## 4 New Skyhook and Displacement-Velocity Control Method  
### Design and Analysis of the New Skyhook Control Algorithm  
A new SH control algorithm is proposed to address the large vibration transmissibility at and around the natural resonance frequencies of the suspension system. The concept of this algorithm was first considered in our previous study [23]. Similar to the conventional SH control, the damping coefficient for the new SH control, \(c_{n\_SH}\), depends on the absolute and relative vibration velocities of the carbody. In addition, it also depends on the skyhook damping variable \(V_{sky}\), as shown in Equation (7). The skyhook damping variable is proportional to the skyhook damping constant \(c_{sky}\) and the contribution coefficient \(\alpha\), as given by the equation \(V_{sky}=\alpha c_{sky}\). Note that \(\alpha\geq 1\), and \(\alpha\) increases as the absolute vibration velocity of the carbody increases, and vice versa. When the vibration velocity of the carbody is low, \(V_{sky}\) is low, resulting in a low \(c_{n\_SH}\). By contrast, when the vibration velocity of the carbody is high, \(V_{sky}\) is high, and \(c_{n\_SH}\) is higher than \(c_{SH}\) by a factor of \(\alpha\) under the same vibration velocity of the carbody. When \(\alpha=1\), the new SH control and conventional SH control algorithms are identical. At frequencies equal to or near the resonance frequency of the system, the suspension system oscillates with great intensity, and \(c_{n\_SH}\) is higher than \(c_{SH}\) by a factor of \(\alpha\), which contributes to the suppression of vibrations in a timely manner, making the new SH control algorithm more effective than the conventional SH control algorithm. Note that \(c_{n\_SH}\) is constrained between the upper and lower bounds of the damping coefficient, which is represented by the max-min function in Equation (7). \[c_{n\_SH}= \begin{cases}\max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|V_{sky\frac{ \dot{y}_{1}}{y_{1}-\dot{y}_{2}}}\right|\Bigr{)}\Bigr{)},&\dot{y}_{1}\bigl{(} \dot{y}_{1}-\dot{y}_{2}\bigr{)}\geq 0\\ c_{min},&\dot{y}_{1}\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}<0\end{cases}  \tag{7}\]  
The construction process for the new SH control algorithm can be summarized by three main steps: (1) determining the vibration velocity domain of the carbody through the numerical simulation problem for the railway vehicle model under a passive suspension system; (2) defining the thresholds and regions of the velocity from the velocity domain of the carbody; and (3) determining the contribution coefficients \(\alpha\) for the new control algorithm by using the trial and error method. As shown in Equation (8), this study identified three velocity thresholds, \(T_{1}\) to \(T_{3}\), corresponding to the four velocity regions--\(\left(0-T_{1}\right],\left(T_{1}-T_{2}\right],\left(T_{2}-T_{3}\right],\) and\(\left(T_{3}-\infty\right)\)--and thus, to four values of the contribution coefficient, \(\alpha_{1}\) to \(\alpha_{4}\). \[c_{n\_SH}= \begin{cases}\max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\alpha_{1 }c_{sky}\frac{\dot{y}_{1}}{y_{1}-\dot{y}_{2}}\right|\Bigr{)}\Bigr{)},&\dot{y} _{1}\bigl{|}\leq T_{1}\cap\dot{y}_{1}\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)} \geq 0\\ \max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\alpha_{2}c_{sky}\frac{\dot{y}_ {1}}{y_{1}-\dot{y}_{2}}\right|\Bigr{)}\Bigr{)},&T_{1}<\left|\dot{y}_{1}\right| \leq T_{2}\cap\dot{y}_{1}\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}\geq 0\\ \max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\alpha_{3}c_{sky}\frac{\dot{y}_ {1}}{y_{1}-\dot{y}_{2}}\right|\Bigr{)}\Bigr{)},&T_{2}<\left|\dot{y}_{1}\right| \leq T_{3}\cap\dot{y}_{1}\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}\geq 0\\ \max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\alpha_{4}c_{sky}\frac{\dot{y}_ {1}}{y_{1}-\dot{y}_{2}}\right|\Bigr{)}\Bigr{)},&T_{3}<\left|\dot{y}_{1}\right| \cap\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}