## 2 Definition of the Simulation Problem
### Evaluation of Effectiveness of Vibration Isolation  
The effectiveness of the proposed controller in isolating vibrations was directly evaluated through the absolute displacement, velocity, and acceleration of the train body. Furthermore, the acceleration frequency response characteristics of the carbody were considered to more comprehensively evaluate ride comfort. Because of its simplicity, the vibration transmissibility characteristics \(H(f)\) from the disturbance input excitation of the rail and the wheel to the vibration acceleration of the carbody was used to represent acceleration frequency response characteristics, as shown in Equation (2) [8]. \[H(f)=\sqrt{\frac{\int_{0}^{t}\dot{y}_{1}^{2}dt}{\int_{0}^{t}\dot{y}_{w}^{2}dt}} \tag{2}\]  
In addition, the natural resonance frequencies of the carbody and bogie components were theoretically calculated using Equation (3), referring to Table 1 for the values of the parameters of the railway vehicle's lateral suspension system. The natural resonance frequencies of the carbody and bogie were calculated to be 1.57 and 14.2 Hz, respectively. \[ \begin{cases}\omega_{n}=\sqrt{\frac{k}{m}}\\ f_{n}=\frac{1}{2\pi}\sqrt{\frac{k}{m}}\end{cases}  \tag{3}\]  
3 Conventional Skyhook, Displacement-Velocity, and Skyhook and Displacement-Velocity Control Methods  
The new SH-DV control algorithm was derived by upgrading the conventional SH, DV, and SH-DV control algorithms. First introduced by [PERSON] et al. [11], the SH control algorithm was developed on the basis of the operating principle of a suspension system using an ideal sky damper. SH control is defined by the damping coefficient \(c_{SH}\), which can be altered continuously. Note that \(c_{SH}\) is proportional to the skyhook damping constant \(c_{sky}\) and the ratio of the absolute and relative vibration velocities of the sprung mass part. The activating condition for SH control is given by the product of the absolute and relative vibration velocities of the sprung mass part. In addition, considering the real-world applications of MRDs, the minimum \(c_{min}\) and maximum \(c_{max}\) values of the damping coefficient were introduced; \(c_{min}\) represents the scenario wherein an external magnetic field is absent (i.e., the MRD functions as a hydraulic damper), whereas \(c_{max}\) represents the scenario wherein the MRD operates at maximum capacity under its allowable assembly space constraints. The damping coefficient for SH control, designed in the context of practical application, has been described in another study [22] and can be mathematically represented as Equation (4):  
\[c_{SH}= \begin{cases}\max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\Bigl{|}c_{sky} \frac{\dot{y}_{1}}{y_{1}-\dot{y}_{2}}\Bigr{|}\Bigr{)}\Bigr{)}&\dot{y}_{1} \bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}\geq 0\\ c_{min}&\dot{y}_{1}\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}<0\end{cases}  \tag{4}\]The DV control algorithm was derived from the mathematical analysis of a minimization problem. [PERSON] et al. [8] constructed the DV control algorithm to minimize the acceleration of the carbody by minimizing the displacement and velocity functions of the carbody and bogie, as shown in Equation (5). The damping coefficient for DV control, \(c_{DV}\), is directly proportional to the spring force [i.e., \(F_{s}=k_{s}(y_{1}-y_{2})\)] and inversely proportional to the relative velocity of the carbody [i.e., \(\big{(}\dot{y}_{1}-\dot{y}_{2}\big{)}\)]. The activating condition for DV control is given by the product of the relative displacement and velocity of the carbody. In particular, when the relative displacement and relative velocity have opposite signs, the DV control algorithm activates a damping coefficient, which is given by \(c_{DV}=\big{|}F_{s}/\big{(}\dot{y}_{1}-\dot{y}_{2}\big{)}\big{|}\), implying that the damping force of DV control cancels out the spring force, so that the carbody's acceleration approaches zero. In other scenarios, the DV control algorithm does not activate a damping coefficient. Considering the applications of MRDs, [PERSON] et al.