Design and Analysis of 1.2m SiC Mirror and the Support

International Journal of Material and Mechanical Engineering (IJMME) Volume 3 Issue 4, November 2014 DOI: 10.14355/ijmme.2014.0304.02

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Design and Analysis of 1.2m SiC Mirror and the Support Zhilai Li Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Science, Changchun 130033, China lizl2004@sohu.com Abstract

In order to design a high stiffness and thermal stability 1.2m class space mirror, materials selection, lightweight and the support structure are taken into account in this study. By combining the excellent intrinsic thermo-mechanical properties, silicon carbide (SiC) is chosen as the material of the primary mirror and a new design with sandwich structure was proposed. Through adjusting the parameters of flexures, high stiffness and thermal environment adaptability are ensured. Detailed finite element analysis (FEA) is performed via a MSC/NASTRAN finite element model. Analysis results indicate that the surface figure reaches to 13.6nm rms under gravity and temperature effect, and the fundamental frequency is 130Hz. It can satisfy the system requirements of high stiffness and thermal stability.

Keywords Primary Mirror; Space Optics; Lightweight Mirror; Support

Introduction With the rapid technology development, space telescopes play more and more important roles in earth environment protection, earth observing and defense.1 The light space telescope with great field of view and high resolution has become the hot research point in the field of space optics. In order to obtain high resolution, space telescope general uses the large aperture and long focal length optical system. Off-axis three-mirror anastigmat (TMA) space telescope has become the good choice because of its characteristics such as large field of view, large relative aperture and no central pupil obstructions.2 In an off-axis TMA space telescope with large aperture and long focal length, large aperture space mirror is one of the most important optical elements whose surface figure will directly influence the optical system image quality.

The diameter of the primary mirror is 1.2m. The substrate and its support structure are designed and

analyzed in detail. The analysis includes the surface figure errors induced by gravity, thermal effects and manufacturing and assembly errors. Stiffness of the mirror subassembly and the eigenfrequencies are also analyzed in this paper. Analysis results indicate that the large aperture primary mirror system meet the design index.

Primary Mirror Structural Design Layout Mirror Material Selection The choice of mirror materials is the key factor for space mirror which directly relates to the research project successful or not.

From table 1 we can see that SiC material shows great potential for space mirror due to its combination features of high stiffness, low CET, and good thermal conductivity. SiC allows relatively cheap and quick manufacturing of mirrors because the mirrors can be shaped conventional tools in a milling.3 RB SiC forming processes have been used to produce very large SiC mirrors, at very low costs. Little postmachine is required because of the near-net-shape forming technology. So we choose RB SiC as the material of mirror in this paper.

TABLE 1 COMPARISON OF PROPERTIES OF DIFFERENT MIRROR MATERIALS

Materials

 g/cm3

Al Be Zerodur ULE Si SiC

2.70 1.85 2.53 2.21 2.33 3.20

E MPa 68000 287000 91000 67000 131 400

CET ×10-6/K 22.5 11.4 0.05 0.03 2.6 2.4

 W / (m  K )



1.64 216 1.64 1.31 137 155

0.33 0.043 0.24 0.17 0.42 0.18

Support Points Design When space mirror substrate is designed, the experience formula is presented by Roberts4 about diameter-thickness ratio dr ( D / t ) of circular mirror and the deformation caused by gravity:

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

International Journal of Material and Mechanical Engineering (IJMME) Volume 3 Issue 4, November 2014

3 ga 4 3 gdr 2 D 2  256 E 16 Et 2

(1)

Where the maximum deformation  (  m ) is caused

combined triangular lightweight holes with auxiliary ribs and the other was open back structure. The two mirror substrates geometry model are shown as Fig. 1.

by gravity;  (kg/m3) is density of the materials; g (m/s2) is gravity acceleration; a (m) is radius of the disk and a  D / 2 ; E (MPa) is elastic modulus; t is the thickness of the disk. When mirror meets the requirements of correctly imaging, that is,   0.16  m (  / 4,   0.6328  m ),

the

theoretical

diameter-

thickness ratio of this mirror is 10.9: 1, so the thickness of mirror is designed initially as 110 mm. Among them the thickness of the reinforcement rib was designed as 5 mm, the reflex plate thickness is 6 mm. Hall5 presented the following experience formula about the least support points for solid disk:

N

1.5r 2 t

g E

(2)

Where N is the least support points number; r is the disk radius; t (m) is the disk thickness;  (kg/m3) is the materials density; g (m/s2) is gravity acceleration; E (MPa) is elastic modulus; and  (  m ) is the

maximum deformation caused by self-weight. When   0.16  m (  / 4,   0.6328  m ), the number of the least support points obtained by above equation is N  3.5 . Considering the actual substrate structure and alignment art of the mirror subassembly, threepoint support is the best choice for the primary, but in the substrate design we preliminarily leave six support holes in order to take six points on back support as alternatives to the primary mirror.

Lightweight Design of Mirror Weight is one of important index should be considered when large aperture space mirror is designed. The shape of mirror lightweight holes mainly is triangular, quadrilateral, hexagonal, fanshaped and circular, etc. The interval and layout of the lightweight holes should be considered based on requirement of optical system to the mirror surface figure. Generally speaking, the stiffness of the triangular and honeycomb lightweight hole is highest, and that the manufacturability of circular one is best.

The quadrilateral lightweight hole is applied to rectangle or elliptic mirror and the fan-shaped one is applied to the mirror whose centre has a circular optical aperture. Two kinds of lightweight mirrors structure are designed after some factors such as strength, stiffness; thermal stability and cost are considered. One substrate was semi-open back which

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(a)

(b)

FIG. 1 PRIMARY MIRROR LIGHTWEIGHT STRUCTURE: (a)SEMI-OPEN BACK MIRROR; (b)-OPEN BACK MIRROR.

Through finite element analysis and comparison, the first structure scheme is finally chosen. The weight of substrate is 97.6 kg, and lightweight rate is 78%.

Support Structure Design Support structural design is an important link of the primary mirror subassembly design. In order to make space mirror meet design index requirements we must design reasonable support structure besides substrate design. Usually, If we want ot enhance the stiffness and strength, support structure’s stiffness need to be increased6 But exorbitant support structure stiffness can lead to the ability of mirror structure adapting to the thermal environment to be reduced. Using the flexible supporting technology can solve the problem that surface figure error of the mirror is reduced which introduced by these factors such as over positioning mount large aperture mirror and temperature change.7 So the difficulty of support structure design lies in how to rationally set the parameters of the flexure which can meet the requirements of stiffness, strength, and thermal stability.

(a)

(b)

FIG. 2 PRIMARY MIRROR SUBASSEMBLY GEOMETRY MODEL: (a)-SUBASSEMBLY 3D MODEL; (b)-FLEXIBLE HINGE PART.

A dual-axis circular flexible hinge is chosen as flexure of the support in this paper. The inserts made of invar

International Journal of Material and Mechanical Engineering (IJMME) Volume 3 Issue 4, November 2014

(4J32) are connected with mirror by adhesive bond and one end of the flexible hinge (titanium alloy) is connected with the inserts by the screws and the other is connected with back triangle support plate (SiC/Al) by the screws. The mirror subassembly model is shown as Fig. 2.

1   ( x)

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d2y dx 2

(7)

1

dy 2  2  1  ( dx ) 

Submitting Eq. (7) into Eq. (6) results in: M ( x)  EJ ( x)

d2y dx 2

(8)

1 2 2

dy  1  ( dx ) 

Because only consider numerical relations, and don’t care about the direction, so  can be omitted, and because the corner is very small, so we can think: 2

 dy     1 , Eq. (8) can be simplified as:  dx  FIG. 3 DEFORMATION SCHEMES OF FLEXIBLE HINGE UNDER THE TORQUE MOMENT.

The single-axis flexible hinge would be simplified as shown in Fig. 3. A horizontal torque make flexible hinge small angular displacement  occurred, slope of the tangent line is: tg 

dy dx

(3)

In fact the angular displacement of the flexible is very dy small, we can think:   tg , so the corner is:   . dx The corner of flexible hinge can be thought as the accumulation of many small stretch bending. Set the length of each small stretch for x , the entire corner and deflection of the hinge are given in the following forms: n

   i i 1 n

y   yi i 1

(4)

(9)

On both sides of Eq. (9) are multiplied by dx , and then make integral. The corner relation for the rectangular coordinate is



dy d2y M ( x)   2 dx   dx dx EJ ( x) dx

(10)

Making quadric integral, the deflection can be presented in the following form: y  

 M ( x)  M ( x) 2 dx     dx dx EJ ( x)  EJ ( x) 

(11)

For integral convenient, we converse rectangular coordinate into polar coordinate. dx  Rd sin  , bh3 b(2 R  t  2 R sin  )3  h  2R  t  2R sin  , J ( x)  12 12 So corner can be written as:

(5)

According to the material mechanics, it is known that the tiny part neutral face curvature of the flexible hinge can be written as:

1 M ( x)   ( x) EJ ( x)

M ( x) d 2 y  EJ ( x) dx 2

(6)

Where  ( x) is the curvature of the x section; M ( x) is moment; E is elastic modulus; and J ( x) is moment of inertia. Arbitrary point’s curvature of curve expression y  f ( x) can be written as:

 

12 R sin  M ( x) d Eb(2 R  t  2R sin  )3

(12)

Deflection can be displayed as following expression:   12 R sin  M ( x) y    d  d 3 Eb (2 R  t  2 R sin  )  

(13)

Where R is radius of flexible hinge, t is the thickness of the hinge, b is hinge width, M ( x) is the moment.

In order to see the relationship among the three variables, we use Romberg integral method and get the stiffness under different R and t .

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International Journal of Material and Mechanical Engineering (IJMME) Volume 3 Issue 4, November 2014

Along with the increase of value of t , variation of corner and displacement stiffness presents quadric curves relationship. Stiffness is smaller when R is larger, but the change is not obvious. In the course of actual design, according to the technical requirement the value of R is determined, and then the requirements of thermal dimensional stability would be met through adjust the value of t and the position of flexible hinge in the support structure.

Finite Element Analysis Mirror structure system analysis mainly examines static stiffness and thermal stability of the mirror under the gravity and temperature loads, another analysis target is the dynamic stiffness of mirror subassembly. Using MSC/Patran software, we build the mirror subassembly FE model which is shown as Fig. 4, (a).

subassembly is high enough and can meet the requirement of designed index.

Conclusions A large aperture primary mirror of an off-axis TMA space telescope is designed in detail in this paper. The problem that the surface figure accuracy of large aperture mirror is difficult to ensure in space application environment is solved through setting the flexure in the support structure. FEA results indicate that the surface figure error of the primary mirror is rms 13.6 nm under gravity and 2℃ uniform temperature rise load cases, and the base frequency of mirror subassembly is 130 Hz. Analysis results show that design of the mirror and its support structure is effective, which can meet the requirements of space applications.

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(b)

FIG. 4 (a)-FE MODEL OF SUBASSEMBLY; (b)-DEFORMATION CLOUD OF THE MIRROR UNDER X DIRECTION GRAVITY LOAD.

The deformation cloud of the primary mirror under X direction gravity is shown in Fig. 4, (b). Surface figure analysis results of the mirror introduced by gravity and 2 ℃ uniform temperature rise load cases are listed in Table 2.

TABLE 2.SURFACE FIGURE OF THE MIRROR UNDER GRAVITY AND

rms/(nm) 15.8 9.8 9.7 6.3 13.6

Tilts: RX(″) 2 0.07 0.11 0.02 0.08

Tilts: RY(″) 2 0.08 0.02 0.03 0.10

In the constraint condition, the modal analysis is done on mirror subassembly and first four order eigenfrequencies are extracted by using Lanczos Method. Modal analysis results show that fundamental frequency of subassembly is 130 Hz. Because fundamental frequency is more than 100 Hz, we can think that the dynamic stiffness of mirror

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Zhilai. Li was born in Jilin, China, in 1965, and received his B.Sc in 1986.

Now, he is a professor in Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Science.

His main research interests include space opto-mechanical structure design and precision machinery.