# Time-Delay Vibration Reduction Control of 3-DOF Vehicle Model with Vehicle Seat
## 2 Vehicle Model Establishment
### Establishment of the Equation of Motion
From Equations (6)-(8), the characteristic equation can be obtained as  
\[D(s,\tau)=P(s)+Q(s)e^{-s\tau} \tag{9}\]  
The specific form is  
\[D(s,\tau)=s^{6}+a_{1}s^{5}+a_{2}s^{4}+a_{3}s^{3}+a_{4}s^{2}+a_{5}s+a_{6} \tag{10}\]  
where  
\[ \begin{array}{c}a_{1}=\zeta_{1}+\zeta_{1}\gamma_{1}+\zeta_{2}\gamma_{1}+ \zeta_{2}\gamma_{2}\\ a_{2}=\kappa_{1}\gamma_{1}+\kappa_{1}\gamma_{2}+\kappa_{2}\gamma_{2}+2\zeta_{2 }{}^{2}\gamma_{1}\gamma_{2}+\zeta_{1}\zeta_{2}\gamma_{1}\\ +\zeta_{1}\xi_{2}\gamma_{2}-g_{1}\gamma_{2}e^{-s\tau}+\zeta_{1}\xi_{2}\gamma_{ 1}\gamma_{2}+\\ a_{3}=\zeta_{2}\gamma_{1}-\zeta_{1}\gamma_{1}+\zeta_{2}\gamma_{2}+\kappa_{1} \zeta_{1}\gamma_{1}+\kappa_{1}\zeta_{1}\gamma{2}\\ +\kappa{2}\zeta_{1}\gamma{2}-g_{1}\zeta_{1}\gamma_{2}e^{-s\tau}+2\zeta_{1} \xi_{2}^{2}\gamma_{1}\gamma_{2}\\ +\kappa{1}\zeta_{1}\gamma_{1}\gamma_{2}+2\kappa_{1}\zeta_{2}\gamma_{1}\gamma_ {2}+\kappa_{2}\zeta_{1}\gamma_{1}\gamma_{2}\\ +\kappa_{2}\zeta_{2}\gamma_{1}\gamma_{2}-g_{1}\xi_{1}\gamma_{1}\gamma_{2}e^{- s\tau}\\ a_{4}=\kappa_{1}\gamma_{1}-\gamma_{1}+\kappa_{1}\gamma_{2}+\kappa_{2}\gamma_{2}+2\zeta_{2}{}^{2} \gamma_{1}\gamma_{2}-g_{1}\gamma_{2}e^{-s\tau}\\ -\zeta_{1}\xi_{2}\gamma_{1}\gamma_{2}+\kappa_{1}\kappa_{2}\gamma_{1}\gamma_{2 }+2\kappa_{1}\xi_{1}\zeta_{2}\gamma_{1}\gamma_{2}+\kappa_{2}\xi_{1}\xi_{2} \gamma_{1}\gamma_{2}\\ a_{5}=2\kappa_{1}\zeta_{2}\gamma_{1}\gamma_{2}-\kappa_{1}\xi_{1}\gamma_{1} \gamma_{2}-\zeta_{2}\gamma_{1}\gamma_{2}\\ -\kappa_{2}\zeta_{1}\gamma_{1}\gamma_{2}\ +\kappa_{2}\zeta_{2}\gamma_{1}\gamma_{2}+ \kappa_{1}\kappa_{2}\zeta_{1}\gamma_{1}\gamma_{2}+\kappa_{1}\xi_{1}\zeta_{1} \gamma_{1}\gamma_{2}\mathrm{e}^{-s\tau}\\ \end{array} \]  
\[ \begin{array}{c}a_{6}=g_{1}\gamma_{1}\gamma_{2}\mathrm{e}^{-s\tau}\ -\kappa_{2}\gamma_{1}\gamma_{2}-\kappa_{1}\gamma_{1}\gamma_{2}+\kappa_{1} \kappa_{2}\gamma_{1}\gamma_{2}\\ \end{array} \]  
For all \(\tau\geq 0,\ D(iw,\tau)=0\), there is no real number solution \(w\) if and only if the following polynomial is true:  
\[F(w)=w^{12}+b_{1}w^{10}+b_{2}w^{8}+b_{3}w^{6}+b_{4}w^{4}+b_{5}w^{2}+b_{6} \tag{11}\]  
See Appendix A for \(b_{1},b_{2},\cdots,b_{6}\) in the formula. According to the Strum sequence discrimination method, the Strum sequence \(F(w)\) can be calculated by computing software. \[D_{1},D_{2},D_{3},D_{4},D_{5},D_{6},D_{7},D_{8},D_{9},D_{10},D_{11},D_{12} \tag{12}\]  
In the formula, the number of polynomials \(D_{1},D_{2},D_{3},D_{4},D_{5},D_{6},D_{7},D_{8},D_{9},D_{10},D_{11}\), \(D_{12}\) is relatively high. The Maple program DISCR can be used to complete the program, as shown in Appendix A. To realize polynomial \(F(w)\) without real roots, the sign change of the sequence of \(F(w)\) discriminant should satisfy \(l=2v\) according to the Strum test. Based on the parameters of a certain vehicle, as shown in Table 1, this paper studies the influence of suspension damping and control gain on the system's time-delay stability. It takes \(c_{2}\) as the adjustable damping to draw the values of \(c_{2}\) and g within a certain value range that meet the conditions. The parameter plane satisfying the condition was divided into several regions according to the full delay stability condition, as shown in Figure 3. Different regions are different discriminant sequences, as shown in Table 2.  
**Theorem 1**. : _Assuming that the pure imaginary root of the characteristic equation of system (9) is not a solution of Q(iv) = 0, then [42]: If \(F(w)=0\) has no real root, then the system is fully time-delay stable; that is, the system does not undergo stability switching. If \(F(w)=0\) has only a positive single root \(w\), then the corresponding delay-free system is asymptotically stable. There is some \(\tau_{c}>0\) that makes the system stable in \(\tau>\tau_{c}\). However, it is unstable for all \(\tau>\tau_{c}\); that is, the system has a stable switch. When the corresponding delay-free system is unstable, the system is unstable for all time delays; that is, the system does not undergo stability switching.