## 3 Numerical Examples  
The free harmonic vibrations are defined by the coupled homogeneous Equations (17) and (18). The solution may be expressed in the form as follows:  
\[ \begin{split} u_{x}(x,t)=V_{x}(x)\sin(\omega t),\quad u_{y}(x,t)=V_ {y}(x)\sin(\omega t),\quad u_{z}(x,t)=V_{z}(x)\sin(\omega t),\\ \theta_{y}(x,t)=\Theta_{y}(x)\sin(\omega t),\quad\theta_{z}(x,t)=\Theta_{z}(x) \sin(\omega t),\quad\psi(x,t)=\Psi(x)\sin(\omega t).\end{split}  \tag{21}\]where \(\omega\) is the radian frequency and \(V_{x}\), \(V_{y}\), \(V_{z}\), \(\Theta_{y}\), \(\Theta_{z}\), and \(\Psi\) are the amplitudes of the sinusoidally varying axial motion along the \(x\) direction, the flexural translation in the \(y\) and \(z\) directions, the flexural rotation about the \(y\) and \(z\) axes, and the torsional rotation, respectively. Substituting Equation (21) into homogeneous Equations (17) and (18) yields  
\[EA\frac{d^{2}V_{x}(x)}{dx^{2}}-EAz_{c}\frac{d^{2}\Theta_{z}(x)}{ dx^{2}}-EAy_{c}\frac{d^{2}\Theta_{y}(x)}{dx^{2}}\] \[+\rho A\omega^{2}V_{x}(x)+\mu\left[-z_{c}\omega^{2}\Theta_{z}(x)- y_{c}\omega^{2}\Theta_{y}(x)\right]=0,\] \[-\rho I_{z}\omega^{2}\Theta_{y}(x)-EI_{z}\frac{d^{2}\Theta_{y}(x)} {dx^{2}}-EI_{xy}\frac{d^{2}\Theta_{z}(x)}{dx^{2}}+EAy_{c}\frac{d^{2}V_{x}(x)}{ dx^{2}}\] \[+\mu y_{c}\omega^{2}V_{x}(x)+\rho I_{xy}\omega^{2}\Theta_{z}(x)- k_{y}AG\left(\frac{dV_{y}(x)}{dx}-\Theta_{y}(x)\right)-K_{y}\frac{d^{2}\Psi(x)}{ dx^{2}}=0,\] \[-\rho I_{y}\omega^{2}\Theta_{z}(x)-EI_{y}\frac{d^{2}\Theta_{z}(x )}{dx^{2}}-EI_{xy}\frac{d^{2}\Theta_{y}(x)}{dx^{2}}+EAz_{c}\frac{d^{2}V_{x}(x)} {dx^{2}}\] \[+\mu z_{c}\omega^{2}V_{x}(x)-\rho I_{xy}\omega^{2}\Theta_{y}(x)- k_{z}AG\left(\frac{dV_{z}(x)}{dx}-\Theta_{z}(x)\right)-K_{z}\frac{d^{2}\Psi(x)}{ dx^{2}}=0, \tag{22}\] \[-\mu\omega^{2}V_{y}(x)+\mu\omega^{2}z_{c}\Psi(x)-k_{y}AG\left( \frac{d^{2}V_{y}(x)}{dx^{2}}-\frac{d\Theta_{y}(x)}{dx}\right)+P\left(\frac{d^ {2}V_{y}(x)}{dx^{2}}-z_{c}\frac{d^{2}\Psi(x)}{dx^{2}}\right)=0,\] \[-\mu\omega^{2}V_{z}(x)-\mu\omega^{2}y_{c}\Psi(x)-k_{z}AG\left( \frac{d^{2}V_{z}(x)}{dx^{2}}-\frac{d\Theta_{z}(x)}{dx}\right)+P\left(\frac{d^ {2}V_{z}(x)}{dx^{2}}+y_{c}\frac{d^{2}\Psi(x)}{dx^{2}}\right)=0,\] \[G/\frac{d^{2}\Psi(x)}{dx^{2}}-\rho\Gamma\omega^{2}\frac{d^{2} \Psi(x)}{dx^{2}}-P\left(\frac{I_{z}}{\mu}\frac{d^{2}\Psi(x)}{dx^{2}}-z_{c} \frac{d^{2}V_{y}(x)}{dx^{2}}+y_{c}\frac{d^{2}V_{z}(x)}{dx^{2}}\right)\] \[+\mu\omega^{2}y_{c}V_{z}(x)-\mu\omega^{2}z_{c}V_{y}(x)+I_{s} \omega^{2}\Psi(x)-E\Gamma\frac{d^{4}\Psi(x)}{dx^{4}}+K_{y}\frac{d^{2}\Theta_{ y}(x)}{dx^{2}}+K_{z}\frac{d^{2}\Theta_{z}(x)}{dx^{2}}=0.\]  
In general, it is not possible to derive the closed-form solution for an arbitrary type of the boundary conditions. Only when both ends of the beams are simply supported is it possible to determine the natural frequencies and the natural modes. In the case of a beam with simply supported ends (fork supports at each end which prevent rotation and can warp freely), the end conditions are  
\[+\rho A\omega^{2}V_{x}(x)+\mu\left[-z_{c}\omega^{2}\Theta_{z}(x)- y_{c}\omega^{2}\Theta_{y}(x)\right]=0,\]0  
as provided in [PERSON]'s research [13]. The boundary conditions for a simply supported beam are that one end is fixed while the other end is free. Using Equation (15),  
\[+\rho A\omega^{2}V_{x}(x)+\mu\left[-z_{c}\omega^{2}\Theta_{z}(x)- y_{c}\omega^{2}\Theta_{y}(x)\right]=0,\]1  
and the axial force is  
\[+\rho A\omega^{2}V_{x}(x)+\mu\left[-z_{c}\omega^{2}\Theta_{z}(x)- y_{c}\omega^{2}\Theta_{y}(x)\right]=0,\]2  
It is known that when one end is fixed, the boundary condition \(\delta u_{x}=0\) holds, which implies that \(V_{x}\) remains constant.