# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 3 Applicative and Comparative Analysis for In Situ Measurement of Cylindricity
### Applicative Analysis of Spiral Scan EST  
Theoretically, the spiral scan EST for measuring the cylindrical profile must face two basic queries. One is harmonic suppression and the other is the multi-point straightness EST based on DFT. As stated above, Equation (9) would be true if the transfer function \(\boldsymbol{\Omega}(p)\)\((p=0,1,\cdots,N\times M-1)\) was not \"zero\". However, when \(p=M\), the transfer function should be:  
\[\boldsymbol{\Omega}(M)=\sum_{n=1}^{5}c_{n}\exp(j\varphi_{n})=\sum_{n=1}^{5}c_ {n}\cos(\varphi_{n})+j\sum_{n=1}^{5}c_{n}\sin(\varphi_{n})\equiv 0.\]  
This means that the \(M\)th harmonic suppression happens in this EST, i.e., the \(M\)th harmonic component of the spiral line profile on the cylinder is unable to exactly be determined. In Reference [10], the partial transfer function \(\boldsymbol{\Omega}(p)\)\((p=0,1,\cdots,250)\) was shown as Figure 10.  
A numerical validation procedure has been developed in this study to help qualitative and quantitative evaluation of the cylindrical profile obtained by this EST. Whether the \(M\)th harmonic suppression impacts correct detection of spiral line profile on the cylinder should be verified first. Figure 9: Median line profiles measured by the parallel four-point EST. We created a set of given data to describe a virtual spiral line profile on the cylinder. A series of random data was used to simulate the radial error motions of two imaginary rolling bearings that supported the cylinder to rotate so that the different error motions in each revolution were simulated. A series of the other random data was used to simulate the carriage straightness error motions including yaw and pitch. The outputs of five probes were set according to Equation (8), and then, the procedures of reconstructing the spiral line on the cylinder were carried out according to Equations (9)-(11). As shown in Figure 11, the given spiral line profile on the cylinder only with the median line deviations is similar to Figure 1b. The difference between the measured spiral line profile and given one presents periodicity, which is the 29 th harmonic component of the spiral line profile on the cylinder. Such a difference causes the spiral line profile measured by this EST to slightly shift as a whole, yet the profile of the spiral line is not distorted. Therefore, the \(M\)th harmonic suppression of this EST will not impact the accuracy of measuring the spiral line profile. Figure 11: Comparison of the measured spiral line profile with given one in the case of ideal measurement. Here, angular position of probes \(\varphi_{n}\) (\(n=1,2,\cdots,5\)) is \(0^{\circ}\), \(395^{\circ}\), \(838^{\circ}\), \(1236^{\circ}\), \(1659^{\circ}\), respectively, and pitch of spiral line is \(T=10\) mm. Number of total rotating circles of spiral line is \(M=29\). Figure 10: Transfer function \(\mathbf{\Omega}(p)\) (\(p=0,1,\cdots,250\)) of spiral scan EST. Here, angular position of five probes \(\varphi_{n}\) (\(n=1,2,\cdots,5\)) is \(0^{\circ}\), \(395^{\circ}\), \(838.7^{\circ}\), \(1236.5^{\circ}\), \(1658.8^{\circ}\), respectively, and pitch of spiral line is \(T=10\). Number of total rotating circles of spiral line is \(M=29\)[10]. The measured spiral line profile as shown in Figure 11 is based on a kind of \"ideal measurement\", i.e., based on an assumption that the straightness profile of the spiral line is closed, because the spiral scan EST is based on DFT in theory. In the case of \"ideal measurement\" during numerical validations, the sampling of five probes were carried out under the condition that the starting point and ending point of given spiral line profile were connected. However, it is impossible in actual measurements. If the angular distance between Probes 1 and 5 is \(\varphi=(\varphi_{5}-\varphi_{1})\) and the total rotation angle of the cylinder is \(2\pi M\), Probes 2-4 cannot normally collect data in the angular range from \((2\pi M-\varphi)\) to \(2\pi M\) during actual measurement.