## Appendix A  
\[M_{j}=\left(\begin{array}{cccc}m&0&0&0\\ 0&m&0&0\\ 0&0&\theta_{y}&0\\ 0&0&0&\theta_{x}\end{array}\right)\quad C_{j}=\left(\begin{array}{cccc}d_{Fx} +d_{xx}&d_{Fxy}+d_{xy}&0&0\\ d_{Fyx}+d_{yx}&d_{Fy}+d_{yy}&0&0\\ 0&0&0&\theta_{z}\omega\\ 0&0&-\theta_{z}\omega&0\end{array}\right)_{j}\]  
\[K_{\alpha}=\left(\begin{array}{cccc}k_{1}&0&k_{2}&0\\ 0&k_{1}&0&k_{2}\\ k_{3}&0&k_{4}&0\\ 0&k_{3}&0&k_{4}\end{array}\right)_{j}\quad K_{\beta}=\sum_{p=1}^{4}K_{\beta p}\]  
\[K_{\beta 1}=\left(\begin{array}{cccc}k_{Fx}+k_{xx}&k_{Fxy}+k_{xy}&0&0\\ k_{Fyx}+k_{yx}&d_{Fy}+d_{yy}&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right)_{j}\quad K_{\beta 2}=\left(\begin{array}{cccc}-k_{1}&0&k_{ 5}&0\\ 0&-k_{1}&0&k_{5}\\ -k_{3}&0&k_{6}&0\\ 0&-k_{3}&0&k_{6}\end{array}\right)_{j}\]  
\[K_{\beta 3}=\left(\begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ k_{7}&0&k_{8}&0\\ 0&k_{7}&0&k_{8}\end{array}\right)\quad K_{\beta 4}=\left(\begin{array}{cccc}k_{ 9}&0&k_{10}&0\\ 0&k_{9}&0&k_{10}\\ k_{11}&0&k_{12}&0\\ 0&k_{11}&0&k_{12}\end{array}\right)_{j+1}\]  
\[K_{\varphi}=\sum_{p=1}^{3}K_{\varphi p}\]  
\[K_{\varphi 1}=\left(\begin{array}{cccc}-k_{9}&0&k_{13}&0\\ 0&-k_{9}&0&k_{13}\\ -k_{11}&0&k_{14}&0\\ 0&-k_{11}&0&k_{14}\end{array}\right)_{j+1}\cdot K_{\varphi 2}=\left( \begin{array}{cccc}k_{15}&0&k_{16}&0\\ 0&k_{15}&0&k_{16}\\ k_{17}&0&k_{18}&0\\ 0&k_{17}&0&k_{18}\end{array}\right)\cdot\]  
\[K_{\varphi}=\left(\begin{array}{cccc}-k_{15}&0&k_{19}&0\\ 0&-k_{15}&0&k_{19}\\ k_{20}&0&k_{21}&0\\ 0&k_{20}&0&k_{21}\end{array}\right)\]  
\[k_{1}=-\bigg{(}\frac{2(1-\beta)}{l\beta(1+2\beta)}T_{0}\bigg{)}\quad k_{2}=- \bigg{(}\frac{(2\beta^{2}-\beta+1)T_{0}}{\beta(1+2\beta)}\bigg{)}\]  
\[k_{3}=\frac{6(1-2\beta)El_{x}}{l^{2}(1+2\beta)\mu}+\frac{12(1+\beta^{\prime})El _{rx}}{l^{2}(1-2\beta^{\prime})\mu}\quad k_{4}=\frac{2(1-6\beta)El_{x}}{l(1+2 \beta)\mu}+\frac{6(1+2\beta^{\prime})El_{rx}}{l(1-2\beta^{\prime})\mu}\]  
\[k_{5}=\frac{(2\beta-1)(\beta+1)T_{0}}{\beta(1+2\beta)}\quad k_{6}=\frac{4 El_{ x}}{l(1+2\beta)\mu}+\frac{6 El_{rx}}{l(1-2\beta^{\prime})\mu}\]  
\[k_{7}=\bigg{(}-\frac{l}{\mu}\bigg{)}_{j}\bigg{(}\frac{12 El_{rx}(1+\beta^{ \prime})}{\beta^{3}(1-2\beta^{\prime})}\bigg{)}_{j+1}\quad k_{8}=\bigg{(}- \frac{l}{\mu}\bigg{)}_{j}\bigg{(}\frac{6 El_{rx}(1+2\beta^{\prime})}{l^{2}(1-2 \beta^{\prime})}\bigg{)}_{j+1}\]\[k_{9}=\frac{12 EI_{x}(1-\beta)}{l^{3}(1+2\beta)}+\frac{12 EI_{rx}(1+\beta^{\prime}) }{l^{3}(1-2\beta^{\prime})}\quad k_{10}=\frac{6(1-2\beta)EI_{x}}{l^{2}(1+2\beta) }+\frac{6(1+2\beta^{\prime})EI_{rx}}{l^{2}(1-2\beta^{\prime})}\]  
\[k_{11}=\frac{6 EI_{x}}{l^{2}(1+2\beta)\mu}+\frac{6 EI_{rx}}{l^{2}(1-2\beta^{ \prime})\mu}\quad k_{12}=\frac{4 EI_{x}}{l(1+2\beta)\mu}+\frac{4 EI_{rx}}{l(1-2 \beta^{\prime})\mu}\]  
\[k_{13}=\frac{6 EI_{x}}{l^{2}(1+2\beta)}+\frac{6 EI_{rx}}{l^{2}(1-2\beta^{\prime}) }\quad k_{14}=\frac{2 EI_{x}}{l(1+2\beta)\mu}+\frac{2 EI_{rx}}{l(1-2\beta^{ \prime})\mu}\]  
\[k_{15}=-\bigg{(}\frac{12 EI_{rx}(1+\beta^{\prime})}{l^{3}(1-2\beta^{\prime})} \bigg{)}_{j+2}\quad k_{16}=-\bigg{(}\frac{6 EI_{rx}(1+2\beta^{\prime})}{l^{2}( 1-2\beta^{\prime})}\bigg{)}_{j+2}\]  
\[k_{17}=\bigg{(}-\frac{1}{\mu}\bigg{)}_{j+1}\bigg{(}\frac{6 EI_{rx}}{l^{2}(1-2 \beta^{\prime})}\bigg{)}_{j+2}+\bigg{(}\frac{1}{\mu}\bigg{)}_{j}\bigg{(}\frac {12 EI_{rx}(1+\beta^{\prime})}{l^{3}(1-2\beta^{\prime})}\bigg{)}_{j+1}\]  
\[k_{18}=\bigg{(}-\frac{1}{\mu}\bigg{)}_{j+1}\bigg{(}\frac{4 EI_{rx}}{l(1-2\beta^{ \prime})}\bigg{)}_{j+2}+\bigg{(}-\frac{1}{\mu}\bigg{)}_{j}\bigg{(}\frac{6 EI_{ rx}}{l^{2}(1-2\beta^{\prime})}\bigg{)}_{j+1}\]  
\[k_{19}=\bigg{(}\frac{-6 EI_{rx}}{l^{2}(1-2\beta^{\prime})}\bigg{)}_{j+2}\quad k _{20}=\bigg{(}\frac{1}{\mu}\bigg{)}_{j+1}\bigg{(}\frac{6 EI_{rx}}{l^{2}(1-2 \beta^{\prime})}\bigg{)}_{j+2}\]  
\[k_{21}=\bigg{(}\frac{1}{\mu}\bigg{)}_{j+1}\bigg{(}\frac{-2 EI_{rx}}{l(1-2\beta^ {\prime})}\bigg{)}_{j+2}\]