# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 5 Dynamic Performance Simulation and Analysis
### Simulation Results  
After obtaining the time-delay control parameters under random excitation through the adaptive particle swarm optimization algorithm, according to the differential equation of motion of the vehicle's active suspension system and external excitation input, the passive suspension, the active suspension based on backstepping control, and the external excitation input are established. A simulation model of active suspension with time-delay control and a comparative analysis of its dynamic performance are provided. Under the condition of random excitation, the changed results of root mean square values of passenger acceleration, body acceleration, suspension dynamic deflection, and tire dynamic load are shown in Table 4.  
Figure 6: Comparison chart of suspension performance simulation results under harmonic excitation; (**a**) Passenger vertical acceleration comparison chart; (**b**) vehicle body vertical acceleration comparison chart; (**c**) suspension dynamic deflection comparison chart; (**d**) tire dynamic load comparison chart. By comparing the values in Table 4, it can be seen that vehicle comfort is measured by vehicle body acceleration and passenger acceleration. The two kinds of active suspension are improved to some extent compared to the passive suspension. Two kinds of active control suspension are affected by random excitation. The active suspension eliminates the external force and causes the suspension dynamic to travel a larger distance than the passive suspension dynamic travels. However, it is within the scope of the design [49]. Compared with the passive suspension system and the active suspension based on inverse control, the performance of the active suspension system with time-delay control under random excitation has been significantly improved. For the passive suspension system, the fluctuation range of passenger vertical acceleration is \(0.4647\) m/s\({}^{2}\)\(\sim\)\(-0.4585\) m/s\({}^{2}\), and the fluctuation range of body vertical acceleration is \(0.6239\) m/s\({}^{2}\)\(\sim\)\(-0.6207\) m/s\({}^{2}\). The fluctuation range of suspension dynamic displacement is \(0.00728\) m \(\sim\)\(-0.00788\) m, and the fluctuation range of tire dynamic load is \(238.3\) N \(\sim\)\(-238.6\) N. For the active suspension system based on backstepping control, the fluctuation range of passenger vertical acceleration is \(0.4478\) m/s\({}^{2}\)\(\sim\)\(-0.4457\) m/s\({}^{2}\), and the fluctuation range of body vertical acceleration is \(0.0330\) m/s\({}^{2}\)\(\sim\)\(-0.3309\) m/s\({}^{2}\). The fluctuation range of suspension dynamic displacement is \(0.01522\) m \(\sim\)\(-0.11418\) m, and the fluctuation range of tire dynamic load is \(155.1\) N \(\sim\)\(-154.8\) N. However, the time-delay control damping control strategy adopted in this paper makes the fluctuation range of passenger vertical acceleration \(0.2876\) m/s\({}^{2}\)\(\sim\)\(-0.2768\) m/s\({}^{2}\), and the fluctuation range of vehicle body vertical acceleration \(0.2941\) m/s\({}^{2}\)\(\sim\)\(-0.3144\) m/s\({}^{2}\). Suspension dynamic displacement fluctuation range is \(0.00742\) m \(\sim\)\(-0.007869\) m, and tire dynamic load fluctuation range is \(140.1\) N \(\sim\)\(-135.5\) N. Compared with the performance of the passive suspension, the active suspension control strategy with time-delay control designed in this paper can optimize passengers' vertical acceleration by \(29.99\%\). The body vertical acceleration was optimized by \(47.23\%\), and the tire dynamic load was optimized by \(40.26\%\). Compared with the active suspension based on inverse control, the active suspension control strategy with delay feedback control is designed in this paper. It can optimize passenger vertical acceleration by \(13.63\%\), body vertical acceleration by \(28.38\%\), and tire dynamic load by \(1.42\%\).