## Appendix B Matrix Elements in Equation (28)  
The expressions of matrix \(\underset{(6\times 6)}{\mathbf{B}}\) forms in Equation (28):  
\[\underset{(6\times 6)}{\mathbf{B}} =\begin{bmatrix}\underset{(3\times 3)}{\mathbf{B}_{11}}&\underset{(3 \times 3)}{\mathbf{B}_{12}}\\ \underset{(3\times 3)}{\mathbf{B}_{21}}&\underset{(3\times 3)}{\mathbf{B}_{2 3}}\\ \end{bmatrix}\]  
where  
\[\underset{(3\times 3)}{\mathbf{B}_{11}} =\begin{bmatrix}-EA\lambda_{n}^{3}+\rho A\omega^{2}\lambda_{n}&0& 0\\ -EAy_{c}\lambda_{n}^{3}+\rho Ay_{c}\omega^{2}\lambda_{n}&-k_{y}AG\lambda_{n}&0 \\ -EAz_{c}\lambda_{n}^{3}+\rho Az_{c}\omega^{2}\lambda_{n}&0&-k_{z}AG\lambda_{n} \end{bmatrix}\]  
\[\underset{(3\times 3)}{\mathbf{B}_{12}} =\begin{bmatrix}EAy_{c}\lambda_{n}^{3}+\rho Ay_{c}\omega^{2} \lambda_{n}&EAz_{c}\lambda_{n}^{3}+\rho Az_{c}\omega^{2}\lambda_{n}&0\\ -\rho Iz_{c}\omega^{2}\lambda_{n}+EIz_{c}\lambda_{n}^{3}+k_{y}AG\lambda_{n}&EIz _{y}\lambda_{n}^{3}+\rho Iz_{y}\omega^{2}\lambda_{n}&K_{y}\lambda_{n}^{2}\\ EIz_{y}\lambda_{n}^{3}-\rho Iz_{y}\omega^{2}\lambda_{n}&-\rho I_{y}\omega^{2} \lambda_{n}+EIy\lambda_{n}^{3}+k_{z}AG\lambda_{n}&K_{z}\lambda_{n}^{2}\end{bmatrix}\]  
\[\underset{(3\times 3)}{\mathbf{B}_{21}} =\begin{bmatrix}0&-\rho A\omega^{2}+k_{y}AG\lambda_{n}^{2}-P \lambda_{n}^{2}&0\\ 0&0&-\rho A\omega^{2}+k_{z}AG\lambda_{n}^{2}-P\lambda_{n}^{2}\\ 0&-\rho A\omega^{2}z_{c}-Pz_{c}\lambda_{n}^{2}&\rho A\omega^{2}y_{c}+Py_{c} \lambda_{n}^{2}\end{bmatrix}\]  
\[\underset{(3\times 3)}{\mathbf{B}_{22}} =\begin{bmatrix}-k_{y}AG\lambda_{n}^{2}&0&\rho A\omega^{2}z_{c}+ Pz_{c}\lambda_{n}^{2}\\ 0&-k_{z}AG\lambda_{n}^{2}&-\rho A\omega^{2}y_{c}-Py_{c}\lambda_{n}^{2}\\ -K_{y}\lambda_{n}^{3}&-K_{z}\lambda_{n}^{3}&-GJ\lambda_{n}^{2}+\rho\Gamma \omega^{2}\lambda_{n}^{2}+P(I_{z}+I_{y}A  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ext. Mode** & **Vibration Mode** & **T-V Beam** & **Classical Theory** & **Error (\%)** \\ \hline \(n=1\) & Bending vibration (y) & 92.60 & 95.78 & 3.43 \\  & Bending vibration (z) & 141.80 & 145.75 & 2.79 \\  & Axial vibration & 1609.39 & 1608.57 & 0.05 \\  & Torsional vibration & 2487.69 & 997.17 & 59.92 \\ \hline \(n=2\) & Bending vibration (y) & 336.88 & 381.12 & 13.13 \\  & Bending vibration (z) & 509.32 & 575.94 & 13.08 \\  & Torsional vibration & 2735.56 & 1994.33 & 27.10 \\  & Axial vibration & 3218.77 & 3217.14 & 0.05 \\ \hline \(n=3\) & Bending vibration (y) & 673.49 & 850.09 & 26.22 \\  & Bending vibration (z) & 1001.67 & 1270.70 & 26.86 \\  & Torsional vibration & 3078.47 & 2991.50 & 2.83 \\  & Axial vibration & 4828.16 & 4825.71 & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Natural frequencies (rad/s) of the beam with length \(L=50\) m studied as Example 1.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Ext. Mode** & **Vibration Mode** & **T-V Beam** & **Classical Theory** & **Error (\%)** \\ \hline \(n=1\) & Bending vibration (y) & 83.42 & 84.01 & 0.71 \\  & Bending vibration (z) & 43.88 & 43.98 & 0.23 \\  & Axial vibration & 1609.39 & 1608.57 & 0.05 \\  & Torsional vibration & 10,678.39 & 997.17 & 90.66 \\ \hline \(n=2\) & Bending vibration (y) & 324.75 & 334.69 & 3.06 \\  & Bending vibration (z) & 173.57 & 175.74 & 1.25 \\  & Axial vibration & 3218.77 & 3217.14 & 0.05 \\  & Torsional vibration & 10,972.25 & 1994.33 & 81.82 \\ \hline \(n=3\) & Bending vibration (y) & 701.09 & 748.00 & 6.69 \\  & Bending vibration (z) & 383.51 & 394.68 & 2.91 \\  & Axial vibration & 4828.16 & 4825.71 & 0.05 \\  & Torsional vibration & 11,435.58 & 2991.50 & 73.84 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Natural frequencies (rad/s) of the beam with length \(L\) = 50 m studied as Example 3.