1. Review of the Modal Analysis of the MiniBooNE Horn MH1
Larry Bartoszek, P.E.
1/25/00
BARTOSZEK ENGINEERING

2. Overview 1
The pulsing of the MiniBooNE horn sends a power spectrum of vibrations into the horn’s mechanical structure.
Pulse is 143 microseconds long.
The pulse repeats 10 times in a row, 1/15 sec between each pulse, then the horn is off until 2 seconds from the first pulse in train.
The concern is that the frequencies propagating in the horn might match the natural frequencies of the structure and induce mechanical resonances that overstress the horn.

3. Analysis Outline: Vibration
Two step analysis:
Calculate relative amplitudes of frequencies in the pulse spectrum by Fourier analysis
FEA modal analysis of horn to get natural frequencies and normal modes
With this information we can see the implications for horn design.

4. Comparison of NuMI and MiniBooNE pulse spectra
The pulse width determines the dominant driving frequencies
Numi pulse width = 5*10-3 sec
MiniBooNE pulse width = 1.43*10-4 sec
NUMI has a calculated natural frequency for the inner conductor of 358 Hz in the 3 spider support design
The Fourier analysis of the NUMI pulse structure shows no significant frequency components above 200 Hz.
They should not have a resonance problem with their inner conductor.

5. Chart of all the Fourier Coefficient Amplitudes of the NuMI pulse spectrum

6. MiniBooNE Fourier Analysis
We have a different situation:
Significant frequency components out to >5 KHz

7. Graph of Relative Amplitude of Fourier Coefficient Cn vs. Frequency

8. Refinement to the MiniBooNE Fourier Analysis
The previous analyses repeated the 1/15th second repetitions forever and did not include the time off.
During Run II and MiniBooNe running we will be running up to 8 MiniBooNE Booster cycles under a Main Injector stacking cycle which is 22 Booster cycles long (1.467 sec) for an average rep rate of 5.45 Hz.

9. Combining wave forms together to get the pulse structure
These wave forms were multiplied together to get the time structure of the pulse train.
The Booster rep rate was truncated to 66 msec (instead of its actual 66.6) for simplicity. This shifts the fine structure plot power clusters (shown on slide 11) slightly away from 15 Hz multiples but the real numbers are centered on 15 Hz multiples.
Si = Ci*Pi*Ti

10. Details of the sub-structure of the Fourier spectrum
Within the envelopes shown above, the Fourier spectrum has discrete lines because the current pulse structure is periodic with a super-period of sec.
The line spacing in the Fourier spectrum is 0.68 Hz (1/1.467 sec).
There is significant power in the spectrum in clusters separated by 15 Hz.
The 1/2 width of the power clusters is 22/8 * 0.68 Hz=1.9 Hz.

11. Plot of the fine structure of the Fourier Spectrum
Frequency, Hz

12. Implications of the sub-structure of the spectrum
The natural frequency of the inner conductor should not be close to a multiple of 15 Hz.
If the Q of the horn structure is low enough, details of the spectrum sub-structure may not matter because many frequencies around a given natural frequency could excite resonance
Low Q would mean lower amplitude than in a high Q system, but more frequencies can excite the system

13. MiniBooNE Design Concerns
If we have a natural frequency of the inner conductor under the available driving components, the inner conductor may resonate, causing ringing between pulses
The number of rings between pulses will be limited by the available damping.
Ringing effect may increase the number of cycles, affecting the fatigue life
Ringing stresses should be a smaller component than primary thermal and magnetic stresses (if Q is low enough to eliminate serious resonances,) so cycle life of the horn may not be significantly affected.

14. FEA Modal Analysis Results
The next slide shows the finite element model of horn MH1.
The following slide shows the first four mode shapes for horn MH1 without any spiders on the inner conductor.

15. Finite Element model of Horn, half symmetric

16. First Four Normal Modes of Horn with No Spiders

17. The Influence of Spiders
“Spiders” are attachments between the inner and outer conductor to help stabilize the inner conductor and stiffen it.
We created an FEA model of the horn with three spiders.
One spider is modeled as three thin stiff beams connecting the IC and OC, radially oriented and equally spaced around inner conductor in 120° intervals.

18. Reasons to add spiders There are two reasons to have spiders:
Increase the column buckling strength of the inner conductor
Differential thermal expansion of the inner conductor with respect to the outer puts the IC in compression.
The temperature rise of MH1 is not great enough to warrant this reason to have spiders
Stiffen the IC to raise its natural frequency.

19. Disadvantages of Spiders
The main disadvantage of spiders is that they represent material that can lower the yield of the horn by multiple scattering losses.
The perfect horn has no material in the way of the particles being focused.
They also add components to the horn that may fail, affecting system reliability.

20. First Four Normal Modes of Horn with 3 Spiders

21. Potential Problems with Analysis
The stiffness of the spiders modeled is probably not similar to the stiffness of the spider conceptual design we are developing
The design is not far enough along yet to estimate stiffness.
Some of the mode shapes look like they rely on local deformations in the OC or IC that are probably not physically realistic.
This is made obvious in the animations of the mode shapes.
Changing element types shifts frequencies
It’s hard to know the accuracy of the analysis

22. Conclusions from Fourier Analysis and Modal FEA
We have significant frequency components out to about 5 KHz, spaced very closely in frequency.
Adding three spiders raises the fundamental natural frequency of the inner conductor from ~77 Hz to ~206 Hz
Fundamental of no-spider horn is uncomfortably close to a multiple of 15 Hz.
Spiders are not going to raise the natural frequency beyond the range of the pulse spectrum.
Do they do anything useful to justify the loss in yield?

23. Further considerations
The thermal and magnetic axisymmetric analysis of the horn indicated that the middle of the end cap sees ~60% of its allowable stress from the inner conductor pushing on it.
All of the mode shapes show inflections of the curvature of the end cap when the inner conductor vibrates away from the beam axis.
These inflections will increase the stress level in the end cap.
Is this effect significant enough to want to limit it?

24. Two questions about inner conductor vibrations:
What is the additional stress imposed on the end cap from inner conductor vibrations?
Can spiders limit the additional stress on the end cap from inner conductor bending?
We can’t calculate the amplitude of the oscillations because we don’t know the Q of the horn structure.
We think Q is low because of Al construction.
We can estimate the incremental increase in stress in the end cap just from static stress analysis

25. Next Finite Element Model
The half symmetric model of the horn was re-run as a static stress analysis with the spiders removed.
We knew that the static deflection of the IC without spiders just from gravity was .002 inches.
The inner conductor was displaced by .020 inches from the beam axis and we looked at end cap stresses.
We neglected the influence of inertia and assumed the static stress from this amplitude of deflection matched stress from dynamic deflections

26. FEA model of IC displacement

27. End Cap Stress from IC Displacement
Maximum equivalent stress in end cap is 2.33MPa = 338 psi

28. Summary of results:
Maximum axisymmetric stress intensity in end cap from thermal and magnetic forces is 3.84 ksi (26.5 MPa).
Maximum incremental stress from .020 inch offset of IC is .338 ksi (2.33 MPa)
Allowable stress at center of end cap (corrected for moisture and R) = 6.63 ksi
Without offset: Scalc/Sallow = .58
With offset: Scalc/Sallow = .63
% Increase in ratio = 8.6%

29. Conclusions 1:
The incremental stress increase due to a .020 inch offset is still well under the 97.5% confidence stress allowable at 2e8 cycles
The curve of stress increase vs displacement is probably not linear and we should run a series of larger offsets
Looking at the mode shapes with spiders, it is not clear that they are effective
There is still an inflection in the end cap with spiders
The amplitude of the oscillations is probably less with spiders than without

30. Conclusions 2:
We are designing a spider and putting ports on the outer conductor to allow them to be installed.
We plan to make displacement and frequency measurements on the horn without spiders during testing to determine if amplitudes are large enough to want to install spiders
We then plan to make more measurements to determine if the spiders are effective.