## 4 New Skyhook and Displacement-Velocity Control Method
### Design and Analysis of New Displacement-Velocity Controller  
Similar to the new SH controller, the new DV controller was designed with the objective of mitigating the transmissibility of vibrations with frequencies near the resonance frequency of the bogie and with low frequencies. Unlike the damping coefficient of the conventional DV control algorithm, which depends only on the ratio of relative displacement and the relative velocity of the carbody, the damping coefficient of the new DV controller, \(c_{n\_DV}\), also depends on the stiffness variable \(V_{s}\), as shown in Equation (9). \(V_{s}\) is proportional to the stiffness constant \(k_{s}\) and the contribution coefficient \(\beta\) and can be expressed using the equation \(V_{s}=\beta k_{s}\). Note that \(\beta\geq 1\), and \(\beta\) increases as the absolute oscillation velocity of the carbody increases, and vice versa. When the vibration velocity of the carbody is low, \(c_{n\_DV}\) is also low. By contrast, at higher carbody's vibration velocities, \(c_{n\_DV}\) is higher than \(c_{DV}\), the damping coefficient of the conventional DV controller, by a factor of \(\beta\). When \(\beta=1\), the new DV control algorithm resembles the conventional DV control algorithm. In other cases, similar to the new SH control algorithm, the new DV control algorithm generates higher values of \(c_{n\_DV}\) by a factor of \(\beta\) compared with the damping coefficient of the conventional DV controller. \[c_{n\_DV}= \begin{cases}\max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|V_{s} \frac{(y_{1}-y_{2})}{y_{1}-y_{2}}\right|\Bigr{)}\Bigr{)},&(y_{1}-y_{2})\bigl{(} \dot{y}_{1}-\dot{y}_{2}\bigr{)}<0\\ c_{min},&(y_{1}-y_{2})\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}\geq 0\end{cases}  \tag{9}\]  
Similar to the process adopted by the new SH control algorithm for determining \(\alpha\), the new DV control algorithm uses a three-step trial and error method to obtain five values of the contribution coefficient, \(\beta_{1}\) to \(\beta_{5}\), corresponding to the five velocity regions \(\left( \begin{array}{c}0-L_{1}\end{array} \right),\left( \begin{array}{c}L_{ 1}-L_{2}\end{array} \right),\left( \begin{array}{c}L_{2}-L_{3}\end{array} \right),\left( \begin{array}{c}L_{3}-L_{4}\end{array} \right)\), and \((L_{4}-\infty)\), where \(L_{1}\) to \(L_{4}\) represent four velocity thresholds in Equation (10). \[c_{n\_DV}=\begin{cases}\max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\beta_{1} k_{s}\frac{(y_{1}-y_{2})}{y_{1}-y_{2}}\right|\Bigr{)}\Bigr{)},&\left|\dot{y}_{1} \right|\leq L_{1}\cap\ (y_{1}-y_{2})\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}<0\\ \max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\beta_{2}k_{s}\frac{(y_{1}-y_{2}) }{y_{1}-y_{2}}\right|\Bigr{)}\Bigr{)},&L_{1}<\left|\dot{y}_{1}\right|\leq L_{2} \cap\ (y_{1}-y_{2})\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}<0\\ \max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\beta_{3}k_{s}\frac{(y_{1}-y_{2}) }{y_{1}-y_{2}}\right|\Bigr{)}\Bigr{)},&L_{2}<\left|\dot{y}_{1}\right|\leq L_{3} \cap\ (y_{1}-y_{2})\bigl{(}\dot{y}_{1}-\dot{y}_{2}\bigr{)}<0\\ \max\Bigl{(}c_{min},\min\Bigl{(}c_{max},\left|\beta_{4 and 0.1 m/s, corresponding to five vibration velocity regions of the carbody during the operating process. Contribution coefficients (beta) associated with these five vibration velocity regions are shown in Figure 5, where beta ranges between 1 and 2. When the vehicle body oscillates around the second resonance frequency region, the contribution coefficient increases by a factor of 1.75, leading to a 1.75-fold increase in the instantaneous damping force. Similarly, around the first resonance frequency region, the contribution coefficient increases by a factor of two, causing the instantaneous damping force to rise by two times, compared with the conventional DV control algorithm.