Optimal Design of Cored Mirror Structures

William P. Barnes

doi:10.1364/AO.8.001191

Applied Optics
Vol. 8,
Issue 6,
pp. 1191-1196
(1969)
•https://doi.org/10.1364/AO.8.001191

Optimal Design of Cored Mirror Structures

William P. Barnes

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William P. Barnes, "Optimal Design of Cored Mirror Structures," Appl. Opt. 8, 1191-1196 (1969)

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Abstract

The design and construction of lightweight mirror substrates has been the subject of considerable study in recent years. A consensus has been reached that the best design approach involves two plates connected by an array of ribs and consisting of a single material throughout. Although a single, rib reinforced plate has some advantage over a solid plate in efficient use of material, the two-plate sandwich is even more efficient, and our attention has been focused there. The consensus has not yet progressed to total agreement on preferred rib configurations, thickness, and spacing or preferred plate thicknesses and total thickness. This article attempts to demonstrate that some preferences can be established in general, although modifications may be appropriate for particular applications.

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Configuration	Areas	Central Deflection, w, of Cell Cover Plate Under Uniform Load, q₀, Clamped Edges	Number of Unit Cells, N_A, and Number of Space Lattice Elements, N_S, w₁ = w₂ = w₃, Equal Total Mirror Area
1. Hexagonal
	Unit cell area, $A_{1} = (\sqrt{3} / 2) B_{1}^{2}$	$w_{1} = 0.00111 (q_{0} / D) B_{1}^{4}$
	Rib area, R₁ = (2α − α²)A₁ per cell
	Space lattice area, $S_{1} = (\sqrt{3} / 4) B_{1}^{2}$

2. Square
	Unit cell area, $A_{2} = B_{2}^{2}$	$w_{2} = 0.00126 (q_{0} / D) B_{2}^{4}$	N_A₂/N_A₁ = 0.922
	Rib area, R₂ = (2α − α²)A₂ per cell	for w₂ = w₁, (B₂/B₁) = 0.969	N_S₂/N_S₁, = 0.461
	Space lattice area, $S_{2} = B_{2}^{2}$

3. Triangular
	Unit cell area, $A_{3} = (3 \sqrt{3} / 4) B_{3}^{2}$	$w_{3} = 0.00151 (q_{0} / D) B_{3}^{4}$	N_A₃/N_A₁ = 0.777
	Rib area, R₃ = (2α − α²)A₃ per cell	for w₃ = w₁, (B₃/B₁) = 0.926	N_S₃/N_S₁, = 0.194
	Space lattice area, $S_{3} = (3 \sqrt{3} / 2) B_{3}^{2}$

α	t	m	n	K_approx^a	K_exact	Error, percent
0.02	0.02	2.04082	1.020408	0.28691	0.29077	1.3
0.02	0.1	11.1111	1.11111	0.079703	0.079716	0.01
0.06	0.06	2.12766	1.06383	0.27723	0.28084	1.3
0.06	0.1	3.70370	1.11111	0.19293	0.19381	0.4
0.1	0.1	2.22222	1.11111	0.26730	0.27065	1.2

Configuration	Areas	Central Deflection, w, of Cell Cover Plate Under Uniform Load, q₀, Clamped Edges	Number of Unit Cells, N_A, and Number of Space Lattice Elements, N_S, w₁ = w₂ = w₃, Equal Total Mirror Area
1. Hexagonal
	Unit cell area, $A_{1} = (\sqrt{3} / 2) B_{1}^{2}$	$w_{1} = 0.00111 (q_{0} / D) B_{1}^{4}$
	Rib area, R₁ = (2α − α²)A₁ per cell
	Space lattice area, $S_{1} = (\sqrt{3} / 4) B_{1}^{2}$

2. Square
	Unit cell area, $A_{2} = B_{2}^{2}$	$w_{2} = 0.00126 (q_{0} / D) B_{2}^{4}$	N_A₂/N_A₁ = 0.922
	Rib area, R₂ = (2α − α²)A₂ per cell	for w₂ = w₁, (B₂/B₁) = 0.969	N_S₂/N_S₁, = 0.461
	Space lattice area, $S_{2} = B_{2}^{2}$

3. Triangular
	Unit cell area, $A_{3} = (3 \sqrt{3} / 4) B_{3}^{2}$	$w_{3} = 0.00151 (q_{0} / D) B_{3}^{4}$	N_A₃/N_A₁ = 0.777
	Rib area, R₃ = (2α − α²)A₃ per cell	for w₃ = w₁, (B₃/B₁) = 0.926	N_S₃/N_S₁, = 0.194
	Space lattice area, $S_{3} = (3 \sqrt{3} / 2) B_{3}^{2}$

α	t	m	n	K_approx^a	K_exact	Error, percent
0.02	0.02	2.04082	1.020408	0.28691	0.29077	1.3
0.02	0.1	11.1111	1.11111	0.079703	0.079716	0.01
0.06	0.06	2.12766	1.06383	0.27723	0.28084	1.3
0.06	0.1	3.70370	1.11111	0.19293	0.19381	0.4
0.1	0.1	2.22222	1.11111	0.26730	0.27065	1.2

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Equations (25)

Applied Optics