## 2 The Original Dynamic Model of Gear System
### Actual Tooth Surface Contact Analysis  
#### 2.1.1 Mathematical Modeling of Typical Manufacturing Errors  
In this paper, four kinds of typical manufacturing errors were selected, including shape deviation and slope deviation along the directions of the tooth profile and the lead, respectively shown in Figure 1. These errors were defined, and the limit deviation corresponding to the machining accuracy was given in the standard of ISO 1328-1:2013 [21].  
The common shape deviation mainly included three forms, concave, convex, and wavy, shown in Figure 2. Among them, the wavy deviation was mainly caused by the vibration during processing or the low-hardness tooth surface scuffing. Generally, its value was small and irregular, which had little influence on the performance of gear transmission. Therefore, this paper mainly considered two forms of concave and convex. The shape deviations along the directions of the tooth profile and the lead, respectively, were simulated by the semi-sine function shown in Equations (1) and (2).  
\[\begin{array}{l}\text{Tooth profile}:\left\{\begin{array}{l}\text{ concave form}:\delta_{f\alpha}=E_{f\alpha}\sin\left(\pi\cdot\frac{h-h_{1}}{h_{2}-h_{1}}\right)\;\; \left(E_{f\alpha}\leq 0,\,h\in[h_{1},h_{2}]\right)\\ \text{convex form}:\delta_{f\alpha}=E_{f\alpha}\sin\left(\pi\cdot\frac{h-h_{1}}{h _{2}-h_{1}}\right)-E_{f\alpha}\left(E_{f\alpha}>0,\,h\in[h_{1},h_{2}]\right) \end{array}\right.\\ \text{Lead}:\left\{\begin{array}{l}\text{ concave form}:\delta_{f\beta}=E_{f \beta}\sin\left(\pi\cdot\frac{l-l_{1}}{l-l_{1}}\right)\;\;\left(E_{f\beta}\leq 0,\,l\in[l_{1},l_{2}]\right)\\ \text{convex form}:\delta_{f\beta}=E_{f\beta}\sin\left(\pi\cdot\frac{l-l_{1}}{l -l_{1}}\right)-E_{f\beta}\left(E_{f\beta}>0,\,l\in[l_{1},l_{2}]\right)\end{array}\right. \end{array}\right.\end{array} \tag{1}\]  
where \(\delta_{f\alpha}\) is the shape deviation of tooth profile, \(\delta_{f\beta}\) is the shape deviation of the lead, and \(E_{f\alpha}\) and \(E_{f\beta}\) are the amplitudes of the shape deviation determined by random sampling, according to the gear precision grade; the determination of the parameters \(h_{1}\), \(h_{2}\), \(l_{1}\), and \(l_{2}\) are shown in Figure 3. In this figure, the origin \(o\) of the coordinate system \(oxy\) is the rotation center of the gear, plane \(oxy\) is parallel to the end of the gear, \(h_{1}\) and \(h_{2}\) are the radii of the root and top for the working tooth surface, \(l_{1}\) and \(l_{2}\) are the coordinate values projected on the \(z\)-axis at both ends of the lead, \(H\) is the working tooth height projected onto the axial section, and \(B\) is the tooth width.  
Figure 1: The typical manufacturing errors: (**a**) shape deviation and slope deviation along the direction of the tooth profile, and (**b**) shape deviation and slope deviation along the direction of the lead.  
The slope deviation had different directions, as shown in Figure 4. Due to the change in the direction of the slope deviation, the root or top protruded along the direction of the tooth profile, and one of the two ends protruded along the direction of the lead. The formula of the slope deviation in each direction is also specified in Equation (3).  
\[\left\{\begin{array}{l}\delta_{\text{Ha}}=E_{Ha}\frac{h-h_{1}}{h_{2}-h_{1}}- \frac{E_{H\beta}}{2}\ (h\in[l_{1},h_{2}],\text{ Root protruding}:E_{Ha}\leq 0,\text{ Top protruding}:E_{Ha}>0)\\ \delta_{H\beta}=-E_{H\beta}\frac{l-l_{1}}{l_{2}-l_{1}}+\frac{E_{H\beta}}{2}\ \ (l\in[l_{1},l_{2}],\text{ Helix angle increasing}:E_{H\beta}\leq 0,\text{ Helix angle reducing}:E_{H\beta}>0)\end{array}\right. \tag{3}\]  
where \(\delta_{Ha}\) is the slope deviation of the tooth profile, \(\delta_{H\beta}\) is the slope deviation of the lead, and \(E_{Ha}\) and \(E_{H\beta}\) are the amplitudes of the slope deviation determined by random sampling according to the gear precision grade.