Dynamic analysis of a multi-contact problem with clearances, application to the SFR fuel pins bundle.

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clearances, application to the SFR fuel pins bundle.

Catterou Thomas, Bruno Cochelin, Stéphane Bourgeois, V. Blanc, Guillaume Ricciardi

To cite this version:

Catterou Thomas, Bruno Cochelin, Stéphane Bourgeois, V. Blanc, Guillaume Ricciardi. Dynamic analysis of a multi-contact problem with clearances, application to the SFR fuel pins bundle.. EU-RODYN 2017 - 10th international conference on structural dynamics, Sep 2017, Rome, Italy. �cea-02508894�

pins bundle.

CATTEROU Thomas, PhD student.

thomas.catterou@cea.fr

Bruno Cochelin, Stephane Bourgeois –

LMA Marseille, France

Victor Blanc, Guillaume Ricciardi

– CEA Cadarache, France

September, 11

₂₀₁₇

EURODYN2017 – X

_{international conference on structural}

Goal

Caracterize the non-linear dynamical behavior of the

ASTRID pins bundle with mounting gaps for different

loads (handling, transport, earthquake).

Ensure the integrity of the first

containment barrier.

Introduction

A fuel pin

A large number of pins (217) and localized contact zone

(~15000)

Hypothesis:

o Timoshenko beams o Small deformations

Methodology of numerical method validation

“Toy model” Semi-analytical and experimental validation Simplified model One row of pins

Experimental validation

Full model

Final validation

Numerical method (Cast3M finite element

software)

Explicit integration scheme (central difference method)

Modal analysis without contacts

Resolution of the fundamental equation of the dynamic on

the modal base

Modal recombination for each time step on contact points

to estimate contact forces 𝐹

Sub-structuring for the full model

Semi-analytical method

Modal basis analysis of two basic problems

Resolution of the fundamental equation of the dynamic on

the modal base

Clamped beam colliding on a spring

𝑋

𝑥 , 𝜔

𝑋

𝑥 , 𝜔

ሷ𝑞

+ 2𝜉

𝜔

ሶ𝑞

+ 𝜔

𝑞 = 0

𝑞

(𝑡) = 𝑒

𝐴

cos 𝜔

𝑡 + 𝐵

sin 𝜔

𝑡

𝐴

=

; 𝐵

=

+

𝐴

; 𝜔

= 𝜔

(1 − 𝜉

).

𝑢

𝑥, 𝑡 = Σ𝑋

𝑥 𝑞

(𝑡)

_𝑢

_{𝑥, 𝑡 = Σ𝑋}

_{𝑥 𝑞}

_(𝑡)

Switching time ?

Semi-analytical method

Root finding algorithm to find the switching time

𝑡

= 𝑡

−

𝑓 𝑡_𝑛 −𝑓 𝑡_𝑛−1

(Secant method)

Creation of the solution

Initial conditions

𝑢

, 𝑣

, 𝑡

If 𝑢

𝐿 > 0

Clamped−spring

solution

Clamped-free

solution

If 𝑢

𝐿 < 0

Root finding algorithm

Identification of contact

or take off instant

Loop

Solution building

between two contacts

Results : displacements

Clamped beam colliding on a spring

𝑅

= 10 / 𝑅

= 3000

𝑅

= 300 / 𝑅

= 3000

Analytical

Numerical

Analytical

Numerical

𝑅

= 300 / 𝑅

= 5

𝑅

=

,

Hardness of the contact

𝑅

=

𝑓

𝑓

Validation – Frequency truncation

𝑅

=

𝑓

𝑓

Local error when

𝑡

≈ 𝑡

reference = semi-analytic method

𝑅

=

Validation - Frequency truncation

𝑅

= 10

𝑅

= 300

Analytical

Numerical

Analytical

Numerical

A high frequency truncation is necessary when 𝑅_𝑘 is high.

Validation – Time step

A small time step is needed for a high 𝑅_𝑘

A small time step doesn’t mean a greater accuracy.

Err

= max( Γ𝑛𝑢𝑚 − Γ𝑟𝑒𝑓 ) max Γ_𝑛𝑢𝑚 , Γ_𝑟𝑒𝑓

𝑓

Frequency

truncation

<<

<

𝑑𝑡

Time step

<

<

𝑑𝑥

Spatial

discretization

<

<

∅

Clamped beam colliding on a spring

𝑭𝒓𝒆𝒒𝒖𝒆𝒏𝒄𝒚 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒉𝒐𝒄𝒌 𝒐𝒇 𝒂 𝒎𝒂𝒔𝒔 𝒎 𝑬𝒙𝒑𝒍𝒊𝒄𝒊𝒕 𝒔𝒄𝒉𝒆𝒎𝒆 𝒄𝒐𝒏𝒅𝒊𝒕𝒊𝒐𝒏 𝑨 𝒔𝒎𝒂𝒍𝒍 𝒕𝒊𝒎𝒆 𝒔𝒕𝒆𝒑 𝒅𝒐𝒆𝒔𝒏’𝒕 𝒊𝒏𝒗𝒐𝒍𝒗𝒆 𝒂 𝒈𝒓𝒆𝒂𝒕𝒆𝒓 𝒂𝒄𝒄𝒖𝒓𝒂𝒄𝒚 _{𝑻𝒊𝒎𝒆 𝒔𝒉𝒐𝒄𝒌} 𝑬𝒙𝒑𝒍𝒊𝒄𝒊𝒕 𝒔𝒄𝒉𝒆𝒎𝒆 𝑩𝒆𝒏𝒅𝒊𝒏𝒈 𝒘𝒂𝒗𝒆 𝒑𝒓𝒐𝒑𝒂𝒈𝒂𝒕𝒊𝒐𝒏

Finite element

model

Fuel pins : Timoshenko beams Wrapper tube : Shells

One row modeled (~500 contact zones)

Sectional view of the assembly

Contact zone

WT edges

Fue

l reg

io

n

Application to the SFR

tube bundle

Release of an assembly in bending on a rigid stop

Several phenomena at different time scale WT contact time (~20ms)

WT breathing waves (~15ms)

Compression waves in the bundle (~3ms) Bending waves in the bundle (~5ms)

Local phenomena in a contact zone if there is a gap (<0.5ms)

WT contact time

Compression waves go back and forth

WT « Breathing » modes

Dynamical behavior of a tube bundle

Peak force depending on clearance

clearance

Similar to a Newton’s cradle behavior

Donahue 2008 Hutzler 2004

Conclusion and outlook

Analytical validation of a contact problem

Numerical method choice

Creation of a validity domain of the numerical model

Application to a pin bundle

Several phenomena highligthed.

Beneficial impact of clearance size for an homogeneous distribution.

Outlooks

Study of heterogeneous distribution of clearances. Experimental validation

Annex 1 Timoshenko Beam

Rationale of the selection of timohenko beam.

0 5 10 15 20 25 10 20 36 50 100 (Fr éq -Fr éq 3D )/ Fr éq (% ) L/R POUT TIMO Timo 0,5 COQUE Théorie

• Contact law

Hertzian spring

𝐹 = ෨𝑘𝛿

Linear spring

𝐹 = 𝑘𝛿

Annex 3 Sub structuring

• Sub-structuring

_{Modal analysis}

Φ

, Φ

, …

HT

17 fuel pins

Grid and

assembly foot

Connectivity mode

Ψ

, Ψ

, …

𝑥 = ΣΦ

𝑞

+ Σ𝜓

𝑞

,

𝐵

= [Φ, Ψ]

෡

𝐾 = 𝐵

K𝐵

𝑒𝑡 ෡

𝑀 = 𝐵

𝑀𝐵

Modal analysis on ෡

𝐾 and ෡

𝑀

𝐸𝑖𝑔𝑒𝑛𝑣𝑎𝑙𝑢𝑒𝑠 𝑎𝑛𝑑 𝐸𝑖𝑔𝑒𝑛𝑣𝑒𝑐𝑡𝑜𝑟𝑠 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑜𝑛 𝐵

Modal recombination