## 4 Deviations in the Fixture Positioning and the Resulting Part Deviations
### Fivxture Deviation  
1 Typical structural dimension planes:  
Part positioning adopts the \"3-2-1\" positioning method. \(P_{1}\) and \(P_{2}\) are the theoretical positions of two points; \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) are the actual positions of two points. For the two positioning points \(P_{1}\) and \(P_{2}\) on the \(\sigma\) plane of part 1, the deviations are shown in Figure 5.  
Assume that the angle between the two positioning points \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) and the y axis is \(\delta\), and the angle between the long-axis direction of the slot of the two-way positioning pin \({P_{2}}^{\prime}\) and the y axis is \(\alpha\). The relative deviation of \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) is decomposed into a deviation \(\Delta T^{\prime}\) along the long axis of the slot of the two-way locating long hole pin and a deviation \(\Delta N^{\prime}\) perpendicular to the line connected to \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\). Since the two-way locating long hole pin can be moved in the slot along the long-axis direction, \(\Delta T^{\prime}\) does not affect the positioning accuracy of the fixture, i.e., it does not cause deviation of the part and can be ignored, but \(\Delta N^{\prime}\) will cause the part to produce a rotation deviation on the fixture, and this deviation is rigid. Assume the deviations of points \(P_{1}\) and \(P_{2}\) in the \(\sigma\) plane are \(\Delta x_{P_{1}}\Delta y_{P_{1}}\) and \(\Delta x_{P_{2}}\Delta y_{P_{2}}\), respectively. According to the sine theorem, decomposing the relative deviations \(\Delta x_{P_{2}}\), \(\Delta y_{P_{2}}\) and \(\Delta y_{P_{2}}-\Delta y_{P_{1}}\) of the two locus points into the vertical direction of the line \({P_{1}}^{\prime}{P_{2}}^{\prime}\), we get:  
\[\frac{\Delta N_{1}{}^{\prime}}{sin\delta_{1}}=\frac{\Delta x_{P_{2}}-\Delta x _{P_{1}}}{sin\delta_{2}},\frac{\Delta N_{2}{}^{\prime}}{sin\delta}=\frac{ \Delta y_{P_{2}}-\Delta y_{P_{1}}}{sin\delta_{3}} \tag{8}\]  
\[\Delta N^{\prime}=\langle\alpha\rangle\cdot\Delta N_{1}{}^{\prime}+\left\langle c \left(\alpha+\frac{\pi}{2}\right)\right\rangle\Delta N_{2}{}^{\prime} \tag{9}\]  
In Equations (8) and (9), \(c\left(\alpha+\frac{\pi}{2}\right)\)=\(cos\left(\alpha+\frac{\pi}{2}\right)\), \(\delta_{1}=\frac{\pi}{2}-\alpha\), \(\delta_{2}=\frac{\pi}{2}+\alpha-\delta\), \(\delta_{3}=\frac{\pi}{2}+\delta-\alpha\), the operator \(\langle\alpha\rangle\) denotes \(\frac{c\alpha}{|\alpha|}=\pm 1\). This gives the part rotation deviation \(\Delta\alpha\) due to the two locating pins as:  
\[\Delta\alpha=\frac{\Delta N^{\prime}}{\left\|H_{P_{1},P_{2}}{}^{\prime} \right\|_{2}} \tag{10}\]  
2 Atypical structural dimension planes:  
Assume the deviations of two positioning points \(P_{1}\) and \(P_{2}\) in the \(\varphi\) plane are as shown in Figure 6 and assume the angle between the straight lines connected to the y axis at the actual positions of the two positioning points \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) are \(\theta\). Decompose the relative  
Figure 5: Diagram of the deviation of two positioning points \(P_{1}\) and \(P_{2}\) on the \(\sigma\) plane. deviation of \({P_{1}}^{\prime}\) and \({P_{2}}^{\prime}\) into the deviation \(\Delta T^{\prime\prime}\) along the linear direction of line \(P_{1}P_{2}\) and the deviation \(\Delta N^{\prime\prime}\) in the vertical direction of line \(P_{1}P_{2}\). Suppose the deviations of points \(P_{1}\) and \(P_{2}\) in the \(\varphi\) plane are \(\Delta y_{P_{1}}\), \(\Delta z_{P_{1}}\) and \(\Delta y_{P_{2}}\), \(\Delta z_{P_{2}}\), respectively, and the relative deviations of the two locus points \(P_{1}\) and \(P_{2}\) are \(\Delta y_{P_{2}}-\Delta y_{P_{1}}\) and \(\Delta z_{P_{2}}-\Delta z_{P_{1}}\).