Out-of-plane dynamic analysis of AFG cycloidal beams is investigated by a powerful and efficient approach in the Laplace space. A powerful inverse algorithm is applied to retransfer the results from the Laplace space. The material and geometric properties are graded in the axial direction. The influence of shear deformation is considered in the formulation and the damping model of Kelvin is used in the damped forced vibration analysis. In this paper, the impact of several material variation exponents on the natural frequencies and on the transient response are discussed in detail. The novelty of the presented research is that it deals with the obtaining of out-of-plane free vibration characteristics of cycloidal beams due to the FGM response without calculation of the mode shapes. To validate the accuracy of the present method, the results are compared with the available literature and those of FEM software, ANSYS. Excellent agreement is observed. (C) 2019 Elsevier Ltd. All rights reserved.