Forced vibration and chaotic behavior of membrane under pretension

Feng, Yao; Wu, Jimei; Wang, Xing

doi:10.1063/5.0084181

In the process of transmission, the flexible membrane will be affected by the pretension of the guide roller and the external excitation force of the conductive silver slurry ejected by the nozzle. Because the large deflection deformation of the middle surface of the membrane caused by pretension and external force seriously affects the printing accuracy of the printed products, it is very important to study the geometric nonlinearity of the flexible membrane. In this paper, the transverse forced vibration model of the flexible membrane with pretension is established by Von Karman’s large deflection theory, and its vibration equation is discretized and solved by the Bubnov–Galerkin method and the fourth-order Runge Kutta method. Meanwhile, by analyzing the bifurcation diagram and TLE (Top Lyapunov Exponents) spectrum, the working state of the motion system with the change in parameters is obtained. In addition, the dynamic response of the system under the given parameters is obtained by comparing the phase diagram with the Poincaré mapping diagram.

I. INTRODUCTION

Membranes in roll-to-roll manufacturing equipment are usually transported at a high speed. Affected by the conditions and moving speed of the manufacturing equipment, the membrane will produce transverse vibration in this process, which will affect the printing quality. For flexible electronic printing, the alignment accuracy of the membrane is usually confined to within ±10 µm; even small vibration may reduce its printing quality.¹ The nonlinear vibration of thin membranes has become a bottleneck, restricting production.^2,3

Ma et al.⁴ established the dynamic model of axially moving film with intermediate elastic support, discretized the dynamic equation by the Galerkin method, and calculated the natural frequency of the system at the same time. Wang et al.^5–8 established the dynamic model of a functionally graded thin plate based on D’Alembert’s principle and studied the nonlinear vibration of a functionally graded thin plate system. Ji et al.⁹ studied the formation mechanism of the bifurcation phenomenon of the roll to roll system and obtained the length of the thin plate during the bifurcation. Liang et al.¹⁰ proposed a multi roll coordinated optimization control method and designed a multiple input multiple output linear quadratic integral controller. Zhang and Zhang¹¹ deduced the analytical solutions of free vibration and buckling of nano-shells by using the first-order shear deformation shell theory. Pirmoradian et al.¹² derived the partial differential equation of the transverse motion of a plate by using the Hamiltonian principle and studied the stability of plate vibration. Li et al.^13–15 studied the nonlinear vibration characteristics of building membrane structures. Ma et al.16,17 studied the transverse vibration characteristics of membranes in the roll to roll manufacturing process and proposed a membrane tension estimation method based on transverse vibration frequency measurement. Tang et al.^19–20 studied the stability of the viscoelastic axis and plate and derived the stability conditions of the system. A Chinese scholar He^22–23 used the homotopy perturbation method to solve the traveling wave solutions of nonlinear equations.

The longitudinal deformation of the membrane’s middle surface includes two parts. The first part is the deformation of the membrane’s middle surface caused by external excitation, and the second part is the deformation of the membrane’s middle surface caused by the pretension generated by the guide roller. In our previous research, the deformation of the middle surface of the membrane caused by pretension was not considered, so the focus of this paper is to analyze the nonlinear vibration characteristics of the film under pretension. In this paper, the effects of different parameters on the local and global dynamic characteristics of moving thin membranes are analyzed by combining the bifurcation diagram, TLE spectrum, phase diagram, and Poincaré map.

II. DYNAMIC MODELING

Figure 1 shows the dynamic model, where b is the membrane length, a is the width, h is the thickness, E is the elastic modulus, ρ is the density, v is the motion velocity, and $\bar{p} \cos (\bar{ω} t)$ represents the external excitation force on the membrane’s surface. The internal forces on the membrane include two parts: internal forces N_ox and N_oy caused by pretension and internal forces N_x and N_y caused by external exciting force.

FIG. 1.

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Dynamic model.

Take a micro-unit of vibration on the membrane, as shown in Fig. 2, where ρdxdy is the mass of the micro-element. The expression of the differential equation of the axial motion of the membrane according to D’Alembert’s principle can be written as follows:

\begin{align} ρ (y) (\frac{\partial^{2} \bar{w}}{\partial t^{2}} + \frac{\partial^{2} \bar{w}}{\partial x \partial t} + v^{2} \frac{\partial^{2} \bar{w}}{\partial x^{2}}) - (N_{x}) \frac{\partial^{2} \bar{w}}{\partial x^{2}} - (N_{y}) \frac{\partial^{2} \bar{w}}{\partial y^{2}} + φ \frac{\partial \bar{w}}{\partial t} \\ = \bar{p} \cos (\bar{ω} t) . \end{align}

(1)

FIG. 2.

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Vibrating membrane element.

By adding pretensions N_ox and N_oy on the basis of Eq. (1), the transverse vibration equation of the moving film with pretension can be obtained as

\begin{align} ρ (y) (\frac{\partial^{2} \bar{w}}{\partial t^{2}} + \frac{\partial^{2} \bar{w}}{\partial x \partial t} + v^{2} \frac{\partial^{2} \bar{w}}{\partial x^{2}}) - (N_{x} + N_{o x}) \frac{\partial^{2} \bar{w}}{\partial x^{2}} \\ - (N_{y} + N_{o y}) \frac{\partial^{2} \bar{w}}{\partial y^{2}} + φ \frac{\partial \bar{w}}{\partial t} = \bar{p} \cos (\bar{ω} t), \end{align}

(2)

\frac{\partial^{4} Φ}{\partial x^{4}} + \frac{\partial^{4} Φ}{\partial y^{4}} = E h ({(\frac{\partial^{2} \bar{w}}{\partial x \partial y})}^{2} - \frac{\partial^{2} \bar{w}}{\partial x^{2}} \frac{\partial^{2} \bar{w}}{\partial y^{2}}),

(3)

where $N_{x} = \frac{\partial^{2} Φ}{\partial y^{2}}$ and N_y = $\frac{\partial^{2} Φ}{\partial x^{2}}$ ⁠.

Assuming that the density function $ρ (y)$ changes along the y-axis, we obtain

ρ (y) = [(1 + α) ρ_{0} - 2 α ρ_{0} |\frac{y}{b} - \frac{1}{2}|] .

(4)

By including Eq. (4) into Eq. (2), we obtain

\begin{align} [(1 + α) ρ_{0} - 2 α ρ_{0} |\frac{y}{b} - \frac{1}{2}|] (\frac{\partial^{2} \bar{w}}{\partial t^{2}} + \frac{\partial^{2} \bar{w}}{\partial x \partial t} + v^{2} \frac{\partial^{2} \bar{w}}{\partial x^{2}}) \\ - (\frac{\partial^{2} Φ}{\partial y^{2}} + N_{o x}) \frac{\partial^{2} \bar{w}}{\partial x^{2}} - (\frac{\partial^{2} Φ}{\partial x^{2}} + N_{o y}) \frac{\partial^{2} \bar{w}}{\partial y^{2}} + φ \frac{\partial \bar{w}}{\partial t} \\ = \bar{p} \cos (\bar{ω} t), \end{align}

(5)

\frac{\partial^{4} Φ}{\partial x^{4}} + \frac{\partial^{4} Φ}{\partial y^{4}} = E h ({(\frac{\partial^{2} \bar{w}}{\partial x \partial y})}^{2} - \frac{\partial^{2} \bar{w}}{\partial x^{2}} \frac{\partial^{2} \bar{w}}{\partial y^{2}}) .

(6)

By introducing the dimensionless variables $ξ = \frac{x}{a}$ ⁠, $η = \frac{y}{b}$ ⁠, $w = \frac{\bar{w}}{h}$ ⁠, $r = \frac{a}{b}$ ⁠, $τ = t \sqrt{\frac{E h^{3}}{ρ_{0} a^{4}}}$ ⁠, $γ = φ \sqrt{\frac{a^{4}}{E h^{3} ρ_{0}}}$ ⁠, $c = v \sqrt{\frac{ρ_{0} a^{2}}{E h^{3}}}$ ⁠, $A_{1} = \frac{a^{2} N_{o x}}{E h^{3}}$ ⁠, $p = \frac{a^{4}}{E h^{4}} \bar{p}$ ⁠, and $λ = \frac{N_{o y}}{N_{o x}}$ ⁠, we get the dimensionless form of Eqs. (5) and (6) as follows:

\begin{align} [(1 + α) - 2 α |η - \frac{1}{2}|] (\frac{\partial^{2} w}{\partial τ^{2}} + 2 c \frac{\partial^{2} w}{\partial ξ \partial τ} + c^{2} \frac{\partial^{2} w}{\partial ξ^{2}}) \\ \times - r^{2} (\frac{\partial^{2} f}{\partial η^{2}} \frac{\partial^{2} w}{\partial ξ^{2}} + \frac{\partial^{2} f}{\partial ξ^{2}} \frac{\partial^{2} w}{\partial η^{2}}) - A_{1} (\frac{\partial^{2} w}{\partial ξ^{2}} + λ r^{2} \frac{\partial^{2} w}{\partial η^{2}}) \\ + γ \frac{\partial w}{\partial τ} = p c o s (ω τ), \end{align}

(7)

\frac{\partial^{4} f}{\partial ξ^{4}} + r^{4} \frac{\partial^{4} f}{\partial η^{4}} = r^{2} {(\frac{\partial^{2} w}{\partial ξ \partial η})}^{2} - r^{2} \frac{\partial^{2} w}{\partial ξ^{2}} \frac{\partial^{2} w}{\partial η^{2}} .

(8)

The boundary condition of the moving membrane is simply supported on four sides,

ξ = 0, 1, \frac{\partial^{2} f}{\partial η^{2}} = 1, \frac{\partial^{2} f}{\partial ξ \partial η} = 0, w = 0,

η = 0, 1, \frac{\partial^{2} f}{\partial ξ^{2}} = 1, \frac{\partial^{2} f}{\partial ξ \partial η} = 0, w = 0 .

Using the Bubnov–Galerkin method to discretize Eqs. (7) and (8), it is assumed that the separated variable form of w(ξ, η, τ) and $f (ξ, η, τ)$ is

w (ξ, η, τ) = W (ξ, η) q (τ),

(9)

f (ξ, η, τ) = F (ξ, η) q^{2} (τ) .

(10)

The vibration mode function satisfying the boundary conditions is taken as

W (ξ, η) = \sum_{i = 1}^{M_{i}} \sum_{j = 1}^{M_{j}} \sin (i π ξ) \sin (j π η) .

(11)

Taking i = 1 and j = 1 in Eq. (11), the first-order mode function can be obtained.

By including Eq. (11) into Eq. (8), we obtain

\frac{\partial^{4} F}{\partial ξ^{4}} + r^{4} \frac{\partial^{4} F}{\partial η^{4}} = \frac{r^{2} π^{4}}{2} (\cos (2 π ξ) + \cos (2 π η)) .

(12)

Suppose the solution of Eq. (12) is

F (ξ, η) = α_{1} \cos (2 π ξ) + β_{1} \cos (2 π η) + r_{a} ξ^{3} + r_{b} ξ^{2} + r_{c} η^{3} + r_{d} η^{2} .

(13)

By including Eq. (13) into Eq. (12), we obtain

α_{1} = \frac{r^{2}}{32}, β_{1} = \frac{1}{32 r^{2}} .

Since Eq. (13) satisfies the boundary conditions of simply supported four sides, it can be obtained that r_a = r_c = 0, $r_{b} = \frac{π^{2} r^{2}}{16} + \frac{1}{2}$ ⁠, and $r_{d} = \frac{π^{2}}{16 r^{2}} + \frac{1}{2}$ ⁠.

Then, Eq. (13) can be expressed as

\begin{align} F (ξ, η) & = \frac{r^{2}}{32} \cos (2 π ξ) + \frac{1}{32 r^{2}} \cos (2 π η) + (\frac{π^{2} r^{2}}{16} + \frac{1}{2}) ξ^{2} \\ + (\frac{π^{2}}{16 r^{2}} + \frac{1}{2}) η^{2} . \end{align}

(14)

By integrating formula (7), we can get

\begin{align} \int_{0}^{1} \int_{0}^{1} \{[(1 + α) - 2 α |η - \frac{1}{2}|] (\frac{\partial^{2} q (τ)}{\partial τ^{2}} W (ξ, η) + 2 c \frac{\partial W (ξ, η)}{\partial ξ} \frac{\partial q (τ)}{\partial τ} + c^{2} \frac{\partial^{2} W (ξ, η)}{\partial ξ^{2}} q (τ)) - r^{2} (\frac{\partial^{2} F (ξ, η)}{\partial η^{2}} \frac{\partial^{2} W (ξ, η)}{\partial ξ^{2}})) \\ + ((\frac{\partial^{2} F (ξ, η)}{\partial ξ^{2}} \frac{\partial^{2} W (ξ, η)}{\partial η^{2}}) q^{3} (τ) - A_{1} (\frac{\partial^{2} W (ξ, η)}{\partial ξ^{2}} + λ r^{2} \frac{\partial^{2} W (ξ, η)}{\partial η^{2}}) q (τ) + γ \frac{\partial q (τ)}{\partial τ} W (ξ, η) - p c o s (ω τ)\} W (ξ, η) d ξ d η = 0 . \end{align}

(15)

The vibration equation of the moving membrane can be expressed as

A_{1} \ddot{q} + B_{1} \dot{q} + C_{1} q - D_{1} q^{3} - \cos (ω τ) E_{1} = 0,

(16)

A_{1} = \int_{0}^{1} \int_{0}^{1} [(1 + α) - 2 α |η - \frac{1}{2}|] W^{2} (ξ, η) d ξ d η = \frac{2 π^{2} (2 + α) + 8 α}{16 π^{2}},

\begin{array}{l} B_{1} & = \int_{0}^{1} \int_{0}^{1} [(1 + α) - 2 α |η - \frac{1}{2}|] \\ \times (2 c \frac{\partial W (ξ, η)}{\partial ξ} W (ξ, η) + γ W^{2} (ξ, η)) d ξ d η = \frac{γ}{4}, \end{array}

\begin{array}{l} C_{1} & = \int_{0}^{1} \int_{0}^{1} [(1 + α) - 2 α |η - \frac{1}{2}|] (c^{2} \frac{\partial^{2} W (ξ, η)}{\partial ξ^{2}} - A_{1} (\frac{\partial^{2} W (ξ, η)}{\partial ξ^{2}})) \\ + ((λ r^{2} \frac{\partial^{2} W (ξ, η)}{\partial η^{2}})) W (ξ, η) d ξ d η = \frac{π^{2}}{4} A_{1} (λ r^{2} + 1) - \frac{(1 + α) c^{2} π^{2}}{4}, \end{array}

\begin{array}{l} D_{1} & = \int_{0}^{1} \int_{0}^{1} r^{2} (\frac{\partial^{2} F (ξ, η)}{\partial η^{2}} \frac{\partial^{2} W (ξ, η)}{\partial ξ^{2}} + \frac{\partial^{2} F (ξ, η)}{\partial ξ^{2}} \frac{\partial^{2} W (ξ, η)}{\partial η^{2}}) \\ \times W (ξ, η) d ξ d η = - \frac{3 π^{4} (1 + r^{4}) + 32 π^{2} r^{2}}{64}, \end{array}

E_{1} = \frac{4 p}{π^{2}} .

The following variables are introduced:

X_{1} = q, X_{2} = \dot{q} .

Then, Eq. (16) can be expressed as

\begin{array}{l} {\dot{X}}_{2} & = - \frac{2 γ π^{2}}{2 π^{2} + α π^{2} + 4 α} X_{2} + \frac{2 {\dot{π}}^{2} [π^{2} A_{1} (λ r^{2} + 1) + c^{2} π^{2} (1 + α)]}{2 π^{2} + α π^{2} + 4 α} X_{1} \\ - \frac{3 π^{6} (1 + r^{4}) - 32 π^{4} r^{2}}{16 π^{2} + 8 α π^{2} + 32 α} X_{1}^{3} + \frac{32 p}{2 π^{2} + α π^{2} + 4 α} \cos (ω τ) . \end{array}

III. LYAPUNOV EXPONENT

In order to quantitatively express the separation speed of adjacent points in the mapping, the Lyapunov exponent is introduced. δx in n-dimensional phase space is n-dimensional, and there will be n variable values. When t = 0, take x₀ as the center and δx(x₀, 0) as the radius to make an n-dimensional hypersphere. With the change in time, the n-dimensional hypersphere will deform into an n-dimensional hyper-ellipsoid at time t, and σ_i of Lyapunov exponent σ is

σ_{i} = \lim_{t \to \infty} \frac{1}{t} \ln \frac{|δ x_{i} (x_{0}, t)|}{|δ x_{i} (x_{0}, 0)|} .

(17)

The multidimensional dynamical system has a positive Lyapunov exponent, which indicates that the system is in chaotic motion.

IV. INITIAL VALUE SENSITIVITY ANALYSIS

Given that damping is γ = 0.031 and γ = 1.31 and the other parameters are aspect ratio r = 1.1, density coefficient α = 0.0011, tension ratio λ = 0.03, external excitation amplitude p = 1, pretension A₁ = 0.01, and external excitation frequency ω = 0.5, the time–history diagram and phase diagram of the moving membrane system are shown in Fig. 3, which demonstrate that the trajectory of the flexible electronic membrane changes greatly with the change in the damping coefficient and with the change in time, a completely different state of motion is formed. In addition, Figs. 4 and 5 show the bifurcation diagram and TLE spectrum of the moving membrane system at γ = 0.031 and γ = 1.31, respectively, which illustrate that great changes in the motion state of the moving membrane system have taken place due to the changes in damping. This sensitivity to the initial conditions reflects the strong internal randomness in the dynamic system, which makes it impossible to accurately predict the behavior of the membrane system after long-term motion. Therefore, it is necessary to study the nonlinear vibration of the membrane system.

FIG. 3.

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(a) Time–history chart (γ = 0.031 and γ = 1.31); (b) Poincaré Map (γ = 0.031 and γ = 1.31).

FIG. 4.

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(a) Bifurcation diagram; (b) corresponding TLE spectrum (γ = 0.031, r = 1.1, α = 0.001 1, λ = 0.03, p = 1, A₁ = 0.01, and ω = 0.5).

FIG. 5.

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(a) Bifurcation diagram; (b) corresponding TLE spectrum (γ = 1.31, r = 1.1, α = 0.001 1, λ = 0.03, p = 1, A₁ = 0.01, and ω = 0.5).

V. INFLUENCE OF VARIOUS PARAMETERS ON MEMBRANE MOTION

During the movement of the membrane, it will be affected by pretension A₁, aspect ratio r, density coefficient α, and other factors. With the effects of the above-mentioned factors on the motion state of the membrane analyzed by the bifurcation diagram and TLE spectrum, as well as the parameter conditions given, the phase diagram and Poincaré map of the moving membrane system can describe the motion state of the system more accurately.

A. Influence of pretension A₁ on system motion

When damping γ = 0.31, aspect ratio r = 1.1, external excitation frequency ω = 0.5, density coefficient α = 0.0011, tension ratio λ = 0.03, external excitation amplitude p = 1, and dimensionless velocity c = 0.6, the bifurcation diagram of the pretension A₁ of the moving membrane system changing with the displacement is given in Fig. 6(a), and the corresponding TLE spectrum, in which the variation interval of pretension A₁ is [0,1], is given in Fig. 6(b). It shows that when the pretension A₁ ∈ (0.374, 0.629), the initial state of the moving membrane is the intersection of chaotic motion and periodic motion, and correspondingly, the Lyapunov exponent in the TLE spectrum jumps back and forth in the positive and negative range. While A₁ ∈ (0.374, 0.629), the moving membrane makes the transition from alternating chaotic motion to the motion state of periodic 2 and further to periodic 1 if A₁ is exerted increasingly, where the Lyapunov exponents are both negative and the system is in a stable working state. This indicates that properly increasing the membrane pretension A₁ in the system is conducive to the stable working state of the system.

FIG. 6.

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(a) Multi-initial bifurcation diagram; (b) corresponding TLE spectrum (γ = 0.31, r = 1.1, ω = 0.5, α = 0.001 1, λ = 0.03, p = 1, and c = 0.6).

We continue to take A₁ = 0.314, A₁ = 0.564, and A₁ = 0.746, and analyze the phase diagram and Poincaré maps of the moving membrane, in which the motion of the membrane has experienced chaotic motion (Fig. 7) → period doubling motion (Fig. 8) → period 1 motion (Fig. 9). Figure 8(b) shows the Poincaré map of the moving membrane, analysis of which shows that when the moving membrane is in period doubling motion, the displacement of the membrane system remains unchanged, but the velocity changes, indicating that the moving membrane collides at this time.

FIG. 7.

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(a) Phase diagram; (b) Poincaré map (A₁ = 0.314, γ = 0.31, r = 1.1, ω = 0.5, α = 0.001 1, λ = 0.03, p = 1, and c = 0.6).

FIG. 8.

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(a) Phase diagram; (b) Poincaré map (A₁ = 0.564, γ = 0.31, r = 1.1, ω = 0.5, α = 0.001 1, λ = 0.03, p = 1, and c = 0.6).

FIG. 9.

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(a) Phase diagram; (b) Poincaré map (A₁ = 0.746, γ = 0.31, r = 1.1, ω = 0.5, α = 0.001 1, λ = 0.03, p = 1, and c = 0.6).

B. Influence of aspect ratio r on system motion

When damping γ = 0.051, external excitation frequency ω = 0.5, density coefficient α = 0.21, tension ratio λ = 0.03, external excitation amplitude p = 1, pretension A₁ = 0.2, and dimensionless velocity c = 0.6, Fig. 10(a) shows the bifurcation diagram of the aspect ratio r with displacement. Figure 10(b) shows the corresponding TLE spectrum. The value range of the aspect ratio r is [0,1]. When r ∈ (0,0.095) and (0.272,1), the moving membrane is in the state of alternating chaotic motion and periodic motion with the increase in the aspect ratio r, and the Lyapunov exponent in the corresponding TLE spectrum changes up and down the coordinate axis. When r ∈ (0.096, 0.271), the moving membrane is in chaotic motion, and the Lyapunov exponent in the corresponding TLE spectrum is always positive.

FIG. 10.

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(a) Multi-initial bifurcation diagram; (b) corresponding TLE spectrum (γ = 0.051, ω = 0.5, α = 0.21, λ = 0.03, p = 1, A₁ = 0.2, and c = 0.6).

We continue to analyze Figs. 11(a) and 12(a). Since the moving membrane is in chaotic motion with periodic motion, the closed curve in the phase diagram also changes to an unclosed curve. The corresponding Poincaré map is shown in Figs. 11(b) and 12(b), respectively, which changes from a large number of points to two fixed points. This justifies that the change in r has a great influence on the motion state of the membrane. As the aspect ratio r of the membrane in the printing machine is determined by the guide roller, special attention should be paid to the distance between the guide rollers.

FIG. 11.

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(a) Phase diagram; (b) Poincaré map (r = 0.136, γ = 0.051, ω = 0.5, α = 0.21, λ = 0.03, p = 1, A₁ = 0.2, and c = 0.6).

FIG. 12.

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(a) Phase diagram; (b) Poincaré map (r = 0.798, γ = 0.051, ω = 0.5, α = 0.21, λ = 0.03, p = 1, A₁ = 0.2, and c = 0.6).

C. Influence of density coefficient α on system motion

Given that damping γ = 0.051, aspect ratio r = 0.3, external excitation frequency ω = 1, tension ratio λ = 0.03, external excitation amplitude p = 2.1, pretension A₁ = 0.2, and dimensionless speed c = 0.6, Fig. 13(a) shows the bifurcation diagram of the density coefficient α (subject to [0,1]) with displacement, and Fig. 13(b) shows the corresponding TLE spectrum. When α ∈ (0, 0.168), (0.18, 0.212), and (0.326, 0.513), the moving membrane is in the state of alternating chaotic motion and periodic motion with the increase in the density coefficient α, and the Lyapunov exponent in the corresponding TLE spectrum changes up and down the coordinate axis. When α ∈ (0.513, 1), the moving membrane is in chaotic motion, and the Lyapunov exponent in the corresponding TLE spectrum is always positive. When α ∈ (0.21, 0.329), the moving membrane is in periodic motion, and the Lyapunov exponent in the corresponding TLE spectrum is always negative. On the whole, the moving membrane will eventually go into a chaotic state with the increase in the density coefficient, so it is necessary to control the density coefficient.

FIG. 13.

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(a) Multi-initial bifurcation diagram; (b) corresponding TLE spectrum (γ = 0.051, r = 0.3, ω = 1, λ = 0.03, p = 2.1, A₁ = 0.2, and c = 0.6).

When taking α = 0.314, α = 0.564, and α = 0.746, analyzing the phase diagram and Poincaré map of the moving membrane system shows that the system moves from periodic 2 motion (Fig. 14) → almost periodic motion (Fig. 15) → chaotic motion (Fig. 16). It evolves from two fixed points at the initial time into a limit cycle to more scattered points. With the increase in the density coefficient, the system eventually leads to chaos.

FIG. 14.

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(a) Phase diagram; (b) Poincaré map (α = 0.314, γ = 0.051, r = 0.3, ω = 1, λ = 0.03, p = 2.1, A₁ = 0.2, and c = 0.6).

FIG. 15.

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(a) Phase diagram; (b) Poincaré map (α = 0.564, γ = 0.051, r = 0.3, ω = 1, λ = 0.03, p = 2.1, A₁ = 0.2, and c = 0.6).

FIG. 16.

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(a) Phase diagram; (b) Poincaré map (α = 0.674, γ = 0.051, r = 0.3, ω = 1, λ = 0.03, p = 2.1, A₁ = 0.2, and c = 0.6).

VI. CONCLUSION

By establishing the dynamic equation of the forced vibration flexible electronic membrane, the bifurcation diagram, TLE spectrum, phase diagram, and Poincaré map of the system are studied. The following conclusions are obtained:

By changing the initial parameters in the membrane dynamic equation, it can be found that the displacement state of the membrane in the time history diagram and Poincaré map of the system has changed and the bifurcation diagram has also changed, indicating that the nonlinear system has initial value sensitivity.
As the pretension increases, the moving film is more likely to be in a stable state of motion. This feature should be used in the design process of the printing press, and the pretension should be appropriately increased. The size of the pretension is related to the rotation speed of the roller, and an appropriate rotation speed should be selected.
The aspect ratio of the moving membrane will affect the motion state of the system. With the increase in the aspect ratio, the system will eventually be in the alternating state of chaotic motion and periodic motion. The aspect ratio is determined by the roller of the printing press, so the appropriate aspect ratio should be selected in the design of the printing press.
The density coefficient has a great influence on the movement of the system. As the density coefficient increases, the system will eventually go into chaos, so it is necessary to select a suitable membrane material in the printing process.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant No. 52075435) and the Natural Science Foundation of Shaanxi Province (Grant Nos. 2021JQ-480 and 2020JM-457).

AUTHOR DECLARATIONS

Conflict of Interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

DATA AVAILABILITY

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

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Author(s)

Forced vibration and chaotic behavior of membrane under pretension

I. INTRODUCTION

II. DYNAMIC MODELING

III. LYAPUNOV EXPONENT

IV. INITIAL VALUE SENSITIVITY ANALYSIS

V. INFLUENCE OF VARIOUS PARAMETERS ON MEMBRANE MOTION

A. Influence of pretension A₁ on system motion

B. Influence of aspect ratio r on system motion

C. Influence of density coefficient α on system motion

VI. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

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Forced vibration and chaotic behavior of membrane under pretension Open Access

I. INTRODUCTION

II. DYNAMIC MODELING

III. LYAPUNOV EXPONENT

IV. INITIAL VALUE SENSITIVITY ANALYSIS

V. INFLUENCE OF VARIOUS PARAMETERS ON MEMBRANE MOTION

A. Influence of pretension A1 on system motion

B. Influence of aspect ratio r on system motion

C. Influence of density coefficient α on system motion

VI. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

DATA AVAILABILITY

REFERENCES

Citing articles via

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Forced vibration and chaotic behavior of membrane under pretension

A. Influence of pretension A₁ on system motion