## 2 Angular Indexing Errors Caused by the Coaxial Deviation of the Mandrel  
Figure 1 shows the configuration of a GMM, model 891E (Mahr Company, Gottingen, Germany). The gear to be measured is mounted onto a mandrel, whose two ends are pivoted by an upper center and a lower center, respectively. In this figure, the gear is replaced by a polygon to test the actual angle of rotation. The upper center is fixed to the rail of the column, and the lower center is driven by the spindle of the rotary stage. A clamping module, including the fixture, connecting rod, and plate, tightly fixes the mandrel to the lower center so that all these parts rotate together as a rigid body. When the spindle rotates, the mandrel is simultaneously rotated. The angle of rotation is measured by an encoder underneath the spindle inside the rotary stage. This is the ideal case without any errors mentioned in the last section. Figure 2 shows a schematic diagram of the actual case and shows that the spindle has radial run-out and tilt with respect to the ideal axis of rotation. The mandrel has a coaxiality error with respect to the ideal spindle axis. The spindle errors are generated from the bearing plane from which the radial error motion and tilt error motion of the shaft, or spindle, start [16]. It is obvious that the mandrel will rotate around a cone contour because the tip of the lower center has radial run-out error motion due to combined error sources at the tip's height. This kind of machine has not only spindle errors but also the mandrel's coaxiality error. The actual rotated angle of the mandrel must be deviated from the reading of the encoder. This is the main goal of this study to investigate the difference in between. For simplicity to explain, the coaxiality error and the spindle error are considered separately. Considering that the spindle average line is the same as its ideal axis of rotation and the mandrel average line is the same as it center line, a simplified illustration of coaxial error can be seen in Figure 3, including two angle states of the spindle rotation at \(0^{\circ}\) and \(180^{\circ}\), respectively. It assumes the upper center is off the spindle average line by a distance of \(e\). The moving parts, including the mandrel, polygon, and the clamping module, will be driven to rotate by the spindle. Figure 1: Mahr 891E gear measuring machine. Figure 2: Actual spindle and coaxiality errors existed in the gear measuring machine (GMM). As shown in Figure 3, \(e\) is the radial coaxiality error of the mandrel to the spindle average line and \(\alpha\) is the tilt angle. \(r_{\theta}\) is the variable radius of the contact point between the connecting rod and the plate while rotating. Let the minimum and maximum values of \(r_{\theta}\) be \(r_{\theta}=0^{\circ}\) and \(r_{\theta}=180^{\circ}\), respectively. From the geometrical relationship, Equations (1) and (2) can be obtained. \[r_{\theta=0^{\circ}}=AB-AO=\frac{r_{A}}{\cos\alpha}-h\tan\alpha \tag{1}\]  
\[r_{\theta=180^{\circ}}=A^{\prime}B^{\prime}+A^{\prime}O=\frac{r_{A}}{\cos \alpha}+h\tan\alpha \tag{2}\]  
where \(L\) is the length of the mandrel between two centers, \(r_{A}\) is the distance between the mandrel and connecting rod, and \(h\) is the height of the lower center from the table top. Since \(r_{\theta}\) varies in a sinusoidal form during rotation with a maximum at \(\theta=180^{\circ}\) and a minimum at \(\theta=0^{\circ}\), it can be expressed by  
\[r_{\theta}=\frac{(r_{\theta=180^{\circ}}+r_{\theta=0^{\circ}})}{2}-\frac{(r_ {\theta=180^{\circ}}-r_{\theta=0^{\circ}})}{2}\cos\theta. \tag{3}\]  
Substituting Equations (1) and (2) into (3), where \(\alpha\) is a very small angle, Equation (3) can be simplified to  
\[r_{\theta}=r_{A}-h\tan\alpha\cos\theta. \tag{4}\]  
Let the angular velocity of the spindle be \(\omega\) and the mandrel be \(\omega_{A}\). The velocity of the contact point between the plate and the connecting rod can be obtained either from the mandrel or the spindle.