Abstract

This paper describes the results of an elastic analysis of a 4-m diam mirror. The mirror is solid quartz with a large central hole, flat back, and spherically dished front surface (f/2.75 system). We solved the three-dimensional elastic equations by a technique known as dynamic relaxation programmed for a digital computer. In this way, the resulting deformation of the optical surface is found for any proposed support system at any desired orientation of the mirror.

© 1968 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Couder, Bull. Astron. 7, 14 (1931).
  2. G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
    [CrossRef]
  3. A. S. Day, The Engineer 219, 218 (1965).
  4. J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
    [CrossRef]
  5. A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), 4th ed.

1966

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

1965

A. S. Day, The Engineer 219, 218 (1965).

1954

G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
[CrossRef]

1931

A. Couder, Bull. Astron. 7, 14 (1931).

Cassell, A. C.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

Couder, A.

A. Couder, Bull. Astron. 7, 14 (1931).

Day, A. S.

A. S. Day, The Engineer 219, 218 (1965).

Hobbs, R. E.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

Love, A. E. H.

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), 4th ed.

Otter, J. R. H.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

Schwesinger, G.

G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
[CrossRef]

Bull. Astron.

A. Couder, Bull. Astron. 7, 14 (1931).

J. Opt. Soc. Amer.

G. Schwesinger, J. Opt. Soc. Amer. 44, 417 (1954).
[CrossRef]

Proc. Inst. Civil Eng.

J. R. H. Otter, A. C. Cassell, R. E. Hobbs, Proc. Inst. Civil Eng. 35, 633 (1966).
[CrossRef]

The Engineer

A. S. Day, The Engineer 219, 218 (1965).

Other

A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover Publications, New York, 1944), 4th ed.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

Orthogonal curvilinear corrdinates. Location of normal and shear stresses, and displacements.

Fig. 2
Fig. 2

Handling boundary forces.

Fig. 3
Fig. 3

Mirror schematic showing orthogonal curvilinear elements. All dimensions shown are in centimeters.

Fig. 4
Fig. 4

Horizontal position, two-ring back support at radii of 101.0 cm and 170.0 cm. 63.60% weight on outer ring.

Fig. 5
Fig. 5

Horizontal position, three-point back support near outer edge. (192.7 cm).

Fig. 6
Fig. 6

Same as Fig. 5 but with the three points at 145 cm.

Fig. 7
Fig. 7

Same as Fig. 5 but with the three points near inner edge (75 cm).

Fig. 8
Fig. 8

Vertical position, cosine (push–pull) outside edge support acting through center of gravity.

Fig. 9
Fig. 9

Vertical position, cosine (push only) outside edge support along lower half of mirror, through cg.

Fig. 10
Fig. 10

Vertical position, 1 + cosine (mercury bag) outside edge support, through cg.

Fig. 11
Fig. 11

Vertical position, constant force (band support) over lower half of mirror outside edge.

Fig. 12
Fig. 12

Vertical position, three-point edge support with equal load on the two lower supports.

Fig. 13
Fig. 13

Same as Fig. 12 but with full weight of mirror on single lower support.

Fig. 14
Fig. 14

Vertical position, cosine (push only) edge support with 50% of weight on upper inner edge and 50% on lower outer edge.

Fig. 15
Fig. 15

Same as Fig. 14 but the outer edge support was 1 cm below the cg. and the inner edge support was 1 cm above the cg.

Fig. 16
Fig. 16

45° orientation, back support of Fig. 4 and edge support of Fig. 14.

Fig. 17
Fig. 17

Same as Fig. 16 except for push–pull edge supports of Fig. 8.

Equations (11)

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

h 1 h 2 h 3 [ x 1 ( σ 1 i h 2 h 3 ) + x 2 ( σ 2 i h 1 h 2 ) + x 3 ( σ 3 i h 1 h 2 ) ] + h i j = 1 j i 3 h j [ σ i j x j ( 1 h i ) - σ j j x i ( 1 h j ) ] + F i = 0             i = 1 , 2 , 3
h i 2 = ( x i / y 1 ) 2 + ( x i / y 2 ) 2 + ( x i / y 3 ) 2 .
σ i i = ν E ( 1 + ν ) ( 1 - 2 ν ) ( e 11 + e 22 + e 33 ) + E 1 + ν e i i , σ i j = E 2 ( 1 + ν ) e i j             i j ,
e i i = h i [ u i x i + j = 1 j i 3 h j u j x j ( 1 h i ) ] , e i j = h j h i x j ( h i u i ) + h i h j x i ( h j u j )             i j ,
L ( { σ i j } ) + F i = ρ u ¨ i + c u ˙ i ,
σ x 2 | A σ 2 - σ 1 Δ x 2 , τ x 1 | A τ 2 - τ 1 Δ x 1 , u x 1 | B u 2 - u 1 Δ x 1 , v x 2 | B v 2 - v 1 Δ x 2 , etc .
L ( { σ } ) + F = ρ ( u ˙ 1 - u ˙ 0 ) / Δ t + c ( u ˙ 0 + u ˙ 1 ) / 2.
u ˙ 1 = 1 - k / 2 1 + k / 2 u ˙ 0 + Δ t ρ ( 1 + k / 2 ) [ L ( σ ) + F ] ,
x 1 = arc tan [ 2 ( b 2 - a 2 ) 1 2 ( y 1 2 + y 2 2 ) 1 2 b 2 - a 2 - y 1 2 - y 2 2 - y 3 2 ] , x 2 = [ ( b 2 - a 2 - y 1 2 - y 2 2 - y 3 2 ) 2 + 4 ( b 2 - a 2 ) ( y 1 2 + y 2 2 ) ] 1 2 y 1 2 + y 2 2 + [ ( b 2 - a 2 ) 1 2 + y 3 ] 2 , x 3 = arc tan y 2 / y 1 ,
y 1 = ( b 2 - a 2 ) 1 2 2 x 2 sin x 1 cos x 3 1 + 2 x 2 cos x 1 + x 2 2 , y 2 = ( b 2 - a 2 ) 1 2 2 x 2 sin x 1 sin x 3 1 + 2 x 2 cos x 1 + x 2 2 , y 3 = ( b 2 - a 2 ) 1 2 1 - x 2 2 1 + 2 x 2 cos x 1 + x 2 2 ,
h 1 = ( 1 + 2 x 2 cos x 1 + x 2 2 ) / [ 2 ( b 2 - a 2 ) 1 2 x 2 ] , h 2 = h 1 , h 3 = h 1 / sin x 1 .

Metrics