## 6 Conclusions  
In this paper, the NESI is applied in the seat suspension and vehicle suspension for the HVS system, and the MFAC method based on the genetic algorithm is designed to further enhance the dynamic performance. The dynamic model of the active HVS system coupled with NESI using the MFAC method is established; its dynamic response under pavement random and shock excitations are acquired, and the dynamic performance is evaluated. The effectiveness of the MFAC method is revealed and the influence of the installed position of MFAC on the dynamic performance of the HVS system is studied. The main conclusions are as follows:  
1. The RMS(\(z_{sa}\)), RMS(\(\varphi_{a}\)), RMS(\(z_{aa}\)), RMS(\(J_{ssdf}\)) and RMS(\(J_{sssr}\)) for the active HVS system coupled with NESI using the MFAC method under random pavement excitation are significantly smaller than those of the original HVS system, the HVS system coupled with NESI, and the active HVS system coupled with NESI using the PID control method; the RMS(\(J_{DTLf}\)) and RMS(\(J_{DTLr}\)) are also slightly smaller than those of the other three HVS systems, which reflects the best dynamic performance among the four HVS systems and demonstrates the effectiveness of MFAC method.
2. The peak amplitudes of \(z_{aa}\), \(\varphi_{a}\), \(z_{aa}\), \(J_{ssdf}\) and \(J_{sssr}\) for the active HVS system coupled with NESI using the MFAC method under pavement shock excitation are smaller than those of the other three HVS systems, and the vibration attenuation time also shortens; the peak amplitudes of \(J_{DTLf}\) and \(J_{DTLr}\) are larger than those of the other three HVS systems, while the vibration attenuation time decreases, which also shows the benefit of the MFAC method.
3. When the MFAC is acting on both the seat and vehicle suspensions, the active HVS system obtains the best dynamic performance compared with the active HVS system where the MFAC is acting on the seat suspension, and the MFAC is acting on the front and rear suspensions.  
Thus, the designed MFAC method can effectively improve the dynamic performance of the HVS system coupled with NESI, which provides guiding significance for vehicle vibration attenuation. In future work, the dynamic characteristics of the full-vehicle-seat system coupled with NESI will be analyzed; the MFAC method will be used in the full-vehicle-seat system to further improve the dynamic performance, and the evaluation criteria of vehicle dynamic performance, including vehicle ride comfort, will be further expanded. Furthermore, experimental research will be carried out to verify the validity of the theoretical results.  
Y.Z.: Conceptualization, formal analysis. C.R.: Data curation, writing--original draft. H.M.: Validation, writing--editing. Y.W.: Supervision, writing--review. 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 State Key Laboratory of Mechanical Transmission For Advanced Equipment (Chongqing University) (Grant No. SKLMT-MSKKT-202314), the Major Project of Basic Science (Natural Science) of the Jiangsu Higher Education Institutions (22 KJA410001), the Changzhou Science and Technology Program (CZ20240019) and the project funded by the Youth Talent Cultivation Program of Jiangsu University.  
Not applicable.  
The data that support the findings of this paper are available from the corresponding author upon reasonable request.  
The authors declare no potential conflicts of interest concerning the research, authorship, and/or publication of this article.