# An Experimental Validated Control Strategy of Maglev Vehicle-Bridge Self-Excited Vibration
## 3 Principle of Self-Excited Vibration
### Stability of Vehicle-Bridge Interaction System  
To study the stability of the maglev vehicle-bridge interaction system, the transfer function from disturbance \(F_{\rm d}\) to the displacement \(y_{\rm B}\) of bridge should be calculated again when the displacement of bridge \(y_{\rm B}\) is considered. In light of Figure 2, it gives that  
\[T_{2}(s)=\frac{G_{\rm B}(s)}{1-G_{\rm E}(s)H_{\rm E}(s)-G_{\rm B}(s)H_{\rm B}(s)} \tag{24}\]  
where  
\[\left\{\begin{array}{l}G_{\rm B}(s)=\frac{\sigma}{m_{\rm B}(s^{2}+2k_{\rm B }\omega_{\rm B}s+\omega_{\rm B}^{2})}\\ H_{\rm B}(s)=-\frac{k_{\rm B}k_{\rm F}^{2}}{2 NAs+k_{\rm B}}\cdot\overline{k}_{ \rm P}\end{array}\right. \tag{25}\]  
Combining Equations (21), (24) and (25), the characteristic equation of the interaction system is  
\[a_{5}s^{5}+a_{4}s^{4}+a_{3}s^{3}+a_{2}s^{2}+a_{1}s+a_{0}=0 \tag{26}\]  
where \(a_{5}=2 NAm_{\rm B}m_{\rm E}\), \(a_{4}=m_{\rm B}m_{\rm E}(k_{\rm B}+4 NA_{\rm B}\omega_{\rm B})+k_{\rm B}k_{\rm F }m_{\rm B}k_{\rm a}\), \(a_{3}=m_{\rm B}m_{\rm E}(2k_{\rm B}k_{\rm B}\omega_{\rm B}+2 NA\omega_{\rm B}^ {2})\\ +\ k_{\rm B}k_{\rm F}m_{\rm B}(2k_{\rm a}k_{\rm B}\omega_{\rm B}+k_{\rm d})\), \(a_{2}=k_{\rm B}\omega_{\rm B}^{2}m_{\rm B}m_{\rm E}\ +\ \alpha m_{\rm E}k_{\rm B}k_{\rm F}\overline{k}_{ \rm p}\ +\ k_{\rm B}k_{\rm F}m_{\rm B}\big{(}k_{\rm p}+2k_{\rm d}\xi_{\rm B} \omega_{\rm B}+k_{\rm a}\omega_{\rm B}^{2}\big{)}\), \(a_{1}=k_{\rm B}k_{\rm F}m_{\rm B}\big{(}k_{\rm d}\omega_{\rm B}^{2}+2k_{\rm p }\xi_{\rm B}\omega_{\rm B}\big{)}\), and \(a_{0}=k_{\rm B}k_{\rm F}m_{\rm B}k_{\rm p}\omega_{\rm B}^{2}\). Generally, the roots of Equation (26) are calculable provided that the parameters are definite, which are denoted as \(x_{1,2}=R_{1}\pm jI_{1}\), \(x_{3,A}=R_{2}\pm jI_{2}\) and \(x_{5}=R_{3}\). If the three real parts \(R_{1}\), \(R_{2}\) and \(R_{3}\) all are negative, the interaction system is stable and the self-excited vibration is avoided.  
However, the stability of the interaction system is closely related with the bridge's modal frequency \(\omega_{\rm B}\). To illustrate this, the parameters of interaction system are set as \(k_{\rm p}=1000\), \(k_{\rm d}=30\), \(k_{\rm a}=0.4\), \(k_{\rm B}=30\), \(N=360\), \(A=0.01848\), \(\xi_{\rm B}=0.005\), and \(B_{0}=0.6193\).  
According to Equations (15) and (17), the traditional cascade control is adopted when the parameters \(\overline{k}_{\rm p}\) is set as the same with \(k_{\rm p}\). This is to say, \(\overline{k}_{\rm p}=1000\). In this case, the real parts \(R_{1}\), \(R_{2}\) and \(R_{3}\) associated with the modal frequency \(\omega_{\rm B}\) are shown in Figure 3.  
Figure 3: The real parts of characteristic roots for the standard cascade control.  
According to Figure 3, it can be seen that the real parts of \(R_{2}\) and \(R_{3}\) are negative from 0 to 1000 rad/s. However, the value of \(R_{1}\) is positive when the modal frequency \(\omega_{\rm B}\) belongs to the interval 67.3-118.7 rad/s. This is to say, the interaction system is unstable when the modal frequency \(\omega_{\rm B}\) belongs to the interval 67.3-118.7 rad/s. The self-excited vibration will appear.  
To avoid the self-excited vibration, tuning the control parameters \(k_{\rm p}\), \(k_{\rm d}\), \(k_{\rm a}\) and \(k_{\rm B}\) is an effective technique. However, the ranges of parameters are limited by the performance specification of the levitation subsystem and the noise level of corresponding signals. Hence, a more feasible and robust method should be developed.