Unclas csci 23K 

G3/39 43€15 

SCHOOL OF ENGINEERING 

& ARCHITECTURE 

The Catholic University of America 

Washington, DC 20064 

.. 

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I 
STATIC DEFLECTION CONTROL OF FLEXIBLE BEAMS 

BY PIEZO-ELECTRIC ACTUATORS 

BY 

Dr. MR M.BhZ 

PROFESSOR 

MECHANICAL ENGINEERING DEPARTMENT 

THE CATHOLIC UNIVERSITY OF AMERICA 

WASHINGTON, D.C. 20064 

NASA CONTRACT #30429-D 
ACKNOWEDOEHENT8 

Special thanks are due to the Space Science and Technology 

branch at NASA Goddard Space Flight Center for providing the 

funds necessary to conduct this study under contract No.30429-D. 

Thanks are particularly due to Mr. Phil Studer for his 

invaluable inputs and for his engineering insight that have been 

instrumental in the initiation and implementation of this effort. 

Special thanks are also due to Hr. - Eric Osborne for his 

continuous thought stimulating discussions. 

a 
a 

NONENCLATURE 

b width of beam and piezo-actuator, m 

0 

e 

0 

a 

a 

a 

d electric charge constant of piezo-actuator, m/v 

D distance to neutral axis of composite beam measured from its 

lower edge, m 

Young's modulus of elasticity of piezo-actuator and beam 

respectively, N/m 

Young's modulus of elsticity of element i of the beam, N/m 

E1,2 

Ei 

2 

2 

F vector of external forces and moments 

area moment of inertia of actuator and beam about the 

neutral axis of the composite beam respectively, m 

area moment of inertia of the element i, m 

stiffness matrix of the element i 

112 

I 

4 

4 

Ii 

Ki 

K overall stiffness matrix of beam-actuator system 

length of element i 

external moment acting on ith node of beam, Nm 

piezo-electric moment generated by piezo-film, Nm 

li 

'e i 

Mf 

N number of elements of the beam 

thickness of piezo-actuator and beam respectively, m 182 

t 

U deflect ion cr iter ion 

V voltage applied across the piezo-electric film, volts 

external force acting on the ith element of the beam, N 

the linear translation of node i, m 

'i 

Yi 

Greek Letters 

6i deflection of node i 

6 deflection vector of all nodes of the beam, m or rad. 

a 
piezo-elcctr ic strain in piczo-actuator, m/m 

bending stresses in piezo-actuator and beam respectively, 

' f 

%,2 

or 

N/mZ 

2 piezo-electric stress in actuator, N/m 
e 

ABSTRACT 

a 

e 

a -- 

This study deals with the utilization of piezo-electric 

actuators in controlling the static deformation of flexible 

beams. 

An optimum design procedure is presented to enable the 

selection of the optimal location, thickness and excitation 

voltage of the piezo-electric actuators in a way that would 

minimize the deflection of the beam to which these actuators are 

bonded. 

Numerical examples are presented to illustrate the 

application of the developed optimization procedure in minimizing 

the structural deformation of beams of different materials when 

subjected to different loading and end conditions using ceramic 

or polymeric piezo-electr ic actuators. 

The obtained results emphasize the importance of the devised 

rational procedure in designing beam-actuator systems with 

a minimal elastic distortions. 

e 

e 
0 

I NTRODUcTION 

0 

0 

The construction and operation of large structures which are 

extremely flexible have posed new and challenging problems 

particularly because these structures are intended to provide 

stable bases for observations and communications such as in space 

structures. With strict constraints imposed on the structural 

deformations, it became essential to suppress such deformations 

to a minimum through the use of active control systems of one 

form or another. 

0 

Distinct among the presently available active control 

I systems are those that rely in their operation on piezo-electric 

actuators. Such systems have proven to be exper imentally 

effective in controlling the vibrations of simple structural 

elements such as rectangular beams 11-21 and hollow cylindrical 

masts 131. The effectiveness of these systems is coupled also 

with the light weight, high force and low power consumption 

capabilities of the piezo-electric actuators 11-81. These 

0 features rendered this class of actuators to be an attractive 

~0 

f 

~0 

candidate for controlling structural deformation. 

The present state-of-the-art of this type of actuators has 

been limited to the analysis of their characteristics 19-111 as 

iiifiuensed by their geometrical of operationai conditions. But, 

no attempt has been made towards selecting their optimal 

geometrical parameters or location which are best suited for a 

* Numbers between brackets designate references 
e 

e 

c 

e 

e 

0 

-2- 

particular structure subjected to knovn loading conditions. Such 

a synthesis procedure is essential to the successful integration 

of the actuators into the structure in order to minimize the 

structural deformation. 

It is therefore, the purpose of this study to develop an 

optimum design procedure that would enable the selection, on 

rational basis, of the optimum design parameters and location of 

plezo-electr ic actuators to satisfy certain structural 

deformation requirements. The procedure vi11 take into account 

the effect that the actuators have on changing the elastic and 

inertial properties of the structure to which they are bonded to. 

THE PIEPO-ELECTRIC ACTUATOR BEAN SYSTEM 

A. General Layout 

Figure (1) shows a general layout of a flexible beam (A) 

whose deflection is to be controlled by a piezo-electric actuator 

(B). The beam, under consideration, can generally be made of 

several steps which are not necessarily of the same thickness or 

the same material. The interfacial nodes between the different 

steps can be subjected to external forces, moments or both. 

Further, the degrees of freedom of any node can be limited to 

linear translation, angular rotations or restrained completely 

depending on the nature of support at the node under 

consideration. 

In this study, the beam is assumed to have rectangular cross 

a 
? 

i 

I 

I 

a 

a, 

a, 

x 

Q) 

E 

m 

- 

.- 

Lr, 

n - I 

t j.* 

r + 

Z 

0 

2- 

Q) 

'0 

0 

z 

0 

0 

a, 

a 

I 

0 

N 

a, 

.I 

L 

w 

- 

.- 

a€ 

Y EI 

CI 

YW 

a 

3 

cn 

L 

ii 
e -4- 

r- 

‘e 

e 

e 

section of constant width (b). The beam is considered to deflect 

in the transverse direction due to the flextural action of the 

external forces and moments. 

In Figure (l), the piezo-electric actuator B is shown bonded 

to the clement i of the flexible beam to form a composite beam. 

When an electric field is applied across the film, then it will 

expand if the field is, for example, along the polarization axis 

of the film and will contract if the two were out of phase. The 

expansion or contraction of the film relative to the beam, by 

virtue of the piezo-electric effect, creates longitudinal bending 

stresses in the composite beam which tend to bend the beam in a 

manner very similar to a bimetallic thermostat. 

With proper selection, placement and control -of the 

actuator, it would be possible to generate enough piezo-electric 

bending stresses to counter balance the effect of the external 

forces and moments acting on the beam in a way that minimizes its 

structural deformation. 

B. Model Of An Actuator-Beam Element 

Figure (2) shows a schematic drawing of a piezo-film A 

bonded to an element B of the flexible beam. 

If a voltage v is applied across the film, a piezo-electric 

strain cf is fntrrjduceC in the film and can be conp.;ted from : 

where d is the electric charge constant of the film, m/v 
e 

e 

e 

e 

t 

e 

e 

e 

e 

a 

bl z 

r 

.I 

+ 

U 

Q) 

'13 

c 

0 

n 

FE 

sa E!-p 

cd .I 

I 

n 

OJ 

W 

Q) 

3 

0) 

L 

ii 

e 

. 
-6- 

- 

tl is the thickness of the piezo-actuator, m 

This strain results In a longitudinal stress of given by : 

e 

where is the Young's modulus of elasticity of the film, N/m2 

This, 

axis of the composite beam, given by : 

in turn generates a bending moment Iff, around the neutral 

e 

-(tl+t2-D) 

(3) 

where t2 is the thickness of the beam, m 

b is the width of the beam and the piezo-film, m 

In equation (3), D is the distance of the neutral axis from the 

lower edge of the beam which can be determined by considering the 

force balance in the longitudinal direction X of the beam; or : 

$oldA t $U2dA = 0 

film beam 

(4) 

or 

D 

ydy t E2bI ydy = 0 

-(t +t -D) -(h2-D) 12 

(5) 

where E2 is Young's mndulus of elasticity of the beam 

Equation (5) yields the following expression for D : 
-7- 

e 

e 

Equation (21, (3) and (6) can be combined to determine the 

bending moment Plf generated by the piezo-film on the composite 

beam as follows : 

For this composite beam, it can be easily shown 1121 that it 

has a flextural rigidity (EiIi) given by : 

EiIi = EIIl t EZIZ (8) 

0 

where Il and I are the area moments of inertia of the film and 

the beam about the neutral axis respectively. 

2 

e 

Let us now assume that the composite beam, shown in Figure 

(21, extends a length li between two nodes (i) and (itl). 

Further, it is assumed that the external forces Vi and Vitl as 

well as the external moments Mei and Meitl are acting on the beam 

at nodes i and it1 respectively. Then, the resulting linear and 

angular deformations of the beam yi and 8, as well as yitl and 

at the nodes i and i+l, respectively, can be related to the 'it1 

0 

0 

* 

d 

loads acting on the element as follows [131 : 

12 61i 

61i 41i 2 

-12 -61i 

21i 2 

61i 

Equation (8) can be rewritten as : 

-12 

-61i 

12 

-61i 

- 

61i 

21i 2 

-61i 

2 % 

Fi = Kib, 
-8- 

e 

0 

i 

0 

e 

* 

where Pi is the resultant forces and moments vector acting on 

the beam element i 

is the stiffness matrix of the composite beam element 

i 

is the deflection vector of the nodes bounding the 

beam element 

Ki 

bi 

Equation (9) constitutes the basic finite element model that 

relates the external loads (V and He) and piezo-electric moments 

(PIf) to the deflections (y and 8) of the element as a function of 

its elastic and inertial parameters. 

The equation can be equally used for any element of the beam 

whether it has a piezo-film bonded to it or not. In the latter 

case, PIf is set to zero and flextural rigidity EiIi is set equal 

to that of the flexible beam element under consideration. 

C. Model Of The Overall Actuator-Beam Syatem 

The force-displacement characteristics of the individual 

elements of the beam-actuator system, as given for element i by 

equation (91, are combined to determine the behavior of the 

overall structure. 

The equilibrium conditions of the overall structure will be 

expressed as : 

External forces and moments Forces and moments 

acting on the nodes =xacting on the elements 

of the overall system at these nodes 

or 
-9- 

Nt1 N+l 

* 

e 

F = cFi = Kibi = K6 

if1 1=1 

(11.) 

where K is the overall stiffness matrix of the system (2n92n) 

Bathe and Wilson 1141, Yanq 1151 and Fenner 1161, for 

example, show how to generate the overall matrix K from the 

stiffness matrices Ki of the individual elements. 

In equation (111, the external force vector F and the 

displacement vector 6 are given by : 

and 

FT = VI H1 V2 H, .. .. VN+l str I 

where superscript T denotes the transpose of the vector 

D. Solution Algorithm 

e 

e 

(12) 

For a particular flexible beam configuration, external 

loading conditions and end conditions the effect of applying 

certain voltage V on a piezo-electric actuator of a specific 

thickness t placed at a particular location i on modifying the 

elastic deflection 6 of the beam can be determined by' solving 

equation (11). 

1 

In equation (111, the overall stiffness matrix K is modified 

to account for the end conditions and type of restraints imposed 

on the different nodes. If, for example, a node(j) is simplysupported and is therefore restrained in the Y direction, i.e. 

yj=O, then the off diagonal elements of 2*j row of the overall 
- 10 - 

e 

i 

e 

e 

0 

e 

e 

stiffness matrix K are set equal to zero together vlth the 

corresponding externally applied force. If the jth node is also 

restrained in its angular motion then 8 =O and the off diagonal 

elements of the (2*Jt1) row of the K matrix are eliminated along 

j 

with the corresponding externally applied moment. 

After such a modification process, the elastic deflection 6 

of the beam can be computed from : 

where K-l is the inverse of the overall stiffness matrix 

Figure (3) shows a flowchart of the solution algorithm. 

OPTIXIZATION OF THE ACTUATOR-BEAH 8YBTEH 

The presented analysis algorithm of the actuator-beam 

system is utilized as a basis for the development of the 

optimization procedure for selecting the thickness tl, excitation 

voltage v and location i of the actuator in order to minimize the 

deflection of the beam. 

The optimum design problem is formulated as follows : 

Find thickness tl, voltage v and location i 

t' = c (Yi2 + oi2) N??. 

i=l 

to Minimize 

Such that vi = v 

Mei = ei 

V/tl< v 

* 

i=1,2, ..., Nt1 

i=1,2, ..., Nt1 

i * 

* 
G 

Y 

INPUT 

No. of elements of beam N 

Young's nodulus of element E 

Moment of inertial of element fi 

Element length 1, 

1 

INPUT 

External forces on elements VI 

External moments on elements Mei 

t 

4 

INPUT 

Restraints on each node 

INPUT 

Properties of actuator 

Young's Modulus E 

Charqe constant 4 

INPUT 

Actuator's thickness tl 

Applied voltage v 

Location i 

a 

COMPUTE 

Individual stiffness matrix 

and 

Overall stiffness matrix 

External moment vector by 

MODIFY 

The stiffness matrix to 

account for restraints 

The deflection of 

nodes [Eq.(14)1 

Figure (3) - Flowchart of the analysis algorithm 
e 

i 

- 12 - 

e Such a formulation results in a nonlinear optimization 

0 

i 

problem in 3 design variables tl, v and i which is subjected to 

five inequality constraints. The first inequality constraint 

guards against the application of a high voltage across the 

actuator that may result in its depolirization. The second 

inequality constraint limits the stresses in the piezo-film 

within the acceptable safe limits defined by the allowable stress 

a*. 

The fact that the actuator location is integer makes the 

Optimization problem of the Mixed-Integer Programming (HIP) type. 

e 

e 

A linearization algorithm is used to reduce the problem to a 

series of successive Linear Programs 1171 that are then optimized 

using the Branch and Bound method 118-191 to search for the 

optimal parameters. 

e 

Figure (4) shows a block diagram of the optimization 

algor i the. 

NUKERICAL EXAMPLES 

a The developed optimization procedure is used to optimally 

select, place and control the action of a single piezo-electric 

actuator bonded to a three-element straight beam that is 0.0125m 

wide, 0.0021m thick and 0.15m long. The effect of varying the 
a 

7 

e 

CALL 

The Analysis Algorithm 

0 

a 

0 

0 

P 

I 

I Linearize the Problem 

around (tl, v, i)J I I 

Use Branch and Bound 

a&gorAthm to,Find 4 t # v and i that 

optirnizh the linearized problem 

. 

Figure (4) - Flowchart of the optimization alqorithm 
... 

- 14 - 

-. 

beam loading and end conditions as well as beam and actuator 

materials on the optimal configuration of the system is 

considered in great details. 

* 

e A. Bffcct Of Loading Condition8 

Ji) Transverse Load 

A steel cantilever beam is considered by fixing node 4 of 

the three element beam shown in Figure (5). A load of 0.1N its 

applied to the beam at node 1 perpendicular to its longitudinal 

8 axis. 

Table (1) - Properties of piezo-electric actuators 

a 

If one P2T ceramic actuator, whose properties are given in 

Table (11, is to be used to control the beam deflection, then 

a this actuator can be placed at element 1, 2 or 3. The effect of 
e 

€ 

v) 

0 

0 

E 

v) 

0 

11 

0 

- .I 

- 

E 

Lc) 

0 

Q 
e 

e 

e 

e 

- 16 - 

these three placement strategies on the sum of the deflection 

squared criterion of the beam is shown in Figures (6-a), (6-b) 

and (6-c) respectively. The figures display the iso-contour maps 

of the deflection criterion drawn in the applied voltage v, 

actuator thickness tl plane with the actuator thickness 

normalized with respect to the beam thickness, t2. 

Figure (6-a) indicates that if the actuator is placed at 

element 1, the deflection criterion attains a minimum value of 

0.346~10-~ when the applied voltage is l20v and the actuator 

thickness tl is 0.00021m or tl/t2=0.1. This value of the 

deflection criterion can be reduced further to 0.0995~10-~ when 

16. 

I an optimal actuator, that has a thickness of 0.00425m 

(tl/t2P0.41# is placed at element 2 and the applied voltage is /. controlled at 120 volts as can be seen from Figure (6-b). 

A global minimum of the deflection criterion is obtained 

when the actuator is placed at element 3. The optimum point is 

I 

I constrained as it lies on the maximum applied voltage/unit 

thickness constraint. The optimum thickness of the actuator is 

0.0001051m8 or tl/t2=0.05, and the applied voltage is adjusted at 

80 volts as shown in Figure (6-c). 

A summary of the optimal parameters of the actuator is given 

in Table (2) as a function of the actuator location. The table 

includes also the ratio between the deflection criterion of the 

optimally controlled beam to that of the uncontrolled beam which 

is subfected to the same loading and end conditions. This ratio 

serves as a measure of the effectiveness of the piezo-actuator in 
a 

I 2m.w 

-288.1- 

a 

e) - Actuator at 

1 

a 

? 

i 

E) - Actuator at 

2 

UM 

e 

t 

c) - Actuator at 

3 

a 

a. 

Figure (6) - Iso-contours of the deflection criterion of a 

steel cantilever beam controlled with a PZT 

actuator when subjected to an external end load. 
1- 

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i - 19 - 

controlling the elastic deformation of the beam. 

a 

a 

e 

a 

a 

a 

Table (2) indicates that the ratio of the deflection 

criteria is 1.72, 6.03 and 17.44 when the actuator is bonded to 

elements 1, 2 and 3 respectively. 

Accordingly, the proper placement of the piezo-actuator 

along the beam is very critical in controlling the elastic 

deflection of the beam. With the actuator placed at element 3, it 

would be possible to have an optimally controlled beam that is 

17.44 times less distorted than the uncontrolled beam. 

In order to gain an insight into the effect that the 

actuator location has on the deflection characteristics of the 

beam, let us consider the profiles of the different elastic lines 

obtained by -the optimized actuators. The-linear and angular 

elastic deformation are shown in Figures (7.a) and (7-b) 

respectively for both the uncontrolled and optimally controlled 

beams. It is evident from the figures that placing an optimized 

actuator at element 1 produces the least compensation for the 

external loads and its effects are limited to the first element 

only. However, placing the actuator at the third element, near 

the fixed end, results in considerable improvement that is not 

confined to element 3 but is manifested clearly all over the 

three elements. 

jii) An End Bendinq Moment 

If the considered cantilever beam is subjected to an 

external bending moment of 0.015Nm applied to its free end, then 

e 
'0 

a 

0 

0 

II I 

c 

Figure (7-a) - Effect of the placement strategy of a PZT 

actuator on the linear deflection characteristics 

of a steel cantilever beam subjected to an 

,external load. 

4 

w 

C 

D 

I 

1 

I 

t 

C 

1 

I 

0 

w 

- 

P 

f 

5,611-84 

3,31144 

1,811-04 

-1,291-04 

I I I 

7 ISBE-02 1,131-91 1 I 501-01 

PlSMNCE - N 

Figure (7-b) - Effect of tne placement strategy of a PZT 

actuator on the angular deflection 

characteristics of a steel cantilever beam 

subjected to an external load. 

e 
0 - 21 - 

the effect of placement of the actuator at element 1, 2 or 3 on 

* 

0 

a 

e 

e 

the deflection criterion is as shown in the iao-contour maps of 

Figure (8-a), (8-b) and (8-c) respectively. 

Figure (8-a) indicates that placing the actuator at element 

1 makes the deflection criterion attains a minimum of 7.05~10-~ 

when the applied voltage is 320v and the actuator thickness is 

0.00084m. When the actuator is placed at 2 the deflection 

criterion assumes a much lower minimum value of 2.ll~lO-~. This 

minimum value is obtained by an actuator that is only half as 

thick which is excited by a much lower voltage of only 200 volts 

as compared to the 320 volts required if the optimal actuator is 

placed at element 1. 

Accordingly, the proper placement of the actuator along the 

structure results not only on minimizing the elastic distortion 

of the beam but also in reducing the size and power consumption 

of the actuator. 

If the actuator is placed, however, at 3, the deflection 

criterion starts to increase and even with an optimized actuator 

the best sum of the deflection squared that could be obtained is 

2.8~10-~. 

A summary of the optimum parameters of the actuator along 

with the resulting improvement as measured with reference to the 

uncontrolled beam is given in Table (2). 

The obtained results indicate that in the case of a moment 

applied to the end of the cantilever beam it is found beneficial 
I 

e 

i 

e 

c 

c 

e 

e 

e 

0 

0 

IMUH 

! 0 

! i -=.a 

a) - Actuator at 

1 

b) - Actuator at 

2 

t 

c) - Actuator at 

3 

, .I. t 

0 I 

0.4 1-& I ors 0. I 

-CTUL.TOR,wELW.I TUICYNEICI RP.T.0 - f*'T=? 

Figure (8) - Iso-contours of the deflection criterion of a 

steel cantilever beam controlled with a PZT 

actuator when subjected to an external bending 

moment. 
- 23 - 

'i 

e 

0 

r) 

to place the piezo-actuator-at clement 2 to achieve the minimum 

distortion. This can also be emphasized by considering the linear 

and angular profiles of the elastic lines shown in Figures (9-a) 

and (9-b) respectively for the controlled and uncontrolled beam. 

It should be pointed out here that the obtained optimal 

location of the actuator in case of an end moment is different 

from that obtained when the load is a transverse load applied to 

the beam end. This is inspite of the fact the load produces the 

same moment at the fixed end of the cantilever beam. 

8. Effect Of End Condition8 

Two end conditions, other than the fixed-free condition of 

the cantilever beam, are considered for the three element steel 

beam of -Figure (5). Under these conditions, the beam is subjected 

to same transverse load of 0.1N. 

Ji) Overhunq Beam 

The beam is fixed at node 4 and simply-supported at node 2. 

The load is applied to the beam at its free end. In this case, 

the minimum deflection criterion is obtained when the actuator is 

bonded to element 2. The optimal actuator should have a thickness 

of 0.000336~1 and be subjected to 120 volts. The improvement in 

the deflection criterion with respect to the uncontrolled case is 

8.17 times. 

A summary of the optimal parameters of the actuator is given 

in Table (2) as function of the actuator location. 
,- 

0 D 

c 

t 

I 

E 

C 

T 

I 

0 

w 

- 

8 .Uf- bS 

4.911-45 

1 531-05 

-1 I901-85 

c 

1 I' ..f 

Figure (9-a) - Effect of the placement strategy of a PZT 

actuator on the linear deflection characteristics 

of a steel cantilever beam subjected to an 

external bending moment. 

II 

W 

c l*l2E-Q3 

E 

1 6,211-04 I 

E 

C 

I 1,221-94 I 

0 

W 

i 

-3 a 781-84 - 

r 

I I I I 

d e,@ 1 +@e 3,751-82 7 I 581-02 1,131-01 1,581-@1 0 

DISTANCE - B 

Figure (9-b) - Effect of the placement strategy of a PZT 

actuator on the angular deflection 

characteristics of a steel cantilever beam 

subjected to an external bending moment. 
a 25 - 

a 

Figures (10-a) and (10-b) display the linear and angular 

deformations respectively of the overhung beam with and without 

the actuators. It is evident, from these two figures, that 

placement of the actuator at element 2 results in the minimal 

deflection profiles. 

The three element beam is considered now with its two end 

nodes 1 and 4 are simply-supported. The transverse load is 

applied to the beam at node 2. 

The results of the optimization procedure, summarized in 

Table (23, indicates that the optimal deflection control can be 

achieved vhen the actuator is placed at element 2. The resulting 

improvement over the uncontrolled beam is 8.91 times. This 

requires the actuator to be 0.00336m thick and excited by 80 

volts. 

a 

'0 

Figures (ll-a) and (ll-b) show the corresponding linear and 

angular deformations respectively of the controlled and 

uncontrolled beam. The figures illustrate clearly the 

effectiveness of the optimal actuator 2 in compensating for the 

externally applied load. 

C. Effect Of Beam Material 

The effect of changing the beam material from steel to 

aluminum is considered for the case of a simply-supported beam 

with a transverse load of 0.1N applied at node 2. 
e 

a 

e 

8 

a 

I II 

Figure (10-a) - Effect of the placement strategy of a PZT 

actuator on the linear deflection characteristics 

of a steel overhung beam. 

A 

W 

C 1,251-W 

D 

E 

f 7,301-95 1 

E 

C 

I 2,891-6 1 

0 

W 

-3 121-85 

Figure (10-b) - Effect of the placement strategy of a PZT 

actuator on the angular deflection 

characteristics of a steel overhung beam. 
a 

0 

a 

a 

DISIANCE - II 

Figure (11-a) - Effect of the placement strategy of a PZT 

actuator on the linear deflection characteristics 

of a simply-supported steel beam. 

i 

1 

f 

1 

L 

C 

I 

I 

0 

W 

r 

a 

1 

SIRFLY-SIPPOhlED 3ltl BEAM , PZT LCIUATOk 

fo(ODE2\ : 0,l N 

I . 

Figure (11-b) - Effect of the placement strategy of a PZT 

actuator on the angular deflection 

characteristics of a simply-supported steel beam. 
e - 28 - 

0 

The optimization algorithm yields results that are similar 

to those obtained for the steel beam. The optimum location of the 

actuator is on element 2 and the improvement over the 

uncontrolled beam is 9.10 which is slightly higher than the steel 

beam case. Important here to note that the thickness of the 

optimal actuator is only half that required to control the steel 

beam whereas the applied voltage is of the same magnitude. 

The optimal linear and angular deflection profiles of the 

aluminum beam, which are shovn in Figures (12-a) and (12-b), are 

very similar to those of the steel beam. However, the magnitude 

of these deflections are about 3 times higher than those of the 

steel beam. This difference is attributed mainly to the 

difference in Young's modulus of elasticity of the two 

materials. 

e 

D. Effect Of Actuator Material 

Two actuator materials are considered other than the PZTceramic actuators. These two materials are : 

* 

ji) G1278-Ceramic 

This type of actuator, as indicated in Table (11, has about 

half the Young's modulus of elasticity of the PZT actuators but 

at the same time it has almost twice as much the charge constant. 

These two factor combined result in nearly the same piezoelectric stress and bending moment. 

The output of the optimization algorithm indicates, as 

expected, that the optimal parameters of the G1278 and PZT 
0 

e 

3," 

Figure (12-a) - Effect of the placement strategy of a PZT 

actuator on the linear deflection characteristics 

Qf a simply-supported aluminum beam. 

E 

r I 

i O.@!+Ki 3,751-82 7 581-82 1 I 131-81 1 I 581-01 1 I I 

DISTANCE - a 

Figure (12-b) - Effect of the placement strategy of a PZT 

actuator on the angular deflection 

characteristics of a simply-supported aluminum 

beam. e 

e 
0 - 30 - 

a 

e 

Q 

0 

e 

0 

0 

actuators are nearly the 

clearly emphasized in 

- 

same as shown in Table (21, Thi6 1s 

the resulting linear and angular 

deformations profiles of Figure (13-a) and (13-b) respectively. 

(ii) KYnar-Polymeric 

A kynar actuator is considered as a typical representative 

of the polymeric piezo-electric actuators. This type of actuators 

has very low Young's modulus of elasticity and also very small 

charge constant. Accordingly, it would only be capable of 

producing small piezo-electric moments unless the voltage applied 

across the actuator is increased considerably to counter balance 

the low E and d parameters. 

Considering this actuator on the aluminum beam, the 

optimization routine indicates that the best results are obtained 

with an actuator placed at location 2. This actuator should be 

twice as thick as the optimal PZT actuator and should be excited 

by a voltage which is 2.5 times as high. With these optimal 

parameters, the obtained improvement is only 2.17 times compared 

to 9.10 times for the PZT actuators. 

Figures (14-a), (14-b) and (14-c) display the iso-contour 

maps of the deflection criterion when the kynar actuator is 

placed at element 1, 2 or 3 respectively. These figures emphasize 

the results of the optimization routine as it can be seen that 

the optimal actuator parameters tend to move towards the large 

thickness and large voltage constraint boundaries. 

Figures (15-a) and (15-b) show the linear and angular 

0 
0 

a 

0 

BISIhNQ - II 

Figure (13-a) - Effect of the placement strategy of a G1278 

actuator on the linear deflection characteristics 

of a simply-supported aluminum beam. 

a 

c 

n 

W 

C , UNCOHTROLLD 

n 

-2,861-84 

I I , i O.BB!*88 3 ,151-82 7,501-82 1 I13t-01 1 I501-01 

BISIBHCI: - II 

t 

Figure (13-b) - Effect of the placement strategy of a G1278 

actuator on the angular deflection 

characteristics of a simply-supported aluminum 

beam. 

0 
I 

i! 

0 

I! 

f 

2a.a I 

a 

F 

o 

I! 

J i 

OOPTIP :UM 

OPTIEI 

e OPT IKUN 

a) - Actuator at 

1 

b) - Actuator at 

2 

c) - Actuator at 

3 

1 I I I 

. Ol5 0.1 0.4 1.6 

nc-run-roa/mEn- TUICYNEOB nn-rxa - ~107-2 

Figure (14) - Iso-contours of the deflection criterion of a 

simply-supported aluminum beam controlled with a 

kynar actuator. 
e 

0 

0 

e 

a 

a 

0 

DISTAKI: - R 

Figure (15-a) - Effect of the placement strategy of a kynar 

actuator on the linear deflection characteristics , of a simply-supported aluminum beam. 

L 

W 1165L-W 

C 

I 

I 

C -8.23E-05 

1 

1 

0 

W -2,861-84 

SINFLY-SUPPORTED lLUMINUH BUM, XYMR ACTUATOR 

T(NODE t) : 1.1 N 

Figure (15-b) - Effect of the placement strategy of a kynar 

actuator on the angular deflection 

characteristics of a simply-supported aluminum 

beam. 
0 - 34 - 

0 

e 

deformations respectively of the aluminum beam. The obtained 

results indicate that the optimal deflections, with the actuator 

at 2, are about 4 times higher than those obtained with the 61278 

or P2T actuators. 

It should however be pointed out here that the moxe 

effective deflection control could be obtained by the kynar 

actuator if the applied voltage is increased beyond the set 

limits of the constraints. 

CON CLUS I ON 8 

0 

0 

0 

0 

a 

This study has addressed the important problems of optimal 

selection, placement and control of piezo-electr ic actuators in 

order to minimize the static deflection of elastic beauts under 

different loading conditions. Emphasis has been placed in this 

study on the effect that the actuators has on changing the 

elastic and inertial properties of the structural element to 

which they are bonded to. 

The developed optimum design procedure is applied, with a 

large degree of flexibility, to elastic beam of various 

configurations and subjected to wide variety of end conditions. 

The obtained results suggest the importance of the developed 

strategies in controlling the structural deformations of elastic 

systems. 

Further work is in progress to integrate the presented 

procedure into the optimal control of the dynamics of more 

e 
- 35 - 

complex structures. 

a 

e 

a 
I- * 

a 

- 36 - 

RBFERENCES 

a 

0 

1. Crawley, E.F.@ and J.de Luis, "Use of Piezo-Ceramics as 

distributed actuators in Large Space Structures", Proc. of 

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a 

e 

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e 

13. Pa2, M., "Structural Dynamics : Theory and Computation", 2nd 

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a 

14. Bathe, K.J. and E.L. Wilson, "Numerical Methods in Finite 

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a