Theoretical Derivation of Gauges for Straddle-type Monorail Vehicle

The theoretical model of the straddle-type monorail vehicle bogie is set up and the function of each component is introduced. The lateral and vertical displacements of carbody when the lateral and roll motion of carbody in the same/reverse direction are attained through theoretical derivation, which can be put into the calculation of the vehicle gauge. A computational program is written to simulate the corresponding results under specific parameters. The correctness of the method to calculate the monorail vehicle gauge is verified.


Introduction
With the booming economic development and the acceleration of the urbanization, it is absolutely important to improve the transportation system, especially in the railway transit [1]. As a brand-new type of urban rail transit vehicle, the transportation volume of straddle-type monorail vehicle is moderate, with a small covering area and a strong terrain adaptability, which has wide application prospects [2]. Therefore, numerous studies have been conducted by researchers to explore the dynamic performances of straddle-type monorail vehicles.
Zhang et al. studied the anti-overturning stability of a typical straddle-type monorail vehicle, they derived a formula to describe the relationship between critical lateral force and the pre-load of the stabilizing wheels from the lateral roll equation, which was verified through UM software [3][4]. Zhu et al. put up with an idea that the adjustment of width of monorail beams can polish the dynamic performances of monorail train, through which they attained the deformation equations of the steering wheels and stabilizing wheels in terms of contact relationship, proving that a proper width of monorail beam can greatly decrease the tire deformation on curved track [5]. Jiang et al. established two types of articulated monorail vehicle models. By utilizing genetic algorithm, they compared the dynamic properties between them, reaching a conclusion that the non-bolster type of monorail vehicle is advantageous in passing the radius curve [6][7]. Bao et al. carried out studies concerned with the dynamic response and safety analysis of a monorail vehicle through ANSYS and SIMPACK, proposing that the vibration of the vehicle should be reduced and the track irregularity is negative to the system [8].
Obviously, the dynamic performances of monorail vehicles have been studied deeply by researchers, however, the calculation method of vehicle gauge is still nonstandard, and there is no clearer method for the gauge of straddle monorail vehicles in China. Therefore, this paper combined theoretical derivation with computational simulation to calculate the gauge for monorail vehicles. Firstly, the theoretical model of the monorail vehicle was introduced, with the main structures presented. Then, the lateral and vertical displacements of the carbody under different working conditions were derived based on Ref [9-10]. Finally, a program was written to calculate the corresponding parameters, and the gauge result was acquired, which verified the calculation formulas.  Figure 1 presents a typical straddle-type monorail vehicle that is running on the track in Chongqing city, whose configuration is made up of carbody, bogie and track system.  Figure 2 illustrates a representative structure of straddle-type monorail vehicle bogie, which consists of frame, air spring, traction motor, center pivot, collector and four types of rubber wheels. When vehicle is running on tracks, the running wheel will always be in contact with the top surface of the track beam, decreasing the vertical vibration, while the stabilizing wheel and the guiding wheel can reduce the lateral vibration. In addition, the auxiliary wheel can be utilized as running wheel if the air leakage happens. Since the movement posture of straddle-type monorail vehicles is similar to that of traditional railway vehicles, the calculation formula of dynamic gauges can refer to relevant standards [9-10]. However, as the straddle-type monorail does not use the steel wheel/rail structure, but uses rubber wheels, there is a distinct difference in the calculation of the wheel/rail relationship and parameters.

Lateral and Roll Displacement of Carbody in the Same Direction
When the monorail vehicle runs on the track, the carbody will have offsets mainly in the lateral and vertical direction due to the corresponding forces, which should be considered in the calculation of vehicle gauges. If the lateral displacement and the roll motion of carbody are in the same direction, then the lateral displacement of carbody can be expressed as equation (1).

Lateral and Roll Displacement of Carbody in the Reverse Direction
Similar to section 3.1, when the lateral displacement and the roll motion of carbody are in the reverse direction, then the lateral displacement of carbody can be expressed as equation (8)

Simulation
As the calculation formulas of the carbody displacement are demonstrated in equation (1)-(10), a computational program is written to calculate the corresponding displacements in order to verify the derivation process, and part of the parameters used in the paper are shown in table 1.

Conclusion
Based on the existing calculation method and theoretical model of monorail vehicle, the derivation of the carbody displacement under different working conditions was conducted. The calculation result was attained through a computational program under specific parameters, which satisfied the requirements of the standard and shall be a reference in calculating the gauge of monorail vehicles.

Acknowledgments
This work was supported by the National Key R&D Program (Grant No. 2018YFE0201400).