Smart annular plate

Chapter 2: Design of an extension mode annular PFC actuator

2.4 Smart annular plate

The actuation capability of the present annular PFC actuator is investigated by utilizing it in control of vibration of an annular substrate plate. The patches of the PFC actuator with the shape of the annular-sector are attached to the top surface of the host annular plate as shown in Fig. 2.6. The radial span of the annular plate is equally divided into a number (N_r) of divisions where every division has a radial span of, r (  r (r_or_i) /N_r). Similarly, the circumferential span (2) of the same plate is also equally divided into a number (N_) of divisions where every division has the circumferential span of 



(  2 / N_^).

Corresponding to these divisions, annular-sectors are formed over the plane of the annular plate. The dimensions of every sector along the radial and circumferential directions are, rand 



, respectively. In every annular-sector,

Chapter 2: Design of an extension mode annular PFC actuator

a patch of the present PFC actuator is located with the dimensions of r_p and

p

 along the radial and the circumferential directions, respectively, as shown in Fig. 2.6. The patches are considered to be spaced closely in any of the radial and circumferential directions ((   _p) ,(  r r_p) r). Generally, the extension mode piezoelectric actuators are located around the anti-nodes of a bending mode shape of plates. So, this arrangement is made to provide a large number of patches around every anti-node for any mode shape of vibration of the smart annular plate within a frequency-domain. Now, as the mechanically induced in-plane normal stresses at all anti-nodes do not appear with the same sign at any instant of time, every patch is activated by taking the feedback of local velocity, and it is achieved by locating the velocity sensors at the middle points of the patches.

The middle plane of the host annular plate (Fig. 2.6) is taken as the reference plane and the origin of the cylindrical coordinate system (r z ) is located at the center of the reference annular plane. The overall smart plate is considered as a thin plate. So, the displacement field within the overall plate is defined according to the first-order shear deformation theory (FSDT) as,

( , , , ) 0( , , ) ( , , )

p r

u r z t u r t z r t , ( , , , ) 0( , , ) ( , , ) v rp  z t v r t z_ r t ,

( , , , ) 0( , , )

w rp  z t w r t (2.24)

where, u_p, v_p and w_p are the displacements at any point within the overall plate along the r,  ^and z directions, respectively; u₀, v₀ and w₀ are the displacements at any point over the reference plane along the r,  and z directions, respectively; _r and _ are the rotations of the normal to the reference plane with respect to the  and r axes, respectively. According to the displacement field (Eq. (2.24)), the displacement vector (d_p) and the generalized displacement vector (d_p^g) are written as,

 

^T

p  up vp wp

d ,



0 0 0



^T

g

p

 u v w  

r _

d

^(2.25)

Chapter 2: Design of an extension mode annular PFC actuator

The state of stress and the state of strain at any point within the plate can be written as,

 

^T

b

  

r 



r



,

 

^T

s

 

rz



z



,

 

^T

b

  

r 



r



,

 

^T

s

 

rz



z



(2.26)

According to the displacement field (Eq. (2.24)), the strain-displacement relations and the displacement vector (d_p) at any point within the overall annular plate can be written as,

( )

_p^g

b



bt

 z

br

 L L d

, g

s



s p

 L d

,

( ) ^g

p  t z r p

d T T d ,

1 / ) (1 / )( / ) (1 /

( ) 0 0 0 0

( 0 0 0

)( / ) ( / ) (1 / ) 0 0 0

bt r r

r r

r

r





 

 

   

 

  

 

    

L ,

0 0 0 ( ) 0

0 0 0 (1 )

0 0 0

(1 / )( / ) (1 / )( / ) ( / ) (1 / )

br r

r r r

r

r 



   

 

   

     

 

 

L ,

0 0 ( / ) 1 0

0 0 (1 / )( / ) 0 1

s

r

r 

   

    

L ,

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

t

 

 

  

 

 

T _,

0 0 0 1 0 0 0 0 0 1

r

 

  

 

T (2.27)

The constitutive relations for the host isotropic annular plate can be written as,

k k

b Cb b

  ,

k k

s Cs s

  , k1

Chapter 2: Design of an extension mode annular PFC actuator

2

1 0

1 0

1 1

0 0 2

k b

E v v

v v

 

 

 

    

 

 

C ,

1 2 0 0 1 2 1

k s

E v

 

    C

(2.28)

where, C_b^kand C_s^k(k1) are the stiffness matrices; E and v are Young’s modulus and Poisson’s ratio, respectively.

The electro-elastic properties of the present PFC actuator vary along the radial direction. These varying properties are evaluated by means of segmentation of its RV into the large number of micro-volumes/sub-volumes where the variation of properties along the radial direction appears over the radially distributed homogeneous micro-volumes of different FVFs. Presently, these varying properties across the radial span of the annular PFC actuator are expressed as functions of radial coordinate (r). Accordingly, the constitutive relations of the actuator for its application in active control of a thin annular plate under the assumption of  _z 0 can be written as,

( ) ( )T

k k k

b Cb r b eb r Ez

  ,

k k( )

s Cs r s

  ,

( ) 33( )

k k k

z b b z

D e r   r E ,k2

11 12

12 22

66

( ) ( ) 0

( ) ( ) ( ) 0

0 0 ( )

k k

k k k

b

k

C r C r

r C r C r

C r

 

 

  

 

 

 

C ,

55

44

( ) 0

( )

0 ( )

k k

s k

C r r

C r

 

  

 

 

C ,

31( ) 32

( ) ( ) 0

k k k

b r  e r e r 

e ,

2

11^k( ) 11^k( ) 13^k ( ) 33^k ( ) C r C r  C r C r _,

12^k ( ) 12^k ( ) 13^k ( ) 23^k ( ) 33^k ( ) C r C r  C r C r C r ,

2

22^k ( ) 22^k ( ) 23^k ( ) 33^k ( ) C r C r  C r C r ,

Chapter 2: Design of an extension mode annular PFC actuator

31 13 3

31^k( )r e^k ( )r C^k ( )r e^k3( )r C33^k ( )r

e   ,

32 23 3

32^k( )r e^k ( )r C^k ( )r e^k3( )r C33^k ( )r

e   (2.29)

In Eqs. (2.28) and (2.29), the superscript k denotes the substrate annular plate or the PFC actuator patches according to its value as 1 or 2, respectively; C_b^k( )r and C_s^k( )r are the stiffness matrices; e_b^k( )r is the piezoelectric matrix; C_ij^k( )r ,

k( )

eij r and



_ij^k ( )r are the stiffness, piezoelectric and dielectric coefficients, respectively; D_z^k is the electric displacement along the transverse (z) direction;

Ez is the applied electric field across the thickness (h_a) of the active layers of the annular PFC actuator, and it (E_z) can be expressed as, E_z V h/ _a for an applied voltage (V) across the top and bottom fully electrode surfaces of the active layers.

The overall annular plate is considered to be subjected to a transversely distributed harmonic mechanical load (p r( , , ) t ). For the corresponding vibration of the overall plate, the first variations of its total potential energy (T_p) and the total kinetic energy (T_k) at any instant of time (t) can be written as [384],

  ^{ }

1 1

2 2 T T

0 2 1

o k k

i k k

r h k k h

p r h b b s s h z z k p

k

T ^ dz E D dz Q rd dr



^

 

^



_







    ^{ }



^{ }





 ^,

( ( , , )) /2

p p z h

Q w p r t _{ (2.30)}

   

1

2 T

2

0 1

o k

i k

r h k

k r h p p p p p p

k

T ^ u v w u v w dz rd dr

 ^     





   

^(2.31)

where, ^k is the mass density for the host plate (k1) or the piezoelectric actuator (k2); h_k and h_k1 are the thickness coordinates at the bottom and top surfaces of k^th layer, respectively. For deriving FE model of the overall annular plate, the plane of the plate is discretized using 9-node quadrilateral element. A typical element is in the shape of annular-sector with the edges parallel to the radial and circumferential directions of the overall plate. At any point within an element, the displacement vector (d_p), generalized displacement vector (d^g_p) and

Chapter 2: Design of an extension mode annular PFC actuator

generalized strains (_b,_s) can be written in terms of the shape function matrix ( Np) and the elemental nodal displacement vector (d^e_p) as follows,

g e

p  p p

d N d ,

 

b b

e p

bt z r

 

 B B d ,

e s s p

 B d ,

 

p r

e p

m m

t z

 

d B B d ,

bt  btp

B L ,

br  brp

B L ,

s  s p

B L  ,

m

t  tp

B T ,

m

r  rp

B T

(2.32) Substituting Eqs. (2.28), (2.29) and (2.27) in Eqs. (2.30) and (2.31) and then using Eq. (2.32), the first variations of the total potential energy (T_p^e) and the total kinetic energy (T_k^e) for a typical element can be obtained as,

  

^T

 ^{ }

e e e e e e e

p p b s p E M

T V t







d K K d P P ,

( )

T

e e e e

k p p

 T   d M d

,

       



^{T T T T}



e

e

b

 

A bt b

r

bt



bt b

r

br



br b

r

bt



br b

r

br

dA

e

K B A B B B B B B B B D B

,



^T

  

e

e

s

 

A s s

r

s

dA

e

K B A B

,

   



^{T T}



e

e

E

 

A bt be

r 

br be

r dA

e

P B A B B

,

   

^T

^{ 0 0} ( , , t) 0 0 ^

^T

e

e

M

t  

A p

p r  dA

e

P N

_,

 

^{T 1}

 

^{T 2}

 

^{T 2}

 

^{T 3}

e

e m m m m m m m m

t t t r r t r r e

A m m m m dA





    

M B B B B B B B B

(2.33)

Chapter 2: Design of an extension mode annular PFC actuator

where, A_e is the elemental area. The rigidity matrices (A_b,A_s,B_b,D_b), electro- elastic coupling vectors (A_be,B_be) and mass parameters (m₁,m₂,m₃) appearing in Eq. (2.33) are as follows,

 

^{2 1}

 

1

k k

h k

b h b

k

r ^ r dz







A C ,

 

^{2 1}

 

1

k k

h k

s h s

k

r ^ r dz







A C ,

 

^{2 1}

 

1

k k

h k

b h b

k

r ^ r z dz







B C ,

 

^{2 1}

 

²

1

k k

h k

b h b

k

r ^ r z dz







D C ,

 

¹

 

2

1

k k

h k

be h a b k

r r dz

h









A e ,

 

¹

 

2

1

k k

h k

be h a b k

r r z dz

h









B e ,

 

^{2 1}

 

1

1

k k

h k

k h

m r ^  r dz







^,

 

^{2 1}

 

2

1

k k

h k

k h

m r ^  r z dz







^,

 

^{2 1}

 

²

3

1

k k

h k

k h

m r ^  r z dz







^(2.34)

In Eq. (2.34), the stiffness matrices (Cb^k

 

r ,Cs^k

 

r ) and the mass density (^k

 

r ) are constant quantities for the host isotropic annular plate (k1). The governing equations of motion of the smart annular plate are derived employing Hamilton’s principle as,

 

2 1

t e e 0

k p

t T T dt



(2.35)

Introducing the expressions of T_p^e and T_k^e in Eq. (2.35), the following governing equations of motion for a typical element of the smart annular plate can be obtained,

  ^{ }

e e e e e e e

p b  s p  EV  M t

M d K K d P P

(2.36)

Chapter 2: Design of an extension mode annular PFC actuator

Assembling the elemental governing equations of motion (Eq. (2.36)), the following global equations of motion of the smart annular plate can be obtained,

   

1 np

q q

p b s p E M

q

V t



  





MX K K X P P (2.37)

In Eq. (2.37), M is the global mass matrix; K_b and K_s are the bending and shear counterparts of the global stiffness matrix; P_M( )t is the global nodal mechanical load vector; X_p is the global nodal displacement vector; P_E^q _{is the} electro-elastic coefficient vector for the q^th patch and it is obtained through the assembly of elemental matrices (P_E^e) for the elements within the q^th patch; V^q is the applied voltage to the q^th patch; n_p is the total number of actuator patches.

The external voltage (V^q) is supplied to the q^thpatch according to the negative velocity feedback control strategy. The transverse velocity (w^q) at the middle point of the q^th patch is sensed and it (w^q) is fed back in the form of a voltage (V^q) as,

q

q q

V  k wd ^(2.38)

where, k_d^q is the velocity feedback control-gain for the q^th patch. The transverse velocity (w^q) can be expressed in terms of the global nodal velocity vector (X_p) as follows,

q q

p

w N XT (2.39)

where, N_T^q

is a row matrix that indicates the element of X^p corresponding to the transverse velocity at the middle point of the q^thpatch. Using Eqs. (2.38) and (2.39) in Eq. (2.37), the following FE equations of motion of the smart annular plate can be obtained,

p  p  p  M

 

t

MX CX KX P ,

K = K 

b

 K

s



,

1 np

q q





C C , C^q (P_E^{q q}k_dN_T^q) (2.40)

Chapter 2: Design of an extension mode annular PFC actuator