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NONLINEAR VIBRATIONS AND STABILITY OF SHELLS NONLINEAR VIBRATIONS AND STABILITY OF SHELLS WITH FLUIDWITH FLUID--STRUCTURE INTERACTIONSTRUCTURE INTERACTION
MM arcoarco AmabiliAmabili
Dipartimento di Ingegneria Industriale, Università di Parma Parma, Italy
Dipartimento di Ingegneria Industriale, Università di Parma, Italy
Outline of PresentationOutline of Presentation
••Reduce the order of models for largeReduce the order of models for large--amplitude amplitude (geometrically nonlinear) vibrations of fluid(geometrically nonlinear) vibrations of fluid--filled circular filled circular cylindrical shells by using the proper orthogonal cylindrical shells by using the proper orthogonal decomposition (POD) method decomposition (POD) method
••LargeLarge--amplitude vibrations of rectangular platesamplitude vibrations of rectangular plates
••Flutter of imperfect shells in supersonic flowFlutter of imperfect shells in supersonic flow
••Nonlinear vibrations and stability of circular cyli ndrical Nonlinear vibrations and stability of circular cyli ndrical shells conveying flow shells conveying flow
REDUCEDREDUCED--ORDER MODELS FOR NONLINEAR VIBRATIONSORDER MODELS FOR NONLINEAR VIBRATIONS
OF CYLINDRICAL SHELLS VIA THE PROPEROF CYLINDRICAL SHELLS VIA THE PROPER
ORTHOGONAL DECOMPOSITION METHODORTHOGONAL DECOMPOSITION METHOD
22 2 24 0
2 2 2 2 2
2 22 2 2 20 0
2 2 2
1 1
2
wF F wD w c h w h w f p
R x R x x
w wF w F w
x x x x
∂∂ ∂ ∂ρ∂ ∂ θ ∂ ∂
∂ ∂∂ ∂ ∂ ∂∂ ∂ θ ∂ ∂ θ ∂ ∂ θ ∂ ∂ θ ∂ θ
∇ + + = − + + +
− + + +
& &&
2 2 2 22 2 2 2 2 24 0 0 0
2 2 2 2 2 2 2
1 1 12
w w ww w w w w wF
E h R x R x x x x x x
∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ θ ∂ ∂ θ ∂ ∂ θ ∂ ∂ ∂ θ ∂ ∂ θ
∇ = − + + − + −
DonnellDonnell’’ s nonlinear shallows nonlinear shallow--shell shell theorytheory
with radial geometric imperfections wwith radial geometric imperfections w00
( ) 2 2 2 2 2( ) 0xM D w x w R∂ ∂ ν ∂ ∂ θ = − + =
w = w0 = 0
2 20 0w x∂ ∂ = at x = 0, L
Nx = 0 and v = 0 at x = 0, L
OOutut--ofof--plane boundary conditionsplane boundary conditions
InIn --plane boundary conditionsplane boundary conditions
3 3
, ,1 1
4
(2 1),0 (2 1)1
( , , ) ( ) cos ( ) ( ) sin ( ) sin( )
( ) sin( )
m k n m k n mm k
m mm
w x t A t k n B t k n x
A t x
θ θ θ λ
λ
= =
− −=
= +
+
∑ ∑
∑
Expansion of the rExpansion of the radialadial displacement displacement ww::Conventional Galerkin approachConventional Galerkin approach
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
HarmonicHarmonicexcitationexcitation
eigenmodeseigenmodes16 dofs16 dofs
Driven modeDriven mode
Companion Companion modemode
Circular shellCircular shell
Driven modeDriven mode Companion modeCompanion mode
Axisymmetric modesAxisymmetric modes
1
( , ) ( ) ( )K
i ii
w t q t ϕ=
= ∑ξ ξ
1
( , ) ( ) ( )K
i ii
w t a t ψ=
= ∑ξ ξ
%
PROPER ORTHOGONAL DECOMPOSITION PROPER ORTHOGONAL DECOMPOSITION (POD) METHOD(POD) METHOD
Conventional Galerkin solutionConventional Galerkin solution
POD solutionPOD solution
Basis functionsBasis functions(eigenmodes)(eigenmodes)Generalized coordinatesGeneralized coordinates
dofsdofs
Proper orthogonal coordinatesProper orthogonal coordinates
Proper orthogonal modesProper orthogonal modes
Reduced dofsReduced dofs
( , ) ( , ) ( ) d ( )w t w tΩ
ψ λψ′ ′ ′< > =∫ ξ ξ ξ ξ ξ% %
( )2( ) ( , )w tλ ψ= −ξ ξ%
% Ω∀ ∈ξ
PO modes are extracted PO modes are extracted from temporal snapshots of the response (from from temporal snapshots of the response (from Galerkin solution in the present case)Galerkin solution in the present case)
Minimizing the objective functionMinimizing the objective function
Eigenvalue problemEigenvalue problem
( )( , ) ( , ) ( )w t w t w= −ξ ξ ξ%
zero-mean responsetime-averaging
1
( ) ( )K
i ii
ψ α ϕ=
= ∑ξ ξ
Projection of the proper orthogonal modes on the Galerkin modesProjection of the proper orthogonal modes on the Galerkin modes
1 1 1 1
( ) ( ) ( ) ( ) ( ) d ( )K K K K
i i j k j k i ii j k i
q t q tΩ
ϕ α ϕ ϕ λ α ϕ= = = =
′ ′ ′ =∑ ∑∑ ∑∫ξ ξ ξ ξ ξ% %
( )i i iq q q= −%zerozero--mean response of the imean response of the i--th generalized coordinateth generalized coordinate
λ=A α Bα
( ) ( )ij i j i jA q t q tτ τ= % %
ij i ijB τ δ=
2 ( ) di iΩτ ϕ= ∫ ξ ξ
The final eigenvalue problem isThe final eigenvalue problem is
Eigenvectors: Coefficients of projection Eigenvectors: Coefficients of projection of PO mode on Galerkin modesof PO mode on Galerkin modes
Eigenvalues: Eigenvalues: significance of significance of PO modesPO modes
(1) (1) (1)( ) ( )ij i j i jA q t q tτ τ= % %
(2) (2) (2)( ) ( )ij i j i jA q t q tτ τ= % %
, , , , ,1 1 1 1 0
( , ) ( ) ( ) = ( ) cos ( ) sin ( ) sin( )K K K M N
i j i j i m n i m n i mi j i m n
w t a t a t n n xα ϕ α θ β θ λ= = = = =
= + ∑ ∑ ∑ ∑∑ξ ξ
% %
(1) (2)1 2
(1) (2)
p p= +A AA
A ATTr ( )=A A A
CASE OF WEIGHTED RESPONSESCASE OF WEIGHTED RESPONSES
Proper orthogonal modesProper orthogonal modes
Frobenius norm of AFrobenius norm of A
weightsweights
Numerical ResultsNumerical ResultsL = 0.52 mL = 0.52 mR = 0.1494 mR = 0.1494 mh = 0.519 mmh = 0.519 mmE = 198 * 10E = 198 * 101111 PaPaρρ = 7800 kg/m= 7800 kg/m33
ρρFF = 1000 kg/m= 1000 kg/m3 3 ((water filledwater filled))νν= 0.3= 0.3Fundamental mode (n = 5, m = 1)Fundamental mode (n = 5, m = 1)Harmonic point excitation of 3 N at x = L/2 and Harmonic point excitation of 3 N at x = L/2 and θθθθθθθθ = 0= 0ζζζζζζζζ1,n1,n = 0.0017= 0.0017ωωωωωωωω1,n1,n = 77.64 Hz= 77.64 Hz
Software: Software: AUTOAUTO for bifurcation analysis and continuationfor bifurcation analysis and continuationIMSL Fortran IMSL Fortran –– DIVPAG routineDIVPAG routine
, stable solutions;, stable solutions;
—— —— —— , unstable periodic solutions, unstable periodic solutions
TR: NeimarkTR: Neimark --Sacker (torus) bifurcationsSacker (torus) bifurcations
BP: Pitchfork bifurcationsBP: Pitchfork bifurcations
1,Maximum Amplitude of driven mode ( )nA t
1,Maximum Amplitude of companion mode ( )nB t
Conventional Galerkin Conventional Galerkin results (16 dofs)results (16 dofs)
1,Time response at excitation frequency / 0.99nω ω =
1, ( )nA t
1,2 ( )nA t
1,0( )A t
Harmonic force excitation
Case “a”
1,nTime response at excitation frequency ω/ω = 0.991 corresponding to point "c"
1, ( )nA t
1, ( )nB t
Significance of POD eigenvalues versus the number of proper orthogonal modes; case “a”
1 10.999
K K
i ii iλ λ
= =≥∑ ∑
%
Significance of POD eigenvalues versus the number of proper orthogonal modes; case “c”
POD model POD model (2dofs) (2dofs) versus conventional versus conventional GalerkinGalerkin model, case model, case ““ aa””
Maximum amplitude of vibration versus excitation frequencyMaximum amplitude of vibration versus excitation frequency
POD model (2 POD model (2 dofsdofs))
unstable conventional unstable conventional GalerkinGalerkin solutionsolution (16 dofs)(16 dofs)
stable conventional stable conventional GalerkinGalerkin solutionsolution (16 dofs)(16 dofs)
Branch 2 is not obtainedBranch 2 is not obtained
1,nMaximum amplitude of A (t) versus excitation frequency
POD modelPOD modelss with 2 with 2 and 3 dofand 3 dofs (coincident)s (coincident)
Convergence of the solution with the number of dofConvergence of the solution with the number of dofPOD model, case POD model, case ““ aa””
POD model with 1 POD model with 1 dofdof
POD model versus conventional POD model versus conventional GalerkinGalerkin model, case model, case ““ cc””
stable stable conventional conventional GalerkinGalerkin solutionssolutions
—— —— —— unstable conventional unstable conventional GalerkinGalerkin solutions.solutions.
1,Maximum Amplitude of ( )nA t
1,Maximum Amplitude of ( )nB t
Maximum amplitude of vibration versus excitation frequency
POD model (POD model (33 dofsdofs))
NO PITCHFORK BIFURCATIONS
POD model versus POD model versus GalerkinGalerkin model, case model, case ““ cc”” combined combined ““ --cc””Maximum amplitude of vibration versus
excitation frequency
1,Maximum Amplitude of ( )nA t
stable stable conventional conventional GalerkinGalerkin solutionssolutions
—— —— —— unstable conventional unstable conventional GalerkinGalerkin solutions.solutions.
1,Maximum Amplitude of ( )nB t
POD model (POD model (33 dofsdofs))
PITCHFORK BIFURCATIONS DETECTED
0.97 0.98 0.99 1 1.01ww1,n
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
A1,
nh
0.97 0.98 0.99 1 1.01ww1,n
0
0.2
0.4
0.6
0.8
B1,
nh
1,nTime response at excitation frequency ω/ω = 0.991
1, ( )nA t
1, ( )nB t
1,0( )A t
POD model Conventional Galerkin
Driven mode
Companion mode
1st axisymmetric mode
case “c”
1,n
Case "c" :
time response
at ω/ω = 0.995Harmonic force excitation
1, ( )nA tDriven mode
1, ( )nB tCompanion mode
1,0( )A t1st axisym. mode
ConclusionsConclusions
••Dimension of model for nonlinear vibrations of shells has Dimension of model for nonlinear vibrations of shells has been reduced from 16 to 3 dofs by using the POD methodbeen reduced from 16 to 3 dofs by using the POD method
••An accurate reducedAn accurate reduced--order model has been builtorder model has been built
••The most accurate reducedThe most accurate reduced--order model has been built by order model has been built by using the time response with amplitude modulationsusing the time response with amplitude modulations
Nonlinear Vibrations of Nonlinear Vibrations of Circular Cylindrical PanelsCircular Cylindrical Panels
and and Rectangular PlatesRectangular PlatesMM arco arco AmabiliAmabili
Dipartimento di Ingegneria Industriale, Università di Parma Parco Area delle Scienze 181/A, Parma, 43100 Italy
,0x x xz kε ε= +
,0 z kθ θ θε ε= +
,0x x xz kθ θ θγ γ= +
2
0,0
1
2x
u w w w
x x x xε ∂ ∂ ∂ ∂ = + + ∂ ∂ ∂ ∂
2
0,0
1
2
w w w w
R R R R Rθεθ θ θ θ
∂ ∂ ∂ ∂= + + + ∂ ∂ ∂ ∂
v
0 0,0x
u w w w w w w
R x x R x R x Rθγθ θ θ θ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
v
MM iddleiddle surface strainsurface strain--displacement displacement (Donnell)(Donnell)
Elastic Strain Energy of the PanelElastic Strain Energy of the Panel
u, v, w = displacements in x, u, v, w = displacements in x, θθθθθθθθ and r directions of the mean planeand r directions of the mean plane
2
2x
wk
x
∂= −∂
2
2 2
wk
Rθ θ∂= −
∂2
2x
wk
R xθ θ∂= −∂ ∂
CChangeshangesin the curvature and torsion in the curvature and torsion (Donnell)(Donnell)
MM iddleiddle surface strainsurface strain--displacement displacement (Novozhilov)(Novozhilov)
2 2 2
0,0
1
2x
wu u w w
x x x x x xε
∂∂ ∂ ∂ ∂ ∂ = + + + + ∂ ∂ ∂ ∂ ∂ ∂
v
2 2 2
,0 2
002
1
2
1
w u ww
R R R
w ww w
R
θεθ θ θ θ
θ θ θ
∂ ∂ ∂ ∂= + + + + + − ∂ ∂ ∂ ∂
∂ ∂ ∂+ − + + ∂ ∂ ∂
v vv
vv
,0
0 00
1x
u u u w ww
x R R x x x
w ww ww
x x x
θγθ θ θ θ
θ θ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂= + + + + + − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂∂ ∂ ∂+ − + + ∂ ∂ ∂ ∂ ∂
v v vv
vv
CChangeshangesin the curvature and torsion in the curvature and torsion (Novozhilov)(Novozhilov)
2
2x
wk
x
∂= −∂
2
2 2 2 2
w w uk
R R R x Rθ θ θ∂ ∂ ∂= − − + +
∂ ∂ ∂v
2
22x
w uk
R x R x Rθ θ θ∂ ∂ ∂= − + −∂ ∂ ∂ ∂
v
( )/ 2
0 0 / 2
1d (1 / ) d d
2
L h
S x x x x
h
U x R z R zα
θ θ θ θσ ε σ ε τ γ θ−
= + + +∫ ∫ ∫
( )21x x
Eθσ ε ν ε
ν= +
−
( )21 x
Eθ θσ ε ν ε
ν= +
−
( )2 1x x
Eθ θτ γ
ν=
+
( )
( )
2 2,0 ,0 ,0 ,0 ,02
0 0
32 2
20 0
34
,0 ,0 ,0 ,0 ,020 0
1 12 d d
2 1 2
1 12 d d
2 212 1
1 1d d ( ),
2 26 1
L
S x x x
L
x x x
L
x x x x x x
E hU x R
E hk k k k k x R
E hk k k k k x R O h
R
α
θ θ θ
α
θ θ θ
α
θ θ θ θ θ θ
νε ε ν ε ε γ θν
νν θν
νε ε ν ε ν ε ε θν
− = + + + −
− + + + + −
− + + + + + + −
∫ ∫
∫ ∫
∫ ∫
Elastic strain energyElastic strain energy
Membrane energy
Bending energy
Membrane and Bending energy coupled
Mode Expansion, Kinetic Energy and External LoadsMode Expansion, Kinetic Energy and External Loads2 2 2
0 0
1( ) d d
2
L
S ST h u w x Rα
ρ θ= + +∫ ∫ v& &&
( )0 0
d dL
x rW q u q q w x Rα
θ θ= + +∫ ∫ v
( ) ( ) cos( )
x
r
q q 0
q f R R x x t
θ
δ θ θ δ ω= =
= − −
% %
%
( )/ 2, / 2
cos( )x L
W f t w θ αω= =
= %
u = v = w = w0 = xM = 2 20 0w x∂ ∂ = , at x = 0, L,
u = w = w0 = N Mθ θ= = 2 20 0w∂ ∂θ = , at θ = 0, α,
Simply supported boundary condition with one immovable edgesSimply supported boundary condition with one immovable edges
Single harmonic radial forceSingle harmonic radial force
2 ,1 1
( , , ) ( ) sin ( / ) sin(2 / ),M N
m nm n
u x t u t n m x Lθ π θ α π= =
=∑∑ with odd value of n
,1 1
( , , ) ( ) cos ( / ) sin( / ),M N
m nm n
x t t n m x Lθ π θ α π= =
=∑∑v v with odd values of m and n
,1 1
( , , ) ( ) sin ( / ) sin( / ),M N
m nm n
w x t w t n m x Lθ π θ α π= =
=∑∑ with odd values of m and n
Basis of displacementsBasis of displacements
Panels with point excitation in the middlePanels with point excitation in the middle
only mode with odd number of circumferential and axial halfonly mode with odd number of circumferential and axial half --waveswaves
Geometric imperfectionsGeometric imperfections
0 ,1 1
( , ) sin ( / ) sin( / ),M N
m nm n
w x A n m x Lθ π θ α π= =
= ∑∑% %
Boundary conditionBoundary condition
( )3
20
12(1 )x x
E hM k kθν
ν= + =
−
( )3
20
12(1 ) x
E hM k kθ θ ν
ν= + =
−
( ),0 ,020
1 x
E hNθ θε ν ε
ν= + =
−
(Only in radial direction, associated with zero initial stress)(Only in radial direction, associated with zero initial stress)
d
dS S S
jj j j
T T UQ
t q q q
∂ ∂ ∂− + = ∂ ∂ ∂ &
1,j dofs= K
d( / 4)
dS
S jj
Th L R q
t qρ α
∂ = ∂ &&
&
, , ,1 , 1 , , 1
dofs dofs dofsS
k k i k i k i k l i k lk i k i k lj
Uq f q q f q q q f
q = = =
∂ = + +∂ ∑ ∑ ∑
0S jT q∂ ∂ =
Numerical and Experimental Results for the limit case
R : Rectangular plate→ ∞
Software: Software: AUTOAUTO for bifurcation analysis and for bifurcation analysis and continuation of nonlinear ODEcontinuation of nonlinear ODE
Benchmark caseBenchmark case(Chu & Herrmann , Ribeiro, Kadiri & Benamar) (Chu & Herrmann , Ribeiro, Kadiri & Benamar)
h = 1 mmh = 1 mmL = 300 mmL = 300 mml = 300 mml = 300 mm
Density = 2778 Kg/m^3Density = 2778 Kg/m^3Young Module = 70 GpaYoung Module = 70 Gpa
Poisson Ratio = 0.3Poisson Ratio = 0.3
Boundary Conditions:Boundary Conditions:u = v = w = 0 at x = 0, L and y = 0, lu = v = w = 0 at x = 0, L and y = 0, l
(zero displacements at the edges)(zero displacements at the edges)
Validation of the present model (16 dofs) Validation of the present model (16 dofs)
Ribeiro
Kadiri & BenamarChu & Herrmann
Present model
L
l
h (thickness) = 0.3 mmh (thickness) = 0.3 mmL (x lenght) = 515 mmL (x lenght) = 515 mml (y lenght) = 184 mml (y lenght) = 184 mm
Density = 2700 Kg/m^3Density = 2700 Kg/m^3Young module = 69 GPaYoung module = 69 GPa
Poisson ratio = 0,33 Poisson ratio = 0,33
The experimental aluminum plateThe experimental aluminum plate
Boundary conditions used to simulate the Boundary conditions used to simulate the experimental plate experimental plate
v = 0, Ny = 0
u = 0, Nx = 0
u = 0, Nx = 0
v = 0, Ny = 0
Convergence of foundamental mode Convergence of foundamental mode m = n =1 (halfm = n =1 (half--waves in x and y directions) waves in x and y directions)
3 dofs
27 dofs9 dofs
12 dofs
0
50
100
150
200
250
300
1 2 3 4 5
m
Fre
qu
ency
(H
z)
n = 1
n = 2
n = 3
Comparison between theory and experiments Comparison between theory and experiments (linear results)(linear results)
n = half-waves in y direction
m = half-waves in x direction
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.92 0.96 1.00 1.04 1.08 1.12
ωωωω /ωωωω 1,n
Dis
pla
cem
ent
/ h
0,0005 N
0,001 N
0,005 N
0,012 N
up
down
0,02 N
up
down
0,012 N
NONLINEAR RESPONSE NONLINEAR RESPONSE –– StepStep--sine test sine test (foundamental mode n=1, m=1)(foundamental mode n=1, m=1)
ForceForce
HARDENING TYPE NONLINEARITY
Comparison between experimental and Comparison between experimental and numerical results numerical results
(Excitation force = 0.0005 N, linear case)(Excitation force = 0.0005 N, linear case)
Comparison of theory and experiments Comparison of theory and experiments (Excitation force = 0.02 N, nonlinear case)(Excitation force = 0.02 N, nonlinear case)
Hardening Hardening resultsresults
Different boundary conditionsDifferent boundary conditions
—— •• —— •• —— : free displacements and rot. (experimental case): free displacements and rot. (experimental case)—— —— —— : zero displacements : zero displacements :: zero displacements and rotationszero displacements and rotations
ConclusionsConclusions
••Refined model for nonlinear vibrations Refined model for nonlinear vibrations of circular cylindrical panels and flat of circular cylindrical panels and flat platesplates
••Convergence of solutionConvergence of solution
••Good agreement with experimental Good agreement with experimental resultsresults
••Model suitable for different boundary Model suitable for different boundary conditionsconditions
Nonlinear Nonlinear Supersonic Flutter of Imperfect Supersonic Flutter of Imperfect Circular Cylindrical ShellsCircular Cylindrical Shells
pp M
M
w w
x M a
M
M
w
t
w w
M R1
2
2 1 20
2
20
2 1 21
1 2
1 2 1= −
−∂ +
∂+ −
−LNM
OQP
∂∂
−+−
RSTUVW
∞
∞
γ( )
( )
( )/ /
2 3
0 03 1
( ) ( )1 1 1 1
4 12
w w w ww wp p p M M
x a t x a t
γ γγ ∞∞ ∞
∂ + ∂ ++ ∂ + ∂ = − + + + ∂ ∂ ∂ ∂
M = Mach number
γ= = = = adiabatic esponent
a∞∞∞∞ = free-stream speed of sound
p∞∞∞∞ = free-stream static pressure
LINEARIZED PISTON THEORYLINEARIZED PISTON THEORY
THIRD ORDER PISTON THEORYTHIRD ORDER PISTON THEORY
M > 1.6
0 2.5 5 7.5 10 12.5 15Freestream Static PressureHkPaL0
0.25
0.5
0.75
1
1.25
1.5
1.75
xaM
A 1,n
th
0 2.5 5 7.5 10 12.5 15Freestream Static PressureHkPaL0
0.25
0.5
0.75
1
1.25
1.5
1.75
xaM
B1,
nt
h
1,Maximum Amplitude of ( )nA t
1,Maximum Amplitude of ( )nB t
22 DOF M22 DOF Modelodel with Geometric Imperfectionswith Geometric Imperfections
Onset of Flutter (Hopf bifurcation)Onset of Flutter (Hopf bifurcation)
Travelling Wave FlutterTravelling Wave Flutter
PARTIAL REPRESENTATION OF THE FLUTTERING SHELLPARTIAL REPRESENTATION OF THE FLUTTERING SHELL
4500 Pap∞ =
displacement augmented 1000 times
n = 23
Nonlinear stability of circular cylindrical Nonlinear stability of circular cylindrical shells conveying flowing liquidshells conveying flowing liquid
-0.5-0.25
00.25
0.5
-0.5-0.25
00.25
0.5
0
0.25
0.5
0.75
1
-0.5-0.25
00.25
0.5
-0.5-0.25
00.25
0.5
FluidFluid --Structure Interaction: Potential FlowStructure Interaction: Potential Flow
v = −∇Ψ Ψ Φ= − +U x
∇ = + + + =22
2
2
2 2
2
2
1 10Φ Φ Φ Φ∂
∂∂∂
∂Φ∂
∂∂θx r r r r
Unsteady potential
Steady potentialFluid velocity
Laplace equation
pt
UxF=
∂∂
+∂∂
FHG
IKJρ Φ Φ
∂∂
Φr
w
tU
w
xr R
FHG
IKJ =
∂∂
+∂∂
FHG
IKJ=
pL
m
m R L
m R L tU
xwr R F
n
n= =
′∂∂
+ ∂∂
FHG
IKJρ
πππ
I ( / )
I ( / )
2
Fluid pressure
Boundary condition at shell-fluid interface
(shell periodically supported)
Pressure at interface
mean flow velocity
Divergence (Divergence (BucklBuckl ing) ofing) of Shell Shell by Flowby Flow
Jumps under perturbationJumps under perturbation
Buckled Shell (Divergence Instability due to flow)Buckled Shell (Divergence Instability due to flow)
CoupledCoupled--mode bifurcationmode bifurcation
Mode n = 5
V = 2 (< VV = 2 (< VCRIT. LIN.CRIT. LIN. )) V = 3 (< VV = 3 (< VCRIT. LIN.CRIT. LIN. )) V = 4V = 4
LARGELARGE --AMPLITUDE FORCED VIBRATIONS OF A WATERAMPLITUDE FORCED VIBRATIONS OF A WATER --FILLED FILLED CIRCULAR CYLINDRICAL SHELL WITH IMPERFECTIONS: CIRCULAR CYLINDRICAL SHELL WITH IMPERFECTIONS:
COMPARISON OF THEORY AND EXPERIMENTSCOMPARISON OF THEORY AND EXPERIMENTS
0.97 0.98 0.99 1 1.01ww1,n
0.2
0.4
0.6
0.8
1
A1,
nh
0.97 0.98 0.99 1 1.01ww1, n
0.2
0.4
0.6
0.8
B1,
nh
1,Maximum Amplitude of ( )nA t 1,Maximum Amplitude of ( )nB t
BIFURCATION DIAGRAM: SHELL WITH FLOWING FLOW BIFURCATION DIAGRAM: SHELL WITH FLOWING FLOW UNDER HARMONIC EXCITATION (CHANGING FREQUENCY)UNDER HARMONIC EXCITATION (CHANGING FREQUENCY)
ConclusionsConclusions••It is necessary to study complex nonlinear dynamics and It is necessary to study complex nonlinear dynamics and fluidfluid --structure interaction by using small dimension modelsstructure interaction by using small dimension models
••Model are based on global discretization (hyperModel are based on global discretization (hyper--elements)elements)
••FluidFluid --structure interaction can be described with simplified structure interaction can be described with simplified but accurate models in several applications without CFDbut accurate models in several applications without CFD
••ReducedReduced--order models (POD) can be useful and accurateorder models (POD) can be useful and accurate
