WITH PERFECTLY CLAMPED EDGES

A new two-dimmensional, single series local model of the transverse vi-bration of a multi-layer, one-span sandwich beam composed of isotropic layers with ideally (perfectly) clamped ends is proposed in the paper. The model is derived within the local theory of linear elastodynamics and it is composed of two two-dimmensional fields and of two approxi-mations of three-dimmensional fields satisfying exactly the equations of motion as well as the Saint-Venant compatibility equations of the the-ory. All through-the-thickness boundary conditions of the local theory of elastodynamics as well as all local compatibility equations (for the displacements and stresses) between adjoining layers are fulfiled in the model. Both the cross-sectional warping and the transverse complian-ce(s) in each layer of the beam are taken into account, thus the model is applicable to the classical three-layer sandwich beam and to a multi-layer sandwich or laminated narrow structure. Key words: sandwich beam, perfect clamping, transverse vibration, local model Notations E – Young’s modulus hj – thickness of jth layer of beam L – length of beam Ux, Uy, Uz, ux, uy, uz – displacements in directions x, y, z, respectively ux(j), uy(j), uz(j) – displacements within jth layer t – time x, y, z – space variables X(T) – trigonometric function of variable x X(H) – hyperbolic function of variable x 790 S. Karczmarzyk zj – coordinate of one (upper) surface of jth layer εqr – strain tensor λL, µL – Lame’s parameters µ = µL, µ(j) – shear modulus and shear modulus of jth layer, respectively ν – Poisson’s ratio ρ, ρ(j) – density and density of jth layer, respectively σzz, σzx, σzz(j), σzx(j) – stresses and stresses in jth layer, respectively 1