# Study on Coupled Vertical Vehicle-Bridge Dynamic Performance of Medium and Low-Speed Maglev Train
## 2 Coupled Vehicle-Bridge System Considering Elastic Beam  
As the critical system of maglev trains, the magnetic coupled vehicle-bridge system, determining the safety and stability of normal operation, is the foundation of vehicle suspending on the bridge [19]. The operation diagram of a maglev train on a line is shown in Figure 1. The whole vehicle body is supported by the magnetic levitation force generated by several suspension frames. The force is dynamically adjusted by the suspension controller to keep the gap between the electromagnet in the suspension frame and the track near the expected value. A suspension frame consists of four suspension points, two of which on each side of the vehicle. According to the decoupling analysis of suspension frames, the coupled vehicle-bridge system can be regarded as multiple independent single-point suspension systems. When the train is on the steel beam with a small damping coefficient, whether it is in static levitation or at low speed, the coupled vibration phenomenon can be caused easily [20]. In order to study this phenomenon, a single-point suspension system model considering elastic beam is established, as shown in Figure 2.  
In Figure 2, \(y_{0}\) is the displacement of the electromagnet from the coordinate origin (along the \(OY\) direction), \(L\) is the beam span, \(x_{R}\) is the vertical deformation of the elastic beam (along the \(OX\) direction), \(x_{c}\) is the vertical displacement of the electromagnet, \(x\) is the suspension gap, \(m\) is the mass of electromagnet, \(F_{m}\) is the levitation force, \(u\) is the voltage of electromagnet coil, \(i\) is the control current, \(R\) is the coil resistance, \(N\) is the number of  
Figure 1: Maglev train on an operation line. Figure 2: Single-point suspension system considering elastic beam. winding turns, and \(S\) is the effective pole area of the magnet, while Vacuum permeability is \(\mu_{0}=4\pi\times 10^{-7}\) H/m. The levitation force provided by a single point can be expressed as:  
\[F_{m}(t)=\frac{\mu_{0}N^{2}S}{4}\bigg{(}\frac{i(t)}{x(t)}\bigg{)}^{2}, \tag{1}\]  
where \(x(t)=x_{c}(t)-x_{R}(t)\); the deviation between the vertical displacement of the electro-magnet and the vertical deformation of the beam is the actual suspension clearance. The dynamic equation of a single-point electromagnet can be described as:  
\[m\ddot{x}_{c}(t)=(m+M)g-F_{m}(t), \tag{2}\]  
where \(M\) is the mass of carriage. The voltage equation of electromagnet can be expressed as:  
\[u=Ri(t)+\frac{\mu_{0}N^{2}S}{2x(t)}\frac{\mathrm{d}i(t)}{\mathrm{d}t}-\frac{ \mu_{0}N^{2}S}{2x^{2}(t)}\frac{\mathrm{d}x(t)}{\mathrm{d}t}. \tag{3}\]  
Considering that the track is a simple supported beam, the coordinate \(x_{R}(y,t)\) of any point on the beam satisfies the Euler-Bernoulli dynamic equation [21; 22; 23]:  
\[E_{r}I_{r}\frac{\partial^{4}x_{R}(y,t)}{\partial y^{4}}+\rho_{r}\frac{ \partial^{2}x_{R}(y,t)}{\partial t^{2}}+\varphi_{r}\frac{\partial x_{R}(y,t) }{\partial t}=f(y,t), \tag{4}\]  
where the elastic modulus and section inertia of the beam are \(E_{r}\) and \(I_{r}\) respectively, \(\rho_{r}\) is the linear density of the beam, \(\varphi_{r}\) is the equivalent damping coefficient of the beam, and \(f(y,t)\) is the external force on the beam, which can be approximately expressed as:  
\[f(y,t)=F_{m}\delta(y-y_{0}). \tag{5}\]  
According to the vibration theory of beam, only when the excitation energy is very high can the vibration of high-order mode appear [21]. Therefore, vibration analysis under the first mode is usually considered [22; 23]. In this paper, only the first mode is considered, and the elastic beam model under the first mode is obtained by modal analysis method [24]:  
\[x_{R}(y,t)=\sigma_{1}(y)q_{1}(t), \tag{6}\]  
where \(\sigma_{1}(y)\) is the first mode function, \(q_{1}(t)\) is the time-dependent amplitude of beam in the first mode.