## 2 Mathematical Dynamic Model of the IRW
### Steady States Analysis
Define this decay coefficient as \(\eta\), that is \[\eta=-\frac{\lambda v((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz}+F_{\rm Rz})}{2 bk_{22}} \tag{44}\]  
The steady-states conclusions gather as follows  
\[\ddot{y}=0,\ddot{\psi}=0,\dot{\omega}_{\rm L}=0,\dot{\omega}_{\rm R}=0,\dot{ \psi}=0,\psi=0 \tag{45}\]  
\[y(t)=y(t_{s})\rm exp\bigg{(}-\frac{\lambda v((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz }+F_{\rm Rz})}{2 bk_{22}}(t-t_{s})\bigg{)} \tag{46}\]  
\[K_{\psi}\psi-((m_{\rm C}+2m_{\rm W})g+F_{\rm Lz}+F_{\rm Rz})\lambda b\psi+ \frac{\lambda J_{\rm Wy}}{b}\dot{y}(\omega_{\rm L}+\omega_{\rm R})=0 \tag{38}\]0  
Equation (46) demonstrates that the lateral displacement \(y\) will exponentially converge to zero under the gravitational effects after time \(t_{s}\), which means that the convergence rate decreases with the convergence of \(y\). Thus, when \(y\) is quite close to zero, the convergence rate will be quite low as well. Additionally, the denominator of the decay coefficient \(\eta\) contains the lateral Kalker's creep coefficient \(k_{22}\), of which the magnitude is large, resulting in a small decay coefficient. Additionally, a small wheel tread conicity \(\lambda\) is sometimes adopted. Therefore, although the lateral displacement \(y\) can converge to zero eventually, the convergence rate is very small, which makes \(y\) almost steady in a short duration of time. That is why the running trailer of the IRW is always offset to the center of the railway track, and only if the time duration is long enough, will the lateral displacement \(y\) converge to zero under the gravitational effects eventually. Equation (47) shows a kinematic relation between differential rotational speed \(\Delta\omega\) and lateral displacement \(y\). The coefficient is relevant to the equivalent wheel-rail tread conicity \(\lambda\), the velocity of the IRW \(v\), and the nominal wheel radius \(r_{0}\). This kinematic relation demonstrates the intrinsic law for the offset running phenomenon on the IRW. Due to the freedom of the two wheels in the rotation direction around their axles, the differential rotational speed of the IRW does not have to be zero because the lateral displacement \(y\) can compensate for it. When the steady state of the IRW changes, for example, the differential rotational speed \(\Delta\omega\) changes from \(\Delta\omega_{1}\) to \(\Delta\omega_{2}\), and the lateral displacement \(y\) also changes from \(y_{1}\) to \(y_{2}\) along the kinematic relation described by Equation (47). Then, the IRW will adjust to a new balanced steady state \((y_{2},\Delta\omega_{2})\). Therefore, the freedom of the differential rotational speed on the IRW leads to the freedom of the lateral displacement. That is why the IRW can be balanced at any state pair \((y,\Delta\omega)\) as long as their kinematic relation can satisfy Equation (47). This phenomenon will be demonstrated later in the simulations and experiments. Figure 5 shows three types of steady convergent motions. If the IRW runs along the center of the railway track, the motion is named centered running. If the IRW does not run along the center of the railway track and is with lateral displacement \(y\), according to the analysis before, it will take a long time for the IRW to converge to the center of the railway track. The motion in this long-time duration is named offset running in this paper. In this section, the creep forces of the two wheels are separately calculated for higher accuracy, the gravitational restoring forces are considered, and the gyroscopic effect can be theoretically proved through mechanics analysis. However, whether these three factors can significantly affect the IRW's dynamic characteristics still needs to be verified. In the next section, a classical IRW model is introduced without consideration of the three factors, and an IRW model built in the general multibody system simulation software SIMPACK 2018 is regarded as a criterion. The G2-IRW model deduced in Section 2.4 will be compared with them in detail.