Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties

Olivier Thomas (1) , Stefan Bilbao (2)

1 LMSSC - Laboratoire de Mécanique des Structures et des Systèmes Couplés
2 Edin. - University of Edinburgh

This study is devoted to the derivation of some properties of the von Kármán equations for geometrically nonlinear models of plates, with a boundary of arbitrary shape, for applications to nonlinear vibration and buckling. An intrinsic formulation of the local partial differential equations in terms of the transverse displacement and an Airy stress function as unknowns is provided. Classical homogeneous boundary conditions-with vanishing prescribed forces and displacements-are derived in terms of the Airy stress function in the case of a boundary of arbitrary geometry. A special property of this operator, crucial for some energy-conserving numerical schemes and called "triple self-adjointness", is derived in the case of an edge of arbitrary shape. It is shown that this property takes a simple form for some classical boundary conditions, so that the calculations in some practical cases are also simplified. The applications of this work are either semi-analytical methods of solution, using an expansion of the solution onto an eigenmode basis of the associated linear problem, or special energy-conserving numerical methods.

Domaines: Vibrations [physics.class-ph]

Olivier Thomas, Stefan Bilbao. Geometrically nonlinear flexural vibrations of plates: In-plane boundary conditions and some symmetry properties. Journal of Sound and Vibration, 2008, 315 (3), pp.569-590. ⟨10.1016/j.jsv.2008.04.014⟩. ⟨hal-03177697⟩