# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## 1 Introduction
[30] proposed a new temporal convolutional RNN model and validated its accuracy through different chaotic systems, including a Lorenz system and a Rossler system. In the same year, [PERSON] et al. [31] introduced different levels of noise to chaotic systems, used different prediction models to predict the chaotic systems, and confirmed the robustness of LSTM without teacher forcing in handling complex systems and noise. In 2023, [PERSON] et al. [32] established a five-degree-of-freedom Duffing system and proposed an LSTM encoder-decoder neural network prediction method, proving its capability in predicting chaotic vibrations in discrete systems. In the same year, [PERSON] et al. [33] proposed a robust RNN to reduce prediction errors, which would occur due to noise existing in real-world time series. At the same time, [PERSON] et al. [34] combined the particle swarm optimization (PSO) algorithm with YUKI to propose PSO-YUKI, using a radial basis function (RBF) neural network to predict the double cracking of cantilever carbon fiber-reinforced polymer (CFRP) beams. The results demonstrate that the PSO-YUKI approach has better accuracy and effectiveness in damage prediction compared to the standalone PSO method. In 2024, [PERSON] et al. [35] experimentally demonstrated that near-surface mounted (NSM)-strengthened beams can reduce crack loss and limit frequency changes under bending conditions. Meanwhile, PSO and genetic algorithms (GAs) were used to fine-tune the gradient boosting (GB) model. Data obtained from static tests proved that the improved GB model is more efficient and effective in predicting the reinforced concrete (RC) strain. In 2025, [PERSON] et al. [36] proposed a data-driven model by combining CNN and convolutional LSTM, validating its accuracy and efficiency in predicting chaotic vibrations in a micro-beam system. Recently, [PERSON] et al. [37] used the reptile search algorithm (RSA) to optimize the artificial neural network (ANN) and proposed a hybrid ANN-RSA model. Combining experiments and numerical simulations, they demonstrated the accuracy and computational efficiency of the ANN-RSA. [PERSON] et al. [38] combined analytical models and ANN to propose an algorithm for predicting the deflection of variable cross-section castellated steel beams and verified the correctness of the algorithm through a finite element model. However, so far, no deep learning methods, including RNN, have been applied to predicting the chaotic vibrations of a laminated composite cantilever beam. Traditional numerical simulation methods typically require researchers to possess extensive domain knowledge, as well as the ability to formulate and solve complex differential equations. In contrast, an RNN data-driven model bypasses the need for theoretical modeling by combining representation learning with model training, without relying on explicit physical equations or expert-derived models. The key advantage of RNN lies in its ability to implicitly capture the underlying nonlinear dynamics of complex systems. By learning temporal dependencies directly from historical input-output sequences, RNN offers a promising alternative for predicting chaotic vibrations in cases where physical models are difficult to derive or unavailable. Therefore, it is important to test the feasibility of the RNN data-driven model for predicting chaotic vibrations of a laminated composite cantilever beam. The current structure of this work is organized as follows: In the Section 2, a multi-dimensional nonlinear dynamic system of a laminated composite cantilever beam subjected to periodic excitation is established. In the Section 3, the concept, definition, and modeling process of the RNN data-driven model is introduced. In the Section 4, numerical simulations are conducted using different amplitudes and angular velocities of the evenly distributed load, to evaluate the accuracy of the employed RNN data-driven model in predicting chaotic vibrations of the studied laminated composite cantilever beam.