Ferromagnetic Thin Plate Research Articles

Multiple resonance of the ferromagnetic thin plate with axial velocity subjected to a harmonic excitation in an air-gap magnetic field

Multiple resonance of the ferromagnetic thin plate with axial velocity subjected to a harmonic excitation in an air-gap magnetic field

Combination resonance of a moving ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field

The combination resonance of a moving rectangular ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field is investigated. Initially, the kinetic and potential energies of the system, considering geometrical nonlinearity, are obtained based on Kirchhoff's theory of thin plates. Additionally, the works of external forces are determined using the principle of virtual work. Subsequently, taking into account the nonlinear magnetization phenomenon of ferromagnetic materials, the magnetization force and Lorentz force acting on the moving ferromagnetic plate are calculated according to electromagnetic theory. Finally, the nonlinear magnetoelastic vibration equation is established by applying the Hamiltonian variational principle. The algebraic equation for static deflection and the differential equation for disturbed deflection are derived by applying the Galerkin method to the vibration equation. Then, an analytical solution for the frequency response equation under combination resonance is obtained using the multiple scale method. The proposed model and analytical research methods are validated by comparison with existing literature and numerical methods. Furthermore, the stability of steady-state solutions is determined, and static bifurcation is analyzed through singularity analysis. With the analytical results, numerical examples are plotted with varying frequency tuning values, magnetic field intensities, load amplitudes and load positions. The effects of these parameters on vibration characteristics are examined by comparing amplitude curves of systems with different physical parameters. The results demonstrate significant nonlinear vibration characteristics of the thin plate under the influence of a constant magnetic field and motion effects.

Dual-mode coupling resonance and dynamic stability of axially moving ferromagnetic thin plate strips in alternating magnetic field

This paper investigates the dual-mode coupling resonance and dynamic stability of axially moving ferromagnetic thin plate strips in alternating magnetic field. According to the basic electromagnetic theory and the magnetoelastic mechanics model, Lorentz electromagnetic force and torque generated by eddy current effect and the magnetizing force generated by magnetization are determined for the ferromagnetic plate strip. Introducing Von Karman geometric nonlinearity, the nonlinear governing equation of axially moving ferromagnetic plate strips under the combined action of mechanical load and magnetizing force is established by Hamilton's principle. For the simply supported plate strip, Galerkin method and the multi-scale method are employed to solve the modal coupling resonance dominated by electromagnetic excitation, obtaining the amplitude-frequency response equations. The stability of resonance responses is analyzed by the Lyapunov stability theory. Through parametric research, the amplitude-frequency characteristic curves, bifurcation diagrams and Lyapunov exponent spectra of each mode are plotted; the effect of different parameters on vibration characteristics and dynamic stability under the modal coupling is analyzed for the system. The results show that as one of the modal amplitudes enters the resonance region, the other modal amplitude decays into the "collapse" region due to energy exchange between modes; each modal amplitudes has more complex multi-valued and jumping phenomena for the nonlinear coupling effect. The increase in velocity will affect the stability of system; the enhancement of magnetic field will make the system change into chaos at high velocity.

The harmonic resonance and singularity analysis of bifurcation for the magnetized elastic plate with action of time-varying magnetic potential

This paper deals with the superharmonic resonance of a ferromagnetic thin rectangular plate in the air-gap magnetic field excited by armature magnetic potential. Electromagnetic forces applied on the plate by the air-gap magnetic field cause the plate to transversal vibrate, which affects the air-gap magnetic field in turn, resulting the vibration and magnetic field coupling. According to the basic electromagnetic field theory and considering the magnetoelastic coupling effect, the air-gap magnetic field intensity is obtained by solving the Laplace's equation satisfied the air-gap magnetic boundary conditions. The electromagnetic force model of soft ferromagnetic plates is determined based on theories of electromagnetic and elasticity. According to the large deflection theory of plates, basic energy relationships and variational equations of the elastic plate are given. Eventually, the nonlinear magnetoelastic vibration equation of ferromagnetic thin plates is derived using Hamilton's principle and Galerkin method. The multi-scale method is used to solve the superharmonic resonance to obtain the amplitude-frequency response equation and the stability discriminant of solutions. The topological analysis of amplitude-frequency equation is carried out using singularity theory, and the bifurcation characteristics of systems on physical parameter planes in different regions are obtained according to the transition set. The correctness of analytical solutions is verified by comparison with numerical solutions. Through numerical calculations, curves of the static deflection and equivalent magnetic force of plate with parameters are given, and the amplitude curves, dynamic phase plane trajectories and time history diagrams of system response with changes in electromagnetic and structural parameters are plotted. Results show that both the decrease of armature magnetic potential amplitude and the increases of plate thickness and initial air-gap thickness reduce the static deflection. The increase of armature magnetic potential amplitude increases the equivalent magnetic force. As the decrease of initial air-gap thickness and the increases of armature magnetic potential amplitude and excitation force amplitude, the amplitudes of the upper branch and lower branch curves representing stable solutions decrease and increase, respectively, and the single-value solution region increases.

Simultaneous resonance of an axially moving ferromagnetic thin plate under a line load in a time-varying magnetic field

Simultaneous resonance of an axially moving ferromagnetic thin plate under a line load in a time-varying magnetic field

Magnetic-Solid Coupling Nonlinear Vibration of an Axially Moving Thin Plate Under Air-Gap Magnetic Field

The nonlinear inherent vibration of an axially moving ferromagnetic thin plate under the action of the armature air-gap magnetic field is investigated. Based on the nonlinear elasticity theory, the energy relationship equations of the thin plate are given. Based on the electromagnetic theory and the solution of Laplace’s magnetic potential equation, magnetic force on the ferromagnetic rectangular plate under air-gap magnetic field environment is deduced. The Hamilton variational principle is applied to derive the magneto-solid coupled nonlinear vibration equation of the axially moving ferromagnetic thin plate. The two-degree-of-freedom nonlinear vibration differential equations containing static load terms with boundary conditions of SSSS and SSCC are obtained by discretizing through the Galerkin method. The multi-scale method is applied to solve the second-order approximation for deriving the first two orders’ intrinsic frequency of the nonlinear system. The variation laws of first two orders’ nonlinear inherent vibration with axial velocity, magnetic potential, initial air-gap thickness, and initial value are given through numerical examples, and the comparative analysis is performed. The results show that the first-order and second-order inherent vibration frequencies decrease with the increase of axial velocity and magnetic potential and increase with the increase of initial air-gap magnetic field thickness. Different materials and different boundary conditions have greater influence on the first-order and second-order inherent frequencies, and show obvious nonlinear characteristics. The results can provide references for analyzing and controlling the vibration behavior of moving structures in electromagnetic environment.

Magnetoelastic simultaneous resonance of axially moving plate strip under a line load in stationary magnetic field

This paper studies the primary-internal simultaneous resonance of the moving ferromagnetic thin plate with the action of an alternating line load in a transverse stationary magnetic field. Based on the theory of thin plates, expressions of strain energy and kinetic energy of the thin plate are given. According to the theoretical magnetoelastic model for ferromagnets, the calculation formula for electromagnetic force is derived. The magnetoelastic coupling nonlinear vibration equation is established by using Hamilton’s principle. Duffing-type nonlinear differential equations are obtained for the clamped-hinged plate strip by applying the Galerkin method. The steady-state response of the system with primary resonance and 1:3 internal resonance is given by using the method of multiple scales, and the stability of steady-state solutions are analyzed according to the theory of Lyapunov stability. The primary-internal simultaneous resonance characteristics of the plate strip are presented through numerical examples. Results show that: energy exchange between the first two modes is available and the first mode dominates; jump phenomena exist in amplitude–frequency curves; both the values of steady solutions and the multi-valued regions of amplitude change with varieties of physical parameters; and numerical results agree with analytical results, which verifies the reliability of analytical research method in this paper.

Magnetoelastic primary resonance of an axially moving ferromagnetic plate in an air gap field

The air gap field widely exists in mechatronic systems to realize energy transfer, but the components in such systems are often regarded as rigid bodies. In this paper, elastomers in an air gap field are discussed for the first time. The magnetoelastic primary resonance of an axially moving ferromagnetic plate in an air gap field is investigated. With the magnetic wall armature as the electromagnetic source, the ferromagnetic thin plate moves axially parallel to the armature. An air gap field excited by the armature is distributed in thin air gap space between them. In addition to eddy current effects, the ferromagnetic thin plate in the air gap field will be magnetized to generate a harmonic magnetic force to excite the plate vibration, which results in the system existing nonlinear coupling characteristics between armature, plate deformation and air gap field. Considering the dynamic feedback of thin plate vibration on the air gap field by changing the air gap thickness, the magnetoelastic vibration equation of the plate is derived by the Hamilton's principle. The vibration differential equation for the simply supported plate is obtained via Galerkin method, and the static deflection of the plate under static magnetic load is solved. The multi-scale and Lyapunov methods are utilized to analyze the primary resonance under harmonic magnetic force, thus obtaining the steady-state resonance equation and the stability discriminants of stationary solutions. Through numerical simulations, the curves and three-dimensional surfaces of amplitude varying with different parameters are plotted. The variation of magnetic field strength, magnetic force, and electromagnetic torques under steady-state response is obtained. The results show that amplitude decreases with the increase of initial air gap thickness but increases with the increase of magnetic potential amplitude on the armature. Velocity has a suppression effect on system resonance, and there is an intrinsic coupling mechanism of action and reaction between the thin plate and air gap field.

An I-integral method for the crack-tip intensity factor in ferromagnetic materials with domain switching

An I-integral method is developed to calculate the crack-tip stress intensity factors (SIFs) in the ferromagnetic thin plate subjected to the external loads. The value of I-integral is not affected by the integral area size, the magnetization distributions or domain walls, which shows the applicability to the large-scale domain switching. Based on the stress and magnetic field obtained from phase-field simulations, the I-integral method is successfully employed to obtain the crack-tip intensity factors and decouples different modes of the crack-tip intensity factors in ferromagnetic thin plates with an impermeable edge crack under different mechanical and/or magnetic loadings. The influence of domain switching on the SIFs is investigated for different loadings. It is found that domain switching under a constant stress loading increases the crack-tip intensity factors, whereas domain switching decreases the crack-tip intensity factors under a constant strain loading. The results indicate that the domain switching-induced shielding or anti-shielding of a crack in ferromagnetic materials with an impermeable edge crack is dependent on the loading condition.

Phase field simulations on domain switching-induced toughening in ferromagnetic materials

Abstract A real-space phase field model is employed to investigate the domain switching-induced shielding or anti-shielding of a crack in ferromagnetic materials with single and multi-domain states. Phase field simulations demonstrate that magnetization switching takes place from the crack tip due to the highly concentrated stress in a ferromagnetic thin plate with a stationary crack. Based on the stress obtained from phase field simulations, the crack-tip stress intensity factors (SIFs) are calculated by linear extrapolation in the ferromagnetic thin plate subjected to different values of mechanical loads. The calculation results indicate that domain switching decreases the crack-tip stress intensity factors, resulting in domain-switching toughening in ferromagnetic materials. Furthermore, the domain switching in the multi-domain ferromagnetic plate induces a much larger decrease of stress intensity factor than that in the single-domain one, which suggests that engineering magnetic domains leads to the design of tougher ferromagnetic materials.

Comparison of Three Modeling Methods for Identifying Unknown Magnetization of Ferromagnetic Thin Plate

This study presents three different magnetization models for identifying unknown magnetization of the ferromagnetic thin plate of a ship. First, the forward problem should be solved to accurately predict outboard magnetic fields due to the magnetization distribution estimated at a certain time. To achieve this, three different modeling methods for representing remanent magnetization (i.e., magnetic charge method, magnetic dipole array method, and magnetic moment method) were utilized. Material sensitivity formulas containing the first-order gradient information of an objective function were then adopted for an efficient search of an optimum magnetization distribution on the hull. The validity of the proposed methods was tested with a scale model ship, and field signals predicted from the three different models were thoroughly investigated with reference to the experimental data.

An Energy Method for Analyzing Magnetoelastic Buckling and Bending of Ferromagnetic Plates in Static Magnetic Fields

Based on an energy method analyses of magnetoelastic buckling and bending of ferromagnetic thin plates in transverse, oblique, and longitudinal fields are performed in this paper. A general expression of the critical magnetic field of the plate in a transverse field is presented. It is shown that the critical magnetic field is not only related to the ratio of the thickness to the cantilevered length, but also to the susceptibility and the demagnetizing factors that are primarily dependent on the ratios of the thickness to width and the thickness to the total length. In different cases, main factors that affect the critical magnetic field are discussed. Theoretical predictions agree with experimental results reasonably.

Vibration and dynamic instability of ferromagnetic thin plates in magnetic fields

Vibration and dynamic instability of ferromagnetic thin plates in magnetic fields

Buckling of a ferromagnetic thin plate in a transverse static magnetic field

An energy theory is proposed on the buckling of a ferromagnetic thin plate in a transverse static magnetic field. The analytic solution of critical field for plates is obtained by using variational method. It is shown that the critical magnetic field is not only influenced by the ratio of the thickness to the length, but also by the width, total length and the magnetic susceptibility of the specimen. Theoretical calculations agree with experiments reasonably.

A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields

The experiments of ferromagnetic plates in different magnetic environments exhibit two distinct phenomena, i.e. the magnetoelastic instability of a ferromagnetic plate in transverse magnetic fields, and the increase of natural frequency of a ferromagnetic plate with low susceptibility in an inplane magnetic field. Although these two typical phenomena can be predicted separately by two kinds of theoretical models in which the magnetic forces are formulated by totally different expressions, no theoretical model has been found to commonly describe them. This makes it difficult to predict theoretically magnetoelastic interaction of a ferromagnetic structure in complex magnetic environment. A variational principle, here, is proposed to establish the governing equations of magnetoelastic interaction for soft ferromagnetic thin plate structures under complex magnetic fields. The functional is chosen as the summation of the magnetic energy and the strain energy as well as the external work from applied magnetic fields. From manipulations of the variational principle, the governing equations of the magnetic field and mechanical deformation together with an expression of equivalent magnetic force exerted on the ferromagnetic plates are obtained. It is shown that this theoretical model can commonly characterize the experimental phenomena of the magnetoelastic interaction aforementioned.

A theoretical model of magnetoelastic bucking for soft ferromagnetic thin plates

As an essential model of magnetoelastic interaction between magnetic field and mechanical deformation, the study on magnetoelastic buckling phenomenon of soft ferromagnetic plates in a magnetic environment has been conducted. One of the key steps for the theoretical prediction of the critical magnetic field is how to formulate magnetic force exerted on the magnetized medium. Till today, the theoretical predictions, from theoretical models in publications, of the magnetoelastic buckling of ferromagnetic cantilevered beam-plate in transverse magnetic field are all higher than their experimental data. Sometimes, the discrepancy between them is as high as 100%. In this paper, the macroscope formulation of the magnetic forces is strictly obtained from the microscope Amperion current model. After that, a new theoretical model is established to describe the magnetoelastic buckling phenomenon of ferromagnetic thin plates with geometrically nonlinear deformation in a nonuniform transverse magnetic field. The numerical method for quantitative analysis is employed by combining the finite elemental method for magnetic fields and the finite difference method for deformation of plates. The numerical results obtained from this new theoretical model show that the theoretical predictions of critical values of the buckling magnetic field for the ferromagnetic cantilevered beam-plate are in excellent agreement with their experimental data. By the way, the region of applicability to the Moon-Pao's model, or the couple model, is checked by quantitative results.

Numerical evaluation of natural frequency change of a ferromagnetic plate due to magnetic stiffness effect

This paper describes the numerical simulation of the dynamic behavior of a ferromagnetic thin plate vibration in a magnetic field. The behavior of a ferromagnetic cantilevered plate, which is set in the direction of a uniform magnetic field, shows that its natural frequency increases with the external magnetic induction. Two methods were applied to evaluate this phenomenon. The magnetic field of the deformed shape has been analyzed by 3D analysis and 1D approximation. The natural frequencies were evaluated by the energy method in the 3D case and by a transient response in the 1D case. The results show that the natural frequency change can be explained by the magnetic stiffness effect caused by magnetization including its saturation.

Experimental estimation of electromagnetic forces on non-magnetic and ferromagnetic thin plates

Mechanical vibration is investigated experimentally to evaluate electromagnetic forces on non-magnetic and ferromagnetic thin plates in the impulsive magnetic field. In addition to a Lorentz force due to eddy currents, a drag force is created by magnetization in the ferromagnetic plate. The electromagnetic forces are evaluated from the peak deflection of a cantilevered acrylic beam holding the test piece. Dependence of the forces on the excitation coil current is discussed based on the experimental results.

Dynamic bending of cracked soft ferromagnetic plate in uniform magnetic field

The scattering of time harmonic flexural waves by a through crack in a soft ferromagnetic thin plate under a uniform magnetic field is considered. Fourier transform is employed reducing the magneto-elastodynamic problem to the solution of a Fredholm integral equation of the second kind. Numerical results are obtained for the dynamic moment intensity factor that tends to increase with increasing magnetic induction and effect becomes more and more pronounced as the frequency of the flexural waves is decreased.

Magnetoelastic interaction of a soft ferromagnetic plate with a through crack under bending.

A magneto-elastic analysis is presented for the bending problem of a soft ferromagnetic thin plate of finite width with a through crack under a uniform magnetic field. Fourier transform method is used to solve the magneto-elastic problem and the result is expressed in terms of an integro-differential equation for the deflection. The integro-differential equation is solved by the standard iteration method and the mixed boundary value problem is reduced to the solution of a Fredholm integral equation of the second kind. The moment intensity factor is computed and the influence of the magnetic field on the normalized values is displayed graphically.