Abstract

A method for calculating the actuator influence functions of an active mirror is described. The method is based on a model of the mirror as a thin plate attached by actuators to a very stiff reaction structure. Arbitrary actuator layouts can be handled. The mirror deflections calculated agree well with the results of a finite element analysis of the structure.

© 1991 Optical Society of America

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References

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  1. S. Timoshenko, S. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.
  2. J. E. Nelson, J. Lubiner, T. S. Mast, “Telescope Mirror Supports: Plate Deflections on Point Supports,” Proc. Soc. Photo-Opt. Instrum. Eng. 332, 212–228 (1982).
  3. D.-S. Wan, J. R. P. Angel, R. E. Parks, “Mirror Deflection on Multiple Axial Supports,” Appl. Opt. 28, 354–362 (1989).
    [CrossRef] [PubMed]
  4. D. McFarland, B. L. Smith, W. D. Bernhart, Analysis of Plates (Spartan, New York, 1972), Chap. 2.
  5. G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, MD, 1983).

1989 (1)

1982 (1)

J. E. Nelson, J. Lubiner, T. S. Mast, “Telescope Mirror Supports: Plate Deflections on Point Supports,” Proc. Soc. Photo-Opt. Instrum. Eng. 332, 212–228 (1982).

Angel, J. R. P.

Bernhart, W. D.

D. McFarland, B. L. Smith, W. D. Bernhart, Analysis of Plates (Spartan, New York, 1972), Chap. 2.

Golub, G.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, MD, 1983).

Lubiner, J.

J. E. Nelson, J. Lubiner, T. S. Mast, “Telescope Mirror Supports: Plate Deflections on Point Supports,” Proc. Soc. Photo-Opt. Instrum. Eng. 332, 212–228 (1982).

Mast, T. S.

J. E. Nelson, J. Lubiner, T. S. Mast, “Telescope Mirror Supports: Plate Deflections on Point Supports,” Proc. Soc. Photo-Opt. Instrum. Eng. 332, 212–228 (1982).

McFarland, D.

D. McFarland, B. L. Smith, W. D. Bernhart, Analysis of Plates (Spartan, New York, 1972), Chap. 2.

Nelson, J. E.

J. E. Nelson, J. Lubiner, T. S. Mast, “Telescope Mirror Supports: Plate Deflections on Point Supports,” Proc. Soc. Photo-Opt. Instrum. Eng. 332, 212–228 (1982).

Parks, R. E.

Smith, B. L.

D. McFarland, B. L. Smith, W. D. Bernhart, Analysis of Plates (Spartan, New York, 1972), Chap. 2.

Timoshenko, S.

S. Timoshenko, S. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.

Van Loan, C.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, MD, 1983).

Wan, D.-S.

Woinowski-Krieger, S.

S. Timoshenko, S. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.

Appl. Opt. (1)

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

J. E. Nelson, J. Lubiner, T. S. Mast, “Telescope Mirror Supports: Plate Deflections on Point Supports,” Proc. Soc. Photo-Opt. Instrum. Eng. 332, 212–228 (1982).

Other (3)

D. McFarland, B. L. Smith, W. D. Bernhart, Analysis of Plates (Spartan, New York, 1972), Chap. 2.

G. Golub, C. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, MD, 1983).

S. Timoshenko, S. Woinowski-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959), Chap. 3.

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Figures (3)

Fig. 1
Fig. 1

Model of an actuator attached to a mirror.

Fig. 2
Fig. 2

Plot of the displacement of the mirror. The actuators are located at the solid circles. A force of 10 lb was applied to the right actuator. Rigid body motion has been removed. The units are microinches.

Fig. 3
Fig. 3

Plot of the error between the series expansion and the finite element model. The actuators are located at the solid circles. A force of 10 lb was applied to the right actuator. The units are microinches.

Tables (1)

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Table I Comparison of Methods

Equations (51)

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D = E h 3 12 ( 1 ν 2 ) ,
Δ = 2 x 2 + 2 y 2 .
D Δ 2 W = i = 1 N q i δ ( r p i ) .
M r = 0 for r = a ,
V r t = Q r 1 r M r θ θ = 0 for r = a ,
i = 1 N q i = 0 ,
i = 1 N q i p i = 0 .
Δ = 2 r 2 + 1 r r + 1 r 2 2 θ 2 .
δ ( r p i ) = 1 b i δ ( r b i ) δ ( θ θ i ) ,
( 2 r 2 + 1 r r + 1 r 2 2 θ 2 ) 2 w = i = 1 N q i b i D δ ( r b i ) δ ( θ θ i ) .
M r = D [ 2 w r 2 + ν ( 1 w r r + 1 r 2 2 w θ 2 ) ] .
V r z = D [ r Δ w + ( 1 ν ) 2 θ 2 ( 1 r 2 w r w r 3 ) ] .
j = 1 N q j b j exp ( i θ j ) = 0 .
w ( r , θ ) = m = + w m ( r ) exp ( i m θ ) .
δ ( θ θ i ) = 1 2 π m = + exp [ i m ( θ θ i ) ] .
L m w m = j = 1 N q j 2 π b j exp ( i m θ j ) δ ( r b j ) ,
L m = ( d 2 d r 2 + 1 r d d r m 2 ) 2 .
[ d 2 d r 2 + ν ( 1 r d d r m 2 r 2 ) ] w m | r ¯ a = 0 ,
[ d d r ( d 2 d r 2 + 1 r d d r m 2 r 2 ) ( 1 ν ) m 2 ( 1 r 2 d d r 1 r 3 ) ] w m | r ¯ a = 0 .
L m w m = 0 ,
L m w m = 0 for r b j ,
w m ( b j + ) = w m ( b j ) , w m ( b j + ) = w m ( b j ) , w m ( b j + ) = w m ( b j ) , w m ( b j + ) w m ( b j ) = q j 2 π b j D exp ( i m θ j ) .
m = 0 , w 0 = A 0 + B 0 r 2 + C 0 log ( r ) + D 0 r 2 log ( r ) ,
m = 1 , w 1 = A 1 r + B 1 r 3 + C 1 r 1 + D 1 r log ( r ) ,
m > 1 , w m = A m r m + B m r m + 2 + C m r m + D m r m + 2 .
Δ j A = A ( b j + ) A ( b j )
Δ j A m = Q j m ( m 1 ) b j m , Δ j B m = Q j m ( m + 1 ) b j m 2 , Δ j C m = Q j m ( m + 1 ) b j m , Δ j D m = Q j m ( m 1 ) b j m 2 ,
Q j = q j b j 2 16 π D exp ( i m θ j ) .
Δ j A 1 = q j 4 π D exp ( i θ j ) b j log ( b j ) , Δ j B 1 = q j 16 π D exp ( i θ j ) 1 b j , Δ j C 1 = q j 16 π D exp ( i θ j ) b j 3 , Δ j D 1 = q j 4 π D exp ( i θ j ) b j .
Δ j A 0 = q j 8 π D b j 2 [ log ( b j ) 1 ] , Δ j B 0 = q j 8 π D b j 2 [ log ( b j ) + 1 ] , Δ j C 0 = q j 8 π D b j 2 , Δ j D 0 = q j 8 π D .
m ( m 1 ) ( 1 ν ) a m A m + ( m + 1 ) [ m + 2 ν ( m 2 ) ] a m + 2 B m + m ( m + 1 ) ( 1 ν ) a m C m + ( m 1 ) [ m 2 ν ( m + 2 ) ] a m + 2 D m = 0 ,
m ( m 1 ) ( 1 ν ) a m A m + ( m + 1 ) ( m 4 m ν ) a m + 2 B m m ( m + 1 ) ( 1 ν ) a m C m + ( m 1 ) ( m + 4 m ν ) a m + 2 D m = 0 .
B 0 = 1 2 a 2 ( 1 ν 1 + ν ) C 0 .
B 1 = 1 ν ( 3 + ν ) a 4 C 1 .
V = span { ( 1 , 1 , , 1 ) , ( r x 1 , r x 2 , , r x N ) , ( r y 1 , r y 2 , , r y N ) } , P = the orthogonal projection R N onto V .
k P x = P F .
P q = 0 .
( I P ) x = A q .
k x = q + F .
k ( I P ) x = q + ( I P ) F ,
( I + k A ) q = ( I P ) F .
( I + k A ) ( I P ) x = A ( I P ) F ,
( I P ) x = ( I + k A ) 1 A ( I P ) F .
x = [ ( I + k A ) 1 A ( I P ) + 1 k P ] F .
B = ( I + k A ) 1 A ( I P ) + 1 k P .
A ˜ = A ( I P ) = U ( λ 1 λ 2 λ N 3 0 0 0 ) V t .
B = V ( λ 1 1 + k λ 1 λ 2 1 + k λ 2 k 1 k 1 k 1 ) U t .
A = a 2 N D A .
B = ( D N a 2 + k A ) 1 A ( I P ) + 1 k P .
B = 1 k I ,
B = 1 k P ,

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