## 3 Mathematical Model of the Valve Disc Motion  
The drilling pump operates through the reciprocating motion of pistons, which cyclically alter the pump chamber volume. This volumetric modulation induces systematic pressure fluctuations that drive the sequential operation of suction and discharge valves to enable mud circulation. The governing mechanism involves valve disc displacement, controlled by dynamic force equilibrium among the pressure differential across the valve disc, gravitational force, resistance from the cylindrical springs, and the preload exerted by wave springs [10; 11; 13]. Valve actuation follows a force-dependent regime: opening initiates when the resultant pressure differential force surpasses the cumulative resisting forces, while closure commences under force inversion conditions. The force transmission dynamics are principally dictated by the interfacial contact mechanics between the disc guidance assembly and the \begin{table}
\begin{tabular}{c c} \hline
**Crank radius \(r\) (mm)** & 178 \\ \hline
**Connecting rod length \(l\) (mm)** & 890 \\ \hline
**Strokes \(n\) (rpm)** & 105 \\ \hline
**Cylinder diameter (mm)** & 160 \\ \hline
**Piston stroke (mm)** & 356 \\ \hline
**Valve seat bore diameter (mm)** & 90 \\ \hline
**Cylindrical spring stiffness \(k_{1}\) (N/mm)** & 11 \\ \hline
**Cylindrical spring preload \(F\) (N)** & 110 \\ \hline
**Wave spring stiffness \(k_{2}\) (N/mm)** &
\begin{tabular}{c} 175.0548 (D100) \\ 217.0437 (D110) \\ \end{tabular} \\ \hline
**Wave spring pre-compression (mm)** & \(1\) \\ \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of the pump and valve. Figure 2: Flow passages and cavities of the drilling pump’s hydraulic end. impact-bearing plate, as schematically depicted in Figure 3. The spring force is modeled as a linear function of displacement based on Hooke's law, assuming small deformations and isotropic material properties. Frictional damping effects between the spring coils are neglected. The kinematic behavior of the valve disc can be mathematically described by the following second-order differential equation:  
\[m\frac{d^{2}h}{dt^{2}}=P-T-mg \tag{1}\]  
where \(m\) is the mass of the valve disc, kg; h is the lift of the valve disc, m; \(P\) is the differential pressure force on the surface of the valve disc, N; \(T\) is the spring force on the valve disc, N; and \(g\) is the acceleration of gravity, m/s\({}^{2}\). Based on the operational principle of the secondary cushioned pump valve, the spring force acting on the valve disc is formulated as follows:  
\[T=\left\{ \begin{aligned} T_{s}-T_{w}=k_{1}(h+h_{0})-k_{2}({h^{ \prime}}_{0}-h)&\quad 0\leq h\leq{h^{\prime}}_{0}\\ T_{s}=k_{1}(h+h_{0})&\quad{h^{\prime}}_{0}<h\leq h_{ \text{max}}\end{aligned} \right. \tag{2}\]  
where \(T_{s}\) is the spring force of the cylindrical spring, N; \(T_{w}\) is the spring force of the wave spring, N; \(k_{1}\) is the stiffness of the cylindrical spring, N/m; \(k_{2}\) is the stiffness of wave spring, N/m; \(h_{0}\) is the pre-compression of the cylindrical spring, m; \(h_{0}\)' is the pre-compression of the wave spring, m; \(h\) is the lift of the valve disc, m; and \(h_{\text{max}}\) is the maximum lift of the valve disc, m.  
Substituting Equation (2) into Equation (1), the motion differential equation of the valve disc can be written as follows:  
\[\left\{ \begin{aligned} \frac{d^{2}h}{dt^{2}}=\left[\begin{array}{c}P-mg- \\ k_{2}({h^{\prime}}_{0}-h)\\ -k_{1}(h+h_{0})\end{array}\right]/m&\quad 0\leq h\leq{h^{\prime}}_{0}\\ \frac{d^{2}h}{dt^{2}}=\left[\begin{array}{c}P-mg-\\ k_{1}(h+h_{0})\end{array}\right]/m&\quad{h^{\prime}}_{0}<h\leq h_{\text{max}} \end{aligned} \right. \tag{3}\]  
Figure 3: Schematic force diagram of the drilling pump valve with secondary cushioning mechanism. The velocity of the valve disc can be expressed as the following:  
\[v_{t+\Delta t}=