## 4 New Skyhook and Displacement-Velocity Control Method
### Design and Analysis of the New Skyhook Control Algorithm
 \tag{8}\]  
Using trial and error, we determined three velocity threshold values of the new SH control, i.e., 0.0075, 0.025, and 0.1 m/s (Figure 2a). These thresholds define four vibration velocity regions of the carbody during the operating process: (1) to 0.0075 m/s, assumed to be corresponding to carbody vibration at a high frequency domain (after the second-order resonance frequency, i.e., from H backwards); (2) 0.0075 to 0.025 m/s, assumed for the carbody vibration at a medium frequency domain (after the first-order resonance and before the second-order resonance frequency, i.e., D to E and G to H); (3) 0.025 to 0.1 m/s, assumed for the carbody vibration at around the second-order resonance frequency, i.e., from A forwards, C to D, and E to G; and (4) \(\geq\)0.1 m/s, assumed for the carbody vibration at around the first-order resonance frequency, i.e., A to C. Contribution coefficients associated with these four vibration velocity regions are presented in Figure 2b, ranging between 1 and 8. As a result, when the vehicle body oscillates around the second resonance frequency region, the contribution coefficient increases by a factor of six, leading to a sixfold increase in the instantaneous damping force. Similarly, around the first resonance frequency region, the contribution coefficient increases by a factor of eight, causing the instantaneous damping force to rise by eight times, compared with the conventional SH control algorithm. Both the first and second resonance frequency regions require hard damping (i.e., high damping force) as clearly stated by [24] regarding the ride comfort problem. Therefore, based on the aforementioned rationale, the proposed control method is expected to work effectively in suppressing the undesirable oscillation of the train body. Figure 3 presents a comparison of the transmissibility of vibrations from the rail to the train body under various control methods. Conventional SH control performed better at isolating vibrations than did passive control, but it was ineffective at approximately 14.6 Hz, which is the resonance frequency of the bogie. At low frequencies, which include the resonance frequency of the carbody (1.5 Hz), the absolute vibration velocity of the carbody was often greater than that of the bogie, resulting in the product of the relative and absolute oscillation velocities of the carbody being positive. By contrast, at around the resonance frequency of the bogie, the absolute oscillation velocity of the bogie was greater than the absolute oscillation velocity of the carbody, resulting in the product of the relative and absolute vibration velocities of the carbody usually being negative. This implies that  
the number of instances when the MRD generated damping forces was lower at frequencies near the resonance frequency of the bogie than at other frequencies, leading to reduced vibration isolation performance. The same trend was observed with the new SH control algorithm at frequencies near the resonance frequency of the bogie. Nevertheless, the resulting damping force in this case was larger than that generated through conventional SH control by a factor of \(\alpha\), which helped the MRD generate sufficient energy in a timely manner to suppress the bogie's resonance. Therefore, the new SH control algorithm is more effective over the entire frequency domain, including the resonance frequencies of the bogie and carbody. For a more in-depth assessment of the new SH control algorithm, the absolute displacement, velocity, and acceleration of the train body were compared under various control methods (Figure 4). The displacement of the carbody was much lower under conventional and new SH control than under passive control, especially at low frequencies [Figure 4a]. The velocity of the carbody under conventional SH control was overall lower than that under passive control, except at frequencies near the resonance frequency of the bogie (14.6 Hz).