## 4 Model Establishment and Simulation Analysis of the Electro-Hydraulic Servo Loading System  
### Model Establishment of the Electro-Hydraulic Servo Loading System  
The electro-hydraulic servo loading system adopted the form of a valve-controlled symmetrical cylinder. Assuming that the electro-hydraulic servo valve is a zero-opened and four-through slide valve, the four orifices are matched and symmetrical. The oil supply pressure \(p_{s}\) is constant, the return oil pressure \(p_{0}\) is zero, and the pressure loss in the pipeline and the pipeline dynamics are ignored. The pressure is assumed to be equal everywhere in each working cavity of the hydraulic cylinder. The oil temperature and bulk elastic modulus are considered constant, and both the internal and external leakage of the hydraulic cylinder are laminar flow. Under these assumptions, the flow equation of the control valve, the flow continuity equation of the hydraulic cylinder, and the force balance equation between the hydraulic cylinder and the load are established [31]:  
\[\left\{\begin{array}{l}Q_{L}=K_{q}x_{p}-K_{c}p_{L}\\ Q_{L}=A_{p}\frac{dx_{p}}{dt}+C_{t}p_{L}+\frac{V_{t}}{4\Phi_{t}}\frac{dp_{L}}{dt} \\ F_{g}=A_{p}p_{L}=m\frac{d^{2}x_{p}}{dt^{2}}+B_{p}\frac{dx_{p}}{dt}+Kx_{p}\\ \end{array}\right. \tag{1}\]  
where \(Q_{L}\) is the load flow, \(K_{q}\) is the servo valve flow gain, \(x_{p}\) is the valve core displacement of the servo valve, \(K_{c}\) is the servo valve flow-pressure coefficient, \(p_{L}\) is the load pressure drop, \(A_{p}\) is the effective area of the hydraulic cylinder piston, \(x_{p}\) is the hydraulic cylinder piston displacement, \(C_{t}\) is the total leakage coefficient of the hydraulic cylinder, \(V_{t}\) is the total volume of the hydraulic cylinder oil chamber, \(\beta_{c}\) is the effective bulk elastic modulus, \(F_{g}\) is the output force of the hydraulic cylinder, \(m\) is the load mass, \(B_{p}\) is the viscous friction coefficient, and \(K\) is the elastic load stiffness.(The complete parameter list and definition are shown in Appendix A: Symbol Table)  
Usually, \(B_{p}\) is very small and can be ignored. Laplace transform is performed on the above formula, the transfer function from the valve core displacement of the electro  
Figure 4: Loading structure of the electro-hydraulic servo loading system.  
hydraulic servo valve to the output force of the hydraulic cylinder can be obtained as follows:  
\[\frac{F_{g}}{X_{v}}=\frac{\frac{K_{g}}{K_{ce}}A_{p}\big{(}\frac{m}{K}s^{2}+1\big{)} }{\frac{V_{t}m}{4\beta_{c}K_{ce}}s^{3}+\frac{m}{K}s^{2}+\Big{(}\frac{V_{t}}{4 \beta_{c}K_{ce}}+\frac{A_{p}^{2}}{K_{ce}R}\Big{)}s+1} \tag{2}\]  
where \(X_{v}\) is obtained by the Laplace transform of \(x_{v}\), \(K_{ce}=K_{c}+C_{t}\) and it is the total flow-pressure coefficient.  
The error signal is  
\[U=U_{r}-U_{f} \tag{3}\]  
where \(U_{r}\) is the voltage signal of the given signal, and \(U_{f}\) is the voltage signal of the feedback signal.  
The servo amplifier converts the deviation signal into a current signal and amplifies it, which can be equivalent to a proportional element [32]:  
\[\frac{I}{U}=K_{a} \tag{4}\]  
where \(K_{a}\) is the servo amplifier gain, and \(I\) is the servo amplifier output current signal.  
The electro-hydraulic servo valve can be approximately regarded as a second-order oscillation element:  
\[\frac{X_{v}}{I}=\frac{K_{sv}}{\frac{s^{2}}{\omega_{sv}^{2}}+\frac{2\sigma_{sv} }{\omega_{sv}}+1} \tag{5}\]  
where \(K_{sv}\) is the servo valve gain, \(\omega_{sv}\) is the servo valve natural frequency, and \(\xi_{sv}\) is the servo valve damping ratio.  
The force sensor can be equivalent to a proportional element  
\[\frac{U_{f}}{F_{g}}=K_{fF} \tag{6}\]  
where \(K_{fF}\) is the force sensor gain.  
Based on (1) to (6), the system block diagram is shown in Figure 5.