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Analysis and active control of nonlinear vibration of composite lattice sandwich plates

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Abstract

This paper is devoted to investigate the nonlinear vibration characteristics and active control of composite lattice sandwich plates using piezoelectric actuator and sensor. Three types of the sandwich plates with pyramidal, tetrahedral and Kagome cores are considered. In the structural modeling, the von Kármán large deflection theory is applied to establish the strain–displacement relations. The nonlinear equations of motion of the structures are derived by Hamilton’s principle with the assumed mode method. The nonlinear free and forced vibration responses of the lattice sandwich plates are calculated. The velocity feedback control (VFC) and H∞ control methods are applied to design the controller. The nonlinear vibration responses of the sandwich plates with pyramidal, tetrahedral and Kagome cores are compared. The influences of the ply angle of the laminated face sheets, the thicknesses of the lattice core and face sheets and the excitation amplitude on the nonlinear vibration behaviors of the sandwich plates are investigated. The correctness of the H∞ control algorithm is verified by comparing with the experiment results reported in the literature. The controlled nonlinear vibration response of the sandwich plate is computed and compared with that of the uncontrolled structural system. Numerical results indicate that the VFC and H∞ control methods can effectively suppress the large amplitude vibration of the composite lattice sandwich plates.

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Funding

This study was funded by the National Natural Science Foundation of China (Project-Nos. 11761131006, 11572007, 11802069) and the German Research Foundation (DFG, Project-No. ZH 15/30-1).

Author information

Authors and Affiliations

  1. College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, 150001, China

    Yuyang Chai, Fengming Li & Zhiguang Song

  2. Department of Civil Engineering, University of Siegen, 57068, Siegen, Germany

    Chuanzeng Zhang

Authors
  1. Yuyang Chai
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  2. Fengming Li
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  3. Zhiguang Song
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  4. Chuanzeng Zhang
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Corresponding author

Correspondence to Fengming Li.

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Appendix

Appendix

The matrices M and Kl in Eq. (21) can be written as follows

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{uu} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {{\mathbf{M}}_{vv} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{ww} } & {{\mathbf{M}}_{wx} } & {{\mathbf{M}}_{wy} } \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{wx}^{{\text{T}}} } & {{\mathbf{M}}_{xx} } & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{M}}_{wy}^{{\text{T}}} } & {\mathbf{0}} & {{\mathbf{M}}_{yy} } \\ \end{array} } \right], $$
(42)

where

$$ {\mathbf{M}}_{uu} = (\rho_{c} h_{c} + 2\rho_{f} h_{f} + 2\rho_{p} h_{p} )\int_{0}^{a} {\int_{0}^{b} {{\mathbf{\zeta \zeta }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } , $$
(43)
$$ {\mathbf{M}}_{vv} = (\rho_{c} h_{c} + 2\rho_{f} h_{f} + 2\rho_{p} h_{p} )\int_{0}^{a} {\int_{0}^{b} {{\mathbf{\varsigma \varsigma }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } , $$
(44)
$${\mathbf{M}}_{ww} = (\rho_{c} h_{c} + 2\rho_{f} h_{f} + 2\rho_{p} h_{p} )\int_{0}^{a} {\int_{0}^{b} {{\mathbf{\xi \xi }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } + \left( {\frac{1}{18}\rho_{p} h_{p} h_{c}^{2} - \frac{1}{3}\rho_{p} h_{c} h_{p}^{2} + 2} \right.\rho_{p} h_{p} h_{f}^{2} + 2\rho_{p} h_{f} h_{p}^{2} \left. { + \frac{1}{18}\rho_{f} h_{f} h_{c}^{2} - \frac{1}{3}\rho_{f} h_{c} h_{f}^{2} + \frac{1}{252}\rho_{c} h_{c}^{3} + \frac{2}{3}\rho_{f} h_{f}^{3} + \frac{2}{3}\rho_{p} h_{p}^{3} - \frac{2}{3}\rho_{p} h_{c} h_{f} h_{p} } \right)\int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial {{\varvec{\upxi}}}}}{\partial x}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x} + \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}} \right){\text{d}}x{\text{d}}y} } , $$
(45)
$$ {\mathbf{M}}_{xx} = \left( {\frac{2}{9}\rho_{p} h_{p} h_{c}^{2} + \frac{2}{9}\rho_{f} h_{f} h_{c}^{2} + \frac{17}{{315}}\rho_{c} h_{c}^{3} } \right)\int_{0}^{a} {\int_{0}^{b} {{\mathbf{\eta \eta }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } , $$
(46)
$$ {\mathbf{M}}_{yy} = \left( {\frac{2}{9}\rho_{p} h_{p} h_{c}^{2} + \frac{2}{9}\rho_{f} h_{f} h_{c}^{2} + \frac{17}{{315}}\rho_{c} h_{c}^{3} } \right)\int_{0}^{a} {\int_{0}^{b} {{\mathbf{\chi \chi }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } , $$
(47)
$$ {\mathbf{M}}_{wx} = \left. {\left( {\frac{2}{3}\rho_{p} h_{c} h_{f} h_{p} - \frac{4}{315}\rho_{c} h_{c}^{3} } \right. - \frac{1}{9}\rho_{f} h_{f} h_{c}^{2} + \frac{1}{3}\rho_{f} h_{c} h_{f}^{2} - \frac{1}{9}\rho_{p} h_{p} h_{c}^{2} + \frac{1}{3}\rho_{p} h_{c} h_{p}^{2} } \right)\int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{{\varvec{\upeta}}}^{{\text{T}}} {\text{d}}x{\text{d}}y} } , $$
(48)
$$ {\mathbf{M}}_{wy} = \left. {\left( {\frac{2}{3}\rho_{p} h_{c} h_{f} h_{p} - \frac{4}{315}\rho_{c} h_{c}^{3} } \right. - \frac{1}{9}\rho_{f} h_{f} h_{c}^{2} + \frac{1}{3}\rho_{f} h_{c} h_{f}^{2} - \frac{1}{9}\rho_{p} h_{p} h_{c}^{2} + \frac{1}{3}\rho_{p} h_{c} h_{p}^{2} } \right)\int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{{\varvec{\upchi}}}^{{\text{T}}} {\text{d}}x{\text{d}}y} } , $$
(49)

and

$$ {\mathbf{K}}_{l} = \left[ {\begin{array}{*{20}c} {{\mathbf{K}}_{uu} } & {{\mathbf{K}}_{uv} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {{\mathbf{K}}_{uv}^{{\text{T}}} } & {{\mathbf{K}}_{vv} } & {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{ww} } & {{\mathbf{K}}_{wx} } & {{\mathbf{K}}_{wy} } \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{wx}^{{\text{T}}} } & {{\mathbf{K}}_{xx} } & {{\mathbf{K}}_{xy} } \\ {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{K}}_{wy}^{{\text{T}}} } & {{\mathbf{K}}_{xy}^{{\text{T}}} } & {{\mathbf{K}}_{yy} } \\ \end{array} } \right], $$
(50)

where

$$ {\mathbf{K}}_{uu} = 2A_{16} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upzeta}}}^{{\text{T}}} }}{\partial y} + \frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upzeta}}}^{{\text{T}}} }}{\partial x}} \right){\text{d}}x{\text{d}}y} } + 2A_{66} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upzeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + 2A_{11} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upzeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} }+ 2P_{11} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upzeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + 2P_{66} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upzeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } , $$
(51)
$$ {\mathbf{K}}_{uv} = 2A_{16} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + 2A_{66} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + 2A_{12} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + 2A_{26} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} }+ 2P_{12} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + 2P_{66} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } , $$
(52)
$${\mathbf{K}}_{vv} = 2A_{26} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial {\mathbf{\varsigma }}}}{\partial y}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial x} + \frac{{\partial {\mathbf{\varsigma }}}}{\partial x}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial y}} \right){\text{d}}x{\text{d}}y} } + 2A_{22} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {\mathbf{\varsigma }}}}{\partial y}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + 2A_{66} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {\mathbf{\varsigma }}}}{\partial x}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + 2P_{11} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {\mathbf{\varsigma }}}}{\partial y}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + 2P_{66} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {\mathbf{\varsigma }}}}{\partial x}\frac{{\partial {\mathbf{\varsigma }}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } , $$
(53)
$$ {\mathbf{K}}_{ww} = \frac{8}{9}A_{12} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + \frac{16}{9}A_{16} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } - \frac{8}{3}B_{12} h_{c} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } - \frac{16}{3}B_{16} h_{c} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} }+ \frac{16}{9}A_{26} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}} \right){\text{d}}x{\text{d}}y} } - \frac{16}{3}B_{26} h_{c} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}} \right){\text{d}}x{\text{d}}y} }+ \frac{8}{15}G_{c} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial x}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{8}{15}G_{c} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{32}{9}A_{66} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } - \frac{32}{3}B_{66} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } + 2D_{12} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + 4D_{16} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + 4D_{26} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}} \right){\text{d}}x{\text{d}}y} } + \frac{8}{9}A_{11} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } - \frac{8}{3}B_{11} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } + \frac{8}{9}A_{22} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} }- \frac{8}{3}B_{22} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } + 2D_{22} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } + 8D_{66} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } + 2D_{11} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } - \frac{2}{3}P_{12} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{66} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } - \frac{1}{18}P_{12} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } - \frac{2}{3}P_{11} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } - \frac{2}{3}P_{11} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } - \frac{8}{3}P_{66} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } - \frac{1}{3}P_{12} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + 2P_{12} h_{p} h_{f}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + 2P_{12} h_{f} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{11} h_{p}^{3} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{11} h_{p}^{3} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} }+ \frac{8}{3}P_{66} h_{p}^{3} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } - \frac{1}{3}P_{11} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } + 2P_{11} h_{p} h_{f}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} }+ 2P_{11} h_{f} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } - \frac{1}{3}P_{11} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } + 2P_{11} h_{p} h_{f}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } + 2P_{11} h_{f} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } - \frac{4}{3}P_{66} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } + 8P_{66} h_{p} h_{f}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } + 8P_{66} h_{f} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{12} h_{p}^{3} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }} + \frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}} \right){\text{d}}x{\text{d}}y} } + \frac{1}{18}P_{11} h_{p} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial x^{2} }}{\text{d}}x{\text{d}}y} } + \frac{1}{18}P_{11} h_{p} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial^{2} {{\varvec{\upxi}}}^{{\text{T}}} }}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } , $$
(54)
$${\mathbf{K}}_{wx} = - \frac{4}{9}A_{11} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{4}{9}A_{12} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{8}{9}A_{16} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} }+ \frac{2}{3}B_{11} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{3}B_{12} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{4}{3}B_{16} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} }- \frac{4}{9}A_{16} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{4}{9}A_{26} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{8}{9}A_{66} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}B_{16} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}B_{26} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{4}{3}B_{66} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} }+ \frac{8}{15}G_{c} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{{\varvec{\upeta}}}^{{\text{T}}} {\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{11} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{12} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{4}{3}P_{66} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{2}{9}P_{66} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{1}{9}P_{11} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{1}{9}P_{12} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{1}{3}P_{11} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} }+ \frac{1}{3}P_{12} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{66} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } , $$
(55)
$${\mathbf{K}}_{wy} = - \frac{4}{9}A_{12} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{4}{9}A_{16} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{4}{9}A_{22} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} }- \frac{8}{9}A_{26} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}B_{12} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}B_{22} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{4}{3}B_{26} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{4}{9}A_{26} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{8}{9}A_{66} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} }+ \frac{2}{3}B_{16} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{3}B_{26} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{4}{3}B_{66} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} }+ \frac{8}{15}G_{c} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{{\varvec{\upchi}}}^{{\text{T}}} {\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{12} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{11} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{4}{3}P_{66} h_{c} h_{f} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{2}{9}P_{66} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } - \frac{1}{9}P_{12} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } - \frac{1}{9}P_{11} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{1}{3}P_{12} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} }+ \frac{1}{3}P_{11} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{3}P_{66} h_{c} h_{p}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{\partial x\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } , $$
(56)
$$ {\mathbf{K}}_{xx} = \frac{2}{9}A_{16} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y} + \frac{{\partial {{\varvec{\upeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}} \right){\text{d}}x{\text{d}}y} } + \frac{8}{15}G_{c} h_{c} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{\eta \eta }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{66} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{11} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{11} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{66} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upeta}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } , $$
(57)
$$ {\mathbf{K}}_{xy} = \frac{2}{9}A_{12} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{16} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{26} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{66} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{66} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{12} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } , $$
(58)
$$ {\mathbf{K}}_{yy} = \frac{2}{9}A_{26} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\left( {\frac{{\partial {{\varvec{\upchi}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y} + \frac{{\partial {{\varvec{\upchi}}}}}{\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}} \right){\text{d}}x{\text{d}}y} } + \frac{8}{15}G_{c} h_{c} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{\chi \chi }}^{{\text{T}}} {\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{66} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upchi}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } + \frac{2}{9}A_{22} h_{c}^{2} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upchi}}}}}{\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{11} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upchi}}}}}{\partial y}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial y}{\text{d}}x{\text{d}}y} } + \frac{2}{9}P_{66} h_{c}^{2} h_{p} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upchi}}}}}{\partial x}\frac{{\partial {{\varvec{\upchi}}}^{{\text{T}}} }}{\partial x}{\text{d}}x{\text{d}}y} } . $$
(59)

The nonlinear stiffness matrix Kn is complex, and it can be obtained by performing the variational operations on the nonlinear part of the potential energy Unon, which can be written as follows

$$U_{non} = 2A_{16} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + 2A_{26} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + A_{26} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y + A_{26} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y+ A_{26} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + A_{16} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{s}}{\text{d}}x{\text{d}}y+ A_{22} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + A_{16} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{\text{d}}x{\text{d}}y+ A_{16} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{\text{d}}x{\text{d}}y + A_{12} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{\text{d}}x{\text{d}}y+ A_{12} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + A_{11} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{\text{d}}x{\text{d}}y+ A_{66} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y + \frac{1}{2}A_{12} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y+ 2A_{66} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + 2A_{66} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y+ \frac{1}{4}A_{22} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y + \frac{1}{4}A_{11} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{s}}{\text{d}}x{\text{d}}y+ P_{12} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + P_{11} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{\text{d}}x{\text{d}}y + \frac{1}{2}P_{12} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y + P_{11} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y+ P_{66} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y + P_{12} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}{\text{d}}x{\text{d}}y + \frac{1}{4}P_{11} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial y}{\mathbf{s}}{\text{d}}x{\text{d}}y + \frac{1}{4}P_{11} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{s}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial x}\frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{s}}{\text{d}}x{\text{d}}y+ 2P_{66} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{g}}^{{\text{T}}} \frac{{\partial {{\varvec{\upzeta}}}}}{\partial y}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y + 2P_{66} h_{p} \int_{0}^{a} {\int_{0}^{b} {{\mathbf{r}}^{{\text{T}}} \frac{{\partial {\mathbf{\varsigma }}}}{\partial x}} } \frac{{\partial {{\varvec{\upxi}}}^{{\text{T}}} }}{\partial x}{\mathbf{ss}}^{{\text{T}}} \frac{{\partial {{\varvec{\upxi}}}}}{\partial y}{\text{d}}x{\text{d}}y. $$
(60)

The matrix Ka in Eq. (21) can be expressed as

$$ {\mathbf{K}}_{a} = [\begin{array}{*{20}c} {{\mathbf{K}}_{ua}^{{\text{T}}} } & {{\mathbf{K}}_{va}^{{\text{T}}} } & {{\mathbf{K}}_{wa}^{{\text{T}}} } & {{\mathbf{K}}_{xa}^{{\text{T}}} } & {{\mathbf{K}}_{ya}^{{\text{T}}} } \\ \end{array} ], $$
(61)

where

$$ {\mathbf{K}}_{ua} = e_{31} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upzeta}}}}}{\partial x}} } {\text{d}}x{\text{d}}y,{\mathbf{K}}_{va} = e_{32} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {\mathbf{\varsigma }}}}{\partial y}} } {\text{d}}x{\text{d}}y, $$
(62)
$$ {\mathbf{K}}_{ua} = e_{31} \left( {\frac{1}{2}h_{p} + h_{f} - \frac{1}{6}h_{c} } \right)\int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial x^{2} }}} } {\text{d}}x{\text{d}}y + e_{32} \left( {\frac{1}{2}h_{p} + h_{f} - \frac{1}{6}h_{c} } \right)\int_{0}^{a} {\int_{0}^{b} {\frac{{\partial^{2} {{\varvec{\upxi}}}}}{{\partial y^{2} }}} } {\text{d}}x{\text{d}}y + e_{31} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial x}\frac{{\partial {{\varvec{\upxi}}}^{{\mathbf{T}}} }}{\partial x}} } {\mathbf{s}}{\text{d}}x{\text{d}}y + e_{32} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upxi}}}}}{\partial y}\frac{{\partial {{\varvec{\upxi}}}^{{\mathbf{T}}} }}{\partial y}} } {\mathbf{s}}{\text{d}}x{\text{d}}y, $$
(63)
$$ {\mathbf{K}}_{xa} = \frac{1}{3}e_{31} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upeta}}}}}{\partial x}} } {\text{d}}x{\text{d}}y,{\mathbf{K}}_{ya} = \frac{1}{3}e_{32} h_{c} \int_{0}^{a} {\int_{0}^{b} {\frac{{\partial {{\varvec{\upchi}}}}}{\partial y}} } {\text{d}}x{\text{d}}y. $$
(64)

Finally, the force or load vector on the right-hand side of Eq. (21) can be written as

$$ {\mathbf{F}} = [\begin{array}{*{20}c} 0 & 0 & {{\mathbf{F}}_{w}^{{\text{T}}} } & 0 & 0 \\ \end{array} ], $$
(65)

where

$$ {\mathbf{F}}_{w} = \int_{0}^{a} {\int_{0}^{b} {{\varvec{\upxi}}} } {\text{d}}x{\text{d}}y. $$
(66)

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Chai, Y., Li, F., Song, Z. et al. Analysis and active control of nonlinear vibration of composite lattice sandwich plates. Nonlinear Dyn 102, 2179–2203 (2020). https://doi.org/10.1007/s11071-020-06059-7

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