# Presentation: 591118

## Offloading the SNAP Primary Mirror

**Robert Besuner**
**February 2005**

## Background

**Examine optical surface distortions of the SNAP 2-meter Closed-Back ULE Primary Mirror in gravity with optical axis vertical (or nearly vertical).**
**Employ point-force offloaders to minimize distortions.**
**Number of offloaders ranges from zero to 60.**
**Study primary mirror on either ‘normal’ or ‘inverted’ bipods.**
**Study effects of deviations from ideal offloading:**
**Non-ideal magnitudes.**
**Non-vertical individual offloaders.**
**Not perfectly level test bench.**
**Budget for clean, focused image testing calls for surface distortion of 10nm or less.**
**Interferometric testing would have looser requirements, based on resolution of test equipment.**
**All distortions reported are surface distortions (one half of WFE values).**

## Scope

**Inverted and normal bipods, various numbers of offloaders.**
**Offloader positions are specified manually, then analytic tools are used to optimize magnitudes for minimum surface distortion.**
**Positions are iterated for best results.**
**(See Appendix B for comparison of inverted and normal bipods under lateral loading.)**
**Cases examined:**
**Perfectly vertical optical axis (illustrated in displacement figures).**
**Test bench tilted 0.5****o or 1o relative to gravity.**
**Uniform offloader force magnitude error:  1%, 5%.**
**Monte Carlo analysis of offloader errors (See Appendix A):**
**σ = 1% from ideal magnitude, normal distribution.**
**σ = 2o from vertical, both axes, normal distribution.**
**Additional values tried for normal bipods with 57 offloaders.**
**All results presented are corrected for tilt and piston.**

## Model-Inverted Bipods

**Model-Inverted Bipods**

**Frit-bonded, closed-back ULE.**
**Mass of mirror is 198kg.**

## Model-Normal Bipods

**Frit-bonded, closed-back ULE.**
**Mass of mirror is 204kg.**
**Bipod line-of-force intersections optimized for minimum sensitivity to lateral loads (see Appendix B).**
**12mm front face**
**8mm back face**
**8mm core id and od**
**2.5mm general core**

## Summary (nm rms surf. distortion)

## Normal Bipods, 57 Offloaders vs.Inverted Bipods, 60 Offloaders

**Normal Bipods, 57 OL’s**
**Level bench:  6.8nm rms, 54nm P-V**
**0.5****o ****bench tilt: 7.0nm rms **
**1****o**** bench tilt: 7.4nm rms **
**Monte Carlo:**
**σ = 1%, 2o:**
**99% prob ********19.0nm rms**
**90% prob ********14.5nm rms**
**σ = 1%, 1o:**
**99% prob ********17.9nm rms**
**90% prob ********13.2nm rms**
**σ = 0.5%, 2o:**
**99% prob ********13.8nm rms**
**90% prob ********11.1nm rms**
**σ = 0.5%, 1o:**
**99% prob ********11.2nm rms**
**90% prob ********9.4nm rms**

**Inverted Bipods, 60 OL’s**
**Level bench:  7.4nm rms, 55nm P-V**
**0.5****o ****bench tilt: 32nm rms **
**1****o**** bench tilt: 63nm rms **
**Monte Carlo:**
**σ = 1%, 2o:**
**99% prob ********44nm rms**
**90% prob ********31nm rms**
**σ = 1%, 1o:**
**99% prob ********24nm rms**
**90% prob ********18nm rms**
**σ = 0.5%, 2o:**
**99% prob ********42nm rms**
**90% prob ********32nm rms**
**σ = 0.5%, 1o:**
**99% prob ********23nm rms**
**90% prob ********17nm rms**

## Inverted Bipods, No Offloaders

**Inverted Bipods, No Offloaders**

- bipod pad

**Level bench: 241nm rms, 953nm P-V**
**0.5****o ****bench tilt:  243nm rms **
**1****o**** bench tilt:  249nm rms**

## Inverted Bipods, 3 Offloaders

**Inverted Bipods, 3 Offloaders**

- 122N

- bipod pad

**Level bench: 128nm rms, 600nm P-V**
**0.5****o ****bench tilt:  132nm rms **
**1****o**** bench tilt:  143nm rms **
**Monte Carlo:**
**99% prob. ********135nm rms**
**90% prob. ********131nm rms**

## Inverted Bipods, 6 Offloaders

**Inverted Bipods, 6 Offloaders**

- 185N

- bipod pad

- 61N

**Level bench: 112nm rms, 550nm P-V**
**0.5****o ****bench tilt: 116nm rms **
**1****o**** bench tilt:  128nm rms **
**Monte Carlo:**
**99% prob. ********129nm rms**
**90% prob. ********121nm rms**

## Inverted Bipods, 9 Offloaders

**Inverted Bipods, 9 Offloaders**

- 119N

- bipod pad

- 123N

- 90N

**Level bench: 56nm rms, 270nm P-V**
**0.5****o ****bench tilt:  66nm rms **
**1****o**** bench tilt:  85nm rms **
**Monte Carlo:**
**99% prob. ********84nm rms**
**90% prob. ********73nm rms**

## Inverted Bipods, 60 Offloaders

**Level bench:  7.4nm rms, 55nm P-V**
**0.5****o ****bench tilt: 32nm rms **
**1****o**** bench tilt: 63nm rms **
**Monte Carlo:**
**σ = 1%, 2o:**
**99% prob ********44nm rms**
**90% prob ********31nm rms**
**σ = 1%, 1o:**
**99% prob ********24nm rms**
**90% prob ********18nm rms**
**σ = 0.5%, 2o:**
**99% prob ********42nm rms**
**90% prob ********32nm rms**
**σ = 0.5%, 1o:**
**99% prob ********23nm rms**
**90% prob ********17nm rms**

## Normal Bipods, No Offloaders

**Normal Bipods, No Offloaders**

**Level bench: 757nm rms, 2.9****μ****m P-V**
**Negligible effect from 1****o ****bench tilt**

- bipod pad

## Normal Bipods, 3 Offloaders

- 312N

- bipod pad

**Level bench: 174nm rms, 770nm P-V**
**Negligible effect from 1****o ****bench tilt**
**Monte Carlo:**
**99% prob. ********181nm rms**
**90% prob. ********177nm rms**

## Normal Bipods, 6 Offloaders

**Normal Bipods, 6 Offloaders**

- 159N

- bipod pad

- 177N

**Level bench: 130nm rms, 604nm P-V**
**Negligible effect from 1****o ****bench tilt**
**Monte Carlo:**
**99% prob. ********135nm rms**
**90% prob. ********133nm rms**

## Normal Bipods, 9 Offloaders

**Normal Bipods, 9 Offloaders**

- 212N

- bipod pad

- 160N

- 100N

**Level bench: 114nm rms, 530nm P-V**
**Negligible effect from 1****o ****bench tilt**
**Monte Carlo:**
**99% prob. ********121nm rms**
**90% prob. ********118nm rms**

## Normal Bipods, 12 Offloaders

**Normal Bipods, 12 Offloaders**

- 162N

- bipod pad

- 121N

- 112N

**Level bench: 53nm rms, 279nm P-V**
**Negligible effect from 1****o ****bench tilt**
**Monte Carlo:**
**99% prob. ********63nm rms**
**90% prob. ********58nm rms**

## Normal Bipods, 18 Offloaders

**Normal Bipods, 18 Offloaders**

- 100N

- bipod pad

- 108N

- 93N

- 80N

- 111N

**Level bench: 35nm rms, 230nm P-V**
**Negligible effect from 1****o ****bench tilt**
**Monte Carlo:**
**99% prob. ********47nm rms**
**90% prob. ********41nm rms**

## Normal Bipods, 33 Offloaders

**Normal Bipods, 33 Offloaders**

- Offloader forces range from 26N to 72N.

**Level bench: 14.9nm rms, 98nm P-V**
**0.5****o ****bench tilt:  14.9nm rms **
**1****o**** bench tilt:  15.1nm rms **
**Monte Carlo:**
**99% prob. ********27nm rms**
**90% prob. ********22nm rms**

## Normal Bipods, 45 Offloaders

**Normal Bipods, 45 Offloaders**

**Level bench: 10.3nm rms, 71nm P-V**
**0.5****o ****bench tilt:  10.4nm rms **
**1****o**** bench tilt:  10.6nm rms **
**Monte Carlo:**
**99% prob. ********22nm rms**
**90% prob. ********17nm rms**

- Offloader forces range from 29N to 63N.

## Normal Bipods, 57 Offloaders

**Normal Bipods, 57 Offloaders**

**Level bench:  6.8nm rms, 54nm P-V**
**0.5****o ****bench tilt: 7.0nm rms **
**1****o**** bench tilt: 7.4nm rms **
**Monte Carlo:**
**σ = 1%, 2o:**
**99% prob ********19.0nm rms**
**90% prob ********14.5nm rms**
**σ = 1%, 1o:**
**99% prob ********17.9nm rms**
**90% prob ********13.2nm rms**
**σ = 0.5%, 2o:**
**99% prob ********13.8nm rms**
**90% prob ********11.1nm rms**
**σ = 0.5%, 1o:**
**99% prob ********11.2nm rms**
**90% prob ********9.4nm rms**

- Offloader forces range from 15N to 44N.

## Appendix A-Monte Carlo Analysis

## App. A-Monte Carlo Description (1)

**PM is modeled on bipods.**
**Unit forces are applied one at a time at each offloader position, with zero gravity.**
**Forces are applied along all three axes, one at a time.**
**Applying lateral forces is for sensitivity to force misalignment.**
**Distortions of each point on the optical surface due to each unit force is found from FEA.**
**There’s a coefficient, C****ijk****, for each node (i) of the front face, for each offloader position (j), for each of three offloader force directions (k).**
**These coefficients go into a spreadsheet.**
**FEA is run without offloaders in one g.  Distortions (****δ****i_1g****) ****at each node of front face sheet are tabularized in the spreadsheet.**

## App. A-Monte Carlo Description(2)

**Multipliers are applied to the distortions in proportion to the force at each offloader.**
**Lateral components represent offloader misalignment from gravity.**
**Distortions due to all the offloaders and the non-offloaded case are summed (superposed).**
**δ****net_i**** = Σ****j****(F****x****C****ijx**** + F****y****C****ijy**** + F****z****C****ijz**** + δ****i_1g****)**
**The resulting surface distortions reflect net result of offloading in one g.**
**Iterating surface distortion, while allowing some random deviation from ideal forces, shows effects of random offloading errors (direction and magnitude.)**
***A hugely beneficial side benefit to setting up the problem this way (in a spreadsheet) is that a solver can be used to minimize surface distortion by varying offloading forces.***
**This yields optimized, ideal offloading forces for minimal distortion.**
**Equivalent to making many thousands of distinct FEA runs.**

## App. A-Sample Monte Carlo Results

## Appendix B-Why lateral loads cause much greater surface distortions with inverted bipods than with normal bipods.

## App. B-The Difference 

**Inverted bipods induce significant force couples in mirror under lateral loading.**
**e.g. TM must be on optimally-located, normal bipods.**
**e.g. horizontal-beam PM test scheme required optimized, normal bipods.**
**Approx. lever arm is from force to intersection of strut lines-of-action.**

- Inverted bipods

- Normal bipods (optimized)

## App. B-Front face sheet stresses, 1g lateral

- Inverted bipods

- Normal bipods (optimized)

**Similar maximum stresses, but inverted bipods have much more widespread stresses.**

## App. B-Front face sheet distortions, 1g lateral

**Inverted bipods **
**3600 n****m rms surface distortion, tilt-corrected**
**9.2 μrad tilt**
**NOTE DIFFERNT DISPLACEMENT SCALES**

**Optimized normal bipods **
**81 ****nm rms surface distortion, tilt-corrected**
**0.89 μrad tilt**

## App. B-Optimizing bipod geometry

- inverted bipods, 3600 nm rms

- One-g lateral loading

- optimized normal bipods, 81 nm rms

- previously studied, non-optimized normal bipods, 850 nm rms