ABSTRACT 1. INTRODUCTION

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1 Finite Element Analysis of the GTC Commissioning Instrument Structure A. Farah, J. Godoy, F. Velázquez, C. Espejo, S. Cuevas, Universidad Nacional Autónoma de México; V. Bringas, A. Manzo, L. del Llano, J. L. Sánchez, A. Chavoya, Centro de Ingeniería y Desarrollo Industrial (México); N. Devaney, J. Castro, L. Cavaller, GRANTECAN (Spain). ABSTRACT Under a contract with the GRANTECAN, the Commissioning Instrument (CI) is a project developed by a team of Mexican scientists and engineers from the Instrumentation Department of the Astronomy Institute at the UNAM and the CIDESI Engineering Center. The CI will verify the Gran Telescopio Canarias (GTC) performance during the commissioning phase between First Light and Day One. The design phase is now completed and the project is currently in the manufacturing phase. The CI main goal is to measure the telescope image quality. To obtain a stable high resolution image, the mechanical structures should be as rigid as possible. This paper describes the several steps of the conceptual design and the Finite Element Analysis (FEA) for the CI mechanical structures. A variety of models were proposed. The FEA was useful to evaluate the displacements, shape modes, weight, and thermal expansions of each model. A set of indicators were compared with decision matrixes. The best performance models were subjected to a re-optimization stage. By applying the same decision method, a CI Structure Model was proposed. The FEA results complied with all the instrument specifications. Displacements values and vibration frequencies are reported. Keywords: Conceptual Design, FEA, Specifications, Structural Design. Specifications and requirements. INTRODUCTION The design main goal is to maintain the CCD detector center as stable as possible. Provided by the GRANTECAN Group (see Table ), every specification was satisfied under the environmental requirements (i.e., change of the gravity vector within any nominal direction, a temperature variation of.8ºc). Table shows the CI maximum permissible gravitational strain corresponding to the displacements of the detector center with the gravity vector acting in any direction. The FEA results were supposed to meet the deflection specifications. Besides, when completely assembled, all the CI eigenfrequencies including those of the local modes were to be either above 60 Hz or comprised between 35 and 45 Hz. Table CI Basic Performance Requirements Table Maximum Gravitational Strain Lateral position uncertainty 0.0 arcseconds on the sky Strain Value Repeatability of lateral position 0. arcseconds on the sky Ux (displacement along the X 6 µm Stability of lateral position 0. arcseconds on the sky Uy (displacement along the Y 6 µm Repeatability of axial position 0.0 arcsecs (imaging mode) Uz (displacement along the Z 6 µm

2 . Problem Definition The CI could be considered as a calibration tool consisting of an opto-mechanical system mounted on a positioner located at the Nasmyth rotator. From the mechanical standpoint, the design was meant to avoid the displacement of the ideal image detection position caused by structural bending. Consequently, the ordinary gravitational and thermal CI flexures had to be minimized. To facilitate understanding of the design, a brief description of the CI components is provided. The CI features two main parts the Instrument Box (IB) and the Commissioning Instrument Positioner (CIP). Figure shows a CI general description. The CIP consists of the Main Structure (MS) and the Support Structure (SS), both linked by a turntable. The CIP rotates while the telescope is operating. In the image mode, the CCD image displacement is to be lower than 6 µm. This being the main specification, the minimum strain criterion was applied. The IB is mounted on the Focusing Stage (FS), fixed to the CIP. The CCD controller and an Electronic Cabinet (EC) are linked to the IB, located on the SS. Both are held by the cross rolled bearing. The bearing is fixed eccentrically to the MS.. CONCEPTUALIZATION From the outset, the structural concept was split in three parts: the MS, the SS and the Focusing Stage Support (FSS), each analyzed separately. The MS was attached to the CI rotator flange, simultaneously supporting the turntable, the IB, the SS, and the CCD controller box. The SS was bolted to the turntable also working as a support for the FS, the IB, and the CCD controller. The SS defines an interface between the turntable and the IB. Being a part of the SS, the FSS held the FS. Figure only shows the suitable conceptual models the rest was discarded because of manufacturing difficulty. Figure CIP General Description 3. FINITE ELEMENT ANALYSIS (FEA) The FEA comprises two steps the Concept Analysis, performed with ALGOR (version ), and the Global FEM Analysis, performed with ANSYS (version 5.7 FEA). This section just deals with the first.

3 3. General Considerations We performed the analysis by applying the linear static stress, the linear mode shapes, and the natural frequencies applications. To simplify calculations, the IB and the EC were not considered. Instead, a directional force vector, parallel to gravity, was applied on the control nodes defined by auxiliary beams for weight simulation. Table 3 shows the beams mechanical properties. Table 3 Mechanical Properties Properties Steel (ASTM A36) Auxiliary Material E (N/m ) x0 x0 Poisson Ratio CTE (K - ).7x0 6.7x0 6 ρ (kg/m 3 ) Figure Conceptual Models for the Main Structure (Mi), Support Structure (Si) and Focusing Stage Support (Pi)

4 3. Boundary Conditions For the MS analysis (see Figure 3), the nodes around the flange periphery were assumed to be completely fixed. For the SS simulation, the weight of the IB and its components was applied as a 500 N force. Gravity was assumed to be positioned at 0, 45, 90, and 80. In addition, a total force of 45 N was applied to simulate the weight of both the IB and the FS. Assuming a high bearing stiffness, the SS boundary conditions were grounded on a given set of completely fixed bearing nodes. 3.3 Performance Indicators To obtain the best design solution for the three CI components, several structural tests were applied. Every test included a set of performance indicators (Equations I). The indicators were conceived by applying the minimum strain criterion as shown in the equations below. Table 4 presents the selected indicators as to the load case, and the orientation for each structural concept. The vector displacement values were calculated with the FEA. Three final models were selected for the MS (M3, M & M6). I u * = z weight I = ( u + u u ) weight I = ( u u ) weight x y + z * 3 x + y * I 4 = ( u x + u y ) I6 = ( ux uy ) ( ux uy ) + * weight + (I) I 5 = /u z I ux * weight z 7 = I = [ u y ] [ uy ] * weight 8 u = difference between maximum and minimum bearing displacement along the z-axis (Figure 3.a) u, u, u = CCD control N displacement vector x y z Table 4 Performance Indicators. Load case Displacement Rotation Projection x-y Main Structure (MS) Gravity I I 0º, 45, Thermal I I 90, 80 3 Combined I I 3 Focusing Stage Support (FSS) and Support Structure (SS) Gravity I 3 0º Thermal I 5 I 4 Combined I 3 Gravity I 6 I 3 45º Thermal I 5 I 4 Combined I 3 Gravity I 7 I 8 90º Thermal I 5 I 4 Combined I 3 vertical Gravity Thermal Combined I 5 I 4 In the end, the three best models were subjected to a re-optimization stage. Likewise, the entire process was applied to the SS and the FSS. The best SS options were S4 and S5 and the best FSS option was P.

5 All the resulting models were used to ensemble six virtual CI alternatives. Finally, the CI final design came forth through the merging of M3-MS, the S5-SS, and the P-FSS. As a concluding remark, it should be noticed that the MS bending is the main source of the control node displacement. Still, the displacements were lower than required (see Figure 3). Figure 3 a) Translation and Rotation; b) Plane x-y Projection of the Displacement Vector 4. GLOBAL FINITE ELEMENT MODEL The Global FEM analysis consisted of applying the static, modal, and thermal analysis (ANSYS). The modeled parts were: the flange (F), the MS, the bearing support plate (interface solid), the bearing (B), the interface shell (SS support plate), the SS, and the auxiliary elements (AE). Figure 4 shows the CIP global model. Figure 4 CIP Global FE Model

6 4. FE-Model Description The CI was simulated featuring a circular ribbed flange for component attachment. The flange was made of a 5 mm thick circular plate (Figure 5) and was fixed to the Nasmyth rotator by 36 bolts. It also featured 6 alignment holes and extracting bolts. The attachment flange dimensions were taken from the GTC CI Specifications Document. We modeled the MS (Figure 6) as attached to the CI rotator flange, simultaneously supporting the turntable, the SS, and the CCD controller box. The EC was fixed to the MS avoiding flexures on the turntable frame. Made of W- beams (W6x5 AISC), the MS was coupled to the flange inner part. As for the MS modeling, shell finite elements were utilized. Figure 5 Ribbed Flange Figure 6 Main Structure The MS Bearing Support (MSBS) was modeled as a plate welded to the MS. The bearing was bolted to the MSBS and simulated with solid finite elements. The MS, the MSBS, and the bearing were linked by couple elements. The SS Interface Plate (SSIP) featured a plate welded to the support beams. The plate was simulated using 9.5 mm thick shell finite elements. The SSIP mesh was modified to match the SS beams end points. The SS was made of squared hollow structural sections (S-HSS) and a W-beam (W4X3). The W-beam was selected for better structural performance (Figure 7). In addition, auxiliary finite elements were used. These elements were meant to define the CCD position. The CCD, the IB, and the FS masses were applied on the control nodes defined by these elements. The sectional properties of the beam elements were arbitrary. Also, another auxiliary tool simulated the boundary conditions for the thermal analysis (Figure 8). Figure 7 SS FE Model Figure 8 Virtual Tool (Thermal Analysis)

7 5. FEA RESULTS & CONCLUSIONS 5. Gravitational Strain Thirty six nodes matching an equal number of fixing bolts were used to simulate the flange placement boundary conditions. The simulation prevented the nodes virtual rotation and displacement. (Figure 9). Figure 9 Flange Boundary Conditions Figure 0 Total Displacements The FEA cast the following CIP displacements by aligning gravity with the x, y and z axis (see Figure 0 and Table 5). Table 5 CCD control node displacements Ux (m) Uy (m) Uz (m) UTotal (m) e e e e e e e e e e e e Thermal Strain An auxiliary tool (36 beams) was used to perform the thermal analysis (Figure 8). The beams matched the 36 flange fixing bolts and were united at the flange center. Rotation and displacement were constrained at the joining point. The tool simulated the uniform radial Nasmyth rotator thermal expansion. The thermal expansion gradient for this analysis was.8 ºC. Table 3 shows the virtual tool mechanical properties. Figure shows the thermal strain produced by the above gradient. Table 6 shows the displacements on the CCD control node. Gravity was disregarded. Table 6 CCD control node thermal gradient displacements Ux (m) Uy (m) Uz (m) UTotal (m) e e e e -4

8 Figure Thermal Strain 5.3 Shape Modes Analysis The structural analysis boundary conditions were applied. Table 7 shows the results for the first 4 vibration modes. Table 7 Vibration Modes Number Eigenfrequency (Hz) Vibration Mode 08, MS first circular membrane 3,53 MS global bending around y axis 3 7,67 MS global bending around x axis 4 93,6 SS bending around the x axis 5.4 Final Remarks The conceptual design and all the different analysis applied attained the original specifications. The resulting CIP maximum stress was within permissible limits. A maximum stress of 7.95 MPa was obtained while gravity matched the x-axis direction. The stress occurred on the SS interface plate, at the intersection with the beam nodes. On the other hand, the A36 steel yielding stress is approximately of 00 MPa. Being completely made of this material, the CI structure could not be misshaped above the preset limits. The CI model total mass is 70 kg. Our future work is to check the actual values of the CI structures during mechanical performance. Besides, we will do a statistical comparison between FEA simulations and the actual mechanical performance to confirm the accuracy of our findings. REFERENCES.- Commissioning Instrument Specifications Document, GRANTECAN, (Document number ESP/OPTI/04-R)..- Cuevas S., Commissioning instrument for the Gran Telescopio Canarias, SPIE Conference [ ]. 3.- Espejo C., Gran Telescopio Canarias commissioning instrument optomechanics, SPIE Conference [ ].