## 2 Modeling of the Dynamic System
### Analysis of the LSJ Vibration Mechanism  
We used the simplified physical model developed by [PERSON] et al. [24] to explain the physical essence of the LSJ phenomenon in the yaw system, as shown in Figure 2a, and assumed that the part connecting the yaw lead motor and the load was the active driving first unit, and the shaking part of the load was the passively driven second unit. Note that the auxiliary motor associated with the CLS system is precisely embedded in the driven second unit (see Figure 1b). In Figure 2a, the active part (1) pushes the driven part (2) at rotational speed \(\omega\). \(k_{1}\) is the stiffness coefficient; c\({}_{1}\) is the viscous torsional damping coefficient between friction surfaces and transmission parts. Due to the existence of rotational speed \(\omega\), with the push of the active part, 1, the spring repeats the processes of energy storage and energy release, and the driven part, 2, repeats the processes of acceleration and deceleration, thus forming a circular \"sticky-sliding\" jumping motion, forming a low-speed shaking microvibration, which is the mechanism of the LSJ. Figure 2b is the vibration model of the yaw system.  
Firstly, assuming that the active component (1) slowly moves clockwise for a certain distance, \(s_{0}\), the driving moment generated by the spring due to compression is exactly equal to the maximum static friction moment, \(M_{\rm j}\), experienced by the driven component, 2, i.e., \(k_{1}s_{0}=M_{\rm j}\); then, object 2 begins to move. Assuming that the distance of object 2 moving after time \(t\) is \(s\), the differential equation for the motion of object 2 is:  
\[J\frac{\mathrm{d}^{2}S}{\mathrm{d}t^{2}}+c_{1}\frac{\mathrm{d}S}{\mathrm{d}t}= -M_{\rm d}+k_{1}(s_{0}+R\omega t-s), \tag{1}\]  
where \(J\) is the moment of inertia; \(c_{1}\) is the torsional damping; \(k_{1}(s_{0}+R\omega t-s)\) is the driving moment; and \(M_{\rm d}\) is the dynamic friction moment.  
In order to simplify the analysis, the dynamic friction moment, \(M_{\rm d}\), is regarded as being composed of two parts: the constant component, \(M\), and component \(h\frac{ds}{dt}\) that varies with the rotary speed, namely \(M_{\rm d}=M+h\frac{ds}{dt}\). The static friction moment, \(\Delta M=M_{\rm j}-M=T\Delta\mu\), is substituted into Formula (1) to obtain:  
\[J\frac{\mathrm{d}^{2}S}{\mathrm{d}t^{2}}+(c_{1}+h)\frac{\mathrm{d}S}{\mathrm{ d}t}+k_{1}s=T\Delta\mu+k_{1}R\omega t, \tag{2}\]  
where \(h\) is the damping attenuation coefficient characterized by \(c_{1}+h=2\xi_{1}\omega_{\rm n1}\ J\), herein \(\omega_{\rm n1}=\sqrt{k_{1}/T}\); \(T\Delta\mu\) is the difference between the static friction torque and dynamic friction torque; \(T\) is the positive pressure torque; and \(\Delta\mu\) is the difference between the dynamic friction coefficient and static friction coefficient (DFC/SFC). The values of the DFC/SFC vary within the range of 0.02\(\sim\)0.06, changing with the variation in the lubrication effect. The values of the structural parameters (SPs) can be found in Table 1.  
Figure 2: Models: (**a**) LSJ physical model; (**b**) vibration model of the yaw system.