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Ana Cristina Ana Cristina Ana Cristina Ana Cristina GalucioGalucioGalucioGalucio****JeanJeanJeanJean----FranFranFranFranççççois Deois Deois Deois Deüüüü**, Fran**, Fran**, Fran**, Franççççois Dubois***ois Dubois***ois Dubois***ois Dubois***
*EADS Corporate Research Center France**Structural Mechanics and Coupled Systems Lab., CNAM
***Mathematics Dept., CNAM & Univ. Orsay
DDDDéééérivation fractionnaire en mrivation fractionnaire en mrivation fractionnaire en mrivation fractionnaire en méééécanique canique canique canique –––– EtatEtatEtatEtat----dededede----llll’’’’artartartart et applicationset applicationset applicationset applications
CNAM Paris CNAM Paris CNAM Paris CNAM Paris –––– 17th November, 2006 17th November, 2006 17th November, 2006 17th November, 2006
On the use of On the use of On the use of On the use of fractionalfractionalfractionalfractional derivativederivativederivativederivative operatorsoperatorsoperatorsoperators to to to to describedescribedescribedescribe viscoelasticviscoelasticviscoelasticviscoelastic dampingdampingdampingdamping in structural in structural in structural in structural
dynamicsdynamicsdynamicsdynamics ---- FE formulation of sandwich FE formulation of sandwich FE formulation of sandwich FE formulation of sandwich beamsbeamsbeamsbeamsand approximation of and approximation of and approximation of and approximation of fractionalfractionalfractionalfractional derivativesderivativesderivativesderivatives by by by by
usingusingusingusing the Gthe Gthe Gthe Gαααα schemeschemeschemescheme
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ObjectivesObjectivesObjectivesObjectives
• To implement a fractional derivative model into a finite element code in structural dynamics
• To test a finite difference based on scheme to approximate fractional derivative operators in viscoelastic constitutive equations
viscoelastic core
elastic face
elastic face
Damped oscillator
Sandwich beam
Viscoelastic beam
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OutlineOutlineOutlineOutlineI. I. I. I. Implementation of the fractional Implementation of the fractional Implementation of the fractional Implementation of the fractional ZenerZenerZenerZener model into model into model into model into a FE code a FE code a FE code a FE code –––– application to sandwich beamsapplication to sandwich beamsapplication to sandwich beamsapplication to sandwich beams
• Constitutive equations – fractional Zener model
• Grünwald-Letnikov approximation
• FE formulation & algorithm
• Examples: validation & results
II.TestingII.TestingII.TestingII.Testing the Gthe Gthe Gthe Gαααα schemeschemeschemescheme
• Outline
• Elementary tests
• Error estimates
• The damped oscillator problem
• Viscoelastic cantilever beam
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I. I. I. I. ImplementationImplementationImplementationImplementation of the of the of the of the fractionalfractionalfractionalfractional ZenerZenerZenerZenermodel model model model intointointointo a FE code a FE code a FE code a FE code –––– application to application to application to application to
sandwich sandwich sandwich sandwich beamsbeamsbeamsbeams
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Fractional derivative viscoelastic modelFractional derivative viscoelastic modelFractional derivative viscoelastic modelFractional derivative viscoelastic model
R.L. Bagley and P.J. Torvik, AIAA (1983)
OneOneOneOne----dimensional constitutive equationdimensional constitutive equationdimensional constitutive equationdimensional constitutive equation
In time domain
In frequency domain
⇒⇒⇒⇒ Fractional derivative Zener Model (4 parameters)
with
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Identified model parameters:
Storage modulus Loss factor
Master curves of the ISD112 (3MMaster curves of the ISD112 (3MMaster curves of the ISD112 (3MMaster curves of the ISD112 (3M™™™™) at 27) at 27) at 27) at 27°°°°Fractional model versus classical Zener model
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with
GrGrGrGrüüüünwald approximation & time discretizationnwald approximation & time discretizationnwald approximation & time discretizationnwald approximation & time discretization
with
Discretizing…
J. Padovan, Comp. Mech. (1987)
A. Schmidt and L. Gaul, Proc. 2nd ECCMR (2001)
Variable changing Only one fractional derivative term
Approximation for fractional derivatives
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The structure consists on a three-layer beam ⇒ constrained layer damping treatmentconstrained layer damping treatmentconstrained layer damping treatmentconstrained layer damping treatment
membranebendingshear
membranebending
membranebending
Finite element formulationFinite element formulationFinite element formulationFinite element formulation
a
b
cviscoelastic
elastic
elastic
• Sandwich beam element
• Fractional model implementation
• Dynamic time integration
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SandwichSandwichSandwichSandwich----beam kinematicsbeam kinematicsbeam kinematicsbeam kinematicsMead and Markus, JSV (1969), Trindade, Benjeddou and Ohayon, IJNME (2001)
• Laminated elastic faces: Euler-Bernoulli beam• Viscoelastic core: Timoshenko beam
Constrained layer damping treatmentConstrained layer damping treatmentConstrained layer damping treatmentConstrained layer damping treatment
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Viscoelastic coreViscoelastic coreViscoelastic coreViscoelastic core
“Anelastic displacements” are evaluated as functions of their own time histories
modified stiffness matrixmodified stiffness matrixmodified stiffness matrixmodified stiffness matrix modified loadingmodified loadingmodified loadingmodified loading
Elementary dof vector
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Equation of motionEquation of motionEquation of motionEquation of motion
� For an elastic material (τ = 0), classical elastic stiffness matrix
is obtained
� For the Zener model (α = 1), previous equations correspond to
a backward Euler discretization of the constitutive equations
RemarkRemarkRemarkRemarkssss+ initial conditions
0
0
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Dynamic time integrationDynamic time integrationDynamic time integrationDynamic time integration
Newmark scheme
storage of “anelasticdisplacements”
evaluatedonce
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M. Enelund et B.L. Josefson, AIAA Journal (1997)
Classical algorithmTaylor et al., IJNME (1970)
Fractional derivatives model
Viscoelastic fixed-free bar subjected to unit step load
Validation Validation Validation Validation –––– Example 1Example 1Example 1Example 1
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Validation Validation Validation Validation –––– Example 2Example 2Example 2Example 2
T.M. Chen, IJNME (1995)
Viscoelastic simply-supported Timoshenko beam subjected to a uniform step function load
Validation for α = 1
good agreement with the “hybrid Laplace transform/FE method”
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ResultsResultsResultsResultsSandwich beam subjected to a triangular pulse
loadlength = 280 mm width = 25 mm thickness:• core = 0.2 mm• skin = 1.5 mm
viscoelastic core
elastic faces
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ResultsResultsResultsResults
Calculations performedwith whole history
Remark: Energy balance is strictly zero.
Influence of the time step and truncation
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Existence of a truncation time where the recent history must be stored.
Very accurate solution with a fine time discretization:Nstep = 2500 Nt = 2500
Influence of the time step and truncation
ResultsResultsResultsResults
AA
BC B C
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II. II. II. II. TestingTestingTestingTesting the Gthe Gthe Gthe Gαααα schemeschemeschemescheme
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How to How to How to How to obtainobtainobtainobtain a a a a fractionalfractionalfractionalfractional derivativederivativederivativederivative operatoroperatoroperatoroperator
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The GThe GThe GThe Gαααα schemeschemeschemescheme
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Elementary testsElementary testsElementary testsElementary tests
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Error estimates in L normError estimates in L normError estimates in L normError estimates in L norm∞
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The damped oscillator problemThe damped oscillator problemThe damped oscillator problemThe damped oscillator problem
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AlgorithmAlgorithmAlgorithmAlgorithm
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ResultsResultsResultsResults
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ResultsResultsResultsResults
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ResultsResultsResultsResults
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Extension to Extension to Extension to Extension to viscoelasticviscoelasticviscoelasticviscoelastic beamsbeamsbeamsbeamsConstitutive Constitutive Constitutive Constitutive equationsequationsequationsequations
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ViscoelasticViscoelasticViscoelasticViscoelastic cantilever cantilever cantilever cantilever beambeambeambeam
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AlgorithmAlgorithmAlgorithmAlgorithm
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ResultsResultsResultsResults
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Quelques rQuelques rQuelques rQuelques rééééfffféééérencesrencesrencesrences1. A.C. GALUCIO, J.-F. DEÜ AND R. OHAYON. Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Computational Mechanics, 33:282–291, 2004.
2. A.C. GALUCIO, J.-F. DEÜ AND R. OHAYON. Atténuation des vibrations de structures par traitement piézoélectrique/viscoélastique en utilisant un mode`le a` de´rive´es fractionnaires. Revue Europeénne des Eléments Finis, 13/5-6-7:509–521, 2004.
3. A.C. GALUCIO, J.-F. DEÜ AND R. OHAYON. A fractional derivative viscoelastic model for hybrid active/passive damping treatments in time domain – Application to sandwich beams. Journal of Intelligent Material Systems and Structures, 16:33–45, 2005.
4. A.C. GALUCIO, J.-F. DEÜ , S. MENGUE AND F. DUBOIS. An adaptation of the Gear scheme for fractional derivatives. Computer Methods in Applied Mechanics and Engineering, 195:6073–6085, 2006.
5. J.-F. DEÜ, A.C. GALUCIO, R. OHAYON. Dynamic responses of flexible-link mechanisms with passive/active damping treatment, Accepted for publication on the Special Issue of the International Journal of Computer and Structures, dedicated to the II ECCOMAS Thematic Conference on Smart Structures and Materials, 18-21 July, 2005, Lisbon.
6. A.C. GALUCIO, J.-F. DEÜ , R. OHAYON. Hybrid active-passive damping treatment of sandwich beams in nonlinear dynamics, Journal of Vibration and Control, Accepted for publication.
7. A.C. GALUCIO. Atténuation des réponses transitoires par traitement hybride piézoélectrique/viscoélastique en utilisant un modèle à dérivées fractionnaires (in French), PhD thesis, CNAM Paris, 2004.
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Conclusions and PerspectivesConclusions and PerspectivesConclusions and PerspectivesConclusions and Perspectives
� Sandwich beam with viscoelastic core and elastic faces
� FE implementation of a viscoelastic fractional derivative model
� The Gα scheme is no more costly than the Grünwald-Letnikov one and it appears to be (α+2)-order accurate for power functions
� The combination Newmark- Gα yields a two-order accuracy for the damped oscillator problem
Coming work: to investigate the order of convergence obtained by using the Gα scheme in the case of sandwich beams