# Chaotic Vibration Prediction of a Laminated Composite Cantilever Beam
## 1 Introduction  
Composite materials offer unique advantages, including high strength, light weight, and excellent fatigue resistance. In recent years, the application of composite materials in modern engineering has become increasingly widespread, particularly in high-performance structures such as aerospace [1, 2], transportation [3], and civil engineering [4]. Because of their internal heterogeneity and complex structure, composites often exhibit pronounced nonlinear dynamic characteristics in practical applications. Under complex external loads, these materials frequently display chaotic vibrations as well as other nonlinear dynamic phenomena. As the complexity of engineering structures continues to increase, traditional theoretical and numerical methods encounter significant limitations in solving complex nonlinear dynamic problems. The rapid development of deep learning in recent years has introduced innovative approaches for tackling complex predictive challenges. Notably, deep learning models such as recurrent neural networks (RNNs), long short-term memory (LSTM), and convolutional neural networks (CNNs) have shown remarkable capability in predicting nonlinear dynamic systems. Currently, deep learning methods have been widely applied in different fields including performance prediction of diverse engineering structures [5] and weather forecasting [6, 7], since data-driven models effectively predict the complex behavior of nonlinear systems. However, research on predicting chaotic vibrations of laminated composite cantilever beams remains unexplored. Composite structures are widely applied in engineering and the emerging field of energy harvesting, and their nonlinear dynamic behavior has been studied extensively over the last decades. Early in the 2000s, [PERSON] et al. [8] modeled a cantilever structure as an Euler-Bernoulli beam and verified that nonlinear nonplanar vibrations of the cantilever beam can exhibit chaotic vibrations under conditions of harmonic axial excitation and lateral excitation in the free segment. [PERSON] et al. [9] performed a theoretical analysis of large deflections of composite cantilever beams based on an Euler-Bernoulli beam and validated their findings through experiments. [PERSON] et al. [10] proposed a frequency-variable structure for piezoelectric vibration energy harvesting based on nonlinear magnetic force and piecewise linear force and conducted a detailed discussion on the factors affecting the proposed vibration energy harvesting device through theoretical and experimental analysis. [PERSON] et al. [11] applied first-order shear deformation theory to study the influence of different parameters on the nonlinear dynamic behavior of low-content graphene-reinforced composite plates. [PERSON] et al. [12] derived the multi-dimensional dynamic equation of graphene-reinforced composite cantilever structures through first-order deformation theory and analyzed the influence of nonlinear terms on the dynamic behavior of different vibration modes. [PERSON] et al. [13] studied the nonlinear behavior of composite cantilever beams subject to concentrated loads using Timoshenko beam theory and discussed the influence of temperature and the number of composite layers on the deformation of cantilever beams. [PERSON] et al. [14] employed third-order shear deformation theory and introduced the dynamic stiffness method to investigate the influence of beam length on the free vibrations of laminated composite cantilever beams. [PERSON] et al. [15] proposed a refined third-order shear deformation theory for nonlinear vibrations of cantilever beams and conducted large-amplitude vibration experiments and numerical simulations on self-healing cantilever beams. [PERSON] et al. [16] established a two-dimensional nonlinear dynamic system of a symmetric cross-ply laminated composite cantilever structure, discussing the effects of different types of excitations on the nonlinear vibrations of the structure and discovering the presence of chaotic vibrations in the established system.