## 1 Introduction  
Ultraprecision computer numerical control (CNC) machines, as cutting-edge equipment in the manufacturing industry, have been attracting lots of attention in recent years. However, chatter, a self-excited vibration occurring during the machining process, often causes poor surface quality, higher inaccuracy, and undesirable tool damage [1]. This has motivated many machining researchers over the years to delve into the development of methodologies that warrant the use of chatter suppression high-performance spindles in CNC machines [2], while also improving the stability of the feed system during machining operations. At present, feed systems can be divided into two main types, one is directly powered by the liner motor, the other uses a rotating motor through a ball screw and a nut to drive the system [3]. Among machine tools, ball screw drives have been widely applied for realizing linear motion due to their advantages, which include high transmission efficiency, high rigidity, strong anti-interference capability, etc. [4]. They have become irreplaceable transmission devices in high-end CNC machine tools. Nevertheless, there is a backlash between the ball screw and the nut, and because of this, vibration will be caused and the stiffness of the ball screw is not high enough to achieve high machining accuracy [5]. The backlash should be reduced or eliminated, and this goal is normally achieved through the application of a preload force. With respect to the application of preload force in machine tools, there have been plenty of studies addressing this. In the practical applications, springs are commonly applied in the system to generate a constant preload force. However, the preload force introduced by the above method is not adjustable, limiting the application of the technology. Considering the development of the variable preload force technology, the existing preload mechanismscan be categorized into passive control devices and active control devices on the basis of their control principle [6]. With regard to passive devices, [PERSON] et al. [7] developed the application of Fe-Mn-Si-based shape memory alloys for axial variable preload force of ball bearings. [PERSON] et al. [8] used an eccentric mass device to convert the radial force caused by the centrifugal force into axial force in order to apply the preload force. However, these devices are still unable to achieve arbitrary adjustment of the preload force. Regarding active control devices, [PERSON] et al. [9] applied preload force to eliminate the backlash and enhance the stiffness. [PERSON] et al. [10] used a hydraulic system to exert preload force, and this was able to generate huge force. However, the hydraulic system also had limitations that the equipment structure was complex, the size was relatively large, and the response was is slow [11]. The methods described above introduce a preload force onto the bearing of the spindles. When it comes to milling machines, the ball screw directly affects the quality of the processing owing to the additional presence of backlash. To solve this problem, preload force also needs to be loaded onto the ball screw. In the traditional method, considering a single nut, the great steel ball method, whereby the groove is smaller than the steel ball, is used to achieve preload. The stretching method, in double-nut cases, refers to a technique whereby the two nuts are separated by a thick washer squeezed between them [12]. The above methods do not have the feature of dynamic adjustment and cannot meet the requirement of high precision. In [13], a device was developed using an electromagnet to generate the preload force. However, the size of the magnet, magnetic isolation, and eddy current heating problems hinder its application in engineering practice. In [14], a cylindrical giant magneto strictive actuator was used to exert preload force on a ball screw. The drive coil generates lots of heat, so a cooling system is needed, resulting in large size and heavy mass.