# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## 2 Beam Model Establishment
\begin{array}{l}-I_{0}\frac{d^{2}w_{0}}{dt^{2}}+c_{1}I_{4}\frac{\partial^{2 }}{\partial x^{2}}\left(\frac{d^{2}w_{0}}{dt^{2}}\right)-c_{1}I_{4}\frac{F_{ 11}c_{1}-D_{11}}{\left(A_{55}-6D_{55}c_{1}+9F_{55}c_{1}^{2}\right)}\frac{ \partial^{4}}{\partial x^{4}}\left(\frac{d^{2}w_{0}}{dt^{2}}\right)-c\frac{d \overline{w}_{0}}{dt}+\\ \frac{\Delta_{11}}{2I}\frac{\partial^{2}w_{0}}{\partial x^{2}}\left[\int_{0}^ {l}\left(\frac{\partial w_{0}}{\partial x}\right)^{2}dx\right]-D_{11}\frac{ \partial^{4}w_{0}}{\partial x^{4}}+c_{1}(F_{11}-c_{1}H_{11})\frac{F_{11}c_{1}- D_{11}}{\left(A_{55}-6D_{55}c_{1}^{2}\right)}\frac{\partial^{4}w_{0}}{ \partial x^{5}}+q=0.\end{array}  \tag{13}\]  
Furthermore, the following dimensionless variables are given in order to validate Equation (13):  
\[\overline{t}=\sqrt{\frac{Q_{11}^{(2)}}{I_{0}b!^{4}}t}=\tau t,\ \overline{x}=\frac{x}{I},\ \overline{w}_{0}=\frac{w_{0}}{h},\ \frac{d\overline{w}_{0}}{d\overline{t}}=\frac{1}{\tau h}\frac{dw_{0}}{dt},\ \frac{d^{2} \overline{w}_{0}}{d\overline{t}^{2}}=\frac{1}{\tau^{2}h}\frac{d^{2}w_{0}}{dt^ {2}},\ \overline{c}=c/\left(\frac{Q_{11}^{(2)}}{h\tau}\right), \tag{14}\]  
where \(I\) and \(I_{0}\) are given in Appendix A. With Equation (14) substituted into Equation (13), it is derived that  
\[ \begin{array}{l}-A\frac{d^{2}\overline{w}_{0}}{dt^{2}}+B\frac{\partial^{2}} {\partial\overline{x}}\left(\frac{d^{2}\overline{w}_{0}}{dt^{2}}\right)-C \frac{\partial^{4}}{\partial\overline{x}}\left(\frac{d^{2}\overline{w}_{0}}{ d\overline{t}^{2}}\right)-D\frac{d\overline{w}_{0}}{dt}+F\frac{\partial^{2} \overline{w}_{0}}{\partial\overline{x}^{2}}\left[\int_{0}^{l}\left(\frac{ \partial\overline{w}_{0}}{d\overline{t}}\right)^{2}d\overline{x}\right]\\ \hskip 142.26378 pt-G\frac{\partial^{4}\overline{w}_{0}}{\partial\overline{x}^{2 }}+H\frac{\partial^{2}\overline{w}_{0}}{\partial\overline{x}^{2}}+\overline{ q}=0,\end{array}  \tag{15}\]  
where \(A,\)\(B,\)\(C,\)\(D,\)\(F,\)\(G,\) and \(H\) are given in Appendix A. To better facilitate the expression, \(\overline{w}_{0},\)\(\overline{x},\)\(\overline{t},\) and \(\overline{q}\) are replaced with \(w_{0},\)\(x,\)\(t,\) and \(q,\) respectively. Based on Galerkin discretization [8], \(w_{0}\) can be given as follows:  
\[w_{0}=\sum_{n=1}^{\infty}\varphi_{n}(x)w_{n,1}(t). \tag{16}\]  
Considering the cantilever boundary conditions, \(\varphi_{n}(x)\) is given as follows:  
\[\varphi_{n}(x)=[\text{ch}(\lambda_{n}x)-\cos(\lambda_{n}x)]-\frac{(\text{ch} \lambda_{n}+\cos\lambda_{n})}{(\text{sh}\lambda_{n}+\text{sin}\lambda_{n})}[ \text{sh}(\lambda_{n}x)-\sin(\lambda_{n}x)], \tag{17}\]  
where \(\lambda_{1}=1.875\) and \(\lambda_{2}=4.694\) are given based on second-order Galerkin discretization. For brevity, \(\varphi_{n}(x)\) and \(w_{n,1}(t)\) are replaced with \(\varphi_{n}\) and \(w_{n,1}\) for the rest of the study. Following second-order Galerkin discretization, the response \(w_{\text{P}}\) at \(x_{\text{P}}=0.75\) is given below:  
\[w_{\text{P}}=\sum_{n=1}^{2}\varphi_{n}w_{n,1}=\varphi_{1}w_{1,1}+\varphi_{2}w_{ 2,1}=1.315382461w_{1,1}+0.27008056w_{2,1}, \tag{18}\]  
where \(w_{1,1}\) and \(w_{2,1}\) are given in Appendix A.