## 4 Experiment Verification
### Experiment Results and Discussion
#### 4.2.2 Dynamic Characteristics Experiments  
The dynamic characteristics experiment is conducted to obtain the displacement of spool versus test time of VCM-DHPV under different supply pressures. Figure 12 shows the comparisons of the step response between the simulation results and measurements of the developed VCM-DHPV under different sample pressures. The red dashed lines stand for simulation results and the black solid lines stand for experimental results. It can be seen in Figure 12 that the curve of the simulations changes along with the trend of the experimental results, so the comparisons between the simulation results and measurements resulted in an acceptable agreement. However, as presented in Figure 12, due to the uncertainties of the sensors, some small overshoots can be observed. The opening response times of the on/off valve under different supply pressures by the simulation and experiment are listed in Table 3.  
As presented in Table 3, the deviation of the simulated and measured response time under the supply pressures of 2, 4, 6 and 8 MPa is about 1.6, 1.5, 1.8 and 2.0 ms, respectively, and the deviations increase with the increasing supply pressures. This is because the friction force between the valve core and the valve body increases with the increasing supply pressure in the experiment, leading to the long response time of VCM-DHPV. However, the friction force is neglected in the numerical simulation. As a result, the deviations between the simulation and experiment increase with the increasing supply pressure. These results show that the simulated results provided a reasonable agreement within neglect of uncertainties of the sensors and the friction force between the valve core and the valve body. Therefore, it can be concluded that the numerical simulation and experimental results show the same change tendency and the presented AMESim model is acceptable in analyzing the dynamic characteristics of VCM-DHPV. Figure 11: Flow curve of VCM-DHPV under different supply pressures: (**a**) VCM-DHPV prototype; and (**b**) maximum flow rate. Figure 12: _Cont._  
Meanwhile, the system identification tools in MATLAB (MathWorks, Natick, MA, USA) are employed to identify the system parameters of the developed VCM-DHPV by the simulation and experiment. The transfer function of on/off valve is acquired, and then the dynamic characteristics of the on/off valve via the frequency responses with the full stroke can be drawn in MATLAB. Figure 13 shows the Bode diagram of VCM-DHPV through the simulation and experiment under different supply pressures. The results show that the transfer function of the on/off valve could be simplified to be a second-order oscillating system. Because the damping ratio in the transfer function of the on/off valve are not considered in the simulation (\(\zeta=0\)), the bandwidth of the on/off valve is significantly influenced by the supply pressure, and its amplitude frequency characteristics of the simulations have an obvious pulse. However, due to the friction force and hydraulic leakage, the damping ratio could not be equal to zero in the actual working condition. It is worth noting in Figure 13 that the frequency responses of the on/off valve are quite different when the supply pressures are set as 2.0, 4.0, 6.0 and 8.0 MPa, respectively. Besides, it is illustrated in Figure 13 that the gain margin GM and phase margin PM of VCM-DHPV are both greater than zero. As a result, it can be concluded that VCM-DHPV is a stable minimum-phase system. In addition, it can be seen in Figure 13 that the frequency responses of the simulations and experiments are not identical, and there is a large delay. This is attributed to the delay effect of the position sensor fixed on the spool in experiment process and the friction forces of the two O type rings ignored in the simulation model.