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# Chaotic Motion Analysis for a Coupled Magnetic-Flow-Mechanical Model of the Rectangular Conductive Thin Plate

1
School of Civil Engineering, Anhui Jianzhu University, Hefei, 230601, China

2
Key Laboratory of Intelligent Underground Detection Technology, Anhui Jianzhu University, Hefei, 230601, China

* Corresponding Author: Xiaofang Kang. Email:

(This article belongs to the Special Issue: Vibration Control and Utilization)

*Computer Modeling in Engineering & Sciences* **2023**, *137*(2), 1749-1771. https://doi.org/10.32604/cmes.2023.027745

**Received** 13 November 2022; **Accepted** 13 February 2023; **Issue published** 26 June 2023

## Abstract

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.## Graphic Abstract

## Keywords

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 [1] from the theoretical aspects as follows: One is the structural geometric nonlinear theory, which mainly addresses the static and dynamic analysis of the structure under large deformation [2–4]; The other is the study of surface aerodynamic theory [5], which mainly addresses the boundary condition dependent deformation state aerodynamic calculation method under large deformation conditions of the structure [6]; The third is the study of the structural/aerodynamic interface coupling method [7,8], which mainly investigates the multidimensional interpolation problem applicable to large deformation in space. The problem of subsonic aeroelasticity of plates focuses on the fluid-structure coupling between the structure and the airflow [9,10]. The variety of parameters, such as mass, damping and stiffness of the structure under the action of subsonic airflow affects the critical instability and nonlinear vibration characteristics of the structure [2,11]. The assumption of small deformation in its research is no longer applicable, the equilibrium state of the structure after force deformation presents obvious geometric differences relative to the undeformed structure, and the geometric nonlinear factors caused by the load-bearing and deformation state of the structure make the structural static and dynamic characteristics change, and change the static and dynamic aeroelastic coupling relationship, thus making the research and application of aeroelasticity face new challenges.

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 [12] and the theory of linear electrodynamics in a free moving medium. If the studied elastomer is located in an initially strong magnetic field, mechanical and thermal loads would generate an electromagnetic field while causing a deformation field. The two fields will interact and influence each other and a coupling mechanism will occur. The action of the electromagnetic field on the deformation field is caused by the Lorentz force in the equations of motion [13–15]. The deformation field affects the strength of the magnetic field, the magnetoelastic wave [16] and the propagation velocity of the electromagnetic wave, and the item depends on the displacement velocity of the deformed object in the magnetic field [17]. Extensive research on the theory of magnetoelastic nonlinear problems in electromagnetic structures is important for the dynamic analysis of structural elements at high temperatures [18–20], high pressures and under the action of strong electromagnetic fields. For example, Liu et al. [21,22] performed numerical simulations using the pseudo-arclength continuation algorithm to analyze the effects of external temperature variations, magnetic potential, electrical potential, and excitation amplitude on the nonlinear vibration response of composite cylindrical shells. When the electromagnetic structure is in an applied electromagnetic field environment, on one hand, the electromagnetic structure is deformed by the electromagnetic force [23,24], and on the other hand, the deformation of the structure leads to a change in the electromagnetic field and thus to a change in the distribution of the electromagnetic force. For the current-carrying conductor [25], the electromagnetic force is the Lorentz force. For polarizable or magnetizable electromagnetic dielectric materials, the electromagnetic force is generated by the interaction of the polarization or magnetization [26] with the external electromagnetic field. A fundamental feature of this mutual coupling of the electromagnetic and mechanical fields is the nonlinearity. Even if the electromagnetic and mechanical fields are treated as linear separately, the coupled electromagnetic-elastic mechanical marginal equations are still nonlinear. Therefore, the study of their nonlinear kinematic states has also become an inevitable trend [27–29]. However, studies in either nonlinear electromagnetic elasticity or nonlinear aeroelasticity have focused on their respective areas of expertise without considering the coupling effects of these two cases. The theory is more complex when considering the combined effect of airflow and periodic excitation on the vibration characteristics of a system under the action of a magnetic field, and there are still many issues to be investigated.

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.

2 Differential Equations of Rectangular Conductive Thin Plate under the Action of External Excitation in Magnetic Field

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 Fig. 1. The length, width and thickness of the plate, respectively a, b and h, satisfy that the thickness is much smaller than the minimum value of the length and width. Taking the middle surface of the plate as the XY plane, establish the coordinate system shown in Fig. 1.

The research idea of this paper is shown in Fig. 2 below.

2.1 Four Basic Conditional Assumptions of Thin Plate Theory

When studying the lateral vibration of elastic thin plates, there are four basic assumptions [30]:

(1) 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.

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

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

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

2.2 Stress Strain Relationship

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 Eq. (1), u, v, w are the displacement components of the points in the midplane, t denotes time, and

According to the assumption of the Kirchhoff straight line method,

In Eq. (2),

where

According to the basic assumption of Kirchhoff, using the generalized Hooke’s law [30], the stress can be described as:

In Eq. (3), E is the Young’s modulus of the material, and μ is the Poisson’s ratio. The equation for large deflection bending of the plate can be obtained as:

In Eqs. (4)–(9),

2.3 Preliminary Establishment of Partial Differential Equations of Motion

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 [31], namely:

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 [16]:

In Eqs. (11)–(15),

2.4 Derivation of Basic Theory of Electromagnetic Field

Assuming that the thin plate is a non-polarized, non-magnetized material with good conductivity, the electromagnetic quantity satisfies the Maxwell equation [13]:

The electromagnetic constitutive relation is as follows:

In Eqs. (16) and (17),

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

In Eqs. (18)–(21),

From the Eq. (17), the in-plane induced current can be obtained as:

2.5 Differential Equation of Motion of Rectangular Thin Plate under External Excitation in Transverse Magnetic Field

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

The unit volume electromagnetic force is:

Integrating Eqs. (26)–(28) in z from

Substituting Eqs. (29)–(33) and the corresponding equation into Eqs. (13)–(15), considering the existence of lateral deformation and damping, the following partial differential equation of motion of a rectangular thin plate in a transverse magnetic field environment can be obtained [15]:

In Eqs. (34)–(36),

From the simply supported condition of the four sides of the rectangular thin plate, the method of separation of variables [22,32] is adopted, and the lateral displacement is given as:

Using the mode shape superposition method, the solution

In the above equations,

According to the linear potential theory [1,33], the pneumatic pressure

Substitute Eqs. (35), (36), (38), (39) into Eq. (34), and use Galerkin’s method to integrate [34], the ordinary differential equation is obtained:

The parameters

Let

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

where

Assuming

Let

A hyperbolic saddle point in the q-z plane and two homoclinic orbits [36]:

3.2 Necessary Conditions for Chaos to Exist

When

The generalized Melnikov function

If the Melnikov function has only one simple zero, the Poincaré map of the disturbance system Eq. (48) has a Strange Attractor, and chaotic motion is possible.

With the continuous increase of

4 Results and Analysis of Example

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

4.1 When 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

4.2 When 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

4.3 When the Magnetic Field Distribution

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.

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

(2) The change of magnetic field only changes the value of

(3) 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

Funding Statement: 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).

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

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*Computer Modeling in Engineering & Sciences*,

*137*

*(2)*, 1749-1771. https://doi.org/10.32604/cmes.2023.027745

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*Comput. Model. Eng. Sci.*, vol. 137, no. 2, pp. 1749-1771. 2023. https://doi.org/10.32604/cmes.2023.027745