## 2 Dynamic Response of Truss Structure
### Impedance Matrix of Truss Beam
\begin{bmatrix}B_{1}&B_{2}&B_{3}&B_{4}\\ 1&1&1&1\\ C_{1}&C_{2}&C_{3}&C_{4}\\ D_{1}&D_{2}&D_{3}&D_{4}\end{bmatrix}\begin{bmatrix}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{bmatrix}  \tag{9}\]  
\[ \begin{bmatrix}U_{L}\\ \Phi_{L}\\ M_{L}\\ F_{L}\end{bmatrix} = \begin{bmatrix}B_{1}e^{\lambda_{1}L}&B_{2}e^{\lambda_{2}L} &B_{3}e^{\lambda_{3}L}&B_{4}e^{\lambda_{4}L}\\ e^{\lambda_{1}L}&e^{\lambda_{2}L}&e^{\lambda_{3}L}&e^{\lambda_{4}L}\\ C_{1}e^{\lambda_{1}L}&C_{2}e^{\lambda_{2}L}&C_{3}e^{\lambda_{3}L}&C_{4}e^{ \lambda_{4}L}\\ D_{1}e^{\lambda_{1}L}&D_{2}e^{\lambda_{2}L}&D_{3}e^{\lambda_{3}L}&D_{4}e^{ \lambda_{4}L}\end{bmatrix}\begin{bmatrix}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{bmatrix}  \tag{10}\]  
Combining Equations (9) and (10), eliminating the \(A_{n}\), we can obtain the whole transfer matrix of a single truss beam as follows:  
\[ \begin{bmatrix}U_{0}\\ \Phi_{0}\\ M_{0}\\ F_{0}\end{bmatrix} = \begin{bmatrix}B_{1}&B_{2}&B_{3}&B_{4}\\ 1&1&1&1\\ C_{1}&C_{2}&C_{3}&C_{4}\\ D_{1}&D_{2}&D_{3}&D_{4}\end{bmatrix}\begin{bmatrix}B_{1}e^{\lambda_{1}L}&B_{ 2}e^{\lambda_{2}L}&B_{3}e^{\lambda_{3}L}&B_{4}e^{\lambda_{4}L}\\ e^{\lambda_{1}L}&e^{\lambda_{2}L}&e^{\lambda_{3}L}&e^{\lambda_{4}L}\\ C_{1}e^{\lambda_{1}L}&C_{2}e^{\lambda_{2}L}&C_{3}e^{\lambda_{3}L}&C_{4}e^{ \lambda_{4}L}\\ D_{1}e^{\lambda_{1}L}&D_{2}e^{\lambda_{2}L}&D_{3}e^{\lambda_{3}L}&D_{4}e^{ \lambda_{4}L}\end{bmatrix} ^{-1} \begin{bmatrix}U_{L}\\ \Phi_{L}\\ M_{L}\\ F_{L}\end{bmatrix}  \tag{11}\]  
which can be abbreviated as:  
\[ \begin{bmatrix}\widehat{U}_{0}\\ \widehat{F}_{0}\end{bmatrix} = \begin{bmatrix}T_{11}&T_{11}\\ T_{11}&T_{11}\end{bmatrix}\begin{bmatrix}\widehat{U}_{L}\\ \widehat{F}_{L}\end{bmatrix}  \tag{12}\]  
where \(\widehat{U}_{0}= \begin{bmatrix}U_{0}&\Phi_{0}\end{bmatrix} ^{T}\), \(\widehat{F}_{0}= \begin{bmatrix}M_{0}&F_{0}\end{bmatrix} ^{T}\); according to the relationship between the transfer matrix and impedance matrix [37; 38], we can obtain the impedance matrix of truss beam in the frequency domain as follows:  
\[Z(\omega)= \begin{bmatrix}T_{12}^{-1}T_{11}&-T_{12}^{-1}\\ T_{21}-T_{22}T_{12}^{-1}T_{11}&T_{22}T_{12}^{-1}\end{bmatrix}  \tag{13}\]  
Thus, the flexural vibration impedance matrix in the yoz plane can be written as:  
\[ \begin{bmatrix}F_{y}^{0}\\ M_{x}^{0}\\ F_{y}^{L}\\ M_{x}^{1}\end{bmatrix} = \begin{bmatrix}Z_{yoz}^{11}&Z_{yoz}^{12}&Z_{yoz}^{13}&Z_{yoz }^{14}\\ Z_{yoz}^{21}&Z_{yoz}^{22}&Z_{yoz}^{23}&Z_{yoz}^{24}\\ Z_{yoz}^{31}&Z_{yoz}^{32}&Z_{yoz}^{34}&Z_{yoz}^{34}\\ Z_{yoz}^{41}&Z_{yoz}^{42}&Z_{yoz}^{43}&Z_{yoz}^{44}\\ \end{bmatrix}\begin{bmatrix}U_{y}^{0}\\ \[PERSON]_{y}^{0}\\ U_{x}^{1}\\ U_{x}^{1}\\ \[PERSON]_{y}^{1}\end{bmatrix}  \tag{14}\]  
The flexural vibration impedance in the xoz plane can also be written as:  
\[ \begin{bmatrix}F_{y}^{0}\\ M_{y}^{1}\\ F_{x}^{L}\\ M_{y}^{1}\end{bmatrix} = \begin{bmatrix}[PERSON]_{yoz}^{11}&[PERSON]_{yoz}^{12}&[PERSON]_{yoz}^{13}&[PERSON]_{yoz }^{14}\\ [PERSON]_{yoz}^{21}&[PERSON]_{yoz}^{22}&[PERSON]_{yoz}^{23}&[PERSON]_{yoz}^{24}\\ [PERSON]_{yoz}^{31}&[PERSON]_{yoz}^{32}&[PERSON]_{yoz}^{33}&[PERSON]_{yoz}^{34}\\ [PERSON]_{yoz}^{41}&[PERSON]_{yoz}^{42}&[PERSON]_{yoz}^{43}&[PERSON]_{yoz}^{44}\\ \end{bmatrix}\begin{bmatrix}U_{y}^{0}\\ \Phi_{y}^{0}\\ U_{x}^{1}\\ \Phi_{y}^{1}\end{bmatrix}  \tag{15}\]  
The longitudinal and torsional can be written as:  
\[ \begin{cases}f_{z}-EA\frac{\partial u_{z}}{\partial z}=0\\ \frac{\partial f_{z}}{\partial z}-\rho A\frac{\partial^{2}u_{z}}{\partial t^{2} }=0\\ \end{cases}  \tag{16}\]  
\[ \begin{cases}\frac{\partial m_{z}}{\partial z}-\rho f_{y}\frac{\partial^{2} \varphi_{z}}{\partial t^{2}}=0\\ m_{z}-G\rho f_{y}\frac{\partial\varphi_{z}}{\partial z}=0\\ \end{cases}  \tag{17}\]According to the same method as flexural vibration, we can also obtain the torsional and longitudinal vibration impedance matrix of a single truss element. \[ \begin{bmatrix}F_{Z}^{0}\\ F_{Z}^{0}\end{bmatrix} = \begin{bmatrix}Z_{\text{axial}}^{11}&Z_{\text{axial}}^{12} \\ Z_{\text{axial}}^{21}&Z_{\text{axial}}^{22}\end{bmatrix}\begin{bmatrix}U_{Z}^{0} \\ U_{Z}^{2}\end{bmatrix}  \tag{18}\]  
\[