Nonlinear free and forced vibration analyses of axially functionally gradedEuler-Bernoulli beams with non-uniform cross-section

Abstract

Nonlinear free and forced vibrations of axially functionally graded Euler-Bernoulli beams with non-uniformcross-section are investigated. The beam has immovable, namely clamped-clamped and pinned-pinned boundaryconditions, which leads to midplane stretching in the course of vibrations. Nonlinearities occur in the system dueto this stretching. Damping and forcing terms are included after nondimensionalization. The equations aresolved approximately using perturbation method and mode shapes by differential quadrature method. In thelinear order natural frequencies and mode shapes are computed. In the nonlinear order, some corrections arise tothe linear problem; the effect of these nonlinear correction terms on natural frequency is examined and fre-quency–response curves are drawn to show the unstable regions. In order to confirm the validity, our results arecompared with others available in literature.