The chaotic motion behavior of the rectangular conductive thin plate that is simply supported on four sides by airflow and mechanical external excitation in a magnetic field is studied. According to Kirchhoff’s thin plate theory, considering geometric nonlinearity and using the principle of virtual work, the nonlinear motion partial differential equation of the rectangular conductive thin plate is deduced. Using the separate variable method and Galerkin’s method, the system motion partial differential equation is converted into the general equation of the Duffing equation; the Hamilton system is introduced, and the Melnikov function is used to analyze the Hamilton system, and obtain the critical surface for the existence of chaos. The bifurcation diagram, phase portrait, time history response and Poincaré map of the vibration system are obtained by numerical simulation, and the correctness is demonstrated. The results show that when the ratio of external excitation amplitude to damping coefficient is higher than the critical surface, the system will enter chaotic state. The chaotic motion of the rectangular conductive thin plate is affected by different magnetic field distributions and airflow.

With the advancement and development of modern high-tech, devices with magnetic, electrical, and other materials as structures are frequently used. Rectangular thin plates are widely used in road and bridge construction, machinery industry, ship engineering, aerospace, and other fields. When the system is disturbed by the outside world, it will not only produce periodic linear dynamic behavior, but also show chaotic motion behavior to a large extent, resulting in the failure of the system under repeated loads. At present, there are two very popular directions for the study of nonlinear dynamics of structures such as thin plates at home and abroad. One is the study of nonlinear aeroelastic problems, the other is the study of nonlinear electromagnetic elasticity aspects.

The study of geometric nonlinear aeroelasticity differs from general aeroelasticity [

The theory of electromagnetic elasticity is devoted to the study of the coupling of electromagnetic fields with deformation fields. This theory is basically a coupling of the theory of linear elasticity [

In this paper, the basic assumptions of Kirchhoff’s theory are used, geometric nonlinearities are considered, and the nonlinear equations of motion of a magnetoelastic rectangular thin plate with simple support on four sides are established using the principle of imaginary work. The Hamiltonian is analyzed with the Melnikov function and the conditions under which the motion exhibits chaotic behavior are obtained. The bifurcation diagram, phase portrait, time history response and Poincaré map of the system were simulated with MATLAB software. The effects of the magnetic field environment as well as the airflow on the chaotic motion of the magnetoelastic rectangular thin plate are also analyzed.

Consider a four-sided simply supported rectangular conductive thin plate under the action of airflow and periodic mechanical excitation in the magnetic field environment shown in

The research idea of this paper is shown in

When studying the lateral vibration of elastic thin plates, there are four basic assumptions [

The vertical line segment perpendicular to the mid-plane of the thin plate has no change in its properties and is perpendicular to the deformed mid-plane.

The layers of materials parallel to the middle surface do not have mutual extrusion.

When the plate is bent, the amount of deflection in the z direction changes to zero.

When the plate is bent, there is no expansion and shear deformation at each point in the middle plane of the plate.

According to the elastic deformation theory, when the plate moves, the displacement of each point whose internal distance is z from the mid-plane can be expressed as follows:

In

According to the assumption of the Kirchhoff straight line method,

In

where

According to the basic assumption of Kirchhoff, using the generalized Hooke’s law [

In

In

During the imaginary displacement, the increment of deformation potential energy of the plate is:

The imaginary work done by the external force on the imaginary displacement is:

According to the principle of virtual displacement, the condition for the system to remain stationary is the external force acting on the system, and the sum of the virtual work done on the virtual displacement and the system deformation potential energy is zero [

According to Kirchhoff’s theory, considering the coupling effect of the thin plate under the external excitation and the electromagnetic field, using the principle of virtual work, the following magnetoelastic equation can be obtained as [

In

Assuming that the thin plate is a non-polarized, non-magnetized material with good conductivity, the electromagnetic quantity satisfies the Maxwell equation [

The electromagnetic constitutive relation is as follows:

In

When it is in the motion state under the magnetic field, the electromagnetic quantity in the thin plate can be written as:

In

From the

The vector expression of the Lorenz force acting on a deformed object by an electromagnetic field is:

The unit volume electromagnetic force is:

Integrating

Substituting

In

From the simply supported condition of the four sides of the rectangular thin plate, the method of separation of variables [

Using the mode shape superposition method, the solution

In the above equations,

According to the linear potential theory [

Substitute

The parameters

Let

The damping coefficient and the force coefficient are considered as perturbation terms. Introducing the small parameter

Assuming

Let

A hyperbolic saddle point in the q-z plane and two homoclinic orbits [

When

The generalized Melnikov function

If the Melnikov function has only one simple zero, the Poincaré map of the disturbance system

With the continuous increase of

Numerical simulations are performed using MATLAB software, and the Four-Order Range-Kuttle Method is used for iterative calculations to obtain the bifurcation diagram, phase portrait, Poincaré map and time history response of the system.

The structure and material parameters are selected as:

The surface corresponding to

The motion of the system discussed in this paper is affected by two changing factors, that is, the magnetic field distribution

Selecting

Take

The time history response for the four cases of

The phase portrait for the four cases of

The Poincaré map for the four cases of

Selecting

Take

The time history response for the four cases of

The phase portrait for the four cases of

The Poincaré map for the four cases of

Selecting

Take

The time-history curve for the four cases of

The phase portrait for the four cases of

The Poincaré map for the four cases of

In this paper, the effects of incoming velocity, magnetic field, and periodic mechanical force on the kinematic behavior of a rectangular conductive thin plate are studied. According to Kirchhoff’s thin plate theory, considering the geometric nonlinearity, the nonlinear dynamic equation of the system motion is established by using the principle of virtual work. The Galerkin’s method is used and the Hamiltonian system is introduced to analyze the Hamiltonian system with Melnikov functions to obtain the criterion for the existence of chaos. The bifurcation diagram, time history response, phase portrait and Poincaré map of the system under different magnetic field strengths are obtained through MATLAB simulation, and the chaotic motion of the rectangular conductive thin plate is qualitatively analyzed. Numerical results verify the possibility of chaotic behavior when the structural parameters given by the theoretical analysis satisfy certain conditions.

Based on the theoretical analysis and numerical calculation results, the chaotic motion is related to the incoming velocity and the magnetic field strength. When

The change of magnetic field only changes the value of

With the constant change of the incoming velocity, the motion of the rectangular conductive thin plate will enter an unstable state, resulting in chaotic motion. The increase of the magnetic field strength

This research was funded by the Anhui Provincial Natural Science Foundation (Grant No. 2008085QE245), the Natural Science Research Project of Higher Education Institutions in Anhui Province (2022AH040045), the Project of Science and Technology Plan of Department of Housing and Urban-Rural Development of Anhui Province (2021-YF22).

The authors declare that they have no conflicts of interest to report regarding the present study.