## 1 Introduction  
Single-side cantilevered rotor systems are widely used in various types of rotating machinery, such as flue gas turbines and pumps, and play important roles in the military and civil engineering fields. These systems are mainly composed of an impeller, a shaft, a bearing housing, bearings, a coupling, and a driving end [1]. For the application of this kind of rotating machinery, there are strict requirements on the accuracy of the installation. Therefore, it is important to ensure system stability and safety in order to avoid major accidents. The main threat to the system is vibration [2]. The excitation sources of vibration mainly come from the rotor system and external excitations. The excitation sources of the rotor system include rotor unbalance, misalignment, and friction [3; 4; 5]. External excitations are mainly caused by the impact load and pulsation excitation of the fluid. When the excitation frequencies are close to the natural frequencies of the system, the resonance occurs in the system, causing unpredictable losses. With the rapid development of the single-side cantilevered rotor system, the vibration problems have become increasingly prominent. These problems pose a huge threat to safety production and economic benefits. In actual civil and military equipment, rotor systems often vibrate due to unbalance, resulting in impeller fracture, seal failure, bearing damage, valve failure, pipeline leakage, and loose foundation [6]. These faults cause equipment damage or even major safety accidents, along with unpredictable personal injury to users. So, it is necessary to control the vibration of the unbalanced single-side cantilevered rotor. The vibration mechanism of the unbalanced single-side cantilevered rotor system has already been researched by many scholars. According to the obtained theory, the vibration control of the system is divided into four types: structural modification, vibration absorption, vibration isolation, and vibration damping. At the same time, the vibration control method is divided into two types: active and passive control. Active control refers to the use of the sensor to collect signals and feedback for the controller to drive the damper to reduce vibration. Passive control refers to vibration control that can be performed without additional controllers and input energy [7]. Active control of the unbalanced single-side cantilevered rotor system includes: (1) using active control to change the natural frequency of the system to avoid the resonance region to achieve vibration control [8] such as installing the mass on the shaft end of the system and changing the position of the mass by PD control. However, when the mass block is applied, the unbalanced excitation of the rotor system will increase, causing the shaft to bend [9]; and (2) using active control to apply external force that moves in the opposite direction to the exciting force, thereby achieving good vibration control such as applying external force at the impeller of the system through proportional gain feedback control. But this may result in insufficient installation space and increase the amplitude of the vibration [10]. Also, other researchers used suspension bearings to change the stiffness of the support and isolate the vibration transmission to the bearing to protect it, but the reduced stiffness of the suspension structure limits the range of allowable speeds and cannot be used beyond that range [11]. Therefore, active control has relatively large limitations, and still needs further development for practical applications. Passive control of the system includes: (1) using the squirrel-cage squeeze film damper to reduce vibration, but this takes up a lot of space and has serious nonlinearity, easily causing a nonlinear response.