Axially moving continua such as belt, chain, and conveyer are common transmission components. The study of the vibration response of axially moving continua is an essential topic to understand the fundamentals of vibration and improve the performance of the machines. However, it typically requires more rigorous effort in mathematical derivation to obtain the analytical forced vibration responses of the axially moving continua because of the characteristics of non-self-adjoint equation of motion. The methods utilized to obtain the analytical solutions include the modal analysis, canonical form, wave propagation, Laplace transform, and transfer function. In this review paper, these methods will be reviewed and presented. The advantages and disadvantages of different methodologies are discussed as well.

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