Elastic foundation effects on dynamic characteristics of agglomerated hybrid laminated truncated conical nanocomposite panels and shells

Abstract

The study explores the free vibration characteristics of hybrid laminated nanocomposite truncated conical shells and panels incorporating functionally graded graphene platelet-reinforced (FG-GPLs) and functionally graded carbon nanotube-reinforced (FG-CNTs) materials, advancing understanding in hybrid material applications. The structures are assumed to be supported by a two-parameter elastic foundation, and the study also considers the influence of CNT agglomeration. The estimation of CNT agglomeration effect is performed using a two-parameter agglomeration model based on the Mori–Tanaka approach, considering the random orientation of carbon nanotubes. By employing a combination of various control parameters, it becomes possible to determine the optimal mode for setting the natural frequency of the system. To ensure accurate calculations for both thin and thick shells, a third-order shear deformation theory is employed. The governing equations and boundary conditions are formulated based on Hamilton's principle. Numerical solutions are obtained through the systematic differential quadrature method, wherein the Kronecker delta function is employed. By introducing subtle adjustments to the governing equations, this method effectively minimizes computational volume and complexity, proving particularly advantageous for addressing problems characterized by higher degrees of freedom. To estimate the effective mechanical properties of the CNT-reinforced nanocomposite layers, the rule of mixtures is employed. Meanwhile, the Halpin–Tsai micromechanical model is utilized for calculating the properties of the GPL-reinforced nanocomposite layers. The presented study establishes convergence and accuracy through evaluation, considering various parameters such as CNTs volume fraction, GPLs mass fraction, different distribution patterns, different geometries under various boundary conditions, vertex angle of the cone, agglomeration characteristics of CNTs, and different stiffness of the elastic foundation.

Appendices

Appendix

$$ \begin{aligned} & (\delta v):\left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66} } \right)\frac{\partial u}{{\partial \theta }} + \left( {\frac{1}{r\left( s \right)}H_{12} + \frac{1}{r\left( s \right)}H_{66} } \right)\frac{{\partial^{2} u}}{\partial s\,\partial \theta } \\ & \qquad + H_{66} \frac{{\partial^{2} v}}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66} \frac{\partial v}{{\partial s}} + \frac{1}{{r\left( s \right)^{2} }}H_{22} \frac{{\partial^{2} v}}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66} } \right)v + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44} } \right)\frac{\partial w}{{\partial \theta }} \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{f} } \right)\frac{{\partial \varphi^{s} }}{\partial \theta } + \left( {\frac{1}{r\left( s \right)}H_{12}^{f} + \frac{1}{r\left( s \right)}H_{66}^{f} } \right)\frac{{\partial^{2} \varphi^{s} }}{\partial s\,\partial \theta } \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{g} } \right)\frac{{\partial \psi^{s} }}{\partial \theta } + \left( {\frac{1}{r\left( s \right)}H_{12}^{g} + \frac{1}{r\left( s \right)}H_{66}^{g} } \right)\frac{{\partial^{2} \psi^{s} }}{\partial s\,\partial \theta } \\ & \qquad + H_{66}^{f} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{f} \frac{{\partial \varphi^{\theta } }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{f} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{f} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{f^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{f} } \right)\varphi^{\theta } + H_{66}^{g} \frac{{\partial^{2} \psi^{\theta } }}{{\partial s^{2} }} \\ & \qquad + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{g} \frac{{\partial \psi^{\theta } }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{g} \frac{{\partial^{2} \psi^{\theta } }}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{g} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{g^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{g} } \right)\psi^{\theta } \\ & \quad = {\text{I}}_{11} \frac{{\partial^{2} v}}{{\partial t^{2} }} + {\text{I}}_{f0} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial t^{2} }} + {\text{I}}_{g0} \frac{{\partial^{2} \psi^{\theta } }}{{\partial t^{2} }} \\ \end{aligned} $$

(A.1)

$$ \begin{aligned} & (\delta w): - \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12} \frac{\partial u}{{\partial s}} - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} u \\ & \quad - \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44} } \right)\frac{\partial v}{{\partial \theta }} + H_{55} \frac{{\partial^{2} w}}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{55} \frac{\partial w}{{\partial s}} \\ & \quad + \frac{1}{{r\left( s \right)^{2} }}H_{44} \frac{{\partial^{2} w}}{{\partial \theta^{2} }} - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22} w - \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{f} - H_{55}^{{f^{\prime}}} } \right)\frac{{\partial \varphi^{s} }}{\partial s} \\ & \quad - \left( {\frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{55}^{{f^{\prime}}} } \right)\varphi^{s} - \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{g} - H_{55}^{{g^{\prime}}} } \right)\frac{{\partial \psi^{s} }}{\partial s} \\ & \quad - \left( {\frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{55}^{{g^{\prime}}} } \right)\psi^{s} \\ & \quad - \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{f} - \frac{1}{r\left( s \right)}H_{44}^{{f^{\prime}}} } \right)\frac{{\partial \varphi^{\theta } }}{\partial \theta } \\ & \quad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{g} - \frac{1}{r\left( s \right)}H_{44}^{{g^{\prime}}} } \right)\frac{{\partial \psi^{\theta } }}{\partial \theta } \\ & \quad + K_{w} w - K_{G} \frac{{\partial^{2} w}}{{\partial s^{2} }} = {\text{I}}_{11} \frac{{\partial^{2} w}}{{\partial t^{2} }} \\ \end{aligned} $$

(A.2)

$$ \begin{aligned} & (\delta \varphi^{s} ):H_{11}^{f} \frac{{\partial^{2} u}}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{f} \frac{\partial u}{{\partial s}} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{f} \frac{{\partial^{2} u}}{{\partial \theta^{2} }} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{f} u \\ & \quad + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{f} } \right)\frac{\partial v}{{\partial \theta }} + \left( {\frac{1}{r\left( s \right)}H_{12}^{f} + \frac{1}{r\left( s \right)}H_{66}^{f} } \right)\frac{{\partial^{2} v}}{\partial s\,\partial \theta } \\ & \quad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{f} - H_{55}^{{f^{\prime}}} } \right)\frac{\partial w}{{\partial s}} - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} w + H_{11}^{ff} \frac{{\partial^{2} \varphi^{s} }}{{\partial s^{2} }} \\ & \quad + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{ff} \frac{{\partial \varphi^{s} }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{ff} \frac{{\partial^{2} \varphi^{s} }}{{\partial \theta^{2} }} + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{ff} - H_{55}^{{f^{\prime}f^{\prime}}} } \right)\varphi^{s} \\ & \quad + H_{11}^{fg} \frac{{\partial^{2} \psi^{s} }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{fg} \frac{{\partial \psi^{s} }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{fg} \frac{{\partial^{2} \psi^{s} }}{{\partial \theta^{2} }} \\ & \quad + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{fg} - H_{55}^{{f^{\prime}g^{\prime}}} } \right)\psi^{s} + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{ff} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{ff} } \right)\frac{{\partial \varphi^{\theta } }}{\partial \theta } \\ & \quad + \left( {\frac{1}{r\left( s \right)}H_{12}^{ff} + \frac{1}{r\left( s \right)}H_{66}^{ff} } \right)\frac{{\partial^{2} \varphi^{\theta } }}{\partial s\,\partial \theta } + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} } \right)\frac{{\partial \psi^{\theta } }}{\partial \theta } \\ & \quad + \left( {\frac{1}{r\left( s \right)}H_{12}^{fg} + \frac{1}{r\left( s \right)}H_{66}^{fg} } \right)\frac{{\partial^{2} \psi^{\theta } }}{\partial s\,\partial \theta } = {\text{I}}_{f0} \frac{{\partial^{2} v}}{{\partial t^{2} }} + {\text{I}}_{ff} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial t^{2} }} + {\text{I}}_{fg} \frac{{\partial^{2} \psi^{\theta } }}{{\partial t^{2} }} \\ \end{aligned} $$

(A.3)

$$ \begin{aligned} & (\delta \psi^{s} ):H_{11}^{g} \frac{{\partial^{2} u}}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{g} \frac{\partial u}{{\partial s}} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{g} \frac{{\partial^{2} u}}{{\partial \theta^{2} }} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{g} u \\ & \quad + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{g} } \right)\frac{\partial v}{{\partial \theta }} + \left( {\frac{1}{r\left( s \right)}H_{12}^{g} + \frac{1}{r\left( s \right)}H_{66}^{g} } \right)\frac{{\partial^{2} v}}{\partial s\,\partial \theta } \\ & \quad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{g} - H_{55}^{{g^{\prime}}} } \right)\frac{\partial w}{{\partial s}} - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} w + H_{11}^{fg} \frac{{\partial^{2} \varphi^{s} }}{{\partial s^{2} }} \\ & \quad + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{fg} \frac{{\partial \varphi^{s} }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{fg} \frac{{\partial^{2} \varphi^{s} }}{{\partial \theta^{2} }} + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{fg} - H_{55}^{{f^{\prime}g^{\prime}}} } \right)\varphi^{s} \\ & \quad + H_{11}^{gg} \frac{{\partial^{2} \psi^{s} }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{gg} \frac{{\partial \psi^{s} }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{gg} \frac{{\partial^{2} \psi^{s} }}{{\partial \theta^{2} }} \\ & \quad + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{gg} - H_{55}^{{g^{\prime}g^{\prime}}} } \right)\psi^{s} + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} } \right)\frac{{\partial \varphi^{\theta } }}{\partial \theta } \\ & \quad + \left( {\frac{1}{r\left( s \right)}H_{12}^{fg} + \frac{1}{r\left( s \right)}H_{66}^{fg} } \right)\frac{{\partial^{2} \varphi^{\theta } }}{\partial s\,\partial \theta } + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{gg} - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{gg} } \right)\frac{{\partial \psi^{\theta } }}{\partial \theta } \\ & \quad + \left( {\frac{1}{r\left( s \right)}H_{12}^{gg} + \frac{1}{r\left( s \right)}H_{66}^{gg} } \right)\frac{{\partial^{2} \psi^{\theta } }}{\partial s\,\partial \theta } = {\text{I}}_{g0} \frac{{\partial^{2} v}}{{\partial t^{2} }}{\text{ + I}}_{fg} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial t^{2} }} + {\text{I}}_{gg} \frac{{\partial^{2} \psi^{\theta } }}{{\partial t^{2} }} \\ \end{aligned} $$

(A.4)

$$ \begin{aligned} & (\delta \varphi^{\theta } ):\left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{f} } \right)\frac{\partial u}{{\partial \theta }} + \left( {\frac{1}{r\left( s \right)}H_{12}^{f} + \frac{1}{r\left( s \right)}H_{66}^{f} } \right)\frac{{\partial^{2} u}}{\partial s\,\partial \theta } \\ & \qquad + H_{66}^{f} \frac{{\partial^{2} v}}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{f} \frac{\partial v}{{\partial s}} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{f} \frac{{\partial^{2} v}}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{f} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{f^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{f} } \right)v \\ & \qquad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{f} - \frac{1}{r\left( s \right)}H_{44}^{{f^{\prime}}} } \right)\frac{\partial w}{{\partial \theta }} \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{ff} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{ff} } \right)\frac{{\partial \varphi^{s} }}{\partial \theta } + \left( {\frac{1}{r\left( s \right)}H_{12}^{ff} + \frac{1}{r\left( s \right)}H_{66}^{ff} } \right)\frac{{\partial^{2} \varphi^{s} }}{\partial s\,\partial \theta } \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} } \right)\frac{{\partial \psi^{s} }}{\partial \theta } + \left( {\frac{1}{r\left( s \right)}H_{12}^{fg} + \frac{1}{r\left( s \right)}H_{66}^{fg} } \right)\frac{{\partial^{2} \psi^{s} }}{\partial s\,\partial \theta } \\ & \qquad + H_{66}^{ff} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{ff} \frac{{\partial \varphi^{\theta } }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{ff} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{ff} + \frac{2\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{ff^{\prime}}} - H_{44}^{{f^{\prime}f^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{ff} } \right)\varphi^{\theta } \\ & \qquad + H_{66}^{fg} \frac{{\partial^{2} \psi^{\theta } }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{fg} \frac{{\partial \psi^{\theta } }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{fg} \frac{{\partial^{2} \psi^{\theta } }}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{fg} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{gf^{\prime}}} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{fg^{\prime}}} - H_{44}^{{f^{\prime}g^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{fg} } \right)\psi^{\theta } \\ & \quad = {\text{I}}_{f0} \frac{{\partial^{2} u}}{{\partial t^{2} }}{\text{ + I}}_{ff} \frac{{\partial^{2} \varphi^{s} }}{{\partial t^{2} }} + {\text{I}}_{fg} \frac{{\partial^{2} \psi^{s} }}{{\partial t^{2} }} \\ \end{aligned} $$

(A.5)

$$ \begin{aligned} & (\delta \psi^{\theta } ):\left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{g} } \right)\frac{\partial u}{{\partial \theta }} + \left( {\frac{1}{r\left( s \right)}H_{12}^{g} + \frac{1}{r\left( s \right)}H_{66}^{g} } \right)\frac{{\partial^{2} u}}{\partial s\,\partial \theta } \\ & \qquad + H_{66}^{g} \frac{{\partial^{2} v}}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{g} \frac{\partial v}{{\partial s}} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{g} \frac{{\partial^{2} v}}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{g} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{g^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{g} } \right)v \\ & \qquad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{g} - \frac{1}{r\left( s \right)}H_{44}^{{g^{\prime}}} } \right)\frac{\partial w}{{\partial \theta }} \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} } \right)\frac{{\partial \varphi^{s} }}{\partial \theta } + \left( {\frac{1}{r\left( s \right)}H_{12}^{fg} + \frac{1}{r\left( s \right)}H_{66}^{fg} } \right)\frac{{\partial^{2} \varphi^{s} }}{\partial s\,\partial \theta } \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{gg} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{gg} } \right)\frac{{\partial \psi^{s} }}{\partial \theta } + \left( {\frac{1}{r\left( s \right)}H_{12}^{gg} + \frac{1}{r\left( s \right)}H_{66}^{gg} } \right)\frac{{\partial^{2} \psi^{s} }}{\partial s\,\partial \theta } \\ & \qquad + H_{66}^{fg} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{fg} \frac{{\partial \varphi^{\theta } }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{fg} \frac{{\partial^{2} \varphi^{\theta } }}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{fg} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{gf^{\prime}}} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{fg^{\prime}}} - H_{44}^{{f^{\prime}g^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{fg} } \right)\varphi^{\theta } \\ & \qquad + H_{66}^{gg} \frac{{\partial^{2} \psi^{\theta } }}{{\partial s^{2} }} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{gg} \frac{{\partial \psi^{\theta } }}{\partial s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{gg} \frac{{\partial^{2} \psi^{\theta } }}{{\partial \theta^{2} }} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{gg} + \frac{2\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{gg^{\prime}}} - H_{44}^{{g^{\prime}g^{\prime}}} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{gg} } \right)\psi^{\theta } \\ & \quad = {\text{I}}_{g0} \frac{{\partial^{2} u}}{{\partial t^{2} }}{\text{ + I}}_{fg} \frac{{\partial^{2} \varphi^{s} }}{{\partial t^{2} }} + {\text{I}}_{gg} \frac{{\partial^{2} \psi^{s} }}{{\partial t^{2} }} \\ \end{aligned} $$

(A.6)

Appendix B

$$ \begin{aligned} & (\delta v):\sum\limits_{m = 1}^{{N_{\theta } }} {\sum\limits_{n = 1}^{{N_{s} }} {\left\{ {\left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{12} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right.} \right.} } \\ & \qquad + \left. {\frac{1}{r\left( s \right)}H_{66} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)u_{nm} + \left( {\frac{1}{{r\left( s \right)^{2} }}H_{22} } \right.\delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44} \delta_{{{\text{in}}}} \delta_{jm} \\ & \qquad - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66} \delta_{{{\text{in}}}} \delta_{jm} + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)v_{nm} \\ & \qquad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } \left. { + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right)w_{nm} } \right. + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \qquad + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{12}^{f} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } \left. { + \frac{1}{r\left( s \right)}H_{66}^{f} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } } \right)\varphi_{nm}^{s} \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + } \right.\frac{1}{r\left( s \right)}H_{12}^{g} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } \\ & \qquad + \left. {\frac{1}{r\left( s \right)}H_{66}^{g} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } } \right)\psi_{nm}^{s} + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{f} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{f^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \qquad - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{f} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{f} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} \left. { + H_{66}^{f} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)\varphi_{nm}^{\theta } \\ & \qquad + \left( {\frac{1}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right. - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{g} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} \\ & \qquad - \left. {\frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{g} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{g} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{g} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{\theta } \\ & \quad = {\text{I}}_{11} \frac{{\partial^{2} v_{ij} }}{{\partial t^{2} }}{\text{ + I}}_{f0} \frac{{\partial^{2} \varphi_{ij}^{\theta } }}{{\partial t^{2} }}{\text{ + I}}_{g0} \frac{{\partial^{2} \psi_{ij}^{\theta } }}{{\partial t^{2} }} \\ \end{aligned} $$

(B.1)

$$ \begin{aligned} & (\delta w):\sum\limits_{m = 1}^{{N_{\theta } }} {\sum\limits_{n = 1}^{{N_{s} }} {\left\{ {\left( { - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} \delta_{{{\text{in}}}} \delta_{jm} \left. { - \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} } \right)} \right.} \right.} } u_{nm} \\ & \quad + \left( { - \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } \left. { - \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right)} \right.v_{nm} + \left( {\frac{1}{{r\left( s \right)^{2} }}H_{44} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right. \\ & \quad - \left. {\frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{55} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{55} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} + K_{w} \delta_{{{\text{in}}}} \delta_{jm} - K_{G} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)w_{nm} \\ & \quad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{55}^{{f^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \delta_{jm} } \right. + H_{55}^{{f^{\prime}}} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} \\ & \quad - \left. {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{f} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{s} + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{55}^{{g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad + H_{55}^{{g^{\prime}}} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} \left. { - \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{g} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{s} + \left( { - \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad - \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } \left. { + \frac{1}{r\left( s \right)}H_{44}^{{f^{\prime}}} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right)\varphi_{nm}^{\theta } + \left( { - \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad - \left. {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } \left. { + \frac{1}{r\left( s \right)}H_{44}^{{g^{\prime}}} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right)\psi_{nm}^{\theta } } \right\} = {\text{I}}_{11} \frac{{\partial^{2} w_{ij} }}{{\partial t^{2} }} \\ \end{aligned} $$

(B.2)

$$ \begin{aligned} & (\delta \varphi^{s} ):\sum\limits_{m = 1}^{{N_{\theta } }} {\sum\limits_{n = 1}^{{N_{s} }} {\left\{ {\left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \delta_{jm} } \right.} \right.} } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{f} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{11}^{f} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} \\ & \quad + \left. {\frac{1}{{r\left( s \right)^{2} }}H_{66}^{f} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)u_{nm} + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad + \left. {\frac{1}{r\left( s \right)}H_{12}^{f} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{66}^{f} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } } \right)v_{nm} + \left( { - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad + \left. {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{f} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} - H_{55}^{{f^{\prime}}} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} } \right)w_{nm} + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{ff} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad - \left. {H_{55}^{{f^{\prime}f^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{ff} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{ff} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{11}^{ff} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{s} \\ & \quad + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \delta_{jm} - H_{55}^{{f^{\prime}g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} } \right. + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } \\ & \quad + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{fg} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{11}^{fg} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{s} + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{ff} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad - \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{ff} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{12}^{ff} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} + \frac{1}{r\left( s \right)}H_{66}^{ff} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{\theta } \\ & \quad + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. + \frac{1}{r\left( s \right)}H_{12}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} \\ & \quad + \left. {\left. {\frac{1}{r\left( s \right)}H_{66}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{\theta } } \right\} = {\text{I}}_{f0} \frac{{\partial^{2} v_{ij} }}{{\partial t^{2} }}{\text{ + I}}_{ff} \frac{{\partial^{2} \varphi_{ij}^{\theta } }}{{\partial t^{2} }} + {\text{I}}_{fg} \frac{{\partial^{2} \psi_{ij}^{\theta } }}{{\partial t^{2} }} \\ \end{aligned} $$

(B.3)

$$ \begin{aligned} & (\delta \psi^{s} ):\sum\limits_{m = 1}^{{N_{\theta } }} {\sum\limits_{n = 1}^{{N_{s} }} {\left\{ {\left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \delta_{jm} } \right.} \right.} } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{g} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{11}^{g} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} \\ & \quad + \left. {\frac{1}{{r\left( s \right)^{2} }}H_{66}^{g} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)u_{nm} + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad + \left. {\frac{1}{r\left( s \right)}H_{12}^{g} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{66}^{g} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } } \right)v_{nm} + \left( { - \frac{\cos \left( \alpha \right)\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad + \left. {\frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{12}^{g} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} - H_{55}^{{g^{\prime}}} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} } \right)w_{nm} + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad - \left. {H_{55}^{{f^{\prime}g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{fg} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{11}^{fg} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{s} \\ & \quad + \left( { - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{22}^{gg} \delta_{{{\text{in}}}} \delta_{jm} - H_{55}^{{g^{\prime}g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} } \right. + \frac{1}{{r\left( s \right)^{2} }}H_{66}^{gg} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } \\ & \quad + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{11}^{gg} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{11}^{gg} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{s} + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad - \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{12}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} + \frac{1}{r\left( s \right)}H_{66}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{\theta } \\ & \quad + \left( { - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{gg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } - \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{gg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{12}^{gg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right. \\ & \quad + \left. {\left. {\frac{1}{r\left( s \right)}H_{66}^{gg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{\theta } } \right\} = {\text{I}}_{fg} \frac{{\partial^{2} \varphi_{ij}^{\theta } }}{{\partial t^{2} }}{\text{ + I}}_{g0} \frac{{\partial^{2} v_{ij} }}{{\partial t^{2} }}{\text{ + I}}_{gg} \frac{{\partial^{2} \psi_{ij}^{\theta } }}{{\partial t^{2} }} \\ \end{aligned} $$

(B.4)

$$ \begin{aligned} & (\delta \varphi^{\theta } ):\sum\limits_{m = 1}^{{N_{\theta } }} {\sum\limits_{n = 1}^{{N_{s} }} {\left\{ {\left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right.} \right.} } \\ & \quad + \left. {\frac{1}{r\left( s \right)}H_{12}^{f} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{66}^{f} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } } \right)u_{nm} \\ & \quad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{f} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{f^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{f} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{f} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{f} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)v_{nm} \\ & \quad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{f} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right.\left. { - \frac{1}{r\left( s \right)}H_{44}^{{f^{\prime}}} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right)w_{nm} \\ & \quad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{ff} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{ff} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{12}^{ff} \tilde{A}_{jm}^{\theta } \tilde{A}_{in}^{s} } \right. \\ & \quad + \left. {\frac{1}{r\left( s \right)}H_{66}^{ff} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{s} + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \quad + \left. {\frac{1}{r\left( s \right)}H_{12}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} + \frac{1}{r\left( s \right)}H_{66}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{s} + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{ff} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad + \frac{2\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{ff^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - H_{44}^{{f^{\prime}f^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{ff} \delta_{{{\text{in}}}} \delta_{jm} \\ & \quad + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{ff} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{ff} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{ff} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)\varphi_{nm}^{\theta } \\ & \quad \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{fg} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{gf^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{fg^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \quad - H_{44}^{{f^{\prime}g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \delta_{jm} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } \\ & \quad + \left. {\left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{fg} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{fg} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{\theta } } \right\} = {\text{I}}_{f0} \frac{{\partial^{2} u_{ij} }}{{\partial t^{2} }}{\text{ + I}}_{ff} \frac{{\partial^{2} \varphi_{ij}^{s} }}{{\partial t^{2} }}{\text{ + I}}_{fg} \frac{{\partial^{2} \psi_{ij}^{s} }}{{\partial t^{2} }} \\ \end{aligned} $$

(B.5)

$$ \begin{aligned} & (\delta \psi^{\theta } ):\sum\limits_{m = 1}^{{N_{\theta } }} {\sum\limits_{n = 1}^{{N_{s} }} {\left\{ {\left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right.} \right.} } \\ & \qquad + \left. {\frac{1}{r\left( s \right)}H_{12}^{g} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } + \frac{1}{r\left( s \right)}H_{66}^{g} \tilde{A}_{{{\text{in}}}}^{s} \tilde{A}_{jm}^{\theta } } \right)u_{nm} \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{g} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{g} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \qquad + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{g} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{g} \delta_{jm} \tilde{B}_{in}^{s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)v_{nm} \\ & \qquad + \left( {\frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\cos \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{44}^{g} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } \left. { - \frac{1}{r\left( s \right)}H_{44}^{{g^{\prime}}} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right)} \right.w_{nm} \\ & \qquad + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. + \frac{1}{r\left( s \right)}H_{12}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} \\ & \qquad + \left. {\frac{1}{r\left( s \right)}H_{66}^{fg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\varphi_{nm}^{s} + \left( {\frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{22}^{gg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)^{2} }}H_{66}^{gg} \delta_{{{\text{in}}}} \tilde{A}_{jm}^{\theta } } \right. \\ & \qquad + \left. {\frac{1}{r\left( s \right)}H_{12}^{gg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} + \frac{1}{r\left( s \right)}H_{66}^{gg} \tilde{A}_{jm}^{\theta } \tilde{A}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{s} + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{fg} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \qquad \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{gf^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} + \frac{\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{fg^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - H_{44}^{{f^{\prime}g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - \frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{fg} \delta_{{{\text{in}}}} \delta_{jm} \\ & \qquad + \left. {\frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{fg} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{fg} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{fg} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } } \right)\varphi_{nm}^{\theta } \\ & \qquad + \left( { - \frac{{\cos^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{44}^{gg} \delta_{{{\text{in}}}} \delta_{jm} + \frac{2\cos \left( \alpha \right)}{{r\left( s \right)}}H_{44}^{{gg^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} - H_{44}^{{g^{\prime}g^{\prime}}} \delta_{{{\text{in}}}} \delta_{jm} } \right. \\ & \qquad - \left. {\left. {\frac{{\sin^{2} \left( \alpha \right)}}{{r\left( s \right)^{2} }}H_{66}^{gg} \delta_{{{\text{in}}}} \delta_{jm} + \frac{1}{{r\left( s \right)^{2} }}H_{22}^{gg} \delta_{{{\text{in}}}} \tilde{B}_{jm}^{\theta } + \frac{\sin \left( \alpha \right)}{{r\left( s \right)}}H_{66}^{gg} \delta_{jm} \tilde{A}_{{{\text{in}}}}^{s} + H_{66}^{gg} \delta_{jm} \tilde{B}_{{{\text{in}}}}^{s} } \right)\psi_{nm}^{\theta } } \right\} \\ & \quad = {\text{I}}_{fg} \frac{{\partial^{2} \varphi_{ij}^{s} }}{{\partial t^{2} }}{\text{ + I}}_{g0} \frac{{\partial^{2} u_{ij} }}{{\partial t^{2} }}{\text{ + I}}_{gg} \frac{{\partial^{2} \psi_{ij}^{s} }}{{\partial t^{2} }} \\ \end{aligned} $$

(B.6)