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Aeroelastic-electric flutter characteristics of functionally graded piezoelectric material plates in supersonic airflow

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Abstract

Excellent mechanical and electric properties enable the piezoelectric materials to be widely and dynamic aero-mechanical-electric coupling properties of the piezoelectric plate structures can significantly affect performance of aircrafts flying in high speed. This paper is focused on the flutter behaviors of the functionally graded piezoelectric material (FGPM) plate in supersonic airflow. The first-order shear deformation theory (FSDT) is employed to formulate the energy functional of the FGPM plate with general boundary conditions. The supersonic air flow is taken into account using the supersonic piston theory. The governing equations for the aero-mechanical-electric coupling system are deduced on basis of the Hamilton’s principle. To address the limitation of intricate boundary conditions to the admissible functions, a modified Fourier series is introduced to yield the unified solutions of the FGPM plate subject to supersonic airflow and with arbitrary boundary conditions. Flutter properties of series of FGPM plates are investigated to demonstrate robust ability of the proposed method to accurately yet consistency include the aero-elastic-electric coupling, inhomogeneous material and arbitrary boundary conditions. The influences of the boundary condition, material constituent, external voltage and yawed flow angle on the vibration, and flutter behaviors of the FGPM plate are examined. Adjusting corresponding parameters accordingly can significantly improve the stability of the plate structures subject to supersonic airflow, which provides physical insights into dynamic optimal design of the plate structures.

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Acknowledgements

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Nos. 52374158, 52204143 and 52234005).

Funding

This study was funded by the National Natural Science Foundation of China (52374158, 52204143, 52234005).

Author information

Authors and Affiliations

  1. Shandong University of Science and Technology, Qianwangang Road 579, Qingdao, 266590, China

    Jinpeng Su, Shoubo Jiang & Qiang Zhang

  2. Wuhan Second Ship Design and Research Institute, Wuhan, 43000, China

    Jianhui Wei

  3. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Dongchuan Road 800, Shanghai, 200240, China

    Jinpeng Su

Authors
  1. Jinpeng Su
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Contributions

Jinpeng Su proposed the conceptualization, derived the methodology, prepared riginal draft and acquired funding. Jianhui Wei was responsible for programing, formal analysis and data curation. Shoubo Jiang validated the proposed method, improved the figures and acquired funding. Qiang Zhang improved the table, checked and revised the programs and acquired funding. All authors revised and reviewed the manuscript.

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Correspondence to Shoubo Jiang.

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Appendices

Appendix A

The displacement fields of the FGPM plate are given by follows, where H denotes the trial functions and ql (l = 1, 2,…, 6) represents the unknown coefficient eigenvector.

$$\begin{gathered} u(x,y) = {\varvec{Hq}}_{1} \hfill \\ v(x,y) = {\varvec{Hq}}_{{2}} \hfill \\ w(x,y) = {\varvec{Hq}}_{{3}} \hfill \\ \theta_{x} (x,y) = {\varvec{Hq}}_{{4}} \hfill \\ \theta_{y} (x,y) = {\varvec{Hq}}_{{5}} \hfill \\ \phi_{0} (x,y) = {\varvec{Hq}}_{6} \hfill \\ \end{gathered}$$
(A1)

The trial functions of the FGPM plate are written as

$${\varvec{H}} = \left[ {\varphi_{{1}} (x)\psi_{{1}} (y), \ldots ,\varphi_{{\text{m}}} (x)\psi_{{\text{n}}} (y), \ldots ,\varphi_{{\text{M}}} (x)\psi_{{\text{N}}} (y)} \right]$$
(A2)

The unknown coefficient eigenvectors are given by

$${\varvec{q}}_{1} = [A_{{{11}}} , \ldots ,A_{{{\text{mn}}}} , \ldots ,A_{{{\text{MN}}}} ]$$
(A3)
$${\varvec{q}}_{2} = [B_{{{11}}} , \ldots ,B_{{{\text{mn}}}} , \ldots ,B_{{{\text{MN}}}} ]$$
(A4)
$${\varvec{q}}_{3} = [C_{{{11}}} , \ldots ,C_{{{\text{mn}}}} , \ldots ,C_{{{\text{MN}}}} ]$$
(A5)
$${\varvec{q}}_{4} = [D_{{{11}}} , \ldots ,D_{{{\text{mn}}}} , \ldots ,D_{{{\text{MN}}}} ]$$
(A6)
$${\varvec{q}}_{5} = [E_{{{11}}} , \ldots ,E_{{{\text{mn}}}} , \ldots ,E_{{{\text{MN}}}} ]$$
(A7)
$${\varvec{q}}_{{6}} = [F_{{{11}}} , \ldots ,F_{{{\text{mn}}}} , \ldots ,F_{{{\text{MN}}}} ]$$
(A8)

Appendix B

The detailed expressions in the stiffness, mass and damping matrices of the FGPM plate are written as

$${\varvec{K}} = \left[ {\begin{array}{*{20}c} {{\varvec{K}}_{{{\text{uu}}}} } & {{\varvec{K}}_{{{\text{uv}}}} } & {\varvec{0}} & {{\varvec{K}}_{{{\text{ux}}}} } & {{\varvec{K}}_{{{\text{uy}}}} } \\ {{\varvec{K}}_{{{\text{uv}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{vv}}}} } & {\varvec{0}} & {{\varvec{K}}_{{{\text{vx}}}} } & {{\varvec{K}}_{{{\text{vy}}}} } \\ {\varvec{0}} & {\varvec{0}} & {{\varvec{K}}_{{{\text{ww}}}} } & {{\varvec{K}}_{{{\text{wx}}}} } & {{\varvec{K}}_{{{\text{wy}}}} } \\ {{\varvec{K}}_{{{\text{ux}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{vx}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{wx}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{xx}}}} } & {{\varvec{K}}_{{{\text{xy}}}} } \\ {{\varvec{K}}_{{{\text{uy}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{vy}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{wy}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{xy}}}}^{{\text{T}}} } & {{\varvec{K}}_{{{\text{yy}}}} } \\ \end{array} } \right]$$
(B1)
$${\varvec{K}}_{{{\text{cou}}}} = \left[ \begin{gathered} {\varvec{K}}_{{{\text{ue}}}} \hfill \\ {\varvec{K}}_{{{\text{ve}}}} \hfill \\ {\varvec{K}}_{{{\text{we}}}} \hfill \\ {\varvec{K}}_{{{\text{xe}}}} \hfill \\ {\varvec{K}}_{{{\text{ye}}}} \hfill \\ \end{gathered} \right]$$
(B2)
$$\begin{gathered} {\varvec{K}}_{{{\text{uu}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {A_{11} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + A_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)} \right]} } dxdy \hfill \\ \quad \quad + \int_{0}^{{L_{1} }} {\left( {\left. {k_{{{\text{uy}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = 0}} + \left. {k_{{{\text{uy}}L_{2} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = L_{2} }} } \right)} dx + \int_{0}^{{L_{2} }} {\left( {\left. {k_{{{\text{ux}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = 0}} + \left. {k_{{{\text{uxL}}_{1} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = L_{1} }} } \right)} dy \hfill \\ \end{gathered}$$
(B3)
$${\varvec{K}}_{{{\text{uv}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {A_{12} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + A_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy$$
(B4)
$${\varvec{K}}_{{{\text{ux}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {B_{11} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + B_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)} \right]} } dxdy$$
(B5)
$${\varvec{K}}_{{{\text{uy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {B_{12} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + B_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy$$
(B6)
$$\begin{gathered} {\varvec{K}}_{{{\text{vv}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {A_{22} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + A_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy \hfill \\ \quad \quad + \int_{0}^{{L_{1} }} {\left( {\left. {k_{{{\text{vy}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = 0}} + \left. {k_{{{\text{vy}}L_{2} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = L_{2} }} } \right)} dx + \int_{0}^{{L_{2} }} {\left( {\left. {k_{{{\text{vx}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = 0}} + \left. {k_{{{\text{vx}}L_{1} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = L_{1} }} } \right)} dy \hfill \\ \end{gathered}$$
(B7)
$${\varvec{K}}_{{{\text{vx}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {B_{12} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + B_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)} \right]} } dxdy$$
(B8)
$${\varvec{K}}_{{{\text{vy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {B_{22} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + B_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy$$
(B9)
$$\begin{gathered} {\varvec{K}}_{{{\text{ww}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ \begin{gathered} K_{{\text{s}}} A_{44} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + K_{{\text{s}}} A_{55} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + K_{{\text{s}}} A_{45} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) \hfill \\ + K_{{\text{s}}} A_{45} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) \hfill \\ \end{gathered} \right]} } dxdy \hfill \\ + \int_{0}^{{L_{1} }} {\left( {\left. {k_{{{\text{wy}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = 0}} + \left. {k_{{{\text{wy}}L_{2} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = L_{2} }} } \right)} dx + \int_{0}^{{L_{2} }} {\left( {\left. {k_{{{\text{wx}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = 0}} + \left. {k_{{{\text{wx}}L_{1} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = L_{1} }} } \right)} dy \hfill \\ + \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {\frac{{\lambda D_{{\text{c}}} }}{{L_{1}^{3} }}\left( {{\varvec{H}}^{{\text{T}}} \frac{{\partial {\varvec{H}}}}{\partial x}{\text{cos}}\theta_{{{\text{air}}}} + {\varvec{H}}^{{\text{T}}} \frac{{\partial {\varvec{H}}}}{\partial y}{\text{sin}}\theta_{{{\text{air}}}} } \right)} \right]} } dxdy \hfill \\ \end{gathered}$$
(B10)
$${\varvec{K}}_{{{\text{wx}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {K_{{\text{s}}} A_{55} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} ({\varvec{H}}) + K_{{\text{s}}} A_{45} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} ({\varvec{H}})} \right]} } dxdy$$
(B11)
$${\varvec{K}}_{{{\text{wy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {K_{{\text{s}}} A_{44} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} ({\varvec{H}}) + K_{{\text{s}}} A_{45} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} ({\varvec{H}})} \right]} } dxdy$$
(B12)
$$\begin{gathered} {\varvec{K}}_{{{\text{xx}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {K_{{\text{s}}} A_{55} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}}) + D_{11} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + D_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)} \right]} } dxdy \hfill \\ \quad \quad + \int_{0}^{{L_{1} }} {\left( {\left. {k_{{{\text{xy}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = 0}} + \left. {k_{{{\text{xy}}L_{2} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = L_{2} }} } \right)} dx + \int_{0}^{{L_{2} }} {\left( {\left. {k_{{{\text{xx}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = 0}} + \left. {k_{{{\text{xx}}L_{1} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = L_{1} }} } \right)} dy \hfill \\ \end{gathered}$$
(B13)
$${\varvec{K}}_{{{\text{xy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {K_{s} A_{45} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}}) + D_{12} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + D_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy$$
(B14)
$$\begin{gathered} {\varvec{K}}_{{{\text{yy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {K_{s} A_{44} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}}) + D_{22} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + D_{66} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy \hfill \\ \quad \quad + \int_{0}^{{L_{1} }} {\left( {\left. {k_{yy0} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{y = 0} + \left. {k_{{yyL_{2} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{y = L_{2} }} } \right)} dx + \int_{0}^{{L_{2} }} {\left( {\left. {k_{yx0} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{x = 0} + \left. {k_{{yxL_{1} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{x = L_{1} }} } \right)} dy \hfill \\ \end{gathered}$$
(B15)
$${\varvec{K}}_{{{\text{ue}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {E_{31} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} ({\varvec{H}})} \right]} } dxdy$$
(B16)
$${\varvec{K}}_{{{\text{ve}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {E_{32} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} ({\varvec{H}})} \right]} } dxdy$$
(B17)
$${\varvec{K}}_{{{\text{we}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {E_{15} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + E_{24} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)} \right]} } dxdy$$
(B18)
$${\varvec{K}}_{{{\text{xe}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {E_{{31{\text{z}}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} ({\varvec{H}}) + E_{15} ({\varvec{H}})^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)} \right]} } dxdy$$
(B19)
$${\varvec{K}}_{{{\text{ye}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {E_{{32{\text{z}}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} ({\varvec{H}}) + E_{24} ({\varvec{H}})^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)} \right]} } dxdy$$
(B20)
$$\begin{gathered} {\varvec{K}}_{e} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\left[ {G_{11} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial x}} \right) + G_{22} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right)^{{\text{T}}} \left( {\frac{{\partial {\varvec{H}}}}{\partial y}} \right) + G_{33} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} \right]} } dxdy \hfill \\ + \int_{0}^{{L_{1} }} {\left( {\left. {k_{{\phi {\text{y}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = 0}} + \left. {k_{{\phi {\text{y}}L_{2} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{y}} = L_{2} }} } \right)} dx + \int_{0}^{{L_{2} }} {\left( {\left. {k_{{\phi {\text{x}}0}} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = 0}} + \left. {k_{{\phi {\text{x}}L_{1} }} {\varvec{H}}^{{\text{T}}} {\varvec{H}}} \right|_{{{\text{x}} = L_{1} }} } \right)} dy \hfill \\ \end{gathered}$$
(B21)
$${\varvec{M}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{{\text{uu}}}} } & {\varvec{0}} & {\varvec{0}} & {{\varvec{M}}_{{{\text{ux}}}} } & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{M}}_{{{\text{vv}}}} } & {\varvec{0}} & {\varvec{0}} & {{\varvec{M}}_{{{\text{vy}}}} } \\ {\varvec{0}} & {\varvec{0}} & {{\varvec{M}}_{{{\text{ww}}}} } & {\varvec{0}} & {\varvec{0}} \\ {{\varvec{M}}_{{{\text{ux}}}}^{{\text{T}}} } & {\varvec{0}} & {\varvec{0}} & {{\varvec{M}}_{{{\text{xx}}}} } & {\varvec{0}} \\ {\varvec{0}} & {{\varvec{M}}_{{{\text{vy}}}}^{{\text{T}}} } & {\varvec{0}} & {\varvec{0}} & {{\varvec{M}}_{{{\text{yy}}}} } \\ \end{array} } \right]$$
(B22)
$${\varvec{M}}_{{{\text{uu}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{0} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B23)
$${\varvec{M}}_{{{\text{ux}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{1} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B24)
$${\varvec{M}}_{{{\text{vv}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{0} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B25)
$${\varvec{M}}_{{{\text{vy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{1} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B26)
$${\varvec{M}}_{{{\text{ww}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{0} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B27)
$${\varvec{M}}_{{{\text{xx}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{2} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B28)
$${\varvec{M}}_{{{\text{yy}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {I_{2} ({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B29)
$${\varvec{C}} = \left[ {\begin{array}{*{20}c} {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {{\varvec{C}}_{{{\text{ww}}}} } & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} & {\varvec{0}} \\ \end{array} } \right]$$
(B30)
$${\varvec{C}}_{{{\text{ww}}}} = \int_{0}^{{L_{1} }} {\int_{0}^{{L_{2} }} {\frac{{\sqrt {\mu \lambda \rho^{{\text{u}}} hD_{{\text{c}}} /M_{\infty } } }}{{L_{1}^{2} }}({\varvec{H}})^{{\text{T}}} ({\varvec{H}})} } dxdy$$
(B31)

where

$$\left( {A_{{{\text{ij}}}} ,{\kern 1pt} {\kern 1pt} B_{{{\text{ij}}}} ,{\kern 1pt} {\kern 1pt} D_{{{\text{ij}}}} } \right) = \int_{ - h/2}^{h/2} {\overline{C}_{{{\text{ij}}}} \left( {1,z,z^{2} } \right){\kern 1pt} dz}$$
$$\left( {E_{31} ,{\kern 1pt} E_{{31{\text{z}}}} } \right) = \int_{ - h/2}^{h/2} {\left( {\frac{dg}{{dz}},z\frac{dg}{{dz}}} \right){\kern 1pt} e_{31} dz}$$
$$\left( {E_{{3{2}}} ,{\kern 1pt} E_{{3{2}z}} } \right) = \int_{ - h/2}^{h/2} {\left( {\frac{dg}{{dz}},z\frac{dg}{{dz}}} \right){\kern 1pt} e_{{3{2}}} dz}$$
(B32)
$$\left( {E_{{{24}}} ,{\kern 1pt} E_{{{15}}} } \right) = \frac{{{1 + }K_{s} }}{{2}}\int_{ - h/2}^{h/2} {(e_{24} ,e_{15} ){\kern 1pt} gdz}$$
$$\left( {G_{{{11}}} ,{\kern 1pt} G_{22} ,{\kern 1pt} G_{33} {\kern 1pt} } \right) = \int_{ - h/2}^{h/2} {\left( {gg,gg,{\kern 1pt} {\kern 1pt} \frac{dg}{{dz}}\frac{dg}{{dz}}} \right){\kern 1pt} \overline{\kappa }_{{{\text{ii}}}} dz}$$

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Su, J., Wei, J., Jiang, S. et al. Aeroelastic-electric flutter characteristics of functionally graded piezoelectric material plates in supersonic airflow. Arch Appl Mech (2024). https://doi.org/10.1007/s00419-024-02603-8

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