Free vibration and critical angular velocity of a rotating variable thickness two-directional FG circular microplate

Abstract

In this paper, the free vibration of a rotating variable thickness two-directional FG circular microplate is studied. The governing equations of motion for the microplate are extracted utilizing the Hamiltonian’s principle in conjunction with the first shear deformation theory as well as the modified couple stress theory. The solution of equations is presented utilizing the differential quadrature method. In special cases, the natural frequency results obtained by the reduced form of the proposed formulation are compared with those available in the literature, indicating a very good accuracy. The results reveal that there is a non-proportional relation between the natural frequencies of the microplate and the thickness-variations of the section. In contrast, the critical angular velocity of that is not much sensitive with respect to the thickness variation. Moreover, the analyses indicate the significant impact of the two-directionality-variation of the graded material on the natural frequencies as well as the critical angular velocities. A map on the effects of the two-directionality-variation of the material property on the free vibration of the microplate is presented. The results show that the increase of the size dependency would lead to the reduction of the non-dimensional natural frequency as well as the critical angular velocity.

Appendices

Appendix A

The strain energy, U, in an isotropic linear elastic material would obtain utilizing the modified couple stress theory as follows (Eshraghi et al. 2016; Reddy and Berry 2012; Ke et al. 2012):

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

(31)

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):

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

(32)

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

(33)

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

(34)

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

(35)

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:

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

(36)

Appendix B

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

(37)

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

(38)

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

(39)

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

(40)

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

(41)

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

(42)

where ks = π²/12 is known as the shear correction factor and the constants are defined as follows:

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

(43)

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

(44)

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

(45)

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

(46)

Appendix C

$$ \left[ \begin{aligned} & (A_{11} )\left( {\frac{{\partial^{2} u}}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial u}{\partial r} - \frac{u}{{r^{2} }}} \right) \hfill \\ &\quad + r\frac{{\partial A_{11} }}{\partial r}\frac{\partial u}{\partial r} + \frac{{\partial A_{12} }}{\partial r}u \hfill \\ &\quad + (B_{11} )\left( {\frac{{\partial^{2} \Phi }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial \Phi }{\partial r} - \frac{\Phi }{{r^{2} }}} \right) \hfill \\ &\quad \left( {r\frac{{\partial B_{11} }}{\partial r}\frac{\partial \Phi }{\partial r} + \frac{{\partial B_{12} }}{\partial r}\Phi } \right) \hfill \\ \end{aligned} \right] = \left( {I_{1} \frac{{\partial^{2} u}}{{\partial t^{2} }} + I_{2} \frac{{\partial^{2} \Phi }}{{\partial t^{2} }}} \right) $$

(47)

$$ \left[ \begin{aligned} &\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 \\ &\quad + \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 \\ &\quad - \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 \\ &\quad- \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 \\ &\quad + \frac{{k_{s} }}{r}\left( {A_{55} r\frac{\partial \Phi }{\partial r} + A_{55} \Phi + \frac{{\partial A_{55} }}{\partial r}r\Phi } \right) \hfill \\ &\quad + \frac{S}{4r}\left( {\frac{1}{2}\frac{{\partial^{3} \Phi }}{{\partial r^{3} }} + \frac{1.5}{2r}\frac{{\partial^{2} \Phi }}{{\partial r^{2} }} + \left( {\frac{1.5}{r} - \frac{1}{{r^{2} }}} \right)\frac{\partial \Phi }{\partial r} + \left( {\frac{1}{{r^{3} }} - \frac{3}{{r^{2} }}} \right)\Phi } \right) \hfill \\ &\quad + \left( {\frac{1}{2r}\frac{\partial S}{\partial r}} \right)\left( {\frac{{\partial^{2} \Phi }}{{\partial r^{2} }} + \left( {\frac{1}{r} + 1.5} \right)\frac{\partial \Phi }{\partial r} + \left( {\frac{1.5}{r} - \frac{1}{{r^{2} }}} \right)\Phi } \right) \hfill \\ &\quad + \left( {\frac{1}{2r}\frac{{\partial^{2} S}}{{\partial r^{2} }}} \right)\left( {\frac{1}{2}\frac{\partial \Phi }{\partial r} + \frac{1.5}{r}\Phi } \right) + \frac{\partial }{r\partial r}\left( {N^{Rotation} r\frac{\partial w}{\partial r}} \right) \hfill \\ \end{aligned} \right] = I_{1} \frac{{\partial^{2} w}}{{\partial t^{2} }} $$

(48)

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

(49)

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

(50)

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

(51)

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

(52)

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

(53)

Appendix D

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

(54)

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

(55)

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

(56)