1 Introduction

Circular plates and their composite structures in electromagnetic field are widely used in engineering practice, such as iron cores of motor stator and rotor, electromagnetic sensor components. Coupled vibration of these structures in electromagnetic field is the main factor determining the safe operation of electromechanical systems. Therefore, it is theoretically significant to study the nonlinear vibration of ferromagnetic circular plates in magnetic field, and it can also provide technical support for the dynamic control of mechanical and electrical systems in engineering practice.

Many scholars have studied the magnetoelastic vibration of structures in magnetic field. Hasanyan and Piliposyan^[1] studied the transverse vibration of magnetoelastic thin plates in transverse magnetic field. Zhou and Miya ^[2] applied the theoretical model of magnetoelastic interaction, analyzed the influence of magnetic field on the increase of natural frequency of ferromagnetic plates, and compared the obtained results with the experimental results. Dong^[3] analyzed the three-dimensional free vibration of magnetoelectroelastic circular and annular plates with different boundary conditions by using the Chebyshev-Ritz method. Bagdoev and Vardanyan^[4] used the average method to study the free vibration frequency of conductive plates in transverse magnetic field under a three-dimensional modeling. The vibration frequency of the ferromagnetic beam under the effect of the tilted magnetic field was analyzed by Liang et al.^[5], and they compared it with the previous experimental data. Ding et al.^[6] studied the steady-state periodic response of the forced vibration of a traveling viscoelastic beam under the 3:1 internal resonance condition. A mechanical model of a magnetoelastic buckling beam subjected to an axial external force in a periodic transverse magnetic field was established by Lu et al.^[7], and the resonance of the principal parameters was studied. Wang and Lee^[8-9] studied the dynamic stability of rectangular simply supported plate and beam plate with soft ferromagnetism under a transverse magnetic field. Hu and Wang^[10] and Hu et al.^[11-13] studied the free vibration of a rotating circular plate under the action of static load under the action of the magnetic field, and analyzed the magnetoelastic resonance, parametric vibration, bifurcation, and dynamic stability of a rotating conducting circular plate subjected to alternating loads. Zhou and Wang^[14] and Hu et al.^[15] analyzed the transverse vibration and stability of the axially moving viscoelastic plate, and studied nonlinear parametric vibration and stability of an axially accelerating rectangular thin plate.

In addition, many scholars have studied the magnetoelastic bending problem of plates in magnetic fields. Horiguchi and Shindo ^[16] performed a bending experiment analysis on soft magnetic beam plates and theoretically analyzed that based on the classical plate bending theory of the magnetoelastic interaction in the soft ferromagnetic material. Chen et al. ^[17] gave a three-dimensional analytical solution of a simply supported magnetoelastic circular plate under the uniform load, and analyzed the stability and buckling of the circular plate in the magnetic field. Based on the energy method, Yang et al.^[18] analyzed the magnetoelastic buckling of ferromagnetic thin plates in a static magnetic field based on an energy method. Li et al.^[19] studied the bending problem of simply supported functionally graded piezoelectric plates under the action of an in-plane magnetic field. Zhou et al.^[20], Harik and Zheng^[21], and Zheng and Wang^[22] found that the magnetoelastic buckling problem of ferromagnetic beam plates is nonlinear based on the Moon-Pao model, and they studied the magnetoelastic bending of ferromagnetic plates under the action of oblique magnetic field. Hasanyan et al.^[23] analyzed the problem of buckling instability of magnetoelastic plates caused by current and its effect on the post-buckling behavior.

In this paper, the magnetoelastic combined resonance generated by mechanical force and electromagnetic force for a ferromagnetic circular plate in alternating magnetic field is investigated. The amplitude-frequency response equation and the stability determination for the steady-state solutions of the circular plate are derived, respectively. The effects of different parameters on the vibration characteristics are analyzed by numerical calculations.

2 Magnetoelastic vibration equation of ferromagnetic circular plate 2.1 Kinetic energy and potential energy

Consider an isotropic ferromagnetic circular plate in an alternating transverse magnetic field, where H_n1 is the upper surface magnetic field, H_n2 is the lower surface magnetic field, and subjected to the transverse load P_z. As shown in Fig. 1, a cylindrical coordinate system (r, θ, z) is established on the circular plate, where r, θ, and z are the radial, circumferential, and transverse coordinates, respectively. The thickness of the circular plate is h, the radius is R, and the mass density is ρ.

Ignoring the in-plane displacement, the expressions of the deformation displacement components at any point of the ferromagnetic circular plate are

(1)

where w is the transverse displacement component of the middle plane of the circular plate, and are the angular displacements.

The internal forces in the middle plane are

(2)

where is the tensile stiffness, E is the modulus of elasticity, and ν is the Poisson ratio.

The kinetic energy expression of the system is

(3)

The total potential energy U of the plate during deformation includes two parts: the bending deformation potential energy U₁ and the middle plane strain potential energy U₂. The expression of U₁ is

(4)

where is the Laplace operator, and is the flexural rigidity.

The expression of U₂ is

(5)

where σ_r is the radial stress, σ_θ is the circumferential stress, and τ_rθ is the shear stress.

2.2 Basic theory of magnetic field 2.2.1 Magnet force

Considering the magnetization effect of a ferromagnetic circular plate in a magnetic field, the magnet force in the magnetic medium^[24] is

(6)

where H is the magnetic field intensity vector, μ₀ is the magnetic permeability in vacuum, μ_r is the relative magnetic permeability, χ_m =μ_r -1 is the magnetic susceptibility of the material, and ∇ is the Hamiltonian operator.

According to the magnetoelastic interaction theory model based on the variational principle of energy functional ^[24], the boundary magnet force (the surface force on the upper and lower surfaces) of the ferromagnetic circular plate is obtained as

(7)

where n is the unit normal vector on the surface of ferromagnetic medium.

The surface force F^em of the upper and lower surfaces and the magnet force f^em are simplified to the middle plane to obtain the following formula for calculating the equivalent transverse magnet force:

(8)

where H_n is the magnetic field intensity in the normal direction of the circular plate surface, and H_τ is the magnetic field intensity in the tangential direction of the circular plate surface.

For the case in Fig. 1, where the normal alternating magnetic field is considered only, the magnetic field intensities on the upper and lower surfaces of a circular plate are determined as follows:

where H₀ is the amplitude of the magnetic field intensity, and ω₁ is the frequency of the magnetic field intensity. Then, the magnet force is

(9)

where .

2.2.2 Eddy current

Considering the eddy current effect produced by a ferromagnetic circular plate in a magnetic field, the resulting Lorentz force expression is

where J=σ₀ (V× B) is the current density vector when the electric field strength is ignored, and σ₀ is the conductivity.

Assuming that the magnetic field inside the circular plate is distributed along the z-axis, we can obtain

(10)

Equation (10) is integrated along the thickness. The expressions of the electromagnetic moments m_r and m_θ of the circular plate are

(11)

(12)

2.2.3 Virtual work of electromagnetic force

In the transverse alternating magnetic field, the virtual work expression of the ferromagnetic circular plate determined by the magnet force and the Lorenz electromagnetic force is

(13)

2.3 The establishment of vibration equation by the Hamilton principle

With the Hamilton principle, the integration is performed at fixed times t₁ to t₂ as

(14)

where δU_P is the virtual work done by the lateral excitation force P_z.

Substituting Eqs. (2)-(5), (9), and (11)-(13) into Eq. (14), for the axisymmetrical problem of a circular plate, the nonlinear vibration equation of a ferromagnetic circular plate in a magnetic field can be derived

(15)

3 Solution to the combined resonance problem of the ferromagnetic circular plate 3.1 Vibration differential equation

The excitation force is

(16)

where P₀ is the amplitude of external excitation force, and ω₂ is the frequency of external excitation force.

The displacement solution satisfying the constraint condition of peripheral clamped supports is set as follows:

(17)

Equation (16) and the displacement solution (17) are substituted into the vibration (15). According to the Galerkin method, the nonlinear vibration differential equation of the circular plate is obtained by integrating and dimensionless processing,

(18)

where

3.2 Multi-scale solution

The small parameter ε is introduced to study the problems for the weak nonlinear system with strong excitation vibration. Then, Eq. (18) can be rewritten as

(19)

where .

Using the multi-scale method to solve the problem, and introducing different time variables to discuss the approximate solution, T₀ =τ, T₁ =ετ, we get the first order approximate solution to the vibration equation as follows:

(20)

Substituting Eq. (20) into Eq. (19) and collecting the coefficients of zeroth and first power terms of ε, we get the approximate linear partial differential equations of each order as

(21)

(22)

where , and

Thus, the general expression of Eq. (21) is

(23)

where is the imaginary unit, and c.c. is the sum of conjugate complex numbers on the right-hand side of Eq. (23).

Substituting Eq. (23) into Eq. (22), we get

(24)

After collating and analyzing Eq. (24), we can see that the combined resonance occurs when frequency relations are 1≈ 4Ω₁ ± Ω₂, 1≈ Ω₂-4Ω₁, 1≈ 2Ω₁ ± 2Ω₂, 1≈ 2Ω₂ -2Ω₁, 1≈ 2Ω₁ ± Ω₂, and 1≈ (2Ω₁ ± Ω₂)/2.

The system resonance problem with frequency relation satisfying condition 1≈ 4Ω₁ +Ω₂ is taken as an example. The frequency tuning parameter σ is introduced, 4Ω₁ +Ω₂ =1+εσ. And eliminating the secular term of Eq. (24), we can obtain

(25)

Setting and separating Eq. (25) into its real and imaginary parts, we can obtain

(26)

where

Let a'=γ '=0. Then, we can obtain the amplitude-frequency response equation of the combined resonance of the ferromagnetic circular plate

(27)

When the frequency relation satisfies 1≈ 4Ω₁ -Ω₂ and 1≈ Ω₂-4Ω₁, the amplitude frequency response equation of ferromagnetic circular plate combined resonance is in agreement with Eq. (27).

When the frequency relation satisfies 2Ω₁ ± 2Ω₂ =1+εσ and 2Ω₂ -2Ω₁ =1+εσ, the amplitude frequency response equation of ferromagnetic circular plate combined resonance is in agreement with Eq. (27), where Γ₁ and Γ₂ are

When the frequency relation satisfies 2Ω₁ ± Ω₂ =1+εσ, the system contains parametric vibration. The real and imaginary parts are separated into the following equations:

(28)

where

For a'=γ '=0, the amplitude frequency response equation is

(29)

When the frequency relation satisfies , the real and imaginary parts are separated into the following equations:

(30)

where

For a'=γ '=0, the amplitude frequency response equation is

(31)

3.3 Stability of steady-state motions

According to the Lyapunov stability theory, the stability analysis of the steady state solution to the system is carried out. We introduce disturbance variables a=a₀ +a₁, and γ =γ₀ +γ₁.

Case Ⅰ Frequency relationships satisfy 4Ω₁ ± Ω₂ =1+εσ, Ω₂-4Ω₁ =1+εσ, 2Ω₁ ± 2Ω₂ =1+εσ, and 2Ω₂ -2Ω₁ =1+εσ. The approximation system of the corresponding Eq. (26) near the singular point (a₀, γ₀) is

(32)

Case Ⅱ Frequency relationship satisfies 2Ω₁ ± Ω₂ =1+εσ. The approximation system of the corresponding Eq. (28) near the singular point (a₀, γ₀) is

(33)

Case Ⅲ Frequency relationship satisfies . The approximation system of the corresponding Eq. (30) near the singular point (a₀, γ₀) is

(34)

Then, the characteristic equation is

(35)

where c₁ and c₂ are

Case Ⅰ

Case Ⅱ

Case Ⅲ

Here, Γ₁, Γ₂ under each frequency relationship are consistent with Subsection 3.2.

According to the Routh-Hurwitz criterion, the necessary and sufficient condition for the steady state solution of the system under μ >0 condition is

(36)

In addition, according to the characteristic root judgment formula Δ =c₁²-4c₂, the behavior analysis from the singular point (balance point) shows that

(Ⅰ) When Δ >0, if c₂ >0, the singular point is the node, if c₂ < 0, the singular point is the saddle point.

(Ⅱ) When Δ =0, the singular point is the node.

(Ⅲ) When Δ < 0, the singular point is the focus.

4 Numerical simulations

The research on the soft ferromagnetic circular thin plate is made of martensitic steel under the condition of the clamped boundary in the transverse alternating magnetic field. The values of the physical parameters^[2] are: mass density ρ =7 800 kg/m³, relative permeability μ_r =1 000, Poisson's ratio ν =0.3, Young's modulus E=200 GPa, conductivity σ₀ =2.3× 10⁶ Ω^-1· m^-1. In the calculation, the frequency relation is mainly 4Ω₁ +Ω₂ =1+εσ for the numerical analysis.

Figure 2 shows the amplitude-frequency characteristics of the ferromagnetic circular plate under the conditions of variable thickness, magnetic field intensity, and excitation force. The solid line in the graph corresponds to the stable solutions, and the dotted line corresponds to the unstable solutions (the same below). The curves show that the resonance amplitude value has a multi-valued and jumping property, and it is significantly increased in the resonance region (near εσ ≈ 0). The curves deviate to the right in varying degrees, showing the characteristics of hard spring. As shown in Figs. 2(a) and 2(b), as the thickness of the plate decreases or the magnetic field intensity increases, the degree of shift of the resonance curve to the right increases, and the resonance region becomes wider. In Fig. 2(c), as the excitation force increases, the resonance region becomes wider, and the amplitude decreases when the tuning parameters are the same in the resonance region.

Figure 3 shows the curve of the critical points of resonance bifurcation with tuning parameter and plate thickness by picking up the critical points of the multi-value solutions of the amplitude-frequency characteristic curves in Fig. 2(a). As can be seen from the curve in Fig. 3, when the resonance bifurcation occurs as the tuning parameter increases, the plate thickness decreases correspondingly, and the curve exhibits a nonlinear relationship.

Figure 4 shows the amplitude-frequency characteristics of the ferromagnetic circular plate with different frequency relationships. In Fig. 4, R=0.3 m, h=0.005 m, H₀ =100 A·m^-1, and P₀=3 000 N· m^-2. The figure shows that, the amplitude-frequency curves under different frequency conditions have a substantially similar shape, the amplitude of the resonance region increases significantly, the curves shift to the right, and there are multi-valued and jumping properties.

Figures 5 and 6 show the curves of amplitude versus magnetic field intensity under different tuning parameters and excitation forces. In Fig. 5, the parameters are: R=0.3 m, h=0.005 m, and P₀=3 000 N· m^-2; in Fig. 6, R=0.3 m, h=0.005 m, and εσ =0.02. The curves in Fig. 5 and Fig. 6 are left-right symmetrical with respect to H₀ =0 A· m^-1, and resonance is excited near H₀ =50 A·m^-1, with large amplitudes and multiple values. When the magnetic field intensity increases to a certain value, the multivalued property vanishes and becomes a single value solution. Figure 7 extracts critical points of the multi-valued solutions to the amplitude-magnetic field intensity characteristic curves of a plurality of different excitation amplitudes in Fig. 6. The curve in the figure shows that as the multi-value phenomenon occurs with the increase in the magnetic field intensity, the corresponding magnetic field force decreases, and the curve approximates a linear relationship.

Figures 8 and 9, respectively, show the amplitude versus the amplitude of the excitation force when the tuning parameter and the magnetic field intensity are different. In Fig. 8, R=0.3 m, h=0.00 5 m, and H₀ =150 A· m^-1. In Fig. 9, R=0.3 m, h=0.005 m, and εσ=0.018. The curves in Fig. 8 and Fig. 9 show that the combined resonance of the system can be excited when the amplitude of the excitation force is small, and the solutions are multi-valued when the excitation force is small. With the increase in the excitation force amplitude, the multi-valued solution becomes a single value solution, and the amplitude decreases. The upper and lower two parts of the curve are stable solutions, with unstable solutions in the middle. At the same time, it can be seen from the figures that the resonance amplitude of the upper-branch curves and the upper half of the lower-branch curves increase with the increase in tuning parameters or the decrease in magnetic field intensity, while the amplitude of the lower half of the lower-branch curve decreases. And in Fig. 9, when the excitation amplitude is about P₀=6 900 N· m^-2, the three curves intersect. Figure 10 draws the curve of the critical points of resonance bifurcation with the variation of magnetic field intensity and excitation force by picking up critical points of the amplitude-exciting force characteristic curves with different tuning parameters in Fig. 8. Figure 10 shows that as the multi-value phenomenon occurs with the increase in the excitation force, the corresponding tuning parameters also increase, and the curve exhibits a nonlinear relationship.

Figure 11 verifies the occurrence of the intersection point appears in Fig. 9. It can be seen from the figure that when εσ=0.018, P₀=6 900 N· m^-2, and the magnetic field intensity is small, the curve has a multi-valued phenomenon. When H₀ =60 A· m^-1, it gradually becomes a single-valued solution. And as the magnetic field intensity increases, the curve approximates a horizontal straight line with amplitude of about 0.088. Therefore, there is an intersection point in the curve in Fig. 9.

Figure 12 is the curve of amplitude versus amplitude of magnet force. The parameters in the figure are taken as R=0.3 m, h=0.005 m, εσ =0.05, and P₀ =1 000 N· m^-2. In Fig. 12, the curve shifts to the left, exhibits a soft spring characteristic. The resonance is excited near F_z0=9 000 N· m^-2, and the multi-value solution exists. As the amplitude of magnet force increases, the multi-value disappears and gradually becomes a single-value solution. And when F_z0 =0 N· m^-2, the corresponding H₀ =0 A·m^-1, when F_z0 =5× 10⁴ N· m^-2, and H₀ =± 282.23 A ·m^-1.

Figures 13 and 14 are the curves of magnet force and electromagnetic torque with time and magnetic field intensity. In each figure, R=0.3 m, h=0.005 m, εσ =0.05, and P₀ =1 000 N· m^-2. Figure 13(a) and Fig. 14(a) are the time-history responses of the magnet force and the electromagnetic torque with time. In Fig. 14, r=0 m. It can be seen from the figures that the magnet force and the electromagnetic torque have cyclic changes. Figure 13(b) shows the curve of maximum magnet force versus magnetic field intensity in Fig. 13(a), which shows a nonlinear increasing relationship. Figure 14(b) shows the variation of electromagnetic torque with the magnetic field intensity at the corresponding time, and the curve shows a nonlinear negative growth. Figure 14(c) shows the curve of the maximum electromagnetic torque with the magnetic field intensity in Fig. 14(a), which shows a nonlinear growth, and the value of the electromagnetic torque under the same magnetic field intensity is greater than that in Fig. 14(b). Figure 13(c) shows the change of magnet force with the magnetic field intensity at the corresponding time. The curve shows a nonlinear negative growth, but the value of magnet force is smaller than that of Fig. 13(b).

Figure 15 is a region division diagram of singular point type. The solid blue line in the figure is the Δ =0 curve and the c₂ =0 curve drawn under the corresponding horizontal and vertical coordinates, and the two curves are coincident in the figure. As shown in Fig. 15(a), the inner region of the blue solid coil Δ >0, c₂ < 0, and the singular point analysis shows that the singular points in the region are unstable saddle points. The blue solid outer region Δ < 0 and c₂ >0, it is known that the singular points in the region are stable focuses. Figure 15 is compared with the corresponding amplitude-frequency characteristic curve, the correctness of the singular point analysis is verified by the unstable solution in the blue line region and the stable solution outside the region.

Figure 16 shows the time history diagrams, phase diagrams, and Poincaré diagrams for different tuning parameters. The figure of each group is a full-response diagram drawn by substituting the amplitude into Eq. (23) and solving the analytical solution of Eq. (18). The time history diagram shows that the system motion is periodic. It can be seen from Fig. 16 that when εσ = 0, the phase trajectory is a closed curve, the Poincaré diagram exhibits a scattered 6-point form, and the system is a 6-periodic motion. When εσ increases to 0.05, the Poincaré diagram presents the form of ring point set, and the system becomes quasi-periodic motion. When εσ = 0.1, the phase diagram presents a closed curve with multiple loops, and the motion state is multi-periodic motion. When εσ =0.12, the system changes to the quasi-periodic motion, and then the quasi-periodic motion and multi-periodic motion alternate.

Figure 17 is a comparison of the response of the upper and lower branches of one of the resonance curves in Fig. 2(c). It is found that the system moves reciprocally and periodically. The phase diagram is a multi-circle surround distribution curve. The Poincaré diagram presents a closed curve point set form. So the motion state is quasi-periodic motion. The amplitude of the upper branch response is large and varies greatly with time. The Poincaré diagram point set is in the form of two circles that almost overlap. The response amplitude of the lower branch is small, the phase diagram is dumbbell shape, and the Poincaré diagram point set is in the form of two closed curves docking.

5 Conclusions

In this paper, the combined resonance of a ferromagnetic circular plate in alternating magnetic field under the combination of external excitation force and magnet force is investigated. The amplitude-frequency equations of the combined resonance and stability determination are derived. The amplitude varying with tuning parameters, magnetic field and excitation force curves and system response diagrams are analyzed, respectively. The results can be summarized as follows:

(ⅰ) When the combined resonance occurs in the system, the amplitude in the resonance region increases significantly, and the multi-valued and jump of the resonance curve exist. Moreover, the resonance curve shifts to the right, and as the plate thickness decreases, the magnetic field intensity and the excitation force increase, this shift will be more obvious.

(ⅱ) The resonance is excited when the magnetic field intensity and excitation force are small, and the amplitude is large and multi-valued. The amplitude changes from multi-value solution to single-value solution with the increase of magnetic field intensity and excitation force, and the amplitude decreases gradually. The upper and lower parts of the curve are stable solutions, and the intermediate ones are unstable solutions.

(ⅲ) When the tuning parameter is 0, the system may present the 6-periodic motion, furthermore, with the increase in the tuning parameter, the alternating process from multi-periodic motion to quasi-periodic motion may occur.

Finally, the structural resonance may be restrained or excited by dominating the magnetic field intensity or exciting force, which can provide technical guidance for solving the dynamic problems of mechanical and electrical engineering.