# Comparison of Current Five-Point Cylindicity Error Separation Techniques
## 3 Applicative and Comparative Analysis for In Situ Measurement of Cylindricity
### Applicative Analysis of V-Block Scan EST
It may be found that if \(\alpha=30^{\circ}\) and \(\varphi=90^{\circ}\), the \(p\)th \((p=5,7,11,13,17,\cdots)\) harmonic suppression occurs. Especially, when \(\alpha=\varphi\), full harmonic suppression will occur. Let total number of scanning points per revolution be \(N\) and \(\delta=2\pi/N\), \(\varphi=k\delta\) and \(\alpha=u\delta\), the non-first harmonic suppression of this EST will be avoided provided that \(k\
eq u\), and the greatest common factor of \((k-u)\) and \((N/2-2u)\) as well as \(N\) is 1, e.g., \(\alpha=45^{\circ}\) and \(\varphi=68^{\circ}\). Therefore, this measuring system needs to carefully adjust the angular position of the probe so as to avoid the non-first harmonic suppression and get the transfer function \(\mathbf{\Omega}(p)(p=0,2,\cdots,15)\) as large as possible. The 0 th harmonic component \(\mathbf{R}(z,0)\) denotes the radius deviations of the sequential cross-sectional profiles. From Equation (15), measurement accuracy of the radius deviations depends on:  
\[\mathbf{\Delta}(z,0)=\left\{\mathbf{M}(z,0)+\frac{(L-z)\mathbf{M}(0,0)+z \mathbf{M}(L,0)}{L[\sin(\alpha)-\sin(\varphi)]}\sin(\varphi)\right\} \tag{17}\]  
It is known that \(\mathbf{M}(0,0)\) and \(\mathbf{M}(L,0)\) come from \(\left.e_{\mathit{s}x}(z)\right|_{z=0,L}\) and \(\left.e_{\mathit{s}y}(z)\right|_{z=0,L}\), i.e., the carriage straightness error motions in the \(z=0,L\) cross-sections, in other word, the guideway's slope in the XOZ and YOZ plane. Due to the existence of guideway's errors at the \(z=0,L\) cross-sections, the reconstructed cylindrical profile would produce a conical form. \(\mathbf{M}(z,0)=\mathrm{DFT}\left[e_{\mathit{s}x}(z)\cos(\varphi)+e_{\mathit{ s}y}(z)\sin(\varphi)\right]|_{p=0}\), indicating the 0 th harmonic component of the disturbances  
Figure 13: Non-first harmonic suppressions and the values of angular parameters of \(\varphi\) and \(\alpha\). If transfer function \(\mathbf{\Omega}(p)=0\) at given \(\varphi\) and \(\alpha\), the \(p\)th harmonic suppression occurs. Here, \(\alpha=30^{\circ}\ \sim 60^{\circ}\) and \(\varphi=40^{\circ}\ \sim 90^{\circ}\), harmonic order \(p=2\)–\(50\). from the carriage straightness error motions in the measured cross-section, directly results in the measurement errors in the radius of the reconstructed cylindrical profile, as shown in Figure 1a. In References [12; 13; 17], computer simulations of this EST for measurement of cylindrical profile were conducted to assess the non-first harmonic suppression. The results indicate that the 11 th harmonic component exists when \(\alpha=60^{\circ}\) and \(\varphi=90^{\circ}\) were selected. From Figure 13, the \(p\)th\((p=11,13,23,25,\cdots)\) harmonic suppression occurs in this case. It is a weakness that the straightness deviations of the guideway affect directly measurement accuracy of the cylindrical profile. It yields the necessity to compensate for the carriage straightness deviations in order to improve the measurement accuracy. In the authors' opinion, however, the compensation method should not change the fact that guideway's slopes in XOZ and YOZ planes would make the reconstructed cylindrical profile be a conical form. A measuring system allowing cylindricity measurements by this EST was constructed [12; 13; 17]. Some potential sources of systematic errors, such as, unequal angles of V-blocks, calibration error of the probe, deflection of the probe axis from its nominal orientation were analyzed. A series of experiments were conducted involving the comparison of measurement results of a group of cylindrical elements. The cylindricity deviation of each sample element was determined with two methods: V-block scan EST and high accurate radial method [19]. The conducted tests showed that the difference lies within the interval \(\pm 19\%\) (for a probability level \(p=0.95\)).