## 2 Formulation of the Problem
\right\} \tag{9}\]  
where  
\[L_{11}=A_{118}\frac{d^{2}}{dr^{2}}+A_{11}g^{\prime}\frac{d}{dr}++A_{11}g\frac {1}{r}\frac{d}{dr}+A_{12}g^{\prime}\frac{1}{r}-A_{22}g\frac{1}{r^{2}}+A_{66}g \frac{n^{2}}{r^{2}}+I_{1}\omega^{2}\]  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]  
\[L_{14}=2B_{16}g\frac{n}{r}\frac{d}{dr}+B_{16}g^{\prime}\frac{n}{r}\]  
\[L_{15}=B_{16}g\frac{d^{2}}{dr^{2}}+B_{16}g^{\prime}\frac{d}{dr}-B_{26}g\frac{ 1}{r}\frac{d}{dr}+B_{26}g\frac{1}{r^{2}}+B_{26}g\frac{n^{2}}{r^{2}}-B_{16}g^{ \prime}\frac{1}{r}\]  
\[L_{21}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{22}g \frac{n}{r^{2}}+A_{66}g^{\prime}\frac{n}{r}+A_{66}g\frac{n}{r^{2}}\]  
\[L_{22}=A_{66}g\frac{d^{2}}{dr^{2}}+A_{66}g^{\prime}\frac{d}{dr}+A_{66}g\frac{ 1}{r}\frac{d}{dr}+A_{22}g\frac{n^{2}}{r^{2}}-A_{66}g^{\prime}\frac{1}{r}-A_{6 6}g\frac{1}{r^{2}}+I_{1}\omega^{2}\]  
\[L_{23}=0\]  
\[L_{24}=B_{16}g\frac{d^{2}}{dr^{2}}+B_{16}g^{\prime}\frac{d}{dr}+2B_{16}g\frac{ 1}{r}\frac{d}{dr}+B_{26}g\frac{1}{r}\frac{d}{dr}+B_{26}g^{\prime}\frac{1}{r}+B _{26}g\frac{1}{r^{2}}+B_{26}g\frac{n^{2}}{r^{2}}\]  
\[L_{25}=2B_{26}g\frac{n}{r}\frac{d}{dr}+B_{26}g^{\prime}\frac{n}{r},\ L_{31}=0,\ L_{32}=0\]  
\[L_{33}=KA_{55}g\frac{d^{2}}{dr^{2}}+KA_{55}g^{\prime}\frac{d}{dr}+KA_{55}g \frac{1}{r}\frac{d}{dr}+KA_{448}\frac{n^{2}}{r^{2}}+I_{1}\omega^{2}\]  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]0  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]1  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]2  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]3  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]4  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]5  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]6  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]7  
\[L_{12}=A_{12}g\frac{n}{r}\frac{d}{dr}+A_{66}g\frac{n}{r}\frac{d}{dr}+A_{12}g^{ \prime}\frac{n}{r}-A_{22}g\frac{n}{r^{2}}-A_{66}g\frac{n}{r^{2}},\ L_{13}=0\]8Introducing the nondimensional parameters, \(R=\frac{r-a}{l}\), \(a\leq r\leq b\) and \(R\in[0,1]\), and \(\lambda\ =\ \omega\ l\ \sqrt{\frac{l_{1}}{A_{11}}}\); a frequency parameter, \(\beta=\ \frac{a}{b}\); the radii ratio, \(\gamma\ =\ \frac{h}{r_{a}}\); the ratios of thickness to the radius of inner circle, \(\delta_{k}=\ \frac{h_{k}}{h}\), and the relative layer thickness of the \(k\)-th layer. (10)  
Here, \(h_{k}\) is the thickness of the \(k\)-th layer, \(h\) is the total thickness of the plate, \(r_{a}\) is the radius of inner circular plate and \(A_{11}\) is a standard extensional rigidity coefficient. After nondimensionalizing Equation (9), a new set of differential equations is obtained.