Abstract

Two constraints in the design of a petal tool are, the angles that define it must all be positive, and wear must never be greater than the desired wear. The first constraint is equivalent to that of the positive dwell times of a small solid tool. In view of this foregoing, we present a design of petal tools that are used to generate conic surfaces from their nearest spheres and that correct the profile of a surface that is polished. We study optimal angular sizes of a petal tool, which are found after we use linear programming to calculate the optimal dwell times of a set of complete annular tools placed in different zones of the glass surface. We report numerical results of designed petal tools.

© 2007 Optical Society of America

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References

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  1. R. A. Jones, "Optimization of computer controlled polishing," Appl. Opt. 16, 218-224 (1977).
    [CrossRef] [PubMed]
  2. M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).
  3. Yu. K. Lysyannyy and L. S. Tsesnek, "Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution," Sov. J. Opt. Technol. 40, 446-448 (1973).
  4. Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, and L. N. Khokhlenkov, "The shaping of optical surfaces by the successively corrected mask method," Sov. J. Opt. Technol. 44, 226-227 (1977).
  5. A. Cordero-Dávila, V. Cabrera-Peláez, J. Cuautle-Cortés, J. González-García, C. Robledo-Sánchez, and N. Bautista-Elivar, "Experimental results and wear predictions of petal tools that freely rotate," Appl. Opt. 44, 1434-1441 (2005).
    [CrossRef] [PubMed]
  6. F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).
  7. N. J. Brown, "Computationally directed axisymmetric aspheric figuring," J. Soc. Photo-Opt. Instrum. Eng. 17, 602-620 (1978).
  8. O. D. Macias Bautista and A. Cordero-Dávila, "Pulido de superficies convexas con herramientas de pétalo," in Program of the 47th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 18, 96-97 (2004).
  9. J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, and A. Santiago-Alvarado, "Calculating petal tools using genetic algorithms," Appl. Opt. 45, 6126-6136 (2006).
    [PubMed]
  10. R. E. Wagner and R. R. Shannon, "Fabrication of aspherics using a mathematical model for material removal," Appl. Opt. 13, 1683-1689 (1974).
    [CrossRef] [PubMed]
  11. D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).
  12. R. Aspden, R. McDonough, and F. R. Nitchie, Jr., "Computer assisted optical surfacing," Appl. Opt. 11, 2739-2747 (1972).
    [CrossRef] [PubMed]
  13. R. A. Jones, "Optimization of computer controlled polishing," Appl. Opt. 16, 218-224 (1977).
    [CrossRef] [PubMed]
  14. J. R. Johnson and E. Waluschka, "Optical fabrication--process modeling--analysis tool box," in Advanced Optical Manufacturing and Testing, Gregory M. Sanger, Paul B. Reid, and Lionel R. Baker, eds., Proc. SPIE 1333, 106-117 (1990).
    [CrossRef]
  15. A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).
  16. A. P. Bogdanov and L. S. Tsesnek, "Contact problems in the shaping of optical surfaces," Opt.-Mekh. Prom-st. 46, 8-10 (1979).
  17. A. P. Bogdanov and L. S. Tsesnek, "Solving contact problems in the shaping of optical surfaces," Opt.-Mekh. Prom-st. 46, 18-20 (1979).
  18. A. Santiago-Alvarado, S. Vázquez-Montiel, R. Nivón-Santiago, and C. Castañeda-Roldán, "Uso de programación lineal para conocer los parámetros geométricos de superficies cónicas convexas," Rev. Mex. Fis. 50, 358-365 (2004).
  19. H. A. Taha, Operations Research: An Introduction (Prentice Hall, 2006).
  20. M. Bazaraa, J. Jarvis, and H. Sherali, Linear Programming and Network Flows (Wiley, 1989).
  21. J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, G. Castro-González, A. Santiago-Alvarado, and L. J. Manzano-Sumano, "Design of petal tools based on the dwell-times of annular tools to generate convex surfaces," in Program of the 49th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 20, 126 (2006).

2006 (2)

J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, and A. Santiago-Alvarado, "Calculating petal tools using genetic algorithms," Appl. Opt. 45, 6126-6136 (2006).
[PubMed]

J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, G. Castro-González, A. Santiago-Alvarado, and L. J. Manzano-Sumano, "Design of petal tools based on the dwell-times of annular tools to generate convex surfaces," in Program of the 49th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 20, 126 (2006).

2005 (1)

2004 (2)

O. D. Macias Bautista and A. Cordero-Dávila, "Pulido de superficies convexas con herramientas de pétalo," in Program of the 47th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 18, 96-97 (2004).

A. Santiago-Alvarado, S. Vázquez-Montiel, R. Nivón-Santiago, and C. Castañeda-Roldán, "Uso de programación lineal para conocer los parámetros geométricos de superficies cónicas convexas," Rev. Mex. Fis. 50, 358-365 (2004).

1990 (1)

J. R. Johnson and E. Waluschka, "Optical fabrication--process modeling--analysis tool box," in Advanced Optical Manufacturing and Testing, Gregory M. Sanger, Paul B. Reid, and Lionel R. Baker, eds., Proc. SPIE 1333, 106-117 (1990).
[CrossRef]

1985 (1)

A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).

1979 (2)

A. P. Bogdanov and L. S. Tsesnek, "Contact problems in the shaping of optical surfaces," Opt.-Mekh. Prom-st. 46, 8-10 (1979).

A. P. Bogdanov and L. S. Tsesnek, "Solving contact problems in the shaping of optical surfaces," Opt.-Mekh. Prom-st. 46, 18-20 (1979).

1978 (1)

N. J. Brown, "Computationally directed axisymmetric aspheric figuring," J. Soc. Photo-Opt. Instrum. Eng. 17, 602-620 (1978).

1977 (3)

R. A. Jones, "Optimization of computer controlled polishing," Appl. Opt. 16, 218-224 (1977).
[CrossRef] [PubMed]

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, and L. N. Khokhlenkov, "The shaping of optical surfaces by the successively corrected mask method," Sov. J. Opt. Technol. 44, 226-227 (1977).

R. A. Jones, "Optimization of computer controlled polishing," Appl. Opt. 16, 218-224 (1977).
[CrossRef] [PubMed]

1976 (1)

D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).

1974 (1)

1973 (1)

Yu. K. Lysyannyy and L. S. Tsesnek, "Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution," Sov. J. Opt. Technol. 40, 446-448 (1973).

1972 (1)

1968 (1)

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

1927 (1)

F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).

Appl. Opt. (6)

Bull. Soc. Mex. Fis. Suppl. (1)

J. González-García, A. Cordero-Dávila, I. Leal-Cabrera, C. I. Robledo-Sánchez, G. Castro-González, A. Santiago-Alvarado, and L. J. Manzano-Sumano, "Design of petal tools based on the dwell-times of annular tools to generate convex surfaces," in Program of the 49th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 20, 126 (2006).

J. Soc. Glass Technol. (1)

F. W. Preston, "The theory and design of plate glass polishing machines," J. Soc. Glass Technol. 11, 214-256 (1927).

J. Soc. Photo-Opt. Instrum. Eng. (1)

N. J. Brown, "Computationally directed axisymmetric aspheric figuring," J. Soc. Photo-Opt. Instrum. Eng. 17, 602-620 (1978).

Opt. Eng. (1)

D. J. Bajuk, "Computer controlled generation of rotationally symmetric aspheric surfaces," Opt. Eng. 15, 401-406 (1976).

Opt.-Mekh. Prom-st. (3)

A. P. Bogdanov, "Optimizing the technological process of automated grinding and polishing of high-precision large optical elements with a small tool," Opt.-Mekh. Prom-st. 52, 32-36 (1985).

A. P. Bogdanov and L. S. Tsesnek, "Contact problems in the shaping of optical surfaces," Opt.-Mekh. Prom-st. 46, 8-10 (1979).

A. P. Bogdanov and L. S. Tsesnek, "Solving contact problems in the shaping of optical surfaces," Opt.-Mekh. Prom-st. 46, 18-20 (1979).

Proc. SPIE (1)

J. R. Johnson and E. Waluschka, "Optical fabrication--process modeling--analysis tool box," in Advanced Optical Manufacturing and Testing, Gregory M. Sanger, Paul B. Reid, and Lionel R. Baker, eds., Proc. SPIE 1333, 106-117 (1990).
[CrossRef]

Rev. Mex. Fis. (1)

A. Santiago-Alvarado, S. Vázquez-Montiel, R. Nivón-Santiago, and C. Castañeda-Roldán, "Uso de programación lineal para conocer los parámetros geométricos de superficies cónicas convexas," Rev. Mex. Fis. 50, 358-365 (2004).

Sov. J. Opt. Technol. (3)

M. N. Golovanova, S. S. Kachkin, Ye. I. Krylova, L. S. Tsesnek, and L. I. Shevel'kova, "A method of manufacturing aspherical surfaces which deviate only slightly from the sphere," Sov. J. Opt. Technol. 35, 254-256 (1968).

Yu. K. Lysyannyy and L. S. Tsesnek, "Computation of the contour of a mask tool surface for shaping a concave paraboloid of revolution," Sov. J. Opt. Technol. 40, 446-448 (1973).

Yu. K. Lysyannyy, L. S. Tsesnek, L. N. Gurevich, and L. N. Khokhlenkov, "The shaping of optical surfaces by the successively corrected mask method," Sov. J. Opt. Technol. 44, 226-227 (1977).

Other (3)

O. D. Macias Bautista and A. Cordero-Dávila, "Pulido de superficies convexas con herramientas de pétalo," in Program of the 47th Congreso Nacional de Física de la Sociedad Mexicana de Física, Bull. Soc. Mex. Fis. Suppl. 18, 96-97 (2004).

H. A. Taha, Operations Research: An Introduction (Prentice Hall, 2006).

M. Bazaraa, J. Jarvis, and H. Sherali, Linear Programming and Network Flows (Wiley, 1989).

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Figures (13)

Fig. 1
Fig. 1

Petal tool formed by a set of incomplete annular tools with different angular sizes α i .

Fig. 2
Fig. 2

Set of annular tools working on a workpiece to generate the desired wear. The Polishing tool at the center of the workpiece is considered solid.

Fig. 3
Fig. 3

Amounts of wear to be generated with petal tools: (a) concave wear, (b) convex wear, and (c) concave surface hyperbolization in progress.

Fig. 4
Fig. 4

Base functions generated to calculate dwell times, which generate a concave hyperbolic surface: (a) base function 1, (b) base function 50, (c) base function 101, and (d) 101 base functions together.

Fig. 5
Fig. 5

(a) Dwell times calculated by solving the linear program, (b) angular sizes calculated from the dwell times, (c) petal tool calculated with angular sizes, and (d) wear generated by this tool.

Fig. 6
Fig. 6

(a) Dwell times calculated by solving the linear program with additional constraints, (b) angular sizes calculated from the dwell times, (c) petal tool calculated with angular sizes, and (d) wear generated by this tool.

Fig. 7
Fig. 7

Base functions generated to calculate dwell times, which in turn will generate a hyperbolic convex surface: (a) base function 30, (b) base function 60, (c) base function 90, and (d) 101 base functions together.

Fig. 8
Fig. 8

(a) Dwell times calculated to solve the linear program, (b) angular sizes calculated from dwell times, (c) petal tool calculated with the angular sizes, and (d) amount of wear generated by the tool.

Fig. 9
Fig. 9

(a) Dwell times calculated by solving the linear program with additional constraints, (b) angular sizes calculated from the dwell times, (c) petal tool calculated with the angular sizes, and (d) wear generated by this tool.

Fig. 10
Fig. 10

Base functions generated to calculate dwell times, which in turn will generate a hyperbolic concave surface in progress: (a) base function 10, (b) base function 40, (c) base function 70, and (d) 101 base functions together.

Fig. 11
Fig. 11

Development of the design of a petal tool to generate a real amount of desired wear: (a) Dwell times calculated with linear programming, (b) angular sizes calculated from the dwell times, (c) the petal tool designed, and (d) wear generated by this petal tool.

Fig. 12
Fig. 12

Wear generated by multiplying the base functions by the dwell times calculated with linear programming: (a) without additional constraints and (b) with additional constraints to obtain decreasing dwell times.

Fig. 13
Fig. 13

Wear generated by multiplying base functions by dwell times calculated with linear programming: (a) without additional constraints and (b) with additional constraints to obtain increasing dwell times.

Tables (1)

Tables Icon

Table 1 Parameters Used to Calculate Simulated Wear

Equations (13)

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[ d 11 d 12 d 1 n d 21 d 22 d 2 n d m 1 d m 2 d m n ] ( t 1 t 2 t n ) = ( δ 1 δ 2 δ m ) ,
t j 0 , j = 1 ,   .   .   .   , n ,
D t = δ , t 0 ,
x = i = 1 n | x i | .
D t δ 0 ,
ε := δ D t .
T := { t n | t 0 , ε 0 } .
min ε 1 + ε 2 + K + ε m                     s . a . d 11 t 1 + d 12 t 2 + + d 1 n t n ε 1       = δ 1   M               M     O     M M     d m 1 t 1 + d m 2 t 2 + + d m n t n ε m = δ n   t 1 , t 2 , t n , ε 1 , K , ε m 0.
t 1 t 2 t n ,
min ε 1 + ε 1 + K + ε m s . a . d 11 t 1 + d 12 t 2 + + d 1 n t n ε 1 = δ 1 M M O M M d m 1 t 1 + d m 2 t 2 + + d m n t n ε m = δ m t 1 + t 2 0 t 2 + t 3 0 O M M t n 1 + t n 0 t 1 , t 2 , t n , ε 1 , K , ε m 0.
t k t k + 1 t k + 2 t n k ,
min ε 1 + ε 2 + K + ε m s . a . d 11 t 1 + d 12 t 2 + . . . + d 1 n t n ε 1 = δ 1 M M O M M d m 1 t 1 + d m 2 t 2 + . . . + d m n t n ε m = δ m t k + t k + 1 0 O M M t 1 t 2 . . t n k 1 + t k + 1 t n ε 1 K ε m 0 t 1 , t 2 , . . . t n , ε 1 , K , ε m 0.
h D = i = 1 M t i h b i ,

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