*4.3.1.2 Formulation of y-Stiffener*

**4.3 Dynamic analysis of piezoelectric stiffened composite plates subjected to**

Consider isoparametric piezoelectric laminated stiffened plate with the general coordinate system (x, y, z), in which the x, y plane coincides with the neutral plane of the plate. The top surface and lower surface of the plate are bonded to the piezoelectric patches (actuator and sensor). The plate subjected to the airflow load

The dynamic equations of a finite smart composite plate are written as follows:

*Smart stiffened plate subjected to airflow. (a) Smart stiffened plate and coordinate system and (b) Lamina details.*

� �

*<sup>e</sup>* þ ½ � *KA <sup>e</sup>*

*<sup>e</sup>* <sup>¼</sup> <sup>Ð</sup> *Ve Bnl b* � �*<sup>T</sup>*

*Wxs*ð Þ¼ *<sup>x</sup>*, *<sup>z</sup> wxs*ð Þ *<sup>x</sup> :* (75)

*bb* � � f g*<sup>u</sup> <sup>e</sup>* <sup>¼</sup> f g*<sup>f</sup> <sup>m</sup>*

½ � *<sup>Q</sup> Bnl b*

*<sup>e</sup>* , (74)

� �*dV* are the

*<sup>z</sup>*¼*txs=*2, (76)

*<sup>e</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>n</sup>*<sup>l</sup> bb* � �

element linear mechanical stiffness and nonlinear mechanical stiffness respectively,

*Uxs*ð Þ¼ *x*, *z u*0ð Þþ *x zθxs*ð Þ *x* ,

where x-axis is taken along the stiffener centerline and the z-axis is its upward

*<sup>z</sup>*¼*txs=*2, *wp*

� �

*<sup>z</sup>*¼�*tp=*<sup>2</sup> <sup>¼</sup> *wxs*<sup>j</sup>

*Bxs* ½ �*<sup>T</sup> Dxs* ½ � *Bxs* ½ �*dx*, (77)

If we consider that the x-stiffener is attached to the lower side of the plate, conditions of displacement compatibility along their line of connection can be

*<sup>z</sup>*¼�*tp=*<sup>2</sup> <sup>¼</sup> *<sup>θ</sup>xs*<sup>j</sup>

The element stiffness and mass matrices are defined as follows [2, 15]:

ð

*le*

where tp is the plate thickness and txs is the x-stiffener depth.

*Kxs* ½ �*<sup>e</sup>* ¼

� �*dV*, and *Knl*

*bb* � �

½ � *<sup>Q</sup> <sup>B</sup>*ln *b*

normal. The plate and stiffener element shown in **Figure 11**.

� �

*<sup>e</sup>* is element external mechanical force vector.

**airflow**

*Perovskite and Piezoelectric Materials*

acting (**Figure 10**).

where *K*ln

f g*<sup>f</sup> <sup>m</sup>*

**Figure 10.**

written as:

*up* � �

**210**

*bb* � �

*4.3.1 Formulation of Stiffener:*

*4.3.1.1 Formulation of x-Stiffener*

*<sup>z</sup>*¼�*tp=*<sup>2</sup> <sup>¼</sup> *uxs*j*<sup>z</sup>*¼*txs=*2, *<sup>θ</sup>xp*

*<sup>e</sup>* <sup>¼</sup> <sup>Ð</sup> *Ve B*ln *b* � �*<sup>T</sup>*

½ � *<sup>M</sup> <sup>e</sup>*f g*u*€ *<sup>e</sup>* <sup>þ</sup> ½ � *CA <sup>e</sup>*f g*u*\_ *<sup>e</sup>* <sup>þ</sup> *<sup>K</sup>*ln

The same as for x-stiffener, the element stiffness and mass matrices of the ystiffener are defined as follows:

$$\left[\boldsymbol{K}\_{\mathcal{Y}}\right]\_{\epsilon} = \int\_{\boldsymbol{l}\_{\epsilon}} \left[\boldsymbol{B}\_{\mathcal{Y}}\right]^{T} \left[\boldsymbol{D}\_{\mathcal{Y}}\right] \left[\boldsymbol{B}\_{\mathcal{Y}}\right] d\mathcal{Y},\tag{79}$$

$$\mathbb{E}\left[\mathbf{M}\_{\mathcal{V}}\right]\_{\varepsilon} = \int\_{A\_{\varepsilon}} \left[ P\left( [\mathbf{N}\_{\mathbb{u}^{0}}]^{T}[\mathbf{N}\_{\mathbb{u}^{0}}] + [\mathbf{N}\_{\mathbb{w}}]^{T}[\mathbf{N}\_{\mathbb{w}}] \right) + I\_{\mathbb{x}}\left( \left[\mathbf{N}\_{\theta\_{\mathcal{}}}\right]^{T}\left[\mathbf{N}\_{\theta\_{\mathcal{}}}\right] \right) \right] d\mathcal{A},\tag{80}$$

### *4.3.2 Modeling the effect of aerodynamic pressure and motion equations of the smart composite plate-stiffeners element*

Based on the first order theory, the aerodynamic pressure *lh* and moment *mθ*, can be described as [15–17]:

$$\begin{split} \mathcal{I}\_{w} &= \frac{1}{2} \rho\_{a} (U \cos a)^{2} B \Big[ k H\_{1}^{\*} \frac{\dot{\mathbf{w}}}{U \cos a} + k H\_{2}^{\*} \frac{B \dot{\theta}}{U \cos a} + k^{2} H\_{3}^{\*} \theta \Big] + \frac{1}{2} \mathcal{C}\_{p} \rho\_{a} (U \sin a)^{2}, \\ \mathcal{I}\_{\theta} &= \frac{1}{2} \rho\_{a} (U \cos a)^{2} B^{2} \Big[ k A\_{1}^{\*} \frac{\dot{\mathbf{w}}}{U \cos a} + k A\_{2}^{\*} \frac{B \dot{\theta}}{U \cos a} + k^{2} A\_{3}^{\*} \theta \Big], \end{split} \tag{81}$$

where *k* ¼ *bω=U* is defined as the reduced frequency, ω is the circular frequency of oscillation of the airfoil, U is the wind velocity, B is the half-chord length of the airfoil or half-width of the plate, ρ<sup>a</sup> is the air density and α is the angle of attack.

**Figure 11.** *Modeling of plate and stiffener element.*

The functions *A*<sup>∗</sup> *<sup>i</sup>* ð Þ *<sup>K</sup>* , *<sup>H</sup>*<sup>∗</sup> *<sup>i</sup>* ð Þ *K* are defined as follows:

$$\begin{split} &H\_{1}^{\*}\left(K\right) = -\frac{\pi}{k}F(k), H\_{2}^{\*}\left(K\right) = -\frac{\pi}{4k}\left[1 + F(k) + \frac{2G(k)}{k}\right], \\ &H\_{3}^{\*}\left(K\right) = -\frac{\pi}{2k^{2}}\left[F(k) - \frac{kG(k)}{2}\right], A\_{1}^{\*}\left(K\right) = \frac{\pi}{4k}F(k), \\ &A\_{2}^{\*}\left(K\right) = -\frac{\pi}{16k}\left[1 - F(k) - \frac{2G(k)}{k}\right], A\_{3}^{\*}\left(K\right) = \frac{\pi}{8k^{2}}\left[\frac{k^{2}}{8} + F(k) - \frac{kG(k)}{2}\right], \end{split} \tag{82}$$

where F(k) and G(k) are defined as:

$$\begin{split} F(k) &= \frac{0.500502k^3 + 0.512607k^2 + 0.2104k + 0.021573}{k^3 + 1.035378k^2 + 0.251293k + 0.021508}, \\ G(k) &= -\frac{0.000146k^3 + 0.122397k^2 + 0.327214k + 0.001995}{k^3 + 2.481481k^2 + 0.93453k + 0.089318}. \end{split} \tag{83}$$

Using finite element method, aerodynamic force vector can be described as:

$$\left\{ \left. f \right\}\_{\epsilon}^{\dot{a}\dot{r}} = -\left[ \mathbf{K}^{a\dot{r}} \right]\_{\epsilon} \{ \mu \}\_{\epsilon} - \left[ \mathbf{C}^{a\dot{r}} \right]\_{\epsilon} \{ \dot{u} \}\_{\epsilon} + \left\{ f \right\}\_{\epsilon}^{\mathbf{u}},\tag{84}$$

*<sup>M</sup>*<sup>∗</sup> ½ �f g*u*€ <sup>þ</sup> ð Þ ½ �þ *CR* ½ � *CA* f g*u*\_ <sup>þ</sup> *<sup>K</sup>* <sup>∗</sup> ½ �þ ½ �þ *KA <sup>K</sup>air* f g*<sup>u</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>∗</sup> f g*m*, (89)

The solution of nonlinear Eq. (89) is carried out by using Newmark direct and

A rectangle cantilever laminated composite plate is assumed to be [0°/90°]s with total thickness 4 mm, length of 600 mm and width of 400 mm with three stiffeners along each direction x and y. The geometrical dimension of the stiffener is 5 mm of high and 10 mm of width. The plate and stiffeners are made of graphite/epoxy with mechanical properties: E11 = 181 GPa, E22 = E33 = 10.3 GPa, E12 = 7.17 GPa,

layer made of PZT-5A are: d31 = d32 <sup>=</sup> �<sup>171</sup> � <sup>10</sup>�<sup>12</sup> m/V, d33 = 374 � <sup>10</sup>�<sup>12</sup> m/V, d15 = d24 <sup>=</sup> �<sup>584</sup> � <sup>10</sup>�<sup>12</sup> m/V, G12 = 7.17 GPa, G23 = 2.87 GPa, G32 = 7.17 GPa, <sup>ν</sup>PZT *<sup>=</sup>* 0.3, <sup>ρ</sup>PZT = 7600 kg�m�<sup>3</sup> and thickness tPZT = 0.15876 mm, <sup>ξ</sup> = 0.05, Gv = 0.5, Gd = 15. The effects of the excitation frequency and location of the actuators are presented through a parametric study to examine the vibration shape of the composite plate activated by the surface bonded piezoelectric actuators. The iterative

*History of the plate at a critical airflow velocity Ucr = 30.5 m/s. (a) Displacement response and*

. Material properties for piezoelectric

*bb* <sup>þ</sup> *<sup>K</sup>*n*<sup>l</sup> bb* .

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

where ½ �¼ *CR <sup>α</sup>R*½ �þ *Mbb <sup>β</sup><sup>R</sup> <sup>K</sup>*ln

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

<sup>ν</sup><sup>12</sup> = 0.35, <sup>ν</sup><sup>23</sup> <sup>=</sup> <sup>ν</sup><sup>32</sup> = 0.38, <sup>ρ</sup> = 1600 kg�m�<sup>3</sup>

Newton-Raphson iteration method.

*4.3.4 Numerical applications*

**Figure 12.**

**213**

*(b) Piezoelectric voltage response.*

with *Kair* � � *e* , *Cair* � � *<sup>e</sup>* and f g*<sup>f</sup> <sup>n</sup> <sup>e</sup>* are the aerodynamic stiffness, damping matrices and lift force vector, respectively

$$\left[K\_{\epsilon}^{air}\right] = \rho\_a(U\cos\alpha)^2 \text{Bk}^2 \int\_{A\_{\epsilon}} \left[H\_3^\*\left(k\right) \left[N\_{\text{w}}\right]^T \left[N\_{\text{d\alpha}}\right] + B A\_3^\*\left(k\right) \left[\frac{\partial N\_{\theta y}}{\partial \mathbf{x}}\right]^T \left[N\_{\theta x}\right] \right] dA,\tag{85}$$

$$\left[C\_{\epsilon}^{air}\right] = \rho\_a(U\cos\alpha)Bk \left[\begin{aligned} &\int\_{A\_{\epsilon}} \left(H\_1^\*\left(k\right)\left[N\_{\text{w}}\right]^T \left[N\_{\text{w}}\right] + B H\_2^\*\left(k\right)\left[N\_{\text{w}}\right]^T \left[N\_{\theta x}\right] dA\right) \\ &+ \int\_{A\_{\epsilon}} \left(B A\_1^\*\left(k\right) \left[\frac{\partial N\_{\theta y}}{\partial \mathbf{x}}\right]^T \left[N\_{\text{w}}\right] + B^2 A\_2^\*\left(k\right) \left[\frac{\partial N\_{\theta y}}{\partial \mathbf{x}}\right]^T \left[N\_{\theta x}\right] dA\right) \end{aligned}\right],\tag{86}$$

$$\left\{\left\{f\right\}\_{\epsilon}^{n} = \mathbf{C}\_{p}\rho\_{\mathfrak{a}}\left(\mathbf{U}\sin\mathfrak{a}\right)^{2}\right\}\left[\left[\mathbf{N}\_{w}\right]^{T}d\mathbf{A},\tag{87}$$

where Ae is the element area, [Nw], [Nθ] are the shape functions.

From Eqs. (74) and (84), the governing equations of motion of the smart composite plate-stiffeners element subjected to an aerodynamic force without damping can be derived as:

$$\left( [\mathbf{M}^\*]\_\epsilon \{\ddot{u}\}\_\epsilon + [\mathbf{C}\_A]\_\epsilon \{\dot{u}\}\_\epsilon + \left( [\mathbf{K}^\*]\_\epsilon + [\mathbf{K}\_A]\_\epsilon + [\mathbf{K}^{air}]\_\epsilon \right) \{u\}\_\epsilon = \left\{ f^{\*\*} \right\}\_\epsilon^{m},\tag{88}$$

where *<sup>M</sup>*<sup>∗</sup> ½ �*<sup>e</sup>* <sup>¼</sup> ½ � *<sup>M</sup> <sup>e</sup>* <sup>þ</sup> *Mxs* ½ �*<sup>e</sup>* <sup>þ</sup> *Mys* � � *e* , *<sup>K</sup>* <sup>∗</sup> ½ �*<sup>e</sup>* <sup>¼</sup> *<sup>K</sup>*ln *bb* � � *<sup>e</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>n</sup>*<sup>l</sup> bb* � � *<sup>e</sup>* <sup>þ</sup> *Kxs* ½ �*<sup>e</sup>* <sup>þ</sup> *Kys* � � *e* , *<sup>f</sup>* <sup>∗</sup> f g*<sup>m</sup> <sup>e</sup>* <sup>¼</sup> f g*<sup>f</sup> <sup>m</sup> <sup>e</sup>* <sup>þ</sup> f g*<sup>f</sup> <sup>n</sup> e* .

#### *4.3.3 Governing differential equations for total system*

Finally, the elemental equations of motion are assembled to obtain the open-loop global equation of motion of the overall stiffened composite plate with the PZT patches as follows:

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

$$\left( [\mathbf{M}^\*] \{ \ddot{\boldsymbol{u}} \} + ([\mathbf{C}\_R] + [\mathbf{C}\_A]) \{ \dot{\boldsymbol{u}} \} + \left( [\mathbf{K}^\*] + [\mathbf{K}\_A] + [\mathbf{K}^{a \dot{r}}] \right) \{ \boldsymbol{u} \} = \{ \boldsymbol{f}^\* \}^m,\tag{89}$$

where ½ �¼ *CR <sup>α</sup>R*½ �þ *Mbb <sup>β</sup><sup>R</sup> <sup>K</sup>*ln *bb* <sup>þ</sup> *<sup>K</sup>*n*<sup>l</sup> bb* .

The solution of nonlinear Eq. (89) is carried out by using Newmark direct and Newton-Raphson iteration method.

### *4.3.4 Numerical applications*

The functions *A*<sup>∗</sup>

*Perovskite and Piezoelectric Materials*

<sup>1</sup> ð Þ¼� *<sup>K</sup> <sup>π</sup>*

<sup>3</sup> ð Þ¼� *<sup>K</sup> <sup>π</sup>*

<sup>2</sup> ð Þ¼� *<sup>K</sup> <sup>π</sup>*

with *Kair* � �

*Kair e*

> *Cair e*

*<sup>f</sup>* <sup>∗</sup> f g*<sup>m</sup>*

**212**

*e* , *Cair* � �

� � <sup>¼</sup> *<sup>ρ</sup>a*ð Þ *<sup>U</sup>* cos *<sup>α</sup>* <sup>2</sup>

� � <sup>¼</sup> *<sup>ρ</sup>a*ð Þ *Uc*os*<sup>α</sup> Bk*

damping can be derived as:

*<sup>e</sup>* <sup>¼</sup> f g*<sup>f</sup> <sup>m</sup>*

patches as follows:

where *<sup>M</sup>*<sup>∗</sup> ½ �*<sup>e</sup>* <sup>¼</sup> ½ � *<sup>M</sup> <sup>e</sup>* <sup>þ</sup> *Mxs* ½ �*<sup>e</sup>* <sup>þ</sup> *Mys*

*4.3.3 Governing differential equations for total system*

*<sup>e</sup>* <sup>þ</sup> f g*<sup>f</sup> <sup>n</sup> e* .

and lift force vector, respectively

*H*<sup>∗</sup>

*H*<sup>∗</sup>

*A*<sup>∗</sup>

*<sup>i</sup>* ð Þ *<sup>K</sup>* , *<sup>H</sup>*<sup>∗</sup>

<sup>2</sup>*k*<sup>2</sup> *F k*ð Þ� *kG k*ð Þ

where F(k) and G(k) are defined as:

f g*<sup>f</sup> air*

*<sup>e</sup>* ¼ � *<sup>K</sup>air* � �

*H*<sup>∗</sup>

*H*<sup>∗</sup>

*BA*<sup>∗</sup>

*<sup>e</sup>* and f g*<sup>f</sup> <sup>n</sup>*

*Ae*

Ð *Ae*

þ Ð *Ae*

f g*<sup>f</sup> <sup>n</sup>*

*<sup>M</sup>*<sup>∗</sup> ½ �*e*f g*u*€ *<sup>e</sup>* <sup>þ</sup> ½ � *CA <sup>e</sup>*f g*u*\_ *<sup>e</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>∗</sup> ½ �*<sup>e</sup>* <sup>þ</sup> ½ � *KA <sup>e</sup>* <sup>þ</sup> *<sup>K</sup>air* � �

*Bk*<sup>2</sup> ð

� �

<sup>16</sup>*<sup>k</sup>* <sup>1</sup> � *F k*ð Þ� <sup>2</sup>*G k*ð Þ

<sup>2</sup> ð Þ¼� *<sup>K</sup> <sup>π</sup>*

2

� �

, *A*<sup>∗</sup>

*k*

*F k*ð Þ¼ <sup>0</sup>*:*500502*k*<sup>3</sup> <sup>þ</sup> <sup>0</sup>*:*512607*k*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*2104*<sup>k</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>021573</sup>

*G k*ð Þ¼� <sup>0</sup>*:*000146*k*<sup>3</sup> <sup>þ</sup> <sup>0</sup>*:*122397*k*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*327214*<sup>k</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>001995</sup>

*e*

*<sup>k</sup> F k*ð Þ, *<sup>H</sup>*<sup>∗</sup>

*<sup>i</sup>* ð Þ *K* are defined as follows:

<sup>1</sup> ð Þ¼ *<sup>K</sup> <sup>π</sup>*

, *A*<sup>∗</sup>

*<sup>k</sup>*<sup>3</sup> <sup>þ</sup> <sup>1</sup>*:*035378*k*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*251293*<sup>k</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>021508</sup> ,

Using finite element method, aerodynamic force vector can be described as:

f g*<sup>u</sup> <sup>e</sup>* � *<sup>C</sup>air* � �

<sup>3</sup> ð Þ*<sup>k</sup>* ½ � *Nw <sup>T</sup>*½ �þ *<sup>N</sup>θ<sup>x</sup> BA*<sup>∗</sup>

<sup>1</sup> ð Þ*<sup>k</sup>* ½ � *<sup>N</sup>*<sup>w</sup> *<sup>T</sup>*½ �þ *<sup>N</sup>*<sup>w</sup> *BH*<sup>∗</sup>

<sup>1</sup> ð Þ*<sup>k</sup> <sup>∂</sup>Nθ<sup>y</sup> ∂x* h i*<sup>T</sup>*

*<sup>e</sup>* <sup>¼</sup> *Cpρ*að Þ *<sup>U</sup>* sin *<sup>α</sup>* <sup>2</sup>

From Eqs. (74) and (84), the governing equations of motion of the smart composite plate-stiffeners element subjected to an aerodynamic force without

� �

Finally, the elemental equations of motion are assembled to obtain the open-loop global equation of motion of the overall stiffened composite plate with the PZT

, *<sup>K</sup>* <sup>∗</sup> ½ �*<sup>e</sup>* <sup>¼</sup> *<sup>K</sup>*ln

where Ae is the element area, [Nw], [Nθ] are the shape functions.

� � *e*

<sup>4</sup>*<sup>k</sup>* <sup>1</sup> <sup>þ</sup> *F k*ð Þþ <sup>2</sup>*G k*ð Þ

� �

4*k F k*ð Þ,

<sup>3</sup> ð Þ¼ *<sup>K</sup> <sup>π</sup>*

*<sup>k</sup>*<sup>3</sup> <sup>þ</sup> <sup>2</sup>*:*481481*k*<sup>2</sup> <sup>þ</sup> <sup>0</sup>*:*93453*<sup>k</sup>* <sup>þ</sup> <sup>0</sup>*:*<sup>089318</sup> *:*

*e*

" #

� �

½ �þ *<sup>N</sup>*<sup>w</sup> *<sup>B</sup>*<sup>2</sup>

ð

*Ae*

f g*u*\_ *<sup>e</sup>* <sup>þ</sup> f g*<sup>f</sup> <sup>n</sup>*

*<sup>e</sup>* are the aerodynamic stiffness, damping matrices

<sup>3</sup> ð Þ*<sup>k</sup> <sup>∂</sup>Nθ<sup>y</sup> ∂x* � �*<sup>T</sup>*

> *A*<sup>∗</sup> <sup>2</sup> ð Þ*<sup>k</sup> <sup>∂</sup>Nθ<sup>y</sup> ∂x* h i*<sup>T</sup>*

� �

*e*

*bb* � �

<sup>2</sup> ð Þ*<sup>k</sup>* ½ � *<sup>N</sup>*<sup>w</sup> *<sup>T</sup>*½ � *<sup>N</sup>θ*<sup>x</sup> *dA*

*k*

8*k*<sup>2</sup>

,

*k*2

<sup>8</sup> <sup>þ</sup> *F k*ð Þ� *kG k*ð Þ

" #

2

,

*<sup>e</sup>* , (84)

½ � *Nθ*<sup>x</sup> *dA*

*<sup>e</sup>* , (88)

� � *e* ,

*<sup>e</sup>* þ *Kxs* ½ �*<sup>e</sup>* þ *Kys*

*dA*, (85)

(86)

½ � *Nθ<sup>x</sup>*

½ � *Nw TdA*, (87)

f g*<sup>u</sup> <sup>e</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>∗</sup> f g*<sup>m</sup>*

*<sup>e</sup>* <sup>þ</sup> *<sup>K</sup>*<sup>n</sup>*<sup>l</sup> bb* � � (82)

(83)

A rectangle cantilever laminated composite plate is assumed to be [0°/90°]s with total thickness 4 mm, length of 600 mm and width of 400 mm with three stiffeners along each direction x and y. The geometrical dimension of the stiffener is 5 mm of high and 10 mm of width. The plate and stiffeners are made of graphite/epoxy with mechanical properties: E11 = 181 GPa, E22 = E33 = 10.3 GPa, E12 = 7.17 GPa, <sup>ν</sup><sup>12</sup> = 0.35, <sup>ν</sup><sup>23</sup> <sup>=</sup> <sup>ν</sup><sup>32</sup> = 0.38, <sup>ρ</sup> = 1600 kg�m�<sup>3</sup> . Material properties for piezoelectric layer made of PZT-5A are: d31 = d32 <sup>=</sup> �<sup>171</sup> � <sup>10</sup>�<sup>12</sup> m/V, d33 = 374 � <sup>10</sup>�<sup>12</sup> m/V, d15 = d24 <sup>=</sup> �<sup>584</sup> � <sup>10</sup>�<sup>12</sup> m/V, G12 = 7.17 GPa, G23 = 2.87 GPa, G32 = 7.17 GPa, <sup>ν</sup>PZT *<sup>=</sup>* 0.3, <sup>ρ</sup>PZT = 7600 kg�m�<sup>3</sup> and thickness tPZT = 0.15876 mm, <sup>ξ</sup> = 0.05, Gv = 0.5, Gd = 15. The effects of the excitation frequency and location of the actuators are presented through a parametric study to examine the vibration shape of the composite plate activated by the surface bonded piezoelectric actuators. The iterative

**Figure 12.** *History of the plate at a critical airflow velocity Ucr = 30.5 m/s. (a) Displacement response and (b) Piezoelectric voltage response.*

error of the load ε<sup>D</sup> = 0.02% is chosen. The piezoelectric stiffened composite plate is subjected to the airflow in the positive x direction as shown in **Figure 10a**.

**References**

Limited; 1978

**29**:729

**31**:315

**49**:S276-S280

**215**

Press; 1969

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

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Dynamic response of the piezoelectric stiffened composite plate is shown in **Figure 12**.
