Active Vibration Control of Graphene Platelets Reinforced Porous Nanocomposite Piezoelectric Cantilever Beams

Abstract

The active vibration control of a composite piezoelectric cantilever beam is investigated in this paper. The intermediate layer of the beam is made of the graphene platelets reinforced porous nanocomposite, and the piezoelectric actuator and sensor adhere to the top and bottom surfaces of the intermediate layer. The symmetric distribution (PD-X and PD–O) and uniform distribution (PD-U) of porosity are taken into account. The graphene platelets used to enhance material properties consider three distribution forms, the symmetric pattern (GPL-X and PD–O) and uniform pattern (GPL-U). Computing the effective Young's modulus, Poisson's ratio and mass density by the Halpin–Tsai model and the rule of mixture for all distributions, respectively. Then, the motion equation of the composite piezoelectric cantilever beam is derived using Lagrange's principle by von Karman's nonlinear shear deformation theory. In order to achieve vibration reduction effect, the active vibration control is applied with a velocity feedback control algorithm. Initially, the computed outcomes are cross-referenced with extant literature to validate the accuracy of the employed solution methodologies. Subsequently, an in-depth analysis is undertaken to elucidate the impacts of diOOerse material parameters and feedback control gains on the active vibration control efficacy of the piezoelectric beam reinforced with Graphene Platelets within a porous nanocomposite framework.

Appendix 1

$$\left( {I_{1} ,I_{2} ,I_{3} } \right) = \mathop \smallint \limits_{{ - \frac{h}{2}}}^{{ + \frac{h}{2}}} \rho \left( z \right)\left( {1,z,z^{2} } \right)dz,$$

$$\left( {I_{{11}} ,I_{{22}} ,I_{{33}} } \right) = \mathop \smallint \limits_{{ - \frac{H}{2}}}^{{ - \frac{h}{2}}} \rho \left( p \right)\left( {1,z,z^{2} } \right)dz + \mathop \smallint \limits_{{\frac{h}{2}}}^{{\frac{H}{2}}} \rho \left( p \right)\left( {1,z,z^{2} } \right)dz,$$

$$\left( {A_{{11}} ,B_{{11}} ,D_{{11}} ,A_{{55}} } \right) = \mathop \smallint \limits_{{ - \frac{h}{2}}}^{{ + \frac{h}{2}}} \left( {\frac{E}{{1 - v^{2} }},z\frac{E}{{1 - v^{2} }},z^{2} \frac{E}{{1 - v^{2} }},\frac{E}{{2\left( {1 + v} \right)}}} \right)dz,$$

$$\left( {N_{p} ,M_{p} } \right) = \mathop \smallint \limits_{{\frac{h}{2}}}^{{\frac{H}{2}}} 2e_{{31}} \frac{1}{{h_{p} }}\left( {1,z} \right)dz,$$

$$\begin{array}{*{20}l} {\left( {E_{{11}} ,F_{{11}} ,G_{{11}} ,E_{{55}} } \right) = \mathop \smallint \limits_{{ - \frac{H}{2}}}^{{ - \frac{h}{2}}} \left( {\frac{{E_{p} }}{{1 - v_{p} ^{2} }},z\frac{{E_{p} }}{{1 - v_{p} ^{2} }},z^{2} \frac{{E_{p} }}{{1 - v_{p} ^{2} }},\frac{{E_{p} }}{{2\left( {1 + v_{p} } \right)}}} \right)dz} \hfill \\ {{\text{}} + \mathop \smallint \limits_{{\frac{h}{2}}}^{{\frac{H}{2}}} \left( {\frac{{E_{p} }}{{1 - v_{p} ^{2} }},z\frac{{E_{p} }}{{1 - v_{p} ^{2} }},z^{2} \frac{{E_{p} }}{{1 - v_{p} ^{2} }},\frac{{E_{p} }}{{2\left( {1 + v_{p} } \right)}}} \right)dz,} \hfill \\ \end{array}$$

$${\mathbf{K}}_{\mathbf{a}}^{1}=-\sum\limits_{i=1}^{n}{\int }_{0}^{L}\left({N}_{p}\frac{d{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}}^{\mathbf{T}}}{dx}\right)dx,{\mathbf{K}}_{\mathbf{a}}^{2}=-\sum_{i=1}^{n}{\int }_{0}^{L}\left(\frac{1}{2}{N}_{p}\frac{d{\mathbf{N}}_{\mathbf{m}}^{\mathbf{w}}}{dx}\frac{d{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{w}}}^{\mathbf{T}}}{dx}\mathbf{W}\right)dx,$$

$${\mathbf{K}}_{\mathbf{a}}^{3}=-\sum\limits_{i=1}^{n}{\int }_{0}^{L}\left({M}_{p}\frac{d{{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}}^{\mathbf{T}}}{dx}\right)dx$$

$$\begin{aligned}{\mathbf{M}}^{\mathbf{u}\mathbf{u}}&={\int }_{0}^{L}{I}_{1}{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}}^{\mathbf{T}}dx+\sum_{i=1}^{n}{\int }_{0}^{L}{I}_{11}{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}}^{\mathbf{T}}dx\boldsymbol{ }\boldsymbol{ }{\mathbf{M}}^{\mathbf{u}{\varvec{\upphi}}} \\ & ={\int }_{0}^{L}{I}_{2}{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}{{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}}^{\mathbf{T}}dx+\sum_{i=1}^{n}{\int }_{0}^{L}{I}_{22}{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}{{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}}^{\mathbf{T}}dx \\ & {\mathbf{M}}^{{\varvec{\upphi}}\mathbf{u}}={\int }_{0}^{L}{I}_{2}{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}}^{\mathbf{T}}dx+\sum_{i=1}^{n}{\int }_{0}^{L}{I}_{22}{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{u}}}^{\mathbf{T}}dx {\mathbf{M}}^{{\varvec{\upphi}}{\varvec{\upphi}}} \\ & ={\int }_{0}^{L}{I}_{3}{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}{{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}}^{\mathbf{T}}dx+\sum_{i=1}^{n}{\int }_{0}^{L}{I}_{33}{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}{{\mathbf{N}}_{\mathbf{m}}^{{\varvec{\upphi}}}}^{\mathbf{T}}dx \\ & {\mathbf{M}}^{\mathbf{w}\mathbf{w}}={\int }_{0}^{L}{I}_{1}{\mathbf{N}}_{\mathbf{m}}^{\mathbf{w}}{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{w}}}^{\mathbf{T}}dx+\sum_{i=1}^{n}{\int }_{0}^{L}{I}_{11}{\mathbf{N}}_{\mathbf{m}}^{\mathbf{w}}{{\mathbf{N}}_{\mathbf{m}}^{\mathbf{w}}}^{\mathbf{T}}dx {\mathbf{M}}^{\mathbf{u}\mathbf{w}}=0 {\mathbf{M}}^{\mathbf{w}\mathbf{u}} \\ & =0 {\mathbf{M}}^{\mathbf{w}{\varvec{\upphi}}}=0 {\mathbf{M}}^{{\varvec{\upphi}}\mathbf{w}}=0\end{aligned}$$