# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## 4 Numerical Simulation
### Prediction Corresponding to Different Initial Conditions  
Chaotic systems are known for their extreme sensitivity to initial conditions. Small disturbances in the initial state may lead to completely different trajectories. It is crucial to test the robustness and generalization ability of RNN for predicting chaotic vibrations under changes in initial conditions. Therefore, in this section, the predictive ability of the RNN model for chaotic vibrations is further validated by changing the initial conditions. With \(Q_{1}=4500\) Pa and \(Q_{2}=41\pi\), chaotic vibrations were identified, and the previously trained RNN model was used to predict the system responses under different initial conditions by individually changing one component of the initial conditions, where the initial conditions were \(w_{1,1}(0)=0\), \(w_{1,2}(0)=0\), \(w_{2,1}(0)=0\), and \(w_{2,2}(0)=0\). (a) \(w_{1,1}(0)=0.001\), \(w_{1,2}(0)=0\), \(w_{2,1}(0)=0\), \(w_{2,2}(0)=0\)  
When \(w_{1,1}(0)=0.001\), the vibration of the laminated composite cantilever beam is shown in Figure 29. Chaotic vibration is observed within the time interval \(t\in[0,140]\), and the maximum displacement amplitude reaches approximately 2.0. Figures 30 and 31 present the 2D phase diagram and the first two modal responses, respectively, providing a detailed illustration of the chaotic vibration. Figure 32 illustrates the comparison between the real value and predicted value of the displacement and velocity. A high level of agreement can be observed between the two curves. In addition, Figures 33 and 34 present the 2D and 3D phase diagrams of the chaotic vibration, respectively. The results indicate that the RNN model can accurately predict the chaotic vibration of the laminated composite cantilever beam, even in the presence of initial condition perturbations. Figure 28: Testing result for 3D phase diagram, \(t\in[126,140]\)  
Figure 27: Testing result for 2D phase diagram, \(t\in[126,140]\)Figure 31: The vibrations for the first mode vibration and the second mode vibration: (**a**) \(w_{1,1}\); (**b**) \(w_{2,1}\). Figure 29: The vibration of the cantilever beam at \(x_{\text{P}}\) for \(w_{1,1}(0)=0.01\). (b) \(w_{1,1}(0)=0\), \(w_{1,2}(0)=0.001\), \(w_{2,1}(0)=0\), \(w_{2,2}(0)=0\)  
When \(w_{1,2}(0)=0.001\), the vibration of the laminated composite cantilever beam is shown in Figure 35. Within the time interval \(t\in[0,140]\), the laminated composite cantilever beam exhibits chaotic vibration, with a maximum amplitude of approximately \(2.1\). Figures 36 and 37 display the 2D phase diagram and the first two vibration modes, providing a detailed characterization of the chaotic vibration. Figure 33: Testing result for 2D phase diagram, \(t\in[126,140]\)  
Figure 32: Testing result for \(w_{1,1}(0)=0.001\): (a) displacement comparison; (b) velocity comparison. Figure 38 presents the comparison between the real and predicted displacement and velocity responses of the laminated composite cantilever beam. As shown in Figure 38, the two curves remain highly consistent. Figures 39 and 40 display the 2D and 3D phase diagrams, providing comparative views of the real value and predicted value from different spatial perspectives. These results confirm that the RNN model is capable of accurately predicting the chaotic vibration of the laminated composite cantilever beam. Figure 34: Testing result for 3D phase diagram, \(t\in[126,140]\)  
Figure 35: The vibration of the cantilever beam at \(x_{\text{P}}\) for \(w_{1,2}(0)=0.001\). Figure 28:0 2D phase diagram at \(x_{\text{P}}\) for \(w_{1,2}(0)=0.001\). Figure 28:1 The vibrations for the first mode vibration and the second mode vibration: (**a**) \(w_{1,1}\); (**b**) \(w_{2,1}\). Figure 28:2 Testing result for \(w_{1,2}(0)=0.001\): (**a**) displacement comparison; (**b**) velocity comparison. Figure 44 illustrates the comparison between the real and predicted displacement and velocity responses. As observed in Figure 44, the predicted curves closely match the real curves.