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

*Mxs* ½ �*<sup>e</sup>* ¼

ð

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

*P Nu* ½ � <sup>0</sup> *<sup>T</sup> Nu* ½ �þ<sup>0</sup> ½ � *Nw <sup>T</sup>*½ � *Nw*

*Kys* � � *<sup>e</sup>* ¼ ð

*P Nu* ½ � <sup>0</sup> *<sup>T</sup> Nu* ½ �þ<sup>0</sup> ½ � *Nw <sup>T</sup>*½ � *Nw*

w\_ *<sup>U</sup>* cos *<sup>α</sup>* <sup>þ</sup> *kH*<sup>∗</sup>

> w\_ *<sup>U</sup>* cos *<sup>α</sup>* <sup>þ</sup> *kA*<sup>∗</sup>

unknowns, Iy is the area moment of inertia related to the y-axis and *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup>*

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

*le*

� � <sup>þ</sup> *Ix <sup>N</sup>θ<sup>y</sup>*

*4.3.2 Modeling the effect of aerodynamic pressure and motion equations of the smart*

Based on the first order theory, the aerodynamic pressure *lh* and moment *mθ*, can

*B*\_ *θ <sup>U</sup>* cos *<sup>α</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

> *B*\_ *θ <sup>U</sup>* cos *<sup>α</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

2

2

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.

� �,

� �

� � <sup>þ</sup> *Iy <sup>N</sup><sup>θ</sup><sup>x</sup>* ½ �*<sup>T</sup> <sup>N</sup><sup>θ</sup><sup>x</sup>* ½ � h i � � *dA*, (78)

*k*¼1 Ð *hk*

*Bys* � �*<sup>T</sup> Dys* � � *Bys* � �*dy*, (79)

*Nθ<sup>y</sup>*

h i*<sup>T</sup>*

*H*<sup>∗</sup> 3 *θ*

> *A*<sup>∗</sup> 3 *θ*

þ 1 2

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

,

(81)

� � h i � �*dA*, (80)

*hk*�<sup>1</sup>

*ρkdz*,

with [Bxs] is the strain-displacement relations matrix, [Dxs] is the stress-strain relations matrix and *le* is the element length, *Nu* ½ � <sup>0</sup> , ½ � *Nw and N<sup>θ</sup><sup>x</sup>* ½ � are the shape function matrices relating the primary variables u0, w, x, in terms of nodal

The same as for x-stiffener, the element stiffness and mass matrices of the y-

*Ae*

with ρ<sup>k</sup> is density of kth layer.

*4.3.1.2 Formulation of y-Stiffener*

stiffener are defined as follows:

*Ae*

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

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

be described as [15–17]:

*lw* <sup>¼</sup> <sup>1</sup> 2

*<sup>m</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup> 2

**Figure 11.**

**211**

*Modeling of plate and stiffener element.*

*composite plate-stiffeners element*

*B kH*<sup>∗</sup> 1

> *B*<sup>2</sup> *kA*<sup>∗</sup> 1

*Mys* � � *<sup>e</sup>* ¼ ð

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 acting (**Figure 10**).

**Figure 10.** *Smart stiffened plate subjected to airflow. (a) Smart stiffened plate and coordinate system and (b) Lamina details.*

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

$$\left[\left[M\right]\_{\varepsilon}\{\ddot{u}\}\_{\varepsilon}+\left[C\_{A}\right]\_{\varepsilon}\{\dot{u}\}\_{\varepsilon}\right]+\left(\left[K\_{bb}^{\text{ln}}\right]\_{\varepsilon}+\left[K\_{bb}^{\text{nI}}\right]\_{\varepsilon}+\left[K\_{A}\right]\_{\varepsilon}\right)\{u\}\_{\varepsilon}=\left\{\dot{f}\right\}\_{\varepsilon}^{\text{m}},\tag{74}$$

where *K*ln *bb* � � *<sup>e</sup>* <sup>¼</sup> <sup>Ð</sup> *Ve B*ln *b* � �*<sup>T</sup>* ½ � *<sup>Q</sup> <sup>B</sup>*ln *b* � �*dV*, and *Knl bb* � � *<sup>e</sup>* <sup>¼</sup> <sup>Ð</sup> *Ve Bnl b* � �*<sup>T</sup>* ½ � *<sup>Q</sup> Bnl b* � �*dV* are the

element linear mechanical stiffness and nonlinear mechanical stiffness respectively, f g*<sup>f</sup> <sup>m</sup> <sup>e</sup>* is element external mechanical force vector.

### *4.3.1 Formulation of Stiffener:*

### *4.3.1.1 Formulation of x-Stiffener*

$$\begin{aligned} U\_{\infty}(\mathbf{x}, \mathbf{z}) &= u\_0(\mathbf{x}) + \mathbf{z}\theta\_{\infty}(\mathbf{x}), \\ W\_{\infty}(\mathbf{x}, \mathbf{z}) &= w\_{\infty}(\mathbf{x}). \end{aligned} \tag{75}$$

where x-axis is taken along the stiffener centerline and the z-axis is its upward normal. The plate and stiffener element shown in **Figure 11**.

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 written as:

$$\left.u\_p\right|\_{x=-t\_p/2} = \left.u\_{\infty}\right|\_{x=t\_{\infty}/2}, \left.\theta\_{\mathbf{x}\mathbf{p}}\right|\_{x=-t\_p/2} = \left.\theta\_{\mathbf{x}}\right|\_{x=t\_{\infty}/2}, \left.w\_{\mathbf{p}}\right|\_{x=-t\_p/2} = \left.w\_{\infty}\right|\_{x=t\_{\infty}/2},\tag{76}$$

where tp is the plate thickness and txs is the x-stiffener depth. The element stiffness and mass matrices are defined as follows [2, 15]:

$$[K\_{\infty}]\_{\epsilon} = \int\_{L\_{\epsilon}} [B\_{\infty}]^T [D\_{\infty}] [B\_{\infty}] d\mathfrak{x},\tag{77}$$

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

$$\mathbb{E}\left[\mathbf{M}\_{\text{xc}}\right]\_{\boldsymbol{\epsilon}} = \int\_{\mathbf{A}\_{\boldsymbol{\epsilon}}} \left[ P\left( [\mathbf{N}\_{\text{u}^{0}}]^{T}[\mathbf{N}\_{\text{u}^{0}}] + [\mathbf{N}\_{\text{w}}]^{T}[\mathbf{N}\_{\text{w}}] \right) + I\_{\mathcal{I}}\left( [\mathbf{N}\_{\theta\_{\text{x}}}]^{T}[\mathbf{N}\_{\theta\_{\text{x}}}] \right) \right] d\mathbf{A},\tag{78}$$

with [Bxs] is the strain-displacement relations matrix, [Dxs] is the stress-strain relations matrix and *le* is the element length, *Nu* ½ � <sup>0</sup> , ½ � *Nw and N<sup>θ</sup><sup>x</sup>* ½ � are the shape function matrices relating the primary variables u0, w, x, in terms of nodal

unknowns, Iy is the area moment of inertia related to the y-axis and *<sup>P</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> k*¼1 Ð *hk hk*�<sup>1</sup> *ρkdz*, with ρ<sup>k</sup> is density of kth layer.
