SALT TELESCOPE INSTRUMENT STRUCTURE ANALYSIS

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1 CONTRACT NO.: 5258 TASK NO.: 780 SALT TELESCOPE INSTRUMENT STRUCTURE ANALSIS SAI-RPT /05/01 REVISION 1.0 Prepared by: Swales Aerospace 5050 Powder Mill Road Beltsville, MD 20705

2 DOCUMENT CHANGE RECORD REVISION DESCRIPTION DATE APPROVAL 1.0 Initial Release REV. - SAI-RPT OF 36

3 TABLE OF CONTENTS 1. SCOPE INTRODUCTION DESIGN REQUIREMENTS ANALSIS COLLIMATOR CAMERAS FRAME VERTICAL STRUTS DIAGONAL STRUTS HORIONTAL STRUTS, RAIL SUPPORTS & RAILS MISCELLANEOUS SUPPORT STRUCTURE COMPONENT LUMPED MASSES AND RIGID ELEMENTS SUPPORT RING MASS PROPERTIES COMPARISON DATA PROCESSING RESULTS... ERROR! BOOKMARK NOT DEFINED. 5. SUMMAR/CONCLUSIONS APPENDI A REV. - SAI-RPT OF 36

4 TABLE OF FIGURES & TABLES FIGURE 2.1: SALT INSTRUMENT STRUCTURE... 6 FIGURE 3.1: SALT COORDINATE SSTEM... 7 TABLE 3.2: SALT INSTRUMENT STRUCTURE MASS PROPERTIES REQUIREMENTS... 8 TABLE 3.3: SALT INSTRUMENT STRUCTURE THERMAL AND WIND GUST REQUIREMENTS... 8 TABLE 4.1: NASTRAN DATA DECK FILE SUMMAR FIGURE 4.2: SALT BASELINE STRUCTURE FEM (ALL-UP, CAMERAS AT 0 ) FIGURE 4.3: SALT STRUCTURE FEM (ALL-UP, CAMERAS AT 90 ) FIGURE 4.4: SALT STRUCTURE FEM (VISIBLE-ONL, CAMERAS AT 0 ) FIGURE 4.5: SALT STRUCTURE FEM (VISIBLE-ONL, CAMERAS AT 90 ) FIGURE 4.6: SALT COLLIMATOR STRUCTURE FEM (TOP VIEW) FIGURE 4.7: SALT COLLIMATOR STRUCTURE FEM (ISOMETRIC VIEW) FIGURE 4.8: SALT CAMERA STRUCTURE FEM (TOP VIEW) FIGURE 4.9: SALT CAMERA STRUCTURE FEM (ISOMETRIC VIEW) FIGURE 4.10: SALT FRAME STRUCTURE FEM (TOP VIEW) FIGURE 4.11: SALT FRAME STRUCTURE FEM (ISOMETRIC VIEW) FIGURE 4.12: SALT MAIN STRUCTURE FEM VERTICAL STRUTS (TOP VIEW) FIGURE 4.13: SALT MAIN STRUCTURE FEM VERTICAL STRUTS (ISOMETRIC VIEW) FIGURE 4.14: SALT MAIN STRUCTURE FEM DIAGONAL STRUTS (TOP VIEW) FIGURE 4.15: SALT MAIN STRUCTURE FEM DIAGONAL STRUTS (ISOMETRIC VIEW) FIGURE 4.16: SALT MAIN STRUCTURE FEM FIGURE 4.17: SALT MAIN STRUCTURE FEM FIGURE 4.18: SALT MISCELLANEOUS SUPPORT STRUCTURE FEM (TOP VIEW) FIGURE 4.19: SALT MISCELLANEOUS SUPPORT STRUCTURE FEM (ISOMETRIC VIEW) FIGURE 4.20: SALT COMPONENT LUMPED MASSES AND RIGID ELEMENTS (FRONT VIEW) FIGURE 4.21: SALT COMPONENT LUMPED MASSES AND RIGID ELEMENTS (ISOMETRIC VIEW) FIGURE 4.22: SALT SUPPORT RING FEM (ISOMETRIC VIEW) TABLE 4.23: MASS PROPERTIES COMPARISON PRO/E DESIGN VERSUS FEM REV. - SAI-RPT OF 36

5 1. SCOPE This report is a summary of the PDR analysis results for the SALT instrument structure. The structure and its associated optics and mechanisms are described in detail in SAI-RPT INTRODUCTION The SALT instrument structure consists of two cameras (visible and near IR), a collimator tube and a truss structure. These elements are designed to support all of the mechanisms and optics to stringent deflection requirements. Both camera structures articulate up to 90 degrees for viewing their appropriate targets. The SALT instrument structure is shown in figure 2.1. There are four key configurations of the SALT instrument structure that require evaluation. The configurations are listed below: 1) the all-up structure with both cameras in the 0 position, 2) the all-up structure with both cameras rotated to the 90 position, 3) the visible-only configuration (NIR camera not present) with the camera in the 0 position, and 4) the visible-only configuration with the camera rotated to the 90 position. The SALT instrument structure is mounted to a very tall, stiff support ring at twelve interface points. The support ring structure is provided by the South Africa team and is not considered part of the structure mass budget. However, its flexibility and mass are included along with the instrument structure in the analysis model. The combined structure of instrument structure and support ring is referred to as the instrument structure assembly. REV. - SAI-RPT OF 36

6 Note: SBB is an acronym for Square Box Beam Figure 2.1: SALT Instrument Structure REV. - SAI-RPT OF 36

7 3. DESIGN REQUIREMENTS The instrument structure assembly is constrained at three points (120 degrees apart) at the bottom of the support ring in translation only. These three locations represent the tracker hexapod interface. All four configurations of the instrument structure must meet the requirements for optical error as well as total mass and c.g. for the various 1-G loading configurations. In addition, the structure must be able to withstand the stresses of the 1-G loads. Figure 3.1 defines the SALT coordinate system. The SALT instrument structure is capable of rotating 360 about the -axis (only ±20 during any one observation) and can articulate ±6 about the -axis. Each of the cameras is able to articulate through a 90 range of motion about their pivot bearing axis. However, the cameras are locked in one position during any one observance. The design requirements must be met for the various permutations of range of motion. Figure 3.1: SALT Coordinate System REV. - SAI-RPT OF 36

8 As the instrument structure rotates and articulates, the various key optical components deform in accordance with the flexibility of the structure. The optical error is a measure of the sum deformations of these key components. There is an error value for both the visible and NIR light paths, and each light path has a parallel dispersion value as well as a perpendicular dispersion value. Therefore, for any given camera position, there are four error values that must be determined: visible light path-parallel to dispersion, visible light path-perpendicular to dispersion, NIR light path-parallel to dispersion, and NIR light pathperpendicular to dispersion. Because the cameras can be located in an infinite number of positions, only the 0 and 90 orientations were examined during this phase of the program. The requirement for the optical error is less than 3 micrometers (or microns). The instrument structure mass properties requirements are defined in table 3.1. The structure must withstand the environments detailed in table 3.2. Property Total mass c.g. c.g. c.g. Requirement < 375 kg < 700 mm <± 50 mm <± 50 mm Table 3.2: SALT Instrument Structure Mass Properties Requirements Table 3.3: SALT Instrument Structure Thermal and Wind Gust Requirements REV. - SAI-RPT OF 36

9 The analysis for the thermal and wind gust conditions was not performed during this phase of the program. However, the instrument structure was designed with these conditions in mind. The open truss design of the structure will minimize the effects of wind loading. And, the choice of Invar, with its very low coefficient of thermal expansion, as the material for the entire instrument structure will assure that the thermal loads are minimized. Finally, the structure must be a simple design that is relatively inexpensive to fabricate. REV. - SAI-RPT OF 36

10 4. ANALSIS A NASTRAN finite element model (FEM) was created for each of the four design configurations. The models were created to optimize the instrument structure design and, ultimately, determine the worst case optical error for the set of 1-G load cases. Details of this process are discussed in the Data Processing section. Figure 4.2 shows the baseline FEM (all-up with both cameras at the 0 position) of the instrument structure assembly. Figure 4.3 shows the all-up FEM with both cameras rotated 90. Figures 4.4 and 4.5 show the visible-only models with the visible camera in the 0 and 90 positions, respectively. The baseline FEM is used to show the various sections of the model, broken out for clarity (figures 4.6 through 4.23). Typically, a top view and isometric view are shown for each section. Table 4.1 summarizes the NASTRAN data deck file ID as well as the number of nodes and elements present. Both cameras present Visible camera only Camera(s) at 0 SALT_PDR_9a.dat SALT_PDR_7a.dat Camera(s) at 90 SALT_PDR_10a.dat SALT_PDR_8a.dat Number of nodes Number of elements Table 4.1: NASTRAN Data Deck File Summary The FEMs were built using kg-mm units. The Invar material used for the instrument structure has a modulus of E+5 N/mm^2 and a density of 8.05 kg/mm^3. The Invar material properties for the support ring have a modulus of E+6 N/mm^2 (10 times that of Invar) and a density of 8.05 kg/mm^3. The baseline FEM was created first. The following steps were then taken to migrate from the all-up model to the visible-only model: 1) remove the mirror mechanism lumped mass, but not its rigid element 2) remove the NIR lumped masses and structure 3) remove the NIR rail non-structural mass 4) add the trim weights to the model The following steps were taken to modify models #9 and #7 (cameras in the 0 orientation) to models #10 and #8 (cameras in the 90 orientation): 1) rotate camera elements 90 2) connect the rigid elements that represent the camera/rail interface to the correct nodes 3) move the etalon mechanism masses 205 mm out of the light path (+ direction for the visible camera and - direction for the NIR camera) REV. - SAI-RPT OF 36

11 Figure 4.2: SALT Baseline Structure FEM (All-up, Cameras at 0 ) SALT_PDR_9a.dat REV. - SAI-RPT OF 36

12 Figure 4.3: SALT Structure FEM (All-up, Cameras at 90 ) SALT_PDR_10a.dat REV. - SAI-RPT OF 36

13 Lumped Mass Representing Trim Weights Figure 4.4: SALT Structure FEM (Visible-only, Cameras at 0 ) SALT_PDR_7a.dat REV. - SAI-RPT OF 36

14 Lumped Mass Representing Trim Weights Figure 4.5: SALT Structure FEM (Visible-only, Cameras at 90 ) SALT_PDR_8a.dat REV. - SAI-RPT OF 36

15 COLLIMATOR The collimator cylinder is modeled with 6 mm thick shell elements. The top and bottom flanges are comprised of bar elements. Figures 4.6 and 4.7 show the collimator structure FEM. Figure 4.6: SALT Collimator Structure FEM (Top View) Collimator Top & Bottom Flanges Figure 4.7: SALT Collimator Structure FEM (Isometric View) REV. - SAI-RPT OF 36

16 CAMERAS The visible and NIR camera structures are identical. Most of the camera cylinder is modeled with 1.5 mm thick shell elements. The aft section (near the second camera optics group) is modeled with 7.5 mm thick shell elements. A total of 1.78 kg of non-structural mass has been assigned to the 7.5 mm thick elements to match the mass properties. The fore and aft rings are modeled with bar elements. A 2 mm thick web (shell elements) connects the camera cylinder to a 10 mm support plate (shell elements). A variety of ribs (bar elements) border the shell elements for added stiffness. Bar elements are also used to model the camera support structure to the rails. Two 0.5 kg lumped masses represent the camera/rail pad. A 1 kg lumped masses represent the camera/bearing interface. The cameras support the polarizer mechanisms, the group 1 and group 2 optics, as well as the CCD camera. The structure for the two cameras is displayed in figures 4.8 and 4.9. Figure 4.8: SALT Camera Structure FEM (Top View) REV. - SAI-RPT OF 36

17 Camera/Bearing Interface Camera Group 2 Optics Camera/Rail Pads Camera Group 1 Optics Figure 4.9: SALT Camera Structure FEM (Isometric View) REV. - SAI-RPT OF 36

18 FRAME The frame structure above the collimator supports the NIR mirror, the NIR focus mechanism and the shutter. This structure has been designed with rectangular Invar beams of varying cross-section. All of the beams have been modeled with bars having the same square tube cross-section of mm mm x 2 mm wall thickness. In order to match the frame mass properties, 0.84 kg of non-structural mass has been assigned to the entire frame. Figures 4.10 and 4.11 show the frame structure. Figure 4.10: SALT Frame Structure FEM (Top View) Figure 4.11: SALT Frame Structure FEM (Isometric View) REV. - SAI-RPT OF 36

19 VERTICAL STRUTS The main support structure is comprised of square tubes with cross-sections of 38.1 mm x 38.1 mm x 1 mm wall thickness. This cross-section is referred to as the standard box beam (SBB). The struts are modeled with bar elements and are used to connect the camera rails, pivot bearings, collimator and interface ring together. They are also used to support the etalons and stiffen the collimator. The vertical struts, shown in figures 4.12 and 4.13 connect the rail support structure to the interface ring. Figure 4.12: SALT Main Structure FEM Vertical Struts (Top View) REV. - SAI-RPT OF 36

20 Figure 4.13: SALT Main Structure FEM Vertical Struts (Isometric View) REV. - SAI-RPT OF 36

21 DIAGONAL STRUTS The diagonal struts are shown in figures 4.14 and These struts are the standard box beam cross-section. They connect the pivot bearings and the collimator to the interface ring. Figure 4.14: SALT Main Structure FEM Diagonal Struts (Top View) REV. - SAI-RPT OF 36

22 Figure 4.15: SALT Main Structure FEM Diagonal Struts (Isometric View) REV. - SAI-RPT OF 36

23 HORIONTAL STRUTS, RAIL SUPPORTS & RAILS Figures 4.16 and 4.17 show the horizontal struts, rail supports and rails. The horizontal struts and rail supports are modeled with bar elements that represent the standard box beam. The bottom horizontal struts connect the bottom end of the collimator to the interface ring. The top horizontal struts connect the pivot bearings to the rail supports and the rail supports to each other. The rails are, likewise, modeled with bar elements. However, their cross-section is an I-beam that is 53.4 mm high and 38.1 mm wide. The flange thickness is 4 mm while the web thickness is 6 mm. The rails are connected to the rail supports with rigid elements. Each of the two rails have 1.28 kg of non-structural mass associated with them that represent the articulation mechanism. REV. - SAI-RPT OF 36

24 Figure 4.16: SALT Main Structure FEM Horizontal Struts, Rail Supports & Rails (Top View) REV. - SAI-RPT OF 36

25 Figure 4.17: SALT Main Structure FEM Horizontal Struts, Rail Supports & Rails (Isometric View) REV. - SAI-RPT OF 36

26 MISCELLANEOUS SUPPORT STRUCTURE The miscellaneous support structure, shown in figures 4.18 and 4.19, serves a number of different functions. Bar elements, representing standard box beams, support the etalons and mirror mechanism as well as stiffen the collimator and connect the collimator to the pivot bearings. The remaining structure, represented with bar and plate elements of varying geometry, joins together the aforementioned structure. Note that the pivot bearing housing is assumed to be very rigid and is modeled with rigid elements. Etalon Support Structure Mirror Mechanism Support Structure Figure 4.18: SALT Miscellaneous Support Structure FEM (Top View) REV. - SAI-RPT OF 36

27 Figure 4.19: SALT Miscellaneous Support Structure FEM (Isometric View) Note: Rotated slightly from standard isometric view for clarity REV. - SAI-RPT OF 36

28 COMPONENT LUMPED MASSES AND RIGID ELEMENTS The optic components and mechanisms are represented as lumped mass elements and are connected to the structure with special rigid elements that do not add additional stiffness to the structure (see figures 4.20 and 4.21). The rigid elements connect the lumped mass to its tie-down points in all six degrees of freedom. The masses associated with the cameras are shown again for clarity. Mirror Mechanism Power Supply Box Collimated Lens Group Lens Holder Waveplate Mechanism Slitmask Mechanism Field Lens Figure 4.20: SALT Component Lumped Masses and Rigid Elements (Front View) REV. - SAI-RPT OF 36

29 NIR CCD Camera NIR Filter Mechanism NIR Camera Group 2 Optics NIR Camera Group 1 Optics NIR Polarizer Mechanism NIR VPH Grating Mechanism NIR Focus Mechanism NIR Etalon Mechanism 2 NIR Etalon Mechanism 1 NIR Camera/Bearing Interface NIR Mirror Shutter Visible Etalon Mechanism 1 Visible Etalon Mechanism 2 Visible Filter Mechanism Visible Focus Mechanism Visible VPH Grating Mechanism Visible Camera/Bearing Interface Visible CCD Camera Visible Polarizer Mechanism NIR Camera Group 2 Optics NIR Camera Group 1 Optics Figure 4.21: SALT Component Lumped Masses and Rigid Elements (Isometric View) REV. - SAI-RPT OF 36

30 SUPPORT RING The support ring FEM, shown in figure 4.22, is a 446 mm tall, 2m diameter cylinder. It connects to the instrument structure at twelve locations with rigid elements. Because the support ring design has not been finalized, it was modeled as a shell (shell elements) with a top and bottom flange (bar elements). The shell is modeled as two 1.25 mm thick plates, separated by 50 mm. The top and bottom flanges have identical rectangular cross-sections of 50 mm x 2.7 mm. The support ring material is assumed to have ten times the stiffness of Invar. The support ring was modeled as described with the concurrence of the University of Wisconsin team in lieu of a concrete design. As the design matures, this ring model can be replaced with a more realistic one. The support ring mass is 65 kg. A total of 70 kg of nonstructural mass is associated with the ring. This represents hardware that is to be mounted to the ring. The support ring mass is excluded from the mass properties budget. Figure 4.22: SALT Support Ring FEM (Isometric View) REV. - SAI-RPT OF 36

31 MASS PROPERTIES COMPARISON A comparison of the mass properties (total mass and c.g.) between the PRO/E design model and the FEM is shown in table The items being compared are a single camera and its optics, the main truss structure (no component masses or support ring) and the entire structure (without support ring) for all four configurations. Note that the FEM mass properties do not include the mass of the wiring harnessing and pneumatic tubing (4.755 kg) that is included in the PRO/E model. Also, the FEM mass properties show the VPH grating mechanism total mass to be kg as opposed to kg in the PRO/E model. This results in a total difference of 5.52 kg. These modifications were made to the PRO/E model after the results for the finite element models had been created. For comparison purposes, the FEM mass properties listed in table 4.2 have the additional 5.15 kg of mass included. It can be seen from the table that the FEM mass properties agree very well with the PRO/E model. The design does not meet the total mass requirement of 375 kg. The all-up design is 95 kg above the requirement while the visible-only configuration is 40 kg greater than the requirement. Only one of the four configurations meet the c.g. requirement (the visible-only model with the camera in the 0 orientation). Both of the all-up designs have -c.g. values that are just outside of the requirement. There is no contingency mass included in these results. PRO/E Design Model Finite Element Model Difference Mass c.g. c.g. c.g. Mass c.g. c.g. c.g. Mass c.g. c.g. c.g. Item Description (kg) (mm) (mm) (mm) (kg) (mm) (mm) (mm) (kg) (mm) (mm) (mm) Visible Camera and Optics NIR Camera and Optics Main Structure All-up, cameras at 0 degrees All-up, cameras at 90 degrees Visible-only, camera at 0 degrees Visible-only, camera at 90 degrees Table 4.23: Mass Properties Comparison PRO/E Design versus FEM REV. - SAI-RPT OF 36

32 DATA PROCESSING The instrument structure is capable of rotating 360 about the -axis. But, during any one observation, the structure will not rotate more than ±20 about the -axis, relative to a baseline angle (beta). Also, during any one observation, the structure can articulate ±6 about the -axis. The angle of rotation from the -axis (alpha) and the articulation angle about the -axis (phi) define this 12 cone. While observing, the cameras are locked in one position. The goal of the analysis is to determine, for the entire range of 1-G load vectors, the worst case optical error for each of the four FEM configurations and the two vectors that produce this error. The number of 1-G load vectors depends on the resolution chosen for the angle parameters. The angle increments were chosen based on a trade-off between the difficulty in processing the number of permutations and the accuracy of the analysis. A reasonable compromise was reached by allowing the beta angle increment to be 10 (36 total beta angles). For each of the 36 possible beta angles, there are 9 alpha/phi angle combinations. If the load vector is aligned with the -axis, both alpha and phi are 0. The vector can then rotate 6 (alpha = 6 ) and articulate 360 (phi) in 45 increments. These nine angle combinations are defined as 0/0, 6/0, 6/45,, 6/315. In order to streamline the analysis, the NASTRAN load case sets consist of three unit load cases (1-G in, 1-G in, and 1-G in ) for each of the twelve unique boundary conditions. This results in 36 total unit displacement sets. A FORTRAN 77 program was then written to read in the FEM output, read in the load vectors and calculate the displacements for each model for 324 load cases. The program then calculates the optical error associated with the key component nodes for each of the displacement sets. This is done by performing a matrix multiplication of the key component node displacements with an error coefficient matrix (provided by University of Wisconsin). The sum of the diagonal (or trace) of this matrix multiplication is defined as the vector optical error. The optical error values are then subtracted from those of adjacent vectors to determine which error difference is the greatest. This max difference is the optical error. There are a total of eight coefficient matrices provided for this analysis. They are use in determining the optical error for the visible light path-parallel to dispersion, the visible light path-perpendicular to dispersion, the NIR light path-parallel to dispersion, and the NIR light path- perpendicular to dispersion for the cameras oriented in both the 0 and 90 positions. The key components, and corresponding nodes, used in calculating the optical error values are presented along with the eight coefficient matrices in Appendix A. The optical error results are discussed in the Mechanical Design Summary Document. The image motion for the truss alone is well within specification. For the full-up design, some flexibility in the camera supports and the NIR fold supports will need to be improved to meet the specification. REV. - SAI-RPT OF 36

33 SUMMAR/CONCLUSIONS The PDR analysis results for the SALT instrument structure have been summarized in this report. All of the critical geometry and load cases have been taken into account and analyzed. The results with a realistic support structure for the full two-beam instrument show a heavier than desired weight (470 kg versus 375 kg) and a worst-case image motion due to flexure above the requirement for some celestial tracks. Many analysis iterations have been performed in an attempt to optimize the stiffness of the structure and, in turn, reduce the optical error. As the design of the optics and corresponding structure evolves in the next phase of the program, further optimization will reduce the optical error. Although the weight and optical error requirements have not yet been met, the SALT telescope instrument structure presented in this report is well on its way to being a fully functional structure. REV. - SAI-RPT OF 36

34 APPENDI A Visible Beam - Dispersion Direction - 0 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror Mechanism Vis Focus Mechanism Vis CCD Camera Vis Optics - group # Vis Optics - group # NIR Beam - Dispersion Direction - 0 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror NIR Focus Mechanism NIR CCD Camera NIR Optics - group # NIR Optics - group # Visible Beam - Perpendicular to Dispersion Direction - 0 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror Mechanism Vis Focus Mechanism Vis CCD Camera Vis Optics - group # Vis Optics - group # NIR Beam - Perpendicular to Dispersion Direction - 0 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror NIR Focus Mechanism NIR CCD Camera NIR Optics - group # NIR Optics - group # REV. - SAI-RPT OF 36

35 Visible Beam - Dispersion Direction - 90 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror Mechanism Vis Focus Mechanism Vis CCD Camera Vis Optics - group # Vis Optics - group # NIR Beam - Dispersion Direction - 90 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror NIR Focus Mechanism NIR CCD Camera NIR Optics - group # NIR Optics - group # Visible Beam - Perpendicular to Dispersion Direction - 90 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror Mechanism Vis Focus Mechanism Vis CCD Camera Vis Optics - group # Vis Optics - group # NIR Beam - Perpendicular to Dispersion Direction - 90 Articulation PDR PDR m rad Location Node ID Tx Ty Tz Rx Ry Rz Field Lens Collim Lens Main Group Mirror NIR Focus Mechanism NIR CCD Camera NIR Optics - group # NIR Optics - group # REV. - SAI-RPT OF 36

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