## 4 Numerical Simulation Results
### Prototype Design  
In this subsection, the proposed isolator parameters are chosen using the equations that were derived earlier. One of the significant parameters is the geometrical parameter, in which the proposed isolator can fit under the seat. The geometric values of \(L_{0}\) and \(a_{0}\) are chosen to be 110 mm and 100 mm, respectively, to fit the system under the drivers' seat of B and C- segment cars [33]. The vertical distance was calculated using the Pythagorean theorem:  
\[h_{0}=\sqrt{L_{0}^{2}-a_{0}^{2}}=\sqrt{110^{2}-100^{2}}=45.5\;\text{mm}, \tag{22}\]  
consequently,  
\[\theta=tan^{-1}=\frac{45.5}{100}=24.5^{\circ} \tag{23}\]  
As noted in Section 3, the larger oblique angle yields better isolation. However, a larger oblique angle means a longer vertical spring should be chosen, which is not preferred due to the system's size. Therefore, the obtained oblique angle seems a reasonable choice. Assuming that the mass of the seat is 20 kg and the minimum mass of the driver is 40 kg, the stiffness of the vertical spring \(k_{p}\) will equal 12.85 N/mm. Any additional mass over the design mass will be considered by the air spring (\(k_{\text{as}}\
eq 0\)). To achieve zero stiffness, the geometric ratio \(\lambda\) must equal the stiffness ratio \(\alpha\), as was shown by Equation (15). The initial horizontal adjustment \(x_{h}\) is set to zero, and the stiffness of the oblique spring \(k_{n}\) will equal 64.22 N/mm. Figure 9 shows the proposed isolator prototype. The prototype consists of two oblique springs, a vertical spring, an air spring, linear guide rails, upper and lower frame, and two adjustable threads. The adjustable threads are used as horizontal adjustments to ensure that the geometric ratio \(\lambda\) is always equal to the stiffness ratio \(\alpha\), regardless of the load value. In  
other words, the adjustable threads are used to increase or decrease the distance \(x_{h}\) based on the load variation, and hence the geometry of the system is altered and the geometric ratio is changed. For this study, the proposed system was tested for specific amounts of load and, consequently, a fixed thread was used (constant \(x_{h}\)), as shown in Figure 9. Hence, for the load variation, it is advisable to use a linear actuator to automatically adjust \(x_{h}\), as discussed in Section 3. Regarding the load variation, an air spring is used to support any load that exceeds the limit that the vertical spring can support. According to the air spring manufacturer (1M1A-1 Firestone), the relationship between the weight, the pressure, and the stiffness is linear as long as the air spring is used at its recommended design height [34]. Therefore, the air spring is mounted on an inverted U-shaped frame to ensure that when the air spring is inflated with a pressure related to the weight and reached its recommended height, the QZS mechanism will always reach the equilibrium position, thus achieving the optimum isolation. The linear guide rails in the middle are used to guarantee that the movement is only in the z-direction and prevent any bending moment. The selected parameters of the proposed isolator prototype are shown in Table 1. Note that the mass is the summation of the seat's mass and the driver's minimum expected mass. Based on the theory and the proposed isolator prototype's theoretical parameters, it can be said with certainty that the prototype exhibits the quasi-zero stiffness behaviour. For further validation, force-displacement and stiffness-displacement curves are introduced. Although the prototype is always at the equilibrium position in the static case, the air spring is an important aspect that should be considered, or it will severely affect the performance. With the load variation, the air amount pumped into the air spring changes depending on the linear relationship noted before; this causes a change in the air spring's stiffness, thus changing the stiffness ratio \(\alpha\). To solve this matter, the geometrical ratio \(\lambda\)