## 2 Governing Equations and Boundary Conditions  
Consider the beam model with coupled bending, torsion, and axial deformation shown in Figure 1. Points SC and MC represent the shear center and the centroid of the beam cross-section, respectively. The distances from the shear center to the \(z\)-axis and \(y\)-axis are \(z_{c}\) and \(y_{c}\), respectively. The cross-section is constant along the beam and is asymmetric. The elastic axis, i.e., the locus of the SCs, is the \(x\) axis, while the mass axis, i.e., the locus of the MCs, is at a distance of \(\sqrt{z_{c}^{2}+y_{c}^{2}}\) from the elastic axis. Both the \(z\)-axis and \(y\)-axis were chosen as principal axes for the beam cross-section. When the beam is subjected to dynamic loads, its response characteristics cannot be accurately estimated using the classical Bernoulli-Euler or Timoshenko beam theory. Therefore, it was necessary to establish a calculation method for the dynamic response of the coupled bending-torsion-axial beam. In this analysis, the effects of axial loads, shear deformation, rotational inertia, and warping stiffness were considered. Let \(u_{z}(x,t)\) and \(u_{y}(x,t)\) be the bending deflections in the \(z\)-direction and \(y\)-direction, and \(\psi(x,t)\) the torsional rotation about the \(x\)-axis of the SCs; let \(u_{x}(x,t)\) be the axial deformation along the \(x\)-axis of the beam, \(\theta_{y}(x,t)\) the cross-sectional rotation due to bending in the \(y\)-direction, and \(\theta_{z}(x,t)\) the cross-sectional rotation due to bending in the \(z\)-direction. A constant compression axial force \(P\) is assumed to act through the centroid of the cross-section of the thin-walled beam. \(P\) can be positive or negative, so that tension is included. The excitations that act on the thin-walled beam are represented by the forces \(f_{x}(x,t)\)\(f_{y}(x,t)\), and \(f_{z}(x,t)\) per unit length that is parallel to the \(x\)-axis, the \(y\)-axis, and the \(z\)-axis and applied to the shear center, together with a torque \(m(x,t)\) about the \(x\)-axis, bending moment \(m_{y}(x,t)\) about the \(y\)-axis, and bending moment \(m_{z}(x,t)\) about the \(z\)-axis per unit length, respectively (Figure 1). In the context of small torsion, the classical relationships used in [PERSON]'s model [14; 16] are straightforward. Firstly, the deformations in three directions are represented by  
\[ \begin{split} u_{X}&=u_{x}-z\theta_{z}-y\theta_{y} -\omega\psi^{\prime}\\ u_{Y}&=u_{y}-z\psi\\ u_{Z}&=u_{z}+y\psi\end{split}  \tag{1}\]  
The component deformations for the defined displacement field [13] are given as  
\[ \begin{split}\varepsilon_{xx}&=\frac{\partial u_{X} }{\partial x}=u_{x}^{\prime}-z\theta_{z}^{\prime}-y\theta_{y}^{\prime}- \omega\psi^{\prime\prime}\\ \varepsilon_{xy}&=\frac{\partial u_{Y}}{\partial x }+\frac{\partial u_{X}}{\partial z}=u_{y}^{\prime}-z\psi^{\prime}-\theta_{y} -\psi^{\prime}\frac{\partial\omega}{\partial z}\\ \varepsilon_{xz}&=\frac{\partial u_{Z}}{\partial x }+\frac{\partial u_{X}}{\partial z}=u_{z}^{\prime}+y\psi^{\prime}-\theta_{z} -\psi^{\prime}\frac{\partial\omega}{\partial z}\end{split}  \tag{2}\]  
Furthermore, by integrating the coupled vibration theory of Timoshenko beams with axial vibrations, as outlined by [PERSON] and [PERSON] [19] and [PERSON] et al. [20], and accounting for shear effects, we propose a novel beam model. In this model, the conditions \(u_{y}^{\prime}-\theta_{y}^{\prime}\
eq 0\) and \(u_{z}^{\prime}-\theta_{z}^{\prime}\
eq 0\) hold, enabling it to accurately describe the coupled behavior of bidirectional bending, torsion, and axial vibrations, while also incorporating the influence of axial loads. For a linear elastic material, the stress-strain relationship is defined by [PERSON]'s law:  
\[