## 1 Introduction  
Understanding the dynamic behavior of thin-walled beams is fundamental in structural engineering to ensure safety and functionality. Thin-walled beam structures are extensively used across various engineering fields, including bridges, aircraft wings, high-rise buildings, and frame structures. As modern engineering demands advance, these structures frequently encounter complex dynamic loads, including axial loads, torsion, and bending. Traditional beam theories, such as Bernoulli-Euler and Timoshenko, offer reliable predictions for simpler scenarios but often fall short when addressing the coupled behavior of bending, torsion, and axial vibration under axial loads. Over the years, numerous scholars have explored the coupled bending-torsion vibration theory of beams. For instance, [PERSON] and [PERSON] [1] and [PERSON] and [PERSON] [2] proposed theories for Timoshenko beams under axial loads. [PERSON] et al. [3, 4] further studied these theories considering the coupling stiffness of bending and torsion. However, earlier theories did not account for warping effects. To address this, [PERSON] and [PERSON] [5] included warping effects in their studies. [PERSON] and [PERSON] [6] investigated the coupled vibration theory of the bidirectional bending and torsion of beams. [PERSON] et al. [7, 8, 9, 10, 11] examined the coupled bending-torsion vibration problems of Bernoulli-Euler and Timoshenko beams under axial loads and the stochastic vibration and dynamic response of coupled bending-torsion. [PERSON] et al. [12] also studied the coupled vibration theory of the bidirectional bending and torsion of Bernoulli-Euler beams under axial loads. [PERSON] [13] studied the coupled vibration theory of the bidirectional bending and torsion of Timoshenko beams, and later they studied the coupled vibration theory under axial loads as well [14]. [PERSON] et al. [15] researched the coupled dynamic response of beams under axial loads and the relationship between axial loads and natural frequencies. [PERSON] et al. [16] studied the coupled bending-torsion vibration theory of Vlasov beams. [PERSON] and [PERSON] [17] provided a theoretical framework for the coupled dynamic response of the bidirectional bending and torsion of beams. [PERSON] et al. [18] gave analytical solutions for the coupled bending-torsion vibration of simply supported beams. Despite these advances, most studies have focused primarily on coupled bending-torsion vibration, with limited attention to axial vibration. [PERSON] and [PERSON] [19] proposed a coupled vibration theory of axial and bending vibrations, and then applied this theory to Timoshenko beams [20]. Although traditional beam theories have provided invaluable insight into simplified engineering problems, they are increasingly inadequate for modeling the complex dynamic behavior of modern structures subjected to combined axial, torsional, and bending loads. This paper introduces an advanced beam model that integrates the [PERSON] and [PERSON] theories, offering a comprehensive framework capable of capturing the full spectrum of coupled bending, torsion, and axial vibrations under axial loads. By addressing the limitations of earlier approaches, this model delivers unprecedented accuracy in predicting the dynamic responses of thin-walled beams, offering a powerful analytical tool for the design and safety assessment of engineering structures under multifaceted loading conditions. Utilizing [PERSON]'s principle, we derived a comprehensive set of dynamic equations and boundary conditions, incorporating damping effects. Our findings significantly enhance the understanding of the dynamic behavior of thin-walled beams under complex loads, providing a robust analytical tool for structural design and safety assessment. This study also explores the impact of various parameters, such as axial loads, on the natural frequencies of the beams, offering practical guidance for engineering design.