## Materials and Methods
### Theory of Dynamic System
#### 2.2.1 Test for Force-Displacement Curve  
In order to measure only the spring constant of the suspension, the cylinder piston was moved at a low speed so as not to generate a damping force, and the reaction force in a static state was measured. Assuming that the speed was very low and could be ignored, the suspension reaction force can be expressed as Equation (2). \[F=kx \tag{2}\]  
Assuming that there was no damping force (c\(\dot{x}\)) in a hydro-pneumatic suspension, the hydraulic circuit diagram can be expressed as shown in Figure 2a. In a cylinder with a rod connected in one direction, the cross-sectional areas of the head side and rod side was different. When the hydraulic system was in a steady state, the same pressure was formed all over the interior, and the same vertical pressure was applied to all sides by [PERSON]'s law; thus, the force was generated by the difference in the area of the cylinder. The area of the cylinder is expressed as Equation (3). If the pressure in the hydraulic system was \(P_{0}\) when the initial compression force was \(F_{0}\) and the cylinder was at the initial position \(x_{0}\), the relationship expressed in Equation (4) could be established. \[\Delta A=A_{head}-A_{rod}=\frac{\pi}{4}\left(D_{heat}{}^{2}-D_{rod}{}^{2}\right) \tag{3}\]  
\[F_{0}=kx_{0}=P_{0}\Delta A \tag{4}\]  
where:  
\(\Delta A\): area difference between cylinder head side and rod side, m\({}^{2}\);  
\(A_{land}\): area of cylinder head side, m\({}^{2}\);  
\(A_{rod}\): area of cylinder rod side, m\({}^{2}\);  
\(D_{head}\): diameter of cylinder head side, m;  
\(D_{rod}\): diameter of cylinder rod side, m;  
\(x_{0}\): initial position of cylinder, m;  
\(F_{0}\): initial compression force, N;  
\(P_{0}\): initial pressure of hydraulic system, N/m\({}^{2}\). When a force in the retraction direction was applied to the suspension, the cylinder moved in the retraction direction, as shown in Figure 2b, and the hydraulic oil escaped from the cylinder head chamber and entered the cylinder rod chamber. Because the amount of space required by the hydraulic oil to move around was less than the volume occupied by the cylinder rod, the remaining hydraulic oil flowed into the accumulator. As the accumulator was filled with hydraulic oil, the gas in the accumulator was compressed and the pressure in the hydraulic system increased, which increased the force to support the weight. This change proceeded until the force to press the suspension and the force supported by the pressure and area difference inside the cylinder were in equilibrium. Figure 2: Conceptual diagram of hydraulic oil movement according to cylinder movement direction: (a) neutral (b) compression and (c) extension. When a force in the extension direction was applied to the suspension, the cylinder moved in the extension direction, as shown in Figure 2c, and the hydraulic oil escaped from the cylinder rod chamber and entered the cylinder head chamber. In this case, because the amount of hydraulic oil was less than the volume of the rod, the hydraulic oil in the accumulator escaped and moved to the cylinder head chamber. As the gas inside the accumulator expanded, the pressure in the hydraulic system decreased and then stopped until the force was re-equilibrated. If the process in Figure 2 is expressed as a force-displacement relationship, the results can be expected, as shown in Figure 3. As shown in Equation (2), the ideal linear system had a straight-lined slope, as shown in Figure 3a, and the slope at this time was defined as the spring constant (k). However, the hydro-pneumatic suspension had nonlinear characteristics in the force-displacement relationship, as shown in Figure 3b, owing to the compressibility of air and hydraulic oil. In this case, the spring constant could not be expressed as a constant, and had to be expressed as a function or table. In an actual suspension system, hysteresis occurs because a force that hinders the movement of the piston acts by various friction elements.