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Radial support analysis for large-aperture rotating wedge prism
Optics & Laser Technology 44 (2012) 1881–1888
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Radial support analysis for large-aperture rotating wedge prism Anhu Li a,b,n, Xuchun Jiang a, Jianfeng Sun b, Yongming Bian a, Lijuan Wang b, Liren Liu b a b
College of Mechanical Engineering, Tongji University, 4800 Caoan Road, Shanghai 201804, China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, PO Box 800-211, Shanghai 201800, China
a r t i c l e i n f o
abstract
Article history: Received 10 December 2011 Received in revised form 19 January 2012 Accepted 19 January 2012 Available online 21 February 2012
The different radial support ways for a rotation prism are presented to evaluate the strains and stresses induced by the gravity action and the thermal effects. Due to the large-aperture, non-uniform quality distribution and especially rotation from 01 to 3601, neither the multi-point support way nor the full surface-contact support can well meet the assembly requirements. The paper proposes a novel project of an adjustable partial surface-contact support way to solve this support problem. The maximum strain and stress under this support way both are less than the tolerance limit of the strength assurance regardless of the prism rotating to any position. Meanwhile, the thermally-induced structural analysis is performed to predict some specific work cases, which is much valuable for the improvement of the assembly design. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Prism Radial support Finite element analysis
1. Introduction In the field of inter-satellite communication, how to accurately steer the beam direction is one of the critical issues for the communication link [1,2]. In general, the method of steering the beam deviation includes acousto-optical modulation, electrooptical modulation, and opto-mechanical method. The optomechanical methods usually can be designed as some reflective or refractive optical scanning systems. In comparison with other optical scanning modes, the refraction method usually features a compact structure with high accuracy [3–5]. In our previous research a fine beam scanning device is presented through the refraction project of rotating a pair of circular wedge prisms. Two wedge prisms, with the same geometric parameters, both rotate around the horizontal optical axis to achieve the scanning range 7101 and the scanning precision superior to 50 mrad [6,7]. The scanning mirror, as a key component to perform the scanning motion, is usually mounted by the radial supports. Due to the influence of the gravity action and the temperature change, the deformation of the mirror has a great impact on the beam scanning accuracy, so it is necessary to design a reasonable supporting project for the optical component [8, 9]. For an optical component with a small size, it can be mounted by a semikinematic way or multi-point installation [10]. However, for a large size mirror, especially a small ratio of width to diameter mirror, the influences of gravity and temperature gradient must
n Corresponding author at. Tongji University, College of Mechanical Engineering, 4800 Caoan Road, Shanghai 201804, China E-mail address: yfl[email protected] (A. Li).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2012.01.026
be evaluated early in the support design. Many researchers have developed some effective methods to solve this problem. Vukobratovich [11] used roller chain supports for a large mirror with a horizontal axis and proved its superiority–Malvick [12] analyzed the elastic deformation of two large mirrors due to the weight influence, and proposed an optimization combination of band and point support–Salas [13] used 18 air bags as the supports for the suspension of the primary mirror in the 2.1 m telescope and analyzed the control problems. In general, the radial supporting methods for an optical component include two types of point and surface contact support ways. The point-contact supporting way can be carried out through the metal or plastic material support points, such as bars agencies, connecting rod and plastic ball screw [14], which are reliable, easy to install and adjustable. The surface-contact support ways usually use mercury tube, steel belt and roller chain as the supports [15]. Compared to the point-contact support ways, the surface-contact support ways have less surface deformation and stress concentration especially for a large size mirror. Based on the above research, this paper focuses on the deformation analysis of a prism assembly in a beam scanner proposed in our previous research [7]. The prism is characterized with large diameter, nun-uniform quality distribution, and especially continuously rotating from 01 to 3601. In order to establish an effective support way, we must redesign and evaluate the possible point or surface contact support ways. The paper is organized as follows: first the different combinations of multipoint support are discussed, then the full surface-contact support way is studied, and finally a novel project of an adjustable partial surface-contact support is proposed to predict the optical
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performance changes under the gravity action and the thermal effects.
2. Description of the model In the development of a large-aperture scanner implemented by a pair of rotating prisms, the deformation of the prism is one of the important error factors of affecting the beam direction. The geometric parameters of the presented prism are as follows: diameter F ¼550 mm, wedge angle a ¼101, and thin end thickness D ¼30 mm, which needs to rotate around a horizontal optical axis from 01 to 3601. The assembly consists of a wedge prism, a wedge-shaped block and an inner cell. The prism, together with a wedge-shaped block with a wedge angle 101, is mounted inside the inner cell. In the paper, In order to improve the optical system performance, we respectively discuss the axial deformation and stress concentration of the prism induced by the self-weight under the point or surface contact support ways, as well as the thermal stress effects caused by the internal temperature gradient. The evaluation indexes are PV value (peak valley) and RMS value (root mean square). The component materials are listed in Table 1.
The analysis results of the surface deformation are shown in Table 2. The surface deformation effects under eight-point support are greatly improved compared to the three-point support way. If we take a wavelength l ¼632.8 nm (the same below), according to the Rayleigh criterion, the maximum axial deformation under two support ways both are less than l/4. Fig. 2, respectively gives the equivalent Von Mises stress under two support ways. The allowable stress of K9 glass is 0.343 MPa, while
3. Different support way analysis 3.1. Point-contact support way There are various point-contact support forms possibly served for a large prism. The effective optimized combinations of multipoint contact supports can well minimize the strain and stress values for the same model. In this paper we only select threepoint support and eight-point support as two special cases for analysis, as shown in Fig. 1. We suppose the thin end of the prism upward as 01 position (the same below). Three-point support way is shown in Fig. 1(a). Three supporting points are located in the same section of d1 ¼15 mm away from the plane side of the prism, two subjacent supporting points of which are symmetrical layout with an angle l1 ¼1201, and one upper supporting point is preloaded by F¼ 9.8 N. The project of eight-point support is shown in Fig. 1(b). Four points are placed in the same section of d2 ¼12 mm away from the plane side, and the separation angle among which is l2 ¼451 respect to the horizontal centerline; two points are located in the same section of d3 ¼42 mm away from the plane side, symmetrically arranged about the vertical centerline; two points, close to the thick end of the prism, are respectively arranged on the section of d4 ¼30 mm and d5 ¼89 mm away from the plane side; four support points of them are added to the preload force F ¼9.8 N. In the finite element modeling, the nylon screw heads are used as supporting points, and the screw heads are simplified as the curved boxes of a certain thickness and similar size, which flexibly contact with the prism through the MPC (multi-point constraint) algorithm in software Ansys. The prism and curved boxes all are meshed by Solid95 unit grid with 20 nodes.
Fig. 1. Point support ways (a) three-point support, (b) eight-point support.
Table 2 Comparison between three-point and eight-point support. Terms
Three-point
PV (nm) RMS (nm)
Eight-point
Plane side
Wedge side
Plane side
Wedge side
73.542 26.826
46.545 27.704
24.234 4.898
14.395 5.848
Table 1 Material performance. Material
Prism Plastic screw, arc gasket Inner cell, wedge-shape block
K9 glass Nylon 66 45 steel
Density r (kg m 3)
Elastic modulus E (GPa)
m
Linear expansion coefficient a (1C)
Coefficient of thermal conductivity (W (m K) 1)
2530 1050 7800
81.32 28.3 196
0.209 0.4 0.24
7.5E 6 8.0E 6 11E 6
1.207 0.27 48
Poisson ratio,
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Fig. 3. Finite element model of the prism assembly under the full surface-contact support.
Fig. 2. Von Mises stress in the prism (a) three-point support, (b) eight-point support.
the maximum Von Mises stress 1.20 MPa and 0.424 MPa, respectively under the three-point support way and the eight-point supporting way both are greater than the allowable stress value. Overall, the multi-point contact support ways can meet the tolerance of wave phase error, but it is difficult to meet the strength requirements of the prism. 3.2. Surface-contact support way 3.2.1. Full surface-contact support The assembly model is described in Section 2. Due to the mode symmetry about the principal cross-section of the prism, only a half model is built in order to simplify the solution process, as shown in Fig. 3. In the actual assembly, the components of the prism and the wedge-shaped block, mounted insider a inner cell, is axially fixed by the screws on the circumference, and the inner cell is constrained along the radial direction by a pair of bearings, which are simplified to the rigid constraints in the finite element model. However, the contact relations among the prism, the wedge-shape block and the inner cell all are simulated as the flexible surface contact through the MPC algorithm in software Ansys.
Fig. 4. Strain contour chart along the Z direction (a) in the plane side, (b) in the wedge side.
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Table 3 PV values and RMS values of the axial deformation in two sides. Plane side (nm)
PV RMS
Wedge side (nm)
F ¼550 mm
F1 ¼450 mm
F ¼ 550 mm
F1 ¼ 450 mm
7.343 3.672
5.889 4.236
7.651 3.609
6.646 4.164
Fig. 6. Variation curves of PV and RMS values in the plane side during the prism rotation from 01 to 3601 around the optical axis.
Fig. 5. Von Mises stress in the prism under the full surface-contact support.
Fig. 4 shows the axial strain contours of the plane side and the wedge side in the prism. Due to the nun-uniform quality distribution in two ends of the wedge prism, the maximum strain occurs in the central region, and the minimum strain is found in the bottom near the support points. We take a circular area F1 ¼450 mm as the clear aperture within the full diameter, and respectively calculate the PV and RMS values of the axial deformation in the plane side and the wedge side. The analysis results are given in Table 3. The equivalent Von Mises stress of the prism under the full surface-contact support is viewed in Fig. 5. The maximum stress is 0.0302 MPa, which can meet the strength requirement of K9 glass. According to the above analysis, the axial deformation in the plane side of the prism is PV¼ 0.0116l and RMS¼0.00580l, and the axial deformation in the wedge side of the prism is PV¼0.0121l and RMS¼0.00570l. Within the clear aperture of F1 ¼450 mm, the axial deformation in the plane side is PV¼0.00931l and RMS¼0.00669l, and the axial deformation in the wedge side is PV¼0.0105l and RMS¼0.00658l. These PV values all are less than l/4 according to the Rayleigh criterion. The maximum stress is 0.0302 MPa, far less than the allowable stress of the prism. Compared with the point support means, the surface-contact supporting way can be a good solution to effectively reduce the stress concentration and the deformation in the prism.
whole prism during the rotation process. We use the parametric design language and macro operation in software Ansys to analyze the entire rotation process. First, we set a start position and a rotation step of the prism to study the forced forms in each position, and export the corresponding calculated results. Then we read the result files into a data processing software tool such as Matlab, and iteratively take the PV and RMS values of the surface deformation during the prism rotation process for curve fitting, which are shown in Fig. 6. According to the analysis results, the PV and RMS values occurring in the plane side are both close to symmetry about the 1801 position, which is due to the computational error of ANSYS software. When the prism rotates to 01 position, that is, the thin side upward, the PV and RMS value both reach to the maximum value, respectively close to 7.5 nm and 4 nm. When the prism rotates to 1801 position, that is, the thick side upward, the PV and RMS value both reach the minimum value, respectively close to 2 nm and 0.5 nm.
4. Partial surface-contact support For the prism with large diameter, non-uniform quality distribution and rotation motion, the traditional point support form can not meet the strength requirement, while the full surfacecontact support way can obtain very good support effects in theory, but which is difficult to be carried out. In fact, under the full surface-contact support way, in order to meet the machining and assembly requirements, the inner cell diameter usually margins a clearance of d ¼0.5 mm over the prism diameter, and the contact relation between the prism and the inner cell is partial and changeable during the course of the prism rotation. Therefore, the decentration situation between the prism and the inner cell often occurs, and the jumpiness is inevitable, which will have a certain extent influence on the scanning accuracy of the device. 4.1. Support design
3.2.2. Rotation process analysis When the prism periodically rotates round the optical axis from 01 to 3601, due to the misalignment between the gravity center of the prism assembly and the rotation axis, the rotation of the gravity center will cause the changes of supporting forces. As a result, it will change the strain and stress distribution in the
In order to avoid the stress concentration and eliminate the beating movement during the prism rotation, it is necessary to design a stable surface-contact support way. A novel project is proposed to support the prism by two curved nylon gaskets and two curved elastic steel sheets, as shown in Fig. 7. One end of the
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supports, and subjects to the support force F arising from two elastic steel sheets. The equivalent rigid constraints are added along the axial direction. The finite element model is shown in Fig. 8. In order to evaluate the axial strain and the overall stress in the prism, we extract the axial strain contours in two sides of the prism, as shown in Fig. 9. The maximum strain occurs in the central region, and the minimum strain is found in the bottom near the support points, which conforms to the above analysis in Section 3.2.1 . In Table 4, the PV and RMS values of the axial deformation are given within the clear aperture of 450 mm.
Fig. 7. The adjustable partial surface-contact support way through a combination of elastic steel sheets and nylon gaskets.
Fig. 8. Finite element model under the adjustable partial surface-contact support.
curved elastic steel sheet is fixed in the arc slot of the inner cell, and the other end is floated, its length l ¼100 mm, width b¼50 mm, and thickness h ¼5 mm. The curved nylon gaskets is mounted on the suspended end of the elastics steel sheet, its length l0 ¼50 mm, width b0 ¼ 50 mm, and thickness h0 ¼5 mm. Two nylon gaskets are arranged near the thick end of the prism and symmetrically distributed along the vertical centerline of the prism, the separation angle of which is l3 ¼901. The size selection and layout style are according to the actual situation. If no load exists on the combination of elastic steel sheets and nylon gaskets, there is a clearance between the bottom surface of suspended end of the elastic steel sheet and the upside of the arc slot cut in the inner cell, the size of which is 1 mm. The upside surface of the nylon gasket protrudes 0.5 mm above the inner surface of the inner cell. The suspended end of the elastic steel sheet can be adjusted by some adjustable bolts in order to change the preloads.
Fig. 9. The strain contour chart along the Z direction (a) in the plane side, (b) in the wedge side.
Table 4 The PV values and RMS values of axial deformation in the prism surfaces. Plane side
4.2. Strain and stress analysis According to the above mount design, we select a special support case for analysis when the prism locates in the 01 position, where the prism only relies on two nylon gasket
PV (nm) RMS (nm)
Wedge side
F ¼ 550 mm
F1 ¼450 mm
F ¼ 550 mm
F1 ¼450 mm
37.023 18.472
32.191 21.067
40.950 18.144
27.362 20.743
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Fig. 10. Von Mises stress contour of the prism.
Fig. 10 illuminates the Von Mises stress on both sides of the prism. The stress concentration occurs at the support positions of the nylon gaskets, where the maximum stress is 0.273 MPa. According to the above results, the axial deformation in the plane side of the prism is PV¼ 0.0232l and RMS¼0.0123l, and the axial deformation in the wedge side is PV¼0.0359l and RMS¼ 0.0115l. Within the area of the clear aperture 450 mm, the PV value and the RMS value is, respectively, 0.0198l and 0.0140l in the plane side, and the PV value and RMS value is, respectively, 0.0295l and 0.0131l in the wedge side, both of which can meet the requirements of the Rayleigh criterion. The maximum stress is 0.273 MPa, also less than the allowable stress of the K9 glass. Similarly, the different deformation situations are further investigated when the prism rotates from 01 to 3601. Fig. 11 illuminates the support forms when the prism rotates to two special positions of 901 and 1801. When the prism rotates to the 01 position, it mainly relies on the direct support of the inner cell. When the prism rotates to the 901 position, it collaboratively relies on the support of the inner cell and two nylon gaskets, and at this time the 0.5 mm inter-space between the elastic steel sheets and the inner cell can be eliminated through adjusting the preload F, supposed F¼ 12 N. When the prism rotates to the 1801 position, the clearance 0.5 mm of two nylon gaskets above the upside of the inner cell just fills the inter-space between the prism and the inner cell, and at this time there is no preload in two elastic steel sheets, and the prism is supported only by the inner cell. Due to the same cases at the 901 and 2701 position, only two special work cases are selected to be discussed, the prism located at 901 and 1801 position. These results are shown in Table 5. According to the analysis above, when the prism rotates from 01 to 3601, the strains and stresses generated from the variable support ways all can meet the application requirement. Moreover, the scalability of this kind of a prism is extensively studied under the adjustable partial surface-contact support approach, which is validated to be a good solution to install a prism or mirror with a diameter over 400 mm and non-uniform quality distribution. 4.3. Thermal-structural coupling analysis In fact, the scanning device may be locates at a thermal environment. Due to the low thermal conductivity of optical glass and the non-uniform heat exchange, the thermal stress generated
Fig. 11. Different rotation angles under the adjustable partial surface-contact support. (a) prism at 901, (b) prism at 1801.
Table 5 The axial deformation and Von Mises stress when the prism rotates to 901and 1801. PV (nm)
RMS (nm)
Maximum Von Mises stress (MPa)
Plane side Wedge side Plane side Wedge side 901 6.359 1801 21.303
9.340 21.526
2.455 9.848
1.691 10.607
0.172 0.301
inside the mirror will lead to the uneven thermal expansion, which may be deforms the prism surface and reduces the accuracy of the scanning device [16, 17]. Here we only focus on the optical performance change typically induced by the temperature gradient inside the mirror. We take a special case of the prism locating at 01 position for analysis. The flow of the thermalstructural coupling analysis is as follows. First, the three-dimensional thermal analysis unit Solid90 is adopted corresponding to Solid95 used in structural analysis and perform the steady-state thermal analysis. Setting the environmental reference temperature T¼ 20 1C, the temperature Tþ DT¼20.05 1C is added at the plane side, where DT is the
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Fig. 12. Temperature gradient inside the prism.
temperature change value. As a result, the internal temperature gradient in the prism is shown in Fig. 12. The Ansys command ‘‘ETCHG’’ is used to transfer the threedimensional thermal unit Solid90 into the structural unit Solid95, and then he contact relations and boundary conditions are add into the finite element model. After the results of thermal analysis are input into the structural analysis, the thermal-structural coupling analysis can be further executed. The deformation contour of the wedge side and the Von Mises stress distribution in the whole prism are, respectively extracted to be viewed in Fig. 13. According to the above steps, the deformation and stress are, respectively solved with the incremental temperature change DT, 0.01 1C, 0.02 1C, 0.05 1C and 0.10 1C. Table 6 lists the surface deformation values of the prism in the full diameter 550 mm and the clear aperture 450 mm. When the temperature change is 0.05 1C, the deformation value of the plane side within the clear aperture is PV¼0.25l and RMS¼0.248l, while that of the wedge side is PV¼0.255l and RMS¼0.226, which are close to or over the admissible requirement according to Rayleigh criterion. Obviously, when the scanning device locates at an adiabatic environment, it can avoid the thermal effects, but which is impossible in practice [18, 19]. So it is essential to take some effective temperature-control measures for the mitigation of thermal effects on the device.
5. Conclusions For a specific prism assembly with large-diameter, non-uniform quality distribution and continuously rotating around the horizontal optical axis, the traditional supporting ways are difficult to meet the actual requirements. This paper discusses the advantages and disadvantages of various radial support ways including multi-point and surface-contact supports. As a result, an adjustable partial surface support project is proposed to solve this problem, which is constituted by two pairs of elastic steel
Fig. 13. Thermal analysis (a) deformation contour in the wedge side, (b) Von Mises stress.
Table 6 PV values (nm) and RMS values (nm) of the plane side and the wedge side.
Plane side
PV RMS
Wedge side
PV RMS
F ¼550 mm F1 ¼ 450 mm F ¼550 mm F1 ¼ 450 mm F ¼550 mm F1 ¼ 450 mm F ¼550 mm F1 ¼ 450 mm
0.01 1C
0.02 1C
0.05 1C
0.1 1C
71.51 57.35 42.05 48.04 77.31 53.59 39.36 45.09
106.20 82.49 65.81 75.17 111.53 80.03 60.43 69.49
218.15 158.20 137.24 156.70 214.32 161.25 124.47 142.73
405.46 284.88 256.37 292.67 385.50 296.79 230.92 264.83
sheets and nylon gaskets. By adjusting the preloads of the elastic steel sheets, the inter-space between the inner cell and the prism can be effectively filled to ensure the stability of the assembly. According to the analysis results, the strains and stresses under the variable support ways all are within the strength admission of K9 glass regardless of the prism rotating to any position. Under the adjustable partial surface support method, the thermallyinduced structural analysis is also performed to study the surface deformation of the prism induced by the axial temperature gradient. The calculated results indicate that the temperature change has a great influence on the prism surface shape, and
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some effective control measures must be taken into consideration, which is essential for the system application. Moreover, further study shows the proposed support method is validated for the scalability of this kind of a prism with a diameter over 400 mm and non-uniform quality distribution, which can be widely used into the optical scanning and tracking test fields.
Acknowledgment This work is funded by National Natural Science Foundation of China (No.50805107, No.60807024) and also supported by key laboratory of space laser communication and testing technology, Chinese Academy of Sciences. References [1] Wang LJ, Liu LR, Luan Z, Sun JF, Zhou Y, Liu DA. The mechanical design of the large-optics double-shearing interferometer for the test of the diffractionlimited wave front. Proceedings of SPIE 2008. 7091: 70910S1-7. [2] Wan LY, Li AH, Wang LJ, Luan Z, Liu LR. Design of an optical testbed for in-lab testing and validation for the intersatellite lasercom terminals. Proceedings of SPIE 2005;5892:549–54. [3] Virgil-Florin Duma, Mirela Nicolov. Neutral density filters with Risley prisms: analysis and design. Applied Optics 2009;48:2678–85. [4] Booth DT, Cox Samuel, Robert Berryman. Precision measurements from verylarge scale aerial digital imagery. Environmental Monitoring and Assessment 2006;112:293–307. [5] Sirohi RS, Kothiyal MP. Double wedge plate shearing interferometer for collimation test. Applied Optics 1987;26:4054–6.
[6] Li AH, Liu LR, Sun JF, Zhong XH, Wang LJ, Liu DA, Luan Z. Research on a scanner for tilting orthogonal double prisms. Applied Optics 2006;45: 8063–9. [7] Sun JF, Liu LR, Yun MJ, Wan LY, Zhang ML. The effect of the rotating doubleprism wide-angle laser beam scanner on the beam shape. Optik 2005;116: 553–6. [8] Ferna´ndez EJ, Artal P. Membrane deformable mirror for adaptive optics: performance limits in visual optics. Optics Express 2003;11:1056–69. [9] Vukobratovich D, Fetterhoff KA, Myers JR, Wheelwright PD, Cunnington GR. Bonded mount for small cryogenic optics. Proceedings of SPIE 2000;4131: 228–39. [10] Hatheway AE. Mountings for a four meter glass mirror. Proceedings of SPIE 1990;1303:142–7. [11] Vukobratovich D, Richard RM. Roller chain supports for large optics. Proceedings of SPIE 1991;1396:522–34. [12] Malvick AJ. Theoretical elastic deformations of the Steward Observatory 230-cm and the science center 154-cm mirrors. Applied Optics 1972;11:575–85. [13] Salas L, Gutierrez L, Pedrayes MH. Active primary mirror support for the 2.1-m telescope at the San Pedro Martir Observatory. Applied Optics 1997; 36:3708–16. [14] Malvick AJ. Dynamic relaxation: a general method for determination of elastic deformation of mirrors. Applied Optics 1968;7:2117–21. [15] Gong XF, Cui XQ. Research on mirror lateral support of large astronomical telescope. Proceedings of SPIE 2006. 6148: 61480Y1-6. [16] Li AH, Liu LR, Sun JF, Zhu YJ, Shen QD. Finite element analysis on a circular wedge prism with 400 mm diameter. Optik 2007;118:589–93. [17] Segato E, Da Deppo V, Debei S, Naletto G, Cremonese G, Flamini E. A method for studying the effects of thermal deformations on optical systems for space application. Proceedings of SPIE 2010:769101–4. [18] Giesen P, Folgering E. Design guidelines for thermal stability in optomechanical engineering. Proceedings of SPIE 2003;5176:126–34. [19] Yoder Jr. PR. Improved semi-kinematic mountings for prisms. Proceedings of SPIE 2002;4771:173–9.
Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Radial support analysis for large-aperture rotating wedge prism Anhu Li a,b,n, Xuchun Jiang a, Jianfeng Sun b, Yongming Bian a, Lijuan Wang b, Liren Liu b a b
College of Mechanical Engineering, Tongji University, 4800 Caoan Road, Shanghai 201804, China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, PO Box 800-211, Shanghai 201800, China
a r t i c l e i n f o
abstract
Article history: Received 10 December 2011 Received in revised form 19 January 2012 Accepted 19 January 2012 Available online 21 February 2012
The different radial support ways for a rotation prism are presented to evaluate the strains and stresses induced by the gravity action and the thermal effects. Due to the large-aperture, non-uniform quality distribution and especially rotation from 01 to 3601, neither the multi-point support way nor the full surface-contact support can well meet the assembly requirements. The paper proposes a novel project of an adjustable partial surface-contact support way to solve this support problem. The maximum strain and stress under this support way both are less than the tolerance limit of the strength assurance regardless of the prism rotating to any position. Meanwhile, the thermally-induced structural analysis is performed to predict some specific work cases, which is much valuable for the improvement of the assembly design. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Prism Radial support Finite element analysis
1. Introduction In the field of inter-satellite communication, how to accurately steer the beam direction is one of the critical issues for the communication link [1,2]. In general, the method of steering the beam deviation includes acousto-optical modulation, electrooptical modulation, and opto-mechanical method. The optomechanical methods usually can be designed as some reflective or refractive optical scanning systems. In comparison with other optical scanning modes, the refraction method usually features a compact structure with high accuracy [3–5]. In our previous research a fine beam scanning device is presented through the refraction project of rotating a pair of circular wedge prisms. Two wedge prisms, with the same geometric parameters, both rotate around the horizontal optical axis to achieve the scanning range 7101 and the scanning precision superior to 50 mrad [6,7]. The scanning mirror, as a key component to perform the scanning motion, is usually mounted by the radial supports. Due to the influence of the gravity action and the temperature change, the deformation of the mirror has a great impact on the beam scanning accuracy, so it is necessary to design a reasonable supporting project for the optical component [8, 9]. For an optical component with a small size, it can be mounted by a semikinematic way or multi-point installation [10]. However, for a large size mirror, especially a small ratio of width to diameter mirror, the influences of gravity and temperature gradient must
n Corresponding author at. Tongji University, College of Mechanical Engineering, 4800 Caoan Road, Shanghai 201804, China E-mail address: yfl[email protected] (A. Li).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2012.01.026
be evaluated early in the support design. Many researchers have developed some effective methods to solve this problem. Vukobratovich [11] used roller chain supports for a large mirror with a horizontal axis and proved its superiority–Malvick [12] analyzed the elastic deformation of two large mirrors due to the weight influence, and proposed an optimization combination of band and point support–Salas [13] used 18 air bags as the supports for the suspension of the primary mirror in the 2.1 m telescope and analyzed the control problems. In general, the radial supporting methods for an optical component include two types of point and surface contact support ways. The point-contact supporting way can be carried out through the metal or plastic material support points, such as bars agencies, connecting rod and plastic ball screw [14], which are reliable, easy to install and adjustable. The surface-contact support ways usually use mercury tube, steel belt and roller chain as the supports [15]. Compared to the point-contact support ways, the surface-contact support ways have less surface deformation and stress concentration especially for a large size mirror. Based on the above research, this paper focuses on the deformation analysis of a prism assembly in a beam scanner proposed in our previous research [7]. The prism is characterized with large diameter, nun-uniform quality distribution, and especially continuously rotating from 01 to 3601. In order to establish an effective support way, we must redesign and evaluate the possible point or surface contact support ways. The paper is organized as follows: first the different combinations of multipoint support are discussed, then the full surface-contact support way is studied, and finally a novel project of an adjustable partial surface-contact support is proposed to predict the optical
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performance changes under the gravity action and the thermal effects.
2. Description of the model In the development of a large-aperture scanner implemented by a pair of rotating prisms, the deformation of the prism is one of the important error factors of affecting the beam direction. The geometric parameters of the presented prism are as follows: diameter F ¼550 mm, wedge angle a ¼101, and thin end thickness D ¼30 mm, which needs to rotate around a horizontal optical axis from 01 to 3601. The assembly consists of a wedge prism, a wedge-shaped block and an inner cell. The prism, together with a wedge-shaped block with a wedge angle 101, is mounted inside the inner cell. In the paper, In order to improve the optical system performance, we respectively discuss the axial deformation and stress concentration of the prism induced by the self-weight under the point or surface contact support ways, as well as the thermal stress effects caused by the internal temperature gradient. The evaluation indexes are PV value (peak valley) and RMS value (root mean square). The component materials are listed in Table 1.
The analysis results of the surface deformation are shown in Table 2. The surface deformation effects under eight-point support are greatly improved compared to the three-point support way. If we take a wavelength l ¼632.8 nm (the same below), according to the Rayleigh criterion, the maximum axial deformation under two support ways both are less than l/4. Fig. 2, respectively gives the equivalent Von Mises stress under two support ways. The allowable stress of K9 glass is 0.343 MPa, while
3. Different support way analysis 3.1. Point-contact support way There are various point-contact support forms possibly served for a large prism. The effective optimized combinations of multipoint contact supports can well minimize the strain and stress values for the same model. In this paper we only select threepoint support and eight-point support as two special cases for analysis, as shown in Fig. 1. We suppose the thin end of the prism upward as 01 position (the same below). Three-point support way is shown in Fig. 1(a). Three supporting points are located in the same section of d1 ¼15 mm away from the plane side of the prism, two subjacent supporting points of which are symmetrical layout with an angle l1 ¼1201, and one upper supporting point is preloaded by F¼ 9.8 N. The project of eight-point support is shown in Fig. 1(b). Four points are placed in the same section of d2 ¼12 mm away from the plane side, and the separation angle among which is l2 ¼451 respect to the horizontal centerline; two points are located in the same section of d3 ¼42 mm away from the plane side, symmetrically arranged about the vertical centerline; two points, close to the thick end of the prism, are respectively arranged on the section of d4 ¼30 mm and d5 ¼89 mm away from the plane side; four support points of them are added to the preload force F ¼9.8 N. In the finite element modeling, the nylon screw heads are used as supporting points, and the screw heads are simplified as the curved boxes of a certain thickness and similar size, which flexibly contact with the prism through the MPC (multi-point constraint) algorithm in software Ansys. The prism and curved boxes all are meshed by Solid95 unit grid with 20 nodes.
Fig. 1. Point support ways (a) three-point support, (b) eight-point support.
Table 2 Comparison between three-point and eight-point support. Terms
Three-point
PV (nm) RMS (nm)
Eight-point
Plane side
Wedge side
Plane side
Wedge side
73.542 26.826
46.545 27.704
24.234 4.898
14.395 5.848
Table 1 Material performance. Material
Prism Plastic screw, arc gasket Inner cell, wedge-shape block
K9 glass Nylon 66 45 steel
Density r (kg m 3)
Elastic modulus E (GPa)
m
Linear expansion coefficient a (1C)
Coefficient of thermal conductivity (W (m K) 1)
2530 1050 7800
81.32 28.3 196
0.209 0.4 0.24
7.5E 6 8.0E 6 11E 6
1.207 0.27 48
Poisson ratio,
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Fig. 3. Finite element model of the prism assembly under the full surface-contact support.
Fig. 2. Von Mises stress in the prism (a) three-point support, (b) eight-point support.
the maximum Von Mises stress 1.20 MPa and 0.424 MPa, respectively under the three-point support way and the eight-point supporting way both are greater than the allowable stress value. Overall, the multi-point contact support ways can meet the tolerance of wave phase error, but it is difficult to meet the strength requirements of the prism. 3.2. Surface-contact support way 3.2.1. Full surface-contact support The assembly model is described in Section 2. Due to the mode symmetry about the principal cross-section of the prism, only a half model is built in order to simplify the solution process, as shown in Fig. 3. In the actual assembly, the components of the prism and the wedge-shaped block, mounted insider a inner cell, is axially fixed by the screws on the circumference, and the inner cell is constrained along the radial direction by a pair of bearings, which are simplified to the rigid constraints in the finite element model. However, the contact relations among the prism, the wedge-shape block and the inner cell all are simulated as the flexible surface contact through the MPC algorithm in software Ansys.
Fig. 4. Strain contour chart along the Z direction (a) in the plane side, (b) in the wedge side.
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Table 3 PV values and RMS values of the axial deformation in two sides. Plane side (nm)
PV RMS
Wedge side (nm)
F ¼550 mm
F1 ¼450 mm
F ¼ 550 mm
F1 ¼ 450 mm
7.343 3.672
5.889 4.236
7.651 3.609
6.646 4.164
Fig. 6. Variation curves of PV and RMS values in the plane side during the prism rotation from 01 to 3601 around the optical axis.
Fig. 5. Von Mises stress in the prism under the full surface-contact support.
Fig. 4 shows the axial strain contours of the plane side and the wedge side in the prism. Due to the nun-uniform quality distribution in two ends of the wedge prism, the maximum strain occurs in the central region, and the minimum strain is found in the bottom near the support points. We take a circular area F1 ¼450 mm as the clear aperture within the full diameter, and respectively calculate the PV and RMS values of the axial deformation in the plane side and the wedge side. The analysis results are given in Table 3. The equivalent Von Mises stress of the prism under the full surface-contact support is viewed in Fig. 5. The maximum stress is 0.0302 MPa, which can meet the strength requirement of K9 glass. According to the above analysis, the axial deformation in the plane side of the prism is PV¼ 0.0116l and RMS¼0.00580l, and the axial deformation in the wedge side of the prism is PV¼0.0121l and RMS¼0.00570l. Within the clear aperture of F1 ¼450 mm, the axial deformation in the plane side is PV¼0.00931l and RMS¼0.00669l, and the axial deformation in the wedge side is PV¼0.0105l and RMS¼0.00658l. These PV values all are less than l/4 according to the Rayleigh criterion. The maximum stress is 0.0302 MPa, far less than the allowable stress of the prism. Compared with the point support means, the surface-contact supporting way can be a good solution to effectively reduce the stress concentration and the deformation in the prism.
whole prism during the rotation process. We use the parametric design language and macro operation in software Ansys to analyze the entire rotation process. First, we set a start position and a rotation step of the prism to study the forced forms in each position, and export the corresponding calculated results. Then we read the result files into a data processing software tool such as Matlab, and iteratively take the PV and RMS values of the surface deformation during the prism rotation process for curve fitting, which are shown in Fig. 6. According to the analysis results, the PV and RMS values occurring in the plane side are both close to symmetry about the 1801 position, which is due to the computational error of ANSYS software. When the prism rotates to 01 position, that is, the thin side upward, the PV and RMS value both reach to the maximum value, respectively close to 7.5 nm and 4 nm. When the prism rotates to 1801 position, that is, the thick side upward, the PV and RMS value both reach the minimum value, respectively close to 2 nm and 0.5 nm.
4. Partial surface-contact support For the prism with large diameter, non-uniform quality distribution and rotation motion, the traditional point support form can not meet the strength requirement, while the full surfacecontact support way can obtain very good support effects in theory, but which is difficult to be carried out. In fact, under the full surface-contact support way, in order to meet the machining and assembly requirements, the inner cell diameter usually margins a clearance of d ¼0.5 mm over the prism diameter, and the contact relation between the prism and the inner cell is partial and changeable during the course of the prism rotation. Therefore, the decentration situation between the prism and the inner cell often occurs, and the jumpiness is inevitable, which will have a certain extent influence on the scanning accuracy of the device. 4.1. Support design
3.2.2. Rotation process analysis When the prism periodically rotates round the optical axis from 01 to 3601, due to the misalignment between the gravity center of the prism assembly and the rotation axis, the rotation of the gravity center will cause the changes of supporting forces. As a result, it will change the strain and stress distribution in the
In order to avoid the stress concentration and eliminate the beating movement during the prism rotation, it is necessary to design a stable surface-contact support way. A novel project is proposed to support the prism by two curved nylon gaskets and two curved elastic steel sheets, as shown in Fig. 7. One end of the
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supports, and subjects to the support force F arising from two elastic steel sheets. The equivalent rigid constraints are added along the axial direction. The finite element model is shown in Fig. 8. In order to evaluate the axial strain and the overall stress in the prism, we extract the axial strain contours in two sides of the prism, as shown in Fig. 9. The maximum strain occurs in the central region, and the minimum strain is found in the bottom near the support points, which conforms to the above analysis in Section 3.2.1 . In Table 4, the PV and RMS values of the axial deformation are given within the clear aperture of 450 mm.
Fig. 7. The adjustable partial surface-contact support way through a combination of elastic steel sheets and nylon gaskets.
Fig. 8. Finite element model under the adjustable partial surface-contact support.
curved elastic steel sheet is fixed in the arc slot of the inner cell, and the other end is floated, its length l ¼100 mm, width b¼50 mm, and thickness h ¼5 mm. The curved nylon gaskets is mounted on the suspended end of the elastics steel sheet, its length l0 ¼50 mm, width b0 ¼ 50 mm, and thickness h0 ¼5 mm. Two nylon gaskets are arranged near the thick end of the prism and symmetrically distributed along the vertical centerline of the prism, the separation angle of which is l3 ¼901. The size selection and layout style are according to the actual situation. If no load exists on the combination of elastic steel sheets and nylon gaskets, there is a clearance between the bottom surface of suspended end of the elastic steel sheet and the upside of the arc slot cut in the inner cell, the size of which is 1 mm. The upside surface of the nylon gasket protrudes 0.5 mm above the inner surface of the inner cell. The suspended end of the elastic steel sheet can be adjusted by some adjustable bolts in order to change the preloads.
Fig. 9. The strain contour chart along the Z direction (a) in the plane side, (b) in the wedge side.
Table 4 The PV values and RMS values of axial deformation in the prism surfaces. Plane side
4.2. Strain and stress analysis According to the above mount design, we select a special support case for analysis when the prism locates in the 01 position, where the prism only relies on two nylon gasket
PV (nm) RMS (nm)
Wedge side
F ¼ 550 mm
F1 ¼450 mm
F ¼ 550 mm
F1 ¼450 mm
37.023 18.472
32.191 21.067
40.950 18.144
27.362 20.743
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Fig. 10. Von Mises stress contour of the prism.
Fig. 10 illuminates the Von Mises stress on both sides of the prism. The stress concentration occurs at the support positions of the nylon gaskets, where the maximum stress is 0.273 MPa. According to the above results, the axial deformation in the plane side of the prism is PV¼ 0.0232l and RMS¼0.0123l, and the axial deformation in the wedge side is PV¼0.0359l and RMS¼ 0.0115l. Within the area of the clear aperture 450 mm, the PV value and the RMS value is, respectively, 0.0198l and 0.0140l in the plane side, and the PV value and RMS value is, respectively, 0.0295l and 0.0131l in the wedge side, both of which can meet the requirements of the Rayleigh criterion. The maximum stress is 0.273 MPa, also less than the allowable stress of the K9 glass. Similarly, the different deformation situations are further investigated when the prism rotates from 01 to 3601. Fig. 11 illuminates the support forms when the prism rotates to two special positions of 901 and 1801. When the prism rotates to the 01 position, it mainly relies on the direct support of the inner cell. When the prism rotates to the 901 position, it collaboratively relies on the support of the inner cell and two nylon gaskets, and at this time the 0.5 mm inter-space between the elastic steel sheets and the inner cell can be eliminated through adjusting the preload F, supposed F¼ 12 N. When the prism rotates to the 1801 position, the clearance 0.5 mm of two nylon gaskets above the upside of the inner cell just fills the inter-space between the prism and the inner cell, and at this time there is no preload in two elastic steel sheets, and the prism is supported only by the inner cell. Due to the same cases at the 901 and 2701 position, only two special work cases are selected to be discussed, the prism located at 901 and 1801 position. These results are shown in Table 5. According to the analysis above, when the prism rotates from 01 to 3601, the strains and stresses generated from the variable support ways all can meet the application requirement. Moreover, the scalability of this kind of a prism is extensively studied under the adjustable partial surface-contact support approach, which is validated to be a good solution to install a prism or mirror with a diameter over 400 mm and non-uniform quality distribution. 4.3. Thermal-structural coupling analysis In fact, the scanning device may be locates at a thermal environment. Due to the low thermal conductivity of optical glass and the non-uniform heat exchange, the thermal stress generated
Fig. 11. Different rotation angles under the adjustable partial surface-contact support. (a) prism at 901, (b) prism at 1801.
Table 5 The axial deformation and Von Mises stress when the prism rotates to 901and 1801. PV (nm)
RMS (nm)
Maximum Von Mises stress (MPa)
Plane side Wedge side Plane side Wedge side 901 6.359 1801 21.303
9.340 21.526
2.455 9.848
1.691 10.607
0.172 0.301
inside the mirror will lead to the uneven thermal expansion, which may be deforms the prism surface and reduces the accuracy of the scanning device [16, 17]. Here we only focus on the optical performance change typically induced by the temperature gradient inside the mirror. We take a special case of the prism locating at 01 position for analysis. The flow of the thermalstructural coupling analysis is as follows. First, the three-dimensional thermal analysis unit Solid90 is adopted corresponding to Solid95 used in structural analysis and perform the steady-state thermal analysis. Setting the environmental reference temperature T¼ 20 1C, the temperature Tþ DT¼20.05 1C is added at the plane side, where DT is the
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Fig. 12. Temperature gradient inside the prism.
temperature change value. As a result, the internal temperature gradient in the prism is shown in Fig. 12. The Ansys command ‘‘ETCHG’’ is used to transfer the threedimensional thermal unit Solid90 into the structural unit Solid95, and then he contact relations and boundary conditions are add into the finite element model. After the results of thermal analysis are input into the structural analysis, the thermal-structural coupling analysis can be further executed. The deformation contour of the wedge side and the Von Mises stress distribution in the whole prism are, respectively extracted to be viewed in Fig. 13. According to the above steps, the deformation and stress are, respectively solved with the incremental temperature change DT, 0.01 1C, 0.02 1C, 0.05 1C and 0.10 1C. Table 6 lists the surface deformation values of the prism in the full diameter 550 mm and the clear aperture 450 mm. When the temperature change is 0.05 1C, the deformation value of the plane side within the clear aperture is PV¼0.25l and RMS¼0.248l, while that of the wedge side is PV¼0.255l and RMS¼0.226, which are close to or over the admissible requirement according to Rayleigh criterion. Obviously, when the scanning device locates at an adiabatic environment, it can avoid the thermal effects, but which is impossible in practice [18, 19]. So it is essential to take some effective temperature-control measures for the mitigation of thermal effects on the device.
5. Conclusions For a specific prism assembly with large-diameter, non-uniform quality distribution and continuously rotating around the horizontal optical axis, the traditional supporting ways are difficult to meet the actual requirements. This paper discusses the advantages and disadvantages of various radial support ways including multi-point and surface-contact supports. As a result, an adjustable partial surface support project is proposed to solve this problem, which is constituted by two pairs of elastic steel
Fig. 13. Thermal analysis (a) deformation contour in the wedge side, (b) Von Mises stress.
Table 6 PV values (nm) and RMS values (nm) of the plane side and the wedge side.
Plane side
PV RMS
Wedge side
PV RMS
F ¼550 mm F1 ¼ 450 mm F ¼550 mm F1 ¼ 450 mm F ¼550 mm F1 ¼ 450 mm F ¼550 mm F1 ¼ 450 mm
0.01 1C
0.02 1C
0.05 1C
0.1 1C
71.51 57.35 42.05 48.04 77.31 53.59 39.36 45.09
106.20 82.49 65.81 75.17 111.53 80.03 60.43 69.49
218.15 158.20 137.24 156.70 214.32 161.25 124.47 142.73
405.46 284.88 256.37 292.67 385.50 296.79 230.92 264.83
sheets and nylon gaskets. By adjusting the preloads of the elastic steel sheets, the inter-space between the inner cell and the prism can be effectively filled to ensure the stability of the assembly. According to the analysis results, the strains and stresses under the variable support ways all are within the strength admission of K9 glass regardless of the prism rotating to any position. Under the adjustable partial surface support method, the thermallyinduced structural analysis is also performed to study the surface deformation of the prism induced by the axial temperature gradient. The calculated results indicate that the temperature change has a great influence on the prism surface shape, and
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some effective control measures must be taken into consideration, which is essential for the system application. Moreover, further study shows the proposed support method is validated for the scalability of this kind of a prism with a diameter over 400 mm and non-uniform quality distribution, which can be widely used into the optical scanning and tracking test fields.
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