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This investigation was primarily to establish assessment criteria for axial fluctuations of head sheaves in a multi-rope friction mine hoist. Large axial fluctuating displacements of head sheaves can excite intense transverse vibrations in inclined ropes, which can result in dangerous collision between two adjacent ropes. A coupled dynamic model was established and then validated by experimental tests and numerical simulations. In this case, it can be concluded that double rotational frequency of head sheave is the primary excitation frequency and rope tension can change the resonance frequency range. Finally, to avoid large transverse displacements of inclined ropes in a multi-rope friction hoist, based on response–excitation displacement curves, assessment criteria for axial fluctuating displacements of head sheaves were established. [ABSTRACT FROM AUTHOR]