MKQXF FEA Model Haris Kokkinos www.feacomp.com

Key Features Shell (15mm) Loading plate Collaring Shoe Collaring Key Ground Insulation Trace Filler wedge Aluminum Bars Tapered mid-plane 150 mm apperture 12° taper Haris Kokkinos www.feacomp.com

Shimming Positions Pole shim Pole adjustment shim Mid-plane shim Collar-Yoke radial shim Coil-Collar radial Shim Haris Kokkinos www.feacomp.com

Magnetic Analysis Haris Kokkinos www.feacomp.com

Magnetic Analysis @ 17.3 kA 11.82 T, 140 T/m Haris Kokkinos www.feacomp.com

@ 19.25 kA 13.083 T, 155 T/m Haris Kokkinos www.feacomp.com

Structural Analysis Haris Kokkinos www.feacomp.com

Material Properties St. Steel (5) - Collars St. Steel (10) - Shell E(293K) = 195 Gpa E(4.2K) = 215 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 9.215e-6 St. Steel (10) - Shell E(293K) = 195 Gpa E(4.2K) = 215 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 9.897e-06 Aluminum (55) - Bars E(293K) = 70 Gpa E(4.2K) = 79 Gpa Prxy(293K) = 0.34 Prxy(4.2K) = 0.34 αx= 1.45e-5 St. Steel (52) - LoadingPlates E(293K) = 193 Gpa E(4.2K) = 210 Gpa Prxy(293K) = 0.28 Prxy(4.2K) = 0.28 α= 9.83e-06 St. Steel (11) - Pins E(293K) = 195 Gpa E(4.2K) = 215 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 9.897e-06 Aluminum Bronze (53) - Wedges E(293K) = 110 Gpa E(4.2K) = 120 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 αx= 1.08e-5 Titanium (57) - PoleWedge E(293K) = 130 Gpa E(4.2K) = 130 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 6.03e-06 St. Steel (52) – Collaring Shoe E(293K) = 215 Gpa E(4.2K) = 195 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 9.215e-06 St. Steel (6) - Keys E(293K) = 190 Gpa E(4.2K) = 210 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 9.01e-06 Magnetic Steel (7) - Yoke E(293K) = 210 Gpa E(4.2K) = 225 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 α= 6.894e-06 Coil (51) Ex(293K) = 52 Gpa Ex(4.2K) = 52 GPa Ey(293K) = 44 Gpa Ey(4.2K) = 44 Gpa Gxy(293K) = 21 Gpa Gxy(4.2K) = 21 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 αy= 1.16e-5 αx= 1.07e-5 G10 (56) - Insulation E(293K) = 30 Gpa E(4.2K) = 30 Gpa Prxy(293K) = 0.3 Prxy(4.2K) = 0.3 αx= 2,44e-5 Haris Kokkinos www.feacomp.com

Lorentz Forces Magnetic Mesh Structural Mesh Mapped Forces Haris Kokkinos www.feacomp.com

Parameters’ values used for the results presented below: Haris Kokkinos www.feacomp.com

Azimuthal Stress Coil Under Press, @ 293 K After Collaring, @ 293 K It’s very interesting that the collared is not really affected by friction. Maybe this is due to the small displacement (0.05mm) applied during collaring (compared to the 11T, 0.1mm). But the coil stress is also not really affected by friction in the 11T. So this is consistent. After Shell Welding, @ 293 K @ 4.2 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Azimuthal Stresses Coil Haris Kokkinos www.feacomp.com

Radial Deformation Coil Haris Kokkinos www.feacomp.com

Von-Mises Stress Collars After Shell Welding, @ 293 K @ 4.2 K, 155 T/m @ 4.2 K @ 4.2 K, 140 T/m The areas with stress over 300 Mpa are indicated in grey colour and these are the areas around the keys that typically plastify during the assembly process. Haris Kokkinos www.feacomp.com

Deformation Collars Under Press, @ 293 K After Collaring, @ 293 K After Shell Welding, @ 293 K @ 4.2 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Von-Mises Stress Yoke After Shell Welding, @ 293 K @ 4.2 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Reaction Forces Collar-Yoke After Shell Welding, @ 293 K @ 4.2 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Contact Pressure Half Yokes Tapered After Shell Welding, @ 293 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Contact Pressure Gap Controllers After Shell Welding, @ 293 K @ 4.2 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Azimuthal Stress Shell Detailed View: Mean Azimuthal Stress 155 MPa After Shell Welding, @ 293 K @ 4.2 K @ 4.2 K, 140 T/m @ 4.2 K, 155 T/m Haris Kokkinos www.feacomp.com

Sensitivity Analysis Haris Kokkinos www.feacomp.com

Tolerances Yoke ± 3,8 μm Collar ± 6,3 μm Angle of loading plate ± 0,2° Coil (outside diameter) ± 75 μm The sensitivity analysis is conducted by utilizing the DOE method (Design of Experiments) which determines sampling points on a response surface About 8 input parameters have to be checked. About 100 variations of the model have to be run Up to now, we have solved for 6 input parameters, ,corresponding to 45 variations Haris Kokkinos www.feacomp.com

Tolerances Response Surface Haris Kokkinos www.feacomp.com

Tolerances Sensitivity Chart Haris Kokkinos www.feacomp.com

Conclusions Haris Kokkinos www.feacomp.com

Conclusions New mechanical design concept for a Nb3Sn quadrupole, suitable for industrial production The FEA shows that the assembly is well controlled, with all parts staying generally within the accepted stress levels Coil pre-stress is applied in a well-controlled manner and is easily tuneable with shims The pole-loading concept allows to position the coil’s peak stress at the poles, and the min stress at the mid-plane Assembly is possible with a dipole-type press, due to the tapered-shape mid-plane poles and the dipole-type collars The Al-bars act like gap controllers at RT, while at cold the tapered - horizontally split iron yoke with a closed gap provides a rigid structure The coil is symmetric at operating temperature Robust parametric, FE multi-physics analysis Next steps: 1) Sensitivity analysis 2) Alignment features at the collar – yoke interface 3) Report for phase II Reference: M. Karppinen, “New Mechanical Concept for Nb3Sn Quadrupole”, CERN-ACC-2014-0244. Haris Kokkinos www.feacomp.com

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