# Presentation: 170349

## A Study of LHC Dipole Shape Stability: Mechanical Models and Experimental Investigations

***M. La China***

***Acknowledgments:***
***W. Scandale (CERN), P. Ferracin (LBNL), J. Peña (UCB)***

## OUTLINE

**Cold Mass Geometry**
**Description**
**Production Statistics**
**Dimensional Inspection Procedure**
**Shape Variations under Operating Conditions**
**Analytical Model**
**Description**
**Simulation of Corrective Procedures**
**Study of Vertical Shape**
**Finite Element Model**
**Description**
**Position Inside the Cryostat**
**Inertial Loads During Transport**

## The LHC Main Dipole

**Superconducting (NbTi ) magnet (2in1)**
**Length x Width: 15.2 x 0. 57 m**
**Mass: 28 ton**
**Nominal performance: 8.3 T @ 11.8 kA, 1.9K **
**Longit. & radial discontinuities**

**SC Coils **(NbTi,Cu)

**Stainless steel Collar **(structural properties)

**Iron yoke**** **(magnetic properties)

**Stainless steel Shrinking cylinder**
- (Helium vessel and structural properties)

**Apertures: V1,V2**

- Cold Mass (short prototype)

**3 composite material (GFRE) supports:**

**COLD MASS GEOMETRY**

*1.9 K*
*7.5K*
*57.5 K*
*300 K*

## Cold Mass Geometry

**Structure Geometry (curvature) fundamental for beam dynamics**
**SC Coil Geometry**
**Mechanical Clearance**
**Corrector Positions (sextupoles)**
**Flange positions**
**Bent part: 2812.36 m radius, 5.09 mrad**** ****apical angle.**
**Tolerance: ****±1mm**** ****(bent part)**** ****±0.3mm**** ****(straight part) **

- Aperture axis, horizontal plane (curvature strongly magnified)

**COLD MASS GEOMETRY**

- 408 mm

## Geometric Parameters

**Two main parameters to identify shape correctness:**
**Sag ****(****clearance, coil curvature****)**
**Nominal: 9.14 mm. (********Sag=error on sag)**
**End positions ****(****correctors, flanges****)**
**Nominal: (0,0) in transversal (X,Z) plane. **

**Geometry checks after: ****	****ASSEMBLY ****in industry****, ****	****TRANSPORT ****to ****CERN****,****	****TESTING ****at ****CERN****.**** **

*We consider sextupoles**and flanges solid with end cover.*

**COLD MASS GEOMETRY**

- 408 mm

## Production Statistics

**Flange positions**

**Sag error (********Sag)**

**After the assembly Cold Masses fulfill geometry specifications:**

## Dimensional Inspection Procedure

**Cold Mass shape is measured by a laser tracker and a mole (with reflector) traveling inside the two apertures**

- Laser tracker

- Reflector

**Data from different stations are merged through a least square fit of a set of reference points (Network)**

| Laser tracker accuracy | Distance resolution | Angle resolution | Reproducibility |
| --- | --- | --- | --- |
|  | 1.26 m | 0.14” | ±5 m/m @2 |

**Each aperture is measured from opposite sides: ****total of 4 ‘Stations’ (multi-station procedure)**

## Saw Tooth Effect

**Measurements are analyzed in terms of deviations from theoretical geometry:**

**A saw tooth noise generally affects all the measurement**

**Saw tooth disappears if measurement from opposite stations are plotted separately**

**The curve difference shows a roto-translation b/t measurements**

**The discrepancy b/t curves may be an intrinsic feature of the multi-station procedure**

**OBSERVATI0N**

## Simulation Outcomes

**We propagated the error introduced by Laser tracker accuracy**

**Saw tooth effect is an intrinsic feature of the multi-station procedure**

**We implemented a Montecarlo simulation of the whole measuring procedure**

**0.1mm saw-tooth can be considered acceptable**

**SIMULATION**

## Shape Variations under Operating Conditions

**We monitored Cold Mass deformations and displacements in an LHC full-cell prototype (String2) induced by:**
**Cool Down, Warm up, Energization**
**Quenches**

**Y**

**X**

**Z**

**3 Dipoles equipped with 3 displ. sensors per side: **
**Displ. in X,Z plane**
**Diameter variation**
**Customized sensor (optic):**
**Sensitivity greater than 1·10****-5****m**
**300K thermal gradient **
**Vacuum**

- mirror

- optical head

**300k**

**1.9k**

- String2 top view

- cryostat

- cold mass

## 300-1.9 K Cool Down Effect

**Diameter contraction follows temperature profile**
**Horizontal displacement is small but erratic during transient**
**Periodic small displacement not understood**

**OBSERVATI0N**

*Here shown:*
- MB6, NC-side
- C. Down: Slow
- Tmax=80K

| After Cool Down Summary |  |
| --- | --- |
| X | <0.1 |
| Z | ~0.9 |
| d | ~1.2 |

******Z, ********d match prediction**

## Energization Effect

**No significant displ./def. during or after ramps**

**ENERGIZATION up to ****5****,**** 8****,**** 10.8****,**** 11.85 ****kA**

**Both diameter and horizontal displacement follow temperature ramp**

*Shown here:*
- MB2, NC-side
- Imax:11850 A
- Ramp rate:± 10A/s

**OBSERVATI0N**

## Quench Effect 1/2

**Displacement is only partially related to the current**
**Recovery  (within 0.01 mm) time ~6 hours!**

**Diameter follows current ramp**

*Shown here :*
- MB2, NC-side
- Quench at 11850 A

**QUENCHES provoked at ****6****,**** 9****,**** 11.85 kA**

**OBSERVATI0N**

**Notes:**
As soon as the quench starts and the current is cut by the protection system

## Quench Effect 2/2

**Temperature sensor located near displacement sensor**
**Horizontal displacement is driven by thermal phenomena**

| Quench Summary | average | standard deviation | # of meas. |
| --- | --- | --- | --- |
| X max | 0.07 mm |  0.02 mm | 23 |
| Recovery time | 3.8 hours | 0.8 hours |  |
| Z max | -0.24 mm | 0.03mm | 5 |
| Recovery time | 3.6 hours | 0.7 hours |  |

**OBSERVATI0N**

## Analytical Model

**Description**
**Simulation of corrective procedures**
**Study of vertical shape**

## Analytical Model Description 1/2

**Slim, homogenous 2D beam on three supports**

**INPUT (boundary conditions):**
**distributed load (weight)**
**concentrated loads (moments and forces applied at the extremities)**
**constrain displacements at feet and extremities, rotations at the extremities**

**OUTPUT**
**deformed shape**

**WATCH OUT: ‘beam’ like ‘bar’, NOT like ‘ray’...!**

## Analytical Model Description 2/2

**Analytical Model Description ****2/2**

**Assumptions:****¬**** no Cross-Talk b/t horizontal and vertical planes****¬ ****equivalent homogenous material**** ****(for small ****	****deformations/displacements)**

**Dipole Cold Mass: ****¬ ****3D**** ****¬**** discontinuous****¬**** non-homogenous**
**Beam model:****¬**** 2D****¬**** continuous****¬**** homogenous**** **

- Cold Mass (short prototype)

- Cold Mass cross-section

## Possible Corrective Actions

**Cold Mass shape after****test may vary from****the original one**
**Shape corrections are****achieved by shifting****and blocking the central****support w.r.t. cryostat.**
**Two approaches are possible:****1****) ****‘Push Per Magnet’****: adjust ****each**** magnet to its original shape by a specific support shift. ****2****) ****‘Push Per Firm’****: adjust the ****average**** of a homogenous set of magnets, common support shift within the set.**** **
**To compare the two options we simulated the procedures on all the magnets in the geometry database.**

- Cold Mass inside thecryostat (vacuum vessel)

- Transverse shift

**Notes:**
APPLYING

## Per Magnet Vs. Per Firm

*Dipole: ****2005, ****aperture: ****1, ****plane: ****horizontal***
*target shape: ****original******, ****current shape: ****after tests******, ****final shape:** ****corrected***

*Correction: ****Per Magnet***
*Foot shift:**** tailored***

*Correction: ****Per Firm***
*Foot shift:**** averaged******over all the magnets******in the set***

**SIMULATION**

**Notes:**
Specifically computed for this magnet

## Comparison Results

| Extremity displacements (200 magnets)(limits [mm]: 0.3 mean, 0.5 Std) |  |  |
| --- | --- | --- |
| Correction | Mean [mm] | Std [mm] |
| None | 0.263 | 0.467 |
| Per Magnet | 0.001 | 0.081 |
| Per Firm | 0.008 | 0.371 |

**The correction ‘****Per Magnet****’ removes both the systematic and the random deformation components, expensive procedure.**
**The correction ‘****Per Firm****’ drastically reduces the systematic and slightly reduces the std allowing greater margin of error for the forthcoming production, with a reduced cost.**

**SIMULATI0N**

**Notes:**
La corr per magnet e’ ideale ma costa in tempo e uomini eccessivo

## Study of Vertical Shape

**Fabricated Cold Masses feature a vertical deflection bigger than what expected**
**Possible cause:****Intrinsic shape systematic fault**
**We used the model to remove the effect of gravity and extract the intrinsic Cold Mass shape ****(****effect of gravity is always present in measurements****)****.**

**Notes:**
Systematic shape fault

## Best Fit of vertical deflection

**Effect of gravity is driven by EI (flexural rigidity)**
**We found EI by fitting the model to the vertical shape measurements (set of CM measured on 2&3 supp.)  **
**We subtracted gravity effect and analyzed the intrinsic shapes**

**SIMULATI0N on OBSERVATIONS**

**Notes:**
As the model gives the deflection due to gravity

## Vertical Shape Study Outcomes

**No systematic shape fault evident in vertical plane ****(****31 magnets analyzed****)**
**Equivalent **EI** 20% smaller than what expected****(****effect of tangential sliding b/t laminations currently under investigation****)**** **

## Finite Element Model

**Description**
**Behavior Inside the Cryostat**
**Inertial Loads During Transport**

## FEM Description 1/2

**FEM Description ****1/2**

**No electric parts are modeled**
**No longitudinal or radial ****discontinuity b/t components**
**Foot–cryostat interface:**
**Solid**
**Sliding w/o friction**
**Sliding w/ friction**

**Cryostat**

## FEM Description 2/2

**Collar**** and ****Yoke**** materials are orthotropic: specific********xy****, ********yz**** and ****E****x**** ****to take into account laminations**** **
**Foot**** material is orthotropic: equivalent ****G****zx**** ****and ****G****zy**** to take into account fiber ****orientation**** **
**Azimuthal pre-stress**** in the shrinking ****cylinder is implemented through ****differential thermal contraction:****equivalent ********x****, ********y****, ********z****.**** **
**Mesh density of Cold Mass, Foot****and Cryostat is parametric, ****independent long. azimuth. and radial**

**Z**

**X**

**Y**

**model results**** in terms of displacements are the same for the two different mesh densities**

## Cold Mass/Cryostat behavior

**In operative configuration Cold Mass is not accessible: fine position tuning is done on the cryostat**
**Mechanical behavior of Cold Mass inside the cryostat is crucial (driven by foot/cryostat interface)**
**A specific test was set up to study the Cold Mass response to transversal forces applied in the ends ****(****Cold Masses are connected by the ends****)**** **

- Cryostat

- Cold Mass

- F

- F

## Response to Transversal Force

**We found an elastic linear force/displacement relation in the extremity thus we defined a transversal stiffness (useful to study interaction between adjacent dipoles)**

**By comparison with the FEM model we found that no sliding of the central foot is activated up to 23 kN per side  (i.e. Foot-cryostat friction coefficient >0.16)**
**Friction was under-estimated**

**SIMULATI0N Vs. OBSERVATIONS**

**Measurements match well FEM with central foot blocked**

**Notes:**
...then we compared the measurements with the model and we found a good match only with the model with the central foot blocked. This means...

## Effect of Inertial Loads

**Cryo-Dipole road transport and handling can considerably affect Cold Mass shape and the integrity of critical components (composite support posts)**

| Transport & Handling Specifications |  |  |  |
| --- | --- | --- | --- |
|  | Long (Y) | Lateral (X) | Vertical (Z) |
| Acc. [m/s2] | 4 | 5 | 7 |

**We studied the effect of transverse, longitudinal and vertical inertial loads with different types of transport restraints (to block CM w.r.t cryostat)**
**We checked:**** foot deformations****,**** Cryostat-Cold Mass relative displacement**** and ****stresses in the restraints****.**

**SIMULATION**

- Nominal restraint

- Light restraint

## Cryo-Dipole with RestraintsFEM Models

**Two different friction coefficients in the support/cryostat interface:**
- 0**:good for supports bad for restraints**
** **∞**:bad for support good for restraints**

**Some of the restraints modeled: **

**Ideal restraint (reference)**

**Cremonian type restraint**

**3 Light restraints**

## Inertial Loads Results

**Stiffer restraints relieve the load directly generated by the acceleration (i.e. in the same direction) on the central support versus the lateral ones.**
**Stiffer constraints can induce load distributions characterized by cross-components as seen for statically indeterminate structures**** **
**Constraints that avoid longitudinal rotation of the Cold Mass are preferable (supports undergo guided bending)**

|  | Lat. | Long. | Vert. |
| --- | --- | --- | --- |
| 1 LR |  |  |  |
| Crem |  |  |  |
| 3 LR |  |  |  |
| Ideal |  |  |  |

- Distribution of Tresca equivalent stresses in the light restraints induced by a 4 m/s2 longitudinal acceleration.

- Restraint performance to feet deformation

**SIMULATI0N**

| good |  |  |  | bad |
| --- | --- | --- | --- | --- |

**The end!**

## Production Statistic

**Two main parameters to identify shape correctness**
**Sagitta (clearance, coil curvature)**
**End position (correctors, flanges)**

**COLD MASS GEOMETRY**

**A statistical analysis of the measurement performed by different groups show general agreement in terms of *****p*****, *****q***** and *****h*****.**

|  | Group 1(10 meas.) |  | Group 2 (14 meas.) |  | Group 3 (14 meas.) |  |
| --- | --- | --- | --- | --- | --- | --- |
|  | Avg | Std | Avg | Std | Avg | Std |
| p [rad] | -3.2E-06 | 1.8E-05  | 8.1E-07  | 6.0E-06  | -2.9E-06  | 9.0E-06  |
| q [mm] | -0.04  | 0.24  | -0.09  | 0.10  | 0.04  | 0.12  |
| h [mm] | 0.114  | 0.067  | 0.094  | 0.045  | 0.079  | 0.033  |

**To trace back the source of the saw tooth effect we simulated the multistation measuring procedure**
**By a Montecarlo based simulation we reproduced the propagation of the error, introduced by the laser tracker accuracy, through the whole measuring procedure. **

## Basic principle of the displacement sensor

**Basic principle of the displacement sensor**

** ****Used to monitor deformations in civil structures.**
** ****Double interferometer in tandem configuration, measuring relative deformations of two optical fibers.**

** **Measurement fiber** attached to the structure and **reference fiber** freely installed nearby**

## Displacement Sensor Customization

**Measurement fiber has been replaced by a ****free light beam**** traveling in the vacuum between the cryostat and the dipole   **
**Temperature gradient: ****~300 K**** **
**Cold mass contracts about 20 mm during cool-down: ****long. displacement**
**Cold mass can ****rotate**
**Measurement in ****vacuum**

**Optical design:**
**Mirror reflector: ******** sensitive to rotations**
**Corner-cube reflector: ******** sensitive to translations**
**Double-pass setup****: ******** ****not sensitive to translations and rotations**

## THEORY

**THEORY**

**The sagitta is computed by fitting the geometric measurement in the horizontal plane with a 2****nd**** order polynomial.**

**2****nd**** order polynomial comes from the difference of two circular arcs (shown data have already been processed)**

## Statistical analysis of flexural rigidity 

**To identify gravity effect we must know the flexural rigidity **EI**.**
**We obtained the **EI** of each Cold Mass by comparing (best-fit) the model and the vertical shape measured on 2 and on 3 supp.**
***We cross-checked deflections on two and three supports***

| 31 magnets | 2 Supports |  | 3 Supports |  | UNITS |
| --- | --- | --- | --- | --- | --- |
|  | MEAN | STD | MEAN | STD |  |
| EI | 141 | 16 | 133 | 26 | MPa·m4 |
| ZS1   ZS2   ZS3 | 7.28   7.21   - | 1.19   0.98   - | 1.84    1.31   1.81 | 0.76   0.82   0.86 | m·10-4 |
| XS1   XS2   XS3 | 3.69  11.47   -  | 0.09   0.12   - | 2.44   7.60   12.76 | 0.13   0.21   0.22 | m |

**Flexural rigidity values are ~20-25% smaller than what predicted in previous analyses (based on few magnets).**
**Support positions are compatible with nominal values.**

**Best fit variables are **EI** and support positions (vertical and horizontal) (tot.:5 and 7 parameters)**

**Best fit results**

**Notes:**
We used a set of magnets for which the measurement were taken both on 2 and on 3 supp. And we X-checked the results

**A simple 2D beam model can describe the Cold Mass mechanical behavior. Typical applications:**** ****¬**** ****simulate and compare the efficiency of corrective actions **** ****¬**** interpret experimental data **** ****¬**** deduce mechanical properties**** ****¬**** simulate the final geometry of connected magnets **
**Improvement foreseen:**** ****¬**** 3 dimensions **** ****¬**** effect of laminations****(****A new analytical model is currently being validated****)**** **

## Effect of sliding between laminations

**Effect of bending (slim beams):****sections can rotate and remain normal to mean axis**

**Effect of shear (short beams):**** sections remain parallel to each other**

- Driven by*:** *GxyAs

- Driven by: EyIz

**Cold Mass longitudinally laminated structure makes easier a shear-like mechanism **

## Shape Modifications Observed in Operative Conditions-SUMMARY

|  | CONNECTION |  | NON CONNECTION |  |
| --- | --- | --- | --- | --- |
|  | X | Z | X | Z |
| MB2 | 0.02 | --- | -0.04 | -0.07 |
| MB5 | -0.2 | -0.1 | -0.03 | -0.07 |
| MB6 | -0.055 | --- | --- | --- |

**Thermal cycle effect is generally smaller than 0.1 mm (erratic)**
**Quench effect on magnet geometry is systematic (<0.1 mm) and transitory with a time constant of around 4 hours**
**Energization provokes systematic and very small dipole deformations**
**After the String2 warm-up (and 8 months tests)  all the dipoles regain their pre cool-down position within few hundredths of millimeter except for MB5, connection side.**

**OBSERVATI0N**

## Geometry of Connected Dipoles

**In operating configuration dipoles are connected through flexible bellows**
**Bellows transmit forces between adjacent magnets that can modify the Cold Mass extremity positions**
**We used the model to simulate the final geometry of connected magnets **

## Geometry of Connected Dipoles: Simulation Results

**The transversal forces transmitted by the bellows can reduce the positioning error of magnet extremities****(****25% on the mean value, 10% on the *****rms*****)**

## Dipole deformation due to Lorentz forces: model vs. mesaurements

**Lorentz forces change external dimension of the dipole**
**We analyse cold mass diameter variation Vs. the excitation current **
**Two dipoles monitored in test string during two current tests**