1. 56 MHz SRF Cavity and Helium vessel Design
C. Pai

2. Major Components in the 56 MHz SRF cavity and Cryostat
Nb Cavity/Ti Helium Vessel
Mechanical Tuner
Cryostat Vacuum Chamber

3. Four Concerns 56 MHz RF Cavity Geometry and Concerns Multipact
Cavity geometry: Inside surface
Date: 11/5/08 from X. Chang.
Dimension in Room temperature
Four Concerns
Multipact
Sensitivity
ASME code
Tuning

4. 56 MHz Cavity Drawing, #

5. Final Assembly DWG 71018695 Tuning Plate Bellow Corrugation
Helium Vessel
Cavity Gap
Cooling Channel
Stiffening Bar
SRF Cavity
HOM port
Fundamental Port

6. Weight of Full Cavity with helium vessel
PRO-E model
VOLUME = e+03 INCH^3
SURFACE AREA = e+04 INCH^2
AVERAGE DENSITY = e-01 POUND / INCH^3
MASS = e+02 POUND
By calculation,:
Nb cavity: 510 lb
Ti Helium vessel: 249 lb
Total weight : 759 lb

7. ASME Code section VIII, division 2, Design by Analysis
4 failure modes to be checked for compliance.
1. Protection Against Plastic Collapse
Load includes maximum pressure (External and Internal) and Weight of the Vessel, Check Membrane and bending stress.
2. Protection Against Local Failure
Load includes: Mechanical Load (Primary) and Thermal load (Secondary) At any point,   σ1 + σ2 + σ3 <  4 S Where   σ1, σ2, σ3 are the local primary membrane plus bending principle stresses at any point. S is the allowable stress of material.
For Niobium S=4,666 psi, (2/3 of 7000 psi), 4S= 18,666 psi
3. Protection Against Collapse From Buckling
4. Protection Against Failure From Cyclic Loading, including Ratcheting assessment.

8. Integral RF cavity/Helium Vessel Model
Pressure load: 20 psi
Weight: 760 lb
Temperature: at 4.5 K
a. Membrane Stresses:
Stress result in the mid plane of shell element
b. Membrane plus bending stresses:
Stress results in the top or bottom surface

9. Integral Pressure Vessel, Under Weight + Pressure
Maximum primary membrane Von Mises stress in RF cavity
Nb RF cavity
Maximum Stress: 3,712 psi
Allowable: 4,666 psi (Nb)
(psi)

10. Integral Pressure Vessel, Under Weight + Pressure
Maximum primary membrane von Mises stress in Helium vessel:
Helium vessel:
Maximum Stress: 5,869 psi
Allowable: 23,333 psi (Ti-II)
psi

11. Integral Pressure Vessel, Under Weight + Pressure
Maximum primary membrane stress in the tuner end bellow:
tuner end bellow
Maximum Stress: 4,569 psi
Allowable: 23,333 psi (Ti-II)
(psi)

12. Buckling Analysis, First Mode Shape of Eigen Buckling
Loading Pressure in calculation: 1 psi
Calculated Eigen Buckling Pressure: 279 psi
Multiplication factor: 279/20=13.95
13.95 > 2.5 (design factor), No buckle
ASME Allowable Buckling Pressure:
Design Factor: φ= (2/βcr)
where βcr=.8 for cylinder, φ= (2/βcr)=2.5
Note: 16 Stiffener bars has reinforcing band to increase buckling load

13. Buckling (Eigen value) of RF cavity Head
Loading pressure in calculation, 20 psi
Multiplication factor: 58.1
58.1 > 16.1 (design factor), No buckle
Mode shape
ASME Allowable Buckling Pressure:
Reduction Factor: φ= (2/βcr)
where βcr=.124 for torispherical head, φ= (2/βcr)=16.1

14. The 56 MHz RF cavity Satisfy provisions of paragraph 5.5.2.3, Method A
That fatigue analysis is not required
Condition A: Minimum tensile stress is not exceeding 80,000 psi.
Total expected number of cycles of type (a) plus type (b) plus type (c)
plus type (d) does not exceed 1000.
Type (a): the expected number of full range pressure cycles including start up
and shut down. (4x 20 years)
Type (b): the expected number of operating pressure cycles in which the range
of pressure variation exceeds 20% of design pressure. (0)
Type (c): the effective number of change in metal temperature between any two
adjacent joints. (same as start up, 4x 20 years)               
Type (d): the number of temperature cycles for component involving welds
between material having different coefficient of expansion is that which
causes the value of (alpha1-alpha2) x del T to exceed
(same as start up, 4 x 20 years)
Total cycle number=(a)+(b)+(c)+(d)= 240. The projected cycle number of 56 MHz RF cavity operation won’t exceed  Fatigue analysis is not required. Also, the stresses are all in elastic range there is no ratcheting concern.

15. Various Local features Stress analyses
Local Features of Pressure Vessel
Fundamental Damper Port
Nb Crown Head
Tuning Plate
Fitting
Cooing Channel Bellow
Cavity Support Lug

16. Force in the Fundamental damper port
Internal Helium Pressure: 20 psi
Expanding force in the bellow: 340 lb
End flange pushing force: 555 lb
4. Total Force in the End flange and port: 895 lb
P=20 psi
Total F=895 lb

17. Fundamental port (Nb cavity) Bending stress at bottom surface
S: 6,365 psi
Allowable: 7,000 psi
psi

18. Fundamental Damper port (Helium vessel), Under Weight + Pressure
Membrane stress in the bellow
Maximum Stress: 4,993 psi
Allowable: 23,333 psi (Ti-II)
psi

19. Crown Head Under Pressure Load
Primary membrane plus bending stress (psi)
Maximum Stress: 4795 psi
Allowable: 7,000 psi (Nb)
psi

20. Tuning Plate, Von Mises Stress (push down 1.5 mm + 20 psi in channel)
Max Von Mises stress: 6,980psi
Yield stress of Niobium : 7,000 psi
psi

21. Cooling Chanel Fitting under Pressure Load
Membrane plus bending stress (Von Mises Stress)
Material: Titanium
Max. Von Mises Stress: 331 psi
Allowable: 23,333 psi
psi

22. Cooling Channel Bellow
Material: Titanium
Thickness: “
ID: .50 “
OD: .75 “
Pitch: .075”
Yield Strength of Ti:
Sy=40,000 psi
Allowable :
2/3 of yield:
Sa=26,666 psi
Design based on
Calorstat Formed bellow:
Code # ,

23. Deflect-6 mm and sideway Spring Rate, Lateral
K=F/Δ L
=.382x2 lb/.236”
=3.24 lb/in
sideway

24. Spring Rate, Vertical
K=F/Δ L
=3.468 lb/.1”
=34.68 lb/in

25. Maximum stress under internal 20 psi pressure+ 1.5 mm tuning
Bellow expanding force: 2.1 lb
Tuner force:
F= .195 lb
1.5 mm tuning plus 20 psi,
Membrane plus bending stress =11,225 psi

26. Buckling mode shape (Squirm), 1.5mm tune + 20 psi
P=20 psi
Eigen value: 4.338
Critical Pressure load: psi

27. Large deflection, 20 psi, .25” (6.35mm) sideway
Titanium Gr.2
Sy=40,000
Seqv=35,623 psi

28. Large deflection, 40 psi, .10” (2.54mm) sideway
Titanium Gr.2
Sy=40,000 psi
Seqv=18,366 psi

29. Vessel Support structure Under hanging weight
Primary Membrane stress
Maximum Stress: 3128 psi
Allowable: 23,333 psi (Ti-II)
F=300 lb
From Nitronic Rod
F=300 lb
From Nitronic Rod
psi

30. Vessel Support structure Under hanging weight
Von Mises Stress, Membrane plus bending stress
Max. Von Mises Stress: 4,073 psi
Allowable: 35,000 psi (Ti-II)
psi

31. Frequency Sensitivity And Port Shape effect of SRF Cavity
C. Pai

32. RF Frequency Sensitivity to Helium Pressure Fluctuation
Fluctuation of helium pressure will change the shape of the RF cavity vessel. When the shape of the cavity changes the resonance frequency will change too.
A stiffened RF cavity will reduce its sensitivity to
helium pressure fluctuation.
Parameters used in calculation:
Resonance frequency at Free state:
Hz (calculated by ANSYS model )
Applied Helium Pressure: 1 atm (1.013 bar)
Helium stability: +/- 1 mbar

33. Model Used In RF Frequency Sensitivity Calculation
Finite Element program:
ANSYS Multiphysics
RF Cavity Model
Cavity/helium Vessel Model
Due to structural symmetry,
only 1/16 of the cavity is modeled to calculate the Resonance frequency

34. Electric Field (E) Plot at Free State RF Resonance Frequency
Normalized Electric field (E ) plot at
Resonance Frequency at Free State
Freq (Hz)

35. Magnetic Field (H) Plot at Free State RF Resonance Frequency
Normalized Magnetic field (H ) plot at
Resonance Frequency at Free state
Freq (Hz)

36. Deform Shape of RF Cavity Due to Helium Pressure (1 atm)
Deform shape of the vessel
Unit in meter
Deform shape of the RF cavity
Under 1 atm pressure ( bar)
Unit in meter

37. RF Sensitivity Due To Helium Pressure Fluctuation
Normalized Electric field (E ) plot at
Resonance Frequency, under 1 atm
Freq (Hz)
∆F=
= Hz 1 atm or bar)
Sensitivity=.282 Hz/mbar

38. RF Frequency Change Due To Lorentz Detuning
Lorentz Pressure:
During operation, magnetic field and electric field will produce pressure in the RF cavity wall.
Pressure from Magnetic field:
P= ¼ μoH2 , μo is permeability of vacuum
Direction: push out the RF cavity wall.
Pressure from Electrical field:
P= ¼ εoE2 , εo is permittivity of vacuum
Direction: Pull in the RF cavity wall.
Total Lorentz Pressure: P= ¼ (μoH2 –εoE2)
H Field
E Field

39. Lorentz Pressure Pressure unit: MPa Lorentz Pressure:
P= ¼ (μoH2 –εoE2)
Based on
Max. E =42.5 MV/m
Max. H= 85 KA/m
Min. P= MPa
Pull in
Max. P= MPa
Push out
Pressure unit: MPa

40. Deform Shape of RF Cavity Due to Lorentz Pressure
Deform shape of the vessel
Deform shape of the RF cavity
Under Lorentz pressure
Unit in meter

41. RF Frequency Change Due To Lorentz Detuning
Normalized Electrical field (E ) plot at
Resonance Frequency Under Lorentz Pressure
Freq Hz
Frequency Change
ΔF= Hz
(Reduced)

42. Cavity Port shape effect 1. HOM port 2. Fundamental Ort

43. 56 MHz RF cavity Assembly, section view

44. Free State, Electric Field with damper ports, normalized value
FREQUENCY : Hz

45. Free State, Magnetic Field with end ports, based on normalized electric field
Frequency : ,126, Hz
Max.: H=

46. Original shape- Free State, Magnetic Field, without ports
Frequency : ,190, Hz
RF Cavity Frequency Comparison:
Original Shape (No ports):
Frequency : ,190, Hz
With Ports:
Frequency : ,126, Hz
Frequency Change: - 63, Hz
Max. H= .0020

47. Frequency change due to 8 end ports
RF Cavity Frequency Comparison:
Original Shape (No ports):
Frequency : ,190, Hz
With 8 end Ports:
Frequency : ,126, Hz
Frequency Change: ΔF=- 63, Hz

48. 1/16 model, (simulate ½ x 16= 8, Fundamental damper port edge radius:

49. E field: 1/16 model, (simulate ½ x 16= 8 ports), damper port edge radius: .50”
FREQUENCY (HERTZ)= 55,895,247.62
R=.50”

50. H Field, 1/16 model, (simulate ½ x 16= 8 ports), damper port edge radius: .50”
FREQUENCY (HERTZ)= 55,895,247.62 H=
R=.50” H= H=

51. E Field, 1/16 model, no Damper Port (but with end ports)
FREQUENCY (HERTZ)= 56,128,337.68

52. Frequency change due to Fundamental Damper Port
(estimate based on a result of 8 damper ports)
RF Cavity Frequency Comparison:
No fundamental damper ports but with 8 end ports:
Frequency : ,128, Hz
8 fundamental damper ports and 8 end ports:
Frequency : ,895, Hz
Frequency Change: ΔF =- 233, Hz (per 8 port)
Estimate Frequency change by a single damper port:
ΔF=-29,136 Hz

53. 56 MHz cavity Magnetic filed check
Effect of Port Joint Radius In the End plate
Radius: 5 mm and 8 mm

54. Complete End part
Drawing #
(Next page)

55. 5mm radius changes to 8 mm in the port joint
Inner Volume of cavity
5mm changes to 8 mm
radius
3 mm thick cavity vessel

56. Magnetic Field, based on 8mm radius fillet
Max. H:
Max. H:

57. Electric Field, based on 8mm radius fillet
FREQUENCY : Hz
Diff of Freq:
= Hz (Reduced)