Abstract

The addition of an apertured mirror before the imaging lens is proposed as a method of providing feedback in a single-axis resonant fiber scanner. Reflection at the scan extremities generates timing signals interlaced with back-scattered data, and a phase locked loop and a proportional controller then adjust the drive frequency and amplitude. The capture range and stability of the system are examined. Verification is obtained using a confocal scanner based on mechanically biaxial fiber.

© 2014 Optical Society of America

Full Article  |  PDF Article
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References

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  1. K. E. Petersen, “Silicon torsional scanning mirror,” IBM J. Res. Develop. 24(5), 631–637 (1980).
    [Crossref]
  2. H. Toshiyoshi, G. D. J. Su, J. LaCosse, and M. C. Wu, “A surface micromachined optical scanner array using photoresist lenses fabricated by a thermal reflow process,” J. Lightwave Technol. 21(7), 1700–1708 (2003).
    [Crossref]
  3. G. Zhou and F. S. Chau, “Micromachined vibratory diffraction grating scanner for multiwavelength collinear laser scanning,” J. Microelectromech. Syst. 15(6), 1777–1788 (2006).
    [Crossref]
  4. H. Urey, “Torsional MEMS scanner design for high-resolution display systems,” Proc. SPIE 4773, 27–37 (2002).
    [Crossref]
  5. M.-H. Kiang, O. Solgaard, R. S. Muller, and K. Y. Lau, “Micromachined polysilicon microscanners for barcode readers,” IEEE Photon. Technol. Lett. 8(12), 1707–1709 (1996).
    [Crossref]
  6. H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
    [Crossref]
  7. W. Piyawattanametha, R. P. J. Barretto, T. H. Ko, B. A. Flusberg, E. D. Cocker, H. Ra, D. Lee, O. Solgaard, and M. J. Schnitzer, “Fast-scanning two-photon fluorescence imaging based on a microelectromechanical systems two- dimensional scanning mirror,” Opt. Lett. 31(13), 2018–2020 (2006).
    [Crossref] [PubMed]
  8. L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27(9), 724–726 (1991).
    [Crossref]
  9. D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
    [Crossref]
  10. F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
    [Crossref] [PubMed]
  11. B. A. Flusberg, J. C. Jung, E. D. Cocker, E. P. Anderson, and M. J. Schnitzer, “In vivo brain imaging using a portable 3.9 gram two-photon fluorescence microendoscope,” Opt. Lett. 30(17), 2272–2274 (2005).
    [Crossref] [PubMed]
  12. J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
    [Crossref] [PubMed]
  13. E. C. Haight and W. W. King, “Stability of nonlinear oscillations of an elastic rod,” J. Acoust. Soc. Am. 52(3B), 899–911 (1972).
    [Crossref]
  14. Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
    [Crossref]
  15. D. A. Roberts and R. R. A. Syms, “1D and 2D laser line scan generation using a fiber optic resonant scanner,” SPIE Proc. 4075, 62–73 (2000).
  16. R. D. Birch, D. N. Payne, and M. P. Varnham, “Fabrication of polarization-maintaining fibers using gas-phase etching,” Electron. Lett. 18(24), 1036–1038 (1982).
    [Crossref]

2009 (1)

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

2006 (2)

2005 (1)

2004 (1)

Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
[Crossref]

2003 (2)

H. Toshiyoshi, G. D. J. Su, J. LaCosse, and M. C. Wu, “A surface micromachined optical scanner array using photoresist lenses fabricated by a thermal reflow process,” J. Lightwave Technol. 21(7), 1700–1708 (2003).
[Crossref]

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

2002 (1)

H. Urey, “Torsional MEMS scanner design for high-resolution display systems,” Proc. SPIE 4773, 27–37 (2002).
[Crossref]

2001 (1)

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

1999 (1)

D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
[Crossref]

1996 (1)

M.-H. Kiang, O. Solgaard, R. S. Muller, and K. Y. Lau, “Micromachined polysilicon microscanners for barcode readers,” IEEE Photon. Technol. Lett. 8(12), 1707–1709 (1996).
[Crossref]

1991 (1)

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27(9), 724–726 (1991).
[Crossref]

1982 (1)

R. D. Birch, D. N. Payne, and M. P. Varnham, “Fabrication of polarization-maintaining fibers using gas-phase etching,” Electron. Lett. 18(24), 1036–1038 (1982).
[Crossref]

1980 (1)

K. E. Petersen, “Silicon torsional scanning mirror,” IBM J. Res. Develop. 24(5), 631–637 (1980).
[Crossref]

1972 (1)

E. C. Haight and W. W. King, “Stability of nonlinear oscillations of an elastic rod,” J. Acoust. Soc. Am. 52(3B), 899–911 (1972).
[Crossref]

Anderson, E. P.

Aoki, Y.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Asaoka, N.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Barretto, R. P. J.

Birch, R. D.

R. D. Birch, D. N. Payne, and M. P. Varnham, “Fabrication of polarization-maintaining fibers using gas-phase etching,” Electron. Lett. 18(24), 1036–1038 (1982).
[Crossref]

Chau, F. S.

G. Zhou and F. S. Chau, “Micromachined vibratory diffraction grating scanner for multiwavelength collinear laser scanning,” J. Microelectromech. Syst. 15(6), 1777–1788 (2006).
[Crossref]

Cocker, E. D.

Denk, W.

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

Fee, M. S.

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

Flusberg, B. A.

Fujimori, O.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Giniunas, L.

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27(9), 724–726 (1991).
[Crossref]

Greenberg, D. S.

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

Grossmann, S.

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

Haight, E. C.

E. C. Haight and W. W. King, “Stability of nonlinear oscillations of an elastic rod,” J. Acoust. Soc. Am. 52(3B), 899–911 (1972).
[Crossref]

Helmchen, F.

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

Holmes, A. S.

D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
[Crossref]

Imai, M.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Isokawa, T.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Jung, J. C.

Juskaitis, R.

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27(9), 724–726 (1991).
[Crossref]

Katashiro, M.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Kerr, J. N. D.

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

Kiang, M.-H.

M.-H. Kiang, O. Solgaard, R. S. Muller, and K. Y. Lau, “Micromachined polysilicon microscanners for barcode readers,” IEEE Photon. Technol. Lett. 8(12), 1707–1709 (1996).
[Crossref]

King, W. W.

E. C. Haight and W. W. King, “Stability of nonlinear oscillations of an elastic rod,” J. Acoust. Soc. Am. 52(3B), 899–911 (1972).
[Crossref]

Ko, T. H.

LaCosse, J.

Lau, K. Y.

M.-H. Kiang, O. Solgaard, R. S. Muller, and K. Y. Lau, “Micromachined polysilicon microscanners for barcode readers,” IEEE Photon. Technol. Lett. 8(12), 1707–1709 (1996).
[Crossref]

Lee, D.

Matsumoto, K.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Miyajima, H.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Muller, R. S.

M.-H. Kiang, O. Solgaard, R. S. Muller, and K. Y. Lau, “Micromachined polysilicon microscanners for barcode readers,” IEEE Photon. Technol. Lett. 8(12), 1707–1709 (1996).
[Crossref]

Ogata, M.

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Payne, D. N.

R. D. Birch, D. N. Payne, and M. P. Varnham, “Fabrication of polarization-maintaining fibers using gas-phase etching,” Electron. Lett. 18(24), 1036–1038 (1982).
[Crossref]

Petersen, K. E.

K. E. Petersen, “Silicon torsional scanning mirror,” IBM J. Res. Develop. 24(5), 631–637 (1980).
[Crossref]

Piyawattanametha, W.

Ra, H.

Reinhall, P. G.

Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
[Crossref]

Roberts, D. A.

D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
[Crossref]

Sawinski, J.

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

Schnitzer, M. J.

Seibel, E. J.

Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
[Crossref]

Shatalin, S. V.

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27(9), 724–726 (1991).
[Crossref]

Smithwick, Q. Y. J.

Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
[Crossref]

Solgaard, O.

Su, G. D. J.

Syms, R. R. A.

D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
[Crossref]

Tank, D. W.

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

Toshiyoshi, H.

Urey, H.

H. Urey, “Torsional MEMS scanner design for high-resolution display systems,” Proc. SPIE 4773, 27–37 (2002).
[Crossref]

Vagners, J.

Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
[Crossref]

Varnham, M. P.

R. D. Birch, D. N. Payne, and M. P. Varnham, “Fabrication of polarization-maintaining fibers using gas-phase etching,” Electron. Lett. 18(24), 1036–1038 (1982).
[Crossref]

Wallace, D. J.

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

Wu, M. C.

Yeatman, E. M.

D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
[Crossref]

Zhou, G.

G. Zhou and F. S. Chau, “Micromachined vibratory diffraction grating scanner for multiwavelength collinear laser scanning,” J. Microelectromech. Syst. 15(6), 1777–1788 (2006).
[Crossref]

Electron. Lett. (3)

L. Giniunas, R. Juskaitis, and S. V. Shatalin, “Scanning fiber-optic microscope,” Electron. Lett. 27(9), 724–726 (1991).
[Crossref]

D. A. Roberts, R. R. A. Syms, A. S. Holmes, and E. M. Yeatman, “Dual numerical aperture confocal operation of a moving fiber bar-code reader,” Electron. Lett. 35(19), 1656–1658 (1999).
[Crossref]

R. D. Birch, D. N. Payne, and M. P. Varnham, “Fabrication of polarization-maintaining fibers using gas-phase etching,” Electron. Lett. 18(24), 1036–1038 (1982).
[Crossref]

IBM J. Res. Develop. (1)

K. E. Petersen, “Silicon torsional scanning mirror,” IBM J. Res. Develop. 24(5), 631–637 (1980).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M.-H. Kiang, O. Solgaard, R. S. Muller, and K. Y. Lau, “Micromachined polysilicon microscanners for barcode readers,” IEEE Photon. Technol. Lett. 8(12), 1707–1709 (1996).
[Crossref]

J. Acoust. Soc. Am. (1)

E. C. Haight and W. W. King, “Stability of nonlinear oscillations of an elastic rod,” J. Acoust. Soc. Am. 52(3B), 899–911 (1972).
[Crossref]

J. Dyn. Syst. Meas. Control (1)

Q. Y. J. Smithwick, P. G. Reinhall, J. Vagners, and E. J. Seibel, “A nonlinear state-space model of a resonating single-fiber scanner for tracking control: theory and experiment,” J. Dyn. Syst. Meas. Control 126(1), 88–101 (2004).
[Crossref]

J. Lightwave Technol. (1)

J. Microelectromech. Syst. (2)

G. Zhou and F. S. Chau, “Micromachined vibratory diffraction grating scanner for multiwavelength collinear laser scanning,” J. Microelectromech. Syst. 15(6), 1777–1788 (2006).
[Crossref]

H. Miyajima, N. Asaoka, T. Isokawa, M. Ogata, Y. Aoki, M. Imai, O. Fujimori, M. Katashiro, and K. Matsumoto, “A MEMS electromagnetic optical scanner for a commercial confocal laser scanning microscope,” J. Microelectromech. Syst. 12(3), 243–251 (2003).
[Crossref]

Neuron (1)

F. Helmchen, M. S. Fee, D. W. Tank, and W. Denk, “A miniature head-mounted two-photon microscope. high-resolution brain imaging in freely moving animals,” Neuron 31(6), 903–912 (2001).
[Crossref] [PubMed]

Opt. Lett. (2)

Proc. Natl. Acad. Sci. U.S.A. (1)

J. Sawinski, D. J. Wallace, D. S. Greenberg, S. Grossmann, W. Denk, and J. N. D. Kerr, “Visually evoked activity in cortical cells imaged in freely moving animals,” Proc. Natl. Acad. Sci. U.S.A. 106(46), 19557–19562 (2009).
[Crossref] [PubMed]

Proc. SPIE (1)

H. Urey, “Torsional MEMS scanner design for high-resolution display systems,” Proc. SPIE 4773, 27–37 (2002).
[Crossref]

Other (1)

D. A. Roberts and R. R. A. Syms, “1D and 2D laser line scan generation using a fiber optic resonant scanner,” SPIE Proc. 4075, 62–73 (2000).

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Figures (6)

Fig. 1
Fig. 1 Resonant fiber scanner, a) system b) ideal variations of drive, response and signal c) controller.
Fig. 2
Fig. 2 Theoretical open-loop frequency dependence of a) amplitude and b) phase, for ζ = 0.005.
Fig. 3
Fig. 3 a) and b) Convergence of frequency error Δfn/Δf0, for initial errors of Δf0/ΔfC = 0.0625 and 0.5, assuming ζ = 0.005, ymax/yL = 2; c) Convergence of voltage error, assuming ϕS = 0.2π.
Fig. 4
Fig. 4 Breadboard scanner: a) schematic and b) experimental rig.
Fig. 5
Fig. 5 Responses of bow-tie fibers with principle axes a) at ± 45° and b) and c) at 0° to vertical.
Fig. 6
Fig. 6 Time variation of drive and signal, for a) misaligned system, b) start and c) end of control loop.

Equations (12)

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

ω 0 n = ( α n / L 2 ) ( E I / ρ A )
y 0 = v 0 C / { ( ω 01 2 ω 2 ) + 2 j ζ ω ω 01 }
ϕ tan 1 ( ζ ω 01 / Δ ω )
ϕ ϕ s Δ ω / ( ζ ω 01 )
v = a / cos ( Δ ϕ )
4 ζ 2 ω 01 4 / { ( ω 01 2 ω 2 ) 2 + 4 ζ 2 ω 2 ω 01 2 } > ( y L / y max ) 2
Δ ω C / ω 01 = ζ { ( y max / y L ) 2 1 }
ω n + 1 = ω n k ω ( ϕ n ϕ S )
Δ ω n + 1 = Δ ω n ( 1 α )
v n + 1 = v n k v { 1 / cos ( Δ ϕ n ) 1 / cos ( Δ ϕ S ) }
v n + 1 = v n ( k v / a ) ( v n v S )
Δ v n + 1 = Δ v n ( 1 β )

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