## 1 Introduction  
Railway vehicles are crucial for economic development and social connectivity because they transport goods and people. With increasing usage, their speed and load-carrying capacity need to be upgraded [1], posing challenges to the quality of train transportation, which is evaluated on the basis of ride comfort and safety. To address these challenges, passive suspension systems are commonly used in railway vehicles, and their structures and underlying principles are continually being optimized. For example, passive inertial suspensions have been used to expand the scope of both vibration isolation ability and road friendliness, however, the adaptation range of such passive suspension systems is relatively low compared with the range of all possible vibration excitation frequencies from the track encountered during the operation of a train [2]. Semiactive suspension systems have been explored as alternatives, offering some advantages, such as (1) being easy to embed into the original passive suspension structure, (2) being easy to implement using simple algorithms of conventional controls, (3) being attainable at a reasonable cost, (4) exhibiting low energy consumption, and (5) demonstrating high effectiveness in vibration isolation under a wide range of vibration frequencies [3; 4; 5; 6; 7]. Of all the devices used in semiactive suspension systems, magnetorheological dampers (MRDs) have garnered widespread attention because of their quick response, reasonable cost, and compact structure [4; 5; 6; 7; 8; 9; 10]. By controlling external magnetic fields, MRDs generate damping forces that vary over a large range depending on dynamic signals from suspension system components. Many control algorithms have been developed for fine-tuning the damping force generated by MRDs. These control algorithms can be divided into two main groups: traditional control (e.g., skyhook (SH) [11], groundhook (GH) [12], acceleration-driven damping (ADD) [13], displacement-velocity (DV) [8], mixed SH-GH [14], mixed SH-ADD [15], mixed SH-DV [8], and advanced hybrid SH-GH control using semiactive inertial suspension to address the control-phase deviation [16]); and modern control (e.g., fuzzy [17] and neural network [18; 19] control). The algorithms in the modern control group are generally more effective in isolating undesirable vibrations of sprung mass parts; however, they require actual data on desirable damping forces generated by MRDs under real-world operating conditions, and obtaining such data is both time-consuming and laborious. By contrast, the algorithms in the traditional control group are easier to implement because they only require the dynamic signals obtained by accelerometers to control the external magnetic field. However, traditional control algorithms have their own limitations. Singular control-type algorithms can effectively attenuate vibrations with frequencies only within a certain range (i.e., low or high-frequency regions). Mixed control type algorithms inherit the effective working region of their singular control components and also expand this region; however, these algorithms are unable to reach the asymptotic effectiveness of their individual singular control components. Moreover, both the traditional singular and mixed control-type algorithms perform inefficietively around the resonance frequency of the sprung and unsprung mass parts [20]. In this study, we propose a new mixed SH-DV control method to isolate undesirable lateral vibrations of the carbody of a railway vehicle, that offers the advantages of a traditional control algorithm while addressing the aforementioned limitations. The proposed method comprises two main stages. First, the conventional singular controllers (SH and DV) are upgraded to improve their vibration isolation performance.