## 6 Conclusions  
Based on the conventional wheel structure, a new vibration-absorbing wheel structure with time-delay feedback control was invented. The time-delay independent stability and time-delay dependent stability of the system are investigated, and the time-delay feedback control parameters are optimized using the PSO algorithm. The proposed new vibration-absorbing wheel structure with time-delay feedback control is proved effective, according to the comprehensive analysis. The following conclusions can be obtained:  
(1) Based on the theory of time-delay independent stability of two-degree-of-freedom vibration system, this paper extends it to the three-degree-of-freedom vibration system. In the range of time-delay independent stability, the system is stable at any value of the time delay. The time-delay independent stability of the system is related to the damping, and the range of time-delay independent stability will increase with the increase in damping. However, in the time-delay dependent region, the stability of the system switches with the increase in time delay and eventually becomes unstable.  
(2) The size of the time delay affects the stability of a system. Time delay can either make a system unstable or make an unstable system stable. Choosing the proper size of time delay can improve the vehicle vibration reduction performance.  
(3) The performance of the new vibration-absorbing wheel structure on the vehicle vibration reduction was analyzed in different external excitation. Compared with the passive suspension, TDVAWS improved the RMS values of SMA and DTD by 92.41% and 54.13% at 11.25 Hz; TDVAWS improved the RMS values of SMA and DTD by 15.21% and 12.52% under random excitation. It can be concluded that the new vibration-absorbing wheel structure with the time-delay feedback control can effectively solve the contradiction between SMA and the DTD, and better improve the ride comfort and handling stability. This can provide a useful reference for further experimental investigation.  
TDVAWS has improved ride comfort and handling stability, but its performance needs further improvement. This needs to be considered, as its time delay varies with time. It means that new optimization methods and solution algorithms need to be created.  
Original draft, K.M.; software, Y.Z., Y.C. ([PERSON]), and P.Z.; writing--review and editing, K.M., Y.C. ([PERSON]), Y.Z. and C.R. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (Grant No. 51275280).  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Performance Indexes** & **Conventional Suspension** & **PVAWS** & **TDVAWS** \\ \hline RMS of SMA \(\mathrm{(m/s^{2})}\) & 1.2342 & 1.1158 & 1.0465 \\ RMS of SD \(\mathrm{(mm)}\) & 11.0682 & 11.0286 & 11.0089 \\ RMS of DTD \(\mathrm{(mm)}\) & 4.0993 & 3.9222 & 3.5861 \\ \hline \hline \end{tabular}
\end{table}
Table 6: RMS of vehicle performance indexes under random excitation.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.