## 1 Introduction  
Multi-support shafting is widely used in aviation and industrial fields, and it has an important impact on the overall performance of the equipment. With the development of high-speed and flexible multi-support shafting, the bending vibration of shafting is becoming more and more serious. At present, there are mainly a few vibration control methods: dynamic balance, passive control and active control [1]. Active control technology attenuates the vibration by actively applying control force. Smart spring support is an active damping device with variable stiffness and damping, which includes the main support (the elastic support of shaft system itself) and the auxiliary support. In regulating the force of input of the piezoelectric actuator on the auxiliary support, friction elements on the main and secondary support generate a positive pressure, the dry friction of which absorbs vibration energy [2]. The vibration reduction technology of the smart spring was first used in helicopter blade vibration control. [PERSON] et al. improved the active pitch link (APL) of the damper and tested the improved damping performance in the rotating tower test in 2013. The results showed that the APL could effectively reduce the vibration response and the transfer power of the blade [3]. [PERSON] et al. placed the smart spring mechanism on the blade load transfer path to achieve the purpose of reducing vibrations and to examine blade stability in the elastodynamic state [4]. [PERSON] et al. conducted research on the dynamic properties of the smart spring on non-rotating helicopter leaves, as well as analysis of the impact of the smart spring on the properties of the models [5]. [PERSON] et al. designed a control scheme to make the stiffness of the smart spring device continuously change and compared the control effect of the state-switching control algorithm [6]. [PERSON] et al. studied the control strategy of the smart spring and carried out the critical vibration control of the transmission shaft system with a smart spring [7]. The influence of support parameters on the vibration characteristics of shafting is mainly focused on the influence of support stiffness and support position. In the study of bearing parameters in support of the vibration characteristics of the system, [PERSON] et al. established the finite element model of the ship shafting bearing base system, analyzed the influence of bearing stiffness at the support and the base stiffness on the vibration transmission characteristics of the system. It was concluded that the change of support stiffness would affect the transverse vibration mode of shafting, and the change of stiffness of the oil-lubricated bearing had little effect on vibration transmission [8]. [PERSON] established a dynamic model of ship propulsion shafting, based on the transfer matrix method, and analyzed the influence of bearing stiffness on the shafting vibration transmission path. The research results showed that the stern bearing stiffness had the greatest impact on the shafting vibration transmission, and the thrust bearing had the least impact [9]. [PERSON] studied the influence of changes in bearing stiffness on the transverse vibration of ship shafting and concluded that the natural frequency of the corresponding direction would be reduced when the bearing stiffness decreased [10]. The influence of bearing stiffness on the critical speed and response characteristics was studied in [11; 12]. [PERSON] et al. studied the influence of dynamic stiffness and the damping coefficients of different bearings on the vibration characteristics of a rotor system [13]. [PERSON] theoretically studied the influence of journal bearings of different lengths on the dynamic characteristics of a rotor-bearing system [14]. [PERSON] studied the influence of bearing stiffness changes at different locations on the vibration characteristics of ship shafting [15]. [PERSON] et al.