## 1 Introduction  
Heavy commercial vehicles, tractors, and construction machinery are obviously excited by vibrations during their operation, due to the poor driving conditions they are in. A vehicle's fires and suspension system can quickly attenuate the high-frequency vibrational excitations that come from the ground. In contrast, low-frequency vibrational excitations that significantly impact the human body are transmitted from the vehicle floor and seat to the driver or passenger. Therefore, during non-road vehicle driving, the driver must withstand low-frequency and high-intensity vibrations for long periods. These kinds of vibrations can affect the driver's health, and even cause driving fatigue, leading to safety accidents [1]. In order to solve this problem, research on reductions in seating system vibrations is important. Seat damping is the simplest, most direct, and most economical method used to meet continually increasing requirements for people's ride comfort in non-road vehicles. Therefore, research on vehicle seat damping is vital for improving the driving comfort of off-road vehicles. Currently, the vibration-reduction methods of vehicle seats mainly include passive vibration reduction, semi-active control vibration reduction, and active control vibration reduction. A passive seat suspension has no feedback control system, but instead relies on the spring and damper of the suspension to attenuate vibrational excitation. The differences between semi-active and passive control are that the stiffness or damping parameters of a semi-active control suspension are variable, and that the feedback control of vibrations can be achieved through the output control forces of stiffness or damping elements. An active suspension also achieves strong damping effects through feedback control. The difference between active and semi-active control is that the control forces of an active control suspension are output by a dedicated actuator. The passive vibration-reduction method is widely used because of its simple structure and low cost. However, the stiffness and damping coefficients of a passive seat cannot change with changing working conditions, and it is difficult to attenuate low-frequency vibrations with a passive vibration-reduction method [2]. The active control method can effectively suppress low-frequency vibrations and adapt to various working conditions; however, active control devices are expensive, and require highly reliable actuators. The semi-active vibration-reduction method has better adaptability than the passive vibration-reduction method to changing conditions, and has lower costs than the active vibration-reduction method. Therefore, it has strong application prospects [3]. A magnetorheological fluid (MRF) damper is a new type of semi-active damper that is based on smart materials. This damper is suitable for vehicle suspensions because of its variable damping coefficient and strong vibration-reduction effect. As a result of these advantages, the magnetorheological fluid damper is widely used in vehicle seat vibration-reduction applications. [PERSON], [PERSON] [4] developed a seat suspension that used an MRF damper combined with a switching adaptive fuzzy controller, and the research results showed that this seat suspension could effectively reduce vibrations transmitted to the driver. [PERSON] [5] developed a novel compact rotary magnetorheological damper with variable stiffness and variable damping characteristics that could reduce the amplitudes of vibrations effectively. [PERSON] [6] developed an innovative seat suspension that operated with an MRF damper combined with a back-stepping sliding mode controller, and the results showed that using this seat suspension could control low-frequency vibrations and achieve better robustness. For commercial and off-road vehicles, driving behaviors such as cornering, acceleration, and obstacle avoidance, as well as other vibrational excitations resulting from harsh working conditions, cause vibrations in multiple directions.