Abstract

We present a ray-tracing-based method for simulation of interference fringe patterns (IFPs) for measuring gear tooth flanks with a two-path interferometer. This simulation method involves two steps. In the first step, the profile of an IFP is achieved by means of ray tracing within the object path of the interferometer. In the second step, the profile of an IFP is filled with interference fringes, according to a set of functions from an optical path length to a fringe gray level. To examine the correctness of this simulation method, simulations are performed for two spur involute gears, and the simulated IFPs are verified by experiments using the actual two-path interferometer built on an optical platform.

© 2010 Optical Society of America

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References

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  1. H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
    [CrossRef]
  2. S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).
  3. S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.
  4. S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
    [CrossRef]
  5. S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
    [CrossRef]
  6. Nigel G. Douglas, Andrew R. Jones, and Frans J. van Hoesel, “Ray-based simulation of an optical interferometer,” J. Opt. Soc. Am. A 12, 124–131 (1995).
    [CrossRef]
  7. Norbert Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A Pure Appl. Opt. 4, S1–S9 (2002).
    [CrossRef]
  8. N. Lindlein, F. Simon, and Johannes Schwider, “Simulation of micro-optical arrays systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
    [CrossRef]
  9. A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of Ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt. 41, 3866–3873 (2002).
    [CrossRef] [PubMed]
  10. F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees coherence through complex optical systems,” Appl. Opt. 48, 1527–1534 (2009).
    [CrossRef] [PubMed]
  11. M. N. Akram, “Simulation and control of narcissus phenomenon using nonsequential ray tracing. 1. Staring camera in 3–5 μm waveband,” Appl. Opt. 49, 964–975 (2010).
    [CrossRef] [PubMed]
  12. D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
    [CrossRef]
  13. J. E. Klein, “Challenges and problems in non-sequential ray tracing,” Proc. SPIE 4442, 60–66 (2001).
    [CrossRef]
  14. J. E. Klein, “Demystifying the sequential ray tracing algorithm,” Proc. SPIE 4769, 67–74 (2002).
    [CrossRef]
  15. S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
    [CrossRef]
  16. S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
    [CrossRef]
  17. http://en.wikipedia.org/wiki/Involute_gear.
  18. J. R. Colbourne, The Geometry of Involute Gears (Springer-Verlag, 1987).
    [CrossRef]
  19. http://en.wikipedia.org/wiki/Involute.

2010 (1)

2009 (3)

F. Riechert, F. Dürr, U. Rohlfing, and U. Lemmer, “Ray-based simulation of the propagation of light with different degrees coherence through complex optical systems,” Appl. Opt. 48, 1527–1534 (2009).
[CrossRef] [PubMed]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
[CrossRef]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
[CrossRef]

2002 (3)

J. E. Klein, “Demystifying the sequential ray tracing algorithm,” Proc. SPIE 4769, 67–74 (2002).
[CrossRef]

Norbert Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A Pure Appl. Opt. 4, S1–S9 (2002).
[CrossRef]

A. Cordero-Dávila, J. Díaz-Anzures, and V. Cabrera-Peláez, “Algorithm for the simulation of Ronchigrams of arbitrary optical systems and Ronchi grids in generalized coordinates,” Appl. Opt. 41, 3866–3873 (2002).
[CrossRef] [PubMed]

2001 (1)

J. E. Klein, “Challenges and problems in non-sequential ray tracing,” Proc. SPIE 4442, 60–66 (2001).
[CrossRef]

2000 (1)

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

1998 (2)

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

N. Lindlein, F. Simon, and Johannes Schwider, “Simulation of micro-optical arrays systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

1996 (1)

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

1995 (2)

Nigel G. Douglas, Andrew R. Jones, and Frans J. van Hoesel, “Ray-based simulation of an optical interferometer,” J. Opt. Soc. Am. A 12, 124–131 (1995).
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

1968 (1)

Akram, M. N.

Cabrera-Peláez, V.

Colbourne, J. R.

J. R. Colbourne, The Geometry of Involute Gears (Springer-Verlag, 1987).
[CrossRef]

Cordero-Dávila, A.

Díaz-Anzures, J.

Douglas, Nigel G.

Dürr, F.

Fang, S.

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

Fang, S. P.

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
[CrossRef]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
[CrossRef]

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

Feder, D. P.

Fujio, H.

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

Jones, Andrew R.

Klein, J. E.

J. E. Klein, “Demystifying the sequential ray tracing algorithm,” Proc. SPIE 4769, 67–74 (2002).
[CrossRef]

J. E. Klein, “Challenges and problems in non-sequential ray tracing,” Proc. SPIE 4442, 60–66 (2001).
[CrossRef]

Komori, M.

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
[CrossRef]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
[CrossRef]

Kubo, A.

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
[CrossRef]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
[CrossRef]

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

Lemmer, U.

Lindlein, N.

N. Lindlein, F. Simon, and Johannes Schwider, “Simulation of micro-optical arrays systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

Lindlein, Norbert

Norbert Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A Pure Appl. Opt. 4, S1–S9 (2002).
[CrossRef]

Mei, X. S.

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
[CrossRef]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
[CrossRef]

Nakai, T.

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

Oyama, K.

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

Riechert, F.

Rohlfing, U.

Saitoh, Y.

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

Schwider, Johannes

N. Lindlein, F. Simon, and Johannes Schwider, “Simulation of micro-optical arrays systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

Simon, F.

N. Lindlein, F. Simon, and Johannes Schwider, “Simulation of micro-optical arrays systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

Suzuki, M.

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

Suzuki, Y.

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

Toya, M.

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

van Hoesel, Frans J.

Yatagai, T.

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

Appl. Opt. (3)

Chin. J. Mech. Eng. (2)

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Ray tracing method for optical system of interferometry measurement used for form deviation of precise complex surface of machine parts,” Chin. J. Mech. Eng. 45(2), 170–177 (2009), in Chinese.
[CrossRef]

S. P. Fang, M. Komori, A. Kubo, and X. S. Mei, “Precision simulation method for images of interference fringe pattern in complex optical system,” Chin. J. Mech. Eng. 45(1), 244–252 (2009), in Chinese.
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

Norbert Lindlein, “Simulation of micro-optical systems including microlens arrays,” J. Opt. A Pure Appl. Opt. 4, S1–S9 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

JSME Int. J. Ser. C (1)

S. P. Fang, H. Fujio, A. Kubo, M. Toya, Y. Suzuki, Y. Saitoh, and M. Suzuki, “New optical system for reducing the interference fringe density in laser interferometric measurement of tooth flank form of a gear,” JSME Int. J. Ser. C 43(2), 455–463 (2000).

Opt. Eng. (1)

N. Lindlein, F. Simon, and Johannes Schwider, “Simulation of micro-optical arrays systems with RAYTRACE,” Opt. Eng. 37, 1809–1816 (1998).
[CrossRef]

Proc. SPIE (2)

J. E. Klein, “Challenges and problems in non-sequential ray tracing,” Proc. SPIE 4442, 60–66 (2001).
[CrossRef]

J. E. Klein, “Demystifying the sequential ray tracing algorithm,” Proc. SPIE 4769, 67–74 (2002).
[CrossRef]

Trans. Jpn. Soc. Mech. Eng. C (3)

H. Fujio, A. Kubo, S. P. Fang, K. Oyama, T. Yatagai, Y. Saitoh, and M. Suzuki, “Measurement of gear tooth form deviation by laser interferometry using CCD image sensor,” Trans. Jpn. Soc. Mech. Eng. C 62, 2422–2430 (1996), in Japanese.
[CrossRef]

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Phase processing method of interferometry measurement for form deviation of the surface of machine parts,” Trans. Jpn. Soc. Mech. Eng. C 61, 106–114 (1995), in Japanese.
[CrossRef]

S. Fang, T. Nakai, A. Kubo, H. Fujio, Y. Saitoh, and M. Suzuki, “Fringe density reduction of relative phase difference in laser interferometry measurement of gear tooth flank,” Trans. Jpn. Soc. Mech. Eng. C 64, 1421–1427 (1998), in Japanese.
[CrossRef]

Other (4)

S. Fang, A. Kubo, H. Fujio, K. Oyama, Y. Saitoh, and M. Suzuki, “Digital phase data processing method for laser interferometry measurement of gear tooth flank,” in Proceedings of VDI International Conference on Gears (VDI Verlag, 1996), Vol. 1230, pp. 1111–1123.

http://en.wikipedia.org/wiki/Involute_gear.

J. R. Colbourne, The Geometry of Involute Gears (Springer-Verlag, 1987).
[CrossRef]

http://en.wikipedia.org/wiki/Involute.

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Figures (10)

Fig. 1
Fig. 1

Involute of a base circle and its normal vector. In gear definitions, α is the pressure angle and inv α means tan α α .

Fig. 2
Fig. 2

Involute in the measurement coordinate system. r 0 is the radius of the pitch circle.

Fig. 3
Fig. 3

Schematic diagram of the two-path interferometric optical system.

Fig. 4
Fig. 4

Direction of the incident light in the measurement coordinate system.

Fig. 5
Fig. 5

Tooth image of a spur involute gear (module is 2.5, number of teeth is 50, and face width is 20 mm ).

Fig. 6
Fig. 6

Zero plane Σ 0 of a zero optical path difference.

Fig. 7
Fig. 7

(a) Simulation of the error-free tooth flank. (b) Simulation of the tooth flank with the 20 μm drum-shaped error.

Fig. 8
Fig. 8

(a) Experimental result of the error-free tooth flank. (b) Experimental result of the tooth flank with the 20 μm drum-shaped error.

Fig. 9
Fig. 9

White and black binary image of the phase differences of the error-free tooth flank.

Fig. 10
Fig. 10

(a) Comparison of the phase levels from the simulation and the experiment for the error-free tooth flank. (b) Comparison of the tooth flank with the 20 μm drum error. Rd, Ro, and Ra are, respectively, the radius of the dedendum circle, the pitch circle, and the addendum circle.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

x 0 = r g · sin θ r g · θ · cos θ y 0 = r g · cos θ + r · g θ · sin θ } ,
θ = side · Θ ,
side = { 1 tooth right flank 1 tooth left flank .
n 0 x = side · cos θ n 0 y = side · sin θ } .
γ = τ + side · ψ side · inv α 0 ,
x = r g · sin ( θ γ ) r g · θ · cos ( θ γ ) y = r g · cos ( θ γ ) + r g · θ · sin ( θ γ ) } ,
n x = side · cos ( θ γ ) n y = side · sin ( θ γ ) } .
a x = side · sin ν a y = 0 a z = cos ν } ,
l 2 ( i , j ) = ( g side · x 0 ( i , j ) ) sin ν + z 0 ( i , j ) cos ν ,
L 2 ( i , j ) = n 0 · l 2 ( i , j ) ,
L 0 ( i , j ) = L 1 ( i , j ) + L 2 ( i , j ) ,
Δ L 0 ( i , j ) = L 0 ( i , j ) L 0 min .
Δ L r ( i , j ) = L r ( i , j ) L r min ,
Δ Φ 1 ( i , j ) = 2 π · Δ L 0 ( i , j ) / λ Δ Φ 2 ( i , j ) = 2 π · Δ L r ( i , j ) / λ } ,
Δ Φ ( i , j ) = Δ Φ 1 ( i , j ) Δ Φ 2 ( i , j ) .
Δ Φ 0 ( i , j ) = Δ Φ ( i , j ) Δ Φ min ,
B ( x , y ) = B max · cos Δ Φ 0 ( x , y ) ,
Δ Φ 0 ( x , y ) = Δ Φ 0 ( x , y ) [ Δ Φ 0 ( x , y ) 2 π ] · 2 π ,
B ( x , y ) = { 255 0 Δ Φ 0 ( x , y ) < π 0 π Δ Φ 0 ( x , y ) < 2 π .

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