FE model of smart annular plate

A comparative study on the smart damping capabilities of cylindrically orthotropic

4.4 FE model of smart annular plate

An FE model of the smart annular plate (Fig. 4.3) is derived in this section according to the aforesaid control strategy. The state of stress and the state of strain at any point within the overall annular plate can be written following Eqs.

(3.16) and (3.17).

 

^T

b

  

r  r



,



_s 

  

_{rz z}



^{T (3.16)}

 

^T

b 

  

r  r



,



s

  

rz z



^{T (3.17)}

Since a thin smart annular plate is considered in the present study, its kinematics of deformation is defined according to the first-order shear deformation theory (FSDT) as given in Eqs. (3.18a)-(3.18c).



, , ,



0



, ,

 

, ,



p r

u r z t u r t z r t ^(3.18a)



, , ,



0



, ,

 

, ,



vp r z t v r t z_ r t ^(3.18b)



, , ,



0



, ,



wp r z t w r t ^(3.18c)

The generalized displacements (u₀,v₀,w₀, _r,_) in the displacement field (Eqs.

(3.18a)-(3.18c)) can be expressed in terms of the generalized displacement vector ( )d as,

T

0 0 0

{u v w

 

_{r }}

d ^(4.1)

According to the displacement field (Eqs. (3.18a)-(3.18c)), the strain- displacement relations are given in Eqs. (3.20)-(3.21). The same relations can also be written as follows,

T

o 0 1 0

b b b, s r

w w

z  r ^ r



 

   

          

   

T

o 0 1 1

b ,

u v u v u v

r r  r r r  r

   

    

           



1 1 T

r r r

b r r r r r r

  

     

 

   

    

            

 (4.2)

The constitutive relations for the substrate isotropic plate or the actuator- patches are given in Eqs. (3.22) and (3.24). These constitutive relations can also be expressed in common expressions for both the layers as follows,



_b^k C_{b b}^k



e E_b^k ,



_s^k C_s^k



_s e E^k_s (4.3)

T T

( ) ( ) , 1,2

k k k k

b b s s k







 

 

D e e E (4.4)

where, the superscript k denotes substrate plate or actuator-patch according to its value as 1 or 2, respectively. As the substrate plate is made of an isotropic material, the piezoelectric matrices (e_b^k _and e_s^k ) are the null matrices for the substrate plate (k1). The forms of the stiffness matrices (C_b^k , C_s^k ) of the substrate plate (k1) are given in Eq. (3.23). For the PFC actuator-patches (k 2), the forms of various property matrices can be written as,

^{2 2}

11 12

2 2 2

21 22

2 66

0 0

0 0

b

C C

C C

C

 

 

  

 

 

 

C ,

2 55 2

2 44

0

s 0 C

C

 

  

 

 

C ,

2 11

2 2

12 2 26

0 0

0 0

0 0

r b

e e

e

 

 

  

 

 

 

e

,

2 0 0 352

0 0 0

r s

 e 

  

 

e ,

2 21

2 2

22 2 16

0 0

0 0

0 0

b

e e e



 

 

  

 

 

 

e , ²

2 34

0 0 0

0 0

s e

  

  

 

e ,

2 31

2 2

32

0 0 0 0

0 0 0

z b

e e

 

 

  

 

 

 

e ,

2 2 15

2 24

0 0

0 0

z s

e e

 

  

 

 

e

(4.5) For radially/circumferentially/transversely poled fibers, the corresponding forms of the matrices (e_b^k,e_s^k) in Eq. (4.5) are denoted by the superscript r//z.

Chapter 4: A comparative study… in vibration control of annular plates

The overall plate is considered to operate under a transversely distributed harmonic mechanical load of, p r( , )



e^{j t} ( j 1 ) where  is the operating frequency and p r( , )



is the intensity of the distributed load over the plane (r ) of the plate. For the vibration of the overall plate under this dynamic load, the first variations of total potential energy (



T_p) and total kinetic energy (T_k) of the overall plate at an instant of time (t) can be written according to Eq. (3.27) and (3.28).

 

1 1

2 2 T T T

0 2 1

( ) θ

o k k

i k k

r h k k h

p r h b b s s h k p

k

dz dz Q rd dr

  ^   ^  _



 

     







  

^{   }



^{ D (3.27)}

   

 

2 1 T

2

0 1

θ

k o

i

k

r h k

k r p p p p p p

k h

u v w u v w dz rd dr

  ^    



 

 



 



 

 ^(3.28)

In Eq. (3.27), the term (Q_p) for the mechanical load has the following form for the transversely distributed harmonic load,

Q_p (w pe) ^{j t (4.6)}

Using Eqs. (4.1), (4.2), (4.3) and (4.4) along with the linear relations between electric field components (E_r/E_ /E_z) and applied voltage (V ), Eqs. (3.27)-(3.28) can be expressed as,

o

oT o T o

2 T oT T T

0

( 0)

i

b b b b b b b b b b

r

p r s s s b be b be s se

j t

T V rd dr

w pe





 

     



    

 

 

     

 

 

 

 

     

    

A B B D

A A B A

(4.7)



^T



^θ

e e

o o

e e

i i

e r

k r _^ rd dr

 

 

^{d md} (4.8)

In Eqs. (4.7) and (4.8), the expressions for rigidity matrices (A_b,A_s,B_b,D_b) and mass matrix per unit area (m) are given in Eq. (3.32), and the electro-elastic coupling vectors (A_be, A_se, B_e) are as follows,

1

2,

k k

h k

be 



h ^ b vdzk_

A e G ^{k 1} ₂,

k

h k

se



h ^ s vdzk_

A e G

1

2,

k k

h k

be



h ^ b vzdzk_

B e G (4.9)

where, the gradient vector (G) for different PFC actuators are as follows,

G{G_r 0 0}^T for LCR actuator {0 G_ 0}T



G for LCC actuator

{0 0 G_z}T



G for TCR/TCC actuator

(4.10)

In order to derive FE model of the smart plate, the plane of the annular plate is discretized using nine-node quadrilateral isoparametric elements. The FE mesh is generated by dividing the radial and circumferential spans in such a manner that the edges of every element are in parallel to the radial and circumferential directions. Since the PFC actuators are used in the patch form, the FE mesh is comprised of two kinds of elements having different stacking sequences. One element is only for the substrate plate, and the other one has the actuator layer over the substrate layer. At any point within an element, the generalized displacement vector (d) and strain vectors (



_b^o,_s,_b) can be written in terms of the shape function matrix (N ) and the elemental nodal displacement vector (d^e) as follows,

 e

d Nd

o o

e,

b  b



B d



_sB d_s ^e,



_bB d_ ^e,

o ,

b  bt

B L N B_s L N_br , B_ L N_s ^(4.11) The different operator matrices (L_bt,L_br,L_s) appearing in Eq. (4.11) are given in Eq. (3.21). Using Eq. (4.11), a procedure as given in Section 3.3.1 in Chapter 3 is followed to derive the global equations of motion for the vibration of the overall plate in the form of Eq. (3.38).

4 1

( ) ^q( ^q) ^q ( )

p b s p E M

q

V V t



  





 

M K K P P _(3.38)

In Eq. (3.38), the electro-elastic coupling vector (P_E^e( )V ) is independent of the applied voltage (V ) for the linear relations between the electric field components (E_r/E_/E_z) and voltage (V ). Also, for the consideration of a number (N_p) of actuator-patches and the transversely distributed harmonic load (p r( , )



e^{j t} ), Eq. (3.38) can be rewritten as,

Chapter 4: A comparative study… in vibration control of annular plates

1

( )

Np

q q j t

b s E M

q

V e ^



  





MX K K X F F (4.12)

In Eq. (4.12), X is the global nodal displacement vector; F_M is the global nodal mechanical load-amplitude vector; F_E^q is the global nodal electro-elastic coefficient vector for q^th actuator-patch among the N_p number of actuator- patches; V^q is the applied voltage across the pairs of surface-electrodes of q^th actuator-patch.

Every actuator-patch is equipped with a velocity-sensor at its middle point.

The transverse velocity (w^q) at this middle point is sensed and fed back in the form of voltage (V_q) across the pairs of surface-electrodes of the corresponding (

qth) actuator-patch following the negative velocity feedback control strategy as given in Eq. (3.39).

^{q q}

q d

V  k w ^(3.39)

The velocity (w^q) at the middle point of q^th patch can be expressed in terms of the global nodal velocity vector as given in Eq. (3.40).

q q

T p

w N X (3.40)

Using Eqs. (3.39) by expressing the local velocity (w^q) in terms of the global nodal velocity vector (X ) through Eq. (3.40), Eq. (4.12) can be written as,

1

( )

Np

j t

b s q M

q

e ^



   





MX K K X C X F , C_qF k N_E^{q q}_{d T}^q (4.13) Equation (4.13) expresses the FE equations of motion of the smart annular plate where the electrically induced actuation force is modeled as active damping force using the velocity feedback control strategy. The final forms of the equations of motion can be written as,

^{j t}

Me^

  

MX CX KX F

( _{b s}),

 

K K K

1 Np

q q





C C ^(4.14)