КОНСТРУКЦИЯ МЕХАНИЗМА КРЕПЛЕНИЯ КРУПНОГАБАРИТНОГО ЗЕРКАЛА ДЛЯ ШИРОКОГО ТЕМПЕРАТУРНОГО ДИАПАЗОНА
КОНСТРУКЦИЯ МЕХАНИЗМА КРЕПЛЕНИЯ КРУПНОГАБАРИТНОГО ЗЕРКАЛА ДЛЯ ШИРОКОГО ТЕМПЕРАТУРНОГО ДИАПАЗОНА
SUPPORT MECHANISM DESIGN OF LARGE APERTURE REFLECTIVE MIRROR FOR LARGE TEMPERATURE VARIATIONS
© 2014 г. Haili Hu, Baojun Zuo, Shouqian Chen, Minda Xu, Zhigang Fan School of Astronautics, Harbin Institute of Technology, Harbin, China Е-mail: cn.huhaili@gmail.com
Зеркала большой апертуры используются в низкофоновой наземной моделирующей системе для генерации одиночных целей и многоцелевых ситуаций в инфракрасной области спектра. В статье описана объединенная структура крепления зеркала в виде ремня и системы опор на тыльной стороне и система регулируемых связей, решающих проблему крепления больших зеркал, работающих при перепадах температуры до 100 K. Анализ методом конечных элементов показал, что значение среднеквадратического отклонения формы оптимизированного зеркала будет лучше, чем 1/40 длины волны при действии собственного веса и перепада температур в 100 K. Подгонка полиномами Цернике показала, что функция передачи модуляции многоцелевого имитатора будет не ниже 0,5, а среднеквадратическое отклонение диаметра пятна меньше чем 0,05 мрад. Эти результаты подтверждают эффективность предложенного механизма крепления, тем самым обеспечивая аналитическую основу создания зеркал метрового класса для больших перепадов температуры окружающей среды.
Cold-background multi-target compounding system provides infrared targets for hardware-in-the-loop simulation system, in which large aperture reflective mirrors are employed. In this paper, we proposed a combined belt-back structure and designed the flexible connection to solve support mechanism of large aperture mirrors for 100 K temperature variations. By Finite Element Method analysis, root mean square of optimized mirror was better than λ/40 under self-gravity and 100 K temperature variations. By Zernike polynomial fitting, the modulate transmission function of multitarget compounding system was over 0.5 and root mean square spot diameter was less than 0.05 mrad. Results demonstrated that proposed support mechanism was effective, providing analytical data for 1m level mirror for large environment temperature variations.
OCIS codes: 220.4610, 220.4880
Submitted 03.10.2013
Introduction
With developments of infrared imaging guidance technique, hardware-in-the-loop (HWIL) simulation applying for infrared missiles has attracted lots of attentions [1−3]. Infrared targets and environments simulation have become a critical technique of HWIL simulation [2, 4]. Coldbackground multi-target compounding system is infrared targets simulator of infrared hardware-in-the-loop simulation system. The compounding system provides infrared targets and cold-background (200 K) for infrared missiles to test the identification, guidance and attack properties.
Large aperture reflective mirrors are key elements of multi-target compounding system and surface shape quality primarily affect compounding system resolution. Consequentially, surface shape quality limits test results and realistic applications of infrared HWIL simulation system. If the size and material are arranged reasonably, small aperture elements can be treated as a rigid body. Therefore, surface shape deformation caused by self-gravity is small and gravity influence on surface shape quality can be ignored. However, because reflective mirror is 1m level aperture in multi-target compounding system, reflective mirror has big bending moments [5]. Moreover, for reflective mirror with horizontal
“Оптический журнал”, 81, 4, 2014
31
axis, because axis is vertical during manufacture, high surface quality during usage and manufacture process are both required. More important, compounding system must work effectively under 300 K and 200 K environment temperatures. Therefore, support mechanism of mirrors should not only reduce surface deformation by self-gravity but also discharge the thermal stress to suit for temperature variations.
Support mechanism greatly determines the design, the functioning, the stability, and the operating convenience of optical devices [6]. Well-known support modes of large aperture reflective mirror include multi-point edge support [7], push-pull support [5], belt support [8], back support [9] and so on. Multi-point edge support is more complex and performances are not better than belt support. Push-pull support has much more support points when aperture is over Φ500 mm, which results in difficult alignment and instability. Belt support has simplicity advantage, but sidewise press is larger. Back support produces larger longitudinal press when mirror axis is horizontal.
In this paper, we researched on the support mechanism of 1 m level aperture for 100 K variation environment. We tried to combine belt support and back support together to counteract sidewise and longitudinal deformation of reflective mirror. Based on Finite Element Method (FEM), we designed and optimized the combined belt-back structure to maintain surface shape precision during axis direction changing. Flexible connection was designed for mirror support points to maintain high performance during 100 K temperature variation. By Zernike polynomial fitting, optical qualities of optimized compounding system were achieved.
1. Cold-background multi-target compounding system
Targets of cold-background multi-target compounding system ranged over both the near, mid and long wave infrared wavebands. Main optical elements in compounding system are reflective for achromatic performance. The basic structure of compounding system is a two-mirror threereflection concentric configuration [1, 3], comprised by a concave spherical reflective mirror and dimpled mirror, as shown in Fig. 1. Dimpled mirror is located at focal plane of main mirror, which is a special convex mirror, covered by
small concave mirrors compactly. With the help of dimpled mirror, every beam outgoing from compounding system is broadened [1, 3], and there is an overlapping area at exit pupil among beams. Therefore, the receiver can receive all targets at the same time at exit pupil.
In cold-background multi-target compounding system, entrance pupil distance Li and exit pupil distance L0 are satisfy equation of Li+ L0 = 4f¢0 [3]. We set Li:L0 = 1:1, so entrance pupil distance and exit pupil distance are equal to radius of concave mirror. Detailed system parameters are present in Table 1. Due to resolution requirement, radius of main mirror was 6500 mm. For realistic manufacture, we separated primary mirror into two main mirrors (1 and 2). Theoretically, main mirror apertures are all Φ839 mm and it is set Φ900 mm for actual installments and alignments.
Receiver
Primary mirror
Dimpled mirror
f0¢ Projectors Lp
Li Fig. 1. Diagram of multi-targets compounding system. f¢0 is focal length of concave mirror. Li is entrance pupil distance and Lp is distance between projectors.
Table 1. Parameters of multi-target compounding system
Parameters
Value
Focal length of primary mirror, mm
3250
Field of view, deg
±3
Pupil diameter of projectors, mm
40
Distance between projectors, mm
120
Entrance pupil distance, mm
6500
Exit pupil distance, mm
6500
Pupil diameter of detector, mm
160
32 “Оптический журнал”, 81, 4, 2014
2. Support mechanism design
2.1. Requirements
In compounding system, mirrors 1 and 2 are Φ900 mm with boundary thickness of 100 mm. Mirror axis is horizontal in usage, while mirror axis is vertical during manufacture. Environments conditions include 300 K in air and 200 K in vacuum. Root mean square (RMS) of surface shape is less than λ/40, where λ = 4 μm is the dominant wavelength. As the compounding system must work at vacuum and cold environments, property of mirror support structure should satisfy: (1) great rigidity and intensity; (2) excellent thermal stability; (3) no volatile contamination; (4) reasonable constraint; (5) positioning accurately; (6) easy to alignment and test.
2.2. Mirror material selection
Material selection of large aperture reflective mirror determines mechanical properties, optical manufacture, material stability and so on. SiC and Ceramet has high rigidity and better strength, but with larger coefficient of linear expansion. Moreover, due to high cost of large aperture materials, SiC and Ceramet won’t be used in temperature-variation ground system generally. Comparably, Zerodur has larger stiffness and thermal stability under large temperature variations, which result in suitability for large changing environments. Therefore, Zerodur is excellent materials for reflective mirror in ground temperature-variation condition [8]. Material properties of Zerodur are shown in Table 2.
2.3. Support mode
Belt support is first choice for reflective mirror with horizontal axis. Mirror is positioning on the structure by flexible steel belt itself and Mirror will fall at center of belt due to gravity. Gravity of mirror is distributed uniformly by support area connected to belt. As illustrated by Fig. 2, according to static equilibrium, normal press on the mirror is described by [7, 8]
F = G/(2Rdsinϕ),
(1)
where, 2R is diameter of mirror and d is thickness of mirror. G is mirror weight and 2ϕ is the contact angle between belt and mirror.
After calculations, when 2ϕ = π, sidewise deformation of surface shape is smallest [7]. Therefore we chose 2ϕ = π as original position of belt support.
If there is only a steel belt, high weight mirror can’t satisfy requirement of λ/40 due to larger sidewise deformation. Importantly, we added multi-circle multi-point support at back of primary mirror, and support points can balance sidewise pressure of belt. 18 points structure proposed by Handle has best support performance [5, 9]. As illustrated by Fig. 3, 18 points are arranged by two circles. Three adjacent points constructed triangularity. Adjacent two triangularities are connected by a rod. Three rods are connected to main support frame. Radiuses of two circles are determined by
R1 = 0.411R0,
(2)
R2 = 0.859R0,
(3)
where, 2R0 is diameter of main mirror. Original design result of 18 points support is
exhibited in Fig. 4.
Table 2. Material properties of Zerodur
Parameters
Value
Intensity, kg/m3
2500
Modulus of elasticity, GPa
91
Poisson ratio
0.24
Coefficient of linear expansion at 273 K, 10−6/K
0.05
Coefficient of linear expansion at 40 K, 10−6/K
−0.7
Thermal conductivity, W/MK−1
1.5
Specific heat capacity, J/kgK
8.2
Specific stiffness, 104 Nm/g
3.64
“Оптический журнал”, 81, 4, 2014
R 2ϕ
F Fig. 2. Diagram of steel-belt support structure.
33
R0 Linear slot
R1 R2
Fig. 5. Flexible connection design of support points.
Fig. 3. Schematic of back support points distribution.
Mirror
Support poin
X r
θ O
Y
Fig. 6. Calculation of rotation rigidity for flexible connection.
t
Belt
Fig. 4. 3D structure of combined belt-back support.
2.4. Thermal press discharge
Working environments of compounding system is changed from 300 K to 200 K. Coefficients of linear expansion between mirror material and support mechanism can’t be exact same. When working temperature is changed, stresses caused by different coefficients of materials will affect on the surface shape precision greatly. Next, we designed special structure to discharge the thermal stress. Flexible connection structure is an effective form of thermal stress discharge, which can reduce the stress influence of temperature variation [10].
For reflective mirror with Φ900 mm, we designed a flexible connection between mirror and
back support point, as shown in Fig. 5. The flexible connection has two advantages. First one is that the hollow design of support point can reduce the stress caused by different deformations. Second one is that the linear slot can discharge the thermal stress along deformation direction.
Flexible connection is designed by analyzing dependence of rotation rigidity on structure parameters. Calculation method of rotation rigidity for flexible connection is shown in Fig. 6, and expression of rotation rigidity is conveyed by following equation [10]:
M =1
π
ò
0
2r
12r sinα + t -2r sinα
dα,
(4)
where, r is radius of slot, and t is thinnest width of connection.
3. Finite element method and results
For compounding system, we concern the effect of steady uniformity temperature field on support structure. According to thermodynamics
34 “Оптический журнал”, 81, 4, 2014
Original Modeling 3D
structure
model
Finite Analysis
Correction
N
FEM model
If convergent?
End Y
N
Satisfy index?
Thermal analysis Forces analysis
Y Structure Parameter
Fig. 7. Optimize design flow of reflective mirror support.
theory, the strain caused by temperature varia-
tions is expressed by
{ }{ε}T = εx,εy,εz,εxy,εyz,εzx T = { }= α ∆Tx,∆Ty,∆Tz,0,0x,0 T ,
(5)
where, {ε}T is strain vector, ΔTx, ΔTy and ΔTz
are temperature variations of x, y and z direction, α is coefficient of heat transfer.
Relationship between stress and strain is expressed by following equation [11]:
{σ} = D×{ε}T ,
(6)
where, {σ} is stress vector and D is expressed by
D = (1-Eµ(1)(-1-µ)2µ)´
´ êëêêêêêêêêêêéêêêêêêêêêêêê11--1µµ000 µµ
µ 1-µ
1
µ 1-µ
0
0
0
µ 1-µ
µ 1-µ
1
0
0
0
0
0
0 1-2µ
2(1- µ)
0
0
0
0
0
0 1-2µ
2(1- µ)
0
21(1-00000-2µµ)úúúúúúúúúúùúúúûúúúúúúúúúú, (7)
where, μ is poisson ratio and E is modulus of elasticity.
Based on equations (1)−(4), we can figure out the original support structure and flexible connection and then optimize them by FEM. The optimization process is illustrated in Fig. 7. Firstly, construct the 3D support model in FEM software. Secondly, mesh the support model and set actual
boundary conditions and loads. Thirdly, calculate the adjustment value and direction based on precious results. Design, optimize and validate the structure again and again. Finally, assign the accuracy of elements and value of forces until satisfy requirements.
For high accuracy of analyzing results, we choose secondary 3D hypostatic unit for mesh generation, and final model is indicated in Fig. 8. We set high intensity meshes at connection area as revealed in Fig. 8.
FEM results are shown in Figs. 9, 10 and 11. As conveyed in Fig. 9, back support can maintain high precision when axis is vertical. Fig. 10 indicats surface property when axis is horizontal. PV value of surface is less than 0.209 μm and RMS value is 70 nm. Results show that combined belt-back support can achieve better performance for 1m mirror. Fig. 11 is FEM results of mirror under environment temperature of 200 K. PV values of surface shape is less than 0.33 μm and RMS value is better than λ/40. Comparing with Fig. 10, we can reach the conclusion that support system can maintain high surface shape when environment temperature was changed from 300 K to 200 K.
Fig. 8. FEM model of belt-back support.
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FEM data
Zernike polonomial fitting
CODEV
MTF Spot RMS
Fig. 12. Analysis flow of optical performances. Z YX
(а)
Fig. 9. Deformation fringe of reflective mirror with vertical axis when environment
temperature is 300 K.
Y ZX
(b)
Fig. 10. Deformation fringe of reflective mirror with horizontal axis when environment temperature is 300 K.
Y ZX
(c)
Fig. 11. Deformation fringe of reflective mirror with horizontal axis when environment temperature is 200 K.
36
Fig. 13. MTF of multi-target compounding system under different conditions. Ideal condition (a), 300 K environment temperature (b), 200 K environment temperature (c).
“Оптический журнал”, 81, 4, 2014
4. Optical performances demonstration
Finally, according to output data of FEM, we examined the effect of surface shape deformation on the optical performances. Analysis process is indicated in Fig. 12. Firstly, we performed Zernike polynomial fitting for the mirror variation of discrete points outputted from FEM. Secondly, we translated the Zernike coefficients to optical design software to obtain the optical modulate transmission function (MTF) and RMS spot diameter.
Take the projector with max abaxial distance as example to determine the influence of surface precision on system resolution due to gravity and thermal effect. According to method in Fig. 12, we figure out the MTF plots of system under different working condition, as shown in
Table 3. RMS spot diameter of multi-target compounding system under different condi-
tions (mrad)
FOV, deg
−3.0
−1.5
0
1.5 3.0
Ideal 0.0361 0.0363 0.0362 0.0361 0.0366
300 K 0.0388 0.0392 0.0391 0.0399 0.0395
200 K 0.0498 0.0477 0.0488 0.0472 0.0494
Remark. FOV – field of view.
system requirement. Therefore, we can reach the conclusion that designed structure can maintain optical resolution of compounding system.
Conclusions Focusing on the reflective mirror with large
Fig. 13b and 13c. Fig. 13a is MTF plot for ideal aperture, high self-gravity and large temperature
condition. RMS spot diameters for all FOV are variation, we optimized the combined belt-back
shown in Table 3. Compared to ideal condition, support mechanism and designed flexible support
MTF and RMS spot diameter didn’t descend when structure, which maintain mirror surface shape
environment temperature is 300 K. This is due and system resolution. After optimization, RMS
to the fact that the optimized support mechanism can reduce the self-gravity effect on surface shape deformation effectively. For environment of 200 K, MTF of compounding system fall about 0.1 at maximum frequency. However, MTF is still over 0.5 at maximum frequency and RMS spot diameter is less 0.05 mrad, which satisfy
of surface deformation is less λ/40 for two working conditions. By Zernike polynomial fitting, MTF of system is over 0.5 and RMS spot diameter is less than 0.05 mrad. Results demonstrated that proposed support mechanism is reasonable for large environment temperature variations of 1 m level reflective mirror.
* * * * *
ЛИТЕРАТУРА
1. Baessler R.J., Popper H. Infrared Simulation System (IRSS). Phase I // Defense Technical Information Center, 1977.
2. Bailey M., Doerr J. Contributions of Hardware-in-the-Loop Simulations to Navy Test and Evaluation // Proc. SPIE. 1996. V. 2741. P. 33.
3. Hu H.L., Zuo B.J., Chen S. Q. Multi-Target Compounding Technique Based on Dimpled Mirror // Opt. Eng. 2012. V. 51. P. 113001.
4. Cantey T.M., Beasleya D.B., Bendera M. Cold Background, Flight Motion Simulator Mounted, Infrared Scene Projectors Developed for Use in AMRDEC Hardware-in-the-Loop // Proc. SPIE. 2004. V. 5408. P. 96.
5. Yoder P. Opto-Mechanical System Design // Cooperate Marcel Dekker Inc, 1993.
6. 6. Raguzin R M., Zadorin E.Yu. Stability of the Support Structures of Optical Devices // Opt. Zh. 2011. V. 78. P. 32 [J. Opt. Tech. 2011. V. 78(1). P. 25].
7. Xu R.W, Liu L.R., Zhu L. Support Schemes and Thermal Effects Analyses of Large-Aperture Interferometer Mirrors // Proc. SPIE. 2004. V. 5531. P. 441.
8. Yu J., Shen S.D, Pan J.H. Pan Y.J. Manufacture of 1.8M Standard Spherical Mirror // Proc. SPIE. 2012. V. 8415. P. 84151A.
9. Wu X.X., Yang H.B., Zhang J.X. Design of Support System for the Large-aperture Sphere Mirror // Acta Photonica Sinica. 2009. V. 38(1).P. 129.
10. Sun BY. Flexible Regulating Structure of Optical Reflector of Space Remote Sensor // Journal of Harbin Institute of Technology. 2009. V. 41(9). P. 201.
11. Bhavikatti S. Finite Element Analysis // New Age International (P) Ltd., Publishers, 2005.
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SUPPORT MECHANISM DESIGN OF LARGE APERTURE REFLECTIVE MIRROR FOR LARGE TEMPERATURE VARIATIONS
© 2014 г. Haili Hu, Baojun Zuo, Shouqian Chen, Minda Xu, Zhigang Fan School of Astronautics, Harbin Institute of Technology, Harbin, China Е-mail: cn.huhaili@gmail.com
Зеркала большой апертуры используются в низкофоновой наземной моделирующей системе для генерации одиночных целей и многоцелевых ситуаций в инфракрасной области спектра. В статье описана объединенная структура крепления зеркала в виде ремня и системы опор на тыльной стороне и система регулируемых связей, решающих проблему крепления больших зеркал, работающих при перепадах температуры до 100 K. Анализ методом конечных элементов показал, что значение среднеквадратического отклонения формы оптимизированного зеркала будет лучше, чем 1/40 длины волны при действии собственного веса и перепада температур в 100 K. Подгонка полиномами Цернике показала, что функция передачи модуляции многоцелевого имитатора будет не ниже 0,5, а среднеквадратическое отклонение диаметра пятна меньше чем 0,05 мрад. Эти результаты подтверждают эффективность предложенного механизма крепления, тем самым обеспечивая аналитическую основу создания зеркал метрового класса для больших перепадов температуры окружающей среды.
Cold-background multi-target compounding system provides infrared targets for hardware-in-the-loop simulation system, in which large aperture reflective mirrors are employed. In this paper, we proposed a combined belt-back structure and designed the flexible connection to solve support mechanism of large aperture mirrors for 100 K temperature variations. By Finite Element Method analysis, root mean square of optimized mirror was better than λ/40 under self-gravity and 100 K temperature variations. By Zernike polynomial fitting, the modulate transmission function of multitarget compounding system was over 0.5 and root mean square spot diameter was less than 0.05 mrad. Results demonstrated that proposed support mechanism was effective, providing analytical data for 1m level mirror for large environment temperature variations.
OCIS codes: 220.4610, 220.4880
Submitted 03.10.2013
Introduction
With developments of infrared imaging guidance technique, hardware-in-the-loop (HWIL) simulation applying for infrared missiles has attracted lots of attentions [1−3]. Infrared targets and environments simulation have become a critical technique of HWIL simulation [2, 4]. Coldbackground multi-target compounding system is infrared targets simulator of infrared hardware-in-the-loop simulation system. The compounding system provides infrared targets and cold-background (200 K) for infrared missiles to test the identification, guidance and attack properties.
Large aperture reflective mirrors are key elements of multi-target compounding system and surface shape quality primarily affect compounding system resolution. Consequentially, surface shape quality limits test results and realistic applications of infrared HWIL simulation system. If the size and material are arranged reasonably, small aperture elements can be treated as a rigid body. Therefore, surface shape deformation caused by self-gravity is small and gravity influence on surface shape quality can be ignored. However, because reflective mirror is 1m level aperture in multi-target compounding system, reflective mirror has big bending moments [5]. Moreover, for reflective mirror with horizontal
“Оптический журнал”, 81, 4, 2014
31
axis, because axis is vertical during manufacture, high surface quality during usage and manufacture process are both required. More important, compounding system must work effectively under 300 K and 200 K environment temperatures. Therefore, support mechanism of mirrors should not only reduce surface deformation by self-gravity but also discharge the thermal stress to suit for temperature variations.
Support mechanism greatly determines the design, the functioning, the stability, and the operating convenience of optical devices [6]. Well-known support modes of large aperture reflective mirror include multi-point edge support [7], push-pull support [5], belt support [8], back support [9] and so on. Multi-point edge support is more complex and performances are not better than belt support. Push-pull support has much more support points when aperture is over Φ500 mm, which results in difficult alignment and instability. Belt support has simplicity advantage, but sidewise press is larger. Back support produces larger longitudinal press when mirror axis is horizontal.
In this paper, we researched on the support mechanism of 1 m level aperture for 100 K variation environment. We tried to combine belt support and back support together to counteract sidewise and longitudinal deformation of reflective mirror. Based on Finite Element Method (FEM), we designed and optimized the combined belt-back structure to maintain surface shape precision during axis direction changing. Flexible connection was designed for mirror support points to maintain high performance during 100 K temperature variation. By Zernike polynomial fitting, optical qualities of optimized compounding system were achieved.
1. Cold-background multi-target compounding system
Targets of cold-background multi-target compounding system ranged over both the near, mid and long wave infrared wavebands. Main optical elements in compounding system are reflective for achromatic performance. The basic structure of compounding system is a two-mirror threereflection concentric configuration [1, 3], comprised by a concave spherical reflective mirror and dimpled mirror, as shown in Fig. 1. Dimpled mirror is located at focal plane of main mirror, which is a special convex mirror, covered by
small concave mirrors compactly. With the help of dimpled mirror, every beam outgoing from compounding system is broadened [1, 3], and there is an overlapping area at exit pupil among beams. Therefore, the receiver can receive all targets at the same time at exit pupil.
In cold-background multi-target compounding system, entrance pupil distance Li and exit pupil distance L0 are satisfy equation of Li+ L0 = 4f¢0 [3]. We set Li:L0 = 1:1, so entrance pupil distance and exit pupil distance are equal to radius of concave mirror. Detailed system parameters are present in Table 1. Due to resolution requirement, radius of main mirror was 6500 mm. For realistic manufacture, we separated primary mirror into two main mirrors (1 and 2). Theoretically, main mirror apertures are all Φ839 mm and it is set Φ900 mm for actual installments and alignments.
Receiver
Primary mirror
Dimpled mirror
f0¢ Projectors Lp
Li Fig. 1. Diagram of multi-targets compounding system. f¢0 is focal length of concave mirror. Li is entrance pupil distance and Lp is distance between projectors.
Table 1. Parameters of multi-target compounding system
Parameters
Value
Focal length of primary mirror, mm
3250
Field of view, deg
±3
Pupil diameter of projectors, mm
40
Distance between projectors, mm
120
Entrance pupil distance, mm
6500
Exit pupil distance, mm
6500
Pupil diameter of detector, mm
160
32 “Оптический журнал”, 81, 4, 2014
2. Support mechanism design
2.1. Requirements
In compounding system, mirrors 1 and 2 are Φ900 mm with boundary thickness of 100 mm. Mirror axis is horizontal in usage, while mirror axis is vertical during manufacture. Environments conditions include 300 K in air and 200 K in vacuum. Root mean square (RMS) of surface shape is less than λ/40, where λ = 4 μm is the dominant wavelength. As the compounding system must work at vacuum and cold environments, property of mirror support structure should satisfy: (1) great rigidity and intensity; (2) excellent thermal stability; (3) no volatile contamination; (4) reasonable constraint; (5) positioning accurately; (6) easy to alignment and test.
2.2. Mirror material selection
Material selection of large aperture reflective mirror determines mechanical properties, optical manufacture, material stability and so on. SiC and Ceramet has high rigidity and better strength, but with larger coefficient of linear expansion. Moreover, due to high cost of large aperture materials, SiC and Ceramet won’t be used in temperature-variation ground system generally. Comparably, Zerodur has larger stiffness and thermal stability under large temperature variations, which result in suitability for large changing environments. Therefore, Zerodur is excellent materials for reflective mirror in ground temperature-variation condition [8]. Material properties of Zerodur are shown in Table 2.
2.3. Support mode
Belt support is first choice for reflective mirror with horizontal axis. Mirror is positioning on the structure by flexible steel belt itself and Mirror will fall at center of belt due to gravity. Gravity of mirror is distributed uniformly by support area connected to belt. As illustrated by Fig. 2, according to static equilibrium, normal press on the mirror is described by [7, 8]
F = G/(2Rdsinϕ),
(1)
where, 2R is diameter of mirror and d is thickness of mirror. G is mirror weight and 2ϕ is the contact angle between belt and mirror.
After calculations, when 2ϕ = π, sidewise deformation of surface shape is smallest [7]. Therefore we chose 2ϕ = π as original position of belt support.
If there is only a steel belt, high weight mirror can’t satisfy requirement of λ/40 due to larger sidewise deformation. Importantly, we added multi-circle multi-point support at back of primary mirror, and support points can balance sidewise pressure of belt. 18 points structure proposed by Handle has best support performance [5, 9]. As illustrated by Fig. 3, 18 points are arranged by two circles. Three adjacent points constructed triangularity. Adjacent two triangularities are connected by a rod. Three rods are connected to main support frame. Radiuses of two circles are determined by
R1 = 0.411R0,
(2)
R2 = 0.859R0,
(3)
where, 2R0 is diameter of main mirror. Original design result of 18 points support is
exhibited in Fig. 4.
Table 2. Material properties of Zerodur
Parameters
Value
Intensity, kg/m3
2500
Modulus of elasticity, GPa
91
Poisson ratio
0.24
Coefficient of linear expansion at 273 K, 10−6/K
0.05
Coefficient of linear expansion at 40 K, 10−6/K
−0.7
Thermal conductivity, W/MK−1
1.5
Specific heat capacity, J/kgK
8.2
Specific stiffness, 104 Nm/g
3.64
“Оптический журнал”, 81, 4, 2014
R 2ϕ
F Fig. 2. Diagram of steel-belt support structure.
33
R0 Linear slot
R1 R2
Fig. 5. Flexible connection design of support points.
Fig. 3. Schematic of back support points distribution.
Mirror
Support poin
X r
θ O
Y
Fig. 6. Calculation of rotation rigidity for flexible connection.
t
Belt
Fig. 4. 3D structure of combined belt-back support.
2.4. Thermal press discharge
Working environments of compounding system is changed from 300 K to 200 K. Coefficients of linear expansion between mirror material and support mechanism can’t be exact same. When working temperature is changed, stresses caused by different coefficients of materials will affect on the surface shape precision greatly. Next, we designed special structure to discharge the thermal stress. Flexible connection structure is an effective form of thermal stress discharge, which can reduce the stress influence of temperature variation [10].
For reflective mirror with Φ900 mm, we designed a flexible connection between mirror and
back support point, as shown in Fig. 5. The flexible connection has two advantages. First one is that the hollow design of support point can reduce the stress caused by different deformations. Second one is that the linear slot can discharge the thermal stress along deformation direction.
Flexible connection is designed by analyzing dependence of rotation rigidity on structure parameters. Calculation method of rotation rigidity for flexible connection is shown in Fig. 6, and expression of rotation rigidity is conveyed by following equation [10]:
M =1
π
ò
0
2r
12r sinα + t -2r sinα
dα,
(4)
where, r is radius of slot, and t is thinnest width of connection.
3. Finite element method and results
For compounding system, we concern the effect of steady uniformity temperature field on support structure. According to thermodynamics
34 “Оптический журнал”, 81, 4, 2014
Original Modeling 3D
structure
model
Finite Analysis
Correction
N
FEM model
If convergent?
End Y
N
Satisfy index?
Thermal analysis Forces analysis
Y Structure Parameter
Fig. 7. Optimize design flow of reflective mirror support.
theory, the strain caused by temperature varia-
tions is expressed by
{ }{ε}T = εx,εy,εz,εxy,εyz,εzx T = { }= α ∆Tx,∆Ty,∆Tz,0,0x,0 T ,
(5)
where, {ε}T is strain vector, ΔTx, ΔTy and ΔTz
are temperature variations of x, y and z direction, α is coefficient of heat transfer.
Relationship between stress and strain is expressed by following equation [11]:
{σ} = D×{ε}T ,
(6)
where, {σ} is stress vector and D is expressed by
D = (1-Eµ(1)(-1-µ)2µ)´
´ êëêêêêêêêêêêéêêêêêêêêêêêê11--1µµ000 µµ
µ 1-µ
1
µ 1-µ
0
0
0
µ 1-µ
µ 1-µ
1
0
0
0
0
0
0 1-2µ
2(1- µ)
0
0
0
0
0
0 1-2µ
2(1- µ)
0
21(1-00000-2µµ)úúúúúúúúúúùúúúûúúúúúúúúúú, (7)
where, μ is poisson ratio and E is modulus of elasticity.
Based on equations (1)−(4), we can figure out the original support structure and flexible connection and then optimize them by FEM. The optimization process is illustrated in Fig. 7. Firstly, construct the 3D support model in FEM software. Secondly, mesh the support model and set actual
boundary conditions and loads. Thirdly, calculate the adjustment value and direction based on precious results. Design, optimize and validate the structure again and again. Finally, assign the accuracy of elements and value of forces until satisfy requirements.
For high accuracy of analyzing results, we choose secondary 3D hypostatic unit for mesh generation, and final model is indicated in Fig. 8. We set high intensity meshes at connection area as revealed in Fig. 8.
FEM results are shown in Figs. 9, 10 and 11. As conveyed in Fig. 9, back support can maintain high precision when axis is vertical. Fig. 10 indicats surface property when axis is horizontal. PV value of surface is less than 0.209 μm and RMS value is 70 nm. Results show that combined belt-back support can achieve better performance for 1m mirror. Fig. 11 is FEM results of mirror under environment temperature of 200 K. PV values of surface shape is less than 0.33 μm and RMS value is better than λ/40. Comparing with Fig. 10, we can reach the conclusion that support system can maintain high surface shape when environment temperature was changed from 300 K to 200 K.
Fig. 8. FEM model of belt-back support.
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FEM data
Zernike polonomial fitting
CODEV
MTF Spot RMS
Fig. 12. Analysis flow of optical performances. Z YX
(а)
Fig. 9. Deformation fringe of reflective mirror with vertical axis when environment
temperature is 300 K.
Y ZX
(b)
Fig. 10. Deformation fringe of reflective mirror with horizontal axis when environment temperature is 300 K.
Y ZX
(c)
Fig. 11. Deformation fringe of reflective mirror with horizontal axis when environment temperature is 200 K.
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Fig. 13. MTF of multi-target compounding system under different conditions. Ideal condition (a), 300 K environment temperature (b), 200 K environment temperature (c).
“Оптический журнал”, 81, 4, 2014
4. Optical performances demonstration
Finally, according to output data of FEM, we examined the effect of surface shape deformation on the optical performances. Analysis process is indicated in Fig. 12. Firstly, we performed Zernike polynomial fitting for the mirror variation of discrete points outputted from FEM. Secondly, we translated the Zernike coefficients to optical design software to obtain the optical modulate transmission function (MTF) and RMS spot diameter.
Take the projector with max abaxial distance as example to determine the influence of surface precision on system resolution due to gravity and thermal effect. According to method in Fig. 12, we figure out the MTF plots of system under different working condition, as shown in
Table 3. RMS spot diameter of multi-target compounding system under different condi-
tions (mrad)
FOV, deg
−3.0
−1.5
0
1.5 3.0
Ideal 0.0361 0.0363 0.0362 0.0361 0.0366
300 K 0.0388 0.0392 0.0391 0.0399 0.0395
200 K 0.0498 0.0477 0.0488 0.0472 0.0494
Remark. FOV – field of view.
system requirement. Therefore, we can reach the conclusion that designed structure can maintain optical resolution of compounding system.
Conclusions Focusing on the reflective mirror with large
Fig. 13b and 13c. Fig. 13a is MTF plot for ideal aperture, high self-gravity and large temperature
condition. RMS spot diameters for all FOV are variation, we optimized the combined belt-back
shown in Table 3. Compared to ideal condition, support mechanism and designed flexible support
MTF and RMS spot diameter didn’t descend when structure, which maintain mirror surface shape
environment temperature is 300 K. This is due and system resolution. After optimization, RMS
to the fact that the optimized support mechanism can reduce the self-gravity effect on surface shape deformation effectively. For environment of 200 K, MTF of compounding system fall about 0.1 at maximum frequency. However, MTF is still over 0.5 at maximum frequency and RMS spot diameter is less 0.05 mrad, which satisfy
of surface deformation is less λ/40 for two working conditions. By Zernike polynomial fitting, MTF of system is over 0.5 and RMS spot diameter is less than 0.05 mrad. Results demonstrated that proposed support mechanism is reasonable for large environment temperature variations of 1 m level reflective mirror.
* * * * *
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