## 1 Introduction  
The multi-support shafting is a typical rotor system [1]. When the shafting is accelerated and over-critical, the vibration is intensified, which restricts the performance of the tail drive system. Excessive vibration may lead to the failure of system components, reduce the service life of equipment, waste energy, generate excessive noise, damage the health of staff, and even cause major accidents resulting in heavy property loss and casualties. Therefore, it is necessary to suppress the bending vibration of the shafting in the over-critical state [2]. At present, vibration control methods include mainly the following [3]: dynamic balance, passive control, and active control. Active control vibration reduction technology attenuates vibration by actively applying control force. The commonly used active shock absorbers are electromagnetic damper [4], squeeze film damper [5], magnetorheological damper, electrorheological damper, and piezoelectric actuator. Among them, the piezoelectric actuator has the advantages of simple structure, light weight, high control accuracy, and fast response; it has a good application prospect in the field of vibration control, but its actuation displacement is small. [PERSON] et al. were the first to study the vibration control of the piezoelectric actuator in a rotor system. By controlling the output force of the piezoelectric actuator to change the supporting parameters of the bearing pedestal, the vibration of the rotor was reduced [6; 7]. The advantages and disadvantages of active damping feedback control and active stiffness feedback control were analyzed. The experimental results showed that the speed feedback control method can effectively suppress the rotor shaft system [8]. The Smart Spring [9] is an active vibration damping device based on a piezoelectric actuator, which is composed of a main support and an auxiliary support. The main support is the elastic support of shafting itself. By controlling the actuating force of the piezoelectric ceramic actuator on the auxiliary support, positive pressure is generated between friction elements on the main support and the auxiliary support, and vibration energy is consumed through dry friction between friction plates. According to the research on the vibration reduction technology of Smart Spring, [PERSON] of Carlton University of Canada has done a lot of work on the application of the Smart Spring to helicopter blade vibration reduction. In 2012, [PERSON] et al. applied a set of semi-active Smart Spring systems to the vibration reduction of a helicopter rotor system. The control scheme selected the state switching, and the vibration reduction performance of the Smart Spring was verified by experiments [10]. In 2013, [PERSON] et al. improved the active variable pitch pull rod APL (Active pitch link) of the damping device and verified the damping performance of the improved APL in the rotating tower test. The results showed that the APL effectively attenuated the vibration response of the blade and reduced the transmission force [11]. Other scholars on [PERSON]'s team have also done a lot of work on the application of Smart Springs to helicopter blade damping. [PERSON] et al. placed the Smart Spring mechanism on the load transfer path of the blade to achieve the purpose of vibration reduction and studied the stability of the blade in the Elastohydrodynamic state [12]. [PERSON] et al. studied the dynamic characteristics of the Smart Spring installed on the blade of a non-rotating helicopter, analyzed the influence of the Smart Spring on the modal characteristics, and finally established the finite element model of the blade [13]. Some scholars from the Aeronautical Research Institute of Canada also studied the Smart Spring damping technology. [PERSON] et al. established a mathematical model under harmonic excitation and designed an adaptive notch algorithm to reduce vibration on the DSP (Digital Signal Processor) platform. Wind tunnel tests were carried out to verify the damping effect of the Smart Spring [14].