## 2 Governing Equations and Boundary Conditions
\begin{cases}\text{Axial force: }F=-EA(u_{x}^{\prime})+EAz_{c}\theta_{z}^{ \prime}+EAy_{c}\theta_{y}^{\prime},\\ \text{Shear force in y-direction: }S_{y}=P(u_{y}^{\prime}-z_{c}\psi^{\prime})-k_{y}AG(u_{y}^{ \prime}-\theta_{y}),\\ \text{Shear force in z-direction: }S_{z}=P(u_{z}^{\prime}+y_{c}\psi^{\prime})-k_{z}AG(u_{z}^{ \prime}-\theta_{z}),\\ \text{Torque: }T=P\bigg{(}\frac{l_{s}}{m}\psi^{\prime}-z_{c}u_{y}^{\prime}+y_{ c}u_{z}^{\prime}\bigg{)}+E\Gamma\psi^{\prime\prime\prime}-GJ\psi^{\prime}-K_{y} \theta_{y}^{\prime}-K_{z}\theta_{z}^{\prime}-\rho\Gamma\ddot{\psi}^{\prime},\\ \text{Bending moment in y-direction: }M_{y}=-EI_{z}\theta_{y}^{\prime}-EI_{z}\theta_{z}^{ \prime}-K_{y}\psi^{\prime}+EAy_{c}u_{x}^{\prime},\\ \text{Bending moment in z-direction: }M_{z}=-EI_{y}\theta_{z}^{\prime}-EI_{z}\theta_{y}^{ \prime}-K_{z}\psi^{\prime}+EAz_{c}u_{x}^{\prime},\\ \text{Bimoment: }B=-E\Gamma\psi^{\prime\prime}.\end{cases}  \tag{16}\]  
Then, the governing equations of motion are introduced as follows. \[ \begin{cases}f_{x}(x,t)=-EA\frac{\partial^{2}u_{x}(x,t)}{\partial x^{2}}+EAz_{ c}\frac{\partial^{2}\theta_{z}(x,t)}{\partial x^{2}}+EAy_{c}\frac{\partial^{2} \theta_{y}(x,t)}{\partial x^{2}}\\ \qquad\qquad+\rho A\frac{\partial^{2}u_{x}(x,t)}{\partial t^{2}}-\mu \bigg{[}z_{c}\frac{\partial^{2}\theta_{z}(x,t)}{\partial t^{2}}+y_{c}\frac{ \partial^{2}\theta_{y}(x,t)}{\partial t^{2}}\bigg{]},\\ f_{y}(x,t)=\mu\frac{\partial^{2}u_{y}(x,t)}{\partial t^{2}}-\mu z_{c}\frac{ \partial^{2}\psi(x,t)}{\partial t^{2}}-k_{y}AG\bigg{(}\frac{\partial^{2}u_{y} (x,t)}{\partial x^{2}}-\frac{\partial\theta_{y}(x,t)}{\partial x}\bigg{)}\\ \qquad\qquad+p\bigg{(}\frac{\partial^{2}u_{y}(x,t)}{\partial x^{2}}-z_{c} \frac{\partial^{2}\psi(x,t)}{\partial x^{2}}\bigg{)},\\ f_{z}(x,t)=\mu\frac{\partial^{2}u_{z}(x,t)}{\partial t^{2}}+\mu y_{c}\frac{ \partial^{2}\psi(x,t)}{\partial t^{2}}-k_{z}AG\bigg{(}\frac{\partial^{2}u_{z} (x,t)}{\partial x^{2}}-\frac{\partial\theta_{z}(x,t)}{\partial x}\bigg{)}\\ \qquad\qquad+p\bigg{(}\frac{\partial^{2}u_{z}(x,t)}{\partial x^{2}}+y_{c} \frac{\partial^{2}\psi(x,t)}{\partial x^{2}}\bigg{)}.\end{cases}  \tag{17}\]  
Moment equilibrium equations:  
\[ \begin{cases}m_{y}(x,t)=\rho l_{y}\frac{\partial^{2}\theta_{z}(x,t)}{\partial t ^{2}}-EI_{y}\frac{\partial^{2}\theta_{z}(x,t)}{\partial x^{2}}-EI_{y}\frac{ \partial^{2}\theta_{y}(x,t)}{\partial x^{2}}+EAz_{c}\frac{\partial^{2}u_{x}(x, t)}{\partial x^{2}}\\ \qquad\qquad-\mu z_{c}\frac{\partial^{2}u_{x}(x,t)}{\partial t^{2}}+\rho l_{y} \frac{\partial^{2}\theta_{y}(x,t)}{\partial t^{2}}-k_{z}AG\bigg{(}\frac{ \partial u_{z}(x,t)}{\partial x}-\theta_{z}(x,t)\bigg{)}-K_{z}\frac{\partial^{2 }\psi(x,t)}{\partial x^{2}},\\ m_{z}(x,t)=\rho l_{z}\frac{\partial^{2}\theta_{y}(x,t)}{\partial t^{2}}-EI_{z} \frac{\partial^{2}\theta_{y}(x,t)}{\partial x^{2}}-EI_{y}\frac{\partial^{2} \theta_{z}(x,t)}{\partial x^{2}}+EAy_{c}\frac{\partial^{2}u_{x}(x,t)}{\partial x ^{2}}\\ \qquad\qquad-\mu y_{c}\frac{\partial^{2}u_{x}(x,t)}{\partial t^{2}}-\rho l_{y} \frac{\partial^{2}\theta_{z}(x,t)}{\partial t^{2}}-k_{y}AG\bigg{(}\frac{ \partial u_{y}(x,t)}{\partial x}-\theta_{y}(x,t)\bigg{)}-K_{y}\frac{\partial^{2 }\psi(x,t)}{\partial x^{2}},\\ m(x,t)=GJ\frac{\partial^{2}\psi(x,t)}{\partial x^{2}}+\rho\Gamma\frac{ \partial^{4}\psi(x,t)}{\partial x^{2}\partial t^{2}}-P\bigg{(}\frac{l_{s}}{\mu }\frac{\partial^{2}\psi(x,t)}{\partial x^{2}}-z_{c}\frac{\partial^{2}u_{y}(x, t)}{\partial x^{2}}+y_{c}\frac{\partial^{2}u_{z}(x,t)}{\partial x^{2}}\bigg{)}\\ \qquad\qquad-\mu y_{c}\frac{\partial^{2}u_{z}(x,t)}{\partial t^{2}}+\mu z_{c} \frac{\partial^{2}u_{y}(x,t)}{\partial t^{2}}-l_{s}\frac{\partial^{2}\psi(x,t)} {\partial t^{2}}-EI\Gamma\frac{\partial^{4}\psi(x,t)}{\partial x^{4}}\\ \qquad\qquad+K_{y}\frac{\partial^{2}\theta_{y}(x,t)}{\partial x^{2}}+K_{z} \frac{\partial^{2}\theta_{z}(x,t)}{\partial x^{2}}.\end{cases}  \tag{18}\]Considering the damping term, the equations of motion are further modified as  
\[