Smart control of nonlinear vibrations of doubly curved functionally graded laminated composite shells under a thermal environment using 1–3 piezoelectric composites

Abstract

This paper addresses the active control of geometrically nonlinear vibrations of doubly curved functionally graded (FG) laminated composite shells integrated with a patch of active constrained layer damping (ACLD) treatment under the thermal environment. Vertically/obliquely reinforced 1-3 piezoelectric composite (PZC) and active fiber composite (AFC) are used as the materials of the constraining layer of the ACLD treatment. Each layer of the substrate FG laminated composite shell is made of fiber-reinforced composite material in which the fibers are longitudinally aligned in the plane parallel to the top or bottom surface of the layer and the layer is assumed to be graded in the thickness direction by way of varying the fiber orientation angle across its thickness according to a power law. The novelty of the present work is that, unlike the traditional laminated composite shells, the FG laminated composite shells are constructed in such a way that the continuous variation of material properties and stresses across the thickness of the shell is achieved. The Golla-Hughes-McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. Based on the first-order shear deformation theory (FSDT), a finite element (FE) model has been developed to model the open-loop and closed-loop nonlinear dynamics of the overall FG laminated composite shell under a thermal environment. Both symmetric and asymmetric FG laminated composite doubly curved shells are considered for presenting the numerical results. The analysis suggests that the ACLD patch significantly improves the damping characteristics of the doubly curved FG laminated composite shells for suppressing their geometrically nonlinear transient vibrations. It is found that the performance of the ACLD patch with its constraining layer being made of the AFC material is significantly higher than that of the ACLD patch with vertically/obliquely reinforced 1-3 PZC constraining layer. The effects of variation of piezoelectric fiber orientation in both the obliquely reinforced 1-3 PZC and the AFC constraining layers on the control authority of the ACLD patch have also been investigated.

Appendix

The various matrices $\left[ {{\mathbf{Z}}_{1} } \right]$, $\left[ {{\mathbf{Z}}_{2} } \right]$, $\left[ {{\mathbf{Z}}_{3} } \right]$, $\left[ {{\mathbf{Z}}_{4} } \right]$, $\left[ {{\mathbf{Z}}_{5} } \right]$ and $\left[ {{\mathbf{Z}}_{6} } \right]$ appearing in Eq. (6) are given by

$$ \begin{aligned} \left[ {{\mathbf{Z}}_{1} } \right] & = \left[ {\begin{array}{*{20}c} {\left[ {{\bar{\mathbf{Z}}}_{1} } \right]} & {{\tilde{\mathbf{O}}}} & {{\tilde{\mathbf{O}}}} \\ \end{array} } \right],\left[ {{\mathbf{Z}}_{2} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\mathbf{h}}}{2}\left[ {\mathbf{I}} \right]} & {\left[ {{\bar{\mathbf{Z}}}_{2} } \right]} & {{\tilde{\mathbf{O}}}} \\ \end{array} } \right],\left[ {{\mathbf{Z}}_{3} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{\mathbf{h}}}{2}\left[ {\mathbf{I}} \right]} & {{\mathbf{h}}_{{\mathbf{v}}} \left[ {\mathbf{I}} \right]} & {\left[ {{\bar{\mathbf{Z}}}_{3} } \right]} \\ \end{array} } \right], \\ \left[ {{\mathbf{Z}}_{4} } \right] & = \left[ {\begin{array}{*{20}c} {\left[ {{\bar{\mathbf{Z}}}_{4} } \right]} & {{\bar{\mathbf{O}}}} & {{\bar{\mathbf{O}}}} \\ \end{array} {\mathbf{z}}\begin{array}{*{20}c} { {\bar{\mathbf{I}}}} & {{\bar{\mathbf{O}}}} & {{\bar{\mathbf{O}}}} \\ \end{array} } \right],\left[ {{\mathbf{Z}}_{5} } \right] = \left[ { - \begin{array}{*{20}c} {({\mathbf{h}}/2){\mathbf{I}}_{1} } & {\left[ {{\bar{\mathbf{Z}}}_{5} } \right]} & {{\bar{\mathbf{O}}}} \\ \end{array} \frac{{\mathbf{h}}}{2}\begin{array}{*{20}c} { {\bar{\mathbf{I}}}} & {\left( {{\mathbf{z}} - \frac{{\mathbf{h}}}{2}} \right) {\bar{\mathbf{I}}}} & {{\bar{\mathbf{O}}}} \\ \end{array} } \right], \\ \left[ {{\mathbf{Z}}_{6} } \right] & = \left[ {\begin{array}{*{20}c} { - ({\mathbf{h}}/2){\mathbf{I}}_{1} } & { - {\mathbf{h}}_{{\mathbf{v}}} {\mathbf{I}}_{1} } & {\left[ {{\bar{\mathbf{Z}}}_{6} } \right]} \\ \end{array} \frac{{\mathbf{h}}}{2}\begin{array}{*{20}c} { {\bar{\mathbf{I}}}} & {{\mathbf{h}}_{{\mathbf{v}}} {\bar{\mathbf{I}}}} & {\left( {{\mathbf{z}} - {\mathbf{h}}_{{{\mathbf{N}} + 2}} } \right) {\bar{\mathbf{I}}}} \\ \end{array} } \right], \\ \end{aligned} $$

where $\left[ {{\bar{\mathbf{Z}}}_{1} } \right] = \left[ {\begin{array}{*{20}c} {{\mathbf{z\tilde{I}}}} & {{\mathbf{z}}\left[ {{\mathbf{I}}_{2} } \right]} \\ {{\mathbf{O}}_{1} } & 1 \\ \end{array} } \right],\left[ {{\bar{\mathbf{Z}}}_{2} } \right] = \left[ {\begin{array}{*{20}c} {\left( {{\mathbf{z}} - \frac{{\mathbf{h}}}{2}} \right){\tilde{\mathbf{I}}}} & {\left( {{\mathbf{z}} - \frac{{\mathbf{h}}}{2}} \right)\left[ {{\mathbf{I}}_{2} } \right]} \\ {{\mathbf{O}}_{1} } & 1 \\ \end{array} } \right],$,,

$$ \begin{aligned} \left[ {{\bar{\mathbf{Z}}}_{3} } \right] & = \left[ {\begin{array}{*{20}c} {({\mathbf{z}} - {\mathbf{h}}_{{{\mathbf{N}} + 2}} ){\tilde{\mathbf{I}}}} & {({\mathbf{z}} - {\mathbf{h}}_{{{\mathbf{N}} + 2}} )\left[ {{\mathbf{I}}_{2} } \right]} \\ {{\mathbf{O}}_{1} } & 1 \\ \end{array} } \right],\left[ {{\bar{\mathbf{Z}}}_{4} } \right] = \left[ {\begin{array}{*{20}c} {1 - ({\mathbf{z}}/{\mathbf{R}}_{1} )} & 0 \\ 0 & {1 - ({\mathbf{z}}/{\mathbf{R}}_{2} )} \\ \end{array} } \right], \\ \left[ {{\bar{\mathbf{Z}}}_{5} } \right] & = \left[ {\begin{array}{*{20}c} {1 - \left( {{\mathbf{z}} - \frac{{\mathbf{h}}}{2}} \right)/{\mathbf{R}}_{1} } & 0 \\ 0 & {1 - \left( {{\mathbf{z}} - \frac{{\mathbf{h}}}{2}} \right)/{\mathbf{R}}_{2} } \\ \end{array} } \right],\left[ {{\bar{\mathbf{Z}}}_{6} } \right] = \left[ {\begin{array}{*{20}c} {1 - ({\mathbf{z}} - {\mathbf{h}}_{{{\mathbf{N}} + 2}} )/{\mathbf{R}}_{1} } & 0 \\ 0 & {1 - ({\mathbf{z}} - {\mathbf{h}}_{{{\mathbf{N}} + 2}} )/{\mathbf{R}}_{2} } \\ \end{array} } \right], \\ \left[ {\mathbf{I}} \right] & = \left[ {\begin{array}{*{20}c} {{\tilde{\mathbf{I}}}} & {\left[ {{\mathbf{I}}_{2} } \right]} \\ {{\mathbf{O}}_{1} } & 0 \\ \end{array} } \right],{\mathbf{I}}_{1} = \left[ {\begin{array}{*{20}c} {1/{\mathbf{R}}_{1} } & 0 \\ 0 & {1/{\mathbf{R}}_{2} } \\ \end{array} } \right] {\text{and}}\;\left[ {{\mathbf{I}}_{2} } \right] = \left[ {\begin{array}{*{20}c} {1/{\mathbf{R}}_{1} } & {1/{\mathbf{R}}_{2} } & 0 \\ \end{array} } \right]^{{\mathbf{T}}} \\ \end{aligned} $$

${\tilde{\mathbf{O}}}$ ${\bar{\mathbf{O}}}$ and ${\mathbf{O}}_{1}$ being 4 × 4, 2 × 2 and 1 × 3 null matrices respectively and ${\tilde{\mathbf{I}}}$ and ${\bar{\mathbf{I}}}$ are 3 × 3 and 2 × 2 identity matrices respectively. The various submatrices appearing in Eq. (23) are explained as follows:

$$ \begin{aligned} B_{tbi} & = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial x}} & 0 & {\frac{1}{{{\mathbf{R}}_{1} }}} \\ 0 & {\frac{{\partial n_{i} }}{\partial y}} & {\frac{1}{{{\mathbf{R}}_{2} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial y}} & {\frac{{\partial n_{i} }}{\partial x}} & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right], B_{rbi} = \left[ {\begin{array}{*{20}c} {\bar{B}_{rbi} } & {\bar{O}_{1} } & {\bar{O}_{1} } \\ {\bar{O}_{1} } & {\bar{B}_{rbi} } & {\bar{O}_{1} } \\ {\bar{O}_{1} } & {\bar{O}_{1} } & {\bar{B}_{rbi} } \\ \end{array} } \right],\bar{B}_{rbi} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial x}} & 0 & 0 \\ 0 & {\frac{{\partial n_{i} }}{\partial y}} & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial n_{i} }}{\partial y}} & {\frac{{\partial n_{i} }}{\partial x}} & 0 \\ 0 & 0 & 1 \\ \end{array} } \\ \end{array} } \right] , \\ B_{tsi} & = \left[ {\begin{array}{*{20}c} 0 & { - \frac{1}{{{\mathbf{R}}_{1} }}} & {\frac{{\partial n_{i} }}{\partial x}} \\ 0 & { - \frac{1}{{{\mathbf{R}}_{2} }}} & {\frac{{\partial n_{i} }}{\partial y}} \\ \end{array} } \right], B_{rsi} = \left[ {\begin{array}{*{20}c} {\bar{I}_{1} } \\ {\bar{B}_{rsi} } \\ \end{array} } \right],\bar{I}_{1} = \left[ {\begin{array}{*{20}c} {\hat{I}_{1} } & {\tilde{O}_{1} } & {\tilde{O}_{1} } \\ {\tilde{O}_{1} } & {\hat{I}_{1} } & {\tilde{O}_{1} } \\ {\tilde{O}_{1} } & {\tilde{O}_{1} } & {\hat{I}_{1} } \\ \end{array} } \right],\hat{I}_{1} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} } \right] \\ \bar{B}_{rsi} & = \left[ {\begin{array}{*{20}c} {\tilde{B}_{rsi} } & {\tilde{O}_{1} } & {\tilde{O}_{1} } \\ {\tilde{O}_{1} } & {\tilde{B}_{rsi} } & {\tilde{O}_{1} } \\ {\tilde{O}_{1} } & {\tilde{O}_{1} } & {\tilde{B}_{rsi} } \\ \end{array} } \right] , \tilde{B}_{rsi} = \left[ {\begin{array}{*{20}c} 0 & 0 & {\frac{{\partial n_{i} }}{\partial x}} \\ 0 & 0 & {\frac{{\partial n_{i} }}{\partial y}} \\ \end{array} } \right] B_{2i} = \tilde{B}_{rsi} \\ \end{aligned} $$

where $\bar{O}_{1}$ and $\tilde{O}_{1}$ are (4 × 3) and (2 × 3) null matrices, respectively. The various matrices appearing in the elemental stiffness matrices of Eq. (25) are given as

$$ \begin{aligned} &\left[{D_{tb}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{\bar{C}_{b}^{k}} \right]dz + \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{\bar{C}_{b}^{N + 2}} \right] dz, &\left[{D_{trb}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{\bar{C}_{b}^{k}} \right]\left[{Z_{1}} \right]dz + \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{\bar{C}_{b}^{N + 2}} \right]\left[{Z_{3}} \right] dz, \\ &\left[{D_{rrb}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{Z_{1}} \right]^{T} \left[{\bar{C}_{b}^{k}} \right]\left[{Z_{1}} \right]dz + \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{Z_{3}} \right]^{T} \left[{\bar{C}_{b}^{N + 2}} \right]\left[{Z_{3}} \right] dz, \\ &\left[{D_{tbt}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{\bar{C}_{b}^{k}} \right] \left\{{\alpha^{k}} \right\}\Updelta T dz , &\left[{D_{rbt}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{Z_{1}} \right] ^{T} \left[{\bar{C}_{b}^{k}} \right] \left\{{\alpha^{k}} \right\} \Updelta T dz \\ &\left[{D_{ts}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{\bar{C}_{s}^{k}} \right]dz + \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{\bar{C}_{s}^{N + 2}} \right] dz, &\left[{D_{trs}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{\bar{C}_{s}^{k}} \right]\left[{Z_{4}} \right]dz + \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{\bar{C}_{s}^{N + 2}} \right]\left[{Z_{6}} \right] dz, \\ &\left[{D_{rrs}} \right] = \mathop \sum \limits_{k = 1}^{N} \mathop \int \limits_{{h_{k}}}^{{h_{k + 1}}} \left[{Z_{4}} \right]^{T} \left[{\bar{C}_{s}^{k}} \right]\left[{Z_{4}} \right]dz + \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{Z_{6}} \right]^{T} \left[{\bar{C}_{s}^{N + 2}} \right]\left[{Z_{6}} \right] dz, \\ &\left[{D_{tbs}} \right]_{p} = \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{\bar{C}_{bs}^{N + 2}} \right] dz , &\left[{D_{trbs}} \right]_{p} = \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{\bar{C}_{bs}^{N + 2}} \right] \left[{Z_{6}} \right]dz , &\left[{D_{rtbs}} \right]_{p} = \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{Z_{3}} \right]^{T} \left[{\bar{C}_{bs}^{N + 2}} \right] dz \\ &\left[{D_{rrbs}} \right]_{p} = \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} \left[{Z_{3}} \right]^{T} \left[{\bar{C}_{bs}^{N + 2}} \right]\left[{Z_{6}} \right] dz \\ \left[{f_{tbp}} \right] = \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} - \left\{{\bar{e}_{b}^{N + 2}} \right\}/h_{p} dz , &\left[{f_{rbp}} \right] = \mathop \int \limits_{{h_{N + 2}}}^{{h_{N + 3}}} - \left[{Z_{3}} \right]^{T} \left\{{\bar{e}_{b}^{N + 2}} \right\}/h_{p} dz \\ \end{aligned} $$