Vibrational investigation of the spinning bi-dimensional functionally graded (2-FGM) micro plate subjected to thermal load in thermal environment

Abstract

This article presented a numerical method for discovering the free vibration of a spinning bi-dimensional functionally graded materials (FGM) micro circular plate exposed to thermal load based on Winkler–Pasternak foundation for the first time. Thermomechanical properties of bi-dimensional FGM micro plate are supposed to change during the thickness and radius directions of the plate. The small effect is taken into consideration the modified couple stress theory. Linear temperature rise during thickness and radius direction is investigated. First-order shear deformation theory is employed to derive the governing equations and boundary conditions of the bi-dimensional FGM circular plate in the thermal environment via Hamilton’s principle. The differential quadrature method is used to achieve the frequency of bi-dimensional FGM micro circular plate exposed to the thermal environment. A parametric analysis is led to assess the efficacy of Winkler and Pasternak parameters, FG power index, coefficients of bi-dimensional FGM, size dependency, non-dimensional angular velocity, temperature changes and thermal loading on the natural frequencies of bi-dimensional FGM micro plate.

Appendices

Appendix A

The strain energy of isotropic linear elastic parts is obtained using the following equation (Eshraghi et al. 2016; Reddy and Berry 2012; Ke et al. 2012):

$${\text{U = }}\frac{1}{2}\int\limits_{V} {(\sigma :\varepsilon + m:\chi )dV} ,$$

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where ε and σ are the linear strain and the Cauchy stress tensors respectively. χ and m are the symmetric curvature strain and deviatoric part of the couple stress tensors. The four last tensors are defined as follows (Eshraghi et al. 2016; Reddy and Berry 2012; Ke et al. 2012):

$$\sigma = \lambda tr(\varepsilon )I + 2\mu \varepsilon ,$$

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$$\varepsilon = \frac{1}{2}[\nabla u + (\nabla u)^{T} ],$$

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$$m = 2l^{2} \mu \chi ,$$

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$$\chi = \frac{1}{2}[\nabla \varLambda + (\nabla \varLambda )^{T} ],$$

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where λ and μ are the Lame’s constant, u and l are the displacement vector and the material length scale parameter respectively, and Λ is a rotation vector defined by:

$$\varLambda = \frac{1}{2}curl\,u.$$

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Appendix B

$$N_{rr} (r) = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\sigma_{rr} dz = \left( {A_{11} (r)\frac{\partial u}{\partial r} + A_{12} (r)\frac{u}{r}} \right) + \left( {B_{11} (r)\frac{\partial \varPhi }{\partial r} + B_{12} (r)\frac{\varPhi }{r}} \right),}$$

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$$N_{\theta \theta } (r) = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\sigma_{\theta \theta } dz = \left( {A_{12} (r)\frac{\partial u}{\partial r} + A_{11} (r)\frac{u}{r}} \right) + \left( {B_{12} (r)\frac{\partial \varPhi }{\partial r} + B_{11} (r)\frac{\varPhi }{r}} \right)} ,$$

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$$M_{rr} (r) = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\sigma_{rr} zdz = \left( {B_{11} (r)\frac{\partial u}{\partial r} + B_{12} (r)\frac{u}{r}} \right) + \left( {A_{11} (r)\frac{\partial \varPhi }{\partial r} + B_{12} (r)\frac{\varPhi }{r}} \right)} ,$$

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$$M_{\theta \theta } (r) = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\sigma_{\theta \theta } zdz = \left( {B_{12} (r)\frac{\partial u}{\partial r} + B_{11} (r)\frac{u}{r}} \right) + \left( {D_{11} (r)\frac{\partial \varPhi }{\partial r} + D_{12} (r)\frac{\varPhi }{r}} \right)} ,$$

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$$M_{rz} (r) = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\sigma_{rz} dz = k_{s} A_{55} } (r)\left( {\frac{\partial w}{\partial r} + \varPhi } \right),$$

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$$\varOmega_{r\theta } = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {m_{r\theta } dz = \frac{1}{2}S(r,z)\left( {\left( {\frac{\partial \varPhi }{\partial r} - \frac{{\partial^{2} w}}{{\partial r^{2} }}} \right) - \frac{1}{r}\left( {\varPhi - \frac{\partial w}{\partial r}} \right)} \right)} ,$$

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where $k_{s} = \frac{{\pi^{2} }}{12}$ s known as the shear correction factor and the constants are defined as follows:

$$\{ A_{11} ,B_{11} ,D_{11} \} = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\left( {\frac{E(r,z)}{{1 - v^{2} (r,z)}}} \right)} \{ 1,z,z^{2} \} dz,$$

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$$\{ A_{12} ,B_{12} ,D_{12} \} = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\left( {\frac{E(r,z)v(z)}{{1 - v^{2} (r,z)}}} \right)} \{ 1,z,z^{2} \} dz,$$

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$$A_{55} = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\left( {\frac{E(r,z)}{2(1 + v(r,z))}} \right)} dz,$$

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$$S = \int\limits_{{ - \frac{h(r)}{2}}}^{{\frac{h(r)}{2}}} {\left( {\frac{{l^{2} (r,z)E(r,z)}}{2(1 + v(r,z))}} \right)} dz.$$

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Appendix C

$$\left[ \begin{array}{l} (A_{11} )\left( {\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{{r^{2} }}} \right) + \hfill \\ r\frac{{\partial A_{11} }}{\partial r}\frac{\partial u}{\partial r} + \frac{{\partial A_{12} }}{\partial r}u \hfill \\ + (B_{11} )\left( {\frac{{\partial^{2} \varPhi }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \varPhi }{\partial r} - \frac{\varPhi }{{r^{2} }}} \right) \hfill \\ \left( {r\frac{{\partial B_{11} }}{\partial r}\frac{\partial \varPhi }{\partial r} + \frac{{\partial B_{12} }}{\partial r}\varPhi } \right) \hfill \\ \end{array} \right] = \left( {I_{1} \frac{{\partial^{2} u}}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \varPhi }}{{\partial t^{2} }}} \right),$$

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$$\left[ \begin{array}{l} \frac{{k_{s} }}{r}\left( {A_{55} r\frac{{\partial^{2} w}}{{\partial r^{2} }} + A_{55} \frac{\partial w}{\partial r} + \frac{{\partial A_{55} }}{\partial r}r\frac{\partial w}{\partial r}} \right) \hfill \\ + \frac{S}{4r}\left( { - \frac{1}{2}\frac{{\partial^{4} w}}{{\partial r^{4} }} - \frac{1.5}{2r}\frac{{\partial^{3} w}}{{\partial r^{3} }} - \left( {\frac{1.5}{r} - \frac{1}{{r^{2} }}} \right)\frac{{\partial^{2} w}}{{\partial r^{2} }} - \left( {\frac{1}{{r^{3} }} - \frac{3}{{r^{2} }}} \right)\frac{\partial w}{\partial r}} \right) \hfill \\ - \left( {\frac{1}{2r}\frac{\partial S}{\partial r}} \right)\left( {\frac{{\partial^{3} w}}{{\partial r^{3} }} + \frac{1.5}{r}\frac{{\partial^{2} w}}{{\partial r^{2} }} + \left( {\frac{1.5}{r} - \frac{1}{{r^{2} }}} \right)\frac{\partial w}{\partial r}} \right) \hfill \\ - \left( {\frac{1}{2r}\frac{{\partial^{2} S}}{{\partial r^{2} }}} \right)\left( {\frac{1}{2}\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{1.5}{r}\frac{\partial w}{\partial r}} \right) \hfill \\ + \frac{{k_{s} }}{r}\left( {A_{55} r\frac{\partial \varPhi }{\partial r} + A_{55} \varPhi + \frac{{\partial A_{55} }}{\partial r}r\varPhi } \right) \hfill \\ + \frac{S}{4r}\left( {\frac{1}{2}\frac{{\partial^{3} \varPhi }}{{\partial r^{3} }} + \frac{1.5}{2r}\frac{{\partial^{2} \varPhi }}{{\partial r^{2} }} + \left( {\frac{1.5}{r} - \frac{1}{{r^{2} }}} \right)\frac{\partial \varPhi }{\partial r} + \left( {\frac{1}{{r^{3} }} - \frac{3}{{r^{2} }}} \right)\varPhi } \right) \hfill \\ + \left( {\frac{1}{2r}\frac{\partial S}{\partial r}} \right)\left( {\frac{{\partial^{2} \varPhi }}{{\partial r^{2} }} + \left( {\frac{1}{r} + 1.5} \right)\frac{\partial \varPhi }{\partial r} + \left( {\frac{1.5}{r} - \frac{1}{{r^{2} }}} \right)\varPhi } \right) \hfill \\ + \left( {\frac{1}{2r}\frac{{\partial^{2} S}}{{\partial r^{2} }}} \right)\left( {\frac{1}{2}\frac{\partial \varPhi }{\partial r} + \frac{1.5}{r}\varPhi } \right) + \frac{\partial }{r\partial r}\left( {(N^{Rotation} + N^{Thermal} )r\frac{\partial w}{\partial r}} \right) \hfill \\ K_{Winkler} w - K_{Pasternak} \nabla^{2} w \hfill \\ \end{array} \right] = I_{1} \frac{{\partial^{2} w}}{{\partial t^{2} }},$$

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$$\begin{aligned} \left[ \begin{aligned} - k_{s} A_{55} \frac{\partial w}{\partial r} + \left( { - \frac{S}{2}} \right)\left( {\frac{1}{4}\frac{{\partial^{3} w}}{{\partial r^{3} }} + \frac{1}{r}\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{1}{{r^{2} }}\frac{\partial w}{\partial r}} \right) - \left( {\frac{1}{4}\frac{\partial S}{\partial r}} \right)\left( {\frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial w}{\partial r}} \right) \hfill \\ - k_{s} A_{55} \varPhi + D_{11} \left( {\frac{{\partial^{2} \varPhi }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \varPhi }{\partial r} - \frac{1}{{r^{2} }}\varPhi } \right) + \left( {\frac{{\partial D_{11} }}{\partial r}} \right)\left( {\frac{\partial \varPhi }{\partial r}} \right) + \left( {\frac{{\partial D_{12} }}{\partial r}} \right)\left( {\frac{1}{r}\varPhi } \right) \hfill \\ \frac{1}{2}\left\{ {\left( {\frac{S}{4}} \right)\left( {\frac{{\partial^{2} \varPhi }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \varPhi }{\partial r} - \frac{1}{{r^{2} }}\varPhi } \right) + \left( {\frac{\partial S}{\partial r}} \right)\left( {\frac{\partial \varPhi }{\partial r}} \right) + \left( {\frac{\partial S}{\partial r}} \right)\left( {\frac{1}{r}\varPhi } \right)} \right\} \hfill \\ \frac{1}{r}\left\{ {(B_{11} )\left( {r\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{\partial u}{\partial r} - \frac{1}{r}u} \right) + \left( {\frac{{\partial B_{11} }}{\partial r}} \right)\left( {r\frac{\partial u}{\partial r}} \right) + \left( {\frac{{\partial B_{12} }}{\partial r}} \right)(u)} \right\} \hfill \\ \end{aligned} \right] = I_{2} \frac{{\partial^{2} u}}{{\partial t^{2} }} + I_{3} \frac{{\partial^{2} \varPhi }}{{\partial t^{2} }}, \hfill \\ \end{aligned}$$

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$$\left[ {\left( {A_{11} \left( {r\frac{\partial u}{\partial r}} \right) + A_{12} u} \right) + \left( {B_{11} \left( {r\frac{\partial \varPhi }{\partial r}} \right) + B_{12} \varPhi } \right)} \right]\delta \,u = 0\quad {\text{at}}\,{\text{r}} = 0,{\text{R,}}$$

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$$\left[ \begin{array}{l} \frac{S}{4}\left( {\frac{{\partial^{2} \varPhi }}{{\partial r^{2} }} - \frac{{\partial^{3} w}}{{\partial r^{3} }} + \frac{1}{r}\frac{\partial \varPhi }{\partial r} - \frac{1}{r}\frac{{\partial^{2} w}}{{\partial r^{2} }} - \frac{\varPhi }{{r^{2} }} + \frac{1}{{r^{2} }}\frac{\partial w}{\partial r}} \right) \hfill \\ + k_{s} A_{55} \left( {\frac{\partial w}{\partial r} + \varPhi } \right) \hfill \\ + \frac{1}{4}\frac{\partial S}{\partial r}\left( {\frac{\partial \varPhi }{\partial r} - \frac{{\partial^{2} w}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \varPhi }{\partial r}} \right) + \frac{1}{r}\left( { - \frac{\partial w}{\partial r} + \varPhi } \right) \hfill \\ \end{array} \right]\delta w = 0\quad {\text{at}}\,{\text{r}} = 0,{\text{R,}}$$

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$$\left( {\frac{S}{4}\frac{1}{r}\left( {\left( {\frac{\partial \varPhi }{\partial r} - \frac{{\partial^{2} w}}{{\partial r^{2} }}} \right) - \frac{1}{r}\left( {\varPhi - \frac{\partial w}{\partial r}} \right)} \right)} \right)\delta \left( {\frac{\partial w}{\partial r}} \right) = 0\quad {\text{at}}\,{\text{r}} = 0,{\text{R,}}$$

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$$\left( {B_{11} \frac{\partial u}{\partial r} + B_{12} \frac{u}{r} + D_{11} \frac{{\partial {{\Phi }}}}{\partial r} + D_{12} \frac{{{\Phi }}}{r} + \frac{S}{4}\left( {\left( {\frac{{\partial {{\Phi }}}}{\partial r} - \frac{{\partial^{2} w}}{{\partial r^{2} }}} \right) - \frac{1}{r}\left( {{{\Phi }} - \frac{\partial w}{\partial r}} \right)} \right)} \right)\delta {{\Phi }} = 0\quad {\text{at}}\,{\text{r}} = 0,{\text{R}} .$$

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Appendix D

$$\left[ \begin{aligned} (A_{11} )\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} u_{k} + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} u_{k} - \frac{{u_{i} }}{{r_{i}^{2} }}} } } \right) + \hfill \\ \left( {r_{i} \left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} } A_{11} } \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} u_{k} } } \right) + \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} A_{12} } } \right)(u_{i} )} \right. \hfill \\ + (B_{11} )\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} \varPhi_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{\left( 1 \right)} \varPhi_{k} - \frac{{\varPhi_{i} }}{{r_{i}^{2} }}} } \right) + \hfill \\ \left( {r_{i} \left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} } B_{11} } \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} } } \right) + \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} B_{12} } } \right)(\varPhi_{i} )} \right. \hfill \\ \end{aligned} \right] = \omega^{2} (I_{1} u_{i} + I_{2} \varPhi_{i} ),$$

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$$\left[ \begin{aligned} \left( {\frac{S}{{4r_{i} }}} \right)\left[ \begin{aligned} \frac{1}{2}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(3)} \varPhi_{k} + \frac{1.5}{{2r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} \varPhi_{k} (\frac{1.5}{{r_{i} }} - \frac{1}{{r_{i}^{2} }})\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} } } } \hfill \\ + \left( {\frac{1}{{r^{3}_{i} }} - \frac{3}{{r_{i}^{2} }}} \right)\varPhi_{i} - \frac{1}{2}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{\left( 4 \right)} w_{k} } - \frac{1.5}{{2r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(3)} w_{k} } \hfill \\ \left( {\frac{1.5}{{r_{i} }} - \frac{1}{{r_{i}^{2} }}} \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} - } \left( {\frac{1}{{r^{3}_{i} }} - \frac{3}{{r_{i}^{2} }}} \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } \hfill \\ \end{aligned} \right] \hfill \\ + (k_{s} A_{55} )\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} + \sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} + \frac{{\varPhi_{i} }}{{r_{i} }}} } } \right) \hfill \\ \left( {\frac{{k_{s} }}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} A_{55} } )(r_{i} \sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} + r_{i} \varPhi_{i} } } \right) - \hfill \\ (\frac{1}{{2r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} S} \left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(3)} w_{k} } + (\frac{1.5}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} } + \left( {\frac{1.5}{{r_{i} }} - \frac{1}{{r_{i}^{2} }}} \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } \right) + \hfill \\ \left( {\left( {\frac{1}{{2r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} S} } \right)\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} \varPhi_{k} } + \left( {\frac{1}{{r_{i} }} + 1.5} \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} } + \left( {\frac{1.5}{{r_{i} }} - \frac{1}{{r_{i}^{2} }}} \right)\varPhi_{i} } \right)} \right) - \hfill \\ \left( {\left( {\frac{1}{{2r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} S} } \right)\left( {\left( {\frac{1}{2}} \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} } + \frac{1.5}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } \right)} \right) + \hfill \\ \left( {\left( {\left( {\frac{1}{{2r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} S} } \right)\left( {\left( {\frac{1}{2}} \right)\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} } + \left( {\frac{1.5}{{r_{i} }}} \right)\varPhi_{i} } \right)} \right)} \right) + \hfill \\ \frac{1}{{r_{i} }}\left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} ((N^{Rotation} + N^{Thermal} )r\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } \right)} } \right)} \right) \hfill \\ (K_{Winkler} w_{i} ) - \left( {K_{Pasternak} \sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } } \right) \hfill \\ \end{aligned} \right] = \omega^{2} I_{1} w_{i} ,$$

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$$\left[ \begin{aligned} - k_{s} A_{55} \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } \right) - \varPhi_{i} } \right) \hfill \\ \left( { - \frac{S}{2}} \right)\left( {\frac{1}{4}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(3)} w_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} + \frac{1}{{r_{i}^{2} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } } \right) \hfill \\ - \left( {\left( {\frac{1}{4}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} S} } \right)\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} w_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} w_{k} } } \right)} \right) \hfill \\ + \left( {\frac{S}{4}(\frac{1}{2})} \right)\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} \varPhi_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} - \frac{{\varPhi_{i} }}{{r_{i}^{2} }}} } \right) \hfill \\ + \left( {\left( {\frac{1}{2}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} S} } \right)\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} } + \frac{{\varPhi_{i} }}{{r_{i} }}} \right)} \right) \hfill \\ + (D_{11} )\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{\left( 2 \right)} \varPhi_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} - \frac{{\varPhi_{i} }}{{r_{i}^{2} }}} } \right) \hfill \\ + \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} D_{11} } \left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} \varPhi_{k} } } \right)} \right)} \right. + \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} D_{12} } } \right)(\varPhi_{i} )} \right) \hfill \\ + (B_{11} )\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(2)} u_{k} } + \frac{1}{{r_{i} }}\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} u_{k} - \frac{{u_{i} }}{{r_{i}^{2} }}} } \right) \hfill \\ + \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} B_{11} } \left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} u_{k} } } \right)} \right)} \right. + \left( {\left( {\sum\limits_{k = 1}^{{n_{i} }} {C_{ik}^{(1)} B_{12} } (u_{i} )} \right)} \right. \hfill \\ \end{aligned} \right] = \omega^{2} (I_{3} \varPhi_{i} + I_{2} u_{i} ).$$

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