## 1 Introduction
For the purpose of weakening steady flow forces to improve valve performance and operational limits, many scholars have proposed compensation methods for spool steady flow forces in various hydraulic valves. These methods mainly focus on optimizing the design of the spool, valve sleeve, and valve body flow paths, typically by adding or removing certain structures or merely modifying the parameters of the original valve design [3]. [PERSON] and [PERSON] [15] optimized all the geometric parameters of the spool to minimize steady flow forces, significantly enhancing the overall performance of a small hydraulic seat valve. [PERSON] et al. [16] reduced steady flow forces by adjusting the shoulder angle of the spool, thereby changing the outlet jet angle, and demonstrated that this compensation method had a negligible effect on the flow capacity of a servo proportional valve. [PERSON] et al. [17; 18; 19] reduced the steady flow forces in hydraulic slide valves by designing jet guide grooves on the spool shoulder and adopting tapered valve stems. [PERSON] et al. [20] compensated for steady flow forces by altering the angle between the inlet of the valve sleeve and the spool axis, improving the dynamic response of a two-dimensional water hydraulic servo valve. [PERSON] et al. [21; 22] introduced damping flanges and damping tail structures on the spool to compensate for steady flow forces in diverged-flow and converged-flow cartridge proportional valves. [PERSON] et al. [23; 24] redesigned the lateral surface of the spool by cutting notches into the spool end face, significantly reducing the driving force of a direct proportional valve, a method also used by [PERSON] et al. [25] to improve the flow characteristics during the opening phase of a proportional valve. Additionally, [PERSON] et al. [26] introduced extra compensating flow paths within the valve body, causing the flow forces from multiple flow streams on the spool to cancel each other out, increasing the flow range by about 45% without requiring changes to springs or solenoids. Furthermore, some researchers have attempted to utilize, rather than reduce, flow forces. For instance, [PERSON] et al. [27] employed the flow forces on the spool to provide pressure compensation for multi-section proportional directional control valves. Despite numerous studies on the flow forces in hydraulic valves, the detailed mechanism of flow force formation on the valve spool, particularly its fluid dynamic causes,remains an unresolved fundamental issue. The asymmetric phenomenon of steady flow forces in hydraulic slide valves operating in bidirectional throttling mode and its underlying causes remain unclear. Investigating this phenomenon holds significant implications for mitigating steady flow forces in bidirectional throttling mode. The organization of this paper is as follows: In Section 2, we introduce the momentum analysis of the steady flow forces in the slide valve and the flow field simulation methods. Section 3 presents a detailed numerical study on the asymmetry of steady flow force characteristics in the inlet and outlet throttling modes of the slide valve. In Section 4, based on the analysis results, a steady flow force compensation scheme is proposed, and an automatic optimization platform for structural parameters is established. Section 5 describes the design of a steady flow force measurement test bench to validate the effectiveness of the proposed compensation structure. Section 6 provides a summary of the paper.