Fourth-order axially moving systems are common dynamic models, typically represented by axially moving beams. This paper derives closed-form forced vibration solutions for damped fourth-order axially moving systems. Three types of models are considered, including axially moving single-span fourth-order system, axially moving multi-span fourth-order system and axially moving coupled system composed of two fourth-order models. The Green’s function method is used to obtain the steady-state solutions. The Green’s functions of single-span system are obtained by the traditional Laplace transform technique, while those for multi-span and coupled systems are expressed by the Green’s functions of single-span systems and superposition principle. That is, the transverse vibration responses of the axially moving single-span, multi-span and coupled systems can be unified derived from the Green’s function of the single-span system via the superposition principle. This unified analytical framework avoids the complex continuity conditions in multi-span systems required by conventional Green’s function methods, and yields more accurate closed-form solutions than general numerical approaches. The reliability and convenience of the results are verified by several examples. The proposed method is characterized by its clarity, convenience for programming, and valuable for dynamic studies of axially moving systems.
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