Comparative study on shear-mode piezoelectric actuators where, C and e are the effective stiffness and the vector of piezoelectric

Shear-based vibration control of annular sandwich plates using different piezoelectric fiber composites: a

Chapter 5: Comparative study on shear-mode piezoelectric actuators where, C and e are the effective stiffness and the vector of piezoelectric

coefficients of the laminate, respectively; E_z is the electric field across the thickness of the active layers (E_zE_z).

5.3 Smart annular sandwich plate

For shear mode piezoelectric actuation of bending mode of deformation of a plate, the shear piezoelectric actuators are usually embedded around the middle plane of the plate [190]. So, the present annular sandwich plate is constructed by inserting the patches of the shear piezoelectric actuators at the core. The configuration of the overall annular plate appears similar to that presented in Fig. 3.2 and it is reproduced here for better readability.

Fig. 3.2 Schematic diagram of the annular sandwich plate with the embedded shear actuator patches.

Since the laminate of a shear actuator (Fig. 5.2) is made in the form of a rectangular block with the Cartesian material coordinate system (xyz), the patches in the shape of the annular sector (Fig. 3.2) can be cut out from a block

Chapter 5: Comparative study on shear-mode piezoelectric actuators of the laminate similar to that in Fig. 3.3, and the core layer of the sandwich plate can be constructed using these patches. The shear actuation force in every patch acts in the

xz

-plane of its Cartesian material coordinate system (xyz), and this shear actuation force is utilized in counteraction of the mechanically induced transverse shear stress (_rz) in the rz-plane of the annular sandwich plate. This mechanically induced transverse shear stress (_rz) changes its sign along the radial and circumferential directions of the annular plate as it is demonstrated in Figs. 3.4 for two typical bending modes of the annular plate.

Following this variation of sign of the stress (_rz), the sign of the externally applied transverse electric field (E_z) is to be varied over the plane of the annular plate for effective shear actuation, and it is achieved by the variation the applied electric field (E_z) over the patches similar to the previous problem in Chapter 3.

5.4 Properties of constituent materials in the annular sandwich plate

Since the actuator patches possess Cartesian material coordinate system, their properties are transformed for their utilization in the reference cylindrical coordinate system as it is demonstrated in Chapter 3 through Fig. 3.5 and Eq.

(3.2). However, for the use of the patches as actuators, the electric field in actuator patches is always specified. So, it may be observed from Eq. (3.13) that the electric displacement (D_z) has no effect on the motion of the overall annular plate since E_z 0 for the specified E_z at any instant of time. Therefore, the converse piezoelectric constitutive relation of the actuator patches would appear in the formulation of governing equation of the overall annular plate, and its aforesaid transformation with respect to the cylindrical coordinate system yields the following expression,

( )

k k k k

Ez

 C e

  ,k2

T k 

C T CT ,

k 

e Te ^(5.7)

where, k represents the bottom face layer, core layer and top face layer of the sandwich plate according to its value as 1, 2 and 3, respectively;



^k/^k is the stress/strain vector at any point in the k^th layer; C^kis the stiffness matrix of the

Chapter 5: Comparative study on shear-mode piezoelectric actuators

kth layer; e^k is the vector of piezoelectric coefficients associated with the transverse electric field (E_z) in the k^th layer; T is the transformation matrix.

Besides the properties (Eq. (5.7)) of the actuator patches, the constitutive relations for the isotropic face layers (k1,3) and the foam at the core (k2) can be written similar to Eq. (3.3) as,

kCk k

 

, k1, 2,3

(1 ) 0 0 0

(1 ) 0 0 0

(1 ) 0 0 0

0 0 0 0 0

2(1 )

0 0 0 0 0

2(1 )

0 0 0 0 0

2(1 )

k k k

s s s

k k k

s s s

k k k

s s s

k k

k k

k k

C v C v C v

C v C v C v

C v C v C v

E v

E v

E v

  

 



 

 

  

 

 

   

 

 

  

 

 

 

  

 

C ^,

/ (1 )(1 2 )

k k k

CsE v  v (3.3)

where, E^k and v^k are Young’s modulus and Poisson’s ratio, respectively.

5.5 FE model of the annular sandwich plate

The bottom plane of the annular sandwich plate is taken as the reference plane and the origin of the reference cylindrical coordinate system (r z ) is located at the center of the reference annular plane. Within this reference cylindrical coordinate system, the kinematics of deformation of the overall annular plate is defined according to the layer-wise deformation theory as given in Eq. (3.4).

3 0

1

( , , , ) ( , , ) ( ) ( , , )

k k k k

i i

i

u r  z t u r  t z z  r  t



 



^,

3 0

1

( , , , ) ( , , ) ( ) ( , , )

k k k k

i i

i

v r  z t v r  t z z  r t



 



^,

3 0

1

( , , , ) ( , , ) ( ) ( , , )

k k k k

i i

i

w r z t w r t z z r t



 



^,

1^k ( 0^k) z  zz ,

Chapter 5: Comparative study on shear-mode piezoelectric actuators

2 2^k ( 0^k) z  zz ,

3 3^k ( 0^k) z  zz

(3.4)

According to this displacement field, the FE model of the annular sandwich plate is derived following the same procedure as given in section 3.4, and the governing FE equations of motion of the annular sandwich plate can be obtained as,

1 np

s s

M E z

s

E



  



MX KX P P (3.17)

For activating the shear actuator patches at the core of the annular sandwich plate, the electric field to an actuator patch (E_z^s for s^th patch) is supplied according to a shear-based feedback control strategy as proposed in section 3.5, where every shear actuator is activated by the feedback of the time- rate of change of the local slope of bending deformation (Eq. 3.18) of the overall annular plate and the corresponding mathematical expression for the applied electric field (E_z^s) arises as given in Eq. (3.19),

0 0

( ) ( )

( )

o s i s

s

r s

w w

r

  ^

 ^,

( )

s s s

i o

r r r

   _(3.18)

0 0

( ) ( )

s

s d o s i s

z s

E k w w

  r 

 ^(3.19)

0 0

(^ow^s)(ⁱw^s) N_T^sX

(3.20) The transverse velocities at the sensing points are expressed in terms of the global nodal velocity vector (X ) through a transformation row matrix (N_T^s) as given in Eq. (3.20), and then introducing Eqs. (3.19) and (3.20) in Eq. (3.17), the FE equations of motion of the annular sandwich plate can be obtained as given in Eq. (3.21).

M

 

t

  

MX CX KX P ,

1 np

s s s E d T s

k N







^P

C (3.21)

Chapter 5: Comparative study on shear-mode piezoelectric actuators