## 2 The Geometrically Exact Beam Model
### Beam Dynamics and Its Linearization  
The weak form of dynamic equilibrium equation can be expressed as:  
\[\delta\mathcal{W}_{\text{int}}-\delta\mathcal{W}_{\text{ext}}+\delta\mathcal{W}_ {\text{ine}}=0 \tag{29}\]  
with the internal virtual work \(\delta\mathcal{W}_{\text{int}}\), external virtual work \(\delta\mathcal{W}_{\text{ext}}\), and inertial virtual work \(\delta\mathcal{W}_{\text{ine}}\). Denoting \(\rho\) as the mass density of the material, the inertial virtual work \(\delta\mathcal{W}_{\text{ine}}\) is given by:  
\[\delta\mathcal{W}_{\text{ine}}=\iint\rho\delta\mathbf{x}_{p}^{\text{T}}\tilde{ \mathbf{x}}_{\rho}dAd\xi_{1}=A_{\rho}\int_{l}\delta\mathbf{h}^{\text{T}}\left[ \begin{array}{c}\tilde{\mathbf{x}}\\ \mathbf{J}\mathbf{\alpha}+\widetilde{\mathbf{\Omega}}(\mathbf{J}\mathbf{\Omega})\end{array} \right]\text{d}\xi_{1} \tag{30}\]  
where \(A_{\rho}=A\rho\) is the density of the cross-section. \(\mathbf{\alpha}\) and \(\mathbf{\Omega}\) are the angular acceleration and angular velocity of beam cross-section in local frame, respectively. Besides, the notation \(\mathbf{J}\) denotes the cross-sectional moment of inertia with relative to the centroid, giving:  
\[\mathbf{J}=\left[\begin{array}{ccc}I_{0}&0&0\\ 0&I_{33}&-I_{23}\\ 0&-I_{23}&I_{22}\end{array}\right] \tag{31}\]  
Then the generalized mass matrix can be derived by linearization of the inertial virtual work \(\delta\mathcal{W}_{\text{ine}}\), giving:  
\[\Delta(\delta\mathcal{W}_{\text{ine}})=A_{\rho}\int_{l}\delta\mathbf{h}^{\text {T}}\left[\begin{array}{cc}\mathbf{I}_{3}\left(\frac{\text{d}^{2}}{\text{d} \tilde{r}^{2}}\right)&0_{3}\\ \mathbf{0}_{3}&\mathbf{T}^{\text{T}}\left[\begin{array}{c}\left(\mathbf{J} \tilde{\mathbf{\alpha}}-\widetilde{\left(\mathbf{J}\mathbf{\Omega}\right)}\tilde{\mathbf{ \Omega}}+\tilde{\mathbf{\Omega}}\tilde{\mathbf{\Omega}}\right)+\\ \left(\mathbf{J}\tilde{\mathbf{\Omega}}-\widetilde{\left(\mathbf{J}\mathbf{\Omega} \right)}+\tilde{\mathbf{\Omega}}\right)\right)\left(\frac{\text{d}}{\text{d}\tilde {r}}\right)+\mathbf{J}\left(\frac{\text{d}^{2}}{\text{d}\tilde{r}^{2}}\right) \end{array}\right]\mathbf{T}\right]\Delta\mathbf{h}^{\text{*}}\text{d}\xi_{1} \tag{32}\]  
Since the virtual rotational vector \(\delta\mathbf{\varphi}\) is defined in the local frame, it is free from the influence of rigid rotation, which greatly reduces the rotational nonlinearity. Consequently, the parts of the stiffness matrix and the generalized mass matrix, which corresponds to the rotation parameters, are invariable under arbitrary rigid motion. Thereby, the number of updates required during the computation process decreases sharply.  
Similarly, the external virtual work can be computed by:  
\[\delta\mathcal{W}_{\text{ext}}=\int\left[\delta\mathbf{x}^{\text{T}}\mathbf{ F}+\delta\mathbf{\varphi}^{\text{T}}\mathbf{T}^{\text{T}}\mathbf{P}\right]\text{d}\xi_{1} +\delta\mathbf{x}(\xi)^{\text{T}}\tilde{\mathbf{F}}+\delta\mathbf{\varphi}(\xi)^{ \text{T}}\mathbf{T}^{\text{T}}\tilde{\mathbf{P}} \tag{33}\]  
where \(\mathbf{F}\) and \(\tilde{\mathbf{F}}\) represent the distributed force and the concentrated force on the centerline of the beam in the inertial frame, respectively. \(\mathbf{P}\) and \(\mathbf{P}\) represent the distributed moment and the concentrated moment on the cross-section of the beam in the local frame, respectively.