1. Mechanical tolerance analysis for the MQXF prototype structures
MQXF structure workshop
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2. Outline Motivations Methodology and models Tolerance sensitivity study
Worst case study
Summary
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3. Main Goals of the Mechanical Tolerance Analysis
Ensure that the coil preload will be provided properly and evenly during operation:
Ensure the radial contact between the collar and the coil
Impact of the azimuthal tolerances and the how the actual assembly process eliminate this impact.
Ensure that stress in the coils and other parts will not exceed the mechanical limits.
Analyze the coil stress error band in the worst case and individual cases.
Find the allowable tolerance that prevents the pole keys from shear failure.
Give guidance of assembly fiducialization and form the mechanical tolerance requirement for each part.
Understand the impact of the mechanical tolerance and deformation on the field quality.
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4. Tolerance Analysis Overview
Tolerance stack up:
Also as known linear stack-up.
The tolerances of each part will be accumulated towards the final part, and lead to a “fit” problem or redistribute the stress in the assembly.
For the MQXF magnet assembly, we are facing the latter problem.
Tolerance stack up analysis represents the cumulative effect of part tolerance with respect to the assembly requirement. 
The stack-up calculations are usually referred to worst case (linear summation) or statistical method (RSS---Root-Sum-Square).
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5. MQXFS1 assembly process overview
The MQXFA1 assembly iteration would be similar as the MQXFS1.
MQXFS1 assembly includes two separate assembly processes:
coil-pack subassembly
yoke-shell subassembly.
Squareness measurements for the coil-pack and yoke-shell subassemblies will be performed before further assembly.
Coil-pack subassembly is then inserted into the yoke-shell structure.
Pack masters and load keys up according to the measurements, insert the master- key package.
Pressurize bladders to compress the coil-pack and allow shimming the load keys. Equal shim is applied all round the magnet.
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6. Coil-pack assembly (MQXFS1)
Coil-Pack assembly road map
Ground plane insulation layout
Coil CMM measurements
Nominal Radial shims
Nominal pole key shims
Assembly with Fuji paper
No shimming of the pole key
Measurement of pole gap dimensions along the length using measurement pins
Pole SG measurements
Actual radial shims
Actual coil OD / actual coil azimuthal size
Pads assembling
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7. Goal: bring all the coils on the same collar arc length
Shims in the coil-pack assembly (MQXFS1)
Goal: bring all the coils on the same collar arc length
Azimuthal tolerances of the coils and pole keys are absorbed by these shims
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8. Yoke-shell assembly (MQXFS1)
Yoke-shell package is assembled vertically
Each quadrant of yoke lamination structures are placed inside the shell.
Place bladders between yokes and cross support.
Pressurize the bladders to achieve yoke gap of 12 mm.
Azimuthal strain in the shell: 230 +/- 17 μm.
Insert gap keys all round to lock the yoke in.
Gap=12mm
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9. Final assembly (MQXFS1)
Hc1
Squareness measurement
Measurements have been taken at both ends and center along the length.
The major deviation with respect to the nominal size is the “banana shape” of the coil due to heat treatment.
Ly1
10.4MPa
Hc2
Ly2
Bladder
Load key
Alignment key
shims
Master package insertion
According to the squareness measurement, take the minimum as the master-key package assembled thickness to ensure insertion.
The shim thickness between load key is adjusted based on the measurements, the resultant thickness is equal in all quadrants.
Loading:
Bladders are inserted between the iron masters to create a gap to add equal shims to the load keys all round.
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10. Tolerance stack-up in MQXF
Radial tolerance stack-up
The coil is assumed in nominal size.
The radial dimensional chain consists of the ID and OD of the collar, pad, masters, yoke and shell.
The assumptions in this analysis:
Radial tolerances will build up towards the gap between the load key and the master attached top the pad--- consistent with the actual assembly process
Same interference applied to all quadrants --- consistent with the actual assembly process
The thickness of the load keys all round are same and remained unchanged --- consistent with the actual assembly process
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11. Tolerance stack-up in MQXF
Azimuthal tolerance stack-up is absorbed by the shims
In the actual assembling process, we shim the two sides of the pole key and the ID of the collar to make contact at the collar/ polekey and collar/coil interfaces;
The contact on collar/polekey interface is checked filler gauge;
Pressure sensitive films have been used to check the contact pressures on the collar-coil interfaces.
Once the coil-pack assembled with even contact at those interfaces, the shims fill the void between coil, pole key and collar, hereby compensate the size deviation of coil and pole key, and prevent the azimuthal tolerance from propagating out of collar.
Actual tolerance in drawings (profile tolerance)
Collar
Pad
Master (pad)
Master (Yoke)
Yoke
Shell
Tolerance (μm) 25 50
+100
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12. Numerical Models and Methodology
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13. Models overview 2D full size model: 3D model
2D model that allows fast computation of structures modelled with actual tolerances.
Has the feature for asymmetry scenarios such as tolerance applied on only one quadrant.
3D model
Octant model (apply tolerance to all quadrants)
Only used to check the maximum stress in each part with a given tolerance.
All components are in contact using contact elements
Contact172 (2D) / 174 (3D)
Target169 (2D) / 170 (3D)
Baseline Friction coefficient is 0.2
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14. Models’ settings
The models assume the coil in nominal shape (not banana shape) only with profile tolerance.
Both models assume the load key is always in nominal size with an initial clearance of 500 μm to the master (simulate the initial shim thickness between load keys).
Positive tolerance in the model means size increasing, vice versa.
Tolerance applied is 25 μm.
500 μm clearance
Boundary conditions
2D model:
Only constraint on the shell nodes UY=0 at Y=0 and UX=0 at X=0
3D model:
Azimuthal symmetry: at 0 and 45 degree of the assembly
Axial symmetry: Z=0, except for the rod
Rod is pre-tentioned: -810με (MQXFS1) /-1000 με (MQXFA)
UX=0 Y X
UY=0
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15. Tolerance stack-up calculations
The two most common stack-up calculation methods are:
Worst case
∆𝑇= 𝑘=1 𝑛 𝑇 𝑘
The calculation is done by assuming that all the individual tolerances occur at their worst limits or dimensions simultaneously.
RSS (Rot-Sum-Square)
∆𝑇= 𝑘=1 𝑛 𝑇 𝑘 2
The calculation is assuming a normal distribution for component variations.
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16. Cases For the 2D model, with only one quadrant (A) has tolerances :
Only Collar has positive / negative tolerance
Only pad has positive / negative tolerance
Only masters has positive / negative tolerance
Only yoke has positive / negative tolerance
Only shell has positive / negative tolerance
Worst case (positive/negative tolerance)
For both 3D models, check the maximum stress deviation in coils: A 03
104 B D
103 05 C
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17. Tolerance Sensitivity Study
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18. Coil stress variation in 2D solution
σ2p
2D model solutions, tolerance applied in only one quadrant A
Coils reach peak stress after cool-down.
The table lists the stress in the coil 104 in the affected quadrant.
The coil 104 and 03 behave as same; so do coil 103 and 05.
σ2m
σ1p
σ1m
Δσ1p
(MPa)
Δσ1m
Δσ2p
Δσ2m
Collar
Pad
Pmaster
Ymaster
Yoke
Shell
Pos.
Neg.
5.5 5
4.5 25 -
-1.8
-1.5
-1.3
-0.6 6
5.3
4.9
-0.9
4.6
5.8
-2.3 -2
4.2
-2.5
-2.2
-1.6
-0.5
4.4
3.5
-0.7
0.5
0.4
-0.1
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19. Coil stress variation in other quadrant
The coil 103 and 05 have less impact from the tolerance in quadrant A.
Overall trends corresponding to the individual tolerance is same as coil 104 and 03.
Δσ1p
(MPa)
Δσ1m
Δσ2p
Δσ2m
Collar
Pad
Pmaster
Ymaster
Yoke
Shell
Pos.
Neg.
2.6
1.7 1
1.5 25 -
-2.4 -2
-2.1
-1.6
2.4
3.1 2
1.8
-1.1
-2.7
-1.4
-1.5
1.1
2.2 3
-2.5
-1.8
-1.9
3.3
2.1
1.6
1.9
1.2 -1
-0.9
-0.5
-1.2
-0.8
-0.6
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20. Tolerance sensitivity of each part
Tolerance closer to the coil has relatively bigger impact on the stress of the coil in the tolerance quadrant.
The master attach to the pad is slightly more sensitive to the tolerance; the yoke and shell are all less sensitive.
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21. Shear stress deviation in the pole key
If only quadrant A has tolerance, it will break symmetry with same interference all round. The pole keys between the quadrant A and the rest quadrants will consequently suffer additional shear force due to the asymmetry.
ΔU is the additional deformation due to tolerance in the asymmetry model. It cause additional shear force on the pole key.
Modulus of G10 in the azimuthal direction is 30 Gpa.
Tolerance is 25 μm. ΔU
The additional shear stress on the pole key varies with different parts;
Δσxy in the worst case with tolerance stack-up (275 μm) could be about 14 MPa.
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22. Max. stress deviation in structure components
The maximum stress in the structure components spears in excitation. In the analysis, the coils are assumed as in nominal size, that implies the analysis ignores the coil shift due to the broken symmetry. Further calculation will involve detailed coil model with cables to calculate the field with shifted coil.
Tolerance is 25 μm in the model.
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23. Worst Case and RSS
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24. Coil stress in worst case
For 25 μm tolerance on ID and OD of each part, the overall tolerance stack ups is computed as +/- 275 μm by the expression:
The deviation bandwidth of the coil azimuthal stress (the coil in the toleranced quadrant) is about 32 MPa.
∆𝑇= 𝑘=1 𝑛 𝑇 𝑘
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25. Coil stress in RSS
For 25 μm tolerance on ID and OD of each part, the overall tolerance based on the statistical method is computed as +/- 83 μm by the expression:
The deviation bandwidth of the coil azimuthal stress (the coil in the toleranced quadrant) is about 9.6 MPa.
∆𝑇= 𝑘=1 𝑛 𝑇 𝑘 2
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26. Max. coil stress deviation in 3D solution
3D model solutions
This is cross checked in the 3D model with same tolerance on each part.
The maximum stress in coil is located close to the ends.
ΔσMAX
(MPa)
Collar
Pad
Pmaster
Ymaster
Yoke
Shell
Pos.
Neg.
4.3 25 - -5
4.1 -3 6
-10
4.5
-9.5
3.9
-6.6
-4.1
2.2
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27. Analysis Next steps
Model the coil with cable to allow compute the field with deformed coils and tolerances.
Analyze the strain on shell and coil vs. tolerance.
Field quality related to the mechanical tolerance and deformations.
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28. Summary
The actual assembly process could deal with the azimuthal tolerance build ups in the coil-pack assembly.
According to the tolerance sensitivity study, the yoke and shell have the least sensitivity on tolerance. Tolerance on the other parts have influence to the coil stress.
The analysis gives a guidance that estimate the coil stress under a given tolerance.
Keep in mind that this analysis is theoretical. The tolerance impact will strongly dependent on the actual assembly process.
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