## 5 Experimental Verification  
To validate the correctness of the previous analysis on flow force characteristics and to assess the effectiveness of the proposed compensation structure in reducing steady flow force, a measurement experiment of the steady flow force of the slide valve is required. Figure 26 illustrates the hydraulic circuit schematic of the testing system and the setup of the testing device. A low-pressure, high-flow hydraulic pump station is selected based on the pressure and flow conditions defined in the simulations. The hydraulic circuit is \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Optimization Algorithm** & \(\mathbf{\theta_{1}}^{c}\) & \(\mathbf{\theta_{2}}^{c}\) & \(\mathbf{F_{\text{spool}}}\)/N \\ \hline SQP & 7.6 & 33 & 1.46 \\ MIGA & 9.2 & 33.4 & 1.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Two-parameter structure optimization results. Figure 24: Curve of design variables versus objective function. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Optimization Algorithm** & \(\mathbf{\theta_{1}}^{c}\) & \(\mathbf{\theta_{2}}^{c}\) & \(\mathbf{d_{1}}\)/mm & \(\mathbf{F_{\text{spool}}}\)/N \\ \hline SQP & 9.1 & 26 & 11.8 & 0.12 \\ MIGA & 6.8 & 25.6 & 11.8 & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Three-parameter structure optimization results. Figure 25: Convergence curves of two optimization algorithms. configured as an outlet throttling speed control loop, maintaining a certain back pressure at the outlet to adjust the pressure differential between the inlet and outlet. An accumulator is utilized to mitigate pressure fluctuations in the hydraulic oil. A pulse signal generator, in conjunction with a driver, controls a stepper motor, while a ball screw converts the motor's rotation into linear motion to actuate the spool. Force and pressure sensors are employed to measure the oil pressure within the valve chamber and the flow forces exerted on the tested spool. Figure 27 presents a structural diagram, physical image, and actual view of the spool being tested. To facilitate observation of the spool's movement, the tested valve is made of acrylic. To ensure a uniform flow of the hydraulic fluid, the slide valve chamber features a dual inlet and dual outlet arrangement, consistent with the 3D geometric model. To avoid the influence of radial binding forces on the flow force measurement of the spool, two circumferential grooves are machined in the annular chamber to balance the radial pressure on the spool. The spool and valve sleeve are fitted with a clearance fit, and an oil drain port is provided on the valve cover to minimize friction as much as possible. Furthermore, to reduce the pressure losses in the hydraulic oil, pressure measurement holes are directly drilled into the valve body for oil pressure measurement. The spool, designed for verifying the steady flow force characteristics, features a valve stem diameter ranging from 8 to 14 mm, with the original stem diameter being 11 mm. Figure 27: (**a**) Main cross-section of valve under test; (**b**) top view of valve under test; (**c**) spools with different \(\lambda_{1}\) and final compensated spool. Figure 26: (**a**) Hydraulic circuit schematic: 1—hydraulic pump, 2—relief valve, 3—accumulator, 4—solenoid switching valve, 5—valve under test, 6—throttle valve, 7—flow meter. (**b**) Test bench: 1—step motor driver, 2—data acquisition card, 3—pulse generator, 4—hydraulic pumping station, 5—force transducer, 6—step motor, 7—solenoid switching valve, 8—pressure transducer, 9—experimental valve, 10—computer. The experiment measured the impact of the inner annular chamber's radial dimensions on the flow force characteristics, with the corresponding flow force curves illustrated in Figure 28. At a valve opening of 1 mm, there was a good consistency between the experimental results and simulations, with the error increasing as the opening widened, reaching a maximum of 17.55% within the experimental range.