Abstract

In a polishing process the wear is greater at the edge when the tool extends beyond the border of the workpiece. To explain this effect, we propose a new model in which the pressure is higher at the edge. This model is applied to the case of a circular tool that polishes a circular workpiece. Our model correctly predicts that a greater amount of material is removed from the edge of the workpiece.

© 2004 Optical Society of America

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References

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  1. F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 9, 214–256 (1927).
  2. W. Rupp, “Loose abrasive grinding of optical surfaces,” Appl. Opt. 11, 2797–2810 (1972).
    [CrossRef] [PubMed]
  3. R. Aspden, R. McDonough, F. R. Nitchie, “Computer assisted optical surfacing,” Appl. Opt. 11, 2739–2747 (1972).
    [CrossRef] [PubMed]
  4. R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16, 218–222 (1977).
    [CrossRef] [PubMed]
  5. N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
    [CrossRef]
  6. A. S. Savel’ev, A. P. Bogdanov, “Automated polishing of large optical components with a small tool,” Sov. J. Opt. Technol. 52, 294–297 (1985).
  7. R. A. Jones, “Computer controlled optical surfacing with orbital tool motion,” Opt. Eng. 25, 785–790 (1986).
    [CrossRef]
  8. D. W. Small, S. J. Hoskins, “An automated asphere polishing machine,” in Optical Manufacturing, Testing, and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986).
    [CrossRef]
  9. D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).
  10. S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
    [CrossRef] [PubMed]
  11. R. A. Jones, “Fabrication using the computer controlled polisher,” Appl. Opt. 17, 1889–1892 (1978).
    [CrossRef] [PubMed]
  12. T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
    [CrossRef]
  13. R. A. Jones, W. J. Rupp, “Rapid optical fabrication with computer-controlled optical surfacing,” Opt. Eng. 30, 1962–1968 (1991).
    [CrossRef]
  14. R. A. Jones, “Fabrication of a large, thin, off-axis aspheric mirror,” Opt. Eng. 33, 4068–4075 (1994).
    [CrossRef]
  15. R. A. Jones, “Computer controlled polishing of telescope mirror segments,” Opt. Eng. 22, 236–240 (1983).
    [CrossRef]
  16. R. E. Wagner, R. R. Shannon, “Fabrication of aspherics using a mathematical model for material removal,” Appl. Opt. 13, 1683–1689 (1974).
    [CrossRef] [PubMed]
  17. E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
    [CrossRef]
  18. G. R. Fowles, Analytical Mechanics (Wiley, Reading, Mass., 1962).

1994

1993

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
[CrossRef]

1991

R. A. Jones, W. J. Rupp, “Rapid optical fabrication with computer-controlled optical surfacing,” Opt. Eng. 30, 1962–1968 (1991).
[CrossRef]

1986

R. A. Jones, “Computer controlled optical surfacing with orbital tool motion,” Opt. Eng. 25, 785–790 (1986).
[CrossRef]

1985

A. S. Savel’ev, A. P. Bogdanov, “Automated polishing of large optical components with a small tool,” Sov. J. Opt. Technol. 52, 294–297 (1985).

1983

R. A. Jones, “Computer controlled polishing of telescope mirror segments,” Opt. Eng. 22, 236–240 (1983).
[CrossRef]

1978

N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
[CrossRef]

R. A. Jones, “Fabrication using the computer controlled polisher,” Appl. Opt. 17, 1889–1892 (1978).
[CrossRef] [PubMed]

1977

1974

1972

1927

F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 9, 214–256 (1927).

Anderson, D. S.

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Angel, J. R. P.

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Aspden, R.

Bennett, T. L.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
[CrossRef]

Bogdanov, A. P.

A. S. Savel’ev, A. P. Bogdanov, “Automated polishing of large optical components with a small tool,” Sov. J. Opt. Technol. 52, 294–297 (1985).

Brown, N. J.

N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
[CrossRef]

Burge, J. H.

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Cabrera-Pelaez, V. H.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Cordero-Dávila, A.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Cuautle-Cortez, J.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Davison, W. B.

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Davison, W. D.

Derigne, S. T.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Ernsthausen, J. M.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
[CrossRef]

Fowles, G. R.

G. R. Fowles, Analytical Mechanics (Wiley, Reading, Mass., 1962).

Gonzalez-García, J.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Hall, C. A.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
[CrossRef]

Hille, B. B.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Hoskins, S. J.

D. W. Small, S. J. Hoskins, “An automated asphere polishing machine,” in Optical Manufacturing, Testing, and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986).
[CrossRef]

Jones, R. A.

R. A. Jones, “Fabrication of a large, thin, off-axis aspheric mirror,” Opt. Eng. 33, 4068–4075 (1994).
[CrossRef]

R. A. Jones, W. J. Rupp, “Rapid optical fabrication with computer-controlled optical surfacing,” Opt. Eng. 30, 1962–1968 (1991).
[CrossRef]

R. A. Jones, “Computer controlled optical surfacing with orbital tool motion,” Opt. Eng. 25, 785–790 (1986).
[CrossRef]

R. A. Jones, “Computer controlled polishing of telescope mirror segments,” Opt. Eng. 22, 236–240 (1983).
[CrossRef]

R. A. Jones, “Fabrication using the computer controlled polisher,” Appl. Opt. 17, 1889–1892 (1978).
[CrossRef] [PubMed]

R. A. Jones, “Optimization of computer controlled polishing,” Appl. Opt. 16, 218–222 (1977).
[CrossRef] [PubMed]

Ketelsen, D. A.

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Kittrell, W. C.

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Luna-Aguilar, E.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Martin, H. M.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

McDonough, R.

Nagel, R. H.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Nitchie, F. R.

Núñez-Alfonso, M.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Pedrayes-López, M. H.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Porsching, T. A.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
[CrossRef]

Preston, F.

F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 9, 214–256 (1927).

Robledo-Sánchez, C.

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

Rupp, W.

Rupp, W. J.

R. A. Jones, W. J. Rupp, “Rapid optical fabrication with computer-controlled optical surfacing,” Opt. Eng. 30, 1962–1968 (1991).
[CrossRef]

Savel’ev, A. S.

A. S. Savel’ev, A. P. Bogdanov, “Automated polishing of large optical components with a small tool,” Sov. J. Opt. Technol. 52, 294–297 (1985).

Shannon, R. R.

Small, D. W.

D. W. Small, S. J. Hoskins, “An automated asphere polishing machine,” in Optical Manufacturing, Testing, and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986).
[CrossRef]

Trebisky, T. J.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Wagner, R. E.

West, S. C.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Young, R. S.

S. C. West, H. M. Martin, R. H. Nagel, R. S. Young, W. D. Davison, T. J. Trebisky, S. T. Derigne, B. B. Hille, “Practical design and performance of the stressed-lap polishing tool,” Appl. Opt. 33, 8094–8100 (1994).
[CrossRef] [PubMed]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

Appl. Opt.

J. Soc. Glass Technol.

F. Preston, “The theory and design of plate glass polishing machines,” J. Soc. Glass Technol. 9, 214–256 (1927).

Math. Comput. Modell.

T. A. Porsching, C. A. Hall, T. L. Bennett, J. M. Ernsthausen, “A mathematical model of material removal with application to CNC finishing,” Math. Comput. Modell. 18, 25–40 (1993).
[CrossRef]

Opt. Eng.

R. A. Jones, W. J. Rupp, “Rapid optical fabrication with computer-controlled optical surfacing,” Opt. Eng. 30, 1962–1968 (1991).
[CrossRef]

R. A. Jones, “Fabrication of a large, thin, off-axis aspheric mirror,” Opt. Eng. 33, 4068–4075 (1994).
[CrossRef]

R. A. Jones, “Computer controlled polishing of telescope mirror segments,” Opt. Eng. 22, 236–240 (1983).
[CrossRef]

N. J. Brown, “Computationally directed axisymmetric aspheric figuring,” Opt. Eng. 17, 602–620 (1978).
[CrossRef]

R. A. Jones, “Computer controlled optical surfacing with orbital tool motion,” Opt. Eng. 25, 785–790 (1986).
[CrossRef]

Sov. J. Opt. Technol.

A. S. Savel’ev, A. P. Bogdanov, “Automated polishing of large optical components with a small tool,” Sov. J. Opt. Technol. 52, 294–297 (1985).

Other

D. W. Small, S. J. Hoskins, “An automated asphere polishing machine,” in Optical Manufacturing, Testing, and Aspheric Optics, G. M. Sanger, ed., Proc. SPIE645, 66–74 (1986).
[CrossRef]

D. S. Anderson, J. R. P. Angel, J. H. Burge, W. B. Davison, S. T. DeRigne, B. B. Hille, D. A. Ketelsen, W. C. Kittrell, H. M. Martin, R. H. Nagel, T. J. Trebisky, S. C. West, R. S. Young, “Stressed-lap polishing of 3.5-m f/1.5 and 1.8-m f/1.0 mirrors,” in Advanced Optical Manufacturing and Testing II, V. J. Doherty, ed., Proc. SPIE1531, 260–269 (1992).

E. Luna-Aguilar, A. Cordero-Dávila, J. Gonzalez-García, M. Núñez-Alfonso, V. H. Cabrera-Pelaez, C. Robledo-Sánchez, J. Cuautle-Cortez, M. H. Pedrayes-López, “Edge effects with Preston equation,” in Future Giant Telescopes, J. Roger, P. Angel, R. Gilmozzi, eds., Proc. SPIE4840, 598–603 (2002).
[CrossRef]

G. R. Fowles, Analytical Mechanics (Wiley, Reading, Mass., 1962).

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Figures (9)

Fig. 1
Fig. 1

Preston calculated the wear distribution of an actual glass (b) beginning with that of an imaginary larger glass (a), and adding the remaining area (A r ) to a border zone of the actual glass.

Fig. 2
Fig. 2

Parameters of the workpiece and tool when the tool border is beyond the workpiece border. The origin of the reference system is at the center of mass of the tool to establish the equilibrium equation.

Fig. 3
Fig. 3

Intersection region A is divided into two regions, B and C. In region B the pressure is p 0, and in skin region C the pressure increases until p 0 + Δp. The border points are located between R v - s and R v , where R v is the radius of the workpiece and s is the thickness of the skin.

Fig. 4
Fig. 4

We show a plot of skin width (s) as a function of the separation between borders (d).

Fig. 5
Fig. 5

Depiction of the tool travel on the workpiece.

Fig. 6
Fig. 6

Wear versus radial position by use of the skin model with a constant value of the skin width of 1.87 cm. The negative pressures are not avoided.

Fig. 7
Fig. 7

Wear versus radial position by use of the skin model with a constant value of the skin width of 0.5 cm. The negative pressures are avoided.

Fig. 8
Fig. 8

Wear versus radial position by use of the variable skin width. The negative pressures are automatically avoided. We used an oscillation amplitude of 2.5 cm.

Fig. 9
Fig. 9

Wear versus radial position by use of the variable skin width. In this case, the oscillation amplitude is increased to 2.65 cm. As can be observed, the wear increases significantly at the border points.

Equations (27)

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

h=t1t2 Dpvdt,
A px, ydxdy=f0,
A xpx, ydxdy=0,
s=2bdb-d.
A px, ydxdy=B p0dxdy+Cp0+Δpdxdy.
A px, ydxdy=p0Adxdy+Δp A-Bdxdy.
p0Adxdy+ΔpAdxdy-Bdxdy=f0.
p0A xdxdy+ΔpA xdxdy-B xdxdy=0.
IIA=Adxdy,
IIB=Bdxdy,
IxA=A xdxdy,
IxB=B xdxdy,
IIAp0+ΔpIIA-IIB=f0,
IxAp0+ΔpIxA-IxB=0,
p0=f0IxA-IxBIIAIxA-IxB-IxAIIA-IIB,
Δp=-f0IxAIIAIxA-IxB-IxAIIA-IIB.
IIA=IIB,
IxA=0.
p0=f0/IIA,
I= fx, ydxdy.
I=2 x=-b,y=0x=x0,y=b2-x2 fx, ydxdy+2 y=0,x=x0y=y0,x=-xtc+a2-y2 fx, ydxdy,
x0=a2-b2-xtc22xtc,
y0=b2-x021/2.
II=2 x=-b,y=0x=x0,y=b2-x2dxdy+2 y=0,x=x0y=y0,x=-xtc+a2-y2dxdy.
II=x0b2-x021/2+b2arcsinx0b+b2π2-2y0xtc+x0+y0a2-y021/2+a2arcsiny0a.
Ix=2 x=-b,y=0x=x0,y=b2-x2 xdxdy+2 y=0,x=x0y=y0,x=-xtc+a2-y2 xdxdy.
Ix=-23b2-x0231/2+xtc2+a2-x02y0-y033-xtcy0a2-y021/2+a2arcsiny0a.

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