# An Experimental Validated Control Strategy of Maglev Vehicle-Bridge Self-Excited Vibration
## 3 Principle of Self-Excited Vibration
### Principle of Self-Excited Vibration from the Perspective of Energy Interchange  
From the quiescent state to the vibration state, the bridge needs to absorb energy. However, the interaction system only consists of the electromagnetic levitation system and bridge, so the absorbed energy by bridge is from the exportation of levitation system. Therefore, the characteristic of the energy exportation of the levitation system may be decisive for the occurrence of the self-excited vibration. In this section, we try to discuss the principle of self-excited vibration from the perspective of energy interchange between the bridge subsystem and levitation subsystem. It has been found that when the vehicle is suspended on some special bridges, standing still or moving under 10 km/h, the self-excited vibration may appear and grow up continuously. Even so, at the beginning of the self-excited vibration, the amplitude of the vibration is tiny enough. In this case, the interaction system may be seen as quasi-static. When studying the stability of the interaction system around the equilibrium point at the quasi-static states, the analysis process can be simplified by a linearization model without introducing significant errors. Linearizing Equations (8), (10) (13) and (14), the linearized system of frequency domain is given by  
\[\left\{ \begin{array}{l}u(s)=2L_{0}si(s)+2 Ri(s)-2F_{\rm i}sy_{\rm E}(s)+2F_{ \rm i}sy_{\rm B}(s)\\ F_{\rm E}(s)=2F_{\rm i}i(s)-2F_{\rm z}y_{\rm E}(s)+2F_{\rm z}y_{\rm B}(s)\\ F_{\rm E}(s)=-m_{\rm E}s^{2}y_{\rm E}(s)\\ u(s)=k_{\rm c}\big{(}k_{\rm p}+k_{\rm d}s+k_{\rm a}s^{2}\big{)}y_{\rm E}(s)-k_{ \rm p}k_{\rm c}y_{\rm B}(s)-(k_{\rm c}-2R)i(s)\end{array} \right. \tag{27}\]  
where \(L_{0}=0.5\eta_{0}A\Lambda^{2}z_{0}^{-1}\), \(F_{\rm i}=0.5\eta_{0}A\Lambda^{2}i_{0}z_{0}^{-2}\), \(F_{\rm z}=0.5\eta_{0}AN^{2}i_{0}^{2}z_{0}^{-3}\). Eliminating the variables \(u(s)\), \(i(s)\) and \(y_{\rm E}(s)\), the transfer function between \(F_{\rm E}(s)\) and \(\dot{y}_{\rm B}(s)\) is  
\[H(s)=\frac{F_{\rm E}(s)}{y_{\rm B}(s)}=-\frac{\eta_{0}m_{\rm E}s}{\eta_{3}s^{3 }+\eta_{2}s^{2}+\eta_{1}s+\eta_{0}} \tag{28}\]  
where \(\eta_{0}=F_{\rm i}k_{\rm c}\Big{(}k_{\rm p}-i_{0}z_{0}^{-1}\Big{)}\), \(\eta_{1}=F_{\rm i}k_{\rm c}k_{\rm d}\), \(\eta_{2}=0.5m_{\rm E}k_{\rm c}+F_{\rm i}k_{\rm c}k_{\rm a}\), and \(\eta_{3}=m_{\rm E}L_{0}\). The vibration frequency is assumed as \(\omega_{\rm Vib}\) and the velocity of the bridge is defined as \(\dot{y}_{\rm B}(t)=0.1\cos(\omega_{\rm Vib}t+\phi)\). According to Equation (28), the electromagnetic force working on the bridge is  
\[F_{\rm E}(t)=0.1|H(j\omega_{\rm Vib})|\cos(\omega_{\rm Vib}t+\angle H(j\omega_ {\rm Vib})) \tag{29}\]  
Furthermore, the averaged power of the electromagnetic force acting on the bridge is  
\[P_{\rm E}(\omega_{\rm Vib})=\frac{1}{T}\!\int_{0}^{T}F_{\rm E}(\tau)\dot{y}_{ \rm B}(\tau)\mathrm{d}\tau=\frac{1}{200}\mathrm{Re}[H(j\omega_{\rm Vib})] \tag{30}\]  
Supposing the damping of bridge is viscous and linear, the damping force is  
\[F_{\rm D}(\omega_{\rm Vib},t)=2\xi_{k}\omega_{\rm Vib}m_{\rm B}\dot{y}_{\rm B} (\tau) \tag{31}\]  
Herein the averaged power consumed by the damping is \[P_{\rm D}(\omega_{\rm Vib})=\frac{1}{T}\int_{0}^{T}F_{\rm D}(\omega_{\rm Vib},\tau) \dot{y}_{\rm B}(\tau)\mathrm{d}\tau=0.01\xi_{k}\omega_{\rm Vib}m_{\rm B} \tag{32}\]  
The average power accumulated is  
\[P_{\rm B}(\omega_{\rm Vib})=P_{\rm E}(\omega_{\rm Vib})-P_{\rm D}(\omega_{\rm Vib }) \tag{33}\]  
In the normal case of equivalent parameters, the relationships between the vibration frequency and averaged powers are shown in Figure 4.  
According Figure 4, for the crossover frequency \(\omega_{\rm P_{E}}\), \(P_{\rm E}(\omega_{\rm P_{E}})=0\). For any \(\omega_{\rm Vib}<\omega_{\rm P_{E}}\), the averaged power \(P_{\rm E}(\omega_{\rm Vib})\) working on the bridge is negative. This means that the levitation subsystem will absorb the vibration energy of the bridge subsystem when self-excited vibration occurs.