## 2 Modeling of the Dynamic System
### Vibration Mechanism of the Yaw System  
The yaw system, including the tower body of a large wind turbine, can be equivalent to a disk supported by the torsion spring, \(k_{2}\), and torsional vibration damping, \(c_{2}\), as shown in Figure 2b. The total mass of the yaw system disk is \(m\), the moment of inertia is \(I\), and the radius is \(R\). The torque generated by the yaw main motor acting on the disc is \(M=R\times F\). The friction torque between the active yaw system and the tower is \(M_{\rm f}\). \(\theta\) is yaw angle, with equilibrium position \(\frac{d\theta(0)}{dt}=0\) representing the position when the yaw moment just overcomes the friction moment. The differential equation of motion based on the equilibrium position is:  
\[I\frac{d^{2}\theta}{dt^{2}}+c_{2}\frac{d\theta}{dt}+k_{2}\cdot\theta=M-M_{\rm f}, \tag{3}\]  
Assuming that the position where the yaw moment, \(M\), just overcomes the static friction moment, \(M_{\rm f0}\), is the equilibrium point position, then \(M=M_{\rm f0}\), and the friction moment, \(M_{\rm f}\), at any time can be described as an offset equation: \(M_{\rm f}=M_{\rm f0}-\Delta M_{\rm f}\frac{d\theta}{dt}\). By substituting the offset equation into Formula (3), the following formula can be obtained:  
\[I\frac{d^{2}\theta}{dt^{2}}+c_{2}\frac{d\theta}{dt}+k_{2}\cdot\theta=\Delta M _{\rm f}\frac{d\theta}{dt}\approx K_{\Delta}(0)\frac{d\theta}{dt}, \tag{4}\]  
where \(\Delta M_{\rm f}\) is the yaw moment attenuation coefficient; \(K_{\Delta}(0)\) is the slope of \(\Delta M_{\rm f}\) at the coordinate origin. From Equation (4), the free vibration equation of viscous damping can be obtained:  
\[\Big{(}k_{2}/\omega_{\rm n2}^{2}\Big{)}\frac{d^{2}\theta}{dt^{2}}+2\omega_{ \rm n2}(\zeta_{2}-\eta)\frac{d\theta}{dt}+\omega_{\rm n2}^{2}\theta=0, \tag{5}\]  
where \(\omega_{\rm n2}=\sqrt{k_{2}/I}\); \(\eta=K_{\Delta}(0)/\Big{[}2(lk_{2})^{0.5}\Big{]}\); \(K_{\Delta}(0)=\alpha M_{\rm fs}\); and \(M_{\rm fs}=\mu R(m\mathrm{g}+300T_{0}/d)\). Herein, \(\eta\) is the yaw rotational friction instability damping ratio; \(\alpha\) is the change rate of the friction moment attenuation; \(M_{\rm fs}\) is the static friction moment between the yaw system and the upper surface of the yaw gear ring; \(T_{0}\) is the pre-tightening torque of the bolt; and \(d\) is the nominal diameter of the bolt. The values of the SPs can be found in Table 1.  
To solve the differential equations given by Equations (2) and (5), a variable replacement method was developed. In Equation (2), since the rotational speed of active part 1 is expressed as \(\omega=\frac{d\theta}{dt}\), and \(\omega t\) is a nonlinear quantity, we might as well assume that \(x=\theta t\); then, there exists:  
\[\frac{dx}{dt}=\frac{d\theta}{dt}t+\theta=\omega t+\theta, \tag{6}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**SP Items** & **Values** \\ \hline Moment of inertia of object 2, \(J\) & 330.2 kg m\({}^{2}\) \\ Radius, \(R\) & 1.1575 m \\ Positive pressure torque, \(T\) & 200 N m \\ Damping ratio, \(\zeta_{1}\) & 0.05 \\ Difference between DFC/SFC, \(\Delta\mu\) & within [0, 0.1] \\ Spring stiffness, \(k_{1}\) & 10 N m/rad \\ Spring stiffness, \(k_{2}\) & 3.3 \(\times\) 10\({}^{8}\) N m/rad \\ Damping ratio, \(\zeta_{2}\) & 0.04 \\ Moment of inertia, \(I\) & 6603.9 kg m\({}^{2}\) \\ Change rate of friction moment attenuation, \(\alpha\) & 0.6 \\ Static friction coefficient, \(\mu\) & 0.07 \\ Pre-tightening torque of the bolt, \(T_{0}\) & 1000 N m \\ Nominal diameter of the bolt, \(d\) & 0.033 m \\ Total mass of yaw system, \(m\) & 4929 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of structural parameters (SPs).