Design and optimization of the tripod flexure for a 2m lightweight mirror for space application

Ping Jiang; Chuang Xue; Kejun Wang; Xiaoyu Wang; Pingwei Zhou

doi:10.1364/AO.476783

Applied Optics
Vol. 62,
Issue 1,
pp. 217-226
(2023)
•https://doi.org/10.1364/AO.476783

Design and optimization of the tripod flexure for a 2m lightweight mirror for space application

Ping Jiang, Chuang Xue, Kejun Wang, Xiaoyu Wang, and Pingwei Zhou

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Ping Jiang, Chuang Xue, Kejun Wang, Xiaoyu Wang, and Pingwei Zhou, "Design and optimization of the tripod flexure for a 2m lightweight mirror for space application," Appl. Opt. 62, 217-226 (2023)

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Abstract

In order to ensure optimal optical performance, primary mirror assembly must be impervious to environmental influences. These environmental influences include gravity, assembly error, and thermal change, under which external loads are imposed on the mirror. The external loads degrade the mirror surface accuracy and cause misalignment between mirrors. In this paper, a tripod flexure with a flexible hinge is designed to alleviate the influence of the external load on the surface accuracy of a 2 m primary mirror. This structure can effectively release the rotational freedom, provide a certain translational flexibility, and yield high axial stiffness. The axial stiffness is used to increase the frequency of the primary mirror assembly. According to the fast optimization model, the derivation of close form compliance equations is developed to characterize the flexibility, and parameter optimization is done to achieve the maximum performance. Then a finite element analysis and test are used to verify the final design. The results show that the index requirements of the 2 m primary mirror have been met.

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Material	SiC (Mirror)	Invar (Sleeve)	Titanium (Flexure)	Adhesive
Yong’s modulus (Mpa)	330	141	109	0.158
Poisson’s ration	0.25	0.25	0.29	0.49
Density ( ${t / m m}^{3}$ )	$3.05 \times 10^{- 9}$	$8.1 \times 10^{- 9}$	$4.4 \times 10^{- 9}$	$1.3 \times 10^{- 9}$
Coefficient of thermal expansion ( $/^{\circ} C$ )	$2.5 \times 10^{- 6}$	$2.5 \times 10^{- 6}$	$9.1 \times 10^{- 6}$	$3 \times 10^{- 3}$

Optimization Parameters	Initial Value/mm	Variation Range/mm	Optimal Value/mm
Thickness $t$	4.5	4–6	5
Width $w$	20	15–22	20
Length $l$	5	4–7	6
Included angle $α$	60°	50–65	60°

Load Cases	Initial	Optimal
Gravity	$\begin{array}{l} M_{y 1} = 41482.1 N \cdot m m \\ R M S = 4.65 n m \\ u_{z 1} = 12.4 µ m \end{array}$	$\begin{array}{l} M_{y 1} = 41123.7 N \cdot m m \\ R M S = 4.65 n m \\ u_{z 1} = 12.6 µ m \end{array}$
0.1 mm assembly error	$\begin{array}{l} M_{y 1} = 1067.5 N \cdot m m \\ R M S = 3.56 n m \end{array}$	$\begin{array}{l} M_{y 1} = 1028.5 N \cdot m m \\ R M S = 3.43 n m \end{array}$
4°C temperature change	$\begin{array}{l} F_{z 1} = 596.3 N \\ M_{y 1} = 26753.2 N \cdot m m \\ β = \frac{M_{y 1}}{F_{z 1}} = 44.8 \\ R M S = 3.43 n m \end{array}$	$\begin{array}{l} F_{z 1} = 498.8 N \\ M_{y 1} = 22185.4 N \cdot m m \\ β = \frac{M_{y 1}}{F_{z 1}} = 44.5 \\ R M S = 3.4 n m \end{array}$
Frequency	130 Hz	129 Hz

Optimization	Test Location	Simulation Result	Experimental Result	Error
Flexibility $C_{Z - F_{z}}$	0°	0.165 mm	0.158 mm	4.43%
Flexibility $C_{Z - F_{z}}$	46°	0.165 mm	0.157 mm	5.1%
Flexibility $C_{θ_{y} - F_{z}}$	0°	248.8″	$244^{''}$	2%
Flexibility $C_{θ_{y} - F_{z}}$	46°	248.8″	$241^{''}$	3.24%
Flexibility $C_{θ_{y} - M_{y}}$	0°	226.6″	$221. 5^{''}$	2.3%

Material	Test Location	1#	2#	3#
Flexibility $C_{z - F_{z}}$	0°	0.158 mm	0.158 mm	0.159 mm
Flexibility $C_{θ_{y} - F_{z}}$	0°	$244^{''}$	$242^{''}$	$241^{''}$

Loading Direction	X				Y/Z
Frequency range	4–9	9–18	18–36	36–100	4–10	10–32	32–100
Amplitude 0–p	16.7 mm	5.4 g	7.2 g	5.4 g	18.0 mm	7.2 g	5.4 g
Load scan rate	2 oct/min

Loading Direction	X/Y/Z
Frequency range (Hz)	10–50	50–800	50–800
Magnitude 0–p	3 dB/oct	$0.032 g^{2} / H z$	$0.032 g^{2} / H z$
RMS acceleration value	6.34 g
Duration	2 min

Abstract

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Applied Optics