actuator patches

International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

Modeling and analysis of curved beams with debonded piezoelectric sensor=actuator patches Dongchang Sun, Liyong Tong ∗ School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, NSW 2006, Australia Received 25 June 2001; received in revised form 7 March 2002

Abstract In this paper, a mathematical model for thin-walled curved beams with partially debonded piezoelectric actuator=sensor patches is presented for investigating the e+ect of debonding of the actuator=sensor on their open- and closed-loop behaviors. The actuator equations and the sensor equations of the curved beam in perfectly bonded and debonded regions are derived. In the perfect bonding region, the adhesive layer is modeled to carry constant peel and shear stresses; while in the debonding area, it is assumed that there is no peel and shear stress transfer between the host beam and the piezoelectric layer. Both displacement continuity and force equilibrium conditions are imposed at the interfaces between the bonded and debonded regions. Based on the model and the sensing equation of the sensor, a closed-loop vibration control for the curved beams is performed. To obtain the frequency response from the presented model, a solution scheme for solving the complex governing equations is given. Using this model and the solution scheme, the e+ects of the debonding of actuator and sensor patches on open- and closed-loop control are investigated through an example. The results show that edge debonding of the piezoelectric patch has a signi1cant side e+ect on the closed-loop control of the curved beams. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Curved beam; Piezoelectric transducers; Debonding

1. Introduction Vibration and shape control of curved structures using surface bonded or embedded piezoelectric sensors and actuators have been attracted signi1cant attention in recent years. Since Tzou and Gadre [1] and Tzou [2] presented the basic actuator and sensor equations of composite shell structures with piezoelectric layers, great e+orts have been concentrated on the modeling, active control, 1nite ∗

Corresponding author. Tel.: +61-2-9351-6949; fax: +61-2-9351-4841. E-mail address: [email protected] (L. Tong).

0020-7403/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 0 2 ) 0 0 0 5 5 - 3

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element method, sensor=actuator location optimization, experiment as well as computational schemes [3–10]. In previous studies, a fundamental assumption is that the piezoelectric layers=patches are perfectly bonded onto the host structures and the adhesive layers are usually not modeled. However, for some new piezoelectric ceramics, such as single crystals, debonding of piezoelectric ceramics from a host structure may be an important issue for reliable control system design of smart structures due to their high actuation ability. For example, piezoelectric single crystals possess a high strain level up to 1.4% leading to high strain energy density, and can sustain very high electric 1eld up to the order of 10 MV=M without electric breakdown. The high strain induced in a single-crystal actuator creates high di+erential straining in the actuator and high stresses in adhesive along the edge of the actuator, and 1nally debonding from the host structure. This is particularly important when the actuator is subjected to a cyclic electric 1eld, for example, in structural vibration suppression, fatigue failure or debonding may be induced. In addition, the debonding between the piezoelectric patches and the host beam may also be caused by other factors such as initial bonding defects, severe loading and fatigue. The debonding of the piezoelectric actuator or sensor patches will lead to great change of both open-loop and closed-loop properties of the structures. Moreover, the debonding of the actuator or sensor will reduce the control precision or may even result in an unexpected failure of the whole structures. Since the debonding of the piezoelectric patches from the host structures makes the governing equations of the structures much more complicated, there exist less theoretical researches on this 1eld. Wang and Meguid [11] examined the e+ect of interfacial debonding of the actuator patch on stress distribution based on singular integral equations. Tylikowski [12] presented a bending-extensional model of a beam with edge debonded piezoelectric patches by neglecting the debonded part of the actuator. Based on a re1ned higher order 1nite element method (FEM), Seeley and Chattopadhyay [13] developed the FEM models of beams and plates with debonded piezoelectric layers=patches and studied the e+ects of the debonding of the piezoelectric actuators on dynamic properties of the open- and closed-loop systems. They also investigated the e+ects of the actuator debonding on closed-loop control of a composite beam by experiment [14]. Recently, Sun and Tong [15,16] investigated the e+ect of the actuator=sensor debonding on dynamics and closed-loop control of beams with piezoelectric sensors and actuators by a novel model, which takes into account both Iexural and longitudinal displacements of the host beam and piezoelectric layers as well as the peel and shear strains of the adhesive layers. To investigate the e+ects of debonding of piezoelectric patches on vibration control of curved beams, a mathematical model for thin-walled curved beams with debonded piezoelectric patches are presented in this paper. The debonded regions of the piezoelectric are modeled by assuming that the adhesive layer carries no stress in the debonded regions. Moreover, in the interfaces between the bonded and debonded regions, both displacements and forces continuity conditions are imposed. A solution scheme is given to obtain the frequency spectrum of the curved beam for both open- and closed-loop cases. Finally, the e+ects of the debonding of actuator and sensor patches are discussed in a simulation example. The results show that edge debonding of the piezoelectric patch can signi1cantly change the actuating=sensing behaviors of the actuator=sensor, natural frequencies and closed-loop control of the curved beam. It is also shown that a curved beam with small curvature is more sensitive to the debonding of the actuator than that with large curvature.

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

z

1757

Debonding

a

o

c

Actuator

d b

Adhesive layer

Sensor

R1 R3 x R2

Fig. 1. A curved beam with debonded piezoelectric actuator and sensor patches.

2. Modeling of curved beams with debonded actuators=sensors Consider a curved thin host beam, on which several curved piezoelectric patch pairs are bonded as the sensor=actuator pairs, as shown in Fig. 1. There exists a debonding area between the piezoelectric patch and the host beam, which is assumed to occur throughout the width of the beam. It is also assumed that the shear and peel strains in the adhesive layer are constants through its thickness in the perfect bonding areas, and that there is no stress transferring between the host beam and the piezoelectric patch in the debonded areas. The contact and friction between the piezoelectric patch and the host beam in the debonded region are not considered for simplicity. The radii of curvature of the actuator, host beam and sensor are denoted by R1 , R2 and R3 , respectively. The entire composite curved beam can be divided into several segments. In di+erent segments, their governing equations may be di+erent which depend on if the piezoelectric patches are bonded the host beam. For the part including host beam and piezoelectric patches, the following equations of motion can be obtained. 2.1. Governing equations For the thin curved beams, the middle surface strains and curvatures changes are given by wi @ui @2 wi 1 @ui + ; i = − 2 + ; i = 1; 2; 3 (1) @x Ri @x Ri @x and the strain and the longitudinal displacement at an arbitrary point can be expressed as [17,18] i0 =

i (z) = (i0 + z i ); i = 1; 2; 3; 
ui @wi ; − ui (z) = ui + z Ri @x

(2)

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Q1+dQ1 M1 T1 T1+dT1

τ1

Q

σ1 σ1

M1+dM1 τ1

Qv1

Qv1+dQv τ1

σ1

τ1

σ1

Q2+dQ2

M2

ft

fl

T2

T2+dT Q2

σ3

τ3 σ3

M2+dM2 τ3 Qv3+dQv

Qv3 τ3

σ3

M3

σ3

τ3

Q3+dQ3

T3 T3+dT3 Q3

M3+dM3

Fig. 2. Free-body diagram of an in1nitesimal beam element with an upper and lower piezoelectric patch.

where the subscripts 1, 2 and 3 represent the upper piezoelectric layer, the host beam and the lower piezoelectric layer, respectively, and u is the longitudinal displacement in mid-plane, w is the transverse displacement. For the segment including the upper piezoelectric patch, the host beam and the lower piezoelectric patch, the free-body diagram is shown in Fig. 2, and from this 1gure the equations of motion can be derived as follows: 1 bh1 u1; tt = T1; x + Q1 =R1 − b1 ;

(3)

1 bh1 w1;tt = Q1; x − T1 =R1 − b1 ;

(4)

M1; x + b1 h1 =2 − Q1 = 0;

(5)

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2 bh2 u2; tt = T2; x + Q2 =R2 + b1 − b3 + fl (x; t);

(6)

2 bh2 w2; tt = Q2; x − T2 =R2 + b1 − b3 + ft (x; t);

(7)

M2; x + b(1 + 3 )h2 =2 − Q2 = 0;

(8)

3 bh3 u3; tt = T3; x + Q3 =R3 + b3 ;

(9)

3 bh3 w3; tt = Q3; x − T3 =R3 + b3 ;

(10)

M3; x + b3 h3 =2 − Q3 = 0;

(11)

where h denotes the thickness, b is the width of the composite beam,  and  are the shear and peel stress of the adhesive layer, T , Q and M are the axial stress resultant, transverse shear force and bending moment, respectively, fl (x; t) and ft (x; t) are the axial and transverse loads per unit length, and i (i = 1; 2; 3) are the mass densities of the three layers. The stress and moment resultants in Eqs. (3) – (11) are  T = Ei bhi i0 − be31i Vi ;   i i = 1; 2; 3; (12) 3  bh E M = i i ; i i 12 where Ei is the Young’s modulus, e31i is the piezoelectric stress constant (for the host beam, e312 =0), Vi the voltage applied on the three layers along their thickness direction. For the sensor, there is no voltage applied, i.e. V3 = 0. 2.2. Model of the debonding The shear and peel stress in the adhesive layers with debonding can be modeled as 
 1 h1 u1 − u2 1 h2 1 = k1 Gv1 (h1 w1; x + h2 w2; x ) + − u1 + u2 ; 2hv1 hv1 2hv1 R1 R2 
 1 h2 u2 − u3 1 h3 3 = k3 Gv3 (h2 w2; x + h3 w3; x ) + − u2 + u3 ; 2hv3 hv3 2hv3 R2 R3

(13) (14)

1 = k1 Ev1 (1 − 1 )=((1 − 21 )(1 + v1 )hv1 )(w1 − w2 );

(15)

3 = k3 Ev3 (1 − 3 )=((1 − 23 )(1 + 3 )hv3 )(w2 − w3 );

(16)

where hv is the thickness of the adhesive layer, Ev and Gv are the Young’s modulus and shear modulus, respectively,  represents the Poison’s ratio, k1 and k3 are parameters characterizing the

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bonding conditions between the piezoelectric layers and the host beam,   0 debonding; i = 1; 3: ki =  1 perfect bonding;

(17)

In the debonding region between the upper piezoelectric patch and the host beam, ki = 0. Note that the peel stress in the bonding layer is obtained by employing the stress–strain relation under full constraint because its thickness is much smaller than the piezoelectric layer and the host beam. The boundary conditions and the continuity conditions are to be given in Section 3. 2.3. Sensor equation The piezoelectric patch can be used to sense the vibration of the curved beam and the sensed signal is used to perform the active control. The charge accumulated on the patch due to the direct piezoelectric e+ect can be evaluated by b q(t) = e313 [3 (z)|z30 + 3 (z)|z31 ] d x; (18) 2 L3 where e313 is the piezoelectric stress constant, z30 and z31 are the z-coordinates of the sensor, and L3 represents the curved path occupied by the curved sensor. Substituting Eq. (2) into Eq. (18), we have   @u3 w3 + q(t) = d x: (19) be313 @x R3 L3 Di+erentiating the charge with respect to time, the current output of the sensor can be obtained as   2 @ u3 1 @w3 I (t) = d x: (20) be313 + @x@t R3 @t L3 Eqs. (19) and (20) establish the relation of the charge generated from the sensor and the vibration of the beam. 2.4. Control law In closed-loop control, the control voltage on the piezoelectric actuator patch is designed by the following proportional and derivative (PD) control law V1 (t) = −g1 q(t) − g2 I (t);

(21)

where g1 and g2 are control gains, q(t) and I (t) are the charge and current generated by the sensor patch.

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3. Nondimensionalized equations By introducing the following nondimensional notations: x #= ; L t tn = 2 L

Tni =

Ti ; E2 bh2

E2 h22 ; 12 2

uni =

Qi ; E2 bh2

Qni = ui ; h2

wni =

Mni =

Mi ; E2 bh22 =12

wi ; h2

(22)

where L is the total length of the composite beam, Eqs. (1) – (9) can be nondimensionalized as m1 kt2 uN n1 = Tn1; # + &1 Qn1 − n1 ; m1 kt2 wN n1 = Qn1; # − &1 Tn1 − n1 ; 12 Qn1 = 0; (2 m2 kt2 uN n2 = Tn2; # + &2 Qn2 + n1 − n3 + fnl (#; tn ); Mn1; # + 6’1 n1 −

m2 kt2 wN n2 = Qn2; # − &2 Tn2 + n1 − n3 + fnt (#; tn ); 12 Qn2 = 0; (2 m3 kt2 uN n3 = Tn3; # + &3 Qn3 + n3 ; Mn2; # + 6’2 (n1 + n3 ) −

m3 kt2 wN n3 = Qn3; # − &3 Tn3 + n3 ; M3; x + 6’3 3 −

(23)

12 Qn3 = 0; (2

Tni = )i ((i uni; # + *i wni ) − Vni ; Mni = )i ’i ((i *i uni; # −

(i2 wni; ## );

i = 1; 2; 3;

n1 = k1 ra1 [un1 − un2 + 12 ((1 wn1; # + (2 wn2; # ) − 12 (*1 un1 + *2 un2 )]; n3 = k3 ra3 [un2 − un3 + 12 ((2 wn2; # + (3 wn3; # ) − 12 (*2 un2 + *3 un3 )]; e n1 = k1 ra1 (wn1 − wn2 ); e n3 = k3 ra3 (wn2 − wn3 );

where the double dot represents the second derivation with respect to tn , and the parameters are de1ned as -i =

i L2 ; E2

(2 =

h2 ; L

&2 =

L ; R2

)i =

Ei ; E2

’i =

hi ; h2

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’vj =

hvj ; h2

)vj =

Evj ; E2

&1 =

2&2 ; 2 + &2 ((1 + 2(v1 )

*i =

hi = & i (i ; Ri

1 − j ; (1 − 2j )(1 + j )

fnl =

Lfl ; E2 bh2

Vn2 = Vn3 = 0;

&3 =

(i = ’i (2 ;

kvj =

fnt =

e311 V1 ; E2 h2

Vn1 =

2&2 ; 2 − &2 ((3 + 2(v3 )

(vj = ’vj (2 ;

(24)

kgj )vj kvj )vj 1 ; raj = ; raie = ; 2(1 + vj ) (vj (vj



E2 h22 (23 1 kt = 2 = ; L 12 2 12m2

kgj =

Lft ; E2 bh2

mi = (i -i ;

i = 1; 2; 3; j = 1; 3:

The parameters in the 1rst two rows in Eq. (24) can be taken as the independent ones, and others are dependent parameters. Sensor equation: The nondimensionalized sensor equations are given by #2 q(t) = u3n (#2 ) − u3n (#1 ) + &3 w3n d#; qn (tn ) = be313 h3 #1 In (tn ) =

dqn (tn ) = u˙ 3n (#2 ) − u˙ 3n (#1 ) + &3 dtn



#2

#1

w˙ 3n d#:

(25a) (25b)

Control law: The control law in Eq. (21) becomes Vn1 (tn ) = −gn1 qn (tn ) − gn2 In (tn );

(26)

where gn1 and gn2 are the nondimensionalized control gains given by gn1 =

gn1 e311 e313 b ; E2

gn2 =

gn2 e311 e313 bkt : E2

(27)

Next, only the excitation induced by the actuator is considered and the voltage is assumed to be uniformly distributed along the piezoelectric actuator patch. In this case, by introducing the state vector Yi =(uni ; Tnie ; wni ; wni; # ; Qni ; Mni )T ; i =1; 2; 3 (Tnie =Tni +Vni ), the governing equations in Eq. (23) can be written into compact form @Yp = Mp YNp + Ap Yp + Bp Vn ; @#

#a 6 # 6 #b ;

(28)

where Yp = (Y1T ; Y2T ; Y3T )T ∈ R18 is the state vector, Mp ∈ R18×18 is the mass density matrix, Ap ∈ R18×18 is the state matrix, Bp = [0; 0; 0; 0; −&1 ; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0]T is the vector related to

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

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the control voltage on the actuator layer, and Vn (tn ) = Vn1 (tn ) is the control voltage applied on the actuator layer. Similarly, for the host beam itself, its basic equations can also be written into the matrix equation form @Y2 = Mb YN 2 + Ab Y2 ; @#

0 6 # ¡ #a ;

#b ¡ # 6 1;

(29)

where Mb ∈ R6×6 is the mass density matrix, Ab ∈ R6×6 is the state matrix of the host beam. Boundary conditions: For general boundary conditions of the piezoelectric patches and the host beam, they can be expressed as Ca1 Y1 (#a ) + Ca2 Y1 (#b ) = Ca

for actuator;

Cb1 Y2 (0) + Cb2 Y2 (1) = Cb

for host beam;

Cs1 Y3 (#a ) + Cs2 Y3 (#b ) = Cs

for sensor;

(30)

where Ca1 ; Ca2 ; Cb1 ; Cb2 ; Cs1 ; Cs2 are 6 × 6 matrices, Ca ; Cb and Cs are vectors. For the piezoelectric actuator/sensor patch with two free ends, its boundary conditions at each end are       0 1 0 0 0 0 0 0 0 0 0 0 0       0 0 0 0 1 0 0 0 0 0 0 0  Vn              0 0 0 0 0 1 0 0 0 0 0 0 0  ; Ca2 = Cs2 =  ; Ca =  ; Cs = 0: Ca1 = Cs1 =        0 0 0 0 0 0 0 1 0 0 0 0 0        0 0 0 0 0 0 0 0 0 0 1 0 0        0 0 0 0 0 0

0

0 0 0 0 0 1

(31) When an initial transverse force Qn is applied at matrices Cb1 ; Cb2 and Cb can be expressed as    1 0 0 0 0 0 0    0 0 1 0 0 0 0       0 0 0 1 0 0 0    = Cb1 =  ; C b2   0 0 0 0 0 0 0    0 0 0 0 0 0 0    0

0

0

0

0

0

0

the free end of the cantilevered host beam, the 





1

 0   0 ;  0  0 

  0      0   Cb =   : 0    Q   n

0

1

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

(32)

0

Continuity conditions: At the interfaces between the bonded and debonded areas, the continuity conditions of all displacements and stress resultants are imposed, i.e. Ypb = Ypd ;

# = #c

and

# = #d ;

(33)

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where the superscripts b and d represent the bonded and debonded regions, respectively, #c and #d are the coordinates of the interfaces between the perfect bonded and debonded regions. In addition, all the displacements and stress resultants of the host beam at the interfaces between the piezoelectric patches bonded and no patches bonded parts should be identical, which can be expressed as Y2p = Y2n ;

# = #a

and

# = #b ;

(34)

where the superscripts p and n represent the parts of the host beam bonded and nonbonded with piezoelectric patches, respectively, #a and #b are the coordinates of the intersect between the region with and without piezoelectric patches.

4. Solution method When the curved host beam is bonded with piezoelectric patches, its governing equations have di+erent forms in di+erent segment. These governing equations become more complicated when the piezoelectric patches are partially debonded from the host beam. To solve this problem, the partial di+erential equations in di+erent segments are transformed into ordinary di+erential equations by taking Fourier transformation. The obtained ordinary di+erential equations together with the boundary and continuity conditions become a special boundary value problem, which can be solved by the following method. Taking Fourier transformation of Eqs. (28) and (29) with respect to tn , the equations for di+erent part of the beam and their boundary conditions become YP 2; # = A2 YP 2 ;

#1 6 # 6 #a ;

YP ; # = Ap YP + Bp VPn ;

#a 6 # 6 #c ;

YP ; # = Ad YP + Bd VPn ;

#c 6 # 6 #d ;

YP ; # = Ap YP + Bp VPn ;

#d 6 # 6 #b ;

YP 2; # = A2 YP 2 ;

#b 6 # 6 #m ;

Ca1 YP 1 (#a ) + Ca2 YP 1 (#b ) = CP a ; Cb1 YP 2 (0) + Cb2 YP 2 (1) = CP b ; Cs1 YP 3 (#a ) + Cs2 YP 3 (#b ) = CP s ; where A2 = Ab − 52 Mb ; A = Ap − 52 Mp , and ∞ P Yi (#) = Yi (#; tn )e−i5tn dtn ; i = 1; 3 −∞

is the Fourier transformation of Yi , and 5 is the nondimensional frequency.

(35)

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

ξ1 ξ2

ξa

ξc

ξd

ξb

1765

ξm

Fig. 3. Subdivision of the whole beam with debonded piezo-patches.

To solve the boundary problem given in Eq. (35), cut the entire beam into a number of small subintervals, and the dividing points are #1 ; #2 ; : : : ; #a ; : : : ; #c ; : : : ; #d ; : : : ; #b ; : : : ; #m , respectively, as shown in Fig. 3. Note that each joining point between di+erent segments of the beam is one a of the dividing points so that all continuity conditions can be satis1ed. Denoting Sj =YP 1 (#j ); ‘ aT T ‘T S˜ j = YP 2 (#j ); Sj =YP 3 (#j ), Sj = [ Sj ; S˜ j ; Sj ]T (j = 1; 2; : : : ; m) and noting the continuity conditions, the following equations should be satis1ed: 6˜ 1 S˜ 1 − S˜ 2 = 0; 6˜ 2 S˜ 2 − S˜ 3 = 0; .. . 6˜ a−1 S˜ a−1 − S˜ a = 0; 6a Sa + 7a VPn − Sa+1 = 0; 6a+1 Sa+1 + 7a+1 VPn − Sa+2 = 0; .. . 6b−21 Sb−2 + 7b−21 VPn − Sb−1 = 0;

(36)

6b−1 Sb−1 + 7b−1 VPn − Sb = 0; 6˜ b S˜ b − S˜ b+1 = 0; .. . 6˜ m−1 S˜ m−1 − S˜ m = 0; a a Ca1 Sa + Ca2 Sb = CP a ; ‘

‘

Cs1 Sa + Cs2 Sb = CP s ; Cb1 S˜ 1 + Cb2 S˜ m = CP b ; where $˜ j = eA2 (#j −#j−1 ) ; $j = eA(#j −#j−1 ) are transition matrices, %j = vector. Eq. (36) can be rewritten in the following matrix form: PS = F;



# j − #j − 1 0

 eA# d# B ∈ R18 is a

(37)

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D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

where P=   .. .. . .   $˜ 1 −I6       .. ..   . .       . .   . . ˜ a−1 .   0 0 −I $ . 6       .   .   ··· · · · · · · · · · · · · · · · · · · · · · . · · · · · · · · · · · · · · ·       . . P P @Vn @Vn   . .   . $a + %a −I + %a .   @Sa @Sa+1     . . @V   n . .   . $a+1 + %a+1 .   @Sa+1     . .   .. ..         .. .. P P @ V @ V n n   . $b−1 + %b−1 . −I + %b−1 ;    @Sb−1 @Sb   ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···   ···       .. ..   . −I6 . 0 $˜ b 0         . . .. ..           .. ..  . . $˜ m −I6        ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ··· ···   ···       .. .. @CP a @CP a Pa Pa @ C @ C   C 0 C 0 a1 a2 . · · · . ‘ ‘   @Sa+1 @Sb−1   @ Sa @ Sb     . .   . .   0 0 Cs1 0 0 Cs2 . .     .. .. Cb1 . . Cb2

T  .. aT ˜ T ‘T T aT T T T ‘T .. T T T ˜ ˜ ˜ ˜ ˜ S = S1 · · · Sa−1 . S Sa S Sa+1 · · · Sb−1 S Sb S . Sb+1 · · · Sm ; a a b b  T .. .. T T P P F = 0 · · · 0 . 0 0 0 0 · · · 0 0 0 0 . 0 · · · 0 Cs Cb :

(38)

‘ Note that VPn and CP a become functions of Sj (j = a; : : : ; b) when the closed-loop control is performed using Eq. (26). The state at each dividing point can be obtained by solving the simultaneous algebraic equation (36) or Eq. (37).

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

1767

5. Simulation example Consider a curved beam clamped at its left end, on which a pair of piezoelectric patch is bonded as actuator and sensor, respectively, as shown in Fig. 2. The piezoelectric actuator and sensor patches are bonded on the upper and lower surfaces of the host beam and their left ends are 5% away from the clamped end of the beam and cover a portion of 20% of the whole beam. Firstly, we examine the e+ect of the debonding between the actuator layer and the host beam on the actuator behavior by applying a constant voltage on the actuator patch. In this example, the parameters are taken as: &2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’1 = 0:2, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, 1 = 3 = 0:34. When a constant voltage Vn = 1 × 10−5 is exerted on the actuator, the force and the moment distributions as well as the displacements in the host beam induced by the actuator are shown in Fig. 4 for di+erent debonding length of the actuator. Fig. 4a and b shows that the moment and the axial force induced by the constant voltage are uniformly distributed in the area covered by the actuator except the small regions near the ends of the actuator. The edge debonding of the actuator changes the distributions of the moment and axial force in the debonded area in the host beam. It is found that the moment in the debonded area has opposite sign with those in the bonded area. The distortion of the distribution of the axial force in the debonded area can also be observed in Fig. 4b. However, the edge debonding of the actuator does not a+ect the moment and axial force distribution far from the debonded part. The edge debonding of the actuator has little e+ect on the distribution of shear and peel stresses in the adhesive layer between the actuator and the host beam, as shown in Fig. 4c and d, respectively. Fig. 4e and f show that both the axial and transverse displacements caused by the actuator decrease as the debonding length increases, which indicate that the actuating ability is weakened by the debonding. At the same time, the charge generated by the sensor patch is depicted in Fig. 5. Fig. 5 shows that the sensor output decreases as the debonding length of the actuator increases particularly when the thickness ratio of actuator to host beam is small. In other words, the actuator is more sensitive to the debonding when it is much thinner than the host beam. Moreover, it can also be seen from Fig. 5 that the strain induced by the actuator becomes smaller as the thickness ratio of the actuator to the host beam increases. This is because the moment induced by the actuator decreases as the thickness ratio ’1 increases, as shown in Fig. 6. When the edge debonding occurs at the left end of the sensor rather than the actuator, its e+ect on the distributions of the moment and axial force of the host beam are quite di+erent, as shown in Fig. 7. In this case, the absolute value of the moment in the debonded region becomes larger than the perfectly bonded region, as indicated in Fig. 7a. On the contrary, the absolute value of the axial force induced by the actuator in the debonded region in the host beam is smaller than that in the perfectly bonded region. Both the moment and the axial force in the debonded region but far from their two ends in the host beam approach a constant. Although the displacement of the beam becomes larger due to the debonding of the sensor, the charge output generated by the sensor decreases as its debonding length increases since the e+ective length of the sensor is shortened, as shown in Fig. 8. Secondly, we turn to investigate the edge debonding of the actuator on natural frequencies of the curved beam for the cases of di+erent curvatures &2 and Young’s modulus ratios )2 of the actuator and the host beam. Choose ’1 = ’3 = 0:214 and keep other parameter unchanged. When 5%, 10%, 15% and 25% debonding occurs at the left end of the actuator, their e+ect on the 1rst

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D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777 1.E-05

1.E-06 0.E+00

0.E+00

-1.E-06 -2.E-06 perfect bonding

-2.E-05

20% debonding 30% debonding

-3.E-05

perfect bonding

-3.E-06

10% debonding

Nn2

Mn2

-1.E-05

-4.E-06

10% debonding

-5.E-06

20% debonding

-6.E-06

30% debonding

-7.E-06 -8.E-06

-4.E-05

-9.E-06 -1.E-05

-5.E-05 0

0.2

0.4

(a)

0.6

0.8

1

0

0.4

0.6

0.8

1

beam span

9.E-04

2.E-03 perfect bonding 10% debonding 20% debonding 30% debonding

1.E-03

8.E-04 7.E-04 6.E-04 perfect bonding

5.E-04

0.E+00

10% debonding

4.E-04

σn1

5.E-04

τn1

0.2

(b)

beam span

20% debonding

3.E-04

30% debonding

2.E-04

-5.E-04

1.E-04 0.E+00

-1.E-03

-1.E-04 -2.E-03 0.05

0.1

(c)

0.15 actuator span

0.2

-2.E-04 0.05

0.25

0.1

(d)

0.15 beam span

0.2

0.25

2.E-01

5.E-03

perfect bonding

1.E-01

0.E+00

10% debonding

1.E-01

20% debonding 1.E-01

-5.E-03 -1.E-02

wn2

un2

perfect bonding 10% debonding

30% debonding

8.E-02 6.E-02

20% debonding -2.E-02

4.E-02

30% debonding

2.E-02

-2.E-02 0.E+00

-3.E-02

-2.E-02

0

(e)

0.2

0.4 0.6 beam span

0.8

0

1

(f )

0.2

0.4

0.6

0.8

1

beam span

Fig. 4. E+ect of edge debonding of the actuator on distribution of the states of the host beam (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’1 = 0:2, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, Vn = 1 × 10−5 ): (a) moment in the host beam; (b) axial force in the host beam; (c) shear stress in the adhesive layer; (d) peel stress in the adhesive layer; (e) longitudinal displacement; and (f) transverse displacement.

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777 0.0006

ϕ1=0.1 ϕ1=0.5 ϕ1=1.5

0.0005

1769

ϕ1=0.2 ϕ1=1

qn

0.0004 0.0003 0.0002 0.0001 0 0

5

10

15

20

25

30

debond length (%)

Fig. 5. E+ect of edge debonding of the actuator on the output of the sensor (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, Vn = 1 × 10−5 ).

1.00E-05

ϕ1=0.1 ϕ1=1.0

ϕ1=0.2 ϕ1=1.5

ϕ1=0.5

Moment in host beam Mn2

0.00E+00 -1.00E-05 -2.00E-05 -3.00E-05 -4.00E-05 -5.00E-05 -6.00E-05 0

0.05

0.1

0.15 beam span

0.2

0.25

0.3

Fig. 6. Moment distribution in the host beam induced by the actuator (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, Vn = 1 × 10−5 ).

four frequencies of the beam with di+erent curvatures are listed in Table 1. In this case, the edge debonding of the actuator decreases the 1rst four frequencies. When the debonding length is less than 25% of the actuator, the longer the edge debonding of the actuator is, the more remarkable the decrease of the frequencies is, particularly for the fundamental frequency. It can be found from Table 1 that the debonding has stronger e+ect on the natural frequencies of the beam for high modulus ratio of the actuator and the host beam. For example, a 25% debonding results in a 1.5% decrease of the fundamental frequency when the modulus ratio is 0.3, it can lead to a 4% decrease when modulus ratio is 1.5. However, the curvature of the beam hardly a+ects the decrease rates of the natural frequencies due to the edge debonding of the actuator. Table 2 presents the 1rst four natural frequencies obtained by varying the debonding length of the actuator for di+erent thickness ratios of the actuator and the host beam. In this case, take the &2 = 0:3, ’3 = 0:2 and other parameters are the same. Table 2 shows that the natural frequencies

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D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777 1.E-06

1.E-05

0.E+00 0.E+00

-1.E-06 -2.E-06

-2.E-05

perfect bonding

-3.E-06

10% debonding

-4.E-06

N n2

M n2

-1.E-05

20% debonding

-3.E-05

30% debonding

10% debonding 20% debonding

-5.E-06

30% debonding

-6.E-06

-4.E-05

-7.E-06 -8.E-06

-5.E-05

-9.E-06 -1.E-05

-6.E-05 0

(a)

perfect bonding

0.05

0.1

0.15 beam span

0.2

0.25

0

0.05

0.1

(b)

0.15 beam span

0.2

0.25

Fig. 7. E+ect of edge debonding of the sensor on distribution of the states of the host beam (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’1 = 0:2, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, Vn = 1 × 10−5 ): (a) moment in the host beam; and (b) axial force in the host beam.

0.0006

ϕ1=0.1 ϕ1=0.5 ϕ1=1.5

0.0005

ϕ1=0.2 ϕ1=1

qn

0.0004 0.0003 0.0002 0.0001 0 0

5

10

15

20

25

30

debonding length (%)

Fig. 8. E+ect of edge debonding of the sensor on the output of the sensor (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, Vn = 1 × 10−5 ).

generally decrease due to the edge debonding of the actuator. However, it should be noted that the fourth frequency of the beam with low thickness ratio may increase when the debonding length reaches 25%. This is because the 1rst frequency of the debonded part of the actuator is close to the fourth frequency of the composite beam and they interact each other. This phenomenon has been reported for the straight beams in the previous studies [13–16]. It can also be seen that the changes of natural frequencies of the beam due to debonding are more signi1cant for large thickness ratio and modulus ratio than those for the small thickness ratio and modulus ratio. For example, when the thickness ratio is 1.0 and the modulus ratio is 1.5, a 25% edge debonding of the actuator can result in a 8.5% decrease of the fundamental frequency, while only 0.6% decrease when the thickness ratio is 0.1 and the modulus ratio is 0.3.

Debonding &2 = 0 &2 = 0:3 &2 = 0:5 &2 = 1:0 length (%) Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4 )1 = E1 =E2 = 0:3 0 3.867 5 3.854 10 3.842 15 3.831 25 3.810 Change − 1.5

22.375 22.326 22.287 22.254 22.204 − 0.8

60.834 60.777 60.741 60.719 60.701 − 0.2

119.572 119.546 119.539 119.537 119.455 − 0.1

3.874 3.861 3.849 3.838 3.818 − 1.5

22.150 22.101 22.062 22.029 21.978 − 0.8

60.607 60.550 60.514 60.492 60.474 − 0.2

119.363 119.337 119.330 119.328 119.247 − 0.1

3.887 3.874 3.862 3.851 3.830 − 1.5

21.769 21.719 21.682 21.650 21.601 − 0.8

60.216 60.159 60.124 60.103 60.085 − 0.2

119.002 118.977 118.970 118.968 118.888 − 0.1

3.949 3.936 3.924 3.913 3.892 − 1.4

20.255 20.208 20.172 20.141 20.095 − 0.8

58.565 58.512 58.479 58.459 58.442 − 0.2

117.444 117.420 117.414 117.412 117.336 − 0.1

)1 = E1 =E2 = 0:5 0 3.939 5 3.920 10 3.902 15 3.886 25 3.855 Change − 2.1

22.484 22.412 22.354 22.307 22.235 − 1.1

61.025 60.947 60.900 60.871 60.849 − 0.3

120.497 120.468 120.462 120.461 120.397 − 0.1

3.947 3.927 3.910 3.893 3.863 − 2.1

22.259 22.187 22.129 22.082 22.010 − 1.1

60.796 60.719 60.671 60.643 60.620 − 0.3

120.279 120.250 120.244 120.242 120.179 − 0.1

3.960 3.941 3.923 3.907 3.876 − 2.1

21.877 21.805 21.749 21.702 21.630 − 1.1

60.404 60.327 60.280 60.252 60.231 − 0.3

119.911 119.882 119.877 119.875 119.812 − 0.1

4.024 4.004 3.987 3.970 3.939 − 2.1

20.357 20.289 20.234 20.190 20.122 − 1.2

58.747 58.675 58.631 58.605 58.586 − 0.3

118.322 118.306 118.301 118.299 118.236 − 0.1

4.062 4.030 4.003 3.976 3.928 − 3.3

22.670 22.556 22.466 22.392 22.281 − 1.7

61.312 61.206 61.143 61.107 61.081 − 0.4

122.044 122.018 122.015 122.011 121.921 − 0.1

4.070 4.038 4.010 3.984 3.936 − 3.3

22.444 22.329 22.241 22.167 22.057 − 1.7

61.081 60.975 60.912 60.876 60.851 − 0.4

121.811 121.784 121.781 121.777 121.688 − 0.1

4.084 4.052 4.024 3.998 3.949 − 3.3

22.059 21.947 21.858 21.785 21.677 − 1.7

60.687 60.582 60.520 60.485 60.460 − 0.4

121.432 121.405 121.402 121.398 121.309 − 0.1

4.150 4.118 4.090 4.063 4.014 − 3.3

20.530 20.422 20.337 20.268 20.165 − 1.8

59.023 58.925 58.867 58.834 58.813 − 0.4

119.824 119.800 119.797 119.793 119.701 − 0.1

)1 = E1 =E2 = 1:5 0 4.138 5 4.099 10 4.064 15 4.031 25 3.972 Change − 4.0

22.788 22.646 22.535 22.444 22.310 − 2.1

61.474 61.354 61.283 61.243 61.217 − 0.4

123.004 122.982 122.980 122.972 122.853 − 0.1

4.146 4.107 4.072 4.039 3.980 − 4.0

22.561 22.420 22.309 22.219 22.085 − 2.1

61.242 61.121 61.051 61.011 60.985 − 0.4

122.761 122.738 122.736 122.729 122.610 − 0.1

4.160 4.121 4.086 4.054 3.993 − 4.0

22.175 22.035 21.925 21.836 21.705 − 2.1

60.846 60.728 60.658 60.619 60.594 − 0.4

122.376 122.353 122.352 122.344 122.225 − 0.1

4.228 4.189 4.154 4.120 4.059 − 4.0

20.639 20.505 20.401 20.316 20.191 − 2.2

59.179 59.067 59.003 58.967 58.946 − 0.4

120.753 120.733 120.732 120.723 120.599 − 0.1

)1 = E1 =E2 = 1:0 0 5 10 15 25 Change

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

Table 1 E+ect of edge debonding of the actuator on natural frequencies of the beam with di+erent curvatures ()3 = 0:3, (2 = 0:007, ’1 = ’3 = 0:214, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01)

1771

1772 Debonding ’1 = 0:1 ’1 = 0:2 ’1 = 0:5 ’1 = 1:0 length (%) Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4 Mode 1 Mode 2 Mode 3 Mode 4 )1 = E1 =E2 = 0:3 0 3.772 5 3.766 10 3.761 15 3.757 25 3.748 Change − 0.6

22.052 22.03 22.013 21.999 21.976 − 0.3

60.863 60.834 60.816 60.804 60.789 − 0.1

119.574 119.557 119.551 119.546 119.585 0.01

3.851 3.839 3.828 3.817 3.798 − 1.4

22.127 22.082 22.045 22.015 21.968 − 0.7

60.661 60.607 60.573 60.552 60.533 − 0.2

119.387 119.361 119.353 119.351 119.255 − 0.1

4.098 4.063 4.031 4.002 3.948 − 3.7

22.373 22.243 22.141 22.058 21.935 − 2.0

60.114 59.997 59.929 59.891 59.878 − 0.4

119.612 119.588 119.586 119.578 119.443 − 0.1

4.42 4.349 4.287 4.23 4.128 − 6.6

22.664 22.398 22.199 22.042 21.822 − 3.7

59.162 59.004 58.923 58.887 58.873 − 0.5

120.599 120.595 120.58 120.514 120.077 − 0.4

)1 = E1 =E2 = 0:5 0 3.812 5 3.803 10 3.794 15 3.788 25 3.773 Change − 1.0

22.111 22.078 22.051 22.029 21.994 − 0.5

60.976 60.934 60.907 60.89 60.874 − 0.2

120.06 120.037 120.03 120.027 120.094 0.03

3.921 3.903 3.886 3.871 3.842 − 2.0

22.232 22.165 22.111 22.066 21.997 − 1.0

60.846 60.772 60.725 60.697 60.675 − 0.3

120.263 120.233 120.227 120.255 120.163 − 0.08

4.209 4.162 4.121 4.082 4.011 − 4.7

22.546 22.373 22.24 22.132 21.975 − 2.5

60.374 60.237 60.16 60.119 60.095 − 0.5

121.122 121.104 121.102 121.085 120.897 − 0.2

4.511 4.429 4.375 4.292 4.176 − 7.4

22.816 22.507 22.277 22.099 21.85 − 4.2

59.377 59.211 59.129 59.093 59.081 − 0.5

122.007 122.004 121.979 121.891 121.375 − 0.5

3.892 3.877 3.862 3.849 3.824 − 1.8

22.231 22.173 22.127 22.088 22.028 − 0.9

61.185 61.12 61.078 61.052 61.031 − 0.3

121.032 121.003 120.995 120.994 120.901 − 0.1

4.042 4.012 3.985 3.96 3.914 − 3.2

22.414 22.306 22.22 22.15 22.044 − 1.7

61.13 61.027 60.946 60.928 60.902 − 0.4

121.757 121.728 121.724 121.721 121.641 − 0.1

4.346 4.284 4.229 4.178 4.086 − 6.0

22.765 22.536 22.361 22.221 22.021 − 3.3

60.668 60.512 60.426 60.383 60.361 − 0.5

122.984 122.973 122.969 122.935 122.663 − 0.3

4.599 4.506 4.425 4.352 4.222 − 8.2

22.97 22.615 22.355 22.155 21.877 − 4.8

59.583 59.411 59.327 59.292 59.281 − 0.5

123.379 123.375 123.338 123.229 122.635 − 0.6

)1 = E1 =E2 = 1:5 0 3.954 5 3.932 10 3.913 15 3.895 25 3.861 Change − 2.4

22.322 22.245 22.183 22.131 22.053 − 1.2

61.33 61.249 61.198 61.167 61.143 − 0.3

121.767 121.736 121.73 121.729 121.669 − 0.1

4.119 4.081 4.048 4.016 3.958 − 3.9

22.53 22.395 22.289 22.201 22.072 − 2.0

61.292 61.174 61.104 61.064 61.037 − 0.4

122.703 122.677 122.675 122.669 122.561 − 0.1

4.411 4.341 4.279 4.223 4.121 − 6.6

22.873 22.614 22.419 22.264 22.043 − 3.6

60.801 60.636 60.548 60.504 60.483 − 0.5

123.867 123.86 123.852 123.808 123.492 − 0.3

4.637 4.539 4.454 4.377 4.241 − 8.5

23.037 22.663 22.389 22.179 21.89 − 5.0

59.672 59.496 59.412 59.377 59.366 − 0.5

123.955 123.95 123.908 123.79 123.166 − 0.6

)1 = E1 =E2 = 1:0 0 5 10 15 25 Change

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

Table 2 E+ect of edge debonding of the actuator on natural frequencies of the beam with di+erent thickness ratios (&2 = 0:3, )3 = 0:3, (2 = 0:007, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01)

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777

1773

0.008 0.008

0.0005 0.0004 0.0003

0.007 0.006 0.004

0.006

0.0002 0.0001 0

0.002 0.005

0

|q n|

3.5

4

4.5

20 21 22 23 24 25

0.004 0.003

perfect bonding

0.002

15% debonding 0.001 0 0

5

10

15

20

25

30

Ω

Fig. 9. Frequency spectrum of the sensor output (&2 =0:3, )1 =)3 =0:3, (2 =0:007, ’1 =0:2, ’3 =0:2, -1 =-3 =2:87×10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, gn1 = 0; gn2 = 1 × 10−4 ).

Thirdly, we check the e+ect of edge debonding of the actuator on the closed-loop control of the beam. An initial transverse impulse of 1 × 10−8 is imposed at the free end of the beam to produce a transient vibration. Meanwhile, the PD controller given in Eq. (26) is used to perform the active control of the beam and the control gains are gn1 = 0, gn2 = 1 × 10−4 . The frequency responses of the controlled beam with and without debonded actuator to the initial impulse can be obtained to examine the e+ect on the closed-loop control. For the curved beam with perfectly bonded actuator, when take &2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’1 = 0:2 ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, the frequency spectrum of the charge output of the sensor is plotted in Fig. 9. Each peak in the spectrum of the sensor output corresponds to one vibration mode of the controlled beam. When a debonding of the actuator occurs, the frequency spectrum of the sensor output will change. Fig. 9 also presents the frequency spectrum of the sensor output when a 15% debonding locates at the left end of the actuator patch. Both location and peak value of each peak in the spectrum varies due to the actuator debonding, which indicates that the frequency and amplitude of the related mode changes when the debonding occurs. Since the beam is controlled by choosing gn1 = 0 in Eq. (26), the frequencies of the closed-loop system are almost the same as the natural frequencies, which have been shown in Tables 1 and 2 with di+erent debonding lengths. Therefore, we focus on examining amplitude change of the 1rst two modes caused by the actuator debonding. To this end, we de1ne the peak value change rate 8k for the kth mode as 8k =

pkd − pkb × 100%; pkb

(39)

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D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777 140

µ2=0 µ2=0.3 µ2=0.5 µ2=1

50 40 30 20 10

100 80 60 40 20 0

0 0

(a)

µ2=0 µ2=0.3 µ2=0.5 µ2=1

120 peak value change rate (%)

peak value change rate (%)

60

5

10 15 debonding length (%)

20

0

25

(b)

5

10

15

20

25

debonding length (%)

Fig. 10. E+ect of debonding length with di+erent curvatures of the beam ()1 = )3 = 0:3, (2 = 0:007, ’1 = 0:2, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, gn1 = 0; gn2 = 1 × 10−4 ): (a) mode 1; and (b) mode 2.

where pkb and pkd are the peak values of the kth peak in the spectrum of the sensor output for perfectly bonded and debonded cases, respectively. The peak value change rate indicates the amplitude change for each mode in the closed-loop system due to the debonding relative to the perfect bonding case. Fig. 10 presents the e+ect of the actuator debonding on the 1rst two modes with di+erent curvatures. It can be seen that the peak value change rates for the 1rst two modes increase signi1cantly as the deboding length of the actuator increases, which indicates that the edge debonding of actuator patch can signi1cantly weaken the control e+ect. Fig. 10 also shows that the e+ect of the debonding of actuator on the closed-loop control highly depends on the curvatures of the beam. The shallow curved beam is more sensitive to the debonding of the actuator than a deep curved one. For example, for the beam with &2 = 0:5, the peak value change rates for modes 1 and 2 caused by a 25% debonding of the actuator are 13% and 17%, and they are as high as 49% and 131%, respectively, for a straight beam (&2 = 0) caused by a 25% debonding of the actuator. To further examine the e+ect of the edge debonding of the actuator on the vibration control of beam, the peak value change rates for modes 1 and 2 are calculated and plotted in Fig. 11 by varying the debonding length and the modulus ratio )1 of the actuator and the host beam with &2 = 0:3. As shown in Fig. 11, the closed-loop control system becomes more sensitive to the actuator debonding when the modulus ratio of the actuator and the host beam decreases. For example, when there is a 25% debonding of the actuator, the peak value change rates are 18%, 17%, 16% and 14% for mode 1, and are 24%, 23%, 21% and 18% for mode 2 when the )1 is set to be equal to 0.3, 0.5, 1.0 and 1.5, respectively. Fig. 12 depicts the e+ect of debonding of the actuator on modes 1 and 2 of the controlled beam with di+erent thickness ratios of the actuator and the host beam, obtained by taking &2 = 0:3, )1 = 0:3 and ’1 = ’3 . When the thickness ratio is much smaller than 1, namely, the actuator patch is much thinner than the host beam, it has very small e+ect on the increasing trend of the peak value change rate caused by the actuator debonding. However, when the actuator and the host beam have the same thickness, the e+ect of the edge debonding of the actuator on the closed-loop control becomes

D. Sun, L. Tong / International Journal of Mechanical Sciences 44 (2002) 1755 – 1777 25

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Fig. 11. E+ect of debonding on mode 1 and 2 with di+erent modulus ratios of actuator and the beam (&2 = 0:3, )3 = 0:3, (2 =0:007, ’1 =0:2, ’3 =0:2, -1 =-3 =2:87×10−9 , -2 =3:14×10−9 , ’v1 =’v2 =1=14, )v1 =)v3 =0:01, gn1 =0; gn2 =1×10−4 ): (a) mode 1; and (b) mode 2.

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Fig. 12. E+ect of debonding on mode 1 and 2 with di+erent thickness ratios (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, gn1 = 0; gn2 = 1 × 10−4 ): (a) mode 1; and (b) mode 2.

lessened. For the second mode, an edge debonding above 10% of the actuator length decreases the vibration amplitude of the beam. This may be because the active control for the case of ’1 =’3 =1:0 is no longer e+ective for the selected gain factor or becomes unstable. As indicated in Fig. 13, in terms of the peak value change rate, the e+ect of the debonding of the actuator on the active control is not a+ected remarkably by the control gain when the debonding is less than 15%.

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Fig. 13. E+ect of debonding on peak value change rate with di+erent control gains (&2 = 0:3, )1 = )3 = 0:3, (2 = 0:007, ’1 = 0:2, ’3 = 0:2, -1 = -3 = 2:87 × 10−9 , -2 = 3:14 × 10−9 , ’v1 = ’v2 = 1=14, )v1 = )v3 = 0:01, gn1 = 0): (a) mode 1; and (b) mode 2.

6. Conclusions In this paper, a mathematical model for a thin curved beam with partially debonded piezoelectric sensor/actuator patches is presented. The debonded regions are modeled by imposing both displacement and force continuity conditions at an interface between the debonded and nondebonded areas. A numerical method is given to obtain the frequency response of the controlled beam with debonding piezoelectric patches. The e+ects of the debonding of the actuator/sensor on actuating/sensing behaviors, natural frequencies and closed-loop control of the curved beam are investigated in an example. The results show that edge debonding of the actuator/sensor can signi1cantly change internal force distribution in the host beam and weaken the actuating and sensing ability. The edge debonding can also change the natural frequencies particularly for the case of large modulus and thickness ratios of the actuator and the host beam. Moreover, the edge debonding of the piezo-patch has harmful e+ects on the closed-loop control of the beam. It is also shown that the shallow curved beam is more sensitive to the edge debonding of the piezoelectric actuator or sensor patch. Acknowledgements The authors are grateful to the support of the Australia Research Council through a large grant (Grant No. A10009074). References [1] Tzou HS, Gadre M. Theoretical-analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. Journal of Sound and Vibration 1989;132(3):433–50. [2] Tzou HS. A new distributed sensor and actuator theory for intelligent shells. Journal of Sound and Vibration 1992;153(2):335–49.

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[3] Qiu JH, Tani JJ. Vibration control of a cylindrical-shell using distributed piezoelectric sensors and actuators. Journal of Intelligent Material Systems and Structures 1995;6(4):474–81. [4] Sonti VR, Jones JD. Curved piezoactuator model for active vibration control of cylindrical shells. AIAA Journal 1996;34(5):1034–40. [5] Banks HT, Zhang Y. Computational methods for a curved beam with piezoceramic patches. Journal of Intelligent Material Systems and Structures 1997;8(3):260–78. [6] Chandrashekhara K, Smyser CP. Dynamic modeling and neural control of composite shells using piezoelectric devices. Journal of Intelligent Material Systems and Structures 1998;9(1):29–43. [7] Zhou YH, Tzou HS. Active control of nonlinear piezoelectric circular shallow spherical shells. International Journal of Solids and Structures 2000;37(13):1663–77. [8] Chee C, Tong L, Steven G. A buildup voltage distribution (BVD) algorithm for shape control of smart plate structures. Computational Mechanics 2000;26(2):115–28. [9] Shih HR. Distributed vibration sensing and control of a piezoelectric laminated curved beam. Smart Materials and Structures 2000;9(6):761–6. [10] Sun DC, Tong L. Modal control of smart shells by optimized discretely distributed piezoelectric transducers. International Journal of Solids and Structures 2001;38(18):3281–99. [11] Wang XD, Meguid SA. On the behaviour of a thin piezoelectric actuator attached to an in1nite host structure. International Journal of Solids and Structures 2000;37:3231–51. [12] Tylikowski A. E+ects of piezoactuator delamination on the transfer functions of vibration control systems. International Journal of Solids and Structures 2001;38:2189–202. [13] Seeley CE, Chattopadhyay A. Modeling of adaptive composites including debonding. International Journal of Solids and Structures 1999;36:1823–43. [14] Seeley CE, Chattopadhyay A. Experimental investigation of composite beams with piezoelectric actuation and debonding. Smart Materials and Structures 1998;7:502–11. [15] Tong L, Sun DC, Atluri SN. Sensing and actuating behaviors of piezoelectric layers with debonding in smart beams. Smart Materials and Structures 2001;10(4):713–23. [16] Sun DC, Tong L, Atluri SN. E+ects of piezoelectric sensor/actuator debonding on vibration control of smart beams. International Journal of Solids and Structures 2001;38(50 –51):9033–51. [17] Seide P. Small elastic deformations of thin shells. Leyden: Noordho+ International Publishing, 1975. p. 39 – 40. [18] Qatu MS. Theories and analyses of thin and moderately thick laminated composite curved beams. International Journal of Solids and Structures 1993;30(20):2743–56.