## 3 Comparative Analysis of Dynamic Meshing Force  
The dynamic meshing force is composed of contact force and damping force. For the traditional single degree of freedom dynamic model, the formula of dynamic meshing force can be described as:  
\[F_{mesh}=k(t)f(x(t)-e(t))+c_{m}(t)\big{(}\dot{x}(t)-\dot{e}(t)\big{)} \tag{12}\]  
For the new model (the dynamic model considering the meshing state of multiple pairs of teeth), the formula of dynamic meshing force for NCR gears is:  
\[F_{mesh}= \begin{cases}F_{mesh}^{(j-1)}+F_{mesh}^{(j)}&nt_{z}\leq t<nt_{z}+(t _{h}-t_{z})\\ F_{mesh}^{(j)}&nt_{z}+(t_{h}-t_{z})\leq t<nt_{z}+t_{z}\\ F_{mesh}^{(j)}+F_{mesh}^{(j+1)}&nt_{z}+t_{z}\leq t<nt_{z}+t_{h}\end{cases}  \tag{13}\]  
where  
\[\left\{ \begin{array}{c}F_{mesh}^{(j-1)}=k^{(j-1)}(t)f\Big{(}x(t)-e^{(j-1)}( t)\Big{)}+c^{(j-1)}(t)\Big{(}\dot{x}(t)-\dot{e}^{(j-1)}(t)\Big{)}\\ F_{mesh}^{(j)}=k^{(j)}(t)f\Big{(}x(t)-e^{(j)}(t)\Big{)}+c^{(j)}(t)\Big{(}\dot {x}(t)-\dot{e}^{(j)}(t)\Big{)}\\ F_{mesh}^{(j+1)}=k^{(j+1)}(t)f\Big{(}x(t)-e^{(j+1)}(t)\Big{)}+c^{(j+1)}(t) \Big{(}\dot{x}(t)-\dot{e}^{(j+1)}(t)\Big{)}\end{array} \right. \tag{14}\]  
HCR gears have at least two tooth pairs in contact at all times, i.e., contact ratios of 2.0 or more, which can be obtained mainly by adding the addendum and lowering the pressure angle. Compared with NCR gears, the machining process of HCR gears may be more complicated due to the increase of addendum. In addition to traditional machining methods, some new machining methods [22, 24] can also be used to ensure the accuracy of the tooth profile. Obviously, for HCR gears, the dynamic meshing force involves five pairs of teeth, i.e., \(F_{mesh}^{(j-2)},F_{mesh}^{(j-1)},F_{mesh}^{(j)},F_{mesh}^{(j+1)},F_{mesh}^ {(j+2)}\). Here, we will mainly take NCR gears as the object for discussion. It can be seen from Formula (12) that \(e(t)\) used in the traditional model is the comprehensive meshing error, which reflects the overall error of the meshing teeth in the meshing process. In many previous literatures, simple harmonic function is often used to express the transmission (meshing) error. However, there are many kinds of deviation parameters in the gear system and the shape and amplitude of transmission error caused by different deviation parameters are often different. Therefore, the use of simple harmonic function may not fully express the specific characteristics of transmission error. In fact, the tooth contact analysis method (TCA) can be used to obtain the meshing (transmission) error [4] and the solution steps are shown in Figure 4. After constructing the tooth surface mathematical model considering different deviations (errors), by solving the tooth surface meshing equation, the meshing error data at each meshing position in the meshing process can be acquired, and then the single pair of teeth meshing error curve can be drawn. By successively calculating the meshing error of each pair of meshing teeth, we can obtain the meshing error curves of all tooth pairs. If the influence of contact ratio is considered, the comprehensive meshing error curve will be obtained by extracting the superior envelope of the meshing error curve of the former and the latter pair of teeth in the double-tooth meshing region. Here, taking the tooth profile deviation as an example, the difference between comprehensive meshing error and tooth meshing error of a single pair will be analyzed. According to References [16, 17], the tooth profile deviation can be described as:  
\[e_{a}(s)=e_{f}(s)+e_{H}(s)=\frac{f_{fa}}{2}\sin\left(2\pi f_{r}\frac{s-s_{0}}{ s_{f}-s_{0}}\right)+f_{Ha}\frac{s-s_{o}}{s_{f}-s_{o}}\left(s_{0}\leq s\leq s_{f}\right) \tag{15}\]  
where \(f_{fa}\) and \(f_{Ha}\) are profile form deviation and profile slope deviation, respectively (mm); \(f_{r}\) is the number of sine periods over the profile evaluation range; \(s\) is the involute rolling path length over the profile evaluation range (mm); \(s_{o}\) and \(s_{f}\) are the minimum and maximum values of \(s\) over the profile evaluation range, respectively.