# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## 4 Numerical Simulation
### Prediction Corresponding to Different Initial Conditions
In addition, both a 2D phase diagram provided in Figure 45 and a 3D phase diagram provided in Figure 46 are presented to compare the predicted results with the real data. The results show that the RNN model can accurately predict the chaotic vibration of the laminated composite cantilever beam even under perturbed initial conditions. Figure 28:3 2D phase diagram at \(x_{\text{p}}\) for \(w_{2,1}(0)=0.001\). Figure 28:4 The vibrations for the first mode vibration and the second mode vibration: (**a**) \(w_{1,1}\); (**b**) \(w_{2,1}\). (d) \(w_{1,1}(0)=0\), \(w_{1,2}(0)=0\), \(w_{2,1}(0)=0\), \(w_{2,2}(0)=0.001\)  
When \(w_{2,2}(0)=0.001\), the laminated composite cantilever beam exhibits chaotic vibration, as shown in Figure 28:5 Within the dimensionless time interval \(t\in[0,\,140]\), the maximum amplitude reaches approximately \(2.0\). Figures 48 and 49 display the 2D phase diagram and the first two vibration modes of the chaotic vibration, respectively, providing a more detailed illustration of the chaotic vibration of the laminated composite cantilever beam. Figure 28:6 Testing result for \(w_{2,1}(0)=0.001\): (a) displacement comparison; (b) velocity comparison. Figure 28:7 Testing result for 2D phase diagram, \(t\in[126,140]\). Figure 50 presents the comparison between the real and predicted displacement and velocity responses. A comparison of the results reveals a strong agreement between the real and predicted data, demonstrating the high accuracy of the RNN model in capturing both the maximum and minimum values of the chaotic vibration amplitude. Figures 51 and 52 illustrate the comparisons in the 2D and 3D phase diagrams, indicating that the RNN model demonstrates high accuracy in capturing the chaotic vibration of the vibration amplitude. The results are shown in the left panel of Figure 28:8  
Figure 28:9 Testing result for 3D phase diagram, \(t\in[126,140]\)  
Figure 27:0 The vibration of the cantilever beam at \(x_{\text{P}}\) for \(w_{2,2}(0)=0.001\). Table 1 presents a brief summary of the prediction errors of the RNN data-driven model for the above four chaotic vibrations in terms of displacement and velocity. By examining the data in Table 1, it can be observed that the RNN data-driven model consistently achieves small MAE and RMSE values for both displacement and velocity. Hence, these results indicate that RNN demonstrates strong prediction performance for displacement and velocity in chaotic vibrations of a laminated composite beam. Figure 27:1 Testing result for 2D phase diagram, \(t\in[126,140]\). Figure 27:2 Testing result for 3D phase diagram, \(t\in[126,140]\)  
Figure 27:3 Testing result for \(w_{2,2}(0)=0.001\): (**a**) displacement comparison; (**b**) velocity comparison. Table 2 summarizes the prediction performance of the RNN model under different initial conditions, with the external excitation given as 4500 Pa \(\sin(41\pi t)\). In each case, only one component of the initial state is varied, while the others are set to 0. As shown in Table 2, the RNN model maintains low prediction errors across all conditions, with MAE ranging from 0.013 to 0.022 and RMSE ranging from 0.021 to 0.038. These results further demonstrate the robustness and accuracy of the RNN model in predicting the chaotic vibration of the laminated composite cantilever beam. The RNN model does not rely on explicit physical governing equations. Its high accuracy in predicting the chaotic vibrations of the laminated composite cantilever beam indicates that it can effectively capture the intrinsic nonlinear dynamic characteristics of the system. During training, the RNN model receives sequences of displacement and velocity over the previous 10 time steps. This multivariate time-series input contains key informational features of chaotic systems, such as amplitude modulation, phase variation, and nonlinear coupling between state variables. The recurrent structure of the RNN allows it to update its internal state at each time step and gradually accumulate the system's evolution history.