Frequency characteristics of a GPL-reinforced composite microdisk coupled with a piezoelectric layer

Abstract

This is the first research on the frequency analysis of a graphene nanoplatelet composite (GPLRC) microdisk in the framework of a numerical-based generalized differential quadrature method. The stresses and strains are obtained using the higher-order shear deformable theory. Rule of mixture is employed to obtain varying mass density, thermal expansion, and Poisson’s ratio, while module of elasticity is computed by modified Halpin–Tsai model. Governing equations and boundary conditions of the GPLRC microdisk covered with piezoelectric layer are obtained by implementing Hamilton’s principle. Regarding perfect bonding between the piezoelectric layer and core, the compatibility conditions are derived. In addition, due to the existence of piezoelectric layer, Maxwell’s equation is derived. The results show that outer-to-inner ratio of radius ($R_{\text{o}} /R_{\text{i}}$), ratios of length scale and nonlocal to thickness (l/h and $\mu$/h), ratio of piezoelectric to core thickness (h_p/h), applied voltage, and GPL weight fraction (g_GPL) have significant influence on the frequency characteristics of the GPLRC microdisk. Another important consequence is that in addition to the nonlinear indirect effects of applied voltage on the natural frequency of the GPLRC microdisk covered with piezoelectric for each specific value of $R_{\text{o}} /R_{\text{i}}$, the impact of the $R_{\text{o}} /R_{\text{i}}$ on the natural frequency is indirect. A useful suggestion of this research is that, for designing the GPLRC circular microplate at the low value of the $R_{\text{o}} /R_{\text{i}}$ should be more attention to the g_GPL and $R_{\text{o}} /R_{\text{i}}$, simultaneously.

Appendix

The governing equations of the GPLRC microdisk are given as follows:

$$ \begin{aligned} & \delta u^{i}_{0}{:} \\ & \quad (1 - l^{2} \nabla^{2} )\left[ \begin{aligned} \frac{\partial }{\partial R}\left( \begin{aligned} & \left[ {A^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + B^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - D^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,\left[ {A^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + B^{i}_{12} \frac{{u_{1} }}{R} - D^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{31} \phi \\ \end{aligned} \right) \hfill \\ - \frac{1}{R}\left( \begin{aligned} & \left[ {A^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + B^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - D^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,Q^{i}_{22} \left[ {A^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + B^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - D^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] \\ \end{aligned} \right) \hfill \\ \end{aligned} \right] \\ & \quad = (1 - \mu^{2} \nabla^{2} )\left[ {I^{i}_{0} \frac{{\partial^{2} u^{i}_{0} }}{{\partial t^{2} }} + I^{i}_{1} \frac{{\partial^{2} u^{i}_{1} }}{{\partial t^{2} }} - I^{i}_{3} c_{1} \left( {\frac{{\partial^{2} u^{i}_{1} }}{{\partial t^{2} }} + \frac{{\partial^{3} w^{i}_{0} }}{{\partial t^{2} \partial R^{i} }}} \right)} \right];\;i = c,p \\ \end{aligned} $$

(P-1)

$$ \begin{aligned} & \delta w^{i}_{0}{:} \\ & \quad (1 - l^{2} \nabla^{2} )\left[ \begin{aligned} & c_{1} \frac{{\partial^{2} }}{{\partial R^{i2} }}\left( \begin{aligned} & \left[ {D^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,\left[ {D^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{12} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{33} \phi \\ \end{aligned} \right) \\ & \quad - \,c_{1} \frac{\partial }{{R^{i} \partial R^{i} }}\left( \begin{aligned} & \left[ {D^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + Q^{i}_{22} \left[ {D^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] \\ \end{aligned} \right) \\ & \quad + \,\frac{\partial }{{\partial R^{i} }}\left( {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{11} \partial \phi /\partial R} \right) \\ & \quad - \,3c_{1} \frac{\partial }{{\partial R^{i} }}\left( {(C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{12} \partial \phi /\partial R^{i} } \right) - N_{1}^{p} w^{i}_{{0,x^{2} }} \\ \end{aligned} \right] \\ & \quad = (1 - \mu^{2} \nabla^{2} )\left[ {c_{1} I^{i}_{3} \frac{{\partial^{3} u^{i}_{0} }}{{\partial R^{i} \partial t^{2} }} + c_{1} I^{i}_{4} \frac{{\partial^{3} u^{i}_{1} }}{{\partial R^{i} \partial t^{2} }} - I^{i}_{6} c_{1}^{2} \left( {\frac{{\partial^{3} u^{i}_{1} }}{{\partial R^{i} \partial t^{2} }} + \frac{{\partial^{4} w_{0}^{i} }}{{\partial t^{2} \partial R^{2} }}} \right) + \left( {I^{i}_{0} \frac{{\partial^{2} w^{i}_{0} }}{{\partial t^{2} }}} \right)} \right];\;i = c,p \\ \end{aligned} $$

(P-2)

$$ \begin{aligned} & \delta u^{i}_{1}{:} \\ & \quad (1 - l^{2} \nabla^{2} )\left[ \begin{aligned} & \frac{\partial }{{\partial R^{i} }}\left( \begin{aligned} & \left[ {B^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + C^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - E^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,\left[ {B^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + C^{i}_{12} \frac{{u^{i}_{1} }}{{R^{i} }} - E^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{32} \phi \\ \end{aligned} \right) \\ & \quad - \,\frac{{c_{1} \partial }}{{\partial R^{i} }}\left( \begin{aligned} & \left[ {D^{i}_{11} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{11} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{11} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,\left[ {D^{i}_{12} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{12} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] - X_{33} \phi \\ \end{aligned} \right) \\ & \quad - \,\frac{1}{{R^{i} }}\left( \begin{aligned} & \left[ {B^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + C^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - E^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,Q^{i}_{22} \left[ {B^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + C^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - E^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] \\ \end{aligned} \right) \\ & \quad + \,\frac{{c_{1} }}{{R^{i} }}\left( \begin{aligned} & \left[ {D^{i}_{12} \frac{{\partial u^{i}_{0} }}{{\partial R^{i} }} + E^{i}_{12} \frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} - G^{i}_{12} c_{1} \left( {\frac{{\partial u^{i}_{1} }}{{\partial R^{i} }} + \frac{{\partial^{2} w^{i}_{0} }}{{\partial R^{i2} }}} \right)} \right] \\ & \quad + \,Q^{i}_{22} \left[ {D^{i}_{22} \frac{{u^{i}_{0} }}{{R^{i} }} + E^{i}_{22} \frac{{u^{i}_{1} }}{{R^{i} }} - G^{i}_{22} c_{1} \left( {\frac{{u^{i}_{1} }}{{R^{i} }} + \frac{{\partial w^{i}_{0} }}{{R^{i} \partial R^{i} }}} \right)} \right] \\ \end{aligned} \right) \\ & \quad - \,\left[ {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{11} \partial \phi /\partial R^{i} } \right] \\ & \quad + \,3c_{1} \left( {S^{i}_{Rz} = (C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {u^{i}_{1} + \frac{{\partial w^{i}_{0} }}{{\partial R^{i} }}} \right) + X_{12} \partial \phi /\partial R^{i} } \right) \\ \end{aligned} \right]\, \\ & \quad = (1 - \mu^{2} \nabla^{2} )\left[ \begin{aligned} & I^{i}_{1} \frac{{\partial^{2} u^{i}_{0} }}{{\partial t^{2} }} + I^{i}_{2} \frac{{\partial^{2} u^{i}_{1} }}{{\partial t^{2} }} - I^{i}_{4} c_{1} \left( {\frac{{\partial^{2} u^{i}_{1} }}{{\partial t^{2} }} + \frac{{\partial^{3} w_{0}^{i} }}{{\partial t^{2} \partial R^{i} }}} \right) - c_{1} I^{i}_{3} \frac{{\partial^{2} u^{i}_{0} }}{{\partial t^{2} }} \\ & \quad - \,c_{1} I^{i}_{4} \frac{{\partial^{2} u^{i}_{1} }}{{\partial t^{2} }} + I^{i}_{6} c_{1}^{2} \left( {\frac{{\partial^{2} u^{i}_{1} }}{{\partial t^{2} }} + \frac{{\partial^{3} w_{0}^{i} }}{{\partial t^{2} \partial R^{i} }}} \right) \\ \end{aligned} \right];\;\;i = c,p \\ \end{aligned} $$

(P-3)

$$ \begin{aligned} & \delta \phi ; \\ & \quad (1 - l^{2} \nabla^{2} )\left( \begin{aligned} & + X_{31} \frac{{\partial u^{p}_{0} }}{{\partial R^{p} }} + (X_{11} - 3X_{12} )\frac{{\partial^{2} w^{p}_{0} }}{{\partial R^{p2} }} - X_{33} \frac{{\partial^{2} w^{p}_{0} }}{{\partial R^{p2} }} \\ & \quad - \,( - X_{11} + 3X_{12} )\frac{{\partial u^{p}_{1} }}{{\partial R^{p} }} + X_{32} \frac{{\partial u^{p}_{1} }}{{\partial R^{p} }} - X_{33} \frac{{\partial u^{p}_{1} }}{{\partial R^{p} }} \\ & \quad - \,X_{41} \frac{{\partial^{2} \phi }}{{\partial R^{p2} }} + X_{42} \phi \\ \end{aligned} \right) = 0 \\ \end{aligned} $$

(P-4)

The GDQ form of the governing equations of the GPLRC microdisk is given as follows:

$$ \begin{aligned} & \delta u^{i}_{0}{:} \\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \hfill \\ + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ \begin{aligned} & \left( \begin{aligned} & \left[ \begin{aligned} & A^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{0} + B^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} \\ & \quad - \,D^{i}_{11} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{i}_{0} } \right) \\ \end{aligned} \right] \\ & \quad + \,\left[ \begin{aligned} & A^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,D^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] - X_{31} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi_{n} \\ \end{aligned} \right) \\ & \quad - \,\frac{1}{R}\left( \begin{aligned} & \left[ \begin{aligned} & A^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,D^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] \\ & \quad + \,Q^{i}_{22} \left[ \begin{aligned} A^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{0} }}{{R^{2i} {}_{n}}}} + B^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i}_{n} }}} \hfill \\ - D^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i}_{n} }}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{w^{i}_{0} }}{{R^{2i}_{n} }}} \right) \hfill \\ \end{aligned} \right] \\ \end{aligned} \right) \\ \end{aligned} \right] \\ & \quad = \left( {1 - \mu^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \hfill \\ + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ {I^{i}_{0} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{0} } I_{v} + I^{i}_{1} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{1} } I_{v} - I^{i}_{3} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{1} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \omega^{2} w^{i}_{0} } I_{v} } \right)} \right];\;i = c,p \\ \end{aligned} $$

(P-5)

$$ \begin{aligned} & \delta w^{i}_{0}{:} \\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \hfill \\ + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ \begin{aligned} & c_{1} \left( \begin{aligned} & \left[ \begin{aligned} & D^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } u^{i}_{0} + E^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } u^{i}_{1} \\ & \quad - \,G^{i}_{11} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(4)}_{n,v} } w^{i}_{0} } \right) \\ \end{aligned} \right] \\ & \quad + \,\left[ \begin{aligned} & D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] - X_{33} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi \\ \end{aligned} \right) \\ & \quad - \,c_{1} \left( \begin{aligned} & \left[ \begin{aligned} & D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] \\ & \quad + \,Q^{i}_{22} \left[ \begin{aligned} D^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R^{2i}_{n} }} + E^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R^{2i}_{n} }} \hfill \\ - G^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R^{2i}_{n} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R^{2i}_{n} }}} \right) \hfill \\ \end{aligned} \right] \\ \end{aligned} \right) \\ & \quad + \,\left( {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{i}_{0} } \right) + X_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi } \right) \\ & \quad - \,3c_{1} \left( \begin{aligned} & (C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } w^{i}_{0} } \right) \\ & \quad + \,X_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi \\ \end{aligned} \right) - N_{1}^{p} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{i}_{0} \\ \end{aligned} \right] \\ & \quad = \left( {1 - \mu^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } + \hfill \\ \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ \begin{aligned} & c_{1} I^{i}_{3} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \omega^{2} u^{i}_{0} } I_{v} + c_{1} I^{i}_{4} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \omega^{2} u^{i}_{1} } I_{v} \\ & \quad - \,I^{i}_{6} c_{1}^{2} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \omega^{2} u^{i}_{1} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} \omega^{2} w^{i}_{0} } I_{v} } \right) + \left( {I^{i}_{0} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} w^{i}_{0} } I_{v} } \right) \\ \end{aligned} \right];\;i = c,p \\ \end{aligned} $$

(P-6)

$$ \begin{aligned} & \delta u^{i}_{1}{:} \\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \hfill \\ + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ \begin{aligned} & \left( \begin{aligned} & \left[ {B^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{0} + C^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} - E^{i}_{11} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } w^{i}_{0} } \right)} \right] \\ & \quad + \,\left[ \begin{aligned} & B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + C^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,E^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] - X_{32} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi \\ \end{aligned} \right) \\ & \quad - \,c_{1} \left( \begin{aligned} & \left[ {D^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{0} + E^{i}_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} - G^{i}_{11} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } u^{i}_{1} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(3)}_{n,v} } w^{i}_{0} } \right)} \right] \\ & \quad + \left[ \begin{aligned} & D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] - X_{33} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi \\ \end{aligned} \right) \\ & \quad - \,\left( \begin{aligned} & \left[ \begin{aligned} & B^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + C^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,E^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{0} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] \\ & \quad + \,Q^{i}_{22} \left[ {B^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{0} }}{{R^{2i}_{n} }}} + C^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i}_{n} }}} - E^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i}_{n} }}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{w^{i}_{0} }}{{R^{2i}_{n} }}} \right)} \right] \\ \end{aligned} \right) \\ & \quad + \,c_{1} \left( \begin{aligned} & \left[ \begin{aligned} & D^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{0} }}{{R_{n}^{i} }} + E^{i}_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} \\ & \quad - \,G^{i}_{12} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{{u^{i}_{1} }}{{R_{n}^{i} }} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \frac{{w^{i}_{1} }}{{R_{n}^{i} }}} \right) \\ \end{aligned} \right] \\ & \quad + \,Q^{i}_{22} \left[ {D^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{0} }}{{R^{2i}_{n} }}} + E^{i}_{22} \sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i}_{n} }}} - G^{i}_{22} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\frac{{u^{i}_{1} }}{{R^{2i}_{n} }}} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \frac{{w^{i}_{0} }}{{R^{2i}_{n} }}} } \right)} \right] \\ \end{aligned} \right) \\ & \quad - \,\left[ {(A^{i}_{55} - 3C^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {u^{i}_{0} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} w^{i}_{0} } } \right) + X_{11} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi } \right] \\ & \quad + \,3c_{1} \left( {(C^{i}_{55} - 3E^{i}_{55} c_{1} )\left( {\sum\limits_{v = 1}^{{N_{n} }} {u^{i}_{1} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} w^{i}_{0} } } \right) + X_{12} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \phi } \right) \\ \end{aligned} \right]\, \\ & \quad = \left( {1 - \mu^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \hfill \\ + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R^{i} }} \hfill \\ \end{aligned} \right)} \right)\left[ \begin{aligned} & I^{i}_{1} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{0} } I_{v} + I^{i}_{2} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{1} } I_{v} - I^{i}_{4} c_{1} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{1} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \omega^{2} w^{i}_{0} } I_{v} } \right) \\ & \quad - \,c_{1} I^{i}_{3} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{0} } I_{v} - c_{1} I^{i}_{4} \sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{1} } I_{v} \\ & \quad + \,I^{i}_{6} c_{1}^{2} \left( {\sum\limits_{v = 1}^{{N_{n} }} {\omega^{2} u^{i}_{1} } I_{v} + \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} \omega^{2} w^{i}_{0} } I_{v} } \right) \\ \end{aligned} \right];i = c,p \\ \end{aligned} $$

(P-7)

$$ \begin{aligned} & \delta \phi{:} \\ & \quad \left( {1 - l^{2} \left( \begin{aligned} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } + \hfill \\ \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } \frac{1}{{R_{{}}^{i} }} \hfill \\ \end{aligned} \right)} \right)\left( \begin{aligned} & + X_{31} \sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{p}_{0} + (X_{11} - 3X_{12} - X_{33} )\sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } w^{p}_{0} \\ & \quad - \,( - X_{11} + 3X_{12} - X_{32} + X_{33} )\sum\limits_{v = 1}^{{N_{n} }} {C^{(1)}_{n,v} } u^{p}_{1} \\ & \quad - \,X_{41} \sum\limits_{v = 1}^{{N_{n} }} {C^{(2)}_{n,v} } \phi + X_{42} \sum\limits_{v = 1}^{{N_{n} }} \phi I_{v} \\ \end{aligned} \right) = 0 \\ \end{aligned} $$

(P-8)