# Vertical Dynamic Response Prediction of the Electromagnetic Levitation Systems
## 1 Introduction  
The maglev train uses electromagnetic force to support, guide, and propel the vehicle to realize non-contact operation. Therefore, the maglev train has the advantages of low noise, strong climbing ability, small mechanical wear, and low maintenance cost [1; 2], which makes it a new type of green and economical rail transportation system. With the development of maglev technology, the maglev system has been widely used [3]. By keeping the controlled levitation gap between the vehicle and the track at 8-10 mm, the maglev train strictly tracks the space trajectory of the track. However, due to the limited deviation range of the levitation gap, the vehicle/track coupling problem is prominent in the urban maglev systems. Furthermore, due to the guideway structure, manufacture technology, installation error, elastic deformation, thermal expansion, and pier settlement, etc., the track irregularity is inevitable [4]. Under the excitation of vertical track irregularity, the levitation gap deviates from the set gap. Generally, the safe deviation range of the levitation gap is \(\pm\)4 mm. In some extreme conditions, the vehicle may collide with the track, resulting in large noise, even levitation failure, which seriously affects the ride comfort of the maglev train. In practice, it is found that there are apparent dynamic response differences in the levitation system [5]. At the same time, with the increase in the speed of the maglev train, the deviation range of the levitation gap increases significantly. With the speed-up of the urban maglev train, the deviation range of the levitation gap may exceed the safe range, thus affecting the stability of the levitation system. Therefore, the vertical dynamic response analysis of the electromagnetic levitation system needs to be solved urgently. The track irregularity can be divided into particular geometric irregularities and random irregularities. Due to the particular structure of the maglev track, there are typical periodical geometric irregularities, which are related to sleeper pace, F-rail section, and beam span [3], etc., and they are modeled as sinusoid functions [5; 6; 7]. The local geometric irregularities caused by steps [5; 8] at track joints are also included in the specific geometric irregularities. Due to its unique structure, special analysis is needed. As a more general excitation source, the random irregularity is obtained from the existing airport PSD (power spectral density) functions, railway PSD functions [9; 10; 11; 12], or the field measured track irregularity data [4; 13]. By using the numerical time-frequency transfer method (such as trigonometric series method, white noise filtering method [10], and IFFT method [12]), the space trajectory data in time-domain or space-domain is obtained from PSD functions and then used as the excitation source of the dynamic model for dynamic analysis. Researchers usually use the finite element modeling methods [14] or the modal analysis method [15; 16] to analyze the vehicle/bridge coupling problem considering elasticity, which is not considered in this paper. For the maglev vehicle model, researchers have established multi-DOF models, such as the 10-DOF model in [12], 30-DOF model in [17], 34-DOF model in [18], 109-DOF model in [14], 162-DOF model in [19], and 210-DOF model in [20]. A fine model is beneficial to the analysis of the vehicle/bridge coupling problem considering elasticity. However, the increase of freedom leads to a rapid increase in the calculations of solving the partial equations. At the same time, for the magnetic levitation system itself, researchers generally adopt a simplified model and do not consider the dynamic difference. Therefore, we focus on the vertical dynamic response of the electromagnetic levitation system itself. Researchers have done much work to analyze and suppress the dynamic response differences of the electromagnetic levitation system, and to weaken the deviation range of the levitation gap. [PERSON] et al.