## 3 Comparison of the Models  
A type of classical IRW model has been presented in some research papers, which has been adopted to analyze the IRW dynamic characteristics and to design control methods. Concerning yaw suspension, the mostly adopted IRW model in these studies [2; 16; 17] can be expressed as follows  
\[\left\{ \begin{aligned} & m\ddot{y}+\frac{2k_{22}}{v}\dot{y}-2k_{22} \dot{y}=0\\ & I\ddot{\psi}+\left(\frac{2b^{2}k_{11}}{v}+C_{\psi}\right)\dot{ \psi}+K_{\psi}\dot{\psi}-\frac{2 br_{0}k_{11}}{v}\dot{\theta}+\frac{2b\lambda k _{11}}{r_{0}}y=0\\ & I_{\rm W}\ddot{\theta}+\frac{2b^{2}k_{11}}{v}\dot{\theta}-\frac {br_{0}k_{11}}{v}\dot{\psi}-\lambda k_{11}y=0\end{aligned} \right. \tag{48}\]  
where \(m\) is the total mass of the IRW, meaning \(m=m_{\rm C}+2m_{\rm W}\). \(\dot{\theta}\) is defined as half differential rotational speed between the left and right wheels, meaning \(2\dot{\theta}=\omega_{\rm L}-\omega_{\rm R}\). \(I\) is the total inertia moment of the IRW in the yaw direction, meaning \(I=J_{\rm Cz}+2J_{\rm Wz}+2m_{\rm W}b^{2}\). \(I_{\rm W}\) is the inertia moment of the wheel about its rotational axle, meaning \(I_{\rm W}=J_{\rm Wy}\). An IRW model built in the general multibody system simulation software SIMPACK 2018, as shown in Figure 6, is more approximate to the IRW in the real world and regarded as a criterion. The wheels adopt a \(1/10\) linear conicity tread with a \(15\) cm linear section in width, and the rails adopted the UIC60 shape. The wheel-rail contact analysis is demonstrated in Figure 7. A two-point contact between the rail and the wheel happens when \(y\) reaches about \(-32\) mm. In this section, the comparison results among the classical IRW model, the improved G2-IRW model deduced in Section 2.4, and the SIMPACK IRW model will be demonstrated and discussed. All these three simulation models were built with one set of physical parameters, shown in Table 1.  
A number of running conditions were designed to validate the G2-IRW model and analyze the gravitational and gyroscopic effects. We considered the variations in the longitudinal speed, yaw stiffness, and yaw damping, which are listed in Tables 2-4, respectively. We chose nine representative conditions to demonstrate the results, including V1-S1-C1, V1-S6-C6, V1-S10-C10, V5-S1-C1, V5-S6-C6, V5-S10-C10, V10-S1-C1, V10-S6-C6, and V10-S10-C10. A \(5\) mm lateral displacement was given as the system's initial state step input, and then the IRW responses without any extra force and moment input. Figure 6: The IRW model built in the software SIMPACK 2018. (**a**) Back view; (**b**) top view. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Definition** & **Value** \\ \hline \(m_{\text{C}}\) & Carrier mass & 200 kg \\ \(m_{\text{W}}\) & Wheel mass & 75 kg \\ \(J_{\text{C}x}\) & Carrier’s inertia moment in the \(x\) direction & 35 kg\(\cdot\)m\({}^{2}\) \\ \(J_{\text{C}y}\) & Carrier’s inertia moment in the \(y\) direction & 20 kg\(\cdot\)m\({}^{2}\) \\ \(J_{\text{C}z}\) & Carrier’s inertia moment in the \(z\) direction & 30 kg\(\cdot\)m\({}^{2}\) \\ \(J_{\text{W}x}\) & Wheel’s inertia moment in the \(x\) direction & 5 kg\(\cdot\)m\({}^{2}\) \\ \(J_{\text{W}y}\) & Wheel’s inertia moment in the \(y\) direction & 8 kg\(\cdot\)m\({}^{2}\) \\ \(J_{\text{W}z}\) & Wheel’s inertia moment in the \(z\) direction & 5 kg\(\cdot\)m\({}^{2}\) \\ \(b\) & Half lateral distance between the left and right wheels & 0.72 m \\ \(r_{0}\) & Nominal wheel radius & 0.4 m \\ \(\lambda\) & Equivalent wheel–rail tread conicity at the center position & 0.1 \\ \(k_{11}\) & Longitudinal Kalker’s creep coefficient & 4 \(\times\) 10\({}^{6}\) N \\ \(k_{22}\) & Lateral Kalker’s creep coefficient & 6 \(\times\) 10\({}^{6}\) N \\ \(g\) & Gravitational acceleration & 9.81 m/s\({}^{2}\) \\ \(F_{\text{L}z}\) & Extra vertical loads on the left wheel & 50 kN \\ \(F_{\text{R}z}\) & Extra vertical loads on the right wheel & 50 kN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The IRW’s parameters in simulations.