# Non-Contact Measurement of Small-Module Gears Using Optical Coherence Tomography
## 2 Methods and Experiments
### Systematic Evaluation of the Measurement Capabilities of the OCT Gear Measurement  
For a measurement system, uncertainty, repeatability, and reproducibility were often used to evaluate the measurement capability [21]. Taken gear modulus analysis as an example, six  
Figure 9: A 0.2 mm modulus of a miniature copper gear measurement. (**a**) whole teeth profile imaging; (**b**) tooth profile measurement using the minimum five times magnification objective lens of the metallographic microscope; (**c**) photo of the gear body and the OCT scan line 1–1\({}^{\prime}\). measurements for each gear have been carried out to obtain the tooth height (as shown in Table 2) from which the modules can be calculated using Equation (1). The measurement uncertainty can be expressed by using standard deviation. The measurement repeatability \(EV\) is the variation of the measured value that is obtained by the same operator by using the same measuring tool and by measuring the same characteristics of the same sample many times, as described by Equation (10) [22]. \[EV=\left[(\overline{R_{1}}+\overline{R_{2}}+\overline{R_{3}})/3\right]\times K _{1} \tag{10}\]  
where \(\overline{R_{1}}\), \(\overline{R_{2}}\), and \(\overline{R_{3}}\) are the range of repeated measurements averaged across the samples by operators #1, #2, and #3, respectively; \(K_{1}\) is the adjustment factor. When the number of measurements is 3, \(K_{1}=0.5908\). The reproducibility \(AV\) is the variation of the mean value of the same part, measured with the same measuring instrument by different operators when measuring the same characteristics of the same sample, described by Equation (11) [22] as:  
\[AV=\sqrt{\left[(\overline{X}_{\text{max}}-\overline{X}_{\text{min}})\times K _{2}\right]^{2}-EV^{2}/(\text{n}\times\text{r})} \tag{11}\]  
where \(\overline{X}\) is the mean value of the sample average, consisting of replicate data that are obtained by one operator.\(\overline{X}_{\text{max}}\) is the maximum of \(\overline{X}\) among the operators. Similarly, \(\overline{X}_{\text{min}}\) is the minimum of \(\overline{X}\) among operators; \(K_{2}\) is the adjustment factor. When the number of the operators is 3, \(K_{2}=0.5231\), n is the number of samples, and r is the number of measurements. Changes due to instrument variation can be obtained by repeatability values \(EV\), and changes due to human variation can be obtained by reproducibility values \(AV\). We chose 10 gears with modulus of 0.5 mm that represent process changes; three operators repeated three consecutive measurements on the 10 products. The depths of the gears as measured by different operators were achieved as in Figures 10-12 (three measurements for each sample). The dotted line is the limit of the data. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Gear Modulus** & \multicolumn{6}{c}{**Depth of the gear \(h\) of six measurements (mm)**} & \multicolumn{2}{c}{**Mean Value**} & \multicolumn{2}{c}{**Standard**} \\ \cline{2-9}
**(mm)*** & **1** & **2** & **3** & **4** & **5** & **6** & **(mm)** & **Deviation** \\ \hline
0.2 (0.45) & 0.484 & 0.453 & 0.467 & 0.464 & 0.452 & 0.447 & 0.461 & \(5.523\times 10^{-3}\) \\
0.5 (1.125) & 1.114 & 1.120 & 1.111 & 1.108 & 1.120 & 1.105 & 1.113 & \(2.530\times 10^{-3}\) \\
0.8 (1.8) & 1.829 & 1.813 & 1.819 & 1.772 & 1.815 & 1.787 & 1.806 & \(8.843\times 10^{-3}\) \\ \hline \hline \end{tabular}  
* The values in parentheses in the first column are the theoretical values of depth of the gear for the corresponding modulus.  
\end{table} 
Table 2: Uncertainty of the measurement of different modulus gears. Figure 10: The depth of the 10 gears measured by the first operator (three measurements for each sample).