# On-Machine Precision Form Truing and In-Situ Measurement of Resin-Bonded Spherical Diamond Wheel
## 4 Results and Discussion
Along with cut-off frequency increasing, more information of form-truing error in the rotation direction as well as more noise signals will be retained. However, the cut-off frequency needs to be greater than the rotational frequency of the spherical diamond wheel to avoid the radial run-out error signals being filtered out. The rotational frequency \(f_{r}\) of the diamond wheel is expressed as:  
\[f_{r}=\frac{n_{t}}{v_{f}}, \tag{1}\]  
where, \(n_{t}\) is the rotation speed of the spherical diamond wheel (rpm), and \(v_{f}\) is the scanning movement speed in the measurement process (mm/min). According to Equation (1), under the experimental conditions of a grinding wheel rotation speed of 600 rpm and a scanning feed rate of 30 mm/min, the rotational frequency is 20 Hz. As shown in Figure 11c, the 3D (along the generatrix and the rotation direction) form-truing error of the spherical diamond wheel is approximately 5 \(\upmu\)m by setting the cut-off frequency of the low-pass filter to 32 Hz. The form-truing error is distributed symmetrically on both sides of the grinding wheel center. The form accuracy of the spherical diamond wheel can be verified by grinding a test specimen, and the actual radius of the spherical diamond wheel can also be calculated based on dimension error of the ground surface. In this paper, the hot-pressed ZnS with a diameter of 20 mm is selected as the test specimen, which incurs little wear to the spherical diamond wheel. The grinding wheel rotation speed is 12,276 rpm, the feeding speed is 5 mm/min, and the grinding depth is 2 \(\upmu\)m. The grinding surface of the test specimen is a sphere with a target radius of 25 mm in the NC program. As shown in Figure 12a, the ground surface form error is approximately 1.6 \(\upmu\)m without any compensation, which is determined mainly by the form-truing accuracy of the spherical diamond wheel. It can be found from Figure 12b that the surface form error of the test specimen is mainly resulted from low-frequency fluctuations with a space period of about 2 mm, which is equivalent to the low-frequency fluctuation period of the spherical diamond wheel surface profile shown in Figure 11c. The test specimen grinding  
Figure 11: Surface profile of the form-trued spherical diamond wheel: (**a**) measurement results and data fitting, (**b**) original form error signals, (**c**) 3D form-truing error after low-pass filtering. results verified the accuracy of the 3D form-truing error that was obtained by laser scan micrometer non-contact measurement. By comparing the obtained 3D form-truing error and ground surface form error values, it is confirmable that the uncertainty of the measured form-truing error is about 3 \(\upmu\)m for aspherical diamond wheel with a diameter of about 50 mm. In addition, the sphere radius of the test specimen is 24.9896 mm, which is 10.4 \(\upmu\)m less than the target radius. Therefore, the measuring radius of the spherical diamond wheel is 10.4 \(\upmu\)m smaller than the actual value. The uncertainty of the measuring radius is high related to the size and surface state of the grinding wheel, but the measuring error can be compensated by test specimen grinding. In conclusion, the accurate form-truing error as well as the actual size of the spherical diamond wheel can be obtained by combining the laser scan micrometer non-contact measurement and test specimen grinding.