# Modeling and Multi-Objective Optimization Design of High-Speed on/off Valve System
## 2 New High-Speed on/off Valve Structure and System Modeling
### System Dynamic Modeling
The control body, as shown in Figure 4, has a valve ball diameter of \(d_{\text{a}}\), an intermediate pin diameter of \(d_{\text{b}}\), and an aperture diameter of \(d_{\text{c}}\) at the intermediate pin location. \(p_{1}\) is the inlet pressure, and \(p_{2}\) is the pressure behind the valve. Using the momentum theorem, the traditional static hydrodynamic force for a ball valve can be derived as follows:  
\[F_{\text{s}}=\rho q(\beta_{2}v_{2}-\beta_{1}v_{1}) \tag{7}\]  
Here, \(F_{s}\) is the static hydrodynamic force, \(V_{1}\) and \(V_{2}\) are the average flow velocity vectors at the upstream and downstream flow sections, \(\beta_{1}\) and \(\beta_{2}\) are momentum correction coefficients, \(\rho\) is the fluid density, and \(q\) is the flow rate. The HSV model can be expressed in the form of an equation of state shown in Equation (8). \[ \begin{bmatrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\end{bmatrix} = \begin{bmatrix}\frac{\partial i}{\partial\varphi(i,z) }u_{t}-\frac{\partial i}{\partial\varphi(i,z)}rx_{1}-\frac{\partial i}{ \partial z}x_{3}\\ \frac{1}{m}[F_{m}(i,x_{2})-F_{s}(\Delta P,x_{2})-F_{k}(x_{2})-Cu\cdot x_{3}] \end{bmatrix}  \tag{8}\]  
\[y= \begin{bmatrix}1&0&0\end{bmatrix} x \tag{9}\]  
where the state variable represents \([i,x,\dot{x}]^{T}\). Figure 4: Static hydrodynamic force calculation diagram. AMESim, an acronym for an Advanced Modeling Environment for performing simulations of engineering systems, serves as a versatile platform for modeling and simulating complex multidisciplinary systems. Users can employ this unified platform to construct intricate system models encompassing various fields of study and conduct simulation calculations and in-depth analyses. Moreover, AMESim allows for the comprehensive examination of both the steady-state and dynamic performance of individual components or complete systems within the same environment [27]. It particularly excels in simulating and analyzing HSV systems. Regarding the system's architecture, a dynamic simulation model, as depicted in Figure 5, is established. Dedicated subsystems for mechanics, hydraulics, and magnetic fields are individually developed within the system and then integrated to generate the dynamic simulation model of the HSV. For the purpose of simplifying calculations, the dynamic simulation model of the HSV system shown in Figure 5 disregards the impact of coil temperature rise and hydraulic oil temperature during operation. The model encompasses three main systems: magnetic circuit, hydraulic, and mechanical components within the HSV. To establish a more accurate magnetic circuit model, the circuit is divided into four parts based on the simplified system model diagram, with further subdivisions based on the magnetic field distribution. The regions denoted by (d-g) in Figure 5 correspond to the respective labeled regions in Figure 6. DT4 is employed as the magnetic circuit material, and the magnetic resistances of each part can be derived from the B-H curve of the material. To emulate the real operational environment of the HSV, a quantitative pump is utilized as the fluid supply source, operating at a rotational speed of 1500 rev/min. The load is represented by a constant pressure source and an overflow valve, replicating the pressure conditions at ports A and T of the HSV during the hydraulic cylinder's extension and retraction. The mechanical components primarily comprise the moving parts of the HSV, facilitating the simulation of valve core displacement, force status, and mass. The coupling of these three systems yields the dynamic model of the HSV. Figure 5: Dynamic simulation model of the system. The three symbols in the upper left corner represent the gravity, electromagnetic, and oil properties in the simulation.