Nonlinear vibration of a moving flexible printed electron web under multiphysics dynamics

Ying, Shu-Di; Wu, Ji-Mei; Wang, Yan

doi:10.1063/5.0053433

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 U₁, 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 b, respectively, the length and width of the base electron web are e and m, respectively, the electron web thickness is δ, the electrostatic excitation voltage of the electron web is U₀, 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 $\bar{p} \cos (ω t)$ ⁠, 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 $\bar{w} (r, φ, t)$ ⁠, and the equivalent gap between the electron web and the deflection plate is d_e.

FIG. 1.

View large Download slide

Multifield coupling model of the moving printed electron web: (a) physical diagram, (b) schematic diagram, (c) mechanical model, and (d) printed electron web machine.

The capacitance is expressed as

C = (σ_{0} \cdot \iint A \cdot d s) / (d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c})

(1)

where C is the capacitance, σ₀ is the vacuum dielectric constant, σ_M is the electron web permittivity, d₀ 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:

f_{e} = - 0.5 \cdot U_{0}^{2} \cdot \frac{d C}{d (d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c})} .

(2)

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

\begin{align} \frac{d C}{d (d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c})} = [d ((σ_{0} \cdot \iint A \cdot d s) / (d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c}))] / {(d (d_{e} - \bar{w})|}_{{\bar{w}}_{0}} \\ + [d^{2} ((σ_{0} \cdot \iint A \cdot d s) \cdot / (d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c}))] / {(d^{2} (d_{e} - \bar{w})|}_{{\bar{w}}_{0}} \\ \cdot Δ \bar{w} (r, φ, t) + [d^{3} (((σ_{0} \cdot \iint A \cdot d s) / 2!) / (d_{0} - \bar{w} (r, φ, t)))) \\ ((+ (\frac{σ_{0}}{σ_{M}} \cdot h_{c}))] / {(d^{3} (d_{e} - \bar{w})|}_{{\bar{w}}_{0}} \cdot Δ {\bar{w}}^{3} (r, φ, t) + \dots . \end{align}

(3)

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

\begin{gathered} \bar{w} (r, φ, t) = {\bar{w}}_{0} + Δ \bar{w} (r, φ, t), U_{0} = E_{0} + Δ E, \\ f_{e} = f_{e 1} (r, ϕ, t) + f_{e 2} (r, ϕ, t), \end{gathered}

(4)

where ${\bar{w}}_{0}$ is the static small deflection, $Δ \bar{w} (r, φ, t)$ is the dynamic deflection, E₀ is the DC voltage, ΔE 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

\begin{gathered} \begin{aligned} f_{e 2} & = - 0.5 \cdot [{({(E_{0} + Δ E)}^{2} \cdot \frac{d^{2} C}{d {(d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{2}}|}_{{\bar{w}}_{0}}) \\ \cdot Δ \bar{w} (r, φ, t) + (2 E_{0} \cdot Δ E + Δ E^{2}) \\ (\cdot {(\frac{d C}{d (d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}|}_{{\bar{w}}_{0}}], \end{aligned} \\ f_{e 1} = 0.5 E_{0}^{2} \cdot \frac{σ_{0} \cdot \iint A \cdot d s}{{(d_{0} - \bar{w} (r, φ, t) + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{2}} . \end{gathered}

(5)

Substituting Eq. (3) into Eq. (5), the dynamic nonlinear electrostatic field force Δf_e2,1 caused by $Δ \bar{w} (r, φ, t)$ and the dynamic nonlinear electrostatic field force Δf_e2,2 caused by ΔE are obtained. The electroinduced nonlinear electrostatic field force caused by $Δ \bar{w} (r, φ, t)$ is

\begin{align} Δ f_{e 2, 1} & = \frac{E_{0}^{2} \cdot σ_{0} \cdot \iint A \cdot d s}{{(d_{0} - {\bar{w}}_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{3}} \cdot Δ \bar{w} (r, φ, t) + \frac{3 E_{0}^{2} \cdot σ_{0} \cdot Δ {\bar{w}}^{2} (r, φ, t)}{{(d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{3}} \frac{\iint A \cdot d s}{{\bar{w}}_{0}} \cdot (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) \\ \cdot {[1 + (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) + {(\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}})}^{2} + \dots]}^{4} + \frac{6 E_{0}^{2} \cdot σ_{0} \cdot Δ {\bar{w}}^{3} (r, φ, t) \cdot \iint A \cdot d s}{{(d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{4} {\bar{w}}_{0}} \cdot (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) \\ \cdot {[1 + (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) + {(\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}})}^{2} + \dots]}^{5} + \dots . \end{align}

(6)

The alternating voltage is

Δ E = E_{1} \sin (ω t) .

(7)

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 Δf_e2,2 caused by ΔE can be obtained by substituting Eq. (7) into Eq. (5),

f_{e 2, 2} = \frac{E_{1} E_{0} σ_{0} \iint A \cdot d s \cdot \sin (ω t)}{{(d_{0} - {\bar{w}}_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{2}} .

(8)

Adding Eqs. (6) and (8), the total dynamic nonlinear electrostatic force is

\begin{align} f_{e 2} & = \frac{E_{0}^{2} \cdot σ_{0} \cdot \iint A \cdot d s}{{(d_{0} - {\bar{w}}_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{3}} \cdot Δ \bar{w} (r, φ, t) + \frac{3 E_{0}^{2} \cdot σ_{0} \cdot Δ {\bar{w}}^{2} (r, φ, t)}{{(d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{3} {\bar{w}}_{0}} \iint A \cdot d s \cdot (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) \\ \cdot {[1 + (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) + {(\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}})}^{2} + \dots]}^{4} + \frac{6 E_{0}^{2} \cdot σ_{0} \cdot Δ {\bar{w}}^{3} (r, φ, t) \cdot \iint A \cdot d s}{{(d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{4} {\bar{w}}_{0}} \cdot (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) \\ \cdot {[1 + (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) + {(\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}})}^{2} + \dots]}^{5} + \frac{E_{1} E_{0} σ_{0} \iint A \cdot d s}{{(d_{0} - {\bar{w}}_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{2}} \cdot \sin (ω t) + \dots . \end{align}

(9)

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:

\nabla \times {\overset{⇀}{E}}_{2} = - \frac{\partial \overset{⇀}{B}}{\partial t},

(10)

Generalized Ampere’s law:

\nabla \times \overset{⇀}{H} = \overset{⇀}{J} + \frac{\partial \overset{⇀}{D}}{\partial t},

(11)

Gauss’s law:

\nabla \cdot \overset{⇀}{D} = ρ,

(12)

Conservation of magnetic flux:

\nabla \cdot \overset{⇀}{B} = 0,

(13)

Electromagnetic constitutive relationship:

\overset{⇀}{D} = σ_{0} \cdot {\overset{⇀}{E}}_{2}, \overset{⇀}{B} = μ_{0} \cdot \overset{⇀}{H}, \overset{⇀}{J} = σ ({\overset{⇀}{E}}_{2} + \frac{\partial \overset{⇀}{u}}{\partial t} \times \overset{⇀}{B}),

(14)

where the differential operator is $\nabla = {\hat{e}}_{x} \frac{\partial}{\partial x} + {\hat{e}}_{y} \frac{\partial}{\partial y} + {\hat{e}}_{z} \frac{\partial}{\partial z}$ ⁠, and ${\hat{e}}_{x}$ ⁠, ${\hat{e}}_{y}$ ⁠, and ${\hat{e}}_{z}$ represent the unit vectors along the x, y, and z axes, respectively. $\overset{⇀}{u}$ is the internal displacement of the flexible printed electron web in motion, $\overset{⇀}{J}$ is the current density, $\overset{⇀}{B}$ is the magnetic induction intensity, ${\overset{⇀}{E}}_{2}$ is the electric field strength, $\overset{⇀}{H}$ is the magnetic field strength, $\overset{⇀}{D}$ is the electric displacement, ρ is the charge density, μ₀ 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.

FIG. 2.

View large Download slide

Moving flexible printed web with a transmission angle of 0° when the wire is beside the web: (a) magnetic density distribution nephogram, (b) distribution of magnetic density contour, and (c) magnetic density vector diagram.

FIG. 3.

View large Download slide

Moving flexible printed web with a transmission angle of 45 when the wire is beside the web: (a) magnetic density distribution nephogram, (b) distribution of magnetic density contour, and (c) magnetic density vector diagram.

FIG. 4.

View large Download slide

Moving flexible printed web with a transmission angle of 0° when the wire is above the web: (a) magnetic density distribution nephogram, (b) distribution of magnetic density contour, and (c) magnetic density vector diagram.

FIG. 5.

View large Download slide

Moving flexible printed web with a transmission angle of 45° when the wire is above the web: (a) magnetic density distribution nephogram, (b) distribution of magnetic density contour, and (c) magnetic density vector diagram.

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

\begin{gathered} H (H_{i}, H_{j}, H_{k}) = H_{0} (H_{0, i}, H_{0, j}, H_{0, k}) + h (h_{i}, h_{j}, h_{k}), \\ B (B_{i}, B_{j}, B_{k}) = B_{0} (B_{0, i}, B_{0, j}, B_{0, k}) + b (b_{i}, b_{j}, b_{k}), \\ E_{2} (E_{i}, E_{j}, E_{k}) = e (e_{i}, e_{j}, e_{k}), \end{gathered}

(15)

where h, b, and e represent the electromagnetic vector excited by the disturbance, and i, j, and k represent the directions of the x, y, and z axes, respectively. The velocity vectors in each direction are

\begin{gathered} {\overset{⇀}{v}}_{i} = U_{1} + \frac{\partial u_{i}}{\partial t} + U_{1} \cdot \frac{\partial u_{i}}{\partial x}, {\overset{⇀}{v}}_{j} = \frac{\partial u_{j}}{\partial t} + U_{1} \cdot \frac{\partial u_{j}}{\partial x}, \\ {\overset{⇀}{v}}_{k} = \frac{\partial \bar{w}}{\partial t} + U_{1} \cdot \frac{\partial \bar{w}}{\partial x} . \end{gathered}

(16)

Substituting Eq. (15) into Eq. (14), the current density vector gives

\{\begin{gathered} {\overset{⇀}{J}}_{i} = σ ({\overset{⇀}{E}}_{i} + {\overset{⇀}{v}}_{j} \cdot B_{k} - {\overset{⇀}{v}}_{k} \cdot B_{j}), \\ {\overset{⇀}{J}}_{j} = σ ({\overset{⇀}{E}}_{j} + {\overset{⇀}{v}}_{k} \cdot B_{i} - {\overset{⇀}{v}}_{i} \cdot B_{k}) . \end{gathered})

(17)

The Lorentz force is

{\overset{⇀}{f}}_{L} = ρ \overset{⇀}{E} + \overset{⇀}{J} \times \overset{⇀}{B} = f_{L, i} \cdot {\hat{e}}_{x} + f_{L, j} \cdot {\hat{e}}_{y} + f_{L, k} \cdot {\hat{e}}_{z} .

(18)

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

\begin{align} f_{L, k} & = σ B_{0, j} [e_{i} + (\frac{\partial u_{j}}{\partial t} - z \frac{\partial^{2} \bar{w}}{\partial t \partial y} - U_{1} \cdot \frac{\partial u_{j}}{\partial x} - U_{1} z \cdot \frac{\partial^{2} \bar{w}}{\partial x \partial y})) \\ \cdot (B_{0, k} - (\frac{\partial \bar{w}}{\partial t} + U_{1} \cdot \frac{\partial \bar{w}}{\partial x}) \cdot B_{0, j}] \\ - σ [e_{j} + (\frac{\partial \bar{w}}{\partial t}) + U_{1} \cdot \frac{\partial \bar{w}}{\partial x}) \cdot B_{0, i} - (U_{1} + \frac{\partial u_{i}}{\partial t}) - z \frac{\partial^{2} \bar{w}}{\partial t \partial x} \\ (+ (U_{1} \cdot \frac{\partial u_{i}}{\partial x} - z U_{1} \cdot \frac{\partial^{2} \bar{w}}{\partial x^{2}}) \cdot B_{0, k}] \cdot B_{0, i} . \end{align}

(19)

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

\begin{align} F_{L, k} & = ∭ f_{L, k} d V = σ \iint A \cdot d s \cdot δ [e_{i} + (\frac{\partial u_{j}}{\partial t} + U_{1} \cdot \frac{\partial u_{j}}{\partial x})) \\ \cdot (B_{0, k} - (\frac{\partial \bar{w}}{\partial t} + U_{1} \cdot \frac{\partial \bar{w}}{\partial x}) \cdot B_{0, j}] \cdot B_{0, j} - σ \iint A \cdot d s \\ \cdot (δ [e_{j} + (\frac{\partial \bar{w}}{\partial t}) + U_{1} \cdot \frac{\partial \bar{w}}{\partial x}) \cdot B_{0, i}) \\ - ((U_{1} + \frac{\partial u_{i}}{\partial t} + U_{1} \cdot \frac{\partial u_{i}}{\partial x}) \cdot B_{0, z}] \cdot B_{0, i} . \end{align}

(20)

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

\{\begin{aligned} {\bar{ε}}_{r} & = \frac{\partial u_{i}}{\partial r} + \frac{1}{2} \cdot {(\frac{\partial \bar{w}}{\partial r})}^{2} - z \cdot [\frac{\partial^{2} \bar{w}}{\partial^{2} r} - \frac{\sin (2 φ)}{r} \cdot \frac{\partial^{2} \bar{w}}{\partial r \cdot \partial φ}) \\ (+ \frac{\sin^{2} (φ)}{r} \cdot \frac{\partial \bar{w}}{\partial r} + \frac{\sin (2 φ)}{r^{2}} \cdot \frac{\partial \bar{w}}{\partial φ} + \frac{1 - 2 \cos^{2} (φ)}{2 r^{2}} \cdot \frac{\partial^{2} \bar{w}}{\partial φ^{2}}], \\ {\bar{ε}}_{φ} & = \frac{1}{r} \cdot (u_{i} + \frac{\partial u_{j}}{\partial φ}) + \frac{1}{2} \cdot {[\sin (φ) \cdot \frac{\partial \bar{w}}{\partial r} + \frac{1}{r} \cdot \frac{\partial \bar{w}}{\partial φ}]}^{2} \\ - z \cdot [\sin^{2} (φ) \cdot \frac{\partial^{2} \bar{w}}{\partial r^{2}} + \frac{\cos^{2} (φ)}{r} \cdot \frac{\partial \bar{w}}{\partial r} + \frac{\cos^{2} (φ)}{r^{2}} \cdot \frac{\partial^{2} \bar{w}}{\partial φ^{2}}], \\ {\bar{γ}}_{r φ} & = \frac{1}{r} \cdot \frac{\partial u_{i}}{\partial φ} + \frac{\partial u_{j}}{\partial r} - \frac{u_{j}}{r} + \frac{\partial \hat{w}}{\partial r} \cdot (\frac{1}{r} \cdot \frac{\partial \bar{w}}{\partial φ}) - z \cdot [\frac{2 \cos (2 φ)}{r}) \\ \cdot (\frac{\partial^{2} \bar{w}}{\partial r \partial φ} - \sin (2 φ) \cdot \frac{\partial \bar{w}}{\partial r} - \frac{2 \cos (2 φ)}{r^{2}} \cdot \frac{\partial \bar{w}}{\partial φ}], \end{aligned})

(21)

where ${\bar{ε}}_{r}$ ⁠, ${\bar{ε}}_{φ}$ ⁠, and ${\bar{γ}}_{r ϕ}$ 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

\{\begin{aligned} \frac{\partial σ_{r}}{\partial r} + \frac{1}{r} \cdot \frac{\partial τ_{r φ}}{\partial φ} + \frac{σ_{r} - σ_{φ}}{r} + f_{r} = 0, \\ \frac{\partial τ_{r φ}}{\partial r} + \frac{1}{r} \cdot \frac{\partial σ_{φ}}{\partial φ} + \frac{2 τ_{r φ}}{r} + f_{φ} = 0 . \end{aligned})

(22)

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.

Equation (21) is substituted into Eq. (22), and the compatibility equation becomes

\{\begin{aligned} r^{2} \cdot \frac{\partial^{2} F_{Tr}}{\partial r^{2}} + 3 r \cdot \frac{\partial F_{Tr}}{\partial r} + 0.5 E δ \cdot {(\frac{\partial \bar{w}}{\partial r})}^{2} = 0, \\ u_{r} = \frac{r}{E δ} (F_{T φ} - v \cdot F_{Tr}), \end{aligned})

(23)

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

\begin{align} U_{ε_{2}} & = \frac{1}{2} \int \int \int (σ_{r} \cdot {\bar{ε}}_{r} + σ_{φ} \cdot {\bar{ε}}_{φ} + τ_{r φ} \cdot {\bar{γ}}_{r φ}) \cdot r d r d φ d z \\ = 0.5 \cdot δ \int (σ_{r} \cdot {\bar{ε}}_{r} + σ_{φ} \cdot {\bar{ε}}_{φ}) \cdot r d r \int_{φ_{A}}^{φ_{B}} A d φ, \end{align}

(24)

where A is the area constant and φ_A and φ_B are the angles. The physical equation gives

\{\begin{aligned} F_{Tr} = \frac{E \cdot δ}{1 - v^{2}} \cdot ({\bar{ε}}_{r} + v^{2} \cdot {\bar{ε}}_{φ}), \\ F_{T φ} = \frac{E \cdot δ}{1 - v^{2}} \cdot ({\bar{ε}}_{r} + v^{2} \cdot {\bar{ε}}_{φ}), \\ τ_{r φ} = \frac{E \cdot δ}{2 (1 + v^{2})} \cdot {\bar{γ}}_{r φ} = 0 . \end{aligned})

(25)

Substituting Eqs. (21) and (25) into Eq. (24) gives

\begin{align} U_{ε_{2}} & = \frac{π E δ}{1 - v^{2}} \int \{{[\frac{\partial u_{r}}{\partial r} + \frac{1}{2} {(\frac{\partial \bar{w}}{\partial r})}^{2}]}^{2} + \frac{u_{r}^{2}}{r^{2}}) \\ + (2 v \cdot \frac{u_{r}}{r} \cdot [\frac{\partial u_{r}}{\partial r} + 0.5 \cdot {(\frac{\partial \bar{w}}{\partial r})}^{2}]\} \cdot r \cdot d r . \end{align}

(26)

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

\begin{align} T & = \frac{ρ δ}{2} \iint_{Ω_{f}} {[\frac{\partial \bar{w}}{\partial t} + U_{1} \cdot (\cos (φ) \cdot \frac{\partial \bar{w}}{\partial r} - \frac{\sin (φ)}{r} \cdot \frac{\partial \bar{w}}{\partial φ})]}^{2} \cdot d r d φ \\ + \frac{1}{2} \iint_{Ω_{f}} {U_{1}}^{2} \cdot d r d φ . \end{align}

(27)

According to Hamilton’s principle, another functional form gives

δ Π = δ_{Δ} \int_{t_{3}}^{t_{4}} (T - U_{ε 2} + Π_{Γ}) d t + \int_{t_{3}}^{t_{4}} f_{e 2} δ_{Δ} \bar{w} + F_{L, k} δ_{Δ} \bar{w} = 0 .

(28)

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 $R = - q_{0} (e - x) \frac{\partial^{2} \bar{w}}{\partial x^{2}}$ ⁠, where q₀ 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

\{\begin{cases} - ρ_{l} \cdot δ \cdot \{\frac{\partial^{2} \bar{w}}{\partial t^{2}} + 2 U_{1} \cdot \cos (φ) \cdot \frac{\partial^{2} \bar{w}}{\partial r \partial t} - 2 U_{1} \cdot \frac{\sin (φ)}{r} \cdot \frac{\partial^{2} \bar{w}}{\partial φ \partial t} + U_{1}^{2}) \\ \cdot [\cos^{2} (φ) \cdot \frac{\partial^{2} \bar{w}}{\partial r^{2}} - \frac{\sin (2 φ)}{r} \cdot \frac{\partial^{2} \bar{w}}{\partial r \partial φ} + \frac{\sin^{2} (φ)}{r} \cdot \frac{\partial \bar{w}}{\partial r} + \frac{\sin (2 φ)}{r^{2}}) \\ ((\cdot \frac{\partial \bar{w}}{\partial φ} + \frac{\sin^{2} (φ)}{r^{2}} \cdot \frac{\partial^{2} \bar{w}}{\partial φ^{2}}]\} - \frac{1}{r} \cdot \frac{d (r \cdot F_{Tr} \cdot \frac{d \bar{w}}{d r})}{d r} + ϑ \cdot \frac{\partial \bar{w}}{\partial t} \\ = σ \iint A \cdot d s \cdot δ [e_{i} + (\frac{\partial u_{j}}{\partial t} + U_{1} \cdot \frac{\partial u_{j}}{\partial x}) \cdot B_{0, k}) \\ - ((\frac{\partial \bar{w}}{\partial t} + U_{1} \cdot \frac{\partial \bar{w}}{\partial x}) \cdot B_{0, j}] \cdot B_{0, j} - σ \iint A \cdot d s \cdot δ \\ \cdot [e_{j} + (\frac{\partial \bar{w}}{\partial t} + U_{1} \cdot \frac{\partial \bar{w}}{\partial x}) \cdot B_{0, i} - (U_{1} + \frac{\partial u_{i}}{\partial t} + U_{1} \cdot \frac{\partial u_{i}}{\partial x}) \cdot B_{0, z}] \\ \cdot B_{0, i} + \frac{E_{0}^{2} \cdot σ_{0} \cdot \iint A \cdot d s}{{(d_{0} - {\bar{w}}_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{3}} \cdot \bar{w} (r, φ, t) \\ + \frac{3 E_{0}^{2} \cdot σ_{0} \cdot {\bar{w}}^{2} (r, φ, t) \cdot \iint A \cdot d s}{{(d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{3} {\bar{w}}_{0}} \cdot (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) \\ \cdot {[1 + (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}}} \cdot \frac{1}{h_{c}} + {(\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}})}^{2} + \dots)]}^{4} \\ + \frac{6 E_{0}^{2} \cdot σ_{0} \cdot {\bar{w}}^{3} (r, φ, t) \cdot \iint A \cdot d s}{{(d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{4} {\bar{w}}_{0}} \cdot (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) \\ \cdot {[1 + (\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}}) + {(\frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}})}^{2} + \dots]}^{5} \\ + \frac{E_{1} E_{0} σ_{0} \iint A \cdot d s \cdot \sin (ω t)}{{(d_{0} - {\bar{w}}_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c})}^{2}} - q_{0} (e - x) \frac{\partial^{2} \bar{w}}{\partial x^{2}} + \bar{p} \cdot \cos (ω t) + \dots \\ r^{2} \cdot \frac{\partial^{2} F_{Tr}}{\partial r^{2}} + 3 r \cdot \frac{\partial F_{Tr}}{\partial r} + 0.5 E δ \cdot {(\frac{\partial \bar{w}}{\partial r})}^{2} = 0 . \end{cases})

(29)

Introducing a dimensionless quantity into Eq. (29),

\begin{gathered} ς = \frac{r}{a}, χ = \frac{a}{b}, w = \frac{\bar{w}}{δ}, Ω = ω \cdot \sqrt{\frac{ρ_{l} \cdot a^{4}}{E \cdot δ^{3}}}, τ = t \cdot \sqrt{\frac{E \cdot δ^{3}}{ρ_{l} \cdot a^{4}}}, \\ c = U_{1} \cdot \sqrt{\frac{ρ_{l} \cdot a^{2}}{E \cdot δ^{3}}}, ℘ = ϑ \cdot \sqrt{\frac{a^{4}}{ρ_{l} \cdot E \cdot δ^{3}}}, p = \bar{p} \cdot \frac{a^{4}}{E \cdot δ^{4}}, \\ f = \frac{ρ_{l} \cdot b^{2}}{E \cdot δ^{3}} \cdot F_{Tr}, β = σ \cdot δ \cdot B_{0, j}^{2} \cdot \sqrt{\frac{a^{4}}{E δ^{3} ρ_{l}}}, Q = \frac{a^{3} q_{0}}{E δ^{3}}, \\ ℵ = \frac{E_{0}^{2} \cdot σ_{0}}{{\bar{w}}_{0}^{2}} \cdot \frac{a^{4}}{E \cdot δ^{4}}, I = \frac{{\bar{w}}_{0}}{d_{0} + \frac{σ_{0}}{σ_{M}} \cdot h_{c}} . \end{gathered}

(30)

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 $B_{0} (0, B_{0, j}, 0)$ 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 ${\bar{w}}_{0} < δ$ ⁠. 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)

\{\begin{cases} \frac{\partial^{2} w}{\partial τ^{2}} + 2 c \cdot \frac{\partial^{2} w}{\partial ς \partial τ} + c^{2} \cdot \frac{\partial^{2} w}{\partial ς^{2}} - χ^{2} \cdot f \cdot \frac{d^{2} w}{d ς^{2}} - χ^{2} \cdot f \cdot \frac{d w}{d ς} \cdot \frac{1}{ς} \\ - χ^{2} \cdot \frac{d w}{d ς} \cdot \frac{d f}{d ς} + ℘ \cdot \frac{\partial w}{\partial τ} + β \cdot (c \cdot \frac{\partial w}{\partial ς} + \frac{\partial w}{\partial τ}) + (1 - ς) \cdot Q \frac{\partial^{2} w}{\partial ς^{2}} \\ = {(\frac{I}{1 - I})}^{3} \cdot ℵ \cdot w + 6 \cdot I^{5} \cdot ℵ \cdot w^{3} + I \cdot {(\frac{I}{1 - I})}^{2} \cdot ℵ \cdot \cos (Ω τ) \\ + P \cos (Ω τ) ς^{2} \cdot \frac{\partial^{2} f}{\partial ς^{2}} + 3 ς \frac{\partial f}{\partial ς} + \frac{0.5 ρ_{l}}{χ^{2}} {(\frac{\partial w}{\partial ς})}^{2} = 0 \\ ς^{2} \cdot \frac{\partial f}{\partial ς} + (1 - v) \cdot {(\frac{b^{2} \cdot ρ_{l}}{E δ^{3}})}^{2} \cdot ς \cdot f = \frac{b^{2} \cdot ρ_{l}}{δ^{2} \cdot a} \cdot u_{r} . \end{cases})

(31)

The moving flexible printed electron web satisfies the boundary conditions

\{\begin{cases} {(w|}_{ς = a, b} = 0, \\ {(\frac{\partial w}{\partial ς} = \frac{\partial w}{\partial φ}|}_{ς = a, b} = 0, \\ {(ς \cdot \frac{\partial f}{\partial ς} + (1 - v) \cdot f|}_{ς = a, b} = 0 . \end{cases})

(32)

IV. DISCRETIZATION OF THE NONLINEAR VIBRATION EQUATION

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

w (ς, φ, τ) = \sum_{i = 1}^{n} C_{i} (τ) \cdot W_{i} (ς, φ),

(33)

f (ς, φ, τ) = \sum_{i = 1}^{n} C_{i}^{2} \cdot f_{i} (ς, φ),

(34)

where $f (ς, φ, τ)$ is the internal force function, $w (ς, φ, τ)$ is the deflection shape function, and $C_{i} (τ)$ is the time function.

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

A \cdot C_{i}^{''} + μ \cdot C_{i}^{'} + K \cdot C_{i} - Θ \cdot {C_{i}}^{3} = {\bar{P}}_{1} \cdot \cos (Ω τ) .

(35)

The coefficient of Eq. (35) is

\{\begin{cases} A = \sum_{i = 1}^{n} \iint_{s} W_{i}^{2} (ς, φ) d s, \\ μ = \sum_{i = 1}^{n} \iint_{s} 2 c \cdot (\frac{\partial W_{i}}{\partial ς}) \cdot W_{i} (ς, φ) d s + \sum_{i = 1}^{n} \iint_{s} (℘ + β) \\ \cdot W_{i}^{2} (ς, φ) d s, \\ K = \sum_{i = 1}^{n} \iint_{s} c^{2} \cdot \frac{\partial^{2} W_{i}}{\partial ς^{2}} \cdot W_{i} (ς, φ) + β c \cdot \frac{\partial W_{i}}{\partial ς} \cdot W_{i} (ς, φ) \\ + Q \cdot (1 - ς) \cdot \frac{\partial^{2} W_{i}}{\partial ς^{2}} \cdot W_{i} (ς, φ) - {(\frac{I}{1 - I})}^{3} \cdot ℵ \cdot W_{i}^{2} (ς, φ) d s, \\ Θ = \sum_{i = 1}^{n} \sum_{j = 1}^{m} \iint_{s} [f_{j} (ς, φ) \cdot \frac{\partial^{2} W_{i}}{\partial ς^{2}} + \frac{f_{j} (ς, φ)}{ς} \cdot \frac{\partial W_{i}}{\partial ς}) \\ + (\frac{\partial f_{j}}{\partial ς} \cdot \frac{\partial W_{i}}{\partial ς}] \cdot W_{i} (ς, φ) d s \\ + \sum_{i = 1}^{n} \iint_{s} 6 \cdot ℵ \cdot (I^{5}) \cdot W_{i}^{4} (ς, φ) d s, \\ {\bar{P}}_{1} = \sum_{i = 1}^{n} \iint_{s} [p + I {(\frac{I}{1 - I})}^{2} \cdot ℵ] \cdot W_{i} (ς, φ) d s . \end{cases})

(36)

Another form of Eq. (35) is

A \cdot C_{i}^{''} + F_{ν} (C_{i}^{''}, C_{i}^{'}) + F_{ι} (C_{i}, C_{i}^{'}) = {\bar{P}}_{1} \cos (Ω τ),

(37)

where $F_{ν} (C_{i}^{''}, C_{i}^{'})$ is the comprehensive equation of nonlinear inertial force and nonlinear damping force and $F_{ι} (C_{i}, C_{i}^{'})$ is the comprehensive equation of the nonlinear elastic force and nonlinear damping force.

The Fourier expansion of Eq. (37) is

\{\begin{cases} F_{ν} (C_{i}^{''}, C_{i}^{'}) = \frac{G_{0}}{2} + \sum_{n = 1}^{\infty} (G_{n} \cos n Ω τ + D_{n} \sin n Ω τ), \\ F_{ι} (C_{i}, C_{i}^{'}) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos n Ω τ + b_{n} \sin n Ω τ), \end{cases})

(38)

where

\{\begin{aligned} G_{0} & = \frac{2}{T} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) d τ G_{n} \\ = \frac{2}{T} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) \cos n Ω τ d τ, \\ D_{n} & = \frac{2}{T} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) \sin n Ω τ d τ a_{0} \\ = \frac{2}{T} \int_{τ}^{τ + T} F_{ι} (C_{i}, C_{i}^{'}) d τ, \\ a_{n} & = \frac{2}{T} \int_{τ}^{τ + T} F_{ι} (C_{i}, C_{i}^{'}) \cos n Ω τ d τ b_{n} \\ = \frac{2}{T} \int_{τ}^{τ + T} F_{ι} (C_{i}, C_{i}^{'}) \sin n Ω τ d τ . \end{aligned})

(39)

Substituting Eq. (38) into Eq. (37) gives

\begin{align} A \cdot C_{i}^{''} + \frac{G_{0}}{2} + \sum_{n = 1}^{\infty} (G_{n} \cos n Ω τ + D_{n} \sin n Ω τ) \frac{a_{0}}{2} \\ + \sum_{n = 1}^{\infty} (a_{n} \cos n Ω τ + b_{n} \sin n Ω τ) = {\bar{P}}_{1} \cos (Ω τ) . \end{align}

(40)

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

M_{e} C_{i}^{''} + C_{e} C_{i}^{'} + K_{e} C_{i} = {\bar{P}}_{1} \cos (Ω τ + φ),

(41)

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

C_{i} = ψ_{0} + ψ \sin (Ω τ) .

(42)

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

\{\begin{cases} M_{e} = A - [\frac{1}{π \cdot Ω^{2} ψ} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) \cdot \sin (Ω τ) \cdot Ω d τ], \\ C_{e} = \frac{1}{π \cdot Ω ψ} \int_{τ}^{τ + T} [F_{ν} (C_{i}^{''}, C_{i}^{'}) + F_{ι} (C_{i}, C_{i}^{'})] \cdot \cos (Ω τ) \cdot Ω d τ, \\ K_{e} = \frac{1}{π \cdot ψ} \int_{τ}^{τ + T} F_{ι} (C_{i}, C_{i}^{'}) \cdot \sin (Ω τ) \cdot Ω d τ, \\ θ = \arctan \frac{C_{e} Ω}{K_{e} - M_{e} \cdot Ω^{2}} . \end{cases})

(43)

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:

X_{1} = C_{i}, X_{2} = C_{i}^{'}, X = {[\begin{matrix} X_{1} & X_{2} \end{matrix}]}^{T}, U = {\bar{P}}_{1} \cdot \cos (Ω τ + φ) .

(44)

The state control equation:

\{\begin{cases} \dot{X} = [\begin{matrix} 0 \\ - \frac{K_{e} = \frac{1}{π \cdot ψ} \int_{τ}^{τ + T} F_{ι} (C_{i}, C_{i}^{'}) \cdot \sin (Ω τ) \cdot Ω d τ}{A - [\frac{1}{π \cdot Ω^{2} ψ} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) \cdot \sin (Ω τ) \cdot Ω d τ]} \end{matrix}) \\ \begin{gathered} (\begin{matrix} 1 \\ - \frac{\frac{1}{π \cdot Ω ψ} \int_{τ}^{τ + T} [F_{ν} (C_{i}^{''}, C_{i}^{'}) + F_{ι} (C_{i}, C_{i}^{'})] \cdot \cos (Ω τ) \cdot Ω d τ}{A - [\frac{1}{π \cdot Ω^{2} ψ} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) \cdot \sin (Ω τ) \cdot Ω d τ]} \end{matrix}] \\ \cdot X + [\begin{matrix} 0 \\ \frac{1}{A - [\frac{1}{π \cdot Ω^{2} ψ} \int_{τ}^{τ + T} F_{ν} (C_{i}^{''}, C_{i}^{'}) \cdot \sin (Ω τ) \cdot Ω d τ]} \end{matrix}] \cdot U, \\ Y = [\begin{matrix} 1 & 0 \end{matrix}] \cdot X . \end{gathered} \end{cases})

(45)

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 ${\bar{P}}_{1} = 1$ ⁠, 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.

FIG. 6.

View large Download slide

Bifurcation graph of dimensionless velocities and displacements.

FIG. 7.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless velocity of 0.6.

FIG. 8.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless velocity of 1.2.

FIG. 9.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless velocity of 1.3.

FIG. 10.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless velocity of 2.

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,

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 1, ℵ = 1, c = 0.6),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 1, ℵ = 1, c = 1.2),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 1, ℵ = 1, c = 1.3),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 1, ℵ = 1, c = 2) .

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 ${\bar{P}}_{1} = 1$ ⁠, 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,

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 0.05, ℵ = 1, c = 1),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 0.2, ℵ = 1, c = 1),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 0.8, ℵ = 1, c = 1),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 1.8, ℵ = 1, c = 1) .

FIG. 11.

View large Download slide

Bifurcation graph of the dimensionless magnetic field and displacement.

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.

FIG. 12.

View large Download slide

Characteristic graph of nonlinear dynamics with a magnetism of 0.05.

FIG. 13.

View large Download slide

Characteristic graph of nonlinear dynamics with a magnetism of 0.2.

FIG. 14.

View large Download slide

Characteristic graph of nonlinear dynamics with a magnetism of 0.8.

FIG. 15.

View large Download slide

Characteristic graph of nonlinear dynamics with a magnetism of 1.8.

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 ${\bar{P}}_{1} = 1$ ⁠, 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,

(Q = 0.3, β = 0.15),

(Q = 1, β = 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,

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 0.3, β = 0.15, ℵ = 0.5, c = 1),

(℘ = 0.1, {\bar{P}}_{1} = 1, Q = 1, β = 1, ℵ = 0.5, c = 1) .

FIG. 16.

View large Download slide

Bifurcation graph of the dimensionless electrostatic field and displacement.

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.

FIG. 17.

View large Download slide

Characteristic graph of nonlinear dynamics with dimensionless voltage.

FIG. 18.

View large Download slide

Characteristic graph of nonlinear dynamics with dimensionless voltage.

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 ${\bar{P}}_{1} = 1$ ⁠, 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,

(℘ = 0.1, p = 1, Q = 0, β = 1, ℵ = 1, c = 1),

(℘ = 0.1, p = 1, Q = 0.1, β = 1, ℵ = 1, c = 1),

(℘ = 0.1, p = 1, Q = 0.2, β = 1, ℵ = 1, c = 1),

(℘ = 0.1, p = 1, Q = 2, β = 1, ℵ = 1, c = 1) .

FIG. 19.

View large Download slide

Bifurcation graph of the dimensionless electrostatic field and displacement.

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.

FIG. 20.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless follower force of 0.

FIG. 21.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless follower force of 0.1.

FIG. 22.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless follower force of 0.2.

FIG. 23.

View large Download slide

Characteristic graph of nonlinear dynamics with a dimensionless follower force of 2.

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.

REFERENCES

1.

B.

Béguin

and

C.

Breitsamter

, “

Effects of membrane pre-stress on the aerodynamic characteristics of an elasto-flexible morphing wing

,”

Aerosp. Sci. Technol.

37

,

138

–

150

(

2014

).

https://doi.org/10.1016/j.ast.2014.05.005

Google Scholar

Crossref

2.

F.

Aish

,

S.

Joyce

,

S.

Malek

, and

C. J. K.

Williams

, “

The use of aparticle method for the modelling of isotropic membrane stress for the form finding of shell structures

,”

Comput.-Aided Des.

61

,

24

–

31

(

2015

).

https://doi.org/10.1016/j.cad.2014.01.014

Google Scholar

Crossref

3.

S.

Hatami

,

M.

Azhariand

, and

M.

Saadatpour

, “

Exact free vibration of webs moving axially at high speed

,” in

Proceedings of the 15th american conference on Applied mathematics

(

WSEAS

,

2009

), pp.

134

–

139

.

Google Scholar

4.

Y.

Hong

,

W.

Yao

, and

Y.

Xu

, “

Numerical and experimental investigation of wrinkling pattern for aerospace laminated membrane structures

,”

Int. J. Aerospace. Eng.

2017

,

8476041

.

https://doi.org/10.1155/2017/8476041

Crossref

5.

C. D.

Coman

and

A. P.

Bassom

, “

On the nonlinear membrane approximation and edge-wrinkling

,”

Int. J. Solids Struct.

82

,

85

–

94

(

2016

).

https://doi.org/10.1016/j.ijsolstr.2015.11.011

Google Scholar

Crossref

6.

M.

Shao

,

J.

Wu

, and

Y.

Wang

, “

Nonlinear vibration and stability of a moving printing web with variable density based on the method of multiple scales

,”

J. Low. Freq. Noise Vib. Act. Control

38

,

1096

–

1109

(

2019

).

https://doi.org/10.1177/1461348419829371

Google Scholar

Crossref

7.

S.

Ying

,

J.

Wu

, and

Y.

Wang

, “

Nonlinear vibration and chaos of a moving flexible graphene smart electronic web

,”

Res. Phys.

19

,

103513

(

2020

).

https://doi.org/10.1016/j.rinp.2020.103513

Google Scholar

Crossref

8.

W.

Zhang

,

T.

Liu

, and

A.

Xi

, “

Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes

,”

J. Sound. Vib.

423

,

65

–

99

(

2018

).

https://doi.org/10.1016/j.jsv.2018.02.049

Google Scholar

Crossref

9.

A.

Ghobadi

,

Y. T.

Beni

, and

H.

Golestanian

, “

Nonlinear thermo-electromechanical vibration analysis of size-dependent functionally graded flexoelectric nano-plate exposed magnetic field

,”

Arch. Appl. Mech.

90

,

2025

–

2070

(

2020

).

https://doi.org/10.1007/s00419-020-01708-0

Google Scholar

Crossref

10.

Y. D.

Hu

and

W. Q.

Li

, “

Study on primary resonance and bifurcation of a conductive circular plate rotating in air-magnetic fields

,”

Nonlinear. Dyn.

93

,

671

–

687

(

2018

).

https://doi.org/10.1007/s11071-018-4217-y

Google Scholar

Crossref

11.

Y. D.

Hu

and

T.

Wang

,”

Nonlinear free vibration of a rotating circular plate under the static load in magnetic field

,”

Nonlinear. Dyn.

85

,

1825

–

1835

(

2016

).

https://doi.org/10.1007/s11071-016-2798-x

Google Scholar

Crossref

12.

Y. D.

Hu

and

L.

Zhe

, “

Magneto-elastic combination resonance of rotating circular plate with varying speed under alternating load

,”

Int. J. Struct. Stab. Dyn.

18

,

1850032

(

2018

).

https://doi.org/10.1142/S0219455418500323

Google Scholar

Crossref

13.

K.

Ri

,

P.

Han

, and

I.

Kim

, “

Nonlinear forced vibration analysis of composite beam combined with DQFEM and IHB

,”

AIP Adv.

10

,

085112

(

2020

).

https://doi.org/10.1063/5.0015053

Google Scholar

Crossref

14.

Q. C.

Nguyen

and

K.-S.

Hong

, “

Stabilization of an axially moving web via regulation of axial velocity

,”

J. Sound. Vib.

330

,

4676

–

4688

(

2011

).

https://doi.org/10.1016/j.jsv.2011.04.029

Google Scholar

Crossref

15.

K. A.

Mchugh

and

E. H.

Dowell

, “

Nonlinear response of an inextensible, cantilevered beam subjected to a nonconservative follower force

,”

J. Comput. Nonlinear Dyn.

14

,

031004

(

2019

).

https://doi.org/10.1115/1.4042324

Google Scholar

Crossref

16.

Y.

Wang

,

K.

Xie

, and

T.

Fu

, “

A unified modified couple stress model for size-dependent free vibrations of FG cylindrical microshells based on high-order shear deformation theory

,”

Eur. Phys. J. Plus

135

,

71

(

2020

).

https://doi.org/10.1140/epjp/s13360-019-00012-3

Google Scholar

Crossref

17.

K.

Higuchi

and

E. H.

Dowell

, “

Dynamic stability of a rectangular plate with four free edges subjected to a follower force

,”

AIAA. J.

28

,

1300

–

1305

(

2015

).

https://doi.org/10.2514/3.25208

Google Scholar

Crossref

18.

X. X.

Guo

,

Z. M.

Wang

, and

Y.

Wang

, “

Dynamic stability of thermoelastic coupling moving plate subjected to follower force

,”

Appl. Acoust.

72

,

100

–

107

(

2011

).

https://doi.org/10.1016/j.apacoust.2010.10.001

Google Scholar

Crossref

19.

Y.

Huang

and

H.

Li

, “

Chao and period-doubling vibration in superconducting levitation systems

,”

AIP Adv.

10

,

095121

(

2020

).

https://doi.org/10.1063/5.0020899

Google Scholar

Crossref

20.

M.

Soltanrezaee

and

M.

Bodaghi

, “

Nonlinear dynamic stability of piezoelectric thermoelastic electromechanical resonators

,”

Sci. Rep.

10

,

2982

(

2020

).

https://doi.org/10.1038/s41598-020-59836-0

Google Scholar

Crossref

PubMed

21.

J.

Wang

and

T.

Ren

, “

Three-to-one internal resonance in MEMS arch resonators

,”

Sensors

19

,

1888

(

2019

).

https://doi.org/10.3390/s19081888

Google Scholar

Crossref

PubMed

22.

M.

Senthilkumar

and

M. G.

Vasundhara

, “

Electromechanical analytical model of shape memory alloy based tunable cantilevered piezoelectric energy harvester

,”

Int. J. Mech. Mater. Des.

15

,

611

–

627

(

2019

).

https://doi.org/10.1007/s10999-018-9413-x

Google Scholar

Crossref

23.

W.

Zhang

,

R. Q.

Wu

, and

K.

Behdinan

, “

Nonlinear dynamic analysis near resonance of a beam-ring structure for modeling circular truss antenna under time-dependent thermal excitation

,”

Aerosp. Sci. Technol.

86

,

296

–

311

(

2019

).

https://doi.org/10.1016/j.ast.2019.01.018

Google Scholar

Crossref

24.

L.-Q.

Chen

,

X.

Li

, and

Z.-Q.

Lu

, “

Dynamic effects of weights on vibration reduction by a nonlinear energy sink moving vertically

,”

J. Sound Vib.

451

,

99

–

119

(

2019

).

https://doi.org/10.1016/j.jsv.2019.03.005

Google Scholar

Crossref

25.

L.-Q.

Chen

,

L.

Peng

, and

A.-Q.

Zhang

, “

Transverse vibration of viscoelastic Timoshenko beam-columns

,”

J. Vib. Control.

23

,

1572

–

1584

(

2017

).

https://doi.org/10.1177/1077546315596483

Google Scholar

Crossref

26.

M.

Shao

,

J.

Wu

, and

Y.

Wang

, “

Nonlinear forced vibration of a moving paper web with varying density

,”

Adv. Mech. Eng.

11

,

1687814019851004

(

2019

).

https://doi.org/10.1177/1687814019851004

Google Scholar

Crossref

27.

Y.

Chai

and

F. M.

Li

, “

Nonlinear vibration behaviors of composite laminated plates with time-dependent base excitation and boundary conditions

,”

Int. J. Nonlinear Sci. Numer. Simul.

18

,

145

–

161

(

2017

).

https://doi.org/10.1515/ijnsns-2016-0138

Google Scholar

Crossref

28.

D.

Hu

and

C. W.

Liu

, “

Nonlinear vibration of a traveling belt with non-homogeneous boundaries

,”

J. Sound Vib.

424

,

78

–

93

(

2018

).

https://doi.org/10.1016/j.jsv.2018.03.010

Google Scholar

Crossref

29.

Y.

Wang

,

H.

Ding

, and

L.-Q.

Chen

, “

Vibration of axially moving hyperelastic beam with finite deformation

,”

Appl. Math. Modell.

71

,

269

–

285

(

2019

).

https://doi.org/10.1016/j.apm.2019.02.011

Google Scholar

Crossref

30.

C. C.

Lin

and

C. D.

Mote

, “

Equilibrium displacement and stress distribution in a two-dimensional, axially moving web under transverse loading

,”

ASME J. Appl. Mech.

62

,

772

–

779

(

1995

).

https://doi.org/10.1115/1.2897013

Google Scholar

Crossref

2021

Author(s)

Nonlinear vibration of a moving flexible printed electron web under multiphysics dynamics

I. INTRODUCTION

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

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

IV. DISCRETIZATION OF THE NONLINEAR VIBRATION EQUATION

V. DISCUSSION

A. Velocity control nonlinear dynamic

B. Magnetic field control nonlinear dynamic

C. Electrostatic field control, nonlinear dynamic

D. Follower force control, nonlinear dynamic

VI. CONCLUSIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Nonlinear vibration of a moving flexible printed electron web under multiphysics dynamics

I. INTRODUCTION

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

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

IV. DISCRETIZATION OF THE NONLINEAR VIBRATION EQUATION

V. DISCUSSION

A. Velocity control nonlinear dynamic

B. Magnetic field control nonlinear dynamic

C. Electrostatic field control, nonlinear dynamic

D. Follower force control, nonlinear dynamic

VI. CONCLUSIONS

ACKNOWLEDGMENTS

DATA AVAILABILITY

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only