Flexible electronic printing technology is a scientific technology that uses an “ink” material with conductive, dielectric, or semiconductor properties printed on a flexible web substrate to achieve precise preparation of flexible electronic devices, which are widely used in information, energy, medical, and military fields. In the preparation of the printing process of flexible printed electron webs under complex working conditions, the moving web will experience substantial unstable nonlinear dynamic behavior, such as divergence, flutter, bifurcation, and chaos. Accordingly, because of the coupling effects of the complex working conditions of the magnetic field, air and nonlinear electrostatic field forces, it is indispensable to explore the nonlinear dynamic equation of the flexible printed electron web in motion. The theory of multiphysics dynamics establishes a nonlinear vibration equation for the flexible printed electron web under multiphysics conditions. The discrete nonlinear vibration equation of state space equation was obtained by the Bubnov–Galerkin method. Utilizing the Runge–Kutta technique of the fourth-order, Poincaré maps, phase-plane diagrams, power spectra, bifurcation graphs, and time history diagrams of the moving flexible printed electron web were obtained. The influences of the velocity, electrostatic field, magnetic induction intensity, and follower force on the flexible printed electron web were analyzed. In addition, the Ansoft Maxwell finite element simulation software was used to simulate the magnetic field distribution of the moving web during roll-to-roll transmission. This paper determines the stable working range of the moving flexible printed electron web, which provides a theoretical basis for the preparation of flexible printed electronic webs.

## I. INTRODUCTION

Research on high-precision flexible printed electron webs has been focused on and applied to wearable flexible electronic devices, smart sensors, Radio Frequency Identification (RFID) tags, micro-/nanomanufactured electron webs, aerospace applications, and other fields.^{1,2} However, in the production process, the moving flexible printed electron web is inevitably affected by the coupling of friction, air, magnetic fields in the environment, and nonlinear electrostatic field forces generated during the drying and curing of the conductive ink.^{3} These factors cause complex nonlinear dynamic behaviors of the flexible printed electron web in motion. The unstable nonlinear dynamic performances of the flexible printed electron web in motion will cause lateral bending deformation of the web surface and even tear the electron web in serious cases.^{4,5} Therefore, the nonlinear dynamics of the moving flexible printed electron web under the influence of different preparation speeds and multiple field couplings are studied. It is of great importance to obtain a stable working range and improve the production efficiency of the electron web.

In recent years, research on the dynamic stability of axial motion systems has achieved great results, but there are few studies on the nonlinear dynamic stability of moving flexible printed electron webs under the coupling of multiple fields. Shao *et al.*^{6} utilized the D’Alembert principle and von Kármán’s nonlinear theory to establish a moving web nonlinear dynamic system and obtained a stable transmission area of the membrane by a multiscale method. Ying *et al.* used the Melnikov method to obtain the orbital parameter function of the vibration equation of a moving graphene electronic membrane and determined the conditions of a chaos criterion.^{7} Zhang used a multiple-scale method to study the resonance response and chaotic behavior of a cylindrical shell of a composite laminated with a radial pretension film.^{8} Ghobadi established a continuous thermoelectric model based on the Kirchhoff plate theory and the modified flexible electrical theory and investigated the free vibration of nonlinear functional gradient nanoflexible plates subject to a magnetic field.^{9} Based on Kirchhoff’s plate theory and Hamilton’s principle, Hu *et al.* deduced a magnetoelastic nonlinear control equation for a rotating conducting circular plate by using an average algorithm and a multiple scale technique.^{10–12} Ri studied the nonlinear dynamics of beam by using the differential quadrature method and harmonic balance method.^{13} Nguyen and Hong investigated the relationship between the velocity and the stability of thin membranes under simple boundary conditions.^{14} Mchugh and Dowell applied energy methods to study the dynamic stability of a cantilever beam under a nonconservative driving force.^{15} Wang *et al.* utilized a unitized model to study the dynamic behavior of a functionally graded shell.^{16} Higuchi analyzed the dynamic stability of a flexible rectangular plate subject to a follower force through the modal analysis method.^{17} Guo *et al.* investigated the mechanical properties of a thermoelastic coupled rectangular plate under the action of follower force by using the differential quadrature method and obtained the relationship between the plate velocity and the thermoelastic coupling coefficient.^{18} Huang explored the superconducting system of chaos and periodic vibration.^{19} Soltanrezaee and Bodaghi used the Galerkin decomposition method to discretize the piezoelectric plate under the action of an electrostatic field force and obtained a nonlinear solution of the equation by a stepwise linearization method.^{20} Wang and Ren studied the beat frequency phenomenon of carbon nanotubes.^{21} Based on the hypothesis of the Euler–Bernoulli beam,^{22} Senthilkumar gave the exact analytical solution of the piezoelectric energy collector by changing the martensite volume fraction of the Shape memory alloy (SMA). Zhang *et al.* used the Galerkin method to design a circular truss antenna.^{23} Chen *et al.* used a harmonic balance algorithm to verify the beam forced vibration.^{24,25} Shao *et al.* solved the differential equations of the nonlinear system by using the fourth-order Runge–Kutta technique and obtained the stability of the working area of the membrane.^{26} Chai and Li used a multiscale algorithm to explore the nonlinear properties of composite laminates under external excitation and boundary conditions.^{27} Hu and Liu applied the differential and integral quadrature methods (DIQM) method to solve the forced resonance response of the lateral vibration around the balance deformation of the conveyor belt.^{28} Wang and Ding studied the nonlinear behavior of an axially moving hyperelastic beam with simple supports by using the Hamilton principle.^{29}

The dynamic coupling knowledge of multiphysics is used in this paper to study the large-deflection nonlinear dynamic equation of a flexible printed electron web with magneto-aeroelasticity under nonlinear electrostatic excitation. The discrete differential equation is derived by the Bubnov–Galerkin method, the nonlinear vibration differential equation is solved by the fourth-order Runge–Kutta technique, and the effects of the velocity of the moving flexible printed electron web, the electrostatic field, the magnetic induction intensity, and the follower force are analyzed.

## II. DERIVATION OF THE DYNAMICS OF THE MOVING FLEXIBLE PRINTED ELECTRON WEB UNDER A NONLINEAR ELECTROSTATIC FORCE

The model shown in Fig. 1 is the physical model of the nonlinear vibration of a flexible printed electron web with magneto-air–solid coupling under the action of electrostatic force in motion. As shown in Fig. 1(a), the lower fixed plate is the deflection plate for the preparation of the flexible printed electron web, and the upper layer of the plate is a simplified structure of the moving flexible printed electron web. There is air between the electron web and the deflection plate, and the flexible printed electron web has no shear force or bending stiffness. The air and the electron web obey the nonslip boundary condition. The velocity of the electron web along the x axis is *U*_{1}, the unit surface density of the electron web is *ρ*_{l}, the inner to outer diameter of the metal ring electron web is ** a** and

**, respectively, the length and width of the base electron web are**

*b***and**

*e***, respectively, the electron web thickness is**

*m**δ*, the electrostatic excitation voltage of the electron web is

*U*

_{0}, the intensity of the magnetic field along the y axis direction is

*B*

_{0y}, the tensile force per unit length of the substrate web in the y and x directions is

*T*

_{y}and

*T*

_{x}, respectively, the transverse load is $p\u0304cos(\omega t)$, the unit tension of the metal annular electron web along the r direction is q

_{a}and q

_{b}, the lateral large deflection displacement of the electron web is $w\u0304r,\phi ,t$, and the equivalent gap between the electron web and the deflection plate is

*d*

_{e}.

The capacitance is expressed as

where ** C** is the capacitance,

*σ*

_{0}is the vacuum dielectric constant,

*σ*

_{M}is the electron web permittivity,

*d*

_{0}is the initial gap between the electron web and the deflection plate, and

*h*

_{c}is the thickness of the electron web conductive ink. The electrostatic equation is derived from Eq. (1) as follows:

The dynamic capacitance of Eq. (2) is expanded by the Taylor series as

According to Eqs. (2) and (3), the electrostatic field force on the moving flexible printed electron web can be divided into two parts: static and dynamic, which can be expressed as

where $w\u03040$ is the static small deflection, $\Delta w\u0304(r,\phi ,t)$ is the dynamic deflection, *E*_{0} is the DC voltage, Δ*E* is the AC voltage, *f*_{e1} is the electrostatic field force constant, and *f*_{e2} is the dynamic nonlinear electrostatic field force.

Substituting Eq. (4) into Eq. (2), the electrostatic field constant and dynamic electrostatic field force are

Substituting Eq. (3) into Eq. (5), the dynamic nonlinear electrostatic field force Δ*f*_{e2,1} caused by $\Delta w\u0304(r,\phi ,t)$ and the dynamic nonlinear electrostatic field force Δ*f*_{e2,2} caused by Δ*E* are obtained. The electroinduced nonlinear electrostatic field force caused by $\Delta w\u0304(r,\phi ,t)$ is

The alternating voltage is

Because the positive and negative alternating voltages cancel each other out, the influence of the static electric field force is eliminated. The nonlinear dynamic electrostatic field force Δ*f*_{e2,2} caused by Δ*E* can be obtained by substituting Eq. (7) into Eq. (5),

## III. DYNAMIC DIFFERENTIAL EQUATION FOR A MOVING FLEXIBLE PRINTED ELECTRON WEB WITH MAGNETO-AIR–SOLID COUPLING

Maxwell equations for a moving flexible printed electron web in medium are

Faraday’s law of electromagnetic induction:

Generalized Ampere’s law:

Gauss’s law:

Conservation of magnetic flux:

Electromagnetic constitutive relationship:

where the differential operator is $\u2207=e\u0302x\u2202\u2202x+e\u0302y\u2202\u2202y+e\u0302z\u2202\u2202z$, and $e\u0302x$, $e\u0302y$, and $e\u0302z$ represent the unit vectors along the x, y, and z axes, respectively. $u\u21c0$ is the internal displacement of the flexible printed electron web in motion, $J\u21c0$ is the current density, $B\u21c0$ is the magnetic induction intensity, $E\u21c02$ is the electric field strength, $H\u21c0$ is the magnetic field strength, $D\u21c0$ is the electric displacement, *ρ* is the charge density, *μ*_{0} is the vacuum permeability, and *σ* is the electrical conductivity.

Using Maxwell equations, the Ansoft Maxwell software simulates the magnetic field distribution of the moving flexible printed electron web during roll-to-roll transmission. The equipment frequency is 50 Hz, the voltage is 220 V AC, the wire uses pure copper material with a radius of 1 mm, and the base of the flexible printed electron web is polyethylene terephthalate (PET). The ring structure is of pure silver. The length and width of the flexible printed electron web are 50 and 100 mm, respectively, the thickness of the electron web is 0.3 mm, and the inner to outer diameter of the metal ring of the electron web is 30 and 60 mm, respectively. The simulation results are given in the following.

Figures 2 and 3 show the magnetic induction intensity produced by a pure copper wire with a length of 120 mm, perpendicular to the xy plane, and 65 mm away from the origin. It can be seen from the figures that the magnetic density gradually increases along the y axis direction, but the change in the magnetic induction intensity along the electron web thickness direction is not obvious when the angle between the flexible printed electron web and the xy plane is 0° and 45° because the electron web is too thin. Figures 4 and 5 show the pure copper wire 5 mm away from the electron web. When the xy plane is at 0°, the magnetic induction intensity is symmetrical about the coordinate origin. The closer to the inner diameter along the y axis direction, the greater the magnetic induction intensity. The magnetic induction intensity on both sides of the y axis can be ignored. When the xy plane is at 45°, the magnetic density is the largest near the wire, and the magnetic density changes marginally along the thickness of the electron web.

In the process of roll-to-roll transmission, the Lorentz force that affects the moving flexible electron web is mainly determined by the magnetic induction intensity generated by the circuit wire of the device. Therefore, studying the relationship curve of the magnetic induction intensity between different transmission angles and the position of the wire is helpful to control the influence of the magnetic field force on the nonlinear vibration of the flexible printed electron web in motion, thereby improving the stability of the system.

The electromagnetic quantities of the flexible electronics in motion can be written as

where ** h**,

**, and e represent the electromagnetic vector excited by the disturbance, and**

*b***,**

*i***, and k represent the directions of the x, y, and z axes, respectively. The velocity vectors in each direction are**

*j*The Lorentz force is

Substituting Eqs. (16) and (17) into Eq. (18), the unit volume Lorentz force in the z axis direction is obtained as follows:

By integrating within the volume of the moving flexible printed electron web, the total Lorentz force is obtained as follows:

The geometric equation for the large deflection of the electron web is

where $\epsilon \u0304r$, $\epsilon \u0304\phi $, and $\gamma \u0304r\varphi $ are the strains in polar coordinates, and *u*_{i} and *u*_{j} are the internal displacements of the electron web.

The balance equation of the electron web is

Ignoring the physical force, the symmetry yields that *f*_{r} = *f*_{φ} = 0 and *τ*_{rφ} = 0. The internal forces are *δσ*_{r} = *F*_{Tr} and *δσ*_{φ} = *F*_{Tφ}.

where ** E** is the elastic modulus,

*v*is Poisson’s ratio, the bending stiffness and shear force of the electron web are omitted, and the strain energy is

where *A* is the area constant and *φ*_{A} and *φ*_{B} are the angles. The physical equation gives

The total kinetic energy of the moving flexible printed electron web is

According to Hamilton’s principle, another functional form gives

Equations (9), (20), and (27) and the energy equation, including the boundary, are substituted into Eq. (28) to solve for the stationary value. Due to the influence of air, the linear distribution follower force of the moving flexible printed electron web is $R=\u2212q0e\u2212x\u22022w\u0304\u2202x2$, where *q*_{0} is the uniform force constant. Finally, a moving flexible printed electron web nonlinear vibration with magneto-air–solid coupling under the action of a nonlinear electrostatic field is

Introducing a dimensionless quantity into Eq. (29),

Regardless of physical strength and considering that the research object is geometrically symmetrical, the effects of *φ* and *F*_{Tφ} are ignored. Assuming that a moving flexible printed electron paper web only has a magnetic field $B00,B0,j,0$ along the y axis, the disturbance electromagnetic vector is ignored. The amplitudes of the AC and DC voltages generated by friction are equal, and the angular frequency of the AC voltage is equal to that of the external excitation force because the condition is $w\u03040<\delta $. Substituting the dimensionless Eq. (30) into Eq. (29), the nondimensional form of the equation for the magnetic-gas–solid coupling nonlinear vibration of the moving flexible printed electron web under the action of the nonlinear electrostatic field is (the physical degradation can be referred to in Ref. 30)

The moving flexible printed electron web satisfies the boundary conditions

## IV. DISCRETIZATION OF THE NONLINEAR VIBRATION EQUATION

Let the internal force and large deflection functions of the nonlinear vibration equation be

where $f\u03c2,\phi ,\tau $ is the internal force function, $w\u03c2,\phi ,\tau $ is the deflection shape function, and $Ci\tau $ is the time function.

Substituting Eqs. (33) and (34) into Eq. (31) and performing the Galerkin discretization, the nonlinear forced vibration state equation is

The coefficient of Eq. (35) is

Another form of Eq. (35) is

where $F\nu Ci\u2032\u2032,Ci\u2032$ is the comprehensive equation of nonlinear inertial force and nonlinear damping force and $F\iota Ci,Ci\u2032$ is the comprehensive equation of the nonlinear elastic force and nonlinear damping force.

The Fourier expansion of Eq. (37) is

where

Equation (40) is equivalently transformed into a linear vibration equation,

where *M*_{e}, *C*_{e}, and *K*_{e} represent the equivalent mass, damping ratio, and stiffness, respectively. According to equivalent linearization theory, the solution for forced vibration is

Substituting Eq. (42) into Eq. (40) and ignoring the higher harmonic force, the equivalent coefficient of Eq. (41) can be obtained as follows:

Therefore, according to Eqs. (43) and (35), the equivalent linearized equations of the nonlinear dynamics of the flexible printed electron web in motion can be obtained.

Introducing the control state variables:

The state control equation:

## V. DISCUSSION

The fourth-order Runge–Kutta technique is used to compute the state-space equations of the flexible printed electron web with multiple field couplings, including dimensionless external excitation, damping ratio, inner to outer diameter ratio, velocity, electrostatic field, magnetic induction intensity, and follower force. The nonlinear vibration characteristics of the flexible printed electron web were studied by using Poincaré maps, phase-plane diagrams, power spectra, bifurcation graphs, and time history diagrams.

### A. Velocity control nonlinear dynamic

Figures 6–10 show the nonlinear vibration properties of the moving flexible printed electron web as the dimensionless velocity increases. The dimensionless angular frequency is Ω = 1, the original value is $0.01,0$, the external excitation force is $P\u03041=1$, the variable-independent small parameter is $I=0.3$, the Poisson ratio is *v* = 0.15, the inner to outer diameter ratio is *χ* = 0.5, the damping ratio is *℘* = 0.1, the dimensionless follower force is *Q* = 1, the dimensionless magnetic induction intensity is *β* = 1, and the dimensionless alternating voltage is *ℵ* = 1. The power spectrum, phase-plane diagrams, Poincaré maps, bifurcation graphs, and time history diagrams are given in the following.

Figure 6 shows a bifurcation graph with a dimensionless velocity at $0,2$. When 0 ⩽ *c* < 1.3, the bifurcation graph has three curves, explaining that the moving printed web is in periodic motion. When 1.3 ≤ *c* < 1.48, the bifurcation graph has a dense point, which illustrates that the moving printed web is in chaotic motion, and the system will be unstable. With the increase in dimensionless velocity, when 1.48 ≤ *c* ≤ 2, the system enters periodic motion again. In summary, as the velocity increases, the moving flexible printed electron web vibration experiences an alternating process from periodic motion to chaotic motion and then back to periodic motion. To analyze the evolution law of the bifurcation, Figs. 7–10 exhibit the time history diagram, phase-plane diagrams, and Poincaré maps at different dimensionless velocities,

When the dimensionless velocity is 0.6, Fig. 7 exhibits the Poincaré map (b) with three fixed spots, the closed phase-plane diagram (c), and the discrete power spectrum (d). This can explain why at this velocity, the moving flexible printed electron web under the action of the nonlinear electrostatic field operates stably in a triple-period condition. Figure 8 shows that when *c* = 1.2, five discrete points are distributed in the Poincaré map (b), the phase-plane diagram (c) corresponds to the regular drawing, and the power spectrum (d) remains in a discretization state, illustrating that the moving flexible printed electron web is in a fivefold periodic condition. When *c* = 1.3, the power spectrum (d), Poincaré map (b), and phase-plane diagram (c) are continuous spectra, dense point, and breakpoint curve, respectively, in Fig. 9, which indicate that the electron web remains in chaotic motion. When *c* = 2, as shown in Fig. 10, having two spots within the Poincaré map (b), the phase-plane diagram (c) is an enclosed curve, and the power spectrum (d) is divergent, indicating that the moving printed electron web is in a twofold periodic condition.

### B. Magnetic field control nonlinear dynamic

When the dimensionless velocity is *c* = 1, the moving flexible printed electron web initial condition is $0.01,0$, the dimensionless angular frequency is Ω = 1, the external excitation force is $P\u03041=1$, the dimensionless follower force is *Q* = 1, the variable independent small parameter is $I=0.3$, the ratio of damping is *℘* = 0.1, the inner to outer diameter ratio is *χ* = 0.5, the Poisson ratio is *v* = 0.15, and the dimensionless alternating voltage is *ℵ* = 1.

According to Fig. 11, when 0 ≤ *β* < 0.15, the graph of bifurcation of the system has dense spots, which shows that the moving flexible printed electron web will be in an unstable chaotic state in this dimensionless magnetic induction region. When 0.15 ≤ *c* < 2, the global bifurcation graph of the system changes to regular distribution points, which indicates that the system will transition from a chaotic state to a periodic state with increasing dimensionless magnetic induction,

Figures 12–15 show that when *β* = 0.05, the Poincaré map (b) of the moving printed electron web is a disarray of scattered points, while the phase-plane diagram (c) and power spectrum (d) are broken lines and continuous lines, which reflects that the moving printed electron web is in a state of chaos. When *β* = 0.2, the Poincaré map (b), phase-plane diagram (c), and power spectrum (d) are four fixed points, a regular curve, and a discrete spectrum, respectively, which means that the moving printed electron web moves with a fourfold period. When *β* = 0.8, the Poincaré map (b) has approximately three spots, the phase trajectory curve (c) is a regular curve, and the power spectrum (d) is a spectrum of scatter; then, the system is evolving from a fourfold period to a threefold period. When *β* = 1.8, the state of the moving printed electron web is shown in Fig. 15. The system is in a state of periodicity, indicating that the electron web is stable. In conclusion, as the magnetic induction intensity increases, the moving printed web will transition from chaotic to periodic. The stable operation of the moving printed electron web can be realized by properly controlling the magnetic induction intensity.

### C. Electrostatic field control, nonlinear dynamic

The basic parameters are as follows: $0.01,0$ is the initial condition of the moving flexible printed electron web, the external excitation force is $P\u03041=1$, the variable independent small parameter is $I=0.3$, the ratio between the inner to outer diameters of the metal ring electron web is *χ* = 0.5, the Poisson ratio is *v* = 0.15, the dimensionless velocity is *c* = 1, and the damping ratio is *℘* = 0.1,

From the bifurcation graph of Fig. 16, when the dimensionless follower force is *Q* = 0.3 and the dimensionless magnetic induction intensity is *β* = 0.15, no matter how the dimensionless alternating voltage changes, the moving flexible electromagnetic web is always within a certain range of movement displacement, and it is in a chaotic state. When the dimensionless follower force is *Q* = 1 and the dimensionless magnetic induction intensity is *β* = 1, the system is always in a periodic state within the global variation range of the dimensionless alternating voltage,

Figures 17 and 18 correspond to the situation when the dimensionless alternating voltage of Fig. 16 is *ℵ* = 0.5. In Fig. 17, (b) is a Poincaré map with dense spots, (c) is an unclosed phase-plane diagram, and (d) is a continuous power spectrum, explaining that the moving flexible printed electron web is in a chaotic state. According to Fig. 18, we can conclude that with three fixed points in the Poincaré map (b) of the moving flexible printed electron web, the phase-plane diagram (c) is a regular curve, and the power spectrum (d) is a scatter spectrum, reflecting that the web is in periodic motion.

### D. Follower force control, nonlinear dynamic

The basic parameters are as follows: the initial condition is $0.01,0$, the dimensionless velocity is *c* = 1, the external excitation force is $P\u03041=1$, the variable independent small parameter is $I=0.3$, the ratio of inner to outer diameter of the metal ring electron web is *χ* = 0.5, the Poisson ratio is *v* = 0.15, the damping ratio is *℘* = 0.1, the dimensionless magnetic induction intensity of the web is *β* = 1, and the dimensionless alternating voltage of the web is *ℵ* = 1.

In Fig. 19, when 0 ≤ *Q* < 0.19, the bifurcation graph of the moving flexible printed electron web is distributed with dense points, and the system is in a state of chaotic motion. When 0.19 ≤ *Q* ≤ 2, the global bifurcation graph is regularly distributed, indicating that the web is in a periodic motion condition. Thus, the moving flexible printed electron web has experienced a path from chaos to period doubling,

When *Q* = 0 (in Fig. 20), the Poincaré map (b) of the moving electron web is a disarray of scattered points, while the phase-plane diagram (c) and power spectrum (d) are broken lines and continuous lines, respectively, which reflects that the moving electron web is in a state of chaos. When *Q* = 0.1 (in Fig. 21), (b) is a Poincaré map with regular dense spots, (c) is an unclosed trajectory of the phase-plane diagram, and (d) is a continuous power spectrum, explaining that the vibration state of the electron web gradually develops from chaos to periodic movement, but the system is still in chaos. When *Q* = 0.2 (in Fig. 22), the Poincaré map (b) has four fixed points, and the phase-plane diagram (c) and power spectrum (d) are a regular curve and a discrete spectrum, respectively, which show that the flexible printed electron web is in periodic motion under the dimensionless follower force. When *Q* = 2 (in Fig. 23), three discrete points are distributed in the Poincaré map (b), the phase-plane diagram (c) corresponds to a regular curve, and the power spectrum (d) remains in a discrete state, which illustrates that the moving flexible printed electron web is in a threefold periodic condition. In summary, as the dimensionless follower force increases, the moving flexible printed electron web gradually transitions to chaos from multiperiod motion.

## VI. CONCLUSIONS

In this paper, the dynamic nonlinear properties of a moving flexible printed electron web with magnetic-air–solid coupling under the action of nonlinear electrostatic field force were studied. The coupled nonlinear vibration equation was discretized by the Bubnov–Galerkin method, and a discrete nonlinear vibration differential equation was obtained. The fourth-order Runge–Kutta technique was used to solve this discrete coupled nonlinear vibration differential equation. The law of nonlinear vibration of the web under multifield couplings, such as an external excitation force, damping ratio, inner diameter ratio, velocity, electrostatic fields, magnetic induction intensity, and a follower force, was analyzed. The conclusions are summarized as follows:

Distribution law of magnetic field of the flexible printed electron web: according to the magnetic induction intensity diagram, the magnetic density of the moving flexible printed electron web is the largest near the wire, the magnetic density of the remaining parts is small and symmetrically distributed around the wire, and the magnetic density change is weak in the web thickness direction.

Taking the velocity as the variable parameter: when 0 ≤

*c*< 1.3, the moving flexible printed electron web moves within this dimensionless velocity range, and the web is in periodic motion. At 1.3 ≤*c*< 1.48, the flexible printed electron web will be in a state of chaos, and the moving web will diverge and lose stability. When 1.48 ≤*c*≤ 2, the system enters periodic motion again. This reflects that the nonlinear vibration state of the moving flexible printed electron web under the action of the nonlinear electrostatic field force will undergo an alternating process from periodic to chaotic to periodic as the dimensionless velocity increases.Taking the magnetic induction intensity as the variable parameter: when 0 ≤

*β*< 0.15, the global bifurcation graph of the moving web is a disarray of scattered spots. This shows that the moving flexible printed electron web will be in an unstable chaotic state. When 0.15 ≤*c*< 2, it transitions from a chaotic state to a periodic motion state.Taking the alternating voltage as the variable parameter: the chaotic and periodic states and the displacement range of the flexible printed electron web can be controlled in the entire global parameter range by appropriately changing the dimensionless magnetic induction intensity and the follower force.

Taking the follower force as the variable parameter: when 0 ≤

*Q*< 0.19, the moving flexible printed electron web is in a motion of chaos, and when 0.19 ≤*Q*≤ 2, the moving flexible printed electron web is in a periodic condition. At the same time, the magnetic-gas–solid coupling motion of the flexible printed electron web under the action of the nonlinear electrostatic field force changes from chaos to quadruple periodic motion to triple periodic motion with an increase in the dimensionless follower force, indicating that increasing the dimensionless follower force can effectively reduce the probability of chaos and improve the stability of the system.

## ACKNOWLEDGMENTS

This work was supported, in part, by the National Natural Science Foundation of China under Grant No. 52075435, in part, by the Key Scientific Research Project of Shaanxi Provincial Department of Education under Grant No. 20JY054, and, in part, by the Shaanxi Provincial Natural Science Foundation under Grant No. 2021JQ-480.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.