## 4 Numerical Simulation Results
### Numerical Simulation  
In this subsection, displacement transmissibility is adopted to measure the performance of the proposed system under harmonic excitation. There are three types of tests that may be used to investigate the performance of the proposed system under different amplitudes, damping coefficients, and loads. An equivalent linear mass-spring-damper system is introduced as a comparison to the proposed system in all tests. The displacement transmissibility results in this section were simulated using MATLAB/Simulink\({}^{\text{\textregistered}}\) software. Equation (10) along with Equations (13) and (15) were used to find the values of the geometrical and stiffness parameters needed based on the force acting on the system. Furthermore, a linearization toolbox was used to perform the linear analysis of nonlinear models by assigning the base displacement (\(d\)) as an input and the seat displacement (\(x\)) as an output. As noted in Section 3 and according to Equation (11), the natural frequency of the proposed system is not only changing with the chosen parameters but is also changing with the amplitude. Consequently, to verify this point, the effect of increasing the amplitude on the natural frequency is illustrated in Figure 11. Thus, the isolation region is shifted to a larger frequency at a constant load, e.g., \(m=100\) kg. Figure 11 also shows that the damping ratio \(\zeta\) changes with the natural frequency, which gives an indication for not using a constant \(\zeta\). Therefore, in this test, it is assumed that the air spring membrane has a damping coefficient, i.e., \(c=70\) kg/s. Table 2 shows the frequency when the isolation starts (0 dB) at each amplitude. It is good to note that the change in the frequency is linearly proportional to the amplitude, which was given by Equation (11). In contrast, although the amplitude change does not affect the equivalent linear isolator due to its constant natural frequency, it lacks the ability to deal with low frequencies and small amplitudes excitements; this is not the case with the proposed system due to its flexibility. Figure 10: Force-displacement and stiffness–displacement curves for different loads. It is also imperative to perform tests regarding the damping case, especially in the proposed system, where both of the natural frequency \(\omega_{n}\) and the damping coefficient \(c\) can change the damping ratio \(\zeta\). Figure 12 shows a comparison between the proposed system and the equivalent linear system for different damping coefficients. It can be clearly observed that the proposed system has advantages in the considered aspects, where it has a broader frequency range of the isolation along with the low-peak transmissibility. Table 3 shows the frequency values when the isolation starts (0 dB) at each damping coefficient. It can be noted that the natural frequency \(\omega_{n}\) of the proposed system is smaller than the equivalent linear system one. Therefore, this allows the damping ratio \(\zeta\) to be larger at a constant damping coefficient \(c\) and achieving a lower damping frequency \(\omega_{d}\). Another aspect that cannot be overlooked is the effects of the load variation on performance. Therefore, Figure 13 is used to compare performance of the proposed system and the equivalent linear system at different loads. Similar to the previous test, the proposed system has better isolation and lower peak transmissibility. It is noted that the equivalent linear isolator has a big deflection due to the added load, which has a force value that is larger than the vertical spring force \(k_{p}\). In contrast, the proposed system keeps the load at the equilibrium position thanks to the air spring. Table 4 shows the frequency when the isolation starts (0 dB) at each load. It can also be seen that although the vertical stiffness (\(k_{p}\) and \(k_{\text{as}}\)) of the proposed system is larger than the linear isolator stiffness (\(k_{p}\) only), it still yields good results.