Development of a Modal Model of SSR1 Cavity

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Department of Civil and Industrial Engineering Master Degree in Mechanical Engineering

Development of a

Modal Model of the SSR1 Cavity

Candidate:

Riccardo Contini . . . .

Thesis Advisors:

Ing. Bernardo Disma Monelli . . . .

Prof. Marco Beghini

. . . .

Prof. Leonardo Bertini

. . . .

2015-2016 III Degree Session

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Resonators are affected by microphonics, external mechanical vibrations which produce shift in RF frequency and dercrease the beam quality. This report summarizes the results of the modal characterization of the SSR1 niobium cavity supplied by Fermilab. At first the experimental setup used for deter-mining the modal parameters is presented, then the cavity natural frequencies falling in the range 0 ÷530 Hz are reported for a free-free condition. Since a significant portion of the cavity surface cannot be effectively used for deter-mining the cavity vibration response, the identification of the experimental mode shapes has been carried out for the scanned parts only. Therefore, to identify the entire cavity mode shapes has been carried out using a Finite El-ement Modal Model (FEMM) of the of the cavity simulating the experimental free-free conditions. However, the comparison among the experimental and numerical results ha been revealed.

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Sommario

Le cavitá risonanti sono affette da vibrazioni meccaniche esterne, chiamate microphonics che modificano la frequenza RF della cavitá peggiorando qua-litá del fascio. Questo studio sintetizza la caratterizzazione modale del com-ponente SSR1 Niobium Cavity fornita da Fermilab. Inizialmente é descritto il setup sperimentale con il quale sono determinati i parametri modali, a seguire sono riportate le frequenze naturali del componente nel range di fre-quenze 0 ÷530 Hz quando analizzato in una configurazione free-free. Poiché una parte significativa della cavitá non puó essere impiegata nella caratteriz-zazione modale, questa viene condotta solo attraverso le parti scannerizzate. Comunque, un modello agli Elementi Finiti é sviluppato per identificare il comportamento dell’intero componente in condizioni free-free con un buon accordo tra i risultati sperimentali e numerici.

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Acknowledgements

Si ringraziano....

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List of Figures

1.1 A simplified SRF cavity. . . 2

1.2 325 MHz cryomodule. . . 3

1.3 Cavity geometry perturbation. . . 4

2.1 A SSR1 jacketed cavity section view (Fig. 2.1a on page 6) and a translucent SSR1 Niobium Cavity (Fig. 2.1b on page 6) . . 6

2.2 SSR1 bare cavity explodedgeometry. . . 7

2.3 SSR1 dressed cavity exploded view. . . 9

2.4 Input Coupler. . . 10

2.5 The SSR1 cavity tuner. . . 10

2.6 SSR1 Niobium Cavity interfaces with the helium vessel . . . . 12

2.7 SSR1 Cavity interfaces . . . 13

2.8 Photos of the SSR1 cavity being manufactured . . . 14

2.9 SSR1 coldmass. Liquid helium pipeline over the string assembly 15 2.10 Liquid helium pressure fluctuation . . . 15

3.1 A sectioned brazed joint. . . 17

3.2 Measuring Activity . . . 17

3.3 Schematic representation of BPF (Fig. 3.3a on page 18) and SPF (Fig. 3.3b on page 18) . . . 18

3.4 Brick elements . . . 19

3.5 Mesh Cavity . . . 20

4.1 . . . 23

4.2 Experimental Setup Scheme. . . 24

4.3 . . . 25 4.4 . . . 26 4.5 White Noise . . . 27 4.6 NBC DOFs. . . 29 4.7 Measuring Activity . . . 29 4.8 . . . 29 5.1 FRF. . . 31

5.2 1st _{experimental mode shape reconstruction. . . .} ₃₂

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5.3 The 2nd _{experimental mode shape reconstruction. . . .} ₃₂

5.4 The 3rd _{experimental mode shape reconstruction. . . .} ₃₃

5.5 The 4th _{experimental mode shape reconstruction. . . .} ₃₃

5.6 The 5th _{experimental mode shape reconstruction. . . .} ₃₄

5.7 The 6th _{experimental mode shape reconstruction. . . .} ₃₄

5.8 The 7th _{experimental mode shape reconstruction. . . .} ₃₅

5.9 The 8th _{experimental mode shape reconstruction. . . .} ₃₅

5.10 The 9th _{experimental mode shape reconstruction. . . .} ₃₆

5.11 The 10th _{experimental mode shape reconstruction. . . .} ₃₆

5.12 The 11th _{experimental mode shape reconstruction. . . .} ₃₇

5.13 The 12th _{experimental mode shape reconstruction. . . .} ₃₇

5.14 The 13th _{experimental mode shape reconstruction. . . .} ₃₈

5.15 1st _{FEM mode shape section view . . . .} ₄₀

5.16 2nd _{FEM mode shape section view . . . .} ₄₀

5.17 3rd _{FEM mode shape axonometric section view . . . .} ₄₁

5.18 4th _{FEM mode shape section view . . . .} ₄₁

5.19 5th _{FEM mode shape section view . . . .} ₄₁

5.20 6th _{FEM mode shape section view . . . .} ₄₂

5.23 9th _{FEM mode shape section views . . . .} ₄₃

5.24 10th _{FEM mode shape section view . . . .} ₄₃

5.25 11th _{FEM mode shape section view . . . .} ₄₄

5.28 14th _{FEM mode shape section view . . . .} ₄₅

5.29 15th _{FEM mode shape section view . . . .} ₄₅

5.30 Mode Matching . . . 46

5.31 Mode Mismatching . . . 47

6.1 Structured light technic working scheme . . . 49

6.2 Optic Scanner working . . . 49

6.3 The sprayed SSR1 Niobium cavity . . . 50

6.4 An axonometric view of SSR1 post-process elaboration. . . . 52

6.5 A top view of SSR1 cavity post-process elaboration geometry. 52 6.6 Lateral views of ... . . . 53

6.7 Section views . . . 54

6.8 example caption . . . 54

A.1 Aerial View of Fermilab . . . 57

A.2 Project X scheme . . . 58

A.3 RF plants and RF power cost dependences with frequency detuning . . . 58

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B.1 SSR1 Niobium Cavity technical drawing . . . 60 B.2 SSR1 Jacketed Cavity technical drawing . . . 61 C.1 Thermal conductivity dependence on temperature for

Nio-bium 300RR . . . 62 C.2 Helium properties at cryogenic temperatures . . . 63

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List of Tables

2.1 Materials in 325 MHz cryomodule . . . 11

2.2 SSR1 cavity ideal operative conditions . . . 15

5.1 Experimental Frequencies. . . 31

5.2 FEM Numerical Frequencies . . . 39

5.3 Comparison. . . 42

5.4 Sensitivity Analysis. . . 45

E.1 Measured Points coordinates. . . 67

E.2 1st _{to 7}th _{Mode Shape Amplitude . . . .} ₇₀

E.3 7th _{to 13}th _{Mode Shape Amplitude . . . .} ₇₁

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Nomenclature

Abbreviation BP Beam Pipe BP F Bem Pipe Flange SP Side Port

SP F Side Port Flange Greek Symbols

Deformation State in Voight’s form Ω, ω Angular Frequency

Φ Inverse Matrix

σ Traction State in Voight’s form

ζ Damping Factor (relative to critical damiping) qmnij . . .

Indices and Superscripts (. . . )i Related to ith eigenmode

or ith _{component of vector}

(. . . )ij Matrix Element in ith row and jth column

[. . . ]−1 _{Inverse Matrix} Latin Symbols [C] Damping Matrix [K] Stiffness Matrix [M ] Mass Matrix xi

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{¨x(t)} Acceleration { ˙x(t)} Velocity

{f (t)} External Load Vector {x(t)} Displacement

f Frequency j Imaginary Unit p(t) Excitation Signal

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Preface

Finite element analysis is a numerical procedure useful for solving struc-tural mechanics problems. More specifically, it is an analytical method for determining the modal properties of a structure. It is often necessary to val-idate the results from this theoretical prediction with measured data from a modal test. This correlation method is generally an iterative process and involves two major steps. First, the modal parameters, both frequencies and mode shapes, are compared and the differences quantified. Second, ad-justments and modifications are made, usually to the finite element model, to achieve more comparable results. The finite element model can then be used to simulate responses to actual operating environments.

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Introduction

1.1 Aim

Particle accelerators are essential tools for high-energy physics, low-energy to medium- low-energy nuclear physics research and free-electron lasers. They are also used in industry for industrial processes, in medicine for can-cer therapy, for national security, and many additional future accelerator applications are envisioned and under study.

Superconducting radio-frequency (SRF) cavities are crucial accelerating ele-ments of modern high performance particle accelerators. Their dramatically lower electrical losses allow operation at substantially higher duty cycles than conventional copper cavities. In the framework of the so-called Proton Improvement Plan-II (PIP-II), Fermilab is planning to upgrade its accelera-tor complex using SRF technologies to deliver a more powerful and intense proton beam for experiments in neutrino physics. The upgraded complex, by means of the latest advances in particle accelerator technology, will open a path to discovery in neutrino and muon physics, as well as physics beyond the present “Standard Model” that may reveal itself along the way. With a unique proton beam of multi-megawatt power, scientists will perform high precision measurements of rare processes in the hope of uncovering new unexpected phenomena. PIP-II will provide a platform for cutting-edge ac-celerator developments.

1.2 Resonators

A resonator is essentially a closed (or largely closed) metal structure that confines electromagnetic fields in the microwave region of the spectrum. The microwaves bounce back and forth between the walls of the cavity. At the cavity’s resonant frequencies they reinforce to form standing waves in the cavity.

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Figure 1.1: A simplified SRF cavity.

The most common application of superconducting RF is in particle accel-erators. Accelerators typically use resonant RF cavities . Radio-Frequency cavities manipulate charged particles passing through them by application of acceleration voltage. Electromagnetic fields are excited in the cavity by coupling in an RF source with an antenna. When the RF frequency fed by the antenna is the same as that of a cavity mode, the resonant fields build to high amplitudes. Charged particles passing through apertures in the cav-ity are then accelerated by the electric fields and deflected by the magnetic fields.

Limits on operative condition (i.e. thermal load for high duty cycle opera-tion) are a serious limitation of normal conducting cavities and lead to the development of a new concept of resonators, the SRF cavities.

Principle of Physics of a SRF Resonators A simplified diagram of the key elements of an SRF cavity setup is shown in Figure 1.1.

The cavity is immersed in a saturated liquid helium bath. Pumping re-moves helium vapor boil-off and controls the bath temperature. The helium vessel is often pumped to a pressure below helium’s superfluid lambda point to take advantage of the superfluid’s thermal properties. Because super-fluid has very high thermal conductivity, it makes an excellent coolant. In addition, superfluids boil only at free surfaces, preventing the formation of bubbles on the surface of the cavity, which would cause mechanical pertur-bations. An antenna is needed in the setup to couple RF power to the cavity fields and, in turn, any passing particle beam. The cold portions of the setup need to be extremely well insulated, which is best accomplished by a vacuum vessel surrounding the helium vessel and all ancillary cold components.

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1.2.1 SSR1 Cavity

SRF cavities called Single Spoke Resonators of type 1 (SSR1) are fun-damental components of the superconducting linear particle accelerator for the Proton Improvement Plan-II project at Fermilab. In the early of the 2000s, 325MHz SSR1 cavities have been successfully designed and devel-oped. They were optimized for interactions with a proton beam at β = 0.22 and to operate at 325 MHz.

The full SRF cavity containment system is a called cryomodule (Fig. 1.2) and contains eight single spoke resonators cavities SSR1 operating in CW mode at 2 K in stainless steel helium vessels. Each has an integral coarse and fine tuner that operates through a lever system and pushes on the cavity end wall. For ease of maintenance, tuner access covers are incorporated into the helium vessel design.

Figure 1.2: 325 MHz cryomodule.

1.2.2 SRF cavity problem(???)

One of the technical challenges facing operation of SRF cavities is the fact that they are hypersensitive to RF fluctuations. It’s a critical issue if the change in cavity frequency exceeds the bandwidth, which leads to a perturbation in amplitude and phase in the accelerating field. A cavity

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can oscillate mechanically in the acoustic frequency regime due to external excitations (microphonics). These oscillations lead to a deformation of the cavity and finally to shifts of the RF frequency which can be much larger than the bandwidth of the cavity. There are different options to overcome this problem:

• passive

– Reduce pressure variations; – Reduce df /dp;

– Reduce vibration levels;

– Provide more RF power overhead; • active

– Use fast or slow tuner to stabilize dynamically resonant frequency of cavity

Eq. 1.1 analytically shows that the fractional change in a cavity’s fre-quency is proportional to the fractional change in its stored energy (depend-ing from volume, shape, magnetic field H0, electric field E0).

ω − ω0 ω0 ≈ ∆Wm− ∆We Wm− We ≈ R V µ|H0| 2_{− |E} 0|2dv R V µ|H0|2+ |E0|2dv (1.1) (a) (b)

Figure 1.3: Cavity geometry perturbation.

1.3 Modal Analysis

All the ways to overcome the typical problem of SRF cavities have been studied and addressed, wherever possible. The design of the helium vessel was improved to reduce considerably the sensitivity of the cavity resonant frequency to the liquid helium fluctuations (df /dp) and tuning devices were

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developed. Instead pressure variations of liquid helium are estimated as a value of 0.2 mbar for the next generation of accelerator’s superconducting cryomodules. Frequency changes can be only controlled by supplying RF power and it’s not an economically attractive solution, see appendix (ref???). To address the issue of frequency detuning due to microphonics, res-onances of a structure need to be identified and quantified defining the structure’s modal parameters. Resonant vibration is caused by an interac-tion between the inertial and elastic properties of the materials within the structure and contributes to many of the vibration related problems. The liquid helium pumping system is the first external source of microphonics to be taken into account and vibration analysis will be carried on with their operating frequency of 60 Hz.

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Case of Study

In this chapter the SSR1 Cavity (Fig. 2.1a) is described in shapes, pro-cesses and treatment and materials paying particular attention on Niobium Cavity (Fig. 2.1b), the inner component. At the end, the principal source of microphonics, the liquid helium pumping system is introduced.

(a) (b)

Figure 2.1: A SSR1 jacketed cavity section view (Fig. 2.1a) and a translucent SSR1 Niobium Cavity (Fig. 2.1b)

2.1 SSR1 Cavity - Geometry and Components

. The jacketed SSR1 cavity consists of two nested cryogenic pressure vessels: the inner vessel is the superconducting SSR1 cavity, Niobium Cavity (NC), and the outermost vessel is the helium containment vessel, Helium Vessel (HV). SSR1 cavity interfaces the the beam-line , connections along the beam-line are made using diamond seals and flanged joints and bellows.

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SHELL CIRCUMFERENTIALn RIBn(x4) ENDWALL (x2) DONUTnRIB (x2) DAISYnRIB (x12) BEAMnPIPE (x2) BPnFLANGE (x2) SPOKEnCOLLAR (x2) HALFnSPOKE (x2) SPOKEnBEAMnPIPE (x2) VACUUMnPORT nFLANGE TUBEnINLET (x2) COUPLERnPORT FLANGE TRANSITIONnRING Nb-side TRANSITIONnRING Stainless-side

Figure 2.2: SSR1 bare cavity explodedgeometry.

Power Couplers and vacuum system device are mounted on through the side ports flanges.

2.1.1 Niobium Cavity

The Niobium Cavity is the main core of SSR1 Cavity. Its internal vol-ume recreates the theorical RF volvol-ume associated to a 325 MHz frequency. Composed by several parts joined together, after the welding process the cavity is exposed to a tuning process and a chemical washing (thp030) in order to achieve suitable quality. SSR1 cavity take its name from the spoke which spans the inner diameter of the shell and forms two accelerating gaps between the spoke and the endwalls.

Shell

The shell is formed from a plate of 3.15 mm being calendered in order to obtain a cylindrical surface. Previously the edges are prearranged for the welding process and four holes are made: two for the spoke edges and two for the side ports. Lastly the component undergoes to a machine-tool working and a final dimensional inspection.

Spoke

The spoke is the inner component of the resonator and its shape re-minds an hyperboiloid except for the cylindrical hole in which the beam pass through. The spoke is final results of moulding, welding and a machine-tooling processes. Two half spokes and the spoke collars are molded from a 3.15 mm plate while the spoke beam pipe is machine tooled. After that, the half spokes are welded together and drilled in order to prepare the spoke

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beam pipe. At the end the two collars and the spoke beam pipe are welded to the spoke.

Endwall

The shell and the two endwalls (EW) encloses the RF volume into the niobium cavity. Endwall is the final result of a cold-molding and machine-tooling processes of a 3.15 mm plate.

Beam Pipe Port

The Beam Pipe Port (BPP) is the gateway for particle into the resonator. Made by a niobium Beam Pipe Tube (BPT) and a stainless steel Beam Pipe Flange (BPF) brazed together it undergoes to final machine-tooling process in order to realize the diamond seal house. Through the BPPs the SSR1 cavity is joined to beam line guaranteeing the required vacuum pressure. Side Ports

The two side ports (SP) are diametrically opposed on an orthogonal to BL and spoke axis. The input coupler is mounted on one SP while vacuum device is mounted on the other SP.

Ribs

Ribs have the function of stiffening the structure, classification is made on working area:

• Daisy Ribs: the conic surface of the endwall is stiffened by six daisy ribs equally spaced around the BPT.

• Donut Ribs: the plane surface of the endwall is stiffened by ah half donut-like shape element. Fourtyeight 4-mm holes are drilled on the external surface in order to increase the heat exchange surface with liquid helium.

• Circumferential Ribs: fuor ribs are placed on the cylindrical shell in order to decrease radial deformations under pressure condition. Transition Ring

The transition ring (TR) is composed by a Transition Ring Niobium Side (Nb-TR) and a Transition Ring Stainless Steel Side (SS-TR) brazed together. The transition ring is installed onto the donut ring. Subsequently, during jacketing operations, the helium vessel is connected to the ring by welding. A series of holes are drilled on TR allowing the flow of liquid helium

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DOME BS BELLOWS ASSEMBLY PLATE BS ADAPTER RING CBSC: Bellows Side CRSC: Ring Side PLATE RS DOME RS TRANSITION RING SHELL - LOWER TUNER SUPPORT (x2) SUPPORT BASE (x2) SSR1-G3 CAVITY SHELL - UPPER TUBE INLET FLANGE LUG - LIFTING (x8)

Figure 2.3: SSR1 dressed cavity exploded view.

to the beam-pipe area and also facilitating future degreasing and chemistry operations.

2.1.2 Helium Vessel

It is the macro-component full welded around the Niobium Cavity. Made in Stainless steel it has the function of containing the liquid helium (exposi-tion to low temperature and 2-bar internal pressure) and compensating RF frequency shifts through the bellows deformation under the tuner forces.

To allow the frequency tuning of the cavity by pushing on the beam pipe flange, the beam pipe was connected to the helium vessel by a bellows (expansion joint). Structural plates, shells, pipes are put around the cavity to create the helium space.

2.1.3 Input Coupler

The input coupler (Fig. 2.4 on the next page) is a 105-ohm coaxial design that supplies approximately 2 kW CW to each SSR1 cavity. The coupler contains a single warm ceramic window that provides separation of the warm and cold coupler sections. The cold end of the outer conductor is 316L-stainless steel. The warm end is copper with phosphor bronze bellows. A forced-air cooling tube is inserted into the inner conductor after assembly that supplies air to cool the coupler tip. Two bellows accommodate motion due to misalignment and thermal contraction.

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Figure 2.4: Input Coupler.

Figure 2.5: The SSR1 cavity tuner.

2.1.4 Tuner

The tuner operates in order to minimize the range of uncertainty on the final frequency (coarse tuning) and detuning due to microphonics (fine tuning). Tuner has a double lever mechanism that allows the coarse and fine tuning of the cavity. A main arm hinged at one end and connected to the actuation system (a stepper motor) at the other end has a probe that tunes the cavity physically pushing on the beam pipe. The actuation system is also connected to a second arm, hinged at the other end and keeps the piezos in series with the motor.

2.2 Material

The materials used in the cryomodule fixture are collected in Table 2.1 on the following page .The SSR1 bare cavitiy is made by parts made of Niobium 300RR, except for the SPFs and the BPFs made in 316L Stainless Steel as the entire helium vessel. The SSR1 Cavity is mounted on support posts fixed on the strongback, the aluminium-made frame of the cryomodule. Two different stainless steel are used. 316L an 304L are both stainless steels and have the same mass and elastic properties. The difference is in the presence of Molybdenum which changes the magnetic susceptibility

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Material ρ E ν [Kgm−3_] _[GPa] Niobium 300RR 8600 70÷105 0.38 AISI 304L 7810 205 0.3 AISI 316L 7810 205 0.3 642 Bronze 8.53 105 0.35 Al 6060 2700 66 0.33

Table 2.1: Materials in 325 MHz cryomodule

of the 316L. Austenite (γ-iron) is a nonmagnetic phase of iron which can change its crystal structure of due to high-temperatures and be converted to the ferromagnetic martensite or ferrite forms of iron. Due to the intensive welding processes and the proximity with the RF volume, the helium vessel is made in 316L. 304L is used for the other structural components of the cryomodule (i.e. support post) where its magnetic susceptibility can not negatively affect requirement due to its lower cost.

2.3 Joints

Several types of joints can be identified for SSR1 cavity. The following classification is based on joint typology.

2.3.1 Welds

NB - NB : Niobium parts are joined by electron-beam welding. It was not possible to strictly adhere to Code rules in defining the types of welds used for joining the niobium parts because of the stringent requirements for the high-quality surfaces in SRF components. Several welded samples were etched and studied microscopically, their integrity (absence of defects) was verified.

NB - SS : This joints have the function to join niobium and stainless steel parts where welding are not allowed. Surfaces are chemically polished and subjected to a specific developed brazing process with an oxygen-free electrolytic copper. This brazed joints satisfy resistance requirement due to differential thermal shrinkage and helium pressure. Due to the singularity of the process, ASME code is devoid of specific rules; brazed joints are studied microscopically and tested.

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• SP Tube - SP Flange;

• NB transition Ring - SS transition ring;

SS-SS : Helium Vessel is manufactured around the niobium cavity. All TIG welds performed on the Helium Vessel are classifiable as single-welded butt joints with backing strip, and single full fillet lap joints, respectively.

All the NB - NB and SS - SS welds are shown in SSR1 technical draw-ings. Instead, theNB - SS joints are shown in Fig. 2.8 on page 14.

(a) (b)

Figure 2.6: SSR1 Niobium Cavity interfaces with the helium vessel

2.3.2 Bolted Flanges

Vacuum related : The connections are made using aluminum hex dia-mond seals and stainless steel flanged joints and bellows. The use of silicon-bronze screws and the specific hexagonal section of the seal guarantee the beam pipe pressure at cryogenic temperatures (Fig. 2.7a on the following page).

• Helium Pipeline- SPF: • Coupler- SPF:

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Positioning : Each SSR1 Jacketed Cavity is mounted on its support post using Silicon-bronze nuts and bolts, stainless steel flat washers, and Belleville washers. Due to chemical composition, the silicon-bronze bolts have the same temperature coefficient of expansion as copper thus preventing stretching or loosening of bolts when the copper expands and contracts with temperature variations. Each cavity is first aligned and subsequently screws are fastened (Fig. 2.7b).

(a) A section view of the unstrassed di-amond seal

(b) Cavity alignment system

Figure 2.7: SSR1 Cavity interfaces

2.4 Process

2.4.1 Welding

SSR1 cavity are manufactured by forming, machining and welding. Joints require in the SSR1 bare cavity electron-beam welding (some EB welds are full-penetration) and brazing process, and in the SSR1 helium vessel full penetration TIG weldings (Fig. 2.8a on the following page]). The full pen-etration EBW and transition ring and helium vessel weldings makes the RF frequency change due to shape variation. A leak check follows the bare cavity welding process.

2.4.2 Buffered Chemical Polishing

This operation has the scope of preparing the interior surface of the resonator for testing and final assembling. The component is ultrasonically cleaned with Ultra-Pure Water and a degreasing agent. Then it undergoes to a Buffered Chemical Polishing (BCP), where the cavity, sealed from external, is immersed in a bath of UPW while a standard HF : HN O3 : H3P O4(1 :

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1 : 2) acid mixture. Acid is forced to flow into the interior volume of the cavity by pumping. Heat generated by the etching is dissipated through the cavity walls (including the spoke walls) to the continuously cooled water bath. After BCP, the component was moved to the a 10 clean area for High Pressure Rinse (HPR). After completing the HPR, the cavity is left in the class 10 clean area for drainage.

2.4.3 Tuning

The leak check and the chemical processing are two main causes of fre-quency shifts. During the vacuum check the cavity geometry is deformed by the atmospheric pressure, which shrinks the structure inducing perma-nent plastic deformations, even though very small. The bulk BCP removes 350 µm in average from the cavity surface. Welding makes the frequency change as well. All these are causes of frequency deviations; in order to con-trol and compensate for these shifts it is necessary to tune the cavities.The tuning process consists in applying a force on the cavity and then release the pressure coming back to relaxed position. This is done in several steps, gradually, to avoid any abrupt change in the cavity geometry. It receives an inelastic deformation until RF measurements reach the target values. Fig. 2.8b shows the SSR1 tuning machine with a cavity in it.

(a) TIG welding process (b) Tuning machine

Figure 2.8: Photos of the SSR1 cavity being manufactured

2.5 Functions (???)

As discussed in the previous chapter, vibration problem needs to be addressed with particular attention due to negative effects on beam quality. External vibration sources, also called microphonics, are the cryogenic and vacuum systems. Even the cooling circuit of the strongbackis studied in order to estimate its weight on the dynamic problem.

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Figure 2.9: SSR1 coldmass. Liquid helium pipeline over the string assembly t p(t) p0 ∆p ∆p

Figure 2.10: Liquid helium pressure fluctuation

a bayonet (Fig. 2.9 on the following page). The cryogenic circuits is fed through the bayonet and liquid helium is moved by its own pumping system. The helium bath pressure is unstable, even if fluctuations are small they generate small frequency shifts of the cavities. To a first approximation helium pressure on the external surface of Niobium Cavity can be modeled as the sine wave function (Eq. 2.1) shown in Fig. 2.10 .

p(t) = p0+ ∆p sin(2πfPt) (2.1)

The modal analysis will study the structure with tuner not working and in ideal operative conditions (Tab. 2.2).

p0 static pressure 2 bar

∆p pressure fluctuation 0.2 mbar fP pumping frequency 60 Hz

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Dynamic System

In this chapter the dynamic system is analyzed in details. Assumptions will be exposed and motivated and the next to be performed analysis will be introduced. Any dynamic system can be described through Eq. 3.1. The definition of the mass matrix [M ], damping matrix [C] and elastic matrix [K] is based on material properties and geometry (shape and dimension) of the structure.

The complexity of the geometry requires Finite Element methods to perform analysis. The FE model must closely reproduce all of the major physical properties, material properties, and boundary conditions, that the physical SSR1 resonator has.

A sensitivity analysis on elastic modulus of Niobium has been necessary due to mismatched values as exposed in section 2.2

[M ]{¨x}+ [C]{ ˙x} + [K]{x} = {F (t)} (3.1)

3.1 Assumptions

All the dynamic analysis performed have the basic assumption of zero damping. The damping matrix [C] can be neglected due to the low damp-ing factor of the structure. Analysis results will not be affected by this approximation, damped pulse approximates the undamped pulse as shown in Eq. 3.2.

(

ωs= ωnp1 − ζ2

ζ ≈0 (3.2)

In the model, brazed brazed interfaces will not be explicitly modeled, bonded connections will be imposed between niobium part and stainless steel components flanges. This assumption is justified by the real geometry

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Figure 3.1: A sectioned brazed joint.

of the joint. As shown in Fig. 3.1 copper layer is so small thickness that its presence can be neglected.

3.1.1 Geometry

Due to due to macroscopic dimensional differences the geometry of the component, initially built from technic drawings (Appendix. AAAAAAA), has been modified with the new measured values after a data collecting activity (Fig. 4.7 on page 29).

(a) (b) (c)

Figure 3.2: Measuring Activity

The major components, such as shell and endwall, resembled the SSR1 Niobium Cavity accurately; instead beam pipe flanges and side port flanges presented substantial differences. SPFs and BPFs were substituted by new flanges schematically shown in Fig. 3.3 on the following page with red and blue colors. This operation is motivated by the results of a preliminary experimental modal analysis showing these flanges partecipating actively in the first mode shapes.

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(a) (b)

Figure 3.3: Schematic representation of BPF (Fig. 3.3a) and SPF (Fig. 3.3b on the preceding page)

Holes on the donut ribs, BPFs and SPFs would have little effect on the results of this analysis and were therefore not included in the CAD models. Chamfers and fillets with characteristic dimension less than 0.5 mm were drawn as sharp edges.

SSR1 Niobium Cavity are manufactured by electron-beam welding sheets having nominal thickness of 3 mm. Some joints in the SSR1 resonator re-quire full-penetration welding from the non-RF side. Welding issues and the chain intermittent weld between the donut rib and endwall are not modeled so weld beads present uniform wall thickness on CAD geometry.

With the chemical polishing the component undegoes to wall thickness reduction of the RF side surfaces. Although the reduction is monitored at 20 locations using an ultrasonic thickness gauge in on-going operation and improvement in operative procedure, the expected etching of 350 µm is not as uniform as hoped due to difficulty in heat extraction and a uniform acid flow pattern. Implementation of a point-to-point real shell thickness is impossibile so CAD geometry presents a uniform 2.8 mm wall thickness instead of 3.15 mm for RF side surfaces.

3.1.2 Isotropic Homogenous Linear Elastic Material

The SSR1 niobium cavity material can be considered as IOLE materials. SSR1 is not subjected to thermal or thermo-mechanical treatments, such as nitriding or carburizing, and the mechanical properties of niobium and stain-less steel have no directional characteristics, so it is legitimate to consider the material as homogeneous and isotropic. Linear elasticity assumption is valid for the modal analysis. Stress states, in linear relationships with infinitesimal deformation (Eq. 3.3 on the previous page), do not produce yielding.

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~

σ = Q~ (3.3)

3.2 Implementation of the Components

A 3-D structural solid is necessary for modeling the structure. A 8-node element, SOLID185 (Fig. 3.4a), is used. It is defined by eight nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions.

A higher order 3-D solid element, SOLID186 (Fig. 3.4b), should be used in p-refinement analysis due to its quadratic displacement behavior. This element is defined by 20 nodes having three degree freedom per node: translations in the nodal x, y, and z directions.

(a) (b)

Figure 3.4: Brick elements

Each element type has its own shape functions matrix [Ne_{(x, y, z)] which}

relates nodal displacement vector {Ue_{} with (internal point displacement}

vector) {ν}. Deformations evaluated from {ν} by [L]. The element stiffness matrix [D], on material properties and element type and its characteristic size is used in relation with [B] (matrix product of [L] and [N ]) to define [M ] and [K] as shown in Eq. 3.6 on the preceding page and Eq. 3.7 on the previous page. Following further analyses will be performed with the consistent mass matrix.

{ν} =    νx(x, y, z) νy(x, y, z) νz(x, y, z)    = [Ne(x, y, z)]{Ue} (3.4)

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{} =    x y z    = [L]{ν} = [L][N ]{Ue} = [B]{Ue} (3.5) [M ] = Z V [N ]Tρ[N ]dV (3.6) [K] = Z V [B]T[D][B]dV (3.7) The entire geometrical model of the niobium cavity was used for FE analysis. Tetrahedron elements were used for a free mesh of the solid model with an element size of 4 mm chosen after a convergence analysis.

Figure 3.5: Mesh Cavity

3.3 FE Modal Analysis

FE model has to be validated in order to be used for the further analysis in operative conditions. Free-free configurations will be used for the modal testing since it is easy to achieve this type of configuration in practice. However, from the above case study, this configuration is not adequate to provide informative test data to validate mathematical models for certain applications.

Due to assumption of Sec. 3.1 on page 16, the dynamic system is de-scribed by Eq. 3.8 instead of Eq. 3.1 on page 16.

[M ]{¨x}+ [K]{x} = {F (t)} (3.8) In modal analysis external forces {F (t)} are zero. Eq. 3.8 on page 20 becomes Eq. 3.9 on page 20. In Appendix (???) eigenvalues and eigenvectors extraction technics are described.

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[M ]{¨x}+ [K]{x} = 0 (3.9) The entire CAD model was used to perform each analysis because using symmetry to reduce the computational burden is not legitimate for modal analysis.

3.3.1 Free-Free Configuration

In a free-free test configuration, the structure is supported from a suspen-sion system designed so as to ensure that the rigid body’s mode frequencies are at least an order of magnitude lower than the fundamental frequency of the structure.

It is very difficult to provide truly flexible suspension systems for testing of large flexible space structures. In the experimental activity two different suspension systems will be used in order to avoid a boundary participation.

3.3.2 Constraint boundary Configuration

The SSR1 Niobium Cavity has interfaces with the helium vessel in prox-imity of BPFs and SPFs. It is important that their boundary conditions implemented in FEA are carefully chosen and modeled such that important dynamic parameters can be obtained from the test validate the mathemat-ical models.

Experimental tests with constraint structure have difficulties associated with valid constaint model. These difficulties can, in some cases, make the ap-proach impractical. First it is impossible to achieve a truly fixed-base test since the fixture will have some degree of coupling with the test article.

Modal parameters can be obtained using a free-free configuration. These results are sufficient in our analysis. Currently in most cases the validation test is performed in a free-free configuration since it is easy to achieve this boundary condition in reality.

The test data could be contaminated by excitation system. Special at-tention should be paid on the shaker interface and its configuration.

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Experimental Modal Model

The goal of the experimental activity is to characterize the modal behav-ior of the SSR1 Niobium Cavity. By which is possible to match the results from FEM analysis and to validate the FE analysis results. The dynamic properties of the structure will be identified using FRF measurements. FRF is defined as the ratio of the Fourier transform of an output response v(ω) divided by the Fourier transform of the input force F (ω) that caused the output.

F RF = v(ω)

F(ω) (4.1)

The Fourier transform of a time-dependent signal x(t) is defined as in Eq. 4.2 on page 22

Sx(ω) =

Z ∞

t=−∞

x(t)e−jωtdt (4.2) Data collection and analysis will be held with robot and a software in order to overcome the difficulty of acquisition and management.

4.1 Experimental Setup

The equipment used is the typical setup for Doppler Vibrometer method data collection:

• shaker TIRA TV 50018;

• vibrometer OFV-505 Sensor Head; • amplifier;

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• LMS Test.Xpress FFT analyzer;

The free-free condition of the structure has been modeled placing the cavity on an toroidal air chamber, the best compromise between the need of cavity to be sustained and the need of less stiffness possible. The cavity spacial orientation is shown in Figure. y-axis is vertical and the x-axis parallel to ground. The shaker applies a z-axis direction force on the external surface of the donut rib and the vibrometer is mounted on the end effector of an industrial robot in order to save the position of the target point and replicate measurement.

The hardware has two input (velocity and load cell force) and one out-put (shaker alimentation force) as shown in Figure 4.2 on page 24. FRF measurements are made by providing artificial excitation with one shaker, attached to the structure. The shaker is usually attached to the structure using a stinger (long slender rod), so that the shaker will only impart force to the structure along the axis of the stinger, the axis of force measurement. A load cell is then attached between the structure and the stinger to mea-sure the excitation force. At least a 2-channel FFT analyzer and a single axis accelerometer are required to make FRF measurements using a shaker. A true random signal is used as excitation signal for shaker testing with an FFT analyzer. When used in combination with spectrum averaging, random excitation randomly excites the non-linearity’s in a structure, which are then removed by spectrum averaging. Obtaining a set of noise free FRF estimates with no distortion in them is very important for obtaining accurate modal parameters. A true random signal is synthesized with a random number generator, and is an unending (non-repeating) random sequence. The main

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Figure 4.2: Experimental Setup Scheme.

disadvantage of a true random signal is that it is always non-periodic in the sampling window. Therefore, a special time domain window (a Hanning window or one like it), must always be used with true random testing to minimize leakage. Typical true random signals are shown in Figure 12.

Shaker testing tends to lead to higher quality frequency response func-tions (FRF) over greater bandwidths. Using shaker excitafunc-tions, generally there is much better control on the frequency ranges excited as well as the level of force applied to the structure. While the measurements obtained with shaker excitation tend to be of higher quality and more consistent, greater caution must be taken during the setup of a shaker test to obtain these pristine measurements. Various elements of the test setup can con-taminate the FRFs, primarily due to the type of shaker attachment on the structure.

4.1.1 Shaker

The electro-dynamic shaker TIRA TV 50018 takes an input tension com-ing from the amplifier e moves the inner pole where stcom-inger is fixed on. At the heart of the shaker is a coil of wire, suspended in a fixed radial magnetic field genereated by permanent magnet.

When a current is passed through this coil, an axial force is produced in proportion to the current and this is transmitted to a table structure to which the test article may be affixed.

The force provided by the machine is proportional to the magnetic flux passing through the coil, to the current flowing through the coil and to the length of wire within the flux field.

The method of supporting the shaker is another factor that can affect the force imparted to the structure. The main body of the shaker must be isolated from the structure to prevent any reaction forces from being trans-mitted through the base of the shaker back to the structure. This can be accomplished by mounting the shaker on a solid floor and suspending the

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Figure 4.3: structure from above.

The mechanism of the force transducer, called a load cell, functions in a fairly simple manner. When the crystal element is strained, either by tension or compression, it generates a charge proportional to the applied force. In this case, the applied force is from the shaker. However, due to mounting methods discussed earlier, this is not necessarily the force trans-mitted to the structure. The shaker contains a load cell which measures the axial forces on the inner pole.

Stinger

The most common way of attaching a shaker to a test structure is through a stinger. Stingers, also called quills, are typically made of drill or threaded rod. This type of geometry can provide high axial stiffness while attempting to keep bending stiffness to a minimum. While the main purpose of the stinger is to dynamically decouple the shaker from the test structure, this is impossible to fully achieve. Force transducers between the stinger and structure can only decouple the structure in the axial direction of the stinger. Any force acting in any other direction can change the stiffness of the structure, thus having significant effects on measured FRFs. While there is a limit to the ratio of axial stiffness to bending stiffness for any type of stinger, this ratio can be adjusted by changing the effective length. While a longer length will have a lower bending stiffness than a shorter length of equal material and diameter, resonances of the stinger becomes an issue which will contaminate the measured FRFs. And if the stinger is too

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Figure 4.4:

short, dynamic effects of the shaker will be imposed on the dynamics of the structure.

The stinger is the metal based component (usually aluminum alloy) glued on the structure. It has a low bending stiffness regard the axial stiffness in order to act as a filter for moments and transversal loads.

4.1.2 Vibrometer

The OFV-505 is single point laser vibrometer determine vibrational ve-locity and displacement of structures. The sensor head OFV-505 integrates interferometer, laser and imaging optics in a stable housing and it is de-signed to be mounted on the EE of an industrial robot.

The sensor head used belongs to the Laser Doppler Vibrometer (LDV) fam-ily. LDV is a scientific instrument that is used to make non-contact vibration measurements of a surface. The laser beam from the LDV is directed at the surface of interest, and the vibration amplitude and frequency are extracted from the Doppler shift of the reflected laser beam frequency due to the mo-tion of the surface. The output of an LDV is generally a continuous analog voltage that is directly proportional to the target velocity component along the direction of the laser beam.

Excitation Signal Component is excited by the shaker with a force vari-able in time and amplitude. Force has the same shape of the electric signal in input at the shaker which comes from the CPU and undergoes to a trans-formation by the amplifier. The signal belongs to the stochastic excitation family composed by white noise and broad-band noise. White noise is char-acterized by having a constant power density over the whole spectrum and zero correlation between successive values, i.e. the autocorrelation function of a white noise signal is a Dirac impulse. As it is impossible for a sig-nal of finite duration to have a constant power density over all frequencies, white noise is a purely theoretical notion which in practice is approximated

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by broadband noise. Broadband noise is characterized by having a nearly constant power density not over the whole spectrum but over a certain fre-quency range. The stochastic excitation signals have been implemented in Matlab by generating a series of random numbers and filtering the resulting signal by a bandpass elliptic filter as provided by Matlab via the function ellip.

t v(t)

Figure 4.5: White Noise

4.2 Data collection

The target points, established from the preliminar FEM analysis are: • 4 on each Beam Pipe Flange;

• 8 on each Side Port Flange;

4.3 Data Analysis

FRF was defined as a ratio of the Fourier transforms of an output and input signal, is it actually computed differently in all modern FFT analyz-ers. This is done to remove random noise and non-linearity’s (distortion) from the FRF estimates. In shaker testing with fixed input, multiple FRFs are measured, each for multiple outputs which corresponds to measuring elements from a single column of the FRF matrix.The overall structural re-sponse (the solid curve) is in fact, the summation of resonance curves. In other words, the overall response of a structure at any frequency is a sum-mation of responses due to each of its modes. It is also evident that close to the frequency of one of the resonance peaks, the response of one mode will dominate the frequency response.

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The FRF Matrix Model Structural dynamics measurement involves measuring elements of an FRF matrix model for the structure, as shown in Figure 7. This model represents the dynamics of the structure between all pairs of input and output DOFs. The FRF matrix model is a frequency domain representation of a structure’s linear dynamics, where linear spectra (FFTs) of multiple inputs are multiplied by elements of the FRF matrix to yield linear spectra (FFTs) of multiple outputs. FRF matrix columns cor-respond to inputs, and rows corcor-respond to outputs. Each input and output corresponds to a measurement DOF of the test structure.

The majority of modern experimental modal analysis relies upon the appli-cation of a modal parameter estimation (curve fitting) technique to a set of FRF measurements.

Modal Parameters From Curve Fitting Modal parameters are most commonly identified by curve fitting a set of FRFs. (They can also be iden-tified by curve fitting an equivalent set of Impulse Responses, or IRFs). In general, curve fitting is a process of matching a mathematical expression to a set of empirical data points. This is done by minimizing the squared error (or squared difference) between the analytical function and the measured data. An example of FRF curve fitting is shown in Figure 17.

4.3.1 Data Management

creo che il segnale sia una velocit`a e che quindi venga integrato per dare uno spostamento

4.3.2 Curve Fitting

Due to light modal density (coupling), local sDOF method are preferred to mDOF, Global, and Multi-Reference methods in curve fitting for their ease of use. sDOF (Single Degree Of Freedom) method estimates modal parameters one mode at a time instead of simultaneously estimating modal parameters for two or more modes at a time as the other methods.

The sDOF method applied to the FRF data sets is theModal Frequency as Peak Frequency method. The frequency of a resonance peak in the FRF is used as the modal frequency. This peak frequency, which is also dependent on the frequency resolution of the measurements, is not exactly equal to the modal frequency but is a close approximation, especially for lightly damped structures. The resonance peak should appear at the same frequency in almost every FRF measurement. It won’t appear in those measurements corresponding to nodal lines (zero magnitude) of the mode shape.

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x

y

z

Figure 4.6: NBC DOFs.

(a) (b)

Figure 4.7: Measuring Activity

Analog Signal Analog Filter ADC Digital Filter Discrete Data FFT Figure 4.8:

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Free-Free Modal Model

Validation

In this chapter will be shown the results of both FE modal analysis and experimental modal analysis. Results are displayed in separate classification and a subsequent phase of critical analysis matches FEM with experimental modes and validates the FE modal model.

5.1 Experimental Results

Two datasets (shown in Appendix BBBBB) are the result of the exper-imental modal analysis. One dataset collects the measured points coordi-nates, the other collects the maximum amplitude of each target point for extracted each mode shape.

Experimental mode shapes reconstruction is done according to Eq. 5.1 on the previous page where sF ik is the final coordinate of the k-th measured

point for the i-th mode considered, s0 k is the initial coordinate of the k-th

measured point and Asik the i-th mode amplitude for the k-th measured

point along s direction.

sF ik = s0 k+ Asik (5.1)

The use of the robot allows the perpendicularity of the sensor head to surface containing the measured points simplifying data analysis. In fact, only one component of the three defining position of the measured point vary and it is the axis of the local coordinate system parallel to the sensor head orientation. Improving the normality of the sensor head to surface maximize the coherence of data. Due to low damping coefficient eigenvector module is approximately equal to its imaginary part (eq. 5.2 on the preceding page). Sec. of Appendix aaaaaa shows a sample of the matlab code developed for the mode reconstruction.

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Aik= |eigenvectorik| = |realik+ j · immik| ≈ immik (5.2) Frequency [Hz] 100 150 200 250 300 350 400 450 500 550 Amplitude[???] ×10-3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Figure 5.1: FRF.

Fig. 5.1 on page 31 shows the Frequency Response Function of the struc-ture in the 100-550 Hz frequency range. Upper limit of this range has been chosen to 550 Hz due to its proximity with the 9th _{harmonic of the external}

force frequency. Thirteen mode frequencies belong to the range and de-formed shape reconstruction of each mode is shown in the following images. Tab 5.1 on page 31 collects the first thirteen mode frequency fexp and a

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Figure 5.2: 1st _{experimental mode shape reconstruction.}

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Figure 5.4: The 3rd _{experimental mode shape reconstruction.}

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Figure 5.6: The 5th _{experimental mode shape reconstruction.}

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Mode Index Mode Shape Main Features fexp (Hz) 1 Bending SPFs out-of-phase 195.6 2 Bending SPFs in-phase 209 3 Jelly out-of-phase BPFs 298 4 Bending SPFs out-of-phase 313 5 Jelly and bending (out-of-phase) SPFs 328 6 Bending SPFs out-of-phase 195.6 7 Bending SPFs (out-of-phase), jelly BPF (out-of-phase) 360

8 Bending SPFs (in-phase) 366

9 Bending SPFs (out-of-phase) 439 10 Bending SPFs (in-phase), jelly BPFs (in-phase) 472 11 Bending SPFs (out-of-phase), bending BPF 509 12 Bending SPFs (out-of-phase), bending BPFs (out-of-phase) 513 13 Bending BPFs (out-of-phase) 522

Table 5.1: Experimental Frequencies.

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5.2 Numerical Results

Fifteen mode shape are obtained in FE modal analysis. Tab. 5.2 collects the results with frequency fF EM a brief description of main features. For

each mode shape, two results are displayed: total deformation and elastic strain energy.

Mode Index Mode Shape Main Features Frequency fF EM (Hz) 1 Bending SPFs out-of-phase 196.2

2 Bending SPFs in-phase 206

3 Jelly out-of-phase BPFs 294 4 Bending SPFs out-of-phase 309 5 Jelly and bending (out-of-phase) SPFs 312

6 Jelly SPFs in-phase 326

7 Bending SPFs in-phase, bending BPFs in phase 334 8 Bending SPFs, bending spoke-BPFs 371

9 Axial spoke 461

10 Bending spoke 471

11 Bending EW (in-phase) 485

12 Jelly BPFs, bending spoke 493 13 Bending EW (out-of-phase) 514 14 Bending EW (out-of-phase) 516 15 Bending EW (out-of-phase) 517

Table 5.2: FEM Numerical Frequencies

5.3 Comparison

As shown in Fig. 5.30 on page 46, experimental and FEA results in good agreement for the first four modes. Discrepancies start from the 5th _and

above mode shape (Fig. 5.31 on page 47). The first five mode are collected in Tab. 5.3 on page 42 with their respective experimental and FEM frequencies and percentage difference.

Elastic modulus Sensitivity Analysis

Due to mismatching values on elastic modulus of Niobium 330RR (Tab. ?? on page ??) a sensitivity analysis. Several simulations with different elastic modulus for Niobium 300RR performed and results have been collected in Tab. 5.4 on page 45. A 90 GPa value as Niobium elastic modulus has been chosen due to the closeness of FEM frequencies to experimental frequencies.

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(a) Total Deformation (b) Elastic Strain Deformation

Figure 5.15: 1st _{FEM mode shape section view}

Figure 5.16: 2nd _{FEM mode shape section view}

A dependence of the mode shape on the Young modulus has been identified on 7th _{mode shapes and above.}

5.4 Discussion

• wall thickness; • shell geometry; • material properties;

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Figure 5.17: 3rd _{FEM mode shape axonometric section view}

Figure 5.18: 4th _{FEM mode shape section view}

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Mode Index Mode Shape Main Features fexp fF EM ∆f (Hz) (Hz) (%) 1 Bending SPFs out-of-phase 195.6 196.2 -0.3 2 Bending SPFs in-phase 209 206 1.4 3 Jelly out-of-phase BPFs 298 294 1.4 4 Bending SPFs out-of-phase 313 309 1.2 5 Jelly and bending (out-of-phase) SPFs 328 312 4.8

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(c) Total Deformation

Figure 5.23: 9th _{FEM mode shape section views}

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Mode Index Mode Shape Main Features E (GPa) 70 80 90 1 Bending SPFs out-of-phase 173.3 185.2 196.2 2 Bending SPFs in-phase 181.5 194.0 206 3 Jelly BPFs out-of-phase 259 277 294 4 Bending SPFs out-of-phase 273 292 309 5 Jelly and bending SPFs (out-of-phase) 276 295 312 6 Jelly SPFs in-phase 287 307 326

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(a) 1stMode Shape Reconstruction (b) m1

(c) 2ndMode Shape Reconstruction (d) m2

(e) 3rd_{Mode Shape Reconstruction} _{(f) m3}

(g) 4thMode Shape Reconstruction (h) m4

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(a) 5thMode Shape Reconstruction (b) m5

(c) 6th _{Mode Shape Reconstruction} _{(d) m6}

(e) 7th Mode Shape Reconstruction (f) m6

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System Geometry Analysis

This chapter contains: a brief report about the reverse engineering ac-tivity, its results, the description of a new geometric model based on RE results. This activity starts from the inquiry on mode mismatching causes. Investigate on differences between real geometry and FE model geometry will be performed using structured light technic shapes. This technic leads to qualitative and quantitative results. Geometric mismatching will be ana-lyzed and critically implemented in a new CAD geometry of SSR1 Niobium cavity.

The Reverse Engineering activity took place in the CAD Lab of the Depart-ment of Civil and Industrial Engineering and performed by Prof. Razionale and Eng. Paoli with the support of the Post-Doc Neri.

Structured light technic Structured light scan, by its relative high-quality output is considered one of the most reliable techniques to recover ob-ject surfaces. Structured light scan methods typically emitted visible coded light patterns onto static and opaque objects to establish correspondence between a projector and a camera for triangulation as shown in Fig. 6.1 on page 49. In the success of these methods rely on scanning objects with proper reflective surface for visible light produces accurate and high reso-lution 3D point cloud. This approach is suitable for high-resoreso-lution scans of static scenes using a short sequence of time-shifted stripe patterns. A low-cost system composed of multiple cameras and projectors positioned around a central performance area capturing the geometry of component. These method are concentrate on object 3D geometry reconstruction with visible light source.

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Figure 6.1: Structured light technic working scheme

6.1 Experimental Setup

• Structured light scanner1_;

– 2 X ImagingSource industrial cameras (1200x1600): DMK 51BU02; – Optoma Projector (1024x768);

– ScanProbe: UniPi developed 3D cloud reconstruction software; • Rotating table;

• Uniform white coating sprayed on the surface for image quality im-provement (Fig. 6.3 on page 50);

Figure 6.2: Optic Scanner working

1

S. Barone, A. Paoli, A.V. Razionale, “Multiple alignments of range maps by active stereo imaging and global marker framing”, Optics and Lasers in Engineering, 2013, 51(2), 116-127

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Figure 6.3: The sprayed SSR1 Niobium cavity

6.1.1 Reverse Engineering Activity

This activity is divided in two phase: after a first data collection oper-ation which produces different 3D point clouds, the final surface is derived from fitting operations and combination of elaborated surfaces.The final Re-verse Engineering surface is then compared to the the original CAD surface. First of all the component is sprayed with adhesive white coating and lights were turned off in order to reduce the surface’s reflection and ob-tain a sufficiently dark room improving final image quality. The component is placed on the rotating table with BPFs along the vertical axis and the system is calibrated. Rectangular shape light pattern, variable in number, dimension and orientation hit the exposed surface the two cameras acquire two different images. The software reconstruct the 3D point cloud cross-matching the input signal to projector and the output from the two cameras. Several partial acquisitions of the component were performed, pivoting the rotating table about its axis, in order to ensure geometry continuity to the final 3D point cloud corresponding to the complete SSR1 Niobium Cavity except for one endwall and spoke.

Using Geomagic software each surface is built by a fitting operation with a n-order polynomial. The single acquisitions are combined to ensure conti-nuity. Details have a main role in this phase due to their usage in combining surfaces.

The comparison phase is carried out super-imposing the RE surface to the fixed original CAD geometry and minimizing the differences between the two bodies by subsequent roto-translations.

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6.2 Results

The final result consist of two overlapping CAD surfaces. Estimated di-mensional differences are shown through a chromatic scale in the following figures. Gray areas belongs only to the orginal CAD geometry and shall not be considered. Scale values follows the positive outward-pointing normal rule.

Results are affected by a precision’s error of less the 0.1 mm. (COMPLETARE)

Different views show the cross-matched CAD: an axonometric view in Fig.6.4, a top view in Fig.6.5 and fuor lateral views in Fig.6.6.

Red areas on BPFs and SPFs are neglected to due comparison with a pre-liminary CAD with different flange geometry. This result does not affect the geometric analysis since outer dimesions are the same. Fig. 6.7 on page 54 shows the higher discrepancy close to weldings, especially near BPFs and spoke edges.

6.3 Modified Modal Model

The cylindrical shell is modified in order to introduce an harmonic dis-torsion.

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Figure 6.4: An axonometric view of SSR1 post-process elaboration.

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(a) (b)

(c) (d)

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(a) Axonometric view of the cross-matched geometry with section plane

(b) A section view of SSR1 real geometry in respect of CAD

Figure 6.7: Section views

r θ

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Dynamic Behaviour of the

Cavity

7.1 Constrained Modal Model & Results

7.2 Harmic & Results

7.3 Discussion

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Appendices

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Fermilab & PIP-II Project

Proton Improvement Plan-II (PIP-II) is Fermilab’s plan for providing powerful, high-intensity proton beams to the laboratory’s experiments. The increased beam power will position Fermilab as the leading laboratory in the world for accelerator-based neutrino experiments. PIP-II will also pro-vide a flexible platform for further enhancement of the Fermilab accelerator complex (Fig. A.1 on the preceding page) to extend this leadership to the full range of particle physics research based on intense beams in the decades to come.

Auxiliary tasks are reducing technical risks, and obtaining experience in the design and operation of a superconducting proton linac.

Five types of SRF cavities will be used to cover the entire velocity and en-ergy range for the acceleration of protons from 2.1 MeV to 0.8 GeV (Fig. A.2 on page 58). The cavity frequencies and cell configuration were selected to

Figure A.1: Aerial View of Fermilab

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Figure A.2: Project X scheme

maximize acceleration efficiency for each accelerating structure, to minimize the cost of the accelerator and its operation, as well the beam loss. As an immediate goal, Fermilab is focusing on building the PIP-II Injector Exper-iment (PXIE) that will be used as a proof of concept design for the PIP-II front-end components.

Fig. A.3 on page 58 shows the strong dependence of the RF plants and RF power with frequency detuning of cavities. It was calculated using the parameters of the PIP-II superconducting accelerator for an use of 20 years, and costs of RF plant and electric energy equal to 15/W and0.1/kWh, re-spectively. An average of about 1.5 M$ for each single Hz of frequency detuning is estimated, so containing the capital and operational costs in a reasonable range is one of the main PIP-II goals.

Figure A.3: RF plants and RF power cost dependences with frequency de-tuning

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SSR1 Technical Drawings

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Cryogenic Temperatures

Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a characteristic critical temperature. The Niobium has a critical tem-perature of 4K. Fig. C.1 on page 62 shows the thermal conductivity depen-dence on temperature which is related to superconductivity phenomena.

0 50 100 150 200 250 300 0 100 200 300 Temperature (K) Th. Cond. (Wm − 1 K − 1 )

Figure C.1: Thermal conductivity dependence on temperature for Niobium 300RR

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A superfluid is a state of matter in which the matter behaves like a fluid with zero viscosity. Heat is transported, not by heat conduction, but by convection. This kind of heat transport is very effective, so the thermal conductivity of He-II is very much better than the best materials.

(a) Helium Lambda Point (b) Heat Capacity Transfer

Figure C.2: Helium properties at cryogenic temperatures

Even if superfluid helium is used in cryomodules, a pumping system is required in order to control helium temperature.

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Numerical Analysis

D.1 Theory

The dynamics of any system is described by the genera Eq.(D.1). Ma-trices [M ] e [K] are symmetric and real, [C] instead is complex and substan-tially full.

[M ]{¨x}+ [C]{ ˙x} + [K]{x} = 0 (D.1) Modal analysis consists of computing all the eigenvalue-eigenvector pairs associated with mode shapes.

D.1.1 NO Damping

The (D.1) equation is simplified in Eq. (D.2) due to low damping factors. Solution can be found as follows.

[M ]{¨x}+ [K]{x} = 0 (D.2) Substituting the unknown solution and its derivate (Eq.(D.5)) in Eq.(D.2).

{x} = {X}eωt (D.3) { ˙x} = ω{X}eλt (D.4) {¨x}= ω2_{X}eλt _(D.5)

Result of the substitution is Eq.(D.6). This problem is called Eigenvalues Problem

[K] − ω2[M ]{X} = 0 (D.6) det[K] − ω2[M{0} = 0 (D.7)

Development of a Modal Model of SSR1 Cavity

Development of a

Modal Model of the SSR1 Cavity

Acknowledgements

Contents

List of Figures

List of Tables

Nomenclature

Preface

Introduction

1.1

Aim

1.2

Resonators

1.3

Modal Analysis

Case of Study

2.1

SSR1 Cavity - Geometry and Components

2.2

Material

2.3

Joints

2.4

Process

2.5

Functions (???)

Dynamic System

3.1

Assumptions

3.2

Implementation of the Components

3.3

FE Modal Analysis

Experimental Modal Model

4.1

Experimental Setup

4.2

Data collection

4.3

Data Analysis

x

y

z

Free-Free Modal Model

Validation

5.1

Experimental Results

5.2

Numerical Results

5.3

Comparison

5.4

Discussion

System Geometry Analysis

6.1

Experimental Setup

6.2

Results

6.3

Modified Modal Model

Dynamic Behaviour of the

Cavity

7.1

Constrained Modal Model & Results

7.2

Harmic & Results

7.3

Discussion

Appendices

Fermilab & PIP-II Project

SSR1 Technical Drawings

Cryogenic Temperatures

Numerical Analysis

D.1

Theory