# Design and Experimental Study of a Down-Drive Piezoelectric High-Frequency Fatigue Testing Machine
## 3 System Dynamics Modelling and Simulation Analysis  
### Structural Design  
A fatigue tester is a typical example of a resonant system. In order to simplify the calculation process, the test machine was discretized into several centralized mass cells. According to the actual working conditions, select the parts that transmit more force and displacement as the centralized mass unit, ensure that the stiffness within each unit is consistent or similar, and avoid the appearance of relatively moving parts within the same unit. Therefore, when calculating the vibration characteristics of a fatigue testing machine in terms of a mass-spring-damping system, this method not only ensures the accuracy of the calculation but also reduces the cost of the calculation. Table 5 describes the components of the centralized mass element subsystem. Figure 8 shows the spring-mass-damping system model of the fatigue testing machine. Based on the above relationship for each mass unit, the ordinary differential equations are listed:  
\[\left[ \begin{array}{cc}m_{1}&\\ &m_{2}\\ &&m_{3}\end{array} \right]\left\{ \begin{array}{c}\ddot{x}_{1}\\ \ddot{x}_{2}\\ \ddot{x}_{3}\end{array} \right\}+\left[ \begin{array}{cc}c_{1}+c_{2}&-c_{2} \\ -c_{2}&c_{2}+2c_{3}-c_{4}&-c_{4}\\ -c_{4}&c_{4}\end{array} \right]\left\{ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\end{array} \right\}+ \tag{1}\] \[\left[ \begin{array}{cc}k_{1}+k_{2}&-k_{2}\\ -k_{2}&-k_{2}+2k_{3}-k_{4}&-k_{4}\\ -k_{4}&k_{4}\end{array} \right]\left\{ \begin{array}{c}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\end{array} \right\}=\left\{ \begin{array}{c}0\\ 0\\ 0\\ F\sin(wt)\end{array} \right\}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Mass Unit** & **Mass Unit Subunit** & **Structural Schematic** \\ \hline \multirow{4}{*}{M1} & M1.1 Force sensor & \multirow{4}{*}{M1.2 Force sensor substrates} \\  & M1.2 Force sensor substrates \\  & M1.3 Transfer plate \\  & M1.4 Upper fixture \\ \hline \multirow{4}{*}{M2} & M2.1 Specimens & \multirow{4}{*}{M2.2 Bottom fixture} \\  & M2.3 Lower fixed support plate \\  & M2.4 Vibration plate \\  & M2.5 Elastic ring \\ \hline \multirow{4}{*}{M3} & M3.1 Connection shaft & \multirow{4}{*}{M3.2 Mass block} \\  & M3.3 Circling overhead \\ \cline{1-1}  & M3.4 Piezoelectric vibrator \\ \cline{1-1}  & M3.5 Pump \\ \hline \hline \end{tabular}
\end{table} 
Table 5: List of discrete chaos models. where \(k_{1}\) is the transducer stiffness; \(k_{2}\) is the elastic ring stiffness. \(K_{3}\) is the sum of the connecting shaft stiffness and the piezoelectric vibrator stiffness. In fatigue systems, changing the differential equations of a discrete system can be carried out by changing the intrinsic frequency of the system by changing the mass of the mass block, since the mass block is replaceable. Five values were averaged over the 0.999 kg to 3.999 kg range to perform a comprehensive frequency sweep analysis from 0 to 250 Hz. Using this method, it is possible to check the correlation between the mass change and the corresponding intrinsic frequency. The force load-frequency diagrams for the five mass blocks are shown in Figure 9; the greater the mass of the mass blocks, the lower the intrinsic frequency of the system. Figure 8: Model of spring–mass–damping system for fatigue testing machine. Figure 9: Intrinsic frequency of the system for mass. Express the piezoelectric vibrator simple harmonic excitation force \(F(t)\) as:  
\[F(t)=Fe^{i\omega t}, \tag{2}\]  
Then the steady-state response of this system can be expressed as:  
\[x_{1}(t)=X_{1}e^{i\omega t},x_{2}(t)=X_{2}e^{i\omega t},x_{3}(t)=X_{3}e^{i\omega t}, \tag{3}\]  
where \(X_{1}\) and \(X_{2}\) are complex quantities, taking values related to the system parameters, piezoelectric vibrator simple harmonic excitation force. Equations (2) and (3) are brought into the vibrational differential Equation (1) and expressed in matrix form:  
\[