# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 2 Vehicle Model Establishment
### Establishment of the Equation of Motion
\tag{5}\]  
where  
\[A=\left[\begin{array}{ccccc}-\frac{c_{1}}{m_{p}}&\frac{c_{1}}{m_{p}}&0&- \frac{k_{1}}{m_{p}}&\frac{k_{1}}{m_{p}}&0\\ \frac{c_{1}}{m_{s}}&\frac{-c_{1}-c_{2}}{m_{s}}&\frac{c_{2}}{m_{s}}&\frac{k_{1 }}{m_{s}}&\frac{-(k_{1}+k_{2})}{m_{s}}&\frac{k_{2}}{m_{s}}\\ 0&\frac{of occupant acceleration to road speed and the frequency domain response function \(|H2|=\omega\Big{|}\frac{X_{\mathrm{S}}}{X_{r}(\omega)}\Big{|}\) of vehicle acceleration to road speed is obtained. The frequency response function curve is shown in Figure 2.  
It can be seen in Figure 2 that frequency response characteristic curve \(|\)H1 \(|\) of occupant acceleration. When there is no increase in control g = 0 and \(\tau\) = 0, the maximum amplitude-frequency characteristic of the vehicle body is generally about 6 Hz, the maximum amplitude is 15.9, and then the amplitude decreases slowly with the increase in frequency. From the frequency response characteristic curve of body acceleration \(|\)H2 \(|\), it can be seen that when the control g = 0 and \(\tau\) = 0 are not increased, the maximum amplitude-frequency characteristic of the body is generally within 5.9 Hz, and the maximum amplitude is 12.46. When it reaches 64.9 Hz, the body amplitude reaches the second peak value of 8.17 Hz. After that, the amplitude decreases slowly with increasing frequency.  
System stability is a powerful guarantee for the safe and reliable operation of time-delay control. In this paper, the stability switching method is used to analyze the stability of time-delay feedback control [37; 38; 39].  
To facilitate the analysis of system stability, the motion differential of the system, i.e., Equation (4), is normalized.  
To make the analyzed physical quantities have a broader theoretical guiding significance, the following auxiliary variables are introduced:  
\[dt^{*}=\sqrt{\frac{m_{p}}{k_{1}}}dt,\zeta_{1}=\frac{c_{1}}{\sqrt{m_{p}k_{1}}}, \zeta_{2}=\frac{c_{2}}{\sqrt{m_{p}k_{2}}},\gamma_{1}=\frac{m_{p}}{m_{s}}, \gamma_{2}=\frac{m_{p}}{m_{u}},\kappa_{1}=\frac{k_{2}}{k_{1}},\kappa_{2}=\frac {k_{3}}{k_{1}},\]  
\[g_{1}=\frac{\xi}{k_{1}},\tau^{*}=\sqrt{\frac{m_{p}}{k_{1}}}\tau\]  
For the convenience of writing, \(t^{*}\),\(\tau^{*}\) can be marked as \(t\),\(\tau\). Equations (1)-(3) can be simplified to:  
\[\ddot{x}_{p}+\zeta_{1}\big{(}\dot{x}_{p}-\dot{x}_{s}\big{)}+\big{(}x_{p}-x_{s} \big{)}=0 \tag{6}\]  
\[\ddot{x}_{s}-\gamma_{1}\zeta_{1}(\dot{x}_{p}-\dot{x}_{s})-\gamma_{1}\big{(}x_{ p}-x_{s}\big{)}+\gamma_{1}\xi_{2}(\dot{x}_{s}-\dot{x}_{u})\]  
\[+\gamma_{1}\kappa_{1}(x_{s}-x_{u})+\gamma_{1}\xi_{1}x_{u}(t-\tau)=0\]  
\[\ddot{x}_{u}-\gamma_{2}\zeta_{2}(\dot{x}_{s}-\dot{x}_{u})-\gamma_{2}\kappa_{1 }(x_{s}-x_{u})\]  
\[+\gamma_{2}\kappa_{2}(x_{u}-x_{r})-\gamma_{2}g_{1}x_{u}(t-\tau)=0\]  
Figure 2: Frequency response characteristic curve. According to the complete discriminant system of polynomials [40; 41], it is assumed that \(f(x)\) is a polynomial of degree n.  
\(D_{1}(f),D_{2}(f),\cdots,D_{n}(f)\) is its discriminant sequence. Assume that the number of sign changes in the discriminant sequence symbol table is \(v\), where the number of non-zero terms is \(l\). If it satisfies \(D_{l}(f)\
eq 0,D_{m}(f)=0(m>l)\), then the logarithm of the complex root of \(f(x)\) is \(v\);  
The number of distinct real roots of \(f(x)\) is \(l-2v.f(x)\) has multiple roots if and only if \(l<v\). The full-time-delay stability of the suspension system is solved by the full-time-delay stability judgment theorem of the time-delay dynamic system. The condition of full-delay stability for linear dynamical systems with a single delay is that the characteristic polynomial \(D(s,\tau)=P(s)+Q_{i}(-s\tau),i=1,2,\cdots,n\) of the system is stable at \(\tau=0\). The coefficient of the first term of \(P(s)\) is 1, and its degree satisfies \(\deg(p)=n>\deg(Q_{i})\), \(i=1,2,\cdots,n\). In addition, at \(\tau\geq 0\), the characteristic equation of the system \(D(iv,\tau)=0\) has no real solution.