# An Experimental Validated Control Strategy of Maglev Vehicle-Bridge Self-Excited Vibration
## 2 Modeling of Vehicle-Bridge Interaction
### Modeling of Bridge  
Based on the above assumptions, the minimum interaction model is shown in Figure 1. The variables \(y_{\text{B}}\) and \(y_{\text{E}}\) are the vertical displacements of bridge and electromagnet, respectively. The variable \(\delta\) is the levitation gap measured by the gap sensor, and \(m_{\text{E}}\) is the equivalent mass of electromagnet.  
The motion of bridge is described by the following differential equation [4]:  
\[EI_{\text{B}}\frac{\partial^{4}y_{\text{B}}(x,t)}{\partial x^{4}}+\rho_{\text{ B}}\frac{\partial^{2}y_{\text{B}}(x,t)}{\partial t^{2}}=f(x,t) \tag{1}\]where the variable \(x\) is the axial coordinate of the bridge, \(t\) is the time, \(E\)\(I_{\mathrm{B}}\) is the bending rigidity, \(\rho_{\mathrm{B}}\) is the mass per unit length, and \(f(x,t)\) is the electromagnetic force acting on the bridge.  
For a simply supported bridge, the first modal frequency \(\omega_{\mathrm{B}}\) and modal shape functions \(\phi_{\mathrm{B}}(x)\) are  
\[\omega_{\mathrm{B}}=\lambda_{\mathrm{B}}^{2}\sqrt{E\,I_{\mathrm{B}}/\rho_{ \mathrm{B}}} \tag{2}\]  
\[\phi_{\mathrm{B}}(x)=\sin(\lambda_{\mathrm{B}}x) \tag{3}\]  
where \(\lambda_{\mathrm{B}}\) is the space wavelength of the bridge's first modal, and \(\lambda_{\mathrm{B}}=\pi/L_{\mathrm{B}}\). The fact observed from the maglev base of china shows that the self-excited vibration is mainly aroused by the first modal of bridges. Hence, in this section, the first modal of bridge is considered solely. Thus, the solutions of Equation (1) may be expressed as  
\[y_{\mathrm{B}}(x,t)=\phi_{\mathrm{B}}(x)q_{\mathrm{B}}(t) \tag{4}\]  
where \(q_{\mathrm{B}}(t)\) is the time-varying amplitude of the modal displacement. Substituting the Equation (4) into Equation (1), multiplying both sides of the aforementioned resultant equation by \(\phi_{k}(x)\), then integrating both sides from \(0\) to \(L_{\mathrm{B}}\) gives  
\[\ddot{q}_{\mathrm{B}}(t)+2\xi_{\mathrm{B}}\omega_{\mathrm{B}}\dot{q}_{ \mathrm{B}}(t)+\omega_{\mathrm{B}}^{2}q_{\mathrm{B}}(t)=2\rho_{\mathrm{B}}^{- 1}L_{\mathrm{B}}^{-1}\sum_{i=1:n}\phi_{\mathrm{B}}(x_{i})F_{\mathrm{E}i}(t) \tag{5}\]  
where \(n\) is the number of levitation units suspended on a single bridge. Multiplying both sides of the resultant equation by \(\phi_{\mathrm{B}}(x)\) gives  
\[\ddot{y}_{\mathrm{B}}(x,t)+2\xi_{\mathrm{B}}\omega_{\mathrm{B}}\dot{y}_{ \mathrm{B}}(x,t)+\omega_{\mathrm{B}}^{2}y_{\mathrm{B}}(x,t)=2m_{\mathrm{B}}^{ -1}\phi_{\mathrm{B}}(x)\cdot\sum_{i=1:n}\phi_{\mathrm{B}}(x_{i})F_{\mathrm{E}i }(t) \tag{6}\]  
where \(m_{\mathrm{B}}=\rho_{\mathrm{B}}L_{\mathrm{B}}\) is the total mass of bridge. The phases of electromagnetic forces \(F_{\mathrm{E}i}(t)(i=1:n)\) are exactly in concert and the amplitude of \(F_{\mathrm{E}i}(t)\) is proportional to the value of \(\phi_{\mathrm{B}}(x_{i})\) when the self-excited vibration occurs with the first-order modal frequency [18]. As for Equation (6), with regard to the special case \(x=0.5L_{\mathrm{B}}\), it gives that  
\[\ddot{y}_{\mathrm{B}}(t)+2\xi_{\mathrm{B}}\omega_{\mathrm{B}}\dot{y}_{ \mathrm{B}}(t)+\omega_{\mathrm{B}}^{2}y_{\mathrm{B}}(t)=\sigma m_{\mathrm{B}}^ {-1}F_{\mathrm{E}}(t) \tag{7}\]  
where \(\sigma=2\sum_{i=1:n}\phi_{\mathrm{B}}^{2}(x_{i})\), the variable \(y_{\mathrm{B}}(t)\) is the modal displacement and the variable \(F_{\mathrm{E}}(t)\) is the electromagnetic force of the levitation unit at the location of \(x=0.5L_{\mathrm{B}}\). Equation (7) may be considered as the response of bridge reused by the electromagnetic force \(F_{\mathrm{E}}(t)\).