# Optical Inspection System for Gear Tooth Surfaces Using a Projection Moiré Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Projection Moiré Measurement System

#### 2.1. Projection Moiré Fringes

_{1}and G

_{2}, the intensity transmission functions, I

_{G1}(x,y) and I

_{G2}(x,y), can be expressed as follows [8,16]:

_{1}and α

_{2}are the average intensities; b

_{1n}and b

_{1m}denote the grating line profiles, such as square, triangular, and sinusoidal wares. Straight line gratings are used in this study; therefore, the above equations can be simplified and rewritten as follows:

_{1}and λ

_{2}denote the grating period of the two gratings. Equations (7) and (8) can be rewritten as follows:

_{1}+ λ

_{2}) is the average grating period, and λ

_{b}= λ

_{1}λ

_{2}/(λ

_{1}+ λ

_{2}) is the beat wavelength between the two gratings.

#### 2.2. Five-Step Phase-Shifting

_{K}(i,j) represents the intensity at K

^{th}phase shifting (K = 1~5), A(i,j) is the constant term, that is, the intensity bias, B(i,j) is the intensity amplitude, and φ(i,j) is the phase information to be determined at the measurement point.

_{n}is the contour interval (the height between adjacent contour lines in the CCD viewing angle), and P

_{o}is the spacing of the lines perpendicular to the CCD viewing direction. The fringe order N is the following:

#### 2.3. Experimental Framework

#### 2.4. GearTooth Surface Measurement Procedure

- (1)
- Perform the distortion correction of the CCD camera by obtaining the distortion correction coefficients with a grid distortion target, as illustrated in Figure 6a. Perform depth calibration to obtain the calibration coefficient of the magnification along the tooth depth, as illustrated in Figure 6b.
- (2)
- Project the moiré fringes on the gear.
- (3)
- Capture the deformed moiré fringes on the gear tooth surface.
- (4)
- Perform the five-step phase-shifting by moving grating 1 on a linear stage and record the images of moiré fringes at each phase-shifting step. Because the fringe pitch is 806 μm, the translation of the grating for each phase-shifting is 201.5 μm. Figure 7 presents five images of moiré fringes on the gear tooth surface for each phase-shifting step.
- (5)
- Perform the image processing and phase unwrapping of the images of moiré fringes using self-developed Matlab codes. Phase unwrapping is conducted using the branch-cut method, and the noise can be filtered using a Butterworth low-pass filter.
- (6)
- (7)
- Compare and calculate the deviations of the optically measured surface from the CMM-measured reference surface.

## 3. Results

#### 3.1. Involute Profile

#### 3.2. Lead Profile

#### 3.3. Tooth Surface Topology

## 4. Discussion

^{3}was measured, and the deviation was calculated. Figure 14a,b depicts the moiré fringes on the gage block and the reconstructed 3-D profile by our optical method, respectively. In addition, Figure 14c illustrates the height reconstructed by our method and the nominal height of the gage block. The average value of the height deviation is 2.76 μm.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the angle formed by the charge-coupled device (CCD) viewing direction and the projection moiré fringe, and a reference plane for the height estimation.

**Figure 4.**(

**a**) Photo of the 18-tooth spur gear; (

**b**) projected moiré fringes on the gear tooth surface.

**Figure 6.**(

**a**) Distortion coefficient correction with a grid distortion target; (

**b**) calibration coefficient of the magnification along the tooth depth.

**Figure 8.**(

**a**) Wrapped phase map on the gear tooth surface and the region of interest; (

**b**) reconstructed 3-D tooth surface.

**Figure 9.**(

**a**) Tooth surface measured by a commercial coordinate measuring machine (CMM); (

**b**) measuring points on the tooth surface.

**Figure 10.**(

**a**) Illustration of the involute profile measurement at the central cross-section; (

**b**) illustration of the lead (axial) profile measurement on the pitch circle.

**Figure 11.**(

**a**) Radial (involute) profiles measured by the CMM and optical system; (

**b**) deviation of the involute profile of the projection moiré system from the CMM method.

**Figure 12.**(

**a**) Lead profiles measured by the CMM and optical system; (

**b**) deviation of the lead profile measured by the projection moiré method from the CMM method.

**Figure 13.**Spur gear tooth surface: (

**a**) Topology of the gear tooth surface in 10 × 5 grids; (

**b**) deviation of the tooth surface topology measured by the projection moiré system from the CMM method.

**Figure 14.**Gauge block measurements: (

**a**) moiré fringes on the gauge block; (

**b**) reconstructed 3-D profile; (

**c**) optically measured profile and gauge block profile.

Component | Specification | |
---|---|---|

Projection module | Illumination | Halogen source |

Collimator | Trioptics 3-100-093 | |

Gratings (×2) | Pitch: 200 μm | |

Biconvex lens (×2) | f = 75 mm | |

f = 100 mm | ||

Phase-shifting module | Stepping servo motor and driver | Oriental motor AR24SAK-1 |

Linear stage | AFM-40-C5 Repeatability: ±2 μm | |

Control system | LABVIEW codes | |

Image capture module | CCD sensor | Basler acA640-90gm |

Camera | Fujinon | |

Image processing | MATLAB codes |

Parameters | Values |
---|---|

Module (mm) | 2 |

Teeth Number | 18 |

Face Width (mm) | 20 |

Pressure Angle (degrees) | 20 |

Pitch Diameter (mm) | 36 |

Addendum Diameter (mm) | 40 |

Measured Tooth Surface Deviation (μm) | ||||
---|---|---|---|---|

Mean | Maximum | Minimum | Standard Deviation | |

Involute profile | 2.67 | 4.49 | 2.59 × 10^{−2} | 1.44 |

Lead profile | 2.02 | 3.63 | 0.27 | 1.24 |

3-D Topology | 2.81 | 6.17 | 0.41 | 1.31 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chen, Y.-C.; Chen, J.-Y.
Optical Inspection System for Gear Tooth Surfaces Using a Projection Moiré Method. *Sensors* **2019**, *19*, 1450.
https://doi.org/10.3390/s19061450

**AMA Style**

Chen Y-C, Chen J-Y.
Optical Inspection System for Gear Tooth Surfaces Using a Projection Moiré Method. *Sensors*. 2019; 19(6):1450.
https://doi.org/10.3390/s19061450

**Chicago/Turabian Style**

Chen, Yi-Cheng, and Jr-Yi Chen.
2019. "Optical Inspection System for Gear Tooth Surfaces Using a Projection Moiré Method" *Sensors* 19, no. 6: 1450.
https://doi.org/10.3390/s19061450