Task 2.3 : Models validations (UM-LPMM, CRPHT)
Interaction with EADS
Shell finite element and Numerical Algorithm
for vibrations of viscoelastic structures
Geometry and hypothesis
z
elastic he1
viscoelastic hv
elastic
x
he2
Classical laminate theory is used.
Elastic layers are modeled with Love Kirchhoff assumptions.
Reissner/Mindlin theory is used to account of the shear
deformation in the viscoelastic layer.
No slips occurs at the interfaces between layers.
Materials are linear, homogeneous and isotropic.
All points of the elastic layers on a normal have the same rotations.
Sandwich finite element obtained
A triangular sandwich finite element
8 d.o.f / node
Longitudinal displacements of faces, rotations and deflection
Algorithms for complex eigenvalue problem
([K (w)] - w2 [M]) [U] = 0
U :complex eigenmode
w2 :complex eigenvalue w2 2 (1 i)
-QR method - Constant complex modulus
- Low damping
-Asymptotic approach (Ma et He 1992)
-Iterative algorithm (Chen et al. 1999)
-Algorithms developed at LPMM
- Continuation algorithm (Computer & Structures, 2001)
- Iterative algorithms(2003)
Principle of algorithms developed
Continuation algorithm
([K (p)] - w2 [M]) [U] = 0 , [K (w)] =[K(0)]+E(w)[N]
Homotopy technique
([K(0)] + E(p)[N]+p2 [M]) [u] =0 0 1
Asymptotic Numerical Method.
U and p are searched as a truncated integer - power series with
respect to
Set of recrrent linear problems with the same matrix
Continuation procedure
While 1 next step of ANM is needed
* Validation (simple model of viscoelasticity)
Abaqus simulation uses volume elements and MSEC.
Eve simulation using our shell element + ANM.
Real modulus Complex modulus
Free vibrations 4 first bending modes
Damped vibrations, =1 4 first bending modes
*Modeling of the experimental sample.
Characteristic of the viscoelastic material: 3M ISD 112
- Nomograph not precise
enough to extract reliable
Prony’s series.
- Value of the Young’s relaxed
modulus largely varying in
literature. ( from 0.135Mpa
[1] to 1.5MPa [2])
We used a model of ISD 112 at 27 °C to illustrate the
capabilities from our element.
[1] Influences of Higher Order Modeling Techniques on the Analysis of Layered Viscoelastic
Damping Treatments. Austin M. Thesis 1998.
[2] Modeling of Frequency-Dependent Viscoelastic Materials for Active-Passive Vibration Damping.
Trindade M.A., Benjeddou A., Ohayon R. I.J.V.A 2000
* Maxwell’s or ADF Model (Trindade, 2000)
Comparison of numerical results (Abaqus, Eve).
Damped vibrations, Maxwell or ADF model, 4 first bending modes
WP2 Task 2.1 : Dissemination
CRPHT , UM-LPMM
H.Hu, S. Belouettar, E.M. Daya and M. Potier-Ferry,
Evaluation of kinematics formulations for viscoelastically damped sandwich
beams
Journal of Sandwich Structures and Materials, Accepted