## 1 Introduction  
Vibration and noise suffered by passengers during the movement of a vehicle mainly come from the unbalanced reciprocating inertia force when the engine is running and from the impact excitation caused by the contact collision between the tires and the uneven road surface [1, 2]. The vibration caused by the engine excitation is characterized by a high-frequency and low-vibration amplitude, while the vibration caused by an uneven road excitation is characterized by a low-frequency and high-vibration amplitude [3, 4]. To obtain better vibration isolation and noise reduction and support performance, HEMs should have dynamic stiffness related to the frequency and displacement. To isolate the vibration caused by the engine excitation, the dynamic stiffness and damping should be small enough to isolate the vibration transmitted from the powertrain to the chassis for better vibration and noise isolation and ride comfort. To isolate the vibration caused by the excitation from the uneven road, the dynamic stiffness and damping should be large enough so that the HEM in a narrow engine compartment can limit the relative displacement between the powertrain and the chassis to ensure the power and support capacity transmitted from the engine to wheels. A typical passive HEM consists of rubber to support the static load and an inner chamber enclosed by a thin rubber bellow filled with a fluid medium to provide additional dynamic stiffness, as shown in Figure 1. The enclosed inner chamber is mainly separated by hard plastic or metal into three small chambers: an upper chamber, a long inertia track, and a lower chamber. Due to the design of a decoupler between the upper and lower chambers, which can move freely within a restricted displacement range, the fluid medium flows through the decoupler track in the high-frequency range when the displacement is small. In the low-frequency range, when the displacement is large, the fluid medium mainly flows between the upper and lower chambers through the inertia track [5; 6; 7; 8]. A passive HEM cannot actively adjust the dynamic features to meet complex load conditions due to the fixed lumped parameter. The vibration of the engine in its steady-state range can be reduced using passive vibration isolation methods. However, these methods are effective only in a narrow selected frequency range. In practical applications, it is crucial to effectively mitigate the impact of external mechanical vibrations on hydraulic valves. [PERSON] [9] conducted a comprehensive theoretical and experimental analysis to explore the feasibility of employing flexible mounting techniques for valves on vibrating surfaces. This approach involves incorporating materials with well-defined stiffness and damping properties between the valve housing and the vibrating surfaces, thereby reducing vibrations within the valve housing. Active HEMs [10; 11] are primarily developed through the substitution of a decoupler with actuators capable of modifying its real-time performance. However, it is associated with drawbacks such as elevated energy consumption and insufficient reliability. Considering the advantages and disadvantages of passive and active HEMs, semiactive HEMs are preferred due to their high performance and low cost. Semiactive HEMs can control the stiffness of the upper or lower chamber, the damping of the fluid medium, or the geometric characteristics of the fluid trajectory between the upper and lower chambers, such as the length or area of the inertia trajectory [12; 13; 14; 15]. When designing a semiactive HEM, it is straightforward to choose a physical parameter as a variable such as the stiffness of the upper or lower cavity, the length of the inertia track, and the cross-sectional area of the inertia track. Experts [2; 11; 14; 15] can also choose the key dynamic stiffness frequencies as a bridge to dynamic performance. The HEM's peak dynamic stiffness frequency at low frequencies and large amplitudes is generally placed at the wheel  
Figure 1: Typical passive HEM [5].