# Non-Contact Measurement of Small-Module Gears Using Optical Coherence Tomography
## 2 Methods and Experiments
### Measurement of the Geometric Parameters of the Gears
#### 2.1.1 Analysis of Module, Addendum Coefficient, and Top Clearance Coefficient  
The gear module is defined as a basic parameter of gears. The parameters of the gear reference circle diameter, tooth tip circle diameter, tooth root circle diameter, tooth thickness, and tooth width are closely related to the module [16], and they determine the size of the tooth. Therefore, it is very important to map out the gear modulus first when measuring the gear parameters.  
Figure 3a showed an OCT image of a small-module metal copper gear. The figure contained four teeth in the scanning range. Figure 3b,c are the right view and top view of the copper gear measured.  
Figure 1: The schematic diagram of the experimental optical coherence tomography (OCT) system. FC: fiber couple; PC: polarization controller; C: collimator; BS: beamsplitter; VA: variable attenuator; M: mirror; SP: spectrograph; G: galvo mirror.  
Figure 2: Measurement of different tooth profile with different angles of incident light: (**a**) imaging of an initial position of the gear; (**b**) imaging the gear with a rotation angle of \(60^{\circ}\).  
The OCT image was firstly binarized by using the \"im2 bw\" function in MatLab R2014a (MathWorks, United States). Second, the \"imfill\" function was employed to fill the image and to remove internal holes. Finally, the contour of the gear was extracted using the \"bwperim\" function. The edge position of the tooth tip and the tooth root of the gear could be obtained by the slope of the tooth profile, which was shown in Figure 3. The tooth tip was located at the position where the slope of a tooth profile changed from positive to negative, whilst the tooth root was at the position where the slope changed from negative to positive. The depth of the gear \(h\) could be calculated once the tooth tip and root were achieved. In addition, in order to reduce the errors of the location of the tip and root of the tooth, we measured three tooth profiles, and we calculated the average depth of the gear of three teeth to determine a more accurate tooth height. Subsequently, the modulus \(m\) could be obtained as:  
\[m=h/(2{h_{a}}^{*}+c^{*}) \tag{1}\]  
where \({h_{a}}^{*}\) is the addendum coefficient and \(c^{*}\) is the top clearance coefficient. These two normative tooth parameters had been used in Equation (1) (\({h_{a}}^{*}=1.0\), \(c^{*}=0.25\) for normal tooth and \({h_{a}}^{*}=0.8\), \(c^{*}=0.3\) for short tooth) [17].  
Generally, in Table 1, the two columns are the series of modulus values. The values at the second column are for the first series, and those at the third column are for the second series. The modulus in the first series should be used preferentially, followed by the second series; the modulus in parentheses should be used as little as possible [17]. From the OCT tooth profile, we could obtain a tooth height \(h=1.10\) mm. For a normal tooth, the modulus could be obtained as \(m=0.49\) whilst \(m=0.57\) for a short tooth. Therefore, from Table 1, the gear should be a normal tooth gear, where the modulus was 0.5 mm. Comparing with the standard value module series table shown in Table 1, the measured modulus is close to the normative one, and the relative difference is 2%.