# Non-Contact Measurement of Small-Module Gears Using Optical Coherence Tomography
## 2 Methods and Experiments
### Measurement of the Geometric Parameters of the Gears
#### 2.1.3 Calculation of Pressure Angle, Modification Coefficient, and Tooth Thickness  
The values of the standard pressure angles are 20 or 15 degrees [16]. It can be calculated by the value of the common normal length. The common normal length \(W_{k}\) is defined as the straight line distance between the tangential points of two parallel planes and two pairs of tooth surfaces. When using a gear tooth micrometer to carry out the measurement, its two measuring surfaces and involute tooth profile surfaces should be tangential to the location near the gear reference circle. In OCT profile images shown in Figure 5, the common normal length across two teeth was measured by \(W_{k}\), and the common normal length across three teeth by \(W_{k+1}\); so that the pressure angle \(\alpha\) could be calculated by Equation (3) [18]:  
\[\alpha=\arccos[(W_{k+1},-W_{k})/\tau m] \tag{3}\]  
By obtaining OCT images of more teeth numbers, and by measuring the average length of a common normal length between different teeth several times, the accuracy of the common normal length and the pressure angle measurement can be further improved. In our current OCT system, the tooth profiles of six teeth were measured in the experiment simultaneously. The results of \(W_{k+1^{\prime}}=3.745\) mm, and \(W_{k^{\prime}}=2.233\) mm were obtained. Further, the pressure angle of the gear \(\alpha=15.62^{\circ}\) was achieved with a relative difference of 4% between the normal one.  
When cutting a gear, if the gear is not installed in a standard manner and the cutter is kept away from or close to the turning center of the blank wheel  
by changing the relative position of the cutter and the gear being machined is called modification correction. A gear that is processed by this method is called a modified gear, and gears with less than 17 teeth usually need to be processed by modifying. The modification coefficient \(x\) can be obtained by Equation (4) by the length of the common normal line across two teeth \(W_{k^{\prime}}\) and the standard common normal line \(W_{k}\)[18].  
\[x=W_{k^{\prime}}-W_{k}/2m\sin\alpha \tag{4}\]  
\[W_{k}=m\cos\alpha[(k-0.5)\pi+zinv\alpha] \tag{5}\]  
\[inv\alpha=\tan\alpha-\alpha \tag{6}\]  
where \(k\) is the tooth number been selected, and \(inv\alpha\) is the gear involute function. By using Equation (5), we could obtain \(W_{k}=2.364\) mm, therefore, \(x=-0.505\) could also been achieved. The value was consistent with experience of using a negative modification coefficient for big gears (gears with more teeth in the transmission often use the negative modification coefficient \(x\)) during general machining. Furthermore, the tooth thickness \(S\) on the actual reference radius can be determined by the numerical size of the common normal line. By the conversion of the equation as follows [19]:  
\[W_{k}=[(k-1)\pi m+S+mzinv\alpha]\cos\alpha \tag{7}\]  
The tooth thickness can be calculated according to the common normal line:  
\[S=\frac{W_{k}}{\cos\alpha}-mzinv\alpha-(k-1)m\pi \tag{8}\]  
The tooth thickness calculated by the equation was \(S\)\(=0.781\) mm. When the modulus was 0.5 mm, the pitch of tooth was \(p=\pi m=1.57\) mm. It could be seen that the calculation result was very close to half of the pitch, and the results are close to the actual values.