## 2 Dynamic Response of Truss Structure  
### Impedance Matrix of Truss Beam  
The nodal forces, moments and displacements, and rotations for vibration motions are shown in Figure 1. Points at each end have three forces, three moments, and correspondence displacements and rotations. Since this paper is intended to be used for mid- and high-frequency analysis, Timoshenko beam theory, which is considered shear deformation and rotatory, is adopted. The total variable for a single Timoshenko beam truss structure is 12. Figure 1: Notation for forces/moments and displacements/rotations of a beam. Each truss beam has flexural, axial, and torsional vibration governing equations. To obtain the exact solutions in the frequency domain, the impedance matrix should be derived first. For example, the flexural vibration governing equation in the yoz plane can be written as [36]:  
\[\left\{ \begin{matrix}f_{y}-k_{p}GA_{p}\Big{(}\frac{\partial u_{y}}{\partial y}- \varphi_{x}\Big{)}=0\\ m_{x}-EI_{p}\frac{\partial\varphi_{x}}{\partial y}=0\\ \frac{\partial f_{z}}{\partial y}-\rho_{p}A_{p}\frac{\partial^{2}u_{x}}{ \partial t^{2}}=0\\ \frac{\partial u_{x}}{\partial y}+f_{y}-\rho_{p}I_{p}\frac{\partial^{2}u_{x}}{ \partial t^{2}}=0\end{matrix} \right. \tag{1}\]  
By separating variables method, the solution for Equation (1) can be written as:  
\[\left\{ \begin{array}{l}u_{y}(z,t)=U_{y}(z)e^{i\omega t}\\ \varphi_{x}(z,t)=\Phi_{x}(z)e^{i\omega t}\\ f_{y}(z,t)=F_{y}(z)e^{i\omega t}\\ m_{x}(z,t)=M_{x}(z)e^{i\omega t}\end{array} \right. \tag{2}\]  
Submitting Equation (2) into Equation (1), it can be simplified to only one variable \(\Phi_{x}\) in the frequency domain:  
\[\frac{\partial^{4}\Phi_{x}(y)}{\partial y^{4}}+\left(\frac{\rho A}{k_{p}GA}+ \frac{\rho I}{EI_{p}}\right)\omega^{2}\frac{\partial^{2}\Phi_{x}(y)}{\partial y ^{2}}+\left(\frac{\rho A}{k_{p}GA}\frac{\rho I}{EI_{p}}\omega^{4}-\frac{\rho I }{EI_{p}}\omega^{2}\right)\Phi_{x}(y)=0 \tag{3}\]  
The solution to Equation (3) can be expressed as:  
\[\Phi_{x}(y)=\sum_{n=1}^{4}A_{n}e^{\lambda_{n}y} \tag{4}\]  
where \(A_{n}\) represents amplitude constants, which are decided by the boundary condition, and the flexural wave number \(\lambda_{n}\) represents the four complex roots of the following equation:  
\[\lambda^{4}+\left(\frac{\rho A}{k_{p}GA}+\frac{\rho I}{EI_{p}}\right)\omega^{ 2}\lambda^{2}+\left(\frac{\rho A}{k_{p}GA}\frac{\rho I}{EI_{p}}\omega^{4}- \frac{\rho I}{EI_{p}}\omega^{2}\right)=0 \tag{5}\]  
By solving Equation (5), the four wavenumbers of the flexural truss beam can be obtained as:  
\[ \begin{array}{l}\lambda_{n}=\pm\sqrt{\frac{-\pm\sqrt{\alpha^{2}-4\beta}}{2} },n=1,2,3,4\\ \alpha=\left(\frac{\rho A}{k_{p}GA}+\frac{\rho I}{EI_{p}}\right)\omega^{2}, \beta=\frac{\rho A}{k_{p}GA}\frac{\rho I}{EI_{p}}\omega^{4}-\frac{\rho I}{EI_{ p}}\omega^{2}\end{array}  \tag{6}\]  
where substituting four wavenumbers \(\lambda_{n}\) into Equation (2), the functions can be written in the matrix form as follows:  
\[ \begin{bmatrix}U_{y}\\ \Phi_{x}\\ M_{x}\\ F_{y}\end{bmatrix} = \begin{bmatrix}B_{1}e^{\lambda_{1}y}&B_{2}e^{\lambda_{2}y}&B _{3}e^{\lambda_{3}y}&B_{4}e^{\lambda_{4}y}\\ e^{\lambda_{1}y}&e^{\lambda_{2}y}&e^{\lambda_{3}y}&e^{\lambda_{4}y}\\ C_{1}e^{\lambda_{1}y}&C_{2}e^{\lambda_{2}y}&C_{3}e^{\lambda_{3}y}&C_{4}e^{\lambda _{4}y}\\ D_{1}e^{\lambda_{1}y}&D_{2}e^{\lambda_{2}y}&D_{3}e^{\lambda_{3}y}&D_{4}e^{\lambda _{4}y}\end{bmatrix}\begin{bmatrix}A_{1}\\ A_{2}\\ A_{3}\\ A_{4}\end{bmatrix}  \tag{7}\]  
where:  
\[ \begin{array}{l}B_{n}=\frac{1}{\lambda_{n}}\Big{[}1-\frac{EI_{p}}{k_{p}GA} \Big{(}\lambda_{n}^{2}+\frac{\rho I}{EI_{p}}\omega^{2}\Big{)}\Big{]}\\ C_{n}=EI_{p}\lambda_{n}\\ D_{n}=-EI_{p}\Big{(}\lambda_{n}^{2}+\frac{\rho I}{EI_{p}}\omega^{2}\Big{)}\end{array}  \tag{8}\]Up until now, the four amplitude \(A_{n}\) constants are still unknown. If the length of a single truss beam is \(L\), both ends of the truss beam should satisfy the impedance matrix Equation (7), which can be written as:  
\[ \begin{bmatrix}U_{0}\\ \Phi_{0}\\ M_{0}\\ F_{0}\end{bmatrix} =