# Gear Integrated Error Determination Using the Gaussian Template Convolution-Facet Method
## 2 Theory of Gear Integrated Error
### Relationship between Integrated and Various Error Terms  
The gear integrated error includes numerous gear error terms. The pitch deviation, profile deviation, and tangential comprehensive deviation can be separated from the integrated error curve in gear section. The radial runout can be separated from SJZ curve [16].  
As shown in Figure 3, the \(AB\) segment is the overlapping meshing area between the tooth \(i\) and the tooth \(i-1\), the \(BC\) segment is the separate meshing area of the tooth \(i\), and the \(CD\) segment is the overlapping meshing area between the tooth \(i+1\) and the tooth \(i\). Assuming the number of sampling points for each tooth face is \(N\), and if the overlap coefficient is \(\varepsilon\), then the number of sampling points for the \(AC\) section is as follows:  
\[N_{3}=INT(N/\varepsilon) \tag{2}\]  
Figure 3: Any segment on the gear integrated error curve.  
Figure 2: A segment on the integrated error curve in gear bidirectional cross-section. (SJZ curve).  
In the formula, _INT_ represents rounding. The number of sampling points for _AB_ segment is as follows:  
\[N_{4}=N-N_{3} \tag{3}\]  
When \(GIE_{0,j}=GIE_{z,j}\), \(z\) is the number of teeth, and \(GIE_{i,j}\) is the integrated error value of the sampling point \(j\) on the tooth \(i\), and then the longitudinal coordinate values of each point on the outer envelope of the gear integrated error curve are as follows:  
\[L[(i-1)\times N+j]=\left\{\begin{array}{l}\max(GIE_{i-1,j+N_{3}},GIE_{i,j}) \ 1\leq j\leq N_{4}\\ GIE_{i,j}\ N_{4}\leq N_{3}\end{array}\right\}1\leq i\leq z \tag{4}\]  
The envelope of the gear integrated error curve is the total tangential composite deviation curve. The maximum variation of the total tangential composite deviation curve along the ordinate within the range of 0-360\({}^{\circ}\) is \(F_{i}^{\prime}\), and the maximum variation of the ordinate within the range of 360\({}^{\circ}/z\) is \(f_{i}^{\prime}\). The formulas are as follows:  
\[F_{i}^{\prime}=\max L[k]-\min L[k]\ 1\leq k\leq N_{3}\times z \tag{5}\]  
\[f_{ik}^{\prime}=\max[k\times j]-\min L[k\times j]\ 1\leq j\leq N_{3},1\leq k \leq z \tag{6}\]  
\[f_{i}^{\prime}=\max(f_{ik}^{\prime}) \tag{7}\]  
According to the definition, the remaining errors can be obtained:  
\[F_{p}=\max(F_{pi})-\min(F_{pi})\ 1\leq i\leq z \tag{8}\]  
\[\begin{array}{l}f_{pt\max}=\max(f_{pti})\\ f_{pt\min}=\min(f_{pti})\end{array}\right\}\ 1\leq i\leq z \tag{9}\]  
where \(F_{ai}=\max(f_{fi,j})-\min(f_{fi,j})\ 1\leq j\leq N\), then  
\[F_{\max}=\max(F_{xi})\ 1\leq i\leq z \tag{10}\]  
In the formula, \(f_{fi,j}\) is the profile deviation of the sampling point \(j\) on the tooth \(i\); \(f_{pt\max}\) is the maximum value of a single-pitch deviation, and \(f_{pt\min}\) is the minimum value of the single-pitch deviation. \(F_{\max}\) is the maximum total profile deviation, where \(v_{i}=\max|GIE_{i,j}-GIE_{i-1,j+N_{3}}|1\leq j\leq N_{4}\), then  
\[f_{pbi}=v_{i}\times\cos\alpha \tag{11}\]  
\[\begin{array}{l}f_{p\max}=\max(f_{pbi})\\ f_{p\min}=\min(f_{pbi})\end{array}\right\}1\leq i\leq z \tag{12}\]