APPENDIX 4.5.C

CONCEPTUAL DESIGN OF PRIMARY MIRROR SEGMENT SUPPORT SYSTEM OF THE GSMT

POINT DESIGN Report prepared for New Initiatives Office, October 2001.

2

Conceptual Design of Primary Mirror Segment Support System of the GSMT

Point Design

MYUNG CHO October, 2001

AURA New Initiatives Office 30m Telescope Project

NIO-RPT-0006 Revision: B

3

Table of Contents

1. ABSTRACT..................................................................................................... 4 2. INTRODUCTION.............................................................................................. 4 3. AXIAL SUPPORT SYSTEM ............................................................................ 6

3.1 FINITE ELEMENT ANALYSIS OF AXIAL GRAVITY....................................................... 6 4. LATERAL SUPPORT SYSTEM....................................................................... 9

4.1 FINITE ELEMENT ANALYSIS OF LATERAL GRAVITY ................................................ 11 5. SENSITIVITY ANALYSES ............................................................................. 13

5.1 SENSITIVITY IN THE SUPPORT SYSTEMS............................................................. 13 5.2 SENSITIVITY IN THE SEGMENT THICKNESS VARIATION........................................... 14

6. FUTURE WORK ........................................................................................... 15 7. SUMMARY.................................................................................................... 16 8. ACKNOWLEDGEMENTS............................................................................. 17 9. REFERENCES.............................................................................................. 18

4

Conceptual Design of Primary Mirror Segment Support System

of the GSMT Point Design

1. ABSTRACT A conceptual design of the primary mirror segment support system was developed for the Giant Segmented Mirror Telescope (GSMT). A parametric study was performed to minimize the optical surface distortions based on finite element (FE) analysis. The primary mirror segment of a “point design optical system”1 was modeled using the I-DEAS ™ program. An 18-point whiffletree support was optimized for the axial gravity of the segment. This axial support system maintains a surface RMS error of 15 nm over the entire hexagonal surface. Three bi-pod supports, spaced 120 degrees apart radially, were considered for lateral support. The bi-pod has a “push-pull” mechanism and was designed to compensate for the lateral gravity. The line of action of the push-pull forces is in the mid-plane of the segment to avoid the lateral moment from the lateral gravity. This lateral support system yields a surface RMS error of 2.2 nm. Additionally, a sensitivity study was conducted with different segment thickness cases, whiffletree position tolerances and pivot arm locations, and force variations in the lateral bi-pod support. For the sensitivity study in this report, simplified FE models were employed as a first-order approximation. An extensive study will be performed with a full-scale mirror model to determine the details.

2. INTRODUCTION The purpose of this report is to develop a conceptual design of the primary mirror segment support system for the Giant Segmented Mirror Telescope (GSMT). The point design of the GSMT incorporates a 30-meter diameter f/1 segmented primary mirror and a 2-meter diameter secondary mirror. The optical design and projected mechanical properties of the optical elements are discussed in another GSMT report1. The main characteristics are summarized in Table 1. Primary Mirror Diameter: 30 meters Focal Ratio: f/1 Conic constant: -1.00 Type: Segmented, with hexagonal segments Segment Size: 0.665 meter length of each side Segment Thickness: 50 mm Segment Material: Zero expansion glass or glass ceramic (Zerodur, ULE, etc.) Mass of segment: 145 kg (glass only) Number of segments: 618 plus spares

Table 1. The GSMT point design optical and optomechanical parameters.

The optical quality and the mirror segment fabrication tolerances were proposed by the California Extremely Large Telescope (CELT). The CELT’s proposed manufacturing tolerances are listed in Table 2.

5

Optical Figure Accuracy < 20 nm RMS (some active optics warping allowed) Surface Finish < 2 nm RMS Machining tolerances < 0.1 mm Edge bevels < 1 mm

Table 2. The CELT’s manufacturing tolerances

The GSMT primary mirror segments will be off-axis aspheres. The highest asphericity occurs in the segments at the outer edge of the aperture, where the center of the outermost segment is just over 15 meters off axis. However, in this segment support optimization the following design parameters were assumed:

1. The primary segment is flat and has a uniform thickness of 50 mm 2. The segment is an exact hexagon with a size of 1330 mm point to point 3. The segment material is a Zerodur glass ceramic mirror by Schott. Its properties are:

Elastic modulus = 20.3E9 N/m2 , Poissons’s ratio = 0.2, mass density = 2050 kg/m3 4. The segment dimensions are used with those specified in report RPT-GSMT-001 5. It is preferable to avoid boring holes into the segments.

The GSMT segment support system was developed to fulfill the essential design requirements and considerations. The major aspects are: 1. The segment support system should be designed to accommodate the figure accuracy

at Zenith and Horizon at all temperatures within the telescope operating range. Therefore, the support system should operate properly from Zenith to Horizon.

2. It is also designed to be highly reliable under environmental conditions such as temperature, humidity, structural vibration, and earthquake.

3. The support design must be compatible with the capacitive edge position sensors. 4. The system should be relatively simple (preferably to avoid boring holes into the

segments), be lightweight, and be cost effective. 5. For serviceability, it should be relatively easy to remove the segments for recoating.

This means that the support design should be compatible with recoating requirements, for example, anything bonded to the segment should be vacuum compatible.

6. The support should be able to accommodate the effects of wind buffeting. It should have an adequate stiffness to control piston and tilt errors from the wind.

7. The segment support system should have features to meet the positioning requirements for both initial alignment and active alignment in real time. The position control motions should be smooth, free of vibration, and precisely controlled.

8. Stresses in all components of the support system should be maintained within safe working values for all possible combinations of fabrication, erection, operation, and survival conditions. Unless specified otherwise, all stresses shall remain below the Precision Elastic Limit of the material under any combination of operational and environmental loading.

The GSMT mirror segment support optimizations were performed to minimize the optical surface RMS distortions due to the segment gravity orientations. The support system was optimized with 18 axial support points and a lateral support that worked from the back surface (no holes or pockets in the mirror).

6

3. AXIAL SUPPORT SYSTEM A typical GSMT primary mirror segment in the point design was specified as hexagonal, 1.33 m from point to point, and a uniform thickness of 50 mm. Since the mirror segment is not rotationally symmetric, common rule-of-thumb support systems for circular mirrors may not be applicable. There are some design studies to determine the optimum support pattern for the hexagonal segments. A study indicated that a set of 18 axial supports should be adequate for a segment mirror of about 1 m size 5. The GSMT mirror segment is held by a set of 18 axial support points from the back surface. This 18-point support is arranged as a whiffletree kinematic mount system. Two whiffletree support patterns were examined in this study. The first support pattern (PAT_1) is 18 points arranged in a six (6) fold symmetric geometry without a support near the hexagonal corner. In other words, there is not a support along the line from one hexagonal corner to the opposite corner. The other pattern (PAT_2) is in the same symmetric pattern, but a support is located near the hexagonal corner. A schematic sketch of the PAT_2 pattern is shown in Figure 1.

Figure 1. 18-point axial support system arranged in PAT_2 (supports near hexagonal corners)

3.1 FINITE ELEMENT ANALYSIS OF AXIAL GRAVITY A parametric study was performed to minimize the optical surface distortions based on finite element analysis. The primary mirror segment was modeled using the I-DEAS ™ program. Three-dimensional solid elements in I-DEAS™ were used to model the segment. The FE model was established with a 1/12 scale mirror model for the axial support optimization. This symmetric model is valid since the geometry of a segment and its support is symmetric along with the

7

gravitational load. The 1/12 mirror model used in this FE analysis is shown in Figure 2. A local coordinate system was defined such that the X axis is on the line at the center to the hexagonal corner; the Y axis is orthogonal to the X; the Z axis is normal to the XY plane using the right-hand rule. The same coordinate system was applied to the FE analysis for the lateral support.

Figure 2. A 1/12 finite element solid model used in the axial support optimization The support system was optimized with 18 axial support points. A least-square optimization scheme (LSQ) 3 was employed to determine optimum support locations for the minimum surface RMS distortions. During the optimizations, axial support positions were continuously modified to achieve the minimum RMS surface distortion. The LSQ calculates offset moments that can be manipulated for appropriate support positions for the next iteration. The optimum support locations were continuously iterated until the offset moments reach a negligible level. The two support patterns (PAT_1 and PAT_2) were optimized for the axial gravity. The optimum 18-point support system was obtained in PAT_2. This optimum axial support system yields a surface RMS distortion of 15 nm over the entire hexagonal surface. The optical surface distortions (support print-through) from the 1/12 FE model are shown in Figure 3. The surface RMS distortion calculated in PAT_1 indicated a slightly higher distortion level than that from PAT_2. This shows that PAT_1 is more susceptible to deflection since the segment is held without supports near the corners.

8

Figure 3. Optical surface distortion (support print through) from the axial support optimization The whiffletree mount system of PAT_2 is illustrated in Figure 4. This figure is a layout sketch of the whiffletree looking down from the back of the segment. It shows a lack of symmetry. This lack of symmetry may be undesirable due to complicating the system design and increasing hardware costs. The optimization indicated superior results to PAT_2 since the 18-point support pattern in PAT_2 has two supports on a radial line to the hexagonal corners.

Figure 4. A layout sketch of the 18-point whiffletree support system based on PAT_2

9

This pattern works better in controlling the corners. A tradeoff study will be needed in the next design phase to evaluate merits in performance versus potential complications resulting from the asymmetric geometry. Detailed FE analyses of the parts of the axial support system will be in the next design phase. Closed form solutions and approximate hand calculations were made for the hardware for the axial supports. Its main hardware consists of:

A set of six (6) tri-leg units. Each unit has three legs to carry one sixth of the segment weight. Each leg in the unit is different in length to carry proper passive forces. A set of three (3) tie beams. Each beam links between two of the tri-leg units. Both ends of the beam have flex pivots. A main frame. The main frame (box beam structure) at the center serves as a support to carry the segment load to the telescope support frame. It is kinematically mounted at three points on the support frame. A set of 18 support pins. Each of the support pins was attached to the free ends of the tri-leg unit. It has a flexure at the support pad which is glued to the segment.

4. LATERAL SUPPORT SYSTEM The GSMT mirror segment is held by a lateral support from the back surface. It is preferable to avoid boring holes or pockets at the back of the segments. With the lateral support at the back surface of the segment, a moment exists under the lateral gravity due to the axial eccentricity of the lateral force with respect to the center of the gravity plane of the segment. The GSMT lateral support system was designed with a “push-pull” mechanism. The push-pull mechanism can be directly attached to the back of the segment. A bi-pod support was developed to service the features. Three bi-pod supports, spaced 120 degrees apart radially, were considered for the lateral support. An isometric view of the bi-pod support arrangement (looking down from the back surface) is shown in Figure 5.

10

Figure 5. An isometric view of the bi-pod support arrangement (looking down from the back surface)

A set of longitudinal forces of the pods are shown in Figure 6. This bi-pod support as shown is capable of applying the moment compensation forces to the segment weight. The sum of the forces in the bi-pod supports maintains equilibrium with the lateral gravity, and the sum of the moment of the force couple compensates for the lateral moment. The GSMT lateral support system utilizes the three bi-pod supports at 120 degrees apart. The line of action of the push-pull forces is in the mid-plane of the segment to avoid the lateral moment from the lateral gravity. This figure did not illustrate a detail mechanism for kinematic attachment geometries. A relatively high rotational stiffness is required to maintain rotational position of the segment with modest accuracy to achieve optical tolerances. For initial calibration and integration, the lateral support needs a certain level of adjustability to accommodate the rotational adjustment. A detailed analysis of the bi-pod supports will be performed in the next design phase.

11

Figure 6. A set of bi-pod support forces for the lateral support

4.1 FINITE ELEMENT ANALYSIS OF LATERAL GRAVITY One half of the segmented mirror was modeled for the lateral gravity case. This half-model was

Figure 7. A one half finite element solid model used in the lateral support calculations

12

used to take advantage of symmetry in both loading and the structural geometry. Three-dimensional solid elements in I-DEAS™ were used for the mirror. For the bi-pod supports, beam elements were used. The one-half mirror model used in this FE analysis is shown in Figure 7. The same local coordinate system was used in this lateral support analysis as was used in the axial support. The FE model was applied to the lateral gravity element loading and the nodal loads. The nodal loads were applied directly to the free end of the bi-pod elements. A set of push-pull forces were directly applied to the nodal points at the back of the segment. These forces were pre-calculated to compensate for the segment’s lateral gravity and its lateral moment. The line of action of the push-pull forces is in the mid-plane of the segment to avoid the lateral moment due to the lateral gravity. The surface distortion due to the lateral gravity was calculated. The surface distortion for the optimum lateral support was calculated. An RMS surface distortion of 2.2 nm was obtained. A plot of the surface distortion due to the lateral gravity is shown in Figures 8.

Figure 8. Optical surface distortion from the bi-pod lateral support system For lateral gravity analysis in this report, the lateral gravity vector was assumed in the local Y axis normal to the mid plane of the local segment X axis. However, this assumption is only applicable to the segments on the central vertical plane of the entire primary mirror array. All other segments

13

will undergo compound lateral force components. As the primary mirror points from the Zenith towards the Horizon, the lateral gravity vector for these segments rotates about the plane that is skewed with respect to those segments. The gravity force components vary depending upon the location of the segment and the skewedness. For this case, the lateral support should have an advantage in being rotationally symmetric (axisymmetric). Otherwise, it should be designed to work for any gravity orientation with respect to the local optical axis of the segment. The main hardware for the lateral support consists of the following parts:

A bi-pod using a standard steel tubing. One end is connected to the support structure frame, the other is connected to the segment. A lateral force pad (invar) that is glued to the segment with a silicone RTV. It works for the lateral shear force from the support for the horizontal gravity component of the segment lateral weight. Flexures at the both ends of the lateral support tube. It transfers the lateral shear force from the lateral support tube to the segment. The flexures are made from a PH stainless steel.

5. SENSITIVITY ANALYSES

The sensitivity of the primary mirror surface to possible errors in the segment axial and lateral support systems is of interest. The results of this sensitivity analysis are used to establish limits for the possible errors. Tolerances for primary mirror support assembly and piece parts will be developed from the results. A sensitivity study on the GSMT segment was conducted with different segment thickness cases, whiffletree position tolerances and pivot arm locations, and lateral force variations. For this study, simplified FE models were employed as a first-order approximation. The FE model was established with a 1/12 scale mirror model for axial support system tolerances. This symmetric model was capable of predicting some systematic errors. However, it limits many extensive sensitivity cases. A one-half scale model was used for lateral sensitivity cases. An extensive study will be performed with a full-scale mirror model to determine the details.

5.1 SENSITIVITY IN THE SUPPORT SYSTEMS The GSMT segment axial supports consist of a whiffletree system and an active mechanical system. Error sources may result from the following aspects:

fabrication and assembly tolerances in the passive and active components load cell calibration tolerances, finite resolution and non-linearities actuator installation and alignment tolerances non-repeatability in reinstallation of the mirror after recoating or service.

Other errors that are associated with the axial supports may be the result of differential thermal expansion between the mirror segment and the segment support systems. Errors also can be caused by shifting of the mirror relative to the mirror support. Flexures are installed to transmit the force to the mirror at the attachment point to the mirror segment. With any lateral shift the flexure will exert a moment on the mirror. It will also exert a lateral force resisting any shift in mirror position. A few specific error cases were considered to represent the types of errors that may result from errors or tolerances in the individual components of the support systems and from specific position errors. The sensitivity to position errors in the axial support was examined. Effects on the

14

optical surface distortion due to the change in axial support forces were also examined. The surface distortions associated with the axial support system were calculated and the results are listed in Table 3.

CASE Peak-Valley RMS Remark

CASE 1 78 14.7 Optimum axial support

CASE 2 203 50.2 Support force change by 1N

CASE 3 84 16.6 Support position change by 1mm

Table 3. Axial support system sensitivity (units = nm)

CASE 1 is the baseline model used in the axial support optimization. The baseline model is the segment described in Table 1. CASE 2 has the same descriptions as CASE 1 but one of the supports experienced a change in force by 1 N. In CASE 3 one of the supports was misplaced by 1 mm from its optimum location. The sensitivity results indicated that the surface distortions are very sensitive to the axial support force variation. The surface error from the support position change is relatively insensitive compared to the support force change. These two cases (CASE 2 and 3) are commonly combined together because the change in a support location causes a force magnitude variation. The lateral support actuators are similar in design to the axial supports. They are susceptible to the same errors as the axial supports. The surface distortions associated with the lateral support system were calculated and the results are listed in Table 4.

CASE Peak-Valley RMS Remark

CASE 4 61 2.2 Optimum lateral support

CASE 5 61 2.4 Support force change by 1N

Table 4. Lateral support system sensitivity (units = nm)

CASE 4 is the baseline model used in the lateral support calculations. In CASE 5 one of the supports experienced a change in force by 1 N. Additionally, a sensitivity calculation was made to quantify the wind buffeting effect. As a first-order approximation for the point design, a wind speed of 10 m/s outside the dome was assumed. It is approximately equivalent to a nominal mean pressure of 50 pascals for a worst case wind scenario over the segment. The surface RMS distortion of 0.6 nm was calculated due to a uniformly distributed mean pressure of 50 pascals. This is approximately 4% of the axial gravity effect.

5.2 SENSITIVITY IN THE SEGMENT THICKNESS VARIATION The current 18-point support system was optimized based on a hexagonal segment thickness of 50mm. “Plate Theory” indicates that the deformation of a plate is highly proportional to the plate thickness. A self-weight deflection of a common plate is proportional to the thickness squared. A

15

sensitivity study was made with different segment thickness cases. Three different segments of the same size were examined. They are of a thickness of 50 mm, 45 mm, and 55mm, respectively. Each of these segments was modeled in a simplified FE model, a 1/12 symmetric FE model. The surface distortions for the different thickness cases were calculated due to the axial gravity. The results are listed in Table 5.

CASE Thickness Peak-Valley RMS Remark

CASE 1 50mm 78 14.7 Optimum axial support

CASE 6 45mm 89 17.2 Thickness=45mm

CASE 7 55mm 69 12.8 Thickness=55mm

Table 5. Sensitivity on the segment thickness (axial gravity, units = nm)

CASE 6 has the same geometry as CASE 1, but the segment has a different thickness of 45 mm. CASE 7 is for a thickness of 55mm. The results from the segment thickness effect indicated that the surface distortions were relatively insensitive to the thickness variation (approximately a 10% RMS variation to 10% thickness variation). However, the surface distortion follows a square power rule to the thickness. It varies in proportion to a square of the segment thickness.

6. FUTURE WORK This report evaluated the overall performance of the segmented mirror supports. The mirror models used are mainly for optimum support locations. These models also have limited applications and lack full representation of all of the segments. There are many unanswered issues with this first-order calculation. Several aspects for future work are addressed herein. An extensive wind study will be performed to quantify the wind buffeting effect for GSMT. The resulting wind loading from a “fluid dynamic model “ will be coordinated with the segment support structure. In this report, a wind speed of 10 m/s outside the dome was assumed as a first-order approximation. It is equivalent to a nominal mean pressure of 50 pascals for a worst-case wind scenario over the segment. Once the wind buffeting parameter is well defined, wind effects on the segment will be examined with pressure variations (with a static mean pressure). On the lateral support system, all segments except those on the central vertical plane of the mirror array experience compound lateral force components. As the primary mirror points from the Zenith towards the Horizon, the lateral gravity vector for these segments rotates about a plane that is skewed with respect to those segments. The lateral support should be an advantage, being rotationally symmetric. Or the system can be designed to work for any gravity orientation with respect to the local optical axis of the segment. An axial flexure mechanism will be developed to carry the axial segment gravity. It will be adequately flexible to accommodate tilt motions and any environmental loads without introducing optical surface distortion to the segment. Lateral support flexures in the bi-pod should adequately transmit the lateral force to the segment. They transfer the lateral shear force from the lateral support tube to the segment. A detailed flexure calculation is required to validate the concept. The support stiffness calculations will be performed to properly maintain the segment position. The stiffness should be robust enough to accommodate and control the segment motion due to any combination of operational and environmental loading.

16

An active optical system will be developed to improve the image by pistons and tilts of each segment. The active optics mechanism will be integrated in the support system. A moment actuator used by Gran Telescopio Canarias will be examined as a potential actuator candidate. Tolerance analysis with full-size FE models will be conducted to predict the effects of operational or any environmental loading. The full-size models enable calculation of the optical surface distortions such as differential thermal expansion between the mirror and the mirror cell, random positional errors and force errors. This is also necessary due to the non-symmetric nature of the applied forces and moments at individual support locations. Stresses in all support mechanisms should be examined. Flexure stresses should be maintained within safe working values for all possible combinations of fabrication, erection, operation, and survival conditions. All stresses shall remain below the Precision Elastic Limit of the material under any combination of operational and environmental loading.

7. SUMMARY This report describes a first-order approximation of the GSMT mirror segment support system. An 18-point whiffletree support was developed for the axial gravity orientation. This axial support system yields a surface RMS error of 15 nm. For lateral gravity loading, three bi-pod supports and a “push-pull” mechanism spaced 120 degrees apart radially were developed. This bi-pod lateral support system yields a surface RMS error of 2.2 nm. This point design support system will be well within the design requirements. A top view of the support assembly was shown in Figure 9. It shows a layout sketch of the axial whiffletree, the lateral bi-pod support, and the back structure. An isometric view of the support assembly is also shown in Figure 10.

Figure 9. Top view of the segment primary mirror support assembly layout

17

Figure 10. Isometric view of the segment primary mirror support assembly layout

A few sensitivity analyses were performed. The results from the segment thickness effect indicated that the surface distortions were relatively insensitive to the thickness variation (approximately a 10% RMS variation to 10% thickness variation). In addition, surface distortions were examined for changes in the support location and support force. The results indicated that the surface distortions are sensitive to the axial support force variation. The surface error from the support position change shows a relative insensitivity compared to the support force change. These two cases, however, are commonly combined together because the change in a support location causes a force magnitude variation. As addressed in “Future Work”, intensive sensitivity and tolerance studies should be performed along with some hardware tests to validate the support design. This report addressed some outstanding technical aspects and tasks to be studied or performed for a next design phase.

8. ACKNOWLEDGEMENTS

The following individuals of the NIO, Gemini, and NOAO staff contributed to this report: Larry Stepp and Teresa Bippert-Plymate. Their review and comments are appreciated. Rick Robles is also acknowledged for the solid images in this report.

18

9. REFERENCES 1) Oschmann, J., Stepp, L., NIO report RPT-GSMT-001, Point Design for GSMT Optical System, January 2001 2) Stepp, L., NIO report RPT-GSMT-002, Fabrication of GSMT Optics, January 2001 3) Cho, M., Price, R.., Gemini report RPT-O-G0017, Optimization of Support Point Locations

and Force Levels of the Primary Mirror, November 1993 4) Cho, M., Roberts, J.., Gemini report RPT-O-G0021, Response of the Primary Mirror to

Support System Errors, November 1993 5) Gunnels, S, Concept Design Report – Mirror segment support, California Extremely Large

Telescope Report #16, March 2001 6) Schier, J, Summary of the CELT Mirror Segment Actuator Survey, California Extremely Large

Telescope Report #15, February 2001

The New Initiatives Office is a partnership between two divisions of the Association of Universities for Research in Astronomy (AURA), Inc.: the National Optical Astronomy Observatory (NOAO) and the Gemini Observatory. NOAO is operated by AURA under cooperative agreement with the National Science Foundation (NSF). The Gemini Observatory is operated by AURA under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil) and CONICET (Argentina).