1. Cavity support scheme options Thomas Jones 1

2. Introduction Both cavities will be supported by the fundamental power coupler and a number of blade flexures. See 253-meng-fea-013 for a thorough comparison of support options using the RFD cavity. It has been proposed that the coupler and blades have individual bellows for penetration through the OVC and that the supports are then held by a common out of vacuum adjustment plate. 2

3. Analysis - Simplified DQW cavity 1. Only coupler2. Ø4mm Ti Rods 3. Two corner blades (2mm thick) 4. Blades in integration position Blade thickness 30mm 3

4. Analysis Result Again blades can significantly improve performance. Compromising the blade position can double the deformation due to gravity and reduce vibration modes up to 50Hz. Analysis Max Deformation (mm) Max von- Mises stress (MPa) Mode 1 Frequency (Hz) Mode 2 Frequency (Hz) Mode 3 Frequency (Hz) Mode 4 Frequency (Hz) 11.52861.410.911.32159.9 20.08123.514.63233.379.6 30.0168.133.141.357.9171.4 40.03410.825.738.342.2120.5 4

5. Detailed model - Setup Model de-featured to improve analysis efficiency. Tuner assembly removed at this stage as exact tuner configuration unknown and complex. HOM absorbers and FPC left with antennae to resolve modes with their deflection. Material properties applied as appropriate. Fine mesh used on support components. OFHC Copper (Work hardened) 316L SS Grade 2 Ti 316L SS Model fully fixed at base of each ‘foot’. Niobium OFHC Copper (Work hardened) 5

6. Detailed model – Structural Results 6

7. Detailed model – Modal Results Mode 1 – 20.3Hz Sway perpendicular to beam axis. Mode 2 – 25.2Hz Sway in line with beam axis. Mode 3 – 43.1Hz Twist about centre of support structure There are then three modes at 82.1Hz, 82.4Hz and 82.8Hz which are modes of the RF couplers. Note that the 82.8Hz has both the FPC and HOM oscillating. 7

8. Structural Result summary Analysis Max Deformation (mm) Max von- Mises stress (MPa) Mode 1 Frequency (Hz) Mode 2 Frequency (Hz) Mode 3 Frequency (Hz) Mode 4 Frequency (Hz) Detailed model0.04927.720.325.243.182.1 Simplified0.03410.825.738.342.2120.5 Difficult to compare models due to the much increased complexity. Blade thickness of detailed model is 35mm, simplified used 30mm. Deflection is increased by introducing the common adjustment plate. The deflection is ~11µm at the flexure. The stress values are all well within acceptable limits. Due to reduced stiffness of the overall structure the first two vibration modes have reduced. The lower 4 th mode is due to the introduction of the RF couplers. 8

9. Response spectrum Using ‘tdl-1165-meng- cal-0010-v3.0 Cavity support transmissibility’ Calculator outputs the following response spectra for each mode. Maximum amplitude of cavity vibration ~10nm at 43Hz mode. A rough calculation by G.Burt gives 100nm as the limit for vibration amplitude. This calculation should be checked, however, we are currently a factor 10 below. This is based on a ‘worst case’ damping coefficient. 9

10. Increased top plate thickness Analysis Max Deformation (mm) Max von- Mises stress (MPa) Mode 1 Frequency (Hz) Mode 2 Frequency (Hz) Mode 3 Frequency (Hz) Mode 4 Frequency (Hz) Detailed model0.04927.720.325.243.182.1 +10% Thickness0.03935.422.027.543.982.5 10 No significant gain in performance. Mode 1 has increased x 1.5 due to moving closer to a local peak. Mode 3 has reduced by a factor of 2.

11. Conclusion Several options were considered for the DQW support configuration. A flexure blade arrangement again gives significantly improved performance. Moving blades from the optimum position (i.e. in the corners) to the integrated position reduces performance by a factor of 5 for deformation and vibration response. A detailed model was investigated in ANSYS to find accurate modes with blades in integrated position. Note, this was without the tuner assembly which will require a separate study. A transmissibility calculator was developed in EXCEL. This can be used to give displacements by combining ground PSD with modal data. Displacements appear to be well within specification for the given ground data. It is quick to replace this data in Excel therefore we can investigate different vibration scenarios much more efficiently than importing into ANSYS. 11

12. Appendix Validation of analysis using RFD

13. J 0 = Mass x r (distance to pivot) 2 r0.322mdistance to centre of mass m202kgsupported mass G7.50E+10Pashear modulus d6.30E-02mcoupler diameter l3.00E-01mcoupler length t1.50E-03mcoupler wall thickness jpjp 2.7E-07mm 4 polar area moment of inertia j0j0 20.7kg m 2 polar mass moment of inertia ktkt 6.85E+04torsional spring constant of coupler fnfn 9.1HzNatural frequency due to torsion Validation 0.322m

14. Validation FEA First mode @ 7.7 Hz Mostly rotation about coupler axis but with some vertical bending. r0.322mdistance to centre of mass m202kgsupported mass G7.50E+10Pashear modulus d6.30E-02mcoupler diameter l3.00E-01mcoupler length t1.50E-03mcoupler wall thickness jpjp 2.7E-07mm 4 polar area moment of inertia j0j0 20.7kg m 2 polar mass moment of inertia ktkt 6.85E+04torsional spring constant of coupler fnfn 9.1HzNatural frequency due to torsion Difference of 1.4Hz between calculation and FEA. This is due to the FEA model calculating some bending of the shaft in the vertical orientation, whereas the empirical model is based purely upon rotation/torsion of the shaft.