## 4 Results and Discussion
### Analysis of Pressure Response  
Figure 8 shows the curves of the control pressure and the spool displacement. In the initial opening process of the direct operating solenoid valve from 0-0.38 s, the pressure greatly rises up because the spool suddenly moves fast. And the movement of the spool keeps the opening area of the control port almost the same so that the pressure increases smoothly during the next 0.5 s. Finally, the pressure goes down with the closing command and the spool returns back. Figure 7: Comparison of experimental and numerical control pressure in the direct operating solenoid valve. Since the motion of the valve is determined by the resultant force on the valve, all forces acting on the spool as well as the spool displacement are depicted in Figure 9. The spool remains still in the range of 0 to 0.4 s because the increasing magnetic force is smaller than the composition of the spring force and the viscous force. And there is no feedback force due to the closure of the input port. Once the magnetic force is big enough, the spool moves immediately after there appears the feedback force. Further analysis on the forces is conducted combining Figures 9 and 10. There are spring force called preload force before the spool starts to move, and viscous force acting on the spool when without current. The preload force comes from the spring compression. When leak  
Figure 8: Control pressure and displacement of the valve. (**a**) Input signal (**b**) Control pressure (c) Spool displacement. Figure 9: Forces on the spool and valve displacement. sleeve and spool, the viscous force is generated by the leakage from the supply port to the control port. When the electric current is input, the magnetic force grows up and becomes larger than the preload force at 0.32 s.  
As the magnetic force grows large enough to overcome the preload force combined with the viscous force, the force balance is thrown off and the spool starts to move against the spring and the spring is compressed, which causes the spring force to rise. When the supply port is connected to the control port, the fluid flows through to the clutch. Meanwhile, the feedback force appears at 0.41 s to hinder the opening of the control port and changes with the control pressure. At the opening phase of the control port, the feedback force heavily fluctuates due to the sharp increase of the feedback pressure. Since different force changes at different speed, the acceleration is not constant and the displacement of the spool varies. It is worth noting that the magnetic force will vary with the input current during the spool movement and remain the similar variation trend with the control pressure. It can be observed that the active force on the spool is the magnetic force. The passive forces on the spool are the spring force, the viscous force and the feedback force. The motion state of the valve depends on the resultant force acting on the spool and can be divided into three phases: (1) Stationary state. There is only spring force acting on the spool. (2) Moving tendency. The magnetic force, spring force and the viscous force apply on the spool. However, the magnetic force is in the opposite direction to the spring and viscous forces. The control port is still closed. (3) Motion state. The feedback force arises in the feedback chamber since the control port is connected to the supply port. The spool begins to move under the action of the magnetic force, the spring force, the viscous force and the feedback force. The pressure and flow response characteristics are important indicators of the performance of the direct operating solenoid valves. A faster response characteristic is not the better one, because the pressure overshoot will be amplified if the response is too fast. However, the response delay will lead to the reduction of the control accuracy. The response characteristics are mainly dependent on the spool motion state which mostly relies on the resultant force on the valve.