads at higher speeds.

However, the vibration may become prominent

factor in this case. This undesired vibration can

be reduced or eliminated by using active

vibration control. The previous decade has seen a

lot of growth in this field accompanied with

various revolutionary ideas sprouting up and

eventually being applied to practical systems

with amazing adapting capabilities.

Some recent works are reported here. The active

vibration control of simple cantilever beams is

studied in [1]-[5]. Piezoelectric patches as

actuators are mounted on the beams. The system

identification and pole placement control method

is used in [1]. The beam with piezo-patches FE

model of the structure is constructed and the

closed loop control is applied in [2] and [3].

Singh [4] also used the beam with piezo-patches

FE model, but applied modal control strategies.

Xu et al [5] reported results on active vibration

control of cantilever beam type of structures by

using the commercial FE package ANSYS©. The

influence of sensor/actuator location is studied

for cantilever type beam. Karagulle et al [6]

extended the work of Xu et al [5] and proposed

the procedure for simulation of active vibration

control in ANSYS©, for cantilever and plate type

of structures. Lim [7] studied the vibration

control of several modes of a clamped square

plate by locating discrete sensor/actuator devices

at points of maximum strain. Quek et al [8]

presented an optimal placement strategy of

piezoelectric sensor/actuator pairs for the

vibration control of laminated composite plates.

Xianmin et al [9] studied the active vibration

control in a four-bar linkage. Numerical

simulations are reported in all the references




given. Experimental results are also reported in

some studies [1], [2] and [5].

The construction of mathematical model,

analytically for complicated real life dynamic

structures is very difficult and time consuming.

This itself may become constraint in

implementing active vibration control strategies

for real life applications. Therefore in present

study an attempt is made to extract the

mathematical model of the clamped-clamped

beam for designing optimal controller for its

active vibration control, from the results of its

modal analysis done in ANSYS©. The design of

the optimal controller is based on state feed back

control law and uses LQR algorithm in

MATLAB©. The control gains obtained not only

accounts for the magnitude but also the phase of

the control force to be exerted by the two

piezoelectric patches on the beam. These control

gains are implemented for simulating active

vibration control in MATLAB©.

2. MODAL ANALYSIS AND THE STATE

SPACE FORM

The equation of motion of a multi degree of

freedom system under external forces is given by

Fxkxcxm =++ ][][][ &&& . (2.1)

Since [m], [c] and [k] are non-diagonal, the

above expression leads to n coupled second order

differential equations. To uncouple these

equations, first the eigenvalue problem is solved

for Eq.2.1 and the eigenvectors are obtained as

x(1) , x(2) ,… x(n). The modal matrix for the system

is defined as

[xn] = [ x(1) x(2) … x(n) ] . (2.2)

For a multi degree of freedom system with the

assumption of proportional damping (that is, the

damping matrix expressed as a linear

combination of the mass and stiffness matrices

as ][][][ kmc βα += ), the solution of Eq.2.1 can

be expressed as a linear combination of the

normal modes [10] as

)(][)( txxtx pn= , (2.3)

where )(txp is the displacement in principal

coordinates. If the normal modes are normalised

with respect to mass [10], the equation of motion

in the principal coordinates becomes of the form

)()(2)(2

tFtxxtx piipiipiiip =++ ωωζ &&& , (2.4)

where i = 1, 2, …, n and

)(][)( tFxtF Tnp = (2.5)

is the vector of forces in principal coordinates.

Thus, a set of n uncoupled differential equations

of the second order is obtained from the set of n

coupled differential equations of the second

order. These n uncoupled differential equations

of the second order are converted into the state

space form as 2n differential equations of the

first order.

The equations of motion in the state-space form

are

BuAxx +=& , (2.6)

DuCxy += , (2.7)

where x is the state vector

y is the output vector

u is the input vector (control vector)

A is the state matrix

B is the input matrix (control matrix)

C is the output matrix

D is the direct transmission matrix.

Now a general algorithm for analyzing any

vibrating structure using ANSYS©

and

MATLAB© is summarized in Section 3.

3. EXTRACTION OF STATE-SPACE

MODEL FROM FE MODEL

3.1. General Theory The state-space model of the system may be

constructed by using eigenvalues and

eigenvectors as illustrated in this section. The

eigenvalues and eigenvectors normalized with

respect to mass are obtained by performing

modal analysis in ANSYS©, as will be described

in Section 3.2. In order to ease computation, a

reduced model of the system may be constructed

by discarding those modes, which do not

contribute to the response of the system. The

contribution of each mode to the overall response

may be calculated as follows. Taking the Laplace

transform of the equation of motion in principal

coordinates, the transfer function for

displacement of the jth node due to a force

applied at the kth node for the i

th mode

(considering damping) is given by [11].

ki

ji

ii

nkinjijki

F

z

ss

zzz =

++=

22 2 ωζω. (3.1)

This is the contribution to the transfer function zjk

from the ith mode. Summing up all such

contributions from individual modes, we get the

total transfer function as

k

j

ii

nkinjijk

F

z

ss

zzz =

++=∑ 22 2 ωζω

. (3.2)

The dc gain for each mode is defined by putting

s = 0 in Eq.3.1. For the ith mode,

2i

nkinji zzdcgain

ω. (3.3)

The peak gain [11] is obtained a

i

dcgainjpeakgain

ζ2−= (3.4)

wherei

ii ω

βωαζ

2

2+= . (3.5)




Model reduction can now be attempted by

sorting the modal contributions according to

their peak gains. Only those modes, which have

higher values of peak gain, will be retained,

while the rest will be eliminated, thus reducing

the size of state space model of the system. Now,

the matrices A, B, C and D of Eq.2.6 and Eq.2.7

for a system with n modes can be written as

follows [11]

−−

−−=

nnn

A

ωζω

ωζω

2.........

10.........

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

.........2

.........10

2

1121

(3.6)

and

=

pn

p

p

F

F

F

B

...

0

0

2

1

, (3.7)

where Fp = [ Fp1 Fp2 … Fpn ]T (3.8)

is the force vector in principal co-ordinates.

Matrix C depends upon the output that we are

interested in. Since we desire the values of

displacement of the nodes, C is given by

C =

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

...00

...0000

...00

2221

1211

nn

nn

xx

xx

, (3.9)

where xn11 , xn12 , … are the elements of xn , the

modal matrix normalized with respect to mass.

D is the direct transmission matrix. Here,

D = [0]. (3.10)

3.2. Modal Analysis in ANSYS©

As discussed in Section 3.1, the first step in

obtaining the state-space model of the beam is to

find its eigenvalues and eigenvectors normalised

with respect to mass. This can be done by

performing modal analysis in ANSYS©. An

aluminium beam of dimensions (504 × 25.4 ×

0.8) mm3 is used for the analysis [5]. Two

piezoelectric actuators are bonded to its surface.

Actuator dimensions are (76.2 × 25.4 × 0.305) mm

3 and they are located 5mm from both the

ends. The piezoelectric material properties are

defined in ANSYS© as

mp, dens,2,7500

! Density for piezoelectric material

mp,perx,2,15.03e-9

! Permittivity in x direction

mp,pery,2,15.03e-9

! Permittivity in y direction

mp,perz,2,13e-9

! Permittivity in z direction

tb,piez,2

! Define piez. table

tbdata,16,17

! E16 piezoelectric constant

tbdata,14,17 ! E25

tbdata,3,-6.5 ! E31

tbdata,6,-6.5 ! E32

tbdata,9,23.3 ! E33

tb,anel,2

! Define structural table

tbdata,1,126e9,79.5e9,84.1e9

! C11, C12, C13

tbdata,7,126e9,84.1e9 ! C22, C23

tbdata,12,117e9 ! C33

tbdata,16,23.3e9 ! C44

tbdata,19,23e9 ! C55

tbdata,21,23e9 ! C66

Modal analysis of the beam is performed by

using the Block Lanczos method. The range of

frequency is defined as 0 to 50000Hz. The

results of the analysis are written to a file with

the extension “eig”. This file is imported in

MATLAB©. The state-space model may now be

formed as described in the following section.

3.3. State Space Modeling in MATLAB©

For the clamped-clamped beam, node number

555 is defined as both the point of application of

the input force as well as the point where the

output (displacement) is measured. This node is

selected since it is a point of non zero

displacement. Hence all the modes can be

excited or observed at this node. Both input and

output are along the Z axis. Hence the

eigenvectors pertaining to the UZ displacement

are required. The eigenvalues are extracted from

the .eig file exported from ANSYS© using the

file ext56uz.m [11]. This code forms the vector

of eigenvalues and the modal matrix of the

system consisting of the eigenvectors pertaining

to the UZ displacement of all the nodes of the FE

model.

The state-space parameters, i.e., the A, B, C and

D matrices of the system are formulated for the

full model as described in Section 3.1. Then a

state-space system is defined using the “ss”

function in MATLAB© with A, B, C and D as

parameters. The transient responses of the beam

obtained from ANSYS© and MATLAB

© are

shown in Fig.3.1.

Since the results closely match, the state-space

model from MATLAB© accurately represents the

system and hence can be used for further

analysis. Then by using the “bode” function in

MATLAB©, the frequency response of the

system is plotted. The response of the full model

and the modal contributions are shown in

Fig.3.2.




Now, a reduced model state-space system is

constructed by truncating the insignificant

modes. The first five modes sorted according to

their peak gains are selected for forming reduced

model (Section 3.1). Frequency response of the

reduced model is plotted in Fig.3.3.

Fig.3.1 Transient Response of

MATLAB© and ANSYS

© Models

Fig.3.2 Frequency Response of Full Model

Transient responses of the full and reduced

models are then plotted as shown in Fig.3.4. It is

seen that the reduced model response closely

follows the full model response.

Fig.3.3 Frequency Response of Full and

Reduced Model

Fig.3.4 Transient Response of Full &

Reduced Model

4. CONTROLLER (LQR) DESIGN IN

MATLAB© WITH STATE FEEDBACK

For active vibration control both coupled control

and independent control methods can be

employed. The independent control method is

efficient only when few modes are of interest.

Coupled control is desirable when simultaneous

control of multimodes is required which is the

case here under study. Two feedback control

laws are available for coupled control design:

state feedback and output feedback.

For the design of controller state feedback

control law is used here. Linear Quadratic

Regulator (LQR) [12] is employed here to

determine the optimal controller gain and it is

chosen to minimise a quadratic cost or

performance index of the form,

∫∞

+=

0

)(2

1dtRuuQxxJ TT , (4.1)

where Q and R are suitably chosen positive

semidefinite weighting matrices and u is the

control force to be applied.

The selection of these weighting matrices is vital

in the controller design process. The relative

magnitude of Q and R are selected to trade off

requirements on the smallness of the state

against requirements on the smallness of the

control force. Q may be chosen as the Identity

matrix (I). An alternative is to

choose HHQT= , where Hxz = , z being the

desired control objective (in this case the

displacement of node 555), x is the vector of

states and H is the appropriate transformation

matrix [12]. In this case, R is a scalar and hence

the control performance such as settling time can

be tuned through changing the value of R. The

control gain [12] vector K may be determined

using the “lqr” command in MATLAB©. The

control force is then given by

Kxuc −= . (4.2)




Since there are two actuators to apply control

force, the system state equations become

exexcccc uBuBuBAxx +++= 2211& or

exexcc uBxKBKBAx +×+×−= ))(( 2211& and

(4.3)

Cxy = . (4.4)

where the subscript c refers to the controlling

force and ex refers to the exciting force.

For state feedback method, values of all the

states are required to be known for every time

step. Unlike state feedback, for the output

feedback method only some of the states are

required to be known at every time step, but

however, the initial value of all the states must

be known to calculate the control gain [12].

5. SIMULATION OF ACTIVE VIBRATION

CONTROL OF BEAM IN MATLAB©

The state-space models are constructed as

described in Section 3.1. The force matrix is

constructed assuming uniform distribution of

force along the ends of the perfectly bonded

actuators [13]. Reduced model is constructed in

MATLAB© assuming both input excitation and

point of observation, at the node 555 of the

beam. The impulse responses of the uncontrolled

and controlled models of the beam are plotted in

MATLAB© for different values of R with Q = I

in Fig.5.1, Fig.5.2 and Fig.5.3. Then the reduced

controlled model is formed and its transient

response is plotted with that of full model in the

same figures (Fig.5.1, Fig.5.2 and Fig.5.3). Since

the two responses closely match, the reduced

model may be used to design the controller for

active vibration control. In Fig.5.4, Fig.5.5 and

Fig.5.6 the variation of control force for different

values of R with Q = I is shown for both the

controlled full and reduced models.

Fig.5.1 Transient Response of Full and Reduced

Controlled Models for Q=I and R=1e-3





Fig.5.4 Control Force for Full and Reduced

Models for Q=I and R=1e-3








6. CONCLUSION

The mathematical model required for optimally

designing the controller for active vibration

control of clamped-clamped beam beam is

extracted from the results of its modal analysis

done in ANSYS©. It is observed that selection of

weighting matrix R plays important role in

designing optimal controller. When R is

increased, the control energy spent decreased,

but the settling time increased. Reduction in

model size is carried out in MATLAB©. The full

model and reduced models are used for optimal

controller design and their responses are studied.

This study clearly shows that reduced model can

be effectively used for optimal controller design,

which in turn saves computational time. This

study may help for analysing real life dynamic

systems for their active vibration control, right

from their modeling to optimal controller design.

7. REFERENCES

[1] W. J. Manning, A. R. Plummer, and M. C.

Levesley, “Vibration Control of a Flexible

Beam with Integrated Actuators and

Sensors”, Smart Materials and Structures

9, pp. 932-939, 2000

[2] P. Gaudenzi, R. Carbonaro and E. Benzi,

“Control of Beam Vibrations by means of

Piezoelectric Devices: Theory and

Experiments”, Composite Structures, pp.

373-379, 2000

[3] I. Bruant, G. Coffignal, F. Lene and M.

Verge, “Active Control of Beam Structures

with Piezoelectric Actuators and Sensors:

Modeling and Simulation”, Smart

Materials and Structures 10, pp. 404-408,

2000

[4] S. P. Singh, H. S. Pruthi and V. P.

Agarwal, “Efficient Modal Control

Strategies for Active Control of

Vibrations”, Journal of Sound and

Vibrations 262, pp. 563-575, 2003

[5] S. X. Xu and T. S. Koko, “Finite Element

Analysis and Design of Actively

Controlled Piezoelectric Smart Structures”,

Finite Element. Analysis Designs 40, pp.

241-262, 2004

[6] H. Karagulle, L. Malgaca and H. F.

Oktem, “Analysis of Active Vibration

Control in Smart Structures by ANSYS”,

Smart Materials and Structures 13, pp.

661- 667, 2004

[7] Y. H. Lim, “Finite Element Simulation of

Closed Loop Vibration Control of a Smart

Plate under Transient Loading”, Smart

Materials and Structures 12, pp. 272-286,

2003

[8] S. T. Quek, S. Y. Wang and K. K. Ang,

“Vibration Control of Composite Plates via

Optimal Placement of Piezoelectric

Patches”, Journal of Intelligent Material,

Systems and Structures 14, pp. 229-245,

2003

[9] Z. Xianmin, S. Changjian and A. G.

Erdman, “Active vibration controller

design and comparison study of flexible

linkage mechanism systems”, Mechanical

Machine Theory 37, pp. 985-997, 2002

[10] S. S. Rao, Mechanical Vibrations, Third

Edition, Prentice-Hall, Upper Saddle

River, NJ, 1995

[11] M. R. Hatch, Vibration Simulation Using

MATLAB and ANSYS, Chapman and

Hall/ CRC, 2001

[12] F. L. Lewis, V. L. Syrmos, Optimal

Control,A Wiley-Interscience Publication,

John Wiley & Sons Inc., 1995

[13] Crawley E. F. and Anderson E. H.,

“Detailed Models of Piezoceramic

Actuation of Beams”, Journal of

Intelligent Material Systems and

Structures 1(1), pp. 4-25, 1990

93929129 Simulation Study of Active Vibration Control of Clamped Clamped Beam by Using Two...

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