Bidirectional Parametric Vibration Of An Axially Moving Ferroelastic Beam

Axially moving beams are widely used in civil engineering,aerospace engineering,transmission engineering and other fields.In practical engineering,These devices are often in the complex field interaction of force,electricity and magnetic field.Because of the coupling effect between the fields,the axial motion system will have complex vibration behavior.Therefore,the dynamic behavior of axially moving system under multi field coupling effect has become an important research topic.The periodic change of parameters in the system will cause parametric excitation and parametric vibration,resulting in intense resonance phenomenon.In this paper,the bi-directional parametric vibration of an axially moving ferromagnetic elastic beam in a transverse magnetic field is studied.According to the basic theory of elasticity,the expressions of potential energy and kinetic energy of ferromagnetic beam in axial motion are given.According to the classical electromagnetic theory,the expressions of electromagnetic force and magnetic couple are obtained.Then,according to the Hamiltonian variational principle,the nonlinear magnetoelastic two-way coupling vibration equation of ferromagnetic beam in transverse magnetic field is obtained.The parametric vibration of ferromagnetic beam in transverse magnetic field is studied.Under the condition of variable speed,the parameter vibration differential equation of ferromagnetic beam is obtained by the method of separating variables,and the Mathieu vibration equation is obtained by coordinate transformation.The parameter vibration problem of ferromagnetic beam with axial variable speed motion is solved by using flokey theory and average method,and the stability condition of the system is obtained.Through an example,the periodic stability diagram,the stability diagram of different parameters and the vibration response diagram are obtained.The influence of different parameters such as velocity,magnetic induction and axial tension on the stability of the system is analyzed.The main parameter resonance of a ferromagnetic beam in a transverse magnetic field is studied.The nonlinear parametric vibration differential equation of ferromagnetic beam with variable speed is obtained by using Galerkin integral method.The amplitude frequency response equations of axially moving ferromagnetic beam at principal parametric resonance are obtained by using multi-scale method,and its stability is analyzed.Through an example,the amplitude frequency characteristic curve,amplitude velocity,amplitude magnetic induction curve of ferromagnetic beam are obtained,and the vibration response diagram and phase diagram of the stable solution of the main parameter resonance system are given.The main parameter resonance of an axially moving ferromagnetic elastic beam in a variable magnetic field is studied.The nonlinear parametric vibration differential equation of ferromagnetic beam in variable magnetic field is obtained by using the variable separation method.By introducing time scale into the vibration equation,the amplitude-frequency response equation of the system under the resonance state of principal parameters is obtained by using the multi-scale method,and the stability criterion is given.Through numerical calculation,the amplitude-frequency characteristic curve of ferromagnetic beam and the rule curve of system amplitude-variation with parameters are given,and the influence of frequency coordination parameters,magnetic induction intensity,speed and other parameters on the resonance characteristics of main parameters is discussed.