Optimal vibration control of piezolaminated smart beams by the maximum principle

Computers and Structures 89 (2011) 744–749

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Optimal vibration control of piezolaminated smart beams by the maximum principle Ismail Kucuk a,⇑,1, Ibrahim S. Sadek a, Eiman Zeini b, Sarp Adali c a

Department of Mathematics and Statistics, American University of Sharjah, United Arab Emirates Department of Mathematics, University of Toronto, Canada c School of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa b

a r t i c l e

i n f o

Article history: Received 29 June 2010 Accepted 22 February 2011

Keywords: Optimal control Vibrating Beams Piezoelectric patches Maximum principle

a b s t r a c t Active control of a vibrating beam using piezoelectric patch actuators is considered. The specific structure to be studied is an Euler–Bernoulli beam with piezoelectric actuators bonded to the top and bottom surfaces of the beam. The equation of motion includes Heaviside functions and their derivatives due to finite size piezo patches which provide the control force to damp out vibrations. Optimal control theory is formulated with the objective function specified as a weighted quadratic functional of the dynamic responses of the beam which is to be minimized at a specified terminal time. The expenditure of the control forces is included in the objective function as a penalty term. The optimal control law is derived using a maximum principle developed by Sadek et al. [1]. The maximum principle involves a Hamiltonian expressed in terms of an adjoint variable with the state and adjoint variables linked by terminal conditions leading to a boundary-initial-terminal value problem. The explicit solution of the problem is developed using eigenfunction expansions of the state and adjoint variables. The numerical results are given to assess the effectiveness and the capabilities of piezo actuation to damp out the vibrations. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The emergence of new actuator mechanisms associated with smart materials has recently influenced the development of active structural control strategy [2]. In particular, the piezoelectric actuators are used as effective control devices to control structural vibrations and have a wide range of engineering applications. One of the most widely used piezo materials in active control is piezoceramic materials such as PZT for their large bandwidth, mechanical simplicity and mechanical ability to produce forces acting against vibrations in the system. The books by Preumont [3] and Banks et al. [4] as well as the review articles by Trindade and Benjeddou [5], Frecker [6] and Correia et al. [7] provide an overview of these materials and the related modeling and control techniques. The extensive use of piezoelectric polymers or ceramics as actuators and sensors for the control of a structure has been reported widely in the literature which, in particular, include the active vibration control problems for beams [8,9], discs [10], rotating beams [11], rectangular plates [12], shells [13], trusses [14] and sandwich structures [15]. A mathematical model considering ⇑ Corresponding author. E-mail address: [email protected] (I. Kucuk). Visiting Research Associate, School of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa. 1

0045-7949/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2011.02.012

geometrical nonlinear effects and piezoelectric effects for smart beams has been studied in [16] where the authors have developed feedback control laws for nonlinear vibrations. The problem of optimal placement of piezoelectric actuators has been studied in a number of recent studies [17–20]. The equipotentiality condition between the piezoelectric layers has been imposed in [21] and the effect of this condition on frequencies was studied. A short discussion on equipotentiality condition was given in [22] where feedback and feedforward controls were implemented as part of an active–passive control strategy. The above work made use of the feedback control to reduce the vibrations employing sensor-actuator closed loops. A comparison of various control strategies for piezoelectric beams was given in [23] where sensor and actuator piezo patches were used to suppress vibrations. In using a piezoelectric material as an actuator, converse piezoelectric effect activated by an electric field, is employed for inducing mechanical stresses or strains which, in turn, are transformed into control forces or moments by a suitable structural arrangement [24]. The present study takes advantage of the fact that the electric field can be employed as a time-dependent open-loop control variable in suppressing the excessive vibrations. This observation is implemented for the structural control of a vibrating beam with the piezo-electrical material acting as an actuator. The effectiveness of the proposed control mechanism is demonstrated by a numerical example. Distributed vibration control of beams using the piezoelectric effect has been studied by several investigators,

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see for example [25–29] using different types of control strategies. For the optimal control of one-dimensional structures, a maximum principle has been formulated to develop open-loop control strategies [30,31]. In the present study, this theory is applied to obtain the optimal open-loop control of a smart beam where the control force is provided by a piezoelectric patch actuator. The formulation of the piezo-control problem is defined as an optimal control problem where the applied voltage is employed as the control function to damp out excessive vibrations by minimizing the dynamic response of the beam using piezoelectric patches as actuators. The dynamic response is defined as the deflection and the velocity of the beam. The objective function is specified as the sum of the weighted quadratic cost functional of the dynamic response at a specified time and the expenditure of control force between the initial and terminal time. The actuators bonded on the surface of the beam are activated by the time-dependent electric field and apply the control moment. The optimal control law for the beam is derived by introducing the adjoint problem and the related Hamiltonian in the form of a maximum principle. The maximum principle gives a relationship between the optimal control force and the adjoint variable which is coupled to the state variable through terminal conditions. Thus the control problem is formulated as a boundary-initial-terminal value problem the solution of which yields the control function. Explicit solutions are obtained by the use of eigenfunction expansions for the state and adjoint variables for a simply supported beam. The effectiveness of the proposed control mechanism to damp out vibrations is demonstrated by a numerical example.

Eq. (1) is subject to the following boundary and initial conditions: Boundary Conditions (BC’s):

wð0; tÞ ¼ 0 and wðL; tÞ ¼ 0;

ð3Þ

wxx ð0; tÞ ¼ 0 and wxx ðL; tÞ ¼ 0: Initial conditions (IC’s):

wðx; 0Þ ¼ w0 ðxÞ and wt ðx; 0Þ ¼ w1 ðxÞ:

ð4Þ

2.2. Control problem and maximum principle The objective of the optimal control problem is to determine a control function f(t) which minimizes a measure of displacement and velocity at a pre-defined terminal time, tf. The performance index consisting of the dynamic response of the beam and a penalty term on the expenditure of voltage is defined as

J ðf Þ ¼

Z

L

0





l1 w2 ðx; tf Þ þ l2 w2t ðx; tf Þ dx þ

Z

tf

0

l3 f 2 ðtÞdt

ð5Þ

where weighting factors lis satisfy li P 0, i = 1, 2; l1 + l2 > 0; and l3 > 0. In Eq. (5), the first term represents the potential energy of the beam at terminal time and the last term is the measure of the total voltage spent in the control process. The optimal control problem involves the computation of the optimal control f0(t) 2 Uadm = {fjf 2 L2(0, tf)} such that

J ðf 0 Þ 6 J ðf Þ; 8f 2 U adm :

ð6Þ

2. Formulation of the control problem

The solution of the optimization problem (6) is obtained by introducing an adjoint problem with the adjoint variable v(x, t) satisfying the following partial differential equation:

2.1. Equation of motion of a beam with piezo patches

L ½v  ¼ qAv tt þ EIv xxxx ¼ 0:

ð7Þ

with boundary conditions We consider a simply supported beam, as shown in Fig. 1, bonded with piezoactuators on the upper and lower surfaces. The beam has a length L, thickness h, mass density q and cross sectional area A. Eb is Young’s modulus and I is the moment of inertia of the beam. Equation of motion is given in the following form [2]

L½w ¼ Kf ðtÞðH00 ðx  x1 Þ  H00 ðx  x2 ÞÞ;

0 < x < L;

v ð0; tÞ ¼ 0 and v ðL; tÞ ¼ 0; v xx ð0; tÞ ¼ 0 and v xx ðL; tÞ ¼ 0; and terminal conditions

v t ðx; tf Þ ¼

0 < t < tf ; ð1Þ

where w = w(x, t) is the deflection of the beam, H is the Heaviside function and the primes denote differentiation with respect to x. For simplicity, one actuator between x1 and x2 with xi 2 [0, L] for i = 1, 2 is considered. Function, f(t) is the applied voltage on a piezoelectric patch and the constant K is given by

K¼

Eb Id31 j and ha

j¼

12Ea ha ðh þ ha Þ h i; 3 3 2Eb h þ Ea ðh þ 2ha Þ3  h

2l1 2l wðx; t f Þ and v ðx; tf Þ ¼  2 wt ðx; t f Þ: qA qA

L½w ¼ qAwtt þ EIwxxxx :

max Hðx; t; v ðx; tÞ; f ðtÞÞ ¼ Hðx; t; v ðx; tÞ; f o ðtÞÞ

ð2Þ

ð9Þ

Theorem (Maximum Principle). It follows from the convexity of J(f) that the optimal control is unique. Assuming that an optimal control exists, our main objective is to derive a maximum principle that can be used to determine the optimal control. The maximum principle can now be stated as follows: If fo(t) 2 Uadm = {fjf 2 L2(0, tf)}is the optimal control then it satisfies the maximum principle f 2U adm

in which d31 is a piezoelectric constant, Ea is Young’s modulus and ha is the thickness of the actuator; H00 ðx  xi Þ ¼ d0 ðx  xi Þ; i ¼ 1; 2 that is the first derivative of Delta Dirac function and the differential operator is given by

ð8Þ

ð10Þ

where v(x, t) is the solution to the adjoint problem given by Eqs. (7)– (9) and H is a Hamiltonian type-function given by

Hðt; v ; f Þ ¼ RðtÞf ðtÞ þ l3 f 2 ðtÞ;

ð11Þ

in which R(t) = vx(x1, t)  vx(x2, t). Then

J ðf o ðtÞÞ 6 J ðf ðtÞÞ;

8f ðtÞ 2 U adm

ð12Þ

and in which fo(t) is indeed the optimal control function. Proof The change in w and f are introduced as Fig. 1. Beam Diagram with piezoelectric patches.

Dw ¼ wðx; tÞ  wo ðx; tÞ;

and Df ¼ f ðx; tÞ  f o ðx; tÞ

ð13Þ

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in which wo is the optimal displacement and fo is the optimal actuator. It is noted that

Z

Z

L

tf

ðv o L½Dw  DwL ½v o Þdtdx

0

0

Z

¼ qA

Z

L

¼

Z

tf

Z

tf

v

o

Dwtt  Dwv

L



v

o

o tt



L

o

0

Z  i 2 þ l2 w2t ðx; t f Þ  wot ðx; t f Þ dx þ

tf

0

o xxxx

ð23Þ

00

ð14Þ

The Taylor series expansion of w2(x, tf) and w2t ðx; t f Þ around w ðx; t f Þ 2 and wot ðx; t f Þ; respectively, yields 2

2

2

2

w2 ðx; t f Þ  wo ðx; t f Þ ¼ 2wo ðx; tf ÞDwð; t f Þ þ r1 ;

Let

w2t ðx; t f Þ  wot ðx; t f Þ ¼ 2wot ðx; tf ÞDwt ð; tf Þ þ r 2 ;

I1 ¼ qA

Z

tf

0 Z L

0 tf

Z

I2 ¼ EI

Z

L

0





v o Dwtt  Dwv ott

v

o

Dwxxxx  Dwv

0



dtdx;

o xxxx



ð15Þ

dtdx

ð16Þ

I3 ¼

L

K v o Df ðtÞðH00 ðx  x1 Þ  H00 ðx  x2 ÞÞdxdt

ð17Þ

v ot ðx; 0ÞDwðx; 0Þ ¼ 0 since wðx; 0Þ ¼ w0 ðxÞ and wo ðx; 0Þ ¼ w0 ðxÞ; v o ðx; 0ÞDwt ðx; 0Þ ¼ 0 since wt ðx; 0Þ ¼ w1 ðxÞ and wot ðx; 0Þ ¼ w1 ðxÞ; L

DJ ðf ðtÞÞ ¼

DJ ðf ðtÞÞ ¼  K Z þ

Similarly, integrating the second term in (16) four times by parts with respect to x and using boundary conditions (3) and (8), the term becomes

Z

tf

L

0

0



v o Dwxxxx  Dwv oxxxx



dtdx ¼ 0:

ð19Þ

Finally, integrating the right side of Eq. (17) by parts with respect to x leads to

I3 ¼

tf 0



Z

tf

L

Kv 0

0

o 0 x Df ðtÞðH ðx

Z

tf

K



 x1 Þ  H ðx  x2 ÞÞdxdt;

0



v ox ðx2 ; tÞ  v ox ðx1 ; tÞ

0 tf





v ox ðx2 ; tÞ  v ox ðx1 ; tÞ

Df ðtÞdt

2

l3 ðf 2 ðtÞ  f o ðtÞÞdt P 0

ð27Þ

since the first term in (26) is strictly positive. Assuming that RðtÞ ¼ Kðv ox ðx2 ; tÞ  v ox ðx1 ; tÞÞ, the latter inequality can be rewritten as

Z 0

tf

 2 ðRðtÞf ðtÞ þ l3 f 2 ðtÞÞ  ðRðtÞf o ðtÞ þ l3 f o ðtÞÞ dt P 0:

ð28Þ

Recall that the maximum principle is introduced to obtain the optimal function f(t),

  max Hðx; t; v ox ðx; tÞ; f ðtÞÞ ¼ max RðtÞf ðtÞ þ l3 f 2 ðtÞ :

f 2U adm

f o ðtÞ ¼

0

ð20Þ

and after using the boundary conditions (8) and the shift property of the Delta Dirac function, one obtains

I3 ¼

tf

f 2U adm

It immediately follows that the optimal function f(t) is obtained as

K v o Df ðtÞðH0 ðx  x1 Þ  H0 ðx  x2 ÞÞjL0 dt

Z

Z

0

ð18Þ

Z

Z tf  o  ðl1 r 1 þ l2 r 2 Þdx  K v x ðx2 ; tÞ  v ox ðx1 ; tÞ Df ðtÞdt 0 0 Z tf  2 o2 þ l3 f ðtÞ  f ðtÞ dt ð26Þ L

which takes the following inequality form

0

Z

Z

0



Dwðx; t f Þv 0t ðx; t f Þ  Dwðx; 0Þv 0t ðx; 0Þ 0   v 0 ðx; tf ÞDwðx; t f Þ þ v 0 ðx; 0ÞDwt ðx; 0Þ dx Z L   Dwðx; t f Þv 0t ðx; t f Þ  v 0 ðx; tf ÞDwðx; t f Þ dx ¼ 0 Z L   l1 w0 ðx; tf ÞDwðx; tf Þ þ l2 w0t ðx; tf ÞDwt ðx; tf Þ dx: ¼ 2

I2 ¼ EI

2l1 wo ðx; t f ÞDwðx; t f Þ þ l1 r 1 þ 2l2 wot ðx; t f ÞDwt ðx; t f Þ Z tf   2 þ l2 r 2 dx þ l3 f 2 ðtÞ  f o ðtÞ dt ð25Þ

in which ri = ri(x) > 0 for i = 1, 2. Incorporating the Eq. (22) into (25) leads to

one obtains

Z

L

0

Integrating Eq. (15) twice by parts with respect to t and applying terminal conditions given in (9) and the following relations

I1 ¼

Z

0

0

0

and r 2 ¼ 2ðDwt Þ2 þ    > 0:

Substituting the terms in (24) into Eq. (23) leads to

DJ ðf ðtÞÞ ¼

Z

tf

ð24Þ

where

r1 ¼ 2ðDwÞ2 þ    > 0;

and

Z



2

o2

K v Df ðtÞðH ðx  x1 Þ  H ðx  x2 ÞÞdxdt 00



l3 f 2 ðtÞ  f o ðtÞ dt:

 dtdx

0

0

DJ ðf ðtÞÞ :¼ J ðf ðtÞÞ  J ðf o ðtÞÞ Z Lh  2 l1 w2 ðx; tf Þ  wo ðx; tf Þ ¼

dtdx

Dwxxxx  Dwv

0

0

Z



0

0

Z

þ EI

tf

Now, consider the change in the performance index

Df ðtÞdt:

ð21Þ

1 K RðtÞ ¼ ðv o ðx2 ; tÞ  v ox ðx1 ; tÞÞ 2l3 2l3 x

ð29Þ

where v(x, t) is the solution of the adjoint problem introduced in (7) along with (8) and (9). In order to determine the control force fo(t) in Eq. (29), one needs to evaluate the adjoint variable vo(x, tf) in (7) that requires the solution of the optimal state function wo(x, tf) for the system given by (1)–(4) through the mixed terminal conditions (9). h

In the light of Eqs. (15)–(21), the equality in Eq. (14) reads

2

Z

L



0

¼

Z 0

2.3. Solution method



l1 wo ðx; tf ÞDwðx; tf Þ þ l2 wot ðx; tf ÞDwt ðx; tf Þ dx

tf

K



v ox ðx2 ; tÞ  v ox ðx1 ; tÞ



Df ðtÞdt:

ð22Þ

The solution, w(x, t), of equation of the motion given in (1) with the boundary conditions (3) and the initial conditions (4) is sought by using the separation of variables:

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I. Kucuk et al. / Computers and Structures 89 (2011) 744–749

wðx; tÞ ¼

N X

un ðxÞZ n ðtÞ;

ð30Þ

n¼1

Parameter

where the eigenvalue is defined as kn ¼ nLp and the corresponding normalized eigenfunction

un ðxÞ ¼

Table 1 Properties of the beam.

rffiffiffi 2 sinðkn xÞ: L

Value

Unit 10

Young Modulus, E Area moment of inertia, I Density, q Area, A

N/m2 m4 kg/m3 m2

6.9  10 9  1011 2700 1.2  104

Substituting the latter form of w(x, t) into (1) rewrites the motion equation as

(

n¼1

) k4n EI € Z n ðtÞ þ Z ðtÞ un ðxÞ ¼ aðxÞf ðtÞ qA n

ð31Þ

where

aðxÞ ¼

0.01 w(0.5,t)

N X

K ðH0 ðx  x1 Þ  H0 ðx  x2 ÞÞ: qA

0.005

Using the orthogonality property of the eigenfunctions gives rise to the following ordinary differential equation in time

Z€ n ðtÞ þ bn Z n ðtÞ ¼ gn f ðtÞ

ð32Þ

0

0.2

0.4

0.6

0.8

1

t -0.005

where

bn ¼

k4n EI qA

and gn ¼

Z

L

aðxÞ sinðkn xÞdx:

-0.01

0

The solution of (31) is given by

pffiffiffiffiffi pffiffiffiffiffi Z n ðtÞ ¼ c1n cosð bn tÞ þ c2n sinð bn tÞ Z t pffiffiffiffiffi gn sinð bn ðt  sÞÞf ðsÞds þ pffiffiffiffiffi bn 0

where the constants cin,i = 1, 2 are determined by using the initial conditions (4):

c1n ¼

Z

L 0

Z

ð34Þ

L

0

Fig. 2. Controlled and uncontrolled tip deflection for the weighting factors l1 = l2 = 1 and l3 = 0.001.

0.6

w0 ðxÞun ðxÞdx;

1 c2n ¼ pffiffiffiffiffi bn

Uncontrolled Controlled

ð33Þ

w1 ðxÞun ðxÞdx:

0.4

Similarly, the solution of the adjoint Eq. (7) is expressed as

v o ðx; tÞ ¼

N X

0.2

un ðxÞQ n ðtÞ

ð35Þ

n¼1

0

where Qn(t) is given by

pffiffiffiffiffi pffiffiffiffiffi Q n ðtÞ ¼ d1n cosð bn tÞ þ d2n sinð bn tÞ:

After substituting (35) into the optimal function f (t) given in (29), one obtains the optimal solution wo(x, t)

wo ðx; tÞ ¼

N X

un ðxÞZ on ðtÞ

0.4

0.6

0.8

1

t

ð36Þ o

0.2

-0.2

-0.4

ð37Þ

n¼1

-0.6

where

pffiffiffiffiffi pffiffiffiffiffi Z on ðtÞ ¼ c1n cos bn t þ c2n sin bn t Z t( N pffiffiffiffiffi pffiffiffiffiffi X K gn pffiffiffiffiffi  sin bn ðt  sÞ kn ðd1n cos bn s 2l3 bn 0 n¼1 pffiffiffiffiffi o þ d2n sin bn sÞ ðcos ðkn x1 Þ  cos ðkn x2 ÞÞ ds ð38Þ where the constants cin,i = 1, 2 are given in (34) and din,i = 1, 2 are computed by the terminal conditions (9). Inserting the expansions (35) and (37) into the mixed terminal conditions (9) leads to a system of linear modal equations:

Controlled Uncontrolled Fig. 3. Controlled and uncontrolled tip velocity for the weighting factors l1 = l2 = 1 and l3 = 0.001.

d l Q ðt f ; d1n ; d2n Þ ¼ 2 1 Z n ðtf ; d1n ; d2n Þ; dt n qA l d Z ðt ; d ; d Þ: Q n ðt f ; d1n ; d2n Þ ¼ 2 2 qA dt n f 1n 2n

ð39Þ

Substituting the modal time solutions (33) and (36) into the lumped parameter system (32) yields two linear equations in terms of d1n

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I. Kucuk et al. / Computers and Structures 89 (2011) 744–749

Table 2 Values of J E ðf o Þ; J C ðf o Þ; E and the tip deflection and velocity at l1 = l2 = 1 for various values of l3.

l3

J E ðf o Þ

J C ðf o Þ

E

w(1,tf)

wt(1,tf)

0.001 0.01 0.1 1.0 10.0 100.0

0.2701 e6 0.4882 e5 0.1545 e4 0.1906 e3 0.5540 e2 0.1871e1

0.4442 e3 0.2963 e2 0.2590 e1 0.2186 0.6763 0.2287

0.4442 0.2963 0.2590 0.2186 0.6763 e1 0.2287 e2

0.393 e12 0.1279 e11 0.2119 e11 0.2009 e11 0.1990 e11 0.1979 e11

0.8589 e14 0.8550 e13 0.8421 e12 0.7754 e11 0.4313 e10 0.7932 e10

and d2n. After solving the system for d1n and d2n, the adjoint function v0(x, t) is determined from Eq. (35). The solution for fo(t) is given explicitly by Eq. (29) and finally the optimal response of the beam wo(x, t) is obtained from (30). The optimal performance index can be computed from

J ðf o Þ ¼

Z

L

2

2

ðl1 wo ðx; t f Þ þ l2 wot ðx; t f ÞÞdx þ

0

Z 0

tf

2

l3 f o ðtÞdt:

ð40Þ 4. Conclusion

3. Numerical results and discussion The theoretical results developed in the preceding sections are simulated to show the robustness of the technique to overcome excessive vibrations in mechanical systems. For the simulations, the terminal time is taken as tf = 1 s, the length of the beam L = 1 m, K = 0.3 Nm/V. The properties of the beam are shown in Table 1 with the values used taken from [2]. The beam is subject to the following initial conditions:

wðx; 0Þ ¼ 0 and wt ðx; 0Þ ¼ u1 ðxÞ: Although the displacement w(x, t) in (30) is calculated for the first mode N = 1, the method can be used for higher modes. The deflection and velocity of the free-end of the cantilever are presented in Figs. 2 and 3 with the collocated sensor-actuator pair placed at x1 = 0.3 and x2 = 0.7 for the time interval 0 6 t 6 1. The weighting factors are taken as l1 = l2 = 1 and l3 = 0.001. A comparison of the controlled and uncontrolled displacements in Fig. 2 indicates substantial damping as a result of the control. The same observation also applies to controlled and uncontrolled velocities shown in Fig. 3. Next we define the weighted dynamic response J E ðf o Þof the controlled beam given by

J E ðf o Þ ¼

Z

L





l1 w2 ðx; tf Þ þ l2 w2t ðx; tf Þ dx

0

ð41Þ

and the weighted voltage spent in the control process, J C ðf o Þ given by

J C ðf o Þ ¼

Z 0

tf

l3 f 2 ðtÞdt

ð42Þ

noting that J ðf o Þ ¼ J E ðf o Þ þ J C ðf o Þin Eq. (40). Moreover the total voltage spent in the control process is given by

E¼

Z

tf

f ðtÞ2 dt:

to the uncontrolled value of J E ðf ¼ 0Þ ¼ 0:02275. Similarly the tip deflection and velocity at terminal time increases as the penalty on the control functional E increases. It is noted that this increase is not monotonic in the case of the tip deflection as the control process minimizes the functional J ðf Þ as a whole, not the deflection or the velocity at a point.

ð43Þ

0

Table 2 shows the values of these functional at the terminal time tf = 1 for various values of l3 with l1 = l2 = 1. Also shown in Table 2 are the values of the tip deflection and velocity at tf = 1. Substantial decrease in the dynamic response functional J E ðf o Þof the controlled beam is observed from Table 2 noting that for the uncontrolled beam J E ðf ¼ 0Þ ¼ 0:02275. As expected J E ðf o Þ increases as the total control voltage given by E decreases as a result of larger l3 which, in turn, increases the penalty on the expenditure of force given by E in Eq. (43). For l3 ¼ 100; J E ðf o Þ ¼ 0:01871 which is close

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