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This paper presents an analytical study on the free and forced vibration of inhomogeneous Euler–Bernoulli beams containing open edge cracks. The beam is subjected to an axial compressive force and a concentrated transverse load moving along the longitudinal direction. The rotational spring model is used to model the crack causing sectional flexibility. The forced response is determined by employing modal series expansion technique. Analytical solutions of natural frequencies and dynamic deflections are obtained for cantilever, hinged–hinged, and clamped–clamped beams whose material properties follow an exponential through-thickness variation. Numerical results are given in both tabular and graphical forms. The effects of cracks, material property gradient, axial compression, and the speed of the moving load are discussed in detail in the parametric study.