# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## Appendix A  
\[A_{11}\frac{\partial^{2}u_{0}}{\partial x^{2}}+A_{11}\frac{\partial w_{0}}{ \partial x}\frac{\partial^{2}w_{0}}{\partial x^{2}}-I_{0}\frac{d^{2}u_{0}}{dt^ {2}}=0,\] (A1)  
\[ \begin{array}{c}\big{(}-A_{55}+6D_{55}c_{1}-9F_{55}c_{1}^{2}\big{)}\mathbb{O }_{x}+\big{(}D_{11}-2F_{11}c_{1}+H_{11}c_{1}^{2}\big{)}\frac{\partial^{2} \mathbb{O}_{x}}{\partial x^{2}}+\big{(}-A_{55}+6D_{55}c_{1}-9F_{55}c_{1}^{2} \big{)}\frac{\partial w_{0}}{\partial x}\\ +\big{(}-F_{11}c_{1}+H_{11}c_{1}^{2}\big{)}\frac{\partial^{3}w_{0}}{\partial x^ {3}}-K_{2}\frac{d^{2}\mathbb{O}_{x}}{dt^{2}}+c_{1}I_{4}\frac{\partial}{ \partial x}\left(\frac{d^{2}\mathbb{O}_{x}}{dt^{2}}\right)=0,\end{array} \] (A2)  
\[ \begin{array}{c}A_{11}\frac{\partial w_{0}}{\partial x}\frac{\partial^{2}w_{ 0}}{\partial x^{2}}+A_{11}\frac{\partial w_{0}}{\partial x}\frac{\partial^{2} w_{0}}{\partial x^{2}}+\frac{3}{A_{11}}\left(\frac{\partial w_{0}}{\partial x} \right)^{2}\frac{\partial^{2}w_{0}}{\partial x^{2}}+c_{1}(F_{11}-c_{1}H_{11}) \frac{\partial^{3}\mathbb{O}_{x}}{\partial x^{3}}-c_{1}^{2}H_{11}\frac{ \partial^{4}w_{0}}{\partial x^{4}}\\ +\big{(}A_{55}-6c_{1}D_{55}+9c_{1}^{2}F_{55}\big{)}\frac{\partial\mathbb{O}_{ x}}{\partial x}+\big{(}A_{55}-6c_{1}D_{55}+9c_{1}^{2}F_{55}\big{)}\frac{\partial^{2} \mathbb{O}_{x}}{\partial x^{2}}-I_{0}\frac{d^{2}w_{0}}{dt^{2}}\\ +c_{1}^{2}I_{6}\frac{\partial^{2}}{\partial x^{2}}\left(\frac{d^{2}w_{0}}{dt^ {2}}\right)-c_{1}I_{4}\frac{\partial}{\partial x}\left(\frac{d^{2}\mathbb{O}_ {x}}{dt^{2}}\right)+q-c\frac{dw_{0}}{dt}=0,\end{array} \] (A3)  
where \(A_{11}\), \(D_{11}\), \(F_{11}\), \(H_{11}\), \(A_{55}\), \(D_{55}\), \(F_{55}\), \(I_{0}\), \(I_{4}\), \(I_{6}\), \(I_{4}\), and \(K_{2}\), are given below,  
\[ \begin{array}{c}(A_{11},D_{11},F_{11},H_{11})=\sum_{k=1}^{3}\int_{z_{k}}^{z_{ k+1}}Q_{11}^{(k)}\Big{(}1,z^{2},z^{4},z^{6}\Big{)}dz,\end{array} \] (A4)  
\[ \begin{array}{c}(A_{55},D_{55},F_{55})=\sum_{k=1}^{3}\int_{z_{k}}^{z_{k+1}}Q _{13}^{(k)}\Big{(}1,z^{2},z^{4}\Big{)}dz,\end{array} \] (A5)  
\[ \begin{array}{c}(I_{0},I_{4},I_{6})=\sum_{k=1}^{3}\int_{z_{k}}^{z_{k+1}}\rho^ {(k)}\Big{(}1,z^{4},z^{6}\Big{)}dz,\end{array} \] (A6)  
\[ \begin{array}{c}J_{4}=I_{4}-I_{6}c_{1},\end{array} \] (A7)  
\[ \begin{array}{c}K_{2}=\Big{(}I_{2}-2I_{4}c_{1}+I_{6}c_{1}^{2}\Big{)},\end{array} \] (A8)  
and \(k=1,2,3\) represents the layer index. In addition, \(Q_{ij}^{(1)}\), \(Q_{ij}^{(2)}\) and \(Q_{ij}^{(3)}\) are the elastic parameters for the lower layer, middle layer, and upper layer respectively, and \(\rho^{(1)}\), \(\rho^{(2)}\) and \(\rho^{(3)}\) are the densities respectively. \[ \begin{array}{c}I=\int_{\Omega}z^{2}dA_{zy}=\frac{bh^{3}}{12},\ I_{0}=\sum_{k =1}^{3}\int_{z_{k-1}}^{z_{k}}\rho^{(k)}dz.\end{array} \] (A9)  
\[ \begin{array}{c}A=\frac{I_{0}}{\rho h},\ B=\frac{1}{I^{2}}\frac{c_{1}I_{4}}{ \rho h},\ C=\frac{1}{I^{2}}\frac{c_{1}I_{4}}{\rho h}\bigg{(}\frac{F_{11}c_{1}- D_{11}}{A_{55}-6D_{55}c_{1}+9F_{55}c_{1}^{2}}\bigg{)},\ D=\frac{c}{\rho h \tau},\ F=\frac{hA_{11}}{2\rho\tau^{2}h},\ G=\frac{D_{11}}{\rho h\tau^{2}h},\\ \frac{c_{1}(F_{11}-c_{1}H_{11})}{\rho h\tau^{2}h}\bigg{(}\frac{F_{11}c_{1}-D_{11 }}{A_{55}-6D_{55}c_{1}+9F_{55}c_{1}^{2}}\bigg{)};\end{array} \] (A10)  
\[ \begin{array}{c}\dot{w}_{1,1}=w_{1,2}\\ \dot{w}_{1,2}=T_{11}w_{1,2}+T_{12}w_{1,1}+T_{13}w_{2,2}+T_{14}w_{2,1}+T_{15}w_{1,1}^{3}+\\ T_{16}w_{1,1}^{2}w_{2,1}+T_{17}w_{2,1}^{2}w_{1,1}+T_{18}w_{2,1}^{3}+T_{19}q \\ \dot{w}_{2,1}=w_{2,2}\\ \dot{w}_{2,2}=T_{21}w_{1,2}+T_{22}w_{1,1}+T_{23}w_{2,2}+T_{24}w_{2,1}+T_{25}w_{1,1}^{3}\\ +T_{26}w_{1,1}^{2}w_{2,1}+T_{27}w_{2,1}^{2}w_{1,1}+T_{28}w_{2,1}^{3}+T_{29}q \end{array} ,\] (A11)where \(T_{1i}\) and \(T_{2i}\)\((i=1,2,\ldots,9)\) are given below,  
\[ \begin{array}{l}T_{11}=U_{1}S_{1b}+U_{2}S_{2b},\ T_{12}=U_{1}S_{1c}+U_{2}S_{2c},\ T_{13}=U_{1}S_{1e}+U_{2}S_{2e},\\ T_{14}=U_{1}S_{1f}+U_{2}S_{2f},\ T_{15}=U_{1}S_{1g}+U_{2}S_{2g},\ T_{16}=U_{1}S_{1 h}+U_{2}S_{2h},\\ T_{17}=U_{1}S_{1i}+U_{2}S_{2i},\ T_{18}=U_{1}S_{1j}+U_{2}S_{2j},\ T_{19}=U_{1}S_{1 k}+U_{2}S_{2k};\end{array}  \tag{112}\]  
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