## 2 Structure and Analysis of Automatic Adjustable Preload System
### Force Analysis of Automatic Adjustable Preload System  
The ball screw system is used to realize X-axis motion, and the stiffness in the X direction is the lowest [27]. As a result, the equivalent dynamic model of the system on the X-axis is established as shown in Figure 2.  
In Figure 2, \(k_{ba}\), \(C_{ba}\) and \(k_{bb}\), \(C_{bb}\) are the equivalent stiffness and damping of the left and right bearing; \(k_{bsa}\), \(C_{bsa}\), \(k_{bsb}\), \(C_{bsb}\) and \(k_{bsc}\), \(C_{bsc}\) are the equivalent stiffness and damping of the left, middle and right sides of the screw; and \(k_{nut}\) and \(C_{nut}\) represent the equivalent stiffness and damping of the nuts. In the model, the workable is assumed to be a lumped mass element and the effect of servo stiffness is ignored. The dynamic equation of the system can be written as:  
\[m\ddot{x}+c\dot{x}+kx=0 \tag{1}\]  
where \(\ddot{x}\), \(\dot{x}\) and \(x\) are the acceleration, velocity and displacement of the system, respectively. \(m\) is the total mass of the system, \(k\) is the total stiffness coefficient. To simplify the analysis, damping (c) is not considered in this case. Through applying voltage to the three PEAs simultaneously in real time, due to the inverse piezoelectric effect, the electrical energy will be converted into mechanical energy. Then, the preload force is applied to the ball screw. Consequently, the backlash is eliminated, and the stiffness of the mechanism is improved, the vibration is suppressed, and the resulting machining precision is enhanced. The force diagram of the preloaded system is presented in Figure 3. It shows that the actuators exert preload force on the slave nut and the master nut, bringing them into contact with the raceway of the screw, and the directions of the two contact angles are completely opposite. Figure 3: The force diagram of the preloaded system. Figure 2: The equivalent dynamic model of the ball screw system. As shown in Figure 3, \(F_{A}\) and \(F_{B}\) are the normal force of a ball on the raceway in the slave nut and the master nut, respectively; \(F_{E}\) is the preload force exerted by the PEAs in the sleeve of the structure; the initial preload force exerted by the adjustable bolt, \(F_{IA}\) and \(F_{IB}\), on the slave nut and the master nut, respectively, and the values of these are equal; \(F_{a}\) is the axial load; \(F_{p}\) is the total preload force. Furthermore, the ball number at work is z. In addition, it is assumed that the structures of the slave nut and the master nut are the same. When \(F_{a}=0\), the initial preload force is the total preload force, and the formula can be written as:  
\[F_{p}=F_{IA}+F_{E}=F_{IB}+F_{E}=zF_{A}sin\beta cos\alpha=zF_{B}sin\beta cos\alpha \tag{2}\]  
The relationship between the normal force of a ball and the axial force, namely \(F_{p}\), can be expressed as:  
\[F_{A}=F_{B}=\frac{F_{IA}+F_{E}}{zsin\beta cos\alpha}=\frac{F_{IB}+F_{E}}{zsin \beta cos\alpha} \tag{3}\]  
When \(F_{a}\
eq 0\), \(P_{A}\) is the actual axial load on the slave nut, and \(P_{B}\) is the actual axial load on the master nut, the following two equations can be obtained:  
\[P_{A}=F_{IA}+F_{E}+\Delta F\quad P_{B}=F_{IB}+F_{E}-\Delta F \tag{4}\]  
In this system, the two nuts are treated as having identical structure. Therefore, under the action of \(F_{a}\), the force balance condition of the screw can be derived into Equations (5) and (6), below:  
\[P_{A}=F_{a}+P_{B} \tag{5}\]  
\[F_{a}-P_{A}+P_{B}=0 \tag{6}\]  
From Equations (4)-(6), the formula is deduced as:  
\[m\ddot{x}+c\dot{x}+kx=0 \tag{1}\]0  
\(+\Delta F\) and \(-\Delta F\) are the force increment and decrement along with the change in \(F_{a}\). Furthermore, \(\alpha\) and \(\beta\) are the helix angle and the contact angle of the ball screw, respectively.