## 1 Introduction  
### Research Background and Conventional Study Status  
Technological advances are constantly being made to improve the ride comfort of vehicles, and various suspension devices for vehicles are emerging. In particular, dampers, which are part of the vehicle suspension system, play a major role in controlling vehicle vibration. Dampers come in a wide range of shapes and functions, including active and passive types. Many studies have been conducted on active dampers, whose damping force changes depending on the road surface over which the vehicle passes, rather than existing passive dampers, which have limited damper performance. However, fully active dampers are difficult to use based on their feasibility and high price. Therefore, semi-active dampers, which are a compromise between passive and active dampers, are favored in many devices that require vibration attenuation. A magnetorheological (MR) damper, which is a representative semi-active damper, uses an MR fluid, which is an intelligent material. MR fluids exhibit rheological properties whose physical properties change based on a magnetic field, which is a key factor in actively controllingthe damping force of the damper. When a magnetic field is applied, the magnetic MR fluid particles form a chain shape along the direction of the magnetic field, and yield stress is generated because this chain structure must be broken for the fluid to flow. The magnitude of the yield stress is controlled by the magnitude of the applied magnetic field. The efficiency of the MR fluid is very high because the yield stress is generated only by the application of a magnetic field. Therefore, the reaction rate is fast, and very little energy is used. The characteristics of MR dampers that provide effective performance using high efficiency fluids have been confirmed in several studies. [PERSON] et al. considered the performance of an MR damper based on the density, viscosity, and orifice dimensions of the MR fluid [1]. [PERSON] et al. proposed a viscous flow model for an MR fluid and analyzed the velocity distribution and damping performance of MR dampers. And the width dimensions of the annular orifice and pole piece and the action point and self-saturation of the hysteresis curve were considered according to the density change in an MR fluid, and an analytical design method for the magnetic circuit was proposed [2]. [PERSON] et al. considered the increase in the damping force of an MR damper through shape optimization of the piston and pole piece [3]. [PERSON] et al. examined the hysteresis of MR dampers using the cylindrical Bingham model, Bouc-Wen model, and polynomial model, and compared it with the experimental results [4; 5]. [PERSON] et al. considered the riding and handling performance of large buses equipped with air springs and MR dampers [6]. [PERSON] et al. considered the relationship between the damping force response speed of an MR damper and vehicle driving characteristics [7]. [PERSON] et al. proposed a plan for comparing the performance characteristics of MR and passive dampers [8]. Despite these characteristics, MR dampers also have disadvantages. Generally, in structured MR dampers, the total damping force and piston speed are not perfectly linear. Even when the piston speed increases slightly, the damping force increases very rapidly. However, after a certain piston speed is reached, regardless of how much the piston speed increases, the damping force shows a slight increase. Because of the characteristics of these MR dampers, a large damping force can be generated even at very low piston speeds, effectively suppressing the vehicle vibration. However, if the direction of motion of the vehicle changes continuously within this low piston speed range, a sudden change in the damping force occurs. Therefore, when vertical vibration occurs at a low piston speed, the damping force rapidly changes from positive to negative, which significantly affects ride comfort.