## 3 Numerical Discussion  
### Simple Truss Structure  
According to the force balance and geometric conditions at the connection point, the impedance matrix of a single truss can be used to build up the two-dimensional truss structure. For example, the simple truss structure as shown in Figure 2, consisted of five sub-structures (truss A, B, C, D, E) and four connected points (P1/P2/P3/P4). The cross-section of the beam is a pipe, and the outer radius and inner radius are 80 mm and 71 mm, respectively. [PERSON]'s modulus, density, and Poisson's ratio of a truss are 210 GPa, 7800 kg/m\({}^{3}\) and 0.3, respectively.  
There are four connection points, each point connected with two or three beams. According to the assemble rule, the impedance matrix of the above truss structure can be written as follows:  
\[\begin{bmatrix}F_{1}\\ F_{2}\\ F_{3}\\ F_{4}\end{bmatrix}=\begin{bmatrix}Z_{11}^{A}+Z_{11}^{D}&Z_{12}^{A}&0&Z_{12}^{D }\\ Z_{21}^{A}&Z_{22}^{A}+Z_{11}^{B}+Z_{11}^{E}&Z_{12}^{B}&Z_{12}^{E}\\ 0&Z_{21}^{B}&Z_{22}^{B}+Z_{11}^{C}&Z_{12}^{C}\\ Z_{21}^{D}&Z_{21}^{E}&Z_{21}^{C}&Z_{21}^{C}+Z_{22}^{D}+Z_{22}^{E}\\ \end{bmatrix}\begin{bmatrix}V_{1}\\ V_{2}\\ V_{3}\\ V_{4}\end{bmatrix} \tag{24}\]  
where:  
\[\begin{array}{l}F_{i}=\begin{bmatrix}F_{i}^{x}&F_{i}^{y}&F_{i}^{z}&M_{i}^{x}& M_{i}^{y}&M_{i}^{z}\end{bmatrix}^{T}\\ V_{i}=\begin{bmatrix}V_{i}^{x}&V_{i}^{y}&V_{i}^{z}&\hat{\hat{\alpha}}_{i}^{x}& \hat{\hat{\alpha}}_{i}^{y}&\hat{\hat{\beta}}_{i}^{z}\end{bmatrix}^{T}(i=1,2,3,4)\end{array} \tag{25}\]  
The total impedance matrix of this simple two-dimensional truss is 24 \(\times\) 24. By solving Equation (24), we can obtain the dynamic response of the truss structure. When you obtain the dynamic responses of each single truss beam, substituting them into their global impedance matrix, the dynamic forces on the left side of Equation (20) can easily be obtained. To verify the present method, the dynamic results are compared with FEM under a unit harmonic force applied on point P1 in the z-direction. The finite element model is made of the beam element, and the mesh size is 0.01 m. The element number of this simple truss is 541. The frequency analysis step is 1 Hz, and calculation frequency range is 1-1000 Hz.  
The amplitude and imaginary part of dynamic displacement on point P1 in the z-direction are compared in Figure 3. It can be seen that the two results are in good agreement.  
Figure 2: Two-dimensional frame structure.  
The amplitude and imaginary part of displacement of point P1 in Y direction in 1-1000 Hz domain is compared with FEM in Figure 4. We can see that both the amplitude and imaginary parts of this displacement are in good agreement too.  
The vibration power flow is multiplied by the displacement and force, from the energy flow viewpoint in physics. Thus, the correctness of calculation forces also needs to be verified. Real and imaginary parts of the force on point P2 in the Z direction are compared with FEM in Figure 5. It can be seen that both real and imaginary parts of the force are in good agreement with FEM results.  
From the above comparison results, one can find that the impedance synthesis method (ISM) can be used for the truss structure and has high accuracy at mid and high frequency.  
Figure 4: Displacement comparison of ISM and FEM of point P1 in Y direction.  
Figure 5: Force comparison of ISM and FEM of point P2 on the Z direction.  
Figure 3: Displacement comparison of ISM and FEM of point P1 on the Z direction.