A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of general cross-section

Forgit, Cédric; Lemoine, Benoit; Marrec, Loïc Le; Rakotomanana, L.

doi:10.1007/s00707-016-1618-1

Cited by 9 publications

(8 citation statements)

References 42 publications

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“…A natural question that comes to mind is: Are there other types of axisymmetric structures that behave similarly? Rings (curved beams) are conceivable -The recent work [8] tends to prove that sensitivity does not occur for thin rings with circular or square sections.…”

Section: Conclusion: the Leading Role Of The Membrane Operator For Thmentioning

confidence: 99%

High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero

Chaussade-Beaudouin¹,

Dauge²,

Faou³

et al. 2017

Operator Theory: Advances and Applications

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The lowest eigenmode of thin axisymmetric shells is investigated for two physical models (acoustics and elasticity) as the shell thickness (2ε) tends to zero. Using a novel asymptotic expansion we determine the behavior of the eigenvalue λ(ε) and the eigenvector angular frequency k(ε) for shells with Dirichlet boundary conditions along the lateral boundary, and natural boundary conditions on the other parts.First, the scalar Laplace operator for acoustics is addressed, for which k(ε) is always zero. In contrast to it, for the Lamé system of linear elasticity several different types of shells are defined, characterized by their geometry, for which k(ε) tends to infinity as ε tends to zero. For two families of shells: cylinders and elliptical barrels we explicitly provide λ(ε) and k(ε) and demonstrate by numerical examples the different behavior as ε tends to zero.[We observe] a phenomenon which is particular to many deep shells, namely that the lowest natural frequency does not correspond to the simplest natural mode, as is typically the case for rods, beams, and plates.This citation emphasizes that for shells these lowest natural frequencies may hide some interesting "strange" behavior. The expression "deep shell" contrasts with "shallow shells" for which the main curvatures are of same order as the thickness. Typical examples of deep shells are cylindrical shells, spherical caps, or barrels (curved cylinders).In acoustics, driven by the scalar Laplace operator, it is well known that, when Dirichlet conditions are applied on the whole boundary, the first eigenmode is simple in both senses that it is not multiple and that it is invariant by rotation. We will revisit this result, in order to extend it to mixed Dirichlet-Neumann conditions. In contrast to the scalar Laplace operator, the simple behavior of the first eigenmode does not carry over to the vector elliptic system -linear elasticity. Relying on asymptotic formulas exhibited in our previous work [5], we analyze two families of shells already investigated in [2]. Doing that, we can compare numerical results provided by several different models: The exact Lamé model, surfacic models (Love and Naghdi), and our 1D scalar reduction.The first of these families are cylindrical shells. We show that the lowest eigenvalue 1 decays proportionally to the thickness 2ε and that the angular frequency k of its mode tends to infinity like ε −1/4 . The second family is a family of elliptic barrels which we call "Airy barrels". Elliptic means that the two main curvatures (meridian and azimuthal) of the midsurface S are non-zero and of the same sign. Airy barrels are characterized by the following relations:

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Section: Conclusion: the Leading Role Of The Membrane Operator For Thmentioning

confidence: 99%

High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero

Chaussade-Beaudouin¹,

Dauge²,

Faou³

et al. 2017

Operator Theory: Advances and Applications

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“…For further convenience, the system (1) is now reformulated into a dimensionless form, similarly to e.g. [41,42]. To this end, constant characteristic amplitudes A c and I c are introduced.…”

Section: Dimensionless Timoshenko Systemmentioning

confidence: 99%

Enriched finite elements and local rescaling for vibrations of axially inhomogeneous Timoshenko beams

Cornaggia

Darrigrand

Marrec

et al. 2020

Journal of Sound and Vibration

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This work presents a new enriched finite element method dedicated to the vibrations of axially inhomogeneous Timoshenko beams. This method relies on the "half-hat" partition of unity and on an enrichment by solutions of the Timoshenko system corresponding to simple beams with a homogeneous or an exponentially-varying geometry. Moreover, the efficiency of the enrichment is considerably increased by introducing a new formulation based on a local rescaling of the Timoshenko problem, that accounts for the inhomogeneity of the beam. Validations using analytical solutions and comparisons with the classical high-order polynomial FEM, conduced for several inhomogeneous beams, show the efficiency of this approach in the time-harmonic domain. In particular low error levels are obtained over ranges of frequencies varying from a factor of one to thirty using fixed coarse meshes. Possible extensions to the research of natural frequencies of beams and to simulations of transient wave propagation are highlighted.

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“…the deformation gradient can be write as F = Q (I + H) where Q is the rotation of the section, I is the identity tensor, and H = d i · (ε + κ × GM) D i ⊗ D 3 . Up to a rotation, the deformation gradient is entirely defined by the knowledge of H (and then by ε and κ).…”

Section: Deformationmentioning

confidence: 99%

“…This is the case for E if = −1 + 1/ √ 3 (i.e., for ∂ F ∂ = 0). For this critical strain, the natural boundary conditions are no more able to fix the slope of the longitudinal vibration ( ∂Δϕ 3 ∂ S ) and of the transverse rotation ( ∂Δω 1 ∂ S and ∂Δω 2 ∂ S ). In fact the longitudinal force E A ∂Δϕ 3 ∂ S and the bending moment ( E I 1 ∂Δω 1 ∂ S and E I 2 ∂Δω 2 ∂ S ) at the boundaries are zero (as expected for natural boundary condition), but the constraint on the kinematical quantities is no more effective.…”

Section: Instabilitymentioning

confidence: 99%

“…In the large frequency domain, the vibrations are strongly affected by the effective properties of the structure and its boundary conditions [1,2]. Of course, a wide literature exists on the sensitivity of the dynamical response to geometry [3], constitutive laws [4][5][6][7][8] and boundary conditions [9]. However, the theoretical investigation of the effect of a finite transformation has not attracted so much attention.…”

Section: Introductionmentioning

confidence: 99%

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Vibration of a Timoshenko beam supporting arbitrary large pre-deformation

2017

Self Cite

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International audienceWe present an induced geometrically exact theory for the three-dimensional vibration of a beam undergoing finite transformation. The beam model coincides with a curvilinear Cosserat body and may be seen as an extension of the Timoshenko beam model. No particular hypothesis is used for the constitutive laws (in the framework of hyperelasticity), the geometry at rest or boundary conditions. The method leads to a weak formulation of the equations of vibration. The obtained internal energy is symmetric and leads to a self-adjoint operator that casts into a geometrical and material parts. Both may be written explicitly in terms of the finite transformation. The results are applied on the vibration of a beam supporting a finite longitudinal strain. The nonlinear effects according to the pre-stress are explicitly detailed for this example: instability, buckling

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A Timoshenko-like model for the study of three-dimensional vibrations of an elastic ring of general cross-section

Abstract: International audienc

Cited by 9 publications

References 42 publications

High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero

High Frequency Oscillations of First Eigenmodes in Axisymmetric Shells as the Thickness Tends to Zero

Enriched finite elements and local rescaling for vibrations of axially inhomogeneous Timoshenko beams

Vibration of a Timoshenko beam supporting arbitrary large pre-deformation

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