The principal⁃parametric joint resonance of an axially variable⁃speed moving beam with elastic restraint at both ends was studied.Based on Hamilton principle

the magneto⁃elastic nonlinear vibration equation of the beam was derived

in which magnetic field generated by parallel current⁃carrying wires was introduced in modelling.The mode shape function was obtained by means of elastic boundary conditions

and then the Galerkin method was employed to separate time⁃items and space⁃items from each other.The approximate analytical solution and the stability of steady⁃state motion were achieved by multi⁃scale method.In analyzing the computational results

the reliability of the analytical solutions were verified by numerical calculation results corresponding to parameters.In addition

the effects of different parameters on the resonance amplitude and frequency were analyzed.The results show that the stiffnesses of springs and the external excitation force may only affect the amplitude of the system

and the axial velocity

the current and the axial force may lead to the variation of the amplitude and frequency of the resonance

where the stability region increases with the increase of the stiffness of springs.