## 4 Modeling and Simulation
### Mathematical Model Analysis of Accumulator  
When the accumulator is connected to the system, it is generally connected through a section of pipeline, which has a great influence on the characteristics of the accumulator. Therefore, when modeling the accumulator, the connecting pipeline can be considered as a part of the accumulator. In practical application, the connecting pipeline between the accumulator and the suspension cylinder should be as short as possible. The accumulator system diagram is shown as Figure 7.  
In addition, the following assumptions should be made when modeling the accumulator [14; 15]. The charge pressure of the accumulator is a slow process, so the change of gas pressure and volume can be seen as an isothermal process; the discharging process of the accumulator is fast, and the  
Figure 6: The schematic diagram of the improved suspension hydraulic system.  
Figure 7: The accumulator system diagram.  
gas does not exchange heat with the outside world, so it can be considered as an adiabatic process approximately; the compressibility of hydraulic oil is quite different from gas, so it can be ignored; and the flow of hydraulic oil in the accumulator can be regarded as laminar flow. The approximate flow continuity equation of the accumulator can be expressed as follows.  
\[Q_{a}=C_{d}A_{d}\sqrt{\frac{2}{\rho}\Delta P^{\prime}} \tag{15}\]  
where \(Q_{a}\) is the flow into accumulator, \(C_{d}\) is the orifice discharge coefficient, \(A_{d}\) is the orifice area, and \(\Delta P^{\prime}\) is the pressure difference before and after the orifice.  
The motion equation of the hydraulic oil in the connecting pipeline and the accumulator is as follows.  
\[L\frac{dQ_{a}}{dt}+RQ_{a}=P_{1}-P_{a} \tag{16}\]  
where \(L\) is the liquid inductance, \(R\) is the fluid resistance, \(P_{1}\) is the pressure of the pipeline.  
\(L\) and \(R\) are expressed as follows.  
\[L=\rho\frac{l_{a}}{A_{d}} \tag{17}\]  
\[R=\frac{128\mu l_{a}}{\pi{d_{a}}^{2}} \tag{18}\]  
where \(\mu\) is the kinetic viscosity, \(l_{a}\) is the length of the pipeline between accumulator and hydraulic cylinder, \(d_{a}\) is the diameter of the pipeline.  
The main function of the accumulator is to absorb the impact pressure of the system. The type of gas charged in the accumulator is nitrogen. The required volume of the accumulator can be calculated according to the Equation (19).  
\[V_{a}=\frac{0.004 QP_{2}(0.0164l_{a}-t)}{P_{2}-P_{1}} \tag{19}\]  
where \(V_{a}\) is the required volume of the accumulator, \(Q\) is the flow in the pipeline, \(P_{2}\) is the maximum allowable impact pressure, \(t\) is the time from the beginning to the end of impact. After calculation and analysis, the accumulator (075-1315-042-711) of Integral Corporation is selected. As the air spring, the accumulator absorbs the impact pressure, and its spring stiffness \(K\) is equal to the ratio of the force \(\Delta F\) exerted by the volume change of accumulator gas on the suspension cylinder to the displacement \(\Delta x\) of the cylinder. It is shown as follows.  
\[K=\frac{\Delta F}{\Delta x}=\frac{\Delta P^{\prime}\cdot A}{\Delta V^{\prime }/A}=A^{2}\frac{dP}{dV}=A^{2}\frac{RP_{a}V_{2n}^{k}}{V_{u}^{k+1}} \tag{20}\]  
where \(\Delta V^{\prime}\) is the changed volume of the gas in the accumulator, \(V_{1a}\) is the gas volume in the initial state, \(V_{2a}\) is the gas volume in the impact state, \(A\) is the area of the cylinder, \(k\) is the isentropic exponent of the gas, \(P_{a}\) is the pressure of the accumulator at any moment.