**3. Static and dynamic analysis of laminated composite beams with piezoelectric layers**

### **3.1 Displacement and strain**

Based on the first-order shear deformation theory (FSDT), the displacement field at any point of the beam is defined as [1, 2]:

$$\begin{aligned} \mathbf{u}(\mathbf{x}, \mathbf{z}) &= \mathbf{u}\_0(\mathbf{x}) + \mathbf{z}\theta\_\mathcal{\mathcal{Y}}(\mathbf{x}), \\ \mathbf{w}(\mathbf{x}, \mathbf{z}) &= \mathbf{w}\_0(\mathbf{x}), \end{aligned} \tag{10}$$

Substituting Eq. (12) into Eq. (11), we obtain:

*DOI: http://dx.doi.org/10.5772/intechopen.89303*

where n is the element node number.

electric field vector {E}, such that [4, 8–10]:

¼ *B<sup>ϕ</sup>*

defined in terms of nodal variables as:

**3.2 Finite element equations**

where T<sup>e</sup>

**197**

� �f g*<sup>ϕ</sup>* <sup>¼</sup>

Using Hamilton's principle, we have [11–13]:

ð*t*2

*t*1

*<sup>T</sup><sup>e</sup>* <sup>¼</sup> <sup>1</sup> 2 ð

*Ve*

*<sup>ρ</sup>*f g*q*\_ *<sup>T</sup>*

work done by external forces. They are determined by:

where tpk is the thickness of the kth piezoelectric layer.

in which

*Ez <sup>k</sup>* ¼ � *<sup>ϕ</sup><sup>k</sup> tpk* where B½ �¼ <sup>b</sup>

The electric potential is constant over the element surface:

f gε |{z} 2�1

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates*

¼ ½ � Bb |{z} 2�6

d

*<sup>ϕ</sup><sup>k</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1

A voltage ϕ is applied across an actuator of layer thickness tp generates an

f g *Ek* ¼ �∇*ϕ<sup>k</sup>* <sup>¼</sup> 0 0 *<sup>E</sup><sup>z</sup>*

0 0 <sup>1</sup>

Substituting Eq. (19) into Eq. (18), the electric field vector {E} can also be

Using Eqs. (15), and (20), the linear piezoelectric constitutive equations coupling

, Ue are the kinetic and potential energy, respectively and We is the

f gE ¼ � B<sup>ϕ</sup>

the elastic and electric fields will be completely determined by Eqs. (2) and (3).

*tp*1

00 0 00 <sup>1</sup>

00 0

*tp*2

*<sup>T</sup> <sup>e</sup>* � *<sup>U</sup> <sup>e</sup>* � *<sup>W</sup> <sup>e</sup>* ð Þ*dt* <sup>¼</sup> 0, (21)

*<sup>e</sup>* f g*q <sup>e</sup>dV*, (22)

*T*

� �f g<sup>ϕ</sup> e, (20)

qb � � e |fflffl{zfflffl} 6�1

dx <sup>1</sup>

d dx

dx <sup>0</sup> *<sup>z</sup>*

<sup>0</sup> <sup>d</sup>

, (15)

NM � �*:* (16)

*Niϕi*, (17)

*<sup>k</sup>* f g, (18)

*ϕ*1 *ϕ*2

� �, (19)

where u, w denotes the displacements of a point (x, z) in the beam; u0, w0 are the displacements of a point at the beam neutral axis, and θ<sup>y</sup> is the rotation of the transverse normal about the y axis. The bending and shear strains associated with the displacement field in Eq. (10) are defined as:

$$\begin{aligned} \{\mathbf{e}\} = \begin{Bmatrix} \mathbf{e}\_x \\ \mathbf{\dot{y}}\_{xx} \end{Bmatrix} = \left\{ \begin{array}{c} \mathbf{d}\mathbf{u} \\ \mathbf{dx} \\ \mathbf{d}\mathbf{x} + \mathbf{d}\mathbf{x} \end{array} \right\} = \left\{ \begin{array}{c} \mathbf{d}\mathbf{u}\_0 \\ \mathbf{dx} + \mathbf{z}\frac{\mathbf{d}\boldsymbol{\theta}\_x}{\mathbf{dx}} \\ \boldsymbol{\theta}\_\mathbf{\dot{y}} + \frac{\mathbf{d}\mathbf{w}\_0}{\mathbf{dx}} \end{array} \right\} = \left[ \begin{array}{cc} \mathbf{d} \\ \mathbf{dx} \\ \mathbf{0} \\ \mathbf{d} \end{array} \quad \mathbf{z}\frac{\mathbf{d}}{\mathbf{dx}} \right] \left\{ \begin{array}{cc} \mathbf{u}\_0 \\ \mathbf{w}\_0 \\ \boldsymbol{\theta}\_x \end{array} \right\}, \tag{11} \end{aligned} \tag{12}$$

in which εx, γxz are the normal strain, and shear strain, respectively.

Using finite element method, we consider 2-node bending elements with 3 degrees of freedom per node (**Figure 1**).

The displacements of the beam neutral axis are expressed in local coordinate system in the form:

$$\begin{aligned} \{\mathbf{d}\_{\mathbf{0}}\} = \left\{ \begin{matrix} \mathbf{u}\_{0} \\ \mathbf{v}\_{0} \\ \mathbf{0}\_{\mathbf{z}} \end{matrix} \right\} = \left\{ \begin{matrix} [\mathbf{N}^{\mathsf{u}}] \{\mathbf{q}^{\mathsf{u}}\} \\ [\mathbf{N}^{\mathsf{v}}] \{\mathbf{q}^{\mathsf{v}}\} \\ \left[\mathbf{N}^{\mathsf{0}\_{\mathsf{x}}}\right] \{\mathbf{q}^{\mathsf{0}\_{\mathsf{x}}}\} \end{matrix} \right\} = \left[ \mathbf{N}^{\mathsf{M}} \right] \{\mathbf{q}\_{\mathsf{b}}\}\_{\mathsf{e}}, \end{aligned} \tag{12}$$

where {qb}e is the *vector of vector of nodal displacements* of element, [NM] is the matrix mechanical shape functions:

$$\left\{ \mathbf{q}\_{\rm b} \right\}\_{\rm e} = \left\{ \mathbf{q}\_{1} \ \mathbf{q}\_{2} \ \mathbf{q}\_{3} \ \mathbf{q}\_{4} \ \mathbf{q}\_{5} \ \mathbf{q}\_{6} \right\}^{\rm T},\tag{13}$$

$$\underbrace{\begin{bmatrix} \mathbf{N}^{\mathsf{M}} \end{bmatrix}}\_{\mathbf{3}\times\mathbf{6}} = \begin{bmatrix} \begin{bmatrix} \mathbf{N}^{\mathsf{u}} \end{bmatrix} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \begin{bmatrix} \mathbf{N}^{\mathsf{v}} \end{bmatrix} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \begin{bmatrix} \mathbf{N}^{\theta\_{\mathsf{x}}} \end{bmatrix} \end{bmatrix},\tag{14}$$

in which [N<sup>u</sup> ], [N<sup>v</sup> ], [Nθ<sup>z</sup> ] are, in this order, the row vectors of longitudinal, transverse along *y*, and rotation about *z* shape functions.

**Figure 1.** *Two noded beam element.*

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates DOI: http://dx.doi.org/10.5772/intechopen.89303*

Substituting Eq. (12) into Eq. (11), we obtain:

$$\underbrace{\{\mathfrak{e}\}}\_{2\times1} = \underbrace{[\mathbf{B}\_{\mathsf{b}}]}\_{2\times6} \underbrace{\{\mathbf{q}\_{\mathsf{b}}\}}\_{6\times1},\tag{15}$$

$$\text{where } [\mathbf{B}\_{\mathbf{b}}] = \begin{bmatrix} \mathbf{d} & \mathbf{0} & z \frac{\mathbf{d}}{\mathbf{dx}} \\ \mathbf{0} & \mathbf{d} & \mathbf{1} \\ \mathbf{0} & \frac{\mathbf{d}}{\mathbf{dx}} & \mathbf{1} \end{bmatrix} [\mathbf{N}^{\mathbf{M}}].\tag{16}$$

The electric potential is constant over the element surface:

$$
\phi\_k = \sum\_{i=1}^n N\_i \phi\_i,\tag{17}
$$

where n is the element node number.

A voltage ϕ is applied across an actuator of layer thickness tp generates an electric field vector {E}, such that [4, 8–10]:

$$\{E\_k\} = -\nabla \phi\_k = \begin{Bmatrix} \mathbf{0} & \mathbf{0} & E\_k^\varepsilon \end{Bmatrix},\tag{18}$$

in which

**3. Static and dynamic analysis of laminated composite beams with**

Based on the first-order shear deformation theory (FSDT), the displacement

u x, z ð Þ¼ u0ð Þþ x *z*θ*y*ð Þ x ,

where u, w denotes the displacements of a point (x, z) in the beam; u0, w0 are the displacements of a point at the beam neutral axis, and θ<sup>y</sup> is the rotation of the transverse normal about the y axis. The bending and shear strains associated with

> dθ*<sup>z</sup>* dx

9 >>=

>>; ¼

9 >=

� �

>; <sup>¼</sup> NM � � qb

3 7

] are, in this order, the row vectors of longitudinal,

� � e

dw0 dx

du0 dx <sup>þ</sup> <sup>z</sup>

8 >><

>>:

in which εx, γxz are the normal strain, and shear strain, respectively. Using finite element method, we consider 2-node bending elements with 3

> 8 ><

> >:

θ*<sup>y</sup>* þ

The displacements of the beam neutral axis are expressed in local coordinate

Nu ½ � <sup>q</sup><sup>u</sup> f g <sup>N</sup><sup>v</sup> ½ � <sup>q</sup><sup>v</sup> f g N<sup>θ</sup>*<sup>z</sup>* � � q<sup>θ</sup>*<sup>z</sup>* � �

where {qb}e is the *vector of vector of nodal displacements* of element, [NM] is the

<sup>e</sup> ¼ q1 q2 q3 q4 q5 q6 � �<sup>T</sup>

> <sup>N</sup><sup>u</sup> ½ � 0 0 0 N<sup>v</sup> ½ � <sup>0</sup> 0 0N<sup>θ</sup><sup>z</sup>

w x, z ð Þ¼ w0ð Þ <sup>x</sup> , (10)

d

<sup>0</sup> <sup>d</sup>

dx 0 z <sup>d</sup>

dx <sup>1</sup>

dx

8 ><

>:

, (12)

, (13)

<sup>5</sup>, (14)

u0 w0 θ*z*

9 >=

>; ,

(11)

**piezoelectric layers**

*Perovskite and Piezoelectric Materials*

**3.1 Displacement and strain**

f g<sup>ε</sup> <sup>¼</sup> <sup>ε</sup>*<sup>x</sup>*

system in the form:

in which [N<sup>u</sup>

*Two noded beam element.*

**Figure 1.**

**196**

γ*xz* � �

¼

8 >><

>>:

degrees of freedom per node (**Figure 1**).

f g d0 ¼

matrix mechanical shape functions:

], [N<sup>v</sup>

field at any point of the beam is defined as [1, 2]:

the displacement field in Eq. (10) are defined as:

du dx du dz þ

dw dx

> u0 v0 θ*z*

9 >= >; ¼

8 ><

>:

qb � �

> NM � � |ffl{zffl} 3�6

transverse along *y*, and rotation about *z* shape functions.

], [Nθ<sup>z</sup>

¼

2 6 4

9 >>=

>>; ¼

$$E\_k^\* = -\frac{\phi\_k}{t\_{pk}} = \begin{bmatrix} B\_\phi \end{bmatrix} \begin{Bmatrix} \phi \end{Bmatrix} = \begin{bmatrix} 0 & 0 & \frac{1}{t\_{p1}} & 0 & 0 & 0\\ & t\_{p1} & & \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{t\_{p2}} \end{bmatrix}^T \begin{Bmatrix} \phi\_1\\ \phi\_2 \end{Bmatrix},\tag{19}$$

where tpk is the thickness of the kth piezoelectric layer.

Substituting Eq. (19) into Eq. (18), the electric field vector {E} can also be defined in terms of nodal variables as:

$$\{\mathbf{E}\} = -\left[\mathbf{B}\_{\Phi}\right] \{\Phi\}\_{\mathbf{e}},\tag{20}$$

Using Eqs. (15), and (20), the linear piezoelectric constitutive equations coupling the elastic and electric fields will be completely determined by Eqs. (2) and (3).

#### **3.2 Finite element equations**

Using Hamilton's principle, we have [11–13]:

$$\int\_{t\_1}^{t\_2} (T^\epsilon - U^\epsilon - W^\epsilon) dt = 0,\tag{21}$$

where T<sup>e</sup> , Ue are the kinetic and potential energy, respectively and We is the work done by external forces. They are determined by:

$$T^{\epsilon} = \frac{1}{2} \int\_{V\_{\epsilon}} \rho \{\dot{q}\}\_{\epsilon}^{T} \{q\}\_{\epsilon} dV,\tag{22}$$

$$U^{\epsilon} = \frac{1}{2} \int\_{V\_{\epsilon}} \{\varepsilon\}\_{\epsilon}^{T} \{\sigma\}\_{\epsilon} dV,\tag{23}$$

*Ke ϕb* h if g*<sup>q</sup> <sup>e</sup>* � *<sup>K</sup><sup>e</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.89303*

*K<sup>ϕ</sup><sup>b</sup>*

Substituting Eq. (36) into Eq. (35) yields:

� � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>*

� � � � f g*<sup>q</sup>* <sup>¼</sup> f g*<sup>f</sup>* <sup>þ</sup> *Kb<sup>ϕ</sup>*

½ �þ *Kbb Kb<sup>ϕ</sup>*

dynamic equation of motion:

½ � *Mbb* f g€*q* þ ½ �þ *Kbb Kb<sup>ϕ</sup>*

*3.2.3 Free vibration analysis*

and f g *Q <sup>s</sup>* ¼ *K<sup>ϕ</sup><sup>b</sup>*

� � *s*

have:

**199**

*3.2.2 Dynamic analysis*

vector.

Assembling the element equations yields general static equation:

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates*

½ � *Kbb* f g*q* þ *Kb<sup>ϕ</sup>*

where [Kbb], [Kϕϕ] are the overall mechanical stiffness and piezoelectric permittivity matrices respectively; [Kbϕ] and [Kϕb] are the overall mechanical electrical and electrical - mechanical coupling stiffness matrices, respectively, and {q}, {ϕ} are respectively the overall mechanical displacement, and electric potential

Substituting {q} from Eq. (37) into Eq. (36), we obtain the vector {ϕ}.

From Eqs. (25) and (26), assembling the element equations yields general

½ � *Mbb* f g€*q* þ ½ � *Kbb* f g*q* þ *Kb<sup>ϕ</sup>*

� � � � f g*<sup>q</sup>* <sup>¼</sup> f g*<sup>f</sup>* <sup>þ</sup> *Kb<sup>ϕ</sup>*

� � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup><sup>ϕ</sup><sup>b</sup>*

f g*q <sup>s</sup>* ¼ f g*ϕ <sup>s</sup>*

*s*

The beam vibrations induce charges and electric potentials in sensor layers. Therefore, the control system allows current to flow and feeds back to the actuators. In this case, if we apply no external charge Q to a sensor, from Eq. (39), we will

f g*q <sup>s</sup>* is the induced charge due to strain. The operation of the amplified control loop implies, the actuating voltage is

f g*<sup>ϕ</sup> <sup>a</sup>* <sup>¼</sup> *Gd*f g*<sup>ϕ</sup> <sup>s</sup>* <sup>þ</sup> *Gv <sup>ϕ</sup>*\_ � �

*Kϕ<sup>b</sup>*

� � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>*

½ � *Mbb* f gþ €*q* ½ �þ *Kbb Kb<sup>ϕ</sup>*

determined by the following relationship [1, 10, 14]:

For free vibrations, from Eq. (40), the governing equation is:

*Kϕϕ* � ��<sup>1</sup>

*<sup>s</sup> K<sup>ϕ</sup><sup>b</sup>* � � *s*

Substituting Eq. (39) into Eq. (38), we obtain:

*ϕϕ* h if g*<sup>ϕ</sup> <sup>e</sup>* <sup>¼</sup> f g *<sup>Q</sup> <sup>e</sup>*, (34)

� �f g*<sup>ϕ</sup>* <sup>¼</sup> f g*<sup>f</sup>* , (35)

� �f g*<sup>q</sup>* � *<sup>K</sup>ϕϕ* � �f g*<sup>ϕ</sup>* <sup>¼</sup> f g *<sup>Q</sup> :* (36)

� � *Kϕϕ* � ��<sup>1</sup>

� �f g*<sup>q</sup>* � *<sup>K</sup>ϕϕ* � �f g*<sup>ϕ</sup>* <sup>¼</sup> f g *<sup>Q</sup>* , (39)

� � � � f g¼ *<sup>q</sup>* f g<sup>0</sup> *:* (41)

� �f g*<sup>ϕ</sup>* <sup>¼</sup> f g*<sup>f</sup>* , (38)

� � *Kϕϕ* � ��<sup>1</sup>

*:* (42)

, (43)

f g *Q* , (37)

f g *Q* , (40)

$$\boldsymbol{W}^{\epsilon} = \int\_{\boldsymbol{V}\_{\epsilon}} \{\boldsymbol{q}\}\_{\epsilon}^{T} \{\boldsymbol{f}\_{b}\}\_{\epsilon} \boldsymbol{dV} + \int\_{\mathcal{S}\_{\epsilon}} \{\boldsymbol{q}\}\_{\epsilon}^{T} \{\boldsymbol{f}\_{s}\}\_{\epsilon} \boldsymbol{dS} + \{\boldsymbol{q}\}\_{\epsilon}^{T} \{\boldsymbol{f}\_{c}\}\_{\epsilon},\tag{24}$$

in which *f <sup>b</sup>* � � *e* , *fs* � � *e* , *f <sup>c</sup>* � � *<sup>e</sup>* are the body, surface, and concentrated forces acting on the element, respectively. Ve and Se are elemental volume and area. Substituting Eqs. (15), (2), (20), (22), (23), and (24) into Eq. (21), one obtains:

$$\left[\left[\mathcal{M}\_{bb}^{\epsilon}\right]\{\ddot{q}\}\_{\epsilon} + \left[K\_{bb}^{\epsilon}\right]\{q\}\_{\epsilon} + \left[K\_{b\phi}^{\epsilon}\right]\{\phi\}\_{\epsilon} = \{f\}\_{\epsilon},\tag{25}$$

$$
\left[K^{\epsilon}\_{\phi b}\right]\{q\}\_{\epsilon} - \left[K^{\epsilon}\_{\phi \phi}\right]\{\phi\}\_{\epsilon} = \{Q\}\_{\epsilon},\tag{26}
$$

where

$$\text{Element mass matrix: } \left[\boldsymbol{M}\_{bb}^{\epsilon}\right] = \int\_{V\_{\epsilon}} \rho \left[\boldsymbol{N}^{\mathsf{M}}\right]^{T} \left[\boldsymbol{N}^{\mathsf{M}}\right] dV,\tag{27}$$

$$\text{Element mechanical stiffness matrix: } \begin{bmatrix} K\_{bb}^{\epsilon} \end{bmatrix} = \int\_{\mathcal{S}\_{\epsilon}} [B\_b]^T [H] [B\_b] dS,\tag{28}$$

Element mechanical-electrical coupling stiffness matrix:

$$\mathbb{E}\left[K\_{b\phi}^{\epsilon}\right] = \int\_{S\_{\epsilon}} [B\_b]^T [\overline{\varepsilon}] \left[B\_\phi\right] d\mathcal{S},\tag{29}$$

Element electrical-mechanical coupling stiffness matrix:

$$\left[K^{\epsilon}\_{\phi b}\right] = \left[K^{\epsilon}\_{b\phi}\right]^T,\tag{30}$$

Element piezoelectric permittivity matrix:

$$\mathbb{E}\left[K^{\epsilon}\_{\phi\phi}\right] = -\int\_{\mathcal{S}\_{\epsilon}} \left[B\_{\phi}\right]^{T} \left[\overline{p}\right] \left[B\_{\phi}\right] d\mathcal{S},\tag{31}$$

$$\text{where } [H] = \begin{bmatrix} c\_{11} & \mathbf{0} \\ \mathbf{0} & c\_{22} \end{bmatrix}, [\overline{e}] = \begin{bmatrix} e\_{11} & e\_{12} \\ e\_{21} & e\_{22} \end{bmatrix}, [\overline{p}] = \begin{bmatrix} t\_{p1}p\_{11} & \mathbf{0} \\ \mathbf{0} & t\_{p2}p\_{22} \end{bmatrix}, \tag{32}$$

*{ f}e*, *{Q }e* are the applied external load and charge, respectively.

#### *3.2.1 Static analysis*

In the case of beams subjected to static loads, zero acceleration, from Eqs. (25) and (26), we obtain the static equations of the beam as follows:

$$\left[K\_{bb}^{\epsilon}\right]\{q\}\_{\epsilon} + \left[K\_{b\phi}^{\epsilon}\right]\{\phi\}\_{\epsilon} = \{f\}\_{\epsilon},\tag{33}$$

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates DOI: http://dx.doi.org/10.5772/intechopen.89303*

$$
\left[K^{\epsilon}\_{\phi b}\right]\{q\}\_{\epsilon} - \left[K^{\epsilon}\_{\phi b}\right]\{\phi\}\_{\epsilon} = \{Q\}\_{\epsilon},\tag{34}
$$

Assembling the element equations yields general static equation:

$$[K\_{bb}]\{q\} + [K\_{b\phi}]\{\phi\} = \{f\},\tag{35}$$

$$\mathbb{E}\left[K\_{\phi b}\right]\{q\} - \left[K\_{\phi \phi}\right]\{\phi\} = \{Q\}.\tag{36}$$

where [Kbb], [Kϕϕ] are the overall mechanical stiffness and piezoelectric permittivity matrices respectively; [Kbϕ] and [Kϕb] are the overall mechanical electrical and electrical - mechanical coupling stiffness matrices, respectively, and {q}, {ϕ} are respectively the overall mechanical displacement, and electric potential vector.

Substituting Eq. (36) into Eq. (35) yields:

$$\left( \left[ \mathbf{K}\_{bb} \right] + \left[ \mathbf{K}\_{b\phi} \right] \left[ \mathbf{K}\_{\phi\phi} \right]^{-1} \left[ \mathbf{K}\_{\phi b} \right] \right) \{ q \} = \{ f \} + \left[ \mathbf{K}\_{b\phi} \right] \left[ \mathbf{K}\_{\phi\phi} \right]^{-1} \{ \mathbf{Q} \}, \tag{37}$$

Substituting {q} from Eq. (37) into Eq. (36), we obtain the vector {ϕ}.

#### *3.2.2 Dynamic analysis*

*<sup>U</sup> <sup>e</sup>* <sup>¼</sup> <sup>1</sup> 2 ð

*<sup>W</sup> <sup>e</sup>* <sup>¼</sup>

*Perovskite and Piezoelectric Materials*

� � *e* , *fs* � � *e* , *f <sup>c</sup>* � �

in which *f <sup>b</sup>*

where

ð

*Ve* f g*q T <sup>e</sup> f <sup>b</sup>* � � *e dV* þ ð

*M<sup>e</sup> bb* � �f g€*<sup>q</sup> <sup>e</sup>* <sup>þ</sup> *<sup>K</sup><sup>e</sup>*

> *Ke ϕb* h i

Element mass matrix: *Me*

Element mechanical-electrical coupling stiffness matrix:

*Ke bϕ* h i

Element electrical-mechanical coupling stiffness matrix:

*Ke ϕϕ* h i

, ½�¼ *e*

and (26), we obtain the static equations of the beam as follows:

*Ke bb* � �f g*<sup>q</sup> <sup>e</sup>* <sup>þ</sup> *<sup>K</sup><sup>e</sup>*

Element piezoelectric permittivity matrix:

*c*<sup>11</sup> 0 0 *c*<sup>22</sup> � �

where ½ �¼ *H*

*3.2.1 Static analysis*

**198**

¼ ð

*Ke ϕb* h i

> ¼ � ð

*{ f}e*, *{Q }e* are the applied external load and charge, respectively.

*Se*

*Se* ½ � *Bb*

> <sup>¼</sup> *<sup>K</sup><sup>e</sup> bϕ* h i*<sup>T</sup>*

*Bϕ* � �*<sup>T</sup>*

*e*<sup>11</sup> *e*<sup>12</sup> *e*<sup>21</sup> *e*<sup>22</sup> � �

In the case of beams subjected to static loads, zero acceleration, from Eqs. (25)

*bϕ* h i ½ � *p B<sup>ϕ</sup>*

, ½ �¼ *p*

Element mechanical stiffness matrix: *K<sup>e</sup>*

*Ve*

*Se* f g*q T <sup>e</sup> fs* � � *e*

Substituting Eqs. (15), (2), (20), (22), (23), and (24) into Eq. (21), one obtains:

*ϕϕ* h i

> *bb* � � <sup>¼</sup>

*bϕ* h i

ð

*Ve*

ð

*Se* ½ � *Bb*

*bb* � � <sup>¼</sup>

*<sup>T</sup>*½ �*<sup>e</sup> <sup>B</sup><sup>ϕ</sup>*

acting on the element, respectively. Ve and Se are elemental volume and area.

*bb* � �f g*<sup>q</sup> <sup>e</sup>* <sup>þ</sup> *<sup>K</sup><sup>e</sup>*

f g*<sup>q</sup> <sup>e</sup>* � *<sup>K</sup><sup>e</sup>*

f g*<sup>ε</sup> <sup>T</sup>*

*<sup>e</sup>* f g*σ edV*, (23)

*T <sup>e</sup> fc* � � *e*

f g*ϕ <sup>e</sup>* ¼ f g*f <sup>e</sup>*, (25)

*ρ N<sup>M</sup>* � �*<sup>T</sup> N<sup>M</sup>* � �*dV*, (27)

� �*dS*, (29)

, (30)

� �*dS*, (31)

, (32)

*tp*1*p*<sup>11</sup> 0 0 *tp*2*p*<sup>22</sup> � �

f g*ϕ <sup>e</sup>* ¼ f g*f <sup>e</sup>*, (33)

*<sup>T</sup>*½ � *<sup>H</sup>* ½ � *Bb dS*, (28)

f g*ϕ <sup>e</sup>* ¼ f g *Q <sup>e</sup>*, (26)

, (24)

*dS* þ f g*q*

*<sup>e</sup>* are the body, surface, and concentrated forces

From Eqs. (25) and (26), assembling the element equations yields general dynamic equation of motion:

$$\left[\mathbf{M}\_{bb}\right]\{\ddot{q}\} + \left[\mathbf{K}\_{bb}\right]\{q\} + \left[\mathbf{K}\_{b\phi}\right]\{\phi\} = \{f\},\tag{38}$$

$$\left[K\_{\phi b}\right]\{q\} - \left[K\_{\phi \phi}\right]\{\phi\} = \{Q\},\tag{39}$$

Substituting Eq. (39) into Eq. (38), we obtain:

$$\left[\mathbf{M}\_{bb}\right]\{\ddot{q}\} + \left(\left[\mathbf{K}\_{bb}\right] + \left[\mathbf{K}\_{b\phi}\right]\left[\mathbf{K}\_{\phi\phi}\right]^{-1}\left[\mathbf{K}\_{\phi b}\right]\right)\{q\} = \left\{f\right\} + \left[\mathbf{K}\_{b\phi}\right]\left[\mathbf{K}\_{\phi\phi}\right]^{-1}\{Q\},\tag{40}$$

#### *3.2.3 Free vibration analysis*

For free vibrations, from Eq. (40), the governing equation is:

$$\{\mathbf{M}\_{bb}\}\{\ddot{q}\} + \left( \left[ \mathbf{K}\_{bb} \right] + \left[ \mathbf{K}\_{b\phi} \right] \left[ \mathbf{K}\_{\phi\phi} \right]^{-1} \left[ \mathbf{K}\_{\phi b} \right] \right) \{q\} = \{\mathbf{0}\}.\tag{41}$$

The beam vibrations induce charges and electric potentials in sensor layers. Therefore, the control system allows current to flow and feeds back to the actuators. In this case, if we apply no external charge Q to a sensor, from Eq. (39), we will have:

$$\left[\left[K\_{\phi\phi}\right]\_{\mathfrak{s}}\right]^{-1}\left[K\_{\phi b}\right]\_{\mathfrak{s}}\{q\}\_{\mathfrak{s}}=\{\phi\}\_{\mathfrak{s}}.\tag{42}$$

and f g *Q <sup>s</sup>* ¼ *K<sup>ϕ</sup><sup>b</sup>* � � *s* f g*q <sup>s</sup>* is the induced charge due to strain.

The operation of the amplified control loop implies, the actuating voltage is determined by the following relationship [1, 10, 14]:

$$\{\phi\}\_a = \mathcal{G}\_d \{\phi\}\_s + \mathcal{G}\_v \{\dot{\phi}\}\_s,\tag{43}$$

where Gd and Gv are the feedback control gains for displacement and velocity. Substituting Eq. (43) into Eq. (39), the charge in the actuator due to actuator strain in response to the beam vibration modified by control system feedback is:

$$\left( \left[ K\_{\phi b} \right]\_a \{ q \} \_a - \left[ K\_{\phi b} \right]\_a \left( G\_d \{ \phi \} \_s + G\_v \{ \dot{\phi} \} \_s \right) = \{ Q \} \_a. \tag{44}$$

Substituting (42) into (44) leads to:

$$\{Q\}\_{a} = \left[\mathbf{K}\_{\phi b}\right]\_{a} \{q\}\_{a} - \mathbf{G}\_{d} \left[\mathbf{K}\_{\phi \phi}\right]\_{a} \left[\mathbf{K}\_{\phi \phi}\right]\_{s}^{-1} \left[\mathbf{K}\_{\phi b}\right]\_{s} \{q\}\_{s} - \mathbf{G}\_{v} \left[\mathbf{K}\_{\phi \phi}\right]\_{a} \left[\mathbf{K}\_{\phi \phi}\right]\_{s}^{-1} \left[\mathbf{K}\_{\phi b}\right]\_{s} \{\dot{q}\}\_{s}. \tag{45}$$

Substituting Eq. (45) into (40), we obtain:

$$\begin{split} \left[M\_{bb}\right] \left\{\ddot{q}\right\} &+ \left(\left[K\_{bb}\right] + \left[K\_{b\phi}\right] \left[K\_{\phi\phi}\right]^{-1} \left[K\_{\phi b}\right]\right) \left\{q\right\} = \left\{\dot{f}\right\} + \\ &+ \left[K\_{b\phi}\right] \left[K\_{\phi\phi}\right]^{-1} \left(\begin{bmatrix} K\_{\phi b} \end{bmatrix}\_{d} \left\{q\right\}\_{d} - G\_{\nu} \left[K\_{\phi\phi}\right]\_{d} \left[K\_{\phi\phi}\right]\_{s}^{-1} \left[K\_{\phi b}\right]\_{s} \left\{\dot{q}\right\}\_{s} - \right) \\ &- G\_{d} \left[K\_{\phi\phi}\right]\_{d} \left[K\_{\phi\phi}\right]\_{s}^{-1} \left[K\_{\phi b}\right]\_{s} \{q\}\_{s} \end{split} \tag{46}$$

in which {q}s � {q}a � {q} is the beam displacement vector, [Kϕϕ]a = [Kϕϕ]s = [Kϕϕ] is the piezoelectric permittivity matrix, and [Kϕb]a = [Kϕb]s = [Kϕb] is the mechanical-electrical coupling stiffness matrix.

Therefore, Eq. (46) becomes:

$$\begin{split} & \left[ \left[ \boldsymbol{M}\_{bb} \right] \left\{ \ddot{q} \right\} + \left[ \boldsymbol{K}\_{bb} \right] \left\{ q \right\} + \boldsymbol{G}\_{v} \left[ \boldsymbol{K}\_{b\phi} \right] \left[ \boldsymbol{K}\_{\phi\phi} \right]^{-1} \left[ \boldsymbol{K}\_{\phi\phi} \right] \left[ \boldsymbol{K}\_{\phi\phi} \right]^{-1} \left[ \boldsymbol{K}\_{\phi b} \right] \left\{ \dot{q} \right\} + \\ & + \boldsymbol{G}\_{d} \left[ \boldsymbol{K}\_{b\phi} \right] \left[ \boldsymbol{K}\_{\phi\phi} \right]^{-1} \left[ \boldsymbol{K}\_{\phi\phi} \right] \left[ \boldsymbol{K}\_{\phi\phi} \right]^{-1} \left[ \boldsymbol{K}\_{\phi b} \right] \left\{ q \right\} = \left\{ f \right\}. \end{split} \tag{47}$$

In the case of considering the structural damping, the equation of motion of the beam is:

$$\{\mathbf{M}\_{bb}\}\{\ddot{q}\} + ([\mathbf{C}\_A] + [\mathbf{C}\_R])\{\dot{q}\} + K^\*\{q\} = \{f\},\tag{48}$$

where ½ �¼ *CA Gv Kb<sup>ϕ</sup>* � � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup>ϕϕ* � � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>* � � is the active damping matrix, *<sup>K</sup>*<sup>∗</sup> ½ �¼ ½ �þ *Kbb Gd Kb<sup>ϕ</sup>* � � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup>ϕϕ* � � *<sup>K</sup>ϕϕ* � ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>* � � � � is the total of mechanical stiffness matrix and piezoelectric, ½ �¼ *CR αR*½ �þ *Mbb βR*½ � *Kbb* is the overall structural damping matrix, αR, and β<sup>R</sup> are respectively the Rayleigh damping coefficients, which are generally determined by the first and second natural frequencies (ω1, ω2) and ratio of damping ξ, *{f}* is the overall mechanical force vector.

Eq. (48) can be solved by the direct integration Newmark's method.

#### **3.3 Numerical analysis**

An example for free vibration of laminated beam affected by piezoelectric layers is presented here. The beam is made of four layers symmetrically (0°/90°/90°/0°) of epoxy-T300/976 graphite material with 2.5 mm thickness per layer, and with one layer piezo ceramic materials bonded to the top and bottom surfaces, 2.0 mm thickness per layer as shown in **Figure 2** is considered (a = 0.254 m, b = 0.0254 m). The material properties of the piezo ceramic layers and graphite-epoxy are shown in **Table 1**.

**Figures 3** and **4** illustrate the vertical displacement w at the free end of the beam

**Properties PZT G1195 N T300/976**

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates*

*ν*<sup>12</sup> = *ν*<sup>13</sup> = *ν*<sup>23</sup> 0.3 0.3

d31 = d32 (m/V) <sup>254</sup> <sup>10</sup><sup>12</sup> p11 = p22 (F/m) 15.3 <sup>10</sup><sup>9</sup> p33 (F/m) 15.0 <sup>10</sup><sup>9</sup> —

] 0.63 <sup>10</sup><sup>6</sup> 1.50 106

] 0.63 <sup>10</sup><sup>6</sup> 0.09 106

] 0.242 <sup>10</sup><sup>6</sup> 0.071 106

] 0.242 <sup>10</sup><sup>6</sup> 0.025 <sup>10</sup><sup>6</sup>

] 7600 1600

**Case 1:** With structural damping, and without piezoelectric damping (*Gv* = 0,

for two cases:

*Gd* = 0).

**201**

**Figure 3.**

**Figure 2.**

E11 [N/cm2

E22 = E33 [N/cm<sup>2</sup>

G12 = G13 [N/cm2

G23 [N/cm2

ρ [kg/m<sup>3</sup>

**Table 1.**

*Piezoelectric composite cantilever beam.*

*DOI: http://dx.doi.org/10.5772/intechopen.89303*

*Relevant mechanical properties of respective materials.*

*Vertical displacement response (Gv = 0, Gd = 0 Case 1).*

The direct integration Newmark's method is used with parameters α<sup>R</sup> = 0.5, β<sup>R</sup> = 0.25; integral time step Δt = 0.005 s with total time calculated t = 15 s.

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates DOI: http://dx.doi.org/10.5772/intechopen.89303*

#### **Figure 2.**

where Gd and Gv are the feedback control gains for displacement and velocity. Substituting Eq. (43) into Eq. (39), the charge in the actuator due to actuator strain in response to the beam vibration modified by control system feedback is:

> *<sup>s</sup> K<sup>ϕ</sup><sup>b</sup>* � � *s*

*<sup>a</sup>*f g*q <sup>a</sup>* � *Gv Kϕϕ*

*<sup>a</sup> Kϕϕ* � ��<sup>1</sup>

[Kϕϕ]s = [Kϕϕ] is the piezoelectric permittivity matrix, and [Kϕb]a = [Kϕb]s = [Kϕb] is

� ��<sup>1</sup> *Kϕϕ*

In the case of considering the structural damping, the equation of motion of the

� ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>*

� ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>* � � � � is the total of mechanical

An example for free vibration of laminated beam affected by piezoelectric layers is presented here. The beam is made of four layers symmetrically (0°/90°/90°/0°) of epoxy-T300/976 graphite material with 2.5 mm thickness per layer, and with one layer piezo ceramic materials bonded to the top and bottom surfaces, 2.0 mm thickness per layer as shown in **Figure 2** is considered (a = 0.254 m, b = 0.0254 m). The material properties of the piezo ceramic layers and graphite-epoxy are shown

The direct integration Newmark's method is used with parameters α<sup>R</sup> = 0.5,

β<sup>R</sup> = 0.25; integral time step Δt = 0.005 s with total time calculated t = 15 s.

stiffness matrix and piezoelectric, ½ �¼ *CR αR*½ �þ *Mbb βR*½ � *Kbb* is the overall structural damping matrix, αR, and β<sup>R</sup> are respectively the Rayleigh damping coefficients, which are generally determined by the first and second natural frequencies (ω1, ω2)

in which {q}s � {q}a � {q} is the beam displacement vector, [Kϕϕ]a =

� � *Kϕϕ*

� ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>*

� � *Kϕϕ*

� � *Kϕϕ*

Eq. (48) can be solved by the direct integration Newmark's method.

� �

*<sup>s</sup> Kϕ<sup>b</sup>* � � *s* f g*q <sup>s</sup>*

� � *Kϕϕ*

½ � *Mbb* f g€*<sup>q</sup>* <sup>þ</sup> ð Þ ½ �þ *CA* ½ � *CR* f g*q*\_ <sup>þ</sup> *<sup>K</sup>*<sup>∗</sup> f g*<sup>q</sup>* <sup>¼</sup> f g*<sup>f</sup>* , (48)

� ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>*

� �f g*<sup>q</sup>* <sup>¼</sup> f g*<sup>f</sup> :* (47)

� �f gþ *<sup>q</sup>*\_

� � is the active damping matrix,

*<sup>a</sup> Gd*f g*<sup>ϕ</sup> <sup>s</sup>* <sup>þ</sup> *Gv <sup>ϕ</sup>*\_ � �

� �

*s*

f g*q <sup>s</sup>* � *Gv Kϕϕ*

*<sup>a</sup> Kϕϕ* � ��<sup>1</sup>

� �

*<sup>s</sup> Kϕ<sup>b</sup>* � � *s* f g*q*\_ *<sup>s</sup>* �

*<sup>a</sup> Kϕϕ* � ��<sup>1</sup>

¼ f g *Q <sup>a</sup>:* (44)

*<sup>s</sup> K<sup>ϕ</sup><sup>b</sup>* � � *s* f g*q*\_ *<sup>s</sup> :* (45)

> 1 A,

(46)

*K<sup>ϕ</sup><sup>b</sup>* � �

*Perovskite and Piezoelectric Materials*

f g *Q <sup>a</sup>* ¼ *K<sup>ϕ</sup><sup>b</sup>*

� �

½ � *Mbb* f g€*q* þ ½ �þ *Kbb Kb<sup>ϕ</sup>*

þ *Kb<sup>ϕ</sup>* � � *Kϕϕ*

þ*Gd Kb<sup>ϕ</sup>*

where ½ �¼ *CA Gv Kb<sup>ϕ</sup>*

*<sup>K</sup>*<sup>∗</sup> ½ �¼ ½ �þ *Kbb Gd Kb<sup>ϕ</sup>*

**3.3 Numerical analysis**

in **Table 1**.

**200**

beam is:

Therefore, Eq. (46) becomes:

� � *Kϕϕ*

½ � *Mbb* f g€*q* þ ½ � *Kbb* f g*q* þ *Gv Kb<sup>ϕ</sup>*

� ��<sup>1</sup> *Kϕϕ*

� � *Kϕϕ*

� � *Kϕϕ*

� � *Kϕϕ*

� ��<sup>1</sup> *Kϕϕ*

� ��<sup>1</sup> *Kϕϕ*

and ratio of damping ξ, *{f}* is the overall mechanical force vector.

Substituting (42) into (44) leads to:

*<sup>a</sup>*f g*q <sup>a</sup>* � *Gd Kϕϕ*

Substituting Eq. (45) into (40), we obtain:

� � *Kϕϕ*

0 @

� ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>*

the mechanical-electrical coupling stiffness matrix.

*<sup>a</sup>*f g*q <sup>a</sup>* � *Kϕϕ*

� �

� �

*<sup>a</sup> Kϕϕ* � ��<sup>1</sup>

� ��<sup>1</sup> *<sup>K</sup>ϕ<sup>b</sup>* � � � � f g*<sup>q</sup>* <sup>¼</sup> f gþ *<sup>f</sup>*

> �*Gd Kϕϕ* � �

� �

*Piezoelectric composite cantilever beam.*


#### **Table 1.**

*Relevant mechanical properties of respective materials.*

#### **Figure 3.**

*Vertical displacement response (Gv = 0, Gd = 0 Case 1).*

**Figures 3** and **4** illustrate the vertical displacement w at the free end of the beam for two cases:

**Case 1:** With structural damping, and without piezoelectric damping (*Gv* = 0, *Gd* = 0).

The components of the strain vector corresponding to the displacement field

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup>v*<sup>0</sup>

*∂z* þ ∂w *<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup>*w0

0

*u*0 *v*0

<sup>¼</sup> f g *<sup>ε</sup>*<sup>0</sup> <sup>þ</sup> *<sup>z</sup>*f g*<sup>κ</sup>* <sup>¼</sup> *<sup>ε</sup><sup>L</sup>*

>; <sup>¼</sup> *wD* � � � *Is* ½ � � �

� � <sup>þ</sup> *<sup>ε</sup><sup>N</sup>* � � <sup>¼</sup> *<sup>ε</sup><sup>N</sup>*

þ *z*

( )

*∂ ∂x*

<sup>0</sup> *<sup>∂</sup> ∂y*

*<sup>∂</sup><sup>y</sup>* � *<sup>z</sup>*

*∂θ<sup>x</sup> ∂y* ,

þ *z*

*∂θy <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>θ<sup>x</sup> ∂y* � �,

(50)

*<sup>∂</sup><sup>y</sup>* � *<sup>θ</sup>x*,

� *∂ ∂y*

� *∂ ∂y*

*w*<sup>0</sup> *θx θy*

9 >=

� �, (53)

*b* � �,

> 8 ><

> >:

*b*

*<sup>γ</sup>yz* ( ) <sup>¼</sup> f g *<sup>ε</sup><sup>s</sup>* , (54)

0

>; <sup>¼</sup> f g *<sup>ε</sup><sup>s</sup> :* (52)

*θx θy*

¼

(51)

( )

*∂ ∂x*

<sup>0</sup> � *<sup>∂</sup> ∂x*

*<sup>∂</sup><sup>x</sup>* , *<sup>ε</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup><sup>v</sup>*

∂w *<sup>∂</sup><sup>y</sup>* <sup>¼</sup> *<sup>∂</sup>u*<sup>0</sup> *∂y* þ *∂v*<sup>0</sup> *∂x* � �

*Piezoelectric composite plate and coordinate system of the plate (a), and lamina details (b).*

*Static and Dynamic Analysis of Piezoelectric Laminated Composite Beams and Plates*

<sup>þ</sup> *<sup>θ</sup>y*, *<sup>γ</sup>*yz <sup>¼</sup> <sup>∂</sup><sup>v</sup>

*∂ ∂x*

*∂ ∂y*

þ ∂w *∂x* �

> *κx κy κxy*

þ *D<sup>κ</sup>* ½ �

*εx εy γxy* 9 >= >; <sup>¼</sup> *<sup>ε</sup><sup>L</sup> b*

8 ><

>:

8 ><

>:

9 >>=

>>; ¼

> *θx θy*

*wo θx θy*

9 >=

*γxz*

( )

8 >><

>>:

0 1

*<sup>∂</sup><sup>y</sup>* �1 0

(49) are defined as: For the linear strain:

**Figure 5.**

*<sup>ε</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup><sup>u</sup>*

*<sup>γ</sup>*xy <sup>¼</sup> *<sup>∂</sup><sup>u</sup> ∂y* þ *∂v ∂x* � �

*<sup>γ</sup>*xz <sup>¼</sup> <sup>∂</sup><sup>u</sup> *∂z* þ ∂w *<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup>*w0 *∂x*

*εx εy γxy* 9 >>=

>>; ¼

*γxz <sup>γ</sup>yz* ( ) <sup>¼</sup>

8 >><

>>:

**203**

or in the vector form:

8 >><

>>:

¼ *D<sup>ε</sup>* ½ �

*εo x εo y γo xy* 9 >>=

>>; þ *z*

> *u*0 *v*0

*∂ ∂x*

*∂*

and for the nonlinear strain:

( )

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> *<sup>∂</sup>u*<sup>0</sup> *∂x* þ *z ∂θy*

*DOI: http://dx.doi.org/10.5772/intechopen.89303*

**Figure 4.** *Vertical displacement response (Gv = 0.5, Gd = 30* � *Case 2).*

**Case 2:** With structural damping, with piezoelectric damping (*Gv* = 0.5, *Gd* = 30).
