Non-linear forced vibrations of an inhomogeneous layer

Abstract

Abstract The non-linear vibrations of an inhomogeneous soil layer which is subjected to a harmonic motion along its bottom are investigated in this study. The Ramberg–Osgood model is transformed to a suitable form to obtain an analytical solution and it is assumed that the shear modulus of the layer varies with depth. The governing equation is a non-linear partial differential equation. Because of weak non-linearity, the displacement and forcing frequency are expanded into perturbation series by using the Lindstedt–Poincare technique, and it is assumed that the response has the same periodicity as the forcing. Then, the zeroeth and the first order linear equations of motion and boundary conditions are obtained. Different types of solutions are obtained for the zeroeth order equation depending on the inhomogeneity parameter α. The orthogonality condition of Millman–Keller [1] is used to extract secular terms which are important in the resonance region. Then, the variation of the amplitude at the top versus the forcing frequency Ω is investigated for some values of inhomogeneity and perturbation parameters.