## Theoretical Modeling
### Element Stiffness and Mass Matrices
\right]\]  
Descriptions of the symbols in Equation (29) are given in Table 1.  
The element mass matrix in the basic coordinate system is:  
\[\mathbf{m}_{o}=\mathbf{T}_{m}^{T}\mathbf{Q}_{m}^{T}\mathbf{m}_{el}\mathbf{Q}_{m} \mathbf{T}_{m} \tag{30}\]  
where  
\[\mathbf{T}_{m}=\left[ \begin{array}{ccc}\mathbf{t}_{m}&\mathbf{0}_{6\times 6} \\ \mathbf{0}_{6\times 6}&\mathbf{t}_{m}\end{array} \right]\quad\mathbf{t}_{m}=\left[ \begin{array}{cccccc}1&0&0&0&z_{g}&-y_{g}\\ 0&1&0&-z_{c}^{\prime}&0&0\\ 0&0&1&y_{c}^{\prime}&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array} \right]\]  
\[\mathbf{Q}_{m}=\left[ \begin{array}{ccc}\mathbf{q}_{m}&\mathbf{0}_{6\times 6 }\\ \mathbf{0}_{6\times 6}&\mathbf{q}_{m}\end{array} \right]\quad\mathbf{q}_{m}=\left[ \begin{array}{cccccc}1&0&0&0&0&0\\ 0&c(\alpha_{m})&s(\alpha_{m})&0&0&0\\ 0&-s(\alpha_{m})&c(\alpha_{m})&0&0&0\\ 0&0&1&0&0\\ 0&0&0&0&c(\alpha_{m})&s(\alpha_{m})\\ 0&0&0&0&-s(\alpha_{m})&c(\alpha_{m})\end{array} \right]\]  
where \(c(\alpha_{m})=\cos(\alpha_{m})\), \(s(\alpha_{m})\sin(\alpha_{m})\), and \((y_{c}^{\prime},z_{c}^{\prime})\) is the coordinates of shear center (C) in the local reference system for the mass matrix \(G_{xy/z^{\prime}}\). The element mass matrix in local axes can be expressed as:  
\[\mathbf{m}_{el}=\left[ \begin{array}{cc}\mathbf{A}&\mathbf{B}^{T}\\ \mathbf{B}&\mathbf{C}\end{array} \right] \tag{31}\]  
in which  
\[\mathbf{A}=\left[ \begin{array}{cccccc}\frac{\rho AI}{3}&0&0&0&0&0\\ 0&\frac{13\rho AI}{35}&0&\frac{7z_{c}^{\prime}\rho AI}{20}&0&\frac{11\rho AI^{ 2}}{210}\\ 0&0&\frac{13\rho AI}{35}&-\frac{7y_{c}^{\prime}\rho AI}{20}&-\frac{11\rho AI^{ 2}}{210}&0\\ 0&\frac{7z_{c}^{\prime}\rho AI}{20}&-\frac{7y_{c}^{\prime}\rho AI}{20}&\frac{ \rho_{l}!}{3}&\frac{y_{c}^{\prime}\rho AI^{2}}{20}&\frac{z_{c}^{\prime}\rho AI^ {2}}{20}\\ 0&0&-\frac{11\rho AI^{2}}{210}&\frac{y_{c}^{\prime}\rho AI^{2}}{20}&\frac{\rho AI ^{3}}{105}&0\\ 0&\frac{11\rho AI^{2}}{210}&0&\frac{z_{c}^{\prime}\rho AI^{2}}{20}&0&\frac{\rho AI ^{3}}{105}\end{array} \right]\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(EA\) & Axial stiffness \\ \(EI_{y}\) & Bending stiffness around \(y\) axis (flapwise) \\ \(EI_{z}\) & Bending stiffness around \(z\) axis (lagwise) \\ \(EI_{yz}\) & Flap-lag bending stiffness around \(y\) and \(z\) axes \\ \(GJ\) & Torsional stiffness \\ \(AT\) & Coupled axial-torsional stiffness \\ \(FT\) & Coupled flap-torsional stiffness \\ \(LT\) & Coupled lag-torsional stiffness \\ \(l\) & Length of element \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables defining stiffness matrix. \[\mathbf{B}=\left[ \begin{array}{cccccc}\frac{\rho AI}{6}&0&0&0&0&0\\ 0&\frac{9\rho AI}{70}&0&\frac{3z_{c}^{\prime}\rho AI}{20}&0&\frac{13\rho AI^{2}} {420}\\ 0&0&\frac{9\rho AI}{70}&-\frac{3\
u_{c}\rho AI}{20}&-\frac{13\rho AI^{2}}{420}&0 \\ 0&\frac{3z_{c}^{\prime}\rho AI}{20}&-\frac{3\
u_{c}^{\prime}\rho AI}{20}&\frac {\rho_{t}!}{6}&\frac{\
u_{c}\rho AI^{2}}{30}&\frac{z_{c}^{\prime}\rho AI^{2}} {30}\\ 0&0&\frac{13\rho AI^{2}}{420}&-\frac{\
u_{c}^{\prime}\rho AI^{2}}{30}&-\frac{ \rho AI^{3}}{140}&0\\ 0&-\frac{13\rho AI^{2}}{420}&0&-\frac{z_{c}^{\prime}\rho AI^{2}}{30}&0&-\frac {\rho AI^{3}}{140}\end{array} \right]\]  
\[\mathbf{C}=\left[ \begin{array}{cccccc}\frac{\rho AI}{3}&0&0&0&0&0\\ 0&\frac{13\rho AI}{35}&0&\frac{7z_{c}^{\prime}\rho AI}{20}&0&\frac{-11\rho AI^{ 2}}{210}\\ 0&0&\frac{13\rho AI}{35}&-\frac{7\
u_{c}^{\prime}\rho AI}{20}&\frac{11\rho AI^{ 2}}{210}&0\\ 0&\frac{7z_{c}^{\prime}\rho AI}{20}&\frac{-7\
u_{c}^{\prime}\rho AI}{20}&\frac {\rho_{t}!}{3}&\frac{-\
u_{c}^{\prime}\rho AI^{2}}{20}&\frac{-z_{c}^{\prime} \rho AI^{2}}{20}\\ 0&0&\frac{11\rho AI^{2}}{210}&-\frac{-\
u_{c}^{\prime}\rho AI^{2}}{20}&\frac{ \rho AI^{3}}{105}&0\\ 0&\frac{-11\rho AI^{2}}{210}&0&-\frac{-z_{c}^{\prime}\rho AI^{2}}{20}&0&\frac {\rho AI^{3}}{105}\end{array}