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The time-dependent properties of a composite can result in a significant change of stress and strain profiles in a rotating component over a period of time, which is critical in terms of machine performance and durability. A viscoelastic analysis has been developed to investigate creep and stress relaxation in a multilayered composite cylinder subjected to rotation. The analysis accounts for layer-by-layer variation of material properties, composite fiber orientations, temperature, and density gradients through the thickness of cylinders. A closed-form solution based on the corresponding elastic problem is derived for a generalized plane strain state in a thick-walled multilayered cylinder. A Laplace transform is then applied to obtain the numerical solution of the viscoelastic problem. The paper illustrates the derivation of the analytical model and solution procedure. The analysis can be used to investigate the creep behavior of rotating machines constructed with multilayered components. Accordingly, the performance of a rotating machine over a period of time can be predicted and improved through machine design and material selection.