## 5 Conclusions  
This research employs NESIs within a half-vehicle seat system to enhance the performance of the vehicle in terms of vertical and pitching movements; develops a dynamic model for the half-vehicle seat system integrated with NESIs; analyzes its response to suffering from pavement harmonic excitation using the HBM and the PALM; and studies the performance using eight evaluation indexes. The optimal structural parameters of the NESIs are also acquired according to the genetic algorithm. In conclusion, the findings can be summarized as follows:  
(1) The half-vehicle seat system integrated with NESIs is a nonlinear mechanical system; it used eight DOFs and nonlinear cube stiffness terms generated by the NESIs, and the analytical results acquired by the HBM and the PALM match well with the numerical solutions, which validates the applicability of the HBM and the PALM for solving such complex nonlinear systems effectively. (2) For pavement harmonic and random excitations, using a NESI can decrease the resonance peaks and RMS values of \(I_{ssf}\), \(I_{sssr}\), \(J_{sssa}\), \(J_{DTLf}\), and \(J_{DTLr}\) for a half-vehicle seat system, and the resonance peaks and RMS values of the other three acceleration evaluation indexes are larger. (3) The structural parameters of the NESIs are optimized by using the genetic algorithm; compared with the initial NESIs, the RMS(\(z_{sa}\)), RMS(\(\varphi_{a}\)), RMS(\(z_{\text{sat}}\)), and RMS(\(J_{sssa}\)) of the half-vehicle seat system integrated with the optimized NESIs decrease by 23.97%, 27.48%, 23.59%, and 14.29%, respectively; furthermore, RMS(\(J_{ssf}\)), RMS(\(J_{sssr}\)), RMS(\(J_{DTLf}\)) and  
Figure 14: Comparison of \(z_{sa}\) and \(\varphi_{a}\) for the half-vehicle seat system integrated with original and optimized NESIs suffering from pavement random excitation. \begin{table}
\begin{tabular}{c c c c} \hline \hline
\begin{tabular}{c} **Dynamic** \\ **Performance Index** \\ \end{tabular} & **Original** & **NESI** & **Optimized NESI** \\ \hline RMS(\(z_{sa}\)) & 0.4498 ms\({}^{-2}\) & 0.4673 ms\({}^{-2}\) & 0.3533 ms\({}^{-2}\) \\ RMS(\(\varphi_{a}\)) & 0.3336 rads\({}^{-2}\) & 0.3573 rads\({}^{-2}\) & 0.2591 rads\({}^{-2}\) \\ RMS(\(z_{aa}\)) & 0.2802 ms\({}^{-2}\) & 0.2900 ms\({}^{-2}\) & 0.2216 ms\({}^{-2}\) \\ RMS(\(I_{sss}\)) & 0.0056 m & 0.0049 m & 0.0054 m \\ RMS(\(I_{sss}\)) & 0.0059 m & 0.0050 m & 0.0054 m \\ RMS(\(J_{sssa}\)) & 0.0020 m & 0.0014 m & 0.0012 m \\ RMS(\(J_{DTLf}\)) & 536.7476 N & 464.3042 N & 478.9819 N \\ RMS(\(J_{DTLr}\)) & 529.3295 N & 450.4304 N & 459.2332 N \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of the RMS values of the eight evaluation indexes for the half-vehicle seat system integrated with original and optimized NESIs suffering from pavement random excitation. RMS(\(J_{DTL}\)) satisfy the limit conditions; in contrast with the original half-vehicle seat system, the system integrated with the optimized NESIs can improve all eight evaluation indexes. Therefore, substituting the inerter in place of the mass of the NES that constitute the NESI meets lightweight design requirements, and using NESIs in a half-vehicle seat system can improve its dynamic performance, which offers an efficient option for vibration attenuation in vehicle engineering. Conceptualization, formal analysis, Y.Z. Data curation, writing--original draft, C.R. Validation, writing--editing, H.M. Supervision, writing--review, Y.W. All authors have read and agreed to the published version of the manuscript. The research described in this paper is supported by the National Natural Science Foundation of China (Grant No. 12172153, 51805216), the China Postdoctoral Science Foundation (Grant No. 2023M731668), the Major Project of Basic Science (Natural Science) of the Jiangsu Higher Education Institutions (22 KJ(A410001), and a project funded by the Youth Talent Cultivation Program of Jiangsu University. The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy.