Modal analysis of a beam resting on a nonhomogeneous Winkler elastic foundation

Free vibrations of a beam with constant flexural stiffness resting on a nonuniform continuous Winkler elastic foundation are considered. The nonhomogeneity of the foundation is characterized by a variable subgrade modulus, which in the general case is assumed to be an arbitrary continuous function. The exact solution of the fourth-order differential equation of beam vibrations with variable coefficients is derived, and an analytical method for its numerical implementation is proposed. A modal analysis of the beam is carried out for the case where the subgrade modulus varies according to an exponential law. To verify the authors’ method (AM), the obtained results are compared with those of the finite element analysis performed in the LIRA-FEM software package.