# Design and Stability Analysis of Six-Degree-of-Freedom Hydro-Pneumatic Spring Wheel-Leg
## 3 Theoretical Analysis of Hydro-Pneumatic Spring
### Simulation and Analysis of Hydro-Pneumatic Springs  
The specific parameters of hydro-pneumatic springs [30] are presented in Table 1. According to the above derivations and analyses, mathematical expressions for various important parameters of the hydro-pneumatic springs were obtained. The mathematical models of the hydro-pneumatic springs were built in MATLAB/Simulink R2018b (as shown in Figure 5) for simulation and analysis. A sinusoidal excitation signal was used as the input excitation for the hydro-pneumatic spring's stroke in the simulation, and the characteristic curves of the hydro-pneumatic spring were obtained, as shown in Figure 6. In Figure 6a,b, the elastic force and stiffness of the hydro-pneumatic spring exhibit significant nonlinear characteristics. As the piston was compressed continuously, the elastic force and stiffness of the hydro-pneumatic spring increased at an accelerating rate. The above phenomenon manifests in the suspension as follows: When the vehicle is driving on a smooth, flat road, the relative displacement of the piston rod of the hydro-pneumatic spring is small, resulting in a low stiffness. When the vehicle is driving on rough, uneven terrain, the relative displacement of the piston rod is large, leading to a sharp increase in stiffness [32]. The variable stiffness characteristics exhibited by the hydro-pneumatic spring under these two different road conditions ensure that the vehicle can maintain a high driving speed in a changing driving environment. They also demonstrate their significant advantages in regulating vehicle suspension systems. Figure 6c indicates that during the compression stroke, both the check damping holes and the constant-flow damping holes were functional. The damping coefficient of the hydro-pneumatic spring is relatively small, exhibiting a smaller damping force of the hydro-pneumatic spring. During the rebound stroke, the check damping holes closed, leaving only the constant-flow damping holes functional. At this point, the damping coefficient of the hydro-pneumatic spring was high. It indicates a strong damping force that attenuated oscillations and provided vibration damping and cushioning effects. In Figure 6d, it can be seen that the damping of the hydro-pneumatic spring varies linearly with speed. It increases with speed and undergoes a turning point at a speed of 0 m/s. Both the through hole and one-way hole of the compression stroke can pass oil, while the oil of the recovery stroke is only allowed to pass through the through hole. The slope of the damping of the pressure recovery stroke relative to the velocity change is greater than the slope of the compression stroke, which attenuates the oscillation generated during the operation of the hydro-pneumatic spring. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Value** & **Name** & **Value** \\ \hline Hydraulic fluid density \(\rho/(\text{kg}/\text{m}^{3})\) & 850 & Diameter of check valve hole \(d_{d}/(\text{m})\) & 0.003 \\ Initial pressure of gas in accumulator \(P_{gc}/(\text{MPa})\) & 6 & Cross-sectional diameter of pipeline \(D_{p}/(\text{m})\) & 0.012 \\ Initial height of gas in accumulator \(h_{a}/(\text{m})\) & 0.2 & Pipeline length \(l/(\text{m})\) & 0.2 \\ Inner diameter of accumulator \(d/(\text{m})\) & 0.04 & Flow coefficient \(C_{d}\) & 0.61 \\ Inner diameter of piston \(D/(\text{kg}/\text{m}^{3})\) & 40 & Gas polytropic index \(\gamma\)[31] & 1.33 \\ Diameter of damping hole \(d_{c}/(\text{m})\) & 0.005 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the electro-hydraulic proportional hydraulic loading system. Figure 5: Simulation models of hydro-pneumatic springs. The output force of the hydro-pneumatic spring is the resultant force of the elastic force and the damping force, as shown in Figure 7. After the hydro-pneumatic spring moved for one cycle, the damping force curve formed a closed ellipsis. The area enclosed by the damping force curve represents the energy consumed by the hydro-pneumatic spring.