The dynamic stability of the axially moving viscoelastic functionally graded Timoshenko beam in a magnetic field is investigated for the first time in this paper. Based on the generalized Hamiltonian principle and the Kelvin viscoelastic constitutive relationship, the dynamic equations and the corresponding simply supported boundary conditions of the axially moving viscoelastic functionally graded Timoshenko beam in a magnetic field are established. The distance between the true position of the beam neutral plane and the geometric middle plane at one-half of the upper surface of the beam is highlighted. The method of direct multi-scale is utilized to investigate the parameter stability of the beam. The solvability condition and Routh-Hurwitz criterion are used to analyze the sub-harmonic parameter resonance. The effects of some parameters on the resonance instability region are presented. Neglecting variations in magnetic field intensity may result in an overestimation of instability-induced safety risks. Ignoring the changes of the functional gradient indexes leads to underestimation of safety-related risk due to instability. The Galerkin method is introduced to verify the results that calculated by the direct multi-scale method. The influence of the truncation order of the Galerkin process on the accuracy of the numerical results is highlighted. The two results show reasonable agreement. This provides a theoretical basis for the debugging of the stable transfer parameters of functionally graded materials in production.
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