# Design and Experimental Study of a Down-Drive Piezoelectric High-Frequency Fatigue Testing Machine
## 3 System Dynamics Modelling and Simulation Analysis
### Simulation Analysis  
In order to further investigate theoretically the influence of the mass \(M\) of the counter-weight mass block and the stiffness \(k_{2}\) of the elastic ring on the vibration characteristics of the fatigue tester, the relevant parameters were determined, as shown in Table 6.  
According to the above Equation (1), the elastic ring stiffness and mass block mass are used as parameters affecting the intrinsic frequency of the fatigue testing machine, by making \(D_{n}=0\) in Equation (13), the intrinsic frequency of the system can be solved for different parameters, The relationship is shown in Figure 10.  
From Figure 10, it can be seen that for the same counterweight mass block mass, increasing the elastic ring stiffness (corresponding to increasing the thickness of the elastic ring at the flexible place Tb in the simulation analysis) gradually increases the system intrinsic frequency; for the same elastic ring stiffness, increasing the mass of the counterweight mass block (which corresponds to increasing the thickness of counterweight mass block a in the simulation analysis) results in a gradual decrease in the intrinsic frequency of the system, which is in accordance with the facts and with the results of the simulation analysis. The overall analysis of the fatigue testing machine was carried out using the finite element method, and the results are shown in Figure 11. The fatigue tester achieved the optimal solution of the modal vibration pattern at an operating frequency of 187 Hz. At this frequency, the piezoelectric actuator vibrates the fatigue tester vertically up and down with no displacement of the rest of the machine, as shown in Figure 12.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mass Unit**} & \multirow{2}{*}{**Parameter Value/kg**} & \multicolumn{2}{c}{**Rigidity Parameter Value/(\(\times 10^{6}\))(N/m)**} & \multicolumn{2}{c}{**Damping Parameters**} & \multirow{2}{*}{**Parameter Value**} \\ \cline{3-4}  & & & & & \\ \hline \(M_{1}\) & 0.2954 & \(k_{1}\) & 3.33 & \(c_{1}\) & 16.8568 \\ \cline{2-4}  & & \(k_{s}(t=0)\) & 5.43 & \(c_{s}(t)\) & 15.0000 \\ \hline \(M_{2}\) & 0.3159 & \(k_{2}\) & 0.1: 0.1: 1 & \(c_{2}\) & 17.2519 \\ \hline \(M_{3}\) & \(0.9990+M(0:0.2:3)\) & \(k_{3}\) & 6.21 & \(c_{3}\) & 20.8610 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Detailed list of relevant parameters of the fatigue testing machine.  
Figure 11: Cloud diagram of the first twenty orders of modal analysis of the fatigue testing machine.  
Figure 10: Plot of mass block and elastic ring stiffness versus intrinsic frequency of fatigue tester.  
According to the above simulation process, the specimen with the central width \(w\) = 0.8 mm, the thickness of the counterweight mass block \(a\) = 0 mm:2 mm:30 mm, and the thickness of the elastic ring at the flexible place \(b\) = 0.1 mm:0.1 mm:1 mm were chosen, and the three-dimensional relationship between the intrinsic frequency of the fatigue testing machine and the thicknesses of the counterweight mass block \(a\), and the thickness of the elastic ring at the flexible place \(b\) was obtained, which is shown in Figure 13.  
From Figure 13 it can be seen that for the same counterweight mass block thickness \(a\), increase the thickness of the elastic ring flexible place b, and fatigue testing machine system intrinsic frequency will gradually increase; for the same elastic ring flexible thickness b, increase the counterweight mass block thickness a, and fatigue testing machine system intrinsic frequency will gradually decrease.  
Figure 12: Finite element modal analysis of fatigue testing machine.  
Figure 13: System intrinsic frequency versus thickness of counterweight mass block a, thickness of elastic ring at flexibility b.