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Active structural-acoustic control of laminated composite plates using vertically/obliquely reinforced 1–3 piezoelectric composite patch

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Abstract

This article deals with the active structural-acoustic control of thin laminated composite plates using vertically reinforced 1–3 piezoelectric fiber-reinforced composite (PFRC) material for constraining layer of active constrained layer damping (ACLD) treatment. A finite element model is developed for the laminated composite plates integrated with ACLD patches and coupled with acoustic cavity to describe the coupled structural-acoustic behavior of the plates enclosing the cavity. Both in-plane and out of plane actuation of the constraining layer of the ACLD treatment have been utilized for deriving the finite element model. The analysis revealed that the vertical actuation dominates over the in-plane actuation. The performance of PFRC layers of the patches has been investigated for active control of sound radiated from thin symmetric and antisymmetric cross-ply and antisymmetric angle-ply laminated composite plates into the acoustic cavity.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 721302, India

    M. C. Ray & A. Faye

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  1. M. C. Ray
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  2. A. Faye
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Appendix

Appendix

In Eqs. (6) and (7) matrices [Z 1 ], [Z 2 ], [Z 3 ], [Z 4 ], [Z 5 ], and [Z 6 ] are given by

$$ \begin{aligned} \left[ {{\mathbf{Z}}_{{\mathbf{1}}} } \right] & = \left[ {\begin{array}{*{20}c} {\left[ {\bar{{\mathbf{Z}}}_{{\mathbf{1}}} } \right]} & {\tilde{{\mathbf{o}}}} & {\tilde{{\mathbf{o}}}} \\ \end{array} } \right],\,\left[ {{\mathbf{Z}}_{{\mathbf{2}}} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{h/2}}} \right)\tilde{{\mathbf{I}}}} & {\left[ {\bar{{\mathbf{Z}}}_{{\mathbf{2}}} } \right]} & {\tilde{{\mathbf{o}}}} \\ \end{array} } \right],\left[ {{\mathbf{Z}}_{{\mathbf{3}}} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{h/2}}} \right)\tilde{{\mathbf{I}}}} & {{\mathbf{h}}_{\mathbf{v}} \tilde{{\mathbf{I}}}} & {\left[ {\bar{{\mathbf{Z}}}_{{\mathbf{3}}} } \right]} \\ \end{array} } \right], \\ \left[ {{\mathbf{Z}}_{{\mathbf{4}}} } \right] & = \left[ {\begin{array}{*{20}c} {\hat{{\mathbf{I}}}} & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {{\mathbf{z}}\hat{{\mathbf{I}}}} & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } \\ \end{array} } \right],\,\left[ {{\mathbf{Z}}_{{\mathbf{5}}} } \right] = \left[ {\begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {\hat{{\mathbf{I}}}} & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {\left( {{\mathbf{h/2}}} \right)\hat{{\mathbf{I}}}} & {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)\hat{{\mathbf{I}}}} & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } \\ \end{array} } \right], \\ \left[ {{\mathbf{Z}}_{{\mathbf{6}}} } \right] & = \left[ {\begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{o}}} } & {\hat{{\mathbf{I}}}} & {\left( {{\mathbf{h/2}}} \right)\hat{{\mathbf{I}}}} & {{\mathbf{h}}_{\mathbf{v}} \hat{{\mathbf{I}}}} & {\left( {{\mathbf{z}} - {\mathbf{h}}_{{\mathbf{3}}} /{\mathbf{2}}} \right)\hat{{\mathbf{I}}}} \\ \end{array} } \right] \\ \end{aligned} $$

in which

$$ \begin{gathered} \left[ {{\bar{\mathbf{Z}}}_{{\mathbf{1}}} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{z}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{z}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{z}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right] ,\,\left[ {{\bar{\mathbf{Z}}}_{{\mathbf{2}}} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right],\,\left[ {{\tilde{\mathbf{o}}}} \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{o} }}} & {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{o} }}} \\ {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{o} }}} & {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{o} }}} \\ \end{array} } \right] ,\hfill \\ \left[ {{\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{o} }}} \right] = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right] ,\,\left[ {{\bar{\mathbf{Z}}}_{{\mathbf{3}}} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{z}} - {\mathbf{h}}_{{\mathbf{3}}} {\mathbf{/2}}} \right)} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\left( {{\mathbf{z}} - {\mathbf{h}}_{{\mathbf{3}}} {\mathbf{/2}}} \right)} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\left( {{\mathbf{z}} - {\mathbf{h}}_{{\mathbf{3}}} {\mathbf{/2}}} \right)} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right],\,\left[ {{\bar{\mathbf{Z}}}_{{\mathbf{1}}} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{z}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{z}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{z}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right], \hfill \\ \left[ {{\bar{\mathbf{Z}}}_{{\mathbf{2}}} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\left( {{\mathbf{z}} - {\mathbf{h/2}}} \right)} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right],\,\left[ {{\tilde{\mathbf{I}}}} \right] = \left[ {\begin{array}{*{20}c} {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{1}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right] ,\,\left[ {{\hat{\mathbf{I}}}} \right] = \left[ {\begin{array}{*{20}c} {\mathbf{1}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{1}} \\ \end{array} } \right], \hfill \\ \end{gathered} $$

The various submatrices B tbi , B tsi , B rbi , and B rsi appearing in (20) are given by

$$\begin{gathered} {\mathbf{B}}_{\mathbf{tbi}} = \left[{\begin{array}{*{20}c} {\frac{{{\varvec{\partial}}{\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{x}}}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{y}}}} & {\mathbf{0}} \\ {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{y}}}} & {\frac{{{\varvec{\partial}}{\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{x}}}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ \end{array} } \right] ,\,{\mathbf{B}}_{\mathbf{tsi}} = \left[ {\begin{array}{*{20}c}{\mathbf{0}} & {\mathbf{0}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{x}}}} \\ {\mathbf{0}} & {\mathbf{0}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{y}}}} \\ \end{array} } \right] ,\,{\mathbf{B}}_{{\mathbf{rbi}}} = \left[ {\begin{array}{*{20}c} {\bar{{\mathbf{B}}}_{{\mathbf{rbi}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} } & {\bar{{\mathbf{B}}}_{{\mathbf{rbi}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} } & {\bar{{\mathbf{B}}}_{{\mathbf{rbi}}} } \\ \end{array} } \right], \hfill \\ \bar{{\mathbf{B}}}_{{\mathbf{rbi}}} = \left[ {\begin{array}{*{20}c} {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{x}}}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{y}}}} & {\mathbf{0}} \\ {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{y}}}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{x}}}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{n}}_{{\mathbf{i}}} } \\ \end{array} } \right],\,{\mathbf{B}}_{\mathbf{rsi}} = \left[ {\begin{array}{*{20}c} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{I}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{I}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{I}}} } \\ {\bar{{\mathbf{B}}}_{{\mathbf{rsi}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\bar{{\mathbf{B}}}_{\mathbf{rsi}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } \\ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} } & {\bar{{\mathbf{B}}}_{{\mathbf{rsi}}} } \\ \end{array} } \right],\,\left[ {\bar{{\mathbf{B}}}_{{\mathbf{rsi}}} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{x}}}} \\ {\mathbf{0}} & {\mathbf{0}} & {\frac{{{\varvec{\partial}} {\mathbf{n}}_{{\mathbf{i}}} }}{{\varvec{\partial}} {\mathbf{y}}}} \\ \end{array} } \right] {\text{ and }} \hfill \\ \left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{I}}} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{n}}_{{\mathbf{i}}} } & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{n}}_{{\mathbf{i}}} } & {\mathbf{0}} \\ \end{array} } \right] \hfill \\ \end{gathered} $$

where \( {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{{\mathbf{0}}} }} \) and \( {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbf{0}}} }} \) are the (4 × 3) and (2 × 3) null matrices, respectively.

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Ray, M.C., Faye, A. Active structural-acoustic control of laminated composite plates using vertically/obliquely reinforced 1–3 piezoelectric composite patch. Int J Mech Mater Des 5, 123–141 (2009). https://doi.org/10.1007/s10999-008-9089-8

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