# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 5 Dynamic Performance Simulation and Analysis  
### Suspension Dynamic Performance Analysis under Harmonic Excitation Input  
At present, the evaluation index of human comfort is the weighted root mean square value of acceleration proposed by ISO2631 standard. The calculation formula of root mean square value of weighted acceleration is  
\[a_{w}=\left[(1.4a_{xw})^{2}+(1.4a_{yw})^{2}+a_{zw}\right]^{1/2} \tag{22}\]  
where \(a_{xw}\) is the root mean square value of longitudinal acceleration, \(a_{yw}\) is the root mean square value of lateral acceleration, and \(a_{zw}\) is the root mean square value of vertical acceleration. Because this paper mainly evaluates the impact of vertical vibration on comfort, only the root mean square value of vertical vibration weighted acceleration is calculated. The formula for calculating the frequency weighted value \(w(f)\) of the root mean square value of vibration-frequency-weighted acceleration at different frequencies is  
\[w(f)\left\{ \begin{array}{ll}0.5&(0.5,2]\\ f/4&(2,4]\\ 1&(4,12.5]\\ 12.5/f&(12.5,80]\end{array} \right. \tag{23}\]  
Figure 5: Time-delay control parameter-change diagram. (**a**) Time-delay parameter-change diagram under harmonic excitation. (**b**) Time-delay parameter-change graph under random excitation. To verify the feasibility and effectiveness of the optimized time-delay control parameter method and the time-delay feedback-vibration-reduction control method proposed in this paper, the passive suspension, the active suspension based on backstepping control, and the time-delay active suspension with optimal parameter feedback control are combined. The performance indicators of the rack are compared by time-domain simulation, and the results of the simulation comparison are shown in Figure 6[47]. To quantitatively analyze the damping performance of the active suspension with time-delay control, Table 3 lists the passengers of the time-delay active suspension system, passive suspension system, and back-control active suspension system under the optimal parameters. Root mean square value of acceleration, body acceleration, and suspension dynamic deflection, tire dynamic load, are used to calculate the corresponding change in percentage. As can be seen from Figure 6, the performance of the active suspension system with time-delay control under harmonic excitation is significantly improved compared with that of the passive suspension system. For the passive suspension system, the fluctuation range of passenger vertical acceleration is \(3.815\) m/s\({}^{2}\)\(\sim-3.815\) m/s\({}^{2}\), and the fluctuation range of body vertical acceleration is \(5.046\) m/s\({}^{2}\)\(\sim-5.046\) m/s\({}^{2}\). The fluctuation range of the suspension dynamic displacement is \(0.08752\) m \(\sim-0.08752\) m, and the fluctuation range of the tire dynamic load is \(1671\) N \(\sim-1671\) N. Meanwhile, for the active suspension system based on backstepping control, the fluctuation range of passenger vertical acceleration is \(3.344\) m/s\({}^{2}\)\(\sim-3.344\) m/s\({}^{2}\), the fluctuation range of body vertical acceleration is \(2.522\) m/s\({}^{2}\)\(\sim-2.522\) m/s\({}^{2}\), and the fluctuation range of suspension dynamic displacement is \(0.0628\) m \(\sim-0.0628\) m. The range of the tire dynamic load fluctuation is \(1157\) N \(\sim-1157\) N. However, the time-delay control strategy adopted in this paper determines the passenger vertical acceleration, and the vibration of the car body is completely eliminated after the vertical acceleration is stabilized. The suspension-dynamic-displacement fluctuation range is \(0.05043\) m \(\sim-0.05043\) m, tire dynamic load fluctuation range is \(89.77\) N \(\sim-89.77\) N. Compared with the ordinary active-suspension backstepping-control strategy, the time-delay damping control active suspension designed in this paper significantly reduces the fluctuation range of the suspension performance index.