## 7 Conclusions  
This paper reports on basic research investigating the dynamic characteristics of the IRW and its underlying mechanisms. Compared to the previous research, the creepages and creep forces are calculated separately on the wheels of each side. More importantly, the effects of the gravitational restoring forces and the gyroscopic moment are carefully considered and analyzed. An improved IRW model G2-IRW is derived through the Euler-Lagrange theorem, from which the three dynamic characteristics of the IRW, derailment, hunting, and offset running, are discovered and summarized. Simulations and scaled-model experiments verified the existence.  
For an IRW without any yaw suspension, the gravitational restoring moment leads to derailment at low speeds, and the gyroscopic moment contributes to slight self-guidance ability at high speeds. When yaw suspensions are installed, the yaw spring can effectively prevent derailment at low speeds, and the yaw damper can effectively suppress hunting motion at high speeds. However, because the self-guidance ability is relatively slight, the IRW could stay in the offset running state for a long time. The mathematical steady-states analysis demonstrates negative-exponential convergence effects under the act of the gravitational restoring forces and an intrinsic linear relation of the IRW between the lateral displacement and the differential rotational speed.  
The improved mathematical model of the IRW proposed in this paper was built in Simulink, and a model built in SIMPACK 2018 with the same set of model parameters was generated. The simulation results demonstrated that the improved mathematical model G2-IRW is much closer to the SIMPACK model than the classical one. In addition, a scaled roller rig and a scaled IRW were designed and built. All three dynamic characteristics: derailment, hunting and offset running, as well as the intrinsic relation law between the lateral displacement and the differential wheel rotational speed, appeared in the experimental results, through which the validity of the theoretical analysis is verified.  
The following supporting information can be viewed at: [[https://www.youtube.com/watch?v=zikUywBYqPg](https://www.youtube.com/watch?v=zikUywBYqPg)]([https://www.youtube.com/watch?v=zikUywBYqPg](https://www.youtube.com/watch?v=zikUywBYqPg)) or [[https://www.bilibil.com/video/BV1](https://www.bilibil.com/video/BV1) fK411b7 Ga/]([https://www.bilibil.com/video/BV1](https://www.bilibil.com/video/BV1) fK411b7 Ga/), (accessed on 20 December 2023). Video S1: Dynamic characteristics of the IRW for railway vehicles.  
Conceptualization, X.J.; methodology, X.J., Y.Y.; software, X.J.; validation, X.J., Y.Y.; formal analysis, X.J.; investigation, X.J.; resources, X.J. and W.W.; data curation, X.J.; writing--original draft preparation, X.J.; writing--review and editing, X.J., Y.Y. and W.W.; visualization, X.J. and Y.Y.; supervision, W.W.; project administration, W.W.; funding acquisition, W.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 51575294.  
Not applicable.  
Figure 37: LM\({}_{\text{A}}\) nonlinear wheel tread profile in the Chinese railway standard TB/T 449\(-\)2003 [25].  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article and Supplementary Materials.  
**Conflicts of Interest:** The authors declare no conflicts of interest.