# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## 2 Beam Model Establishment  
To demonstrate the feasibility of the RNN data-driven model for predicting chaotic vibrations in a multi-dimensional nonlinear dynamic system of a laminated composite cantilever beam, time-series data of chaotic vibrations are generated through numerical simulations. In this section, the multi-dimensional nonlinear dynamic model of a cantilever beam is established. In Figure 1, a three-layer laminated composite cantilever beam is shown, and a Cartesian coordinate system \(oxyz\) is applied to describe the deformation of the beam. The cross-section of the structure is uniformly rectangular; \(l\), \(b\), and \(h\) represents the total length, breadth, and height of the beam, and each layer of the beam features the same length, breadth, and height; the external excitation, denoted by \(q\), is evenly applied to the beam. Figure 1: The structure of a three-layered laminated composite cantilever beam. In the case of no deformation, the location of any point on the structure is as follows:  
\[\mathbf{r}=x\mathbf{i}+z\mathbf{k}, \tag{1}\]  
where \(\mathbf{i}\) and \(\mathbf{k}\) are the unit vectors along \(x\) and \(z\), respectively. Following [PERSON]'s third-order shear theory [39], the displacement of any point on the beam after deformation is as follows:  
\[\mathbf{R}=\bigg{(}x+u_{0}+z\mathcal{O}_{x}-z^{3}c_{1}\bigg{(}\mathcal{O}_{x}+ \frac{\partial w_{0}}{\partial x}\bigg{)}\bigg{)}\mathbf{i}+(z+w_{0})\mathbf{ k}, \tag{2}\]  
where \(c_{1}=4/\big{(}3h^{2}\big{)}\); \(u_{0}\) and \(w_{0}\) are the displacements of a point on the middle plane of the structure, and \(\mathcal{O}_{x}\) is the curvature according to [PERSON]'s third-order shear theory. According to Equations (1) and (2), the structure's kinetic energy \(T\)[40] is obtained as follows:  
\[T=\int_{V}\frac{1}{2}\rho\bigg{(}\frac{d\mathbf{R}}{dt}\bigg{)}^{2}dV, \tag{3}\]  
where \(\rho\) is the density of the structure. Based on the von Karman type equation [41] and Equation (2), the strains of the beam are derived below:  
\[\varepsilon_{11}=\frac{\partial u_{0}}{\partial x}+\frac{1}{2}\bigg{(}\frac{ \partial w_{0}}{\partial x}\bigg{)}^{2}+z\frac{\partial\mathcal{O}_{x}}{ \partial x}-c_{1}z^{3}\bigg{(}\frac{\partial\mathcal{O}_{x}}{\partial x}+ \frac{\partial^{2}w_{0}}{\partial x^{2}}\bigg{)}, \tag{4}\]  
\[\varepsilon_{13}=\Big{(}1-3c_{1}z^{2}\Big{)}\bigg{(}\mathcal{O}_{x}+\frac{ \partial w_{0}}{\partial x}\bigg{)}, \tag{5}\]  
where \(\varepsilon_{11}\) and \(\varepsilon_{13}\) are the strains along \(x\) and \(z\), respectively. From Equations (4) and (5), the potential energy of the beam \(U\) is obtained as follows:  
\[U=\int_{V}\frac{1}{2}(Q_{11}\varepsilon_{11}\varepsilon_{11}+Q_{13} \varepsilon_{13}\varepsilon_{13})dV, \tag{6}\]  
where \(Q_{11}\) and \(Q_{13}\) are the elastic coefficients along \(x\) and \(z\), respectively. The virtual work \(W\), due to the damping force and \(q\), is given below:  
\[W=-b\int_{0}^{l}c\frac{dw_{0}}{dt}w_{0}dx+b\int_{0}^{l}qw_{0}dx, \tag{7}\]  
where  
\[q=Q_{1}\text{sin}(Q_{2}t), \tag{8}\]  
\(c\) is the damping coefficient, and \(Q_{1}\) and \(Q_{2}\) are the magnitude and angular frequency of the excitation. Following [PERSON]'s principle [42], it is obtained that  
\[\int_{t_{1}}^{t_{2}}(\delta L+\delta W)dt=0, \tag{9}\]  
where \(L\) is given as follows:  
\[\mathbf{r}=x\mathbf{i}+z\mathbf{k}, \tag{1}\]0  
With the introduction of Equations (3), (6) and (7) into Equation (9), the equations of motion of the beam are derived by following the assumption of an orthogonally symmetric laminated beam and presented in the Appendix A, and then, \(u_{0}\) and \(\mathcal{O}_{x}\) are derived by following the previous study [2]:  
\[\mathbf{r}=x\mathbf{i}+z\mathbf{k}, \tag{1}\]1  
\[\mathbf{r}=x\mathbf{i}+z\mathbf{k}, \tag{1}\]2  
With the introduction of Equations (11) and (12) into Equation (A1), it is obtained that  
\[