Chaotic dynamics of flexible graphene electronic membranes with variable density in motion

Ying, Shudi; Wu, Jimei; Cai, Zhiduan; Wang, Yuling

doi:10.1063/5.0226095

With advancements in artificial intelligence and wearable technology, flexible electronic devices characterized by their flexibility and extensibility have found widespread applications in fields such as information technology, healthcare, and military. Printing technology can accurately print a circuit diagram onto a flexible membrane substrate by the pressure transfer of a conductive ink, which makes the large-scale printing of flexible graphene electronic membranes possible. However, during the roll-to-roll printing process used to prepare flexible graphene electron membranes, the density of electron membranes is variable due to the uneven distribution of inkjet-printed circuits, which limits the printing speed of flexible graphene electron membranes. Hence, investigating the dynamic properties of flexible graphene electron materials with different densities is of paramount importance to improve the production efficiency and quality of flexible graphene electron membranes. This paper takes roll-to-roll intelligent graphene electronic membranes as the research object. According to Hamilton’s principle, nonlinear vibration partial differential equations for the motion of flexible graphene electron membranes with varying densities were established and subsequently discretized using the assumed displacement function and the Bubnov–Galerkin method. Through numerical calculations, the simulation results obtained based on the fourth-order Runge–Kutta method and the multiscale algorithm were compared, and the multiscale algorithm was verified to be more correct and effective. The primary resonance amplitude–parameter characteristic curve, along with phase-plane portraits, Poincaré maps, power spectrum, time history plots, and bifurcation diagrams, for the nonlinear behavior of the membrane was obtained. The impacts of the density coefficient, velocity, damping ratio, excitation force, and detuning parameters on the nonlinear primary resonance and chaotic behavior of the moving graphene electron membrane were determined, and the stable operational region was identified, laying a theoretical foundation for the development of flexible graphene electronic membranes.

I. INTRODUCTION

In the large-scale printing of flexible graphene electronic membranes, the areal density of flexible graphene electronic membranes is variable. For example, the presence of inks or lubricants on the surface of a flexible graphene electron membrane will affect the areal density of the graphene electronic membrane. In the inkjet printing process, the inkjet printing circuits distributed on flexible graphene electron membranes are complex, which inevitably leads to a change in the surface density of the flexible graphene electronic membrane. The dynamic behavior of a membrane with a nonuniform areal density changes. At the same time, flexible graphene electronic membranes with high-speed transmission are prone to curling, wrinkling, tearing, and other problems. Therefore, studying the chaos and primary resonance of the nonlinear forced motion of moving flexible graphene electron membranes is crucial for determining the stable working speed range and effectively controlling their nonlinear large deflection vibrations.

Although scholars have performed much research on axial motion systems in recent years, great progress has also been made. Nevertheless, few studies have been conducted on the nonlinear behavior of membranes with varying densities. Qing et al.¹ applied the D’Alembert principle and Galerkin method to study the amplitude–frequency response equation of mobile webs and analyzed the influence of different dimensionless velocities and variable density coefficients on nonlinear vibration from the amplitude–frequency curve. Adhikari and Dash² considered Lagrangian equations, derived nonlinear vibration control equations for the discrete motion of laminates, and used the finite element method to explore the free vibration problem of laminates with different boundaries and crossing angles. Foroutan et al.³ investigated the nonlinear motion and buckling of composite cylindrical panels under nonlinear temperature distributions, using the fourth-order Runge–Kutta method to study the effects of temperature and humidity. Shao et al.^4,5 explored the dynamics of a motion film with variable speed and obtained bifurcation diagrams and Poincaré maps, in addition to the stable region and chaotic region of the system. Ahmadi et al.⁶ studied the nonlinear vibration of functionally graded hyperbolic shallow shells and derived the shell model according to classical plate theory and the nonlinear geometric von Kaman equation. The equation was solved by the perturbation method, and the resonance behavior of the shell was studied. Penna et al.⁷ surveyed the nonlinear free vibration of functionally graded nanobeams in the bending process and investigated the closed-form analytical solutions of the nonlinear natural deflection of the beams with different support types, which provided a theoretical basis for the design of functional beams. Xue et al.⁸ examined the dynamic behavior of Mindlin sheets containing cracks. The Ritz method of special allowable functions and the Galerkin discretization method were used to solve the equations, and the nonlinear dynamic response and in-plane preload parameters of plates with various cracks under transverse harmonic excitation were given. Gholami and Ansari⁹ focused on the nonlinear free vibration of functionally graded microplates. With the help of the generalized differential quadrature method and numerical Galerkin scheme, the influence of the material gradient index and different boundary conditions on the nonlinear free vibration of microplates was studied. Vahidi-Moghaddam et al.¹⁰ explored the nonlinear forced vibration of homogeneous Euler–Bernoulli beams with boundary conditions. Combining nonlocal strain gradient theory, frequency response diagrams were obtained for primary, superharmonic, and subharmonic resonances. The simulation results portray the vibration characteristics of the microbeam. Shan et al.¹¹ analyzed the nonlinear bending and nonlinear free vibration of FG nanobeams and obtained the nonlinear bending approximate solution and the free vibration solution to the model under fixed boundary conditions by the two-step perturbation method. Gao and Shen¹² discussed the use of the virtual incremental variational principle and Lagrangian method to establish geometrically nonlinear finite element equations for piezoelectric smart structures. The geometrically nonlinear transient vibration response was studied. Chen et al.^13,14 studied the nonlinear dynamics of axially moving strings and obtained nonlinear parametric vibration, overall bifurcation, and chaotic behavior. Bakhtiari-Nejad and Nazari¹⁵ computed the nonlinear vibration of viscoelastic laminates. Using the multiscale method and finite difference method, dimensionless nonlinear motion equations were analyzed and solved. The nonlinear natural frequency and modal shape were acquired. Zhang et al.¹⁶ equated rotating blades to rotating cantilevered rectangular plates and studied complex nonlinear vibration and internal resonance problems, and a diagram that characterizes the nonlinear behavior of the moving cantilever plate was obtained. Zhang et al.^17,18 probed the chaotic dynamics of the beam and rectangular plate under lateral and planar excitation. Based on the perturbation method, the resonance torus was obtained. Soni et al.¹⁹ considered the lateral vibration of anisotropic plates in a thermal environment, which could be solved by converting the lateral deflection of the modal function. The law showing that the frequency was affected by the crack length and temperature was obtained. Luo et al.²⁰ studied a nonlinear energy harvester. Wang and Zhang²¹ studied the bending and buckling of 3D graphene foam thin plates. The governing equation was derived using Hamilton’s principle, and the Navier method was employed to find the analytical solution for plate bending. Ding et al.^22,23 analyzed the influence of the free vibration characteristics of a moving system. Based on the generalized Hamiltonian principle, the integral partial differential nonlinear equation of the system was established. The effects of critical speed and vibration frequency were given. Razzak et al.²⁴ used multiple timescale methods to solve strongly nonlinear forced vibration systems. The results showed that the method in the thesis is not only effective for weakly nonlinear damped forced systems but also better for strong nonlinear systems with small but strong damping effects. Shao et al.²⁵ investigated the vibration behavior of webs, employing the Galerkin method to discretize the control equation and using the fourth-order Runge–Kutta technique for computation, ultimately identifying the stable operating range for the moving web.

This paper investigates the nonlinear primary resonance and chaotic behavior of a moving flexible graphene electron membrane, considering factors such as density coefficient, velocity, external damping, excitation force, and harmonic solution parameters using energy method principles. The motion is modeled with a discrete differential equation derived via the Bubnov–Galerkin method and Hamiltonian principle. The effects of these parameters on the nonlinear primary resonance and chaos are analyzed using the multiscale algorithm.

II. DYNAMICAL MODEL

Figure 1 presents the dynamical model of a flexible graphene electron membrane’s motion when stimulated externally. The electron membrane has no bending stiffness or shear stress. Among these parameters, U₁ denotes the speed of the membrane along the x-axis, while the internal and external diameters of the electron membrane are represented by a and b, respectively; the electron membrane thickness is δ; the areal density per unit area of the graphene electronic membrane is ρ₀; $\bar{p} \cos (ω t)$ signifies the external motivating force directed into the electron membrane along the z-axis; the rectangular graphene electronic membrane carrying the coil is stretched to T_x and T_y along the x-axis and y-axis directions, respectively; the tensions per unit length of the membrane along the ρ axis direction are q_a and q_b, respectively; and the lateral vibration displacement of the moving flexible electronic membrane is $w (ρ, φ, t)$ ⁠.

FIG. 1.

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Mechanical model of the moving flexible graphene electron membrane. (a) Sketch of the moving membrane with variable density. (b) Flexible intelligent graphene electronic membrane printing machine.

According to von Karman theory, the geometric equation of the axially transported membrane is formulated as

\{\begin{cases} {\bar{ε}}_{r} = \frac{\partial u}{\partial r} + \frac{1}{2} \cdot {(\frac{\partial w}{\partial r})}^{2} - z \cdot [\frac{\partial^{2} w}{\partial^{2} r} - \frac{\sin (2 φ)}{r}, \cdot \frac{\partial^{2} w}{\partial r \cdot \partial φ} + \frac{\sin^{2} (φ)}{r} \cdot \frac{\partial w}{\partial r} + \frac{\sin (2 φ)}{r^{2}} \cdot \frac{\partial w}{\partial φ} + \frac{1 - 2 \cos^{2} (φ)}{2 r^{2}} \cdot \frac{\partial^{2} w}{\partial φ^{2}}], \\ {\bar{ε}}_{φ} = \frac{1}{r} \cdot (u + \frac{\partial v}{\partial φ}) + \frac{1}{2} \cdot {[\sin (φ) \cdot \frac{\partial w}{\partial r} + \frac{1}{r} \cdot \frac{\partial w}{\partial φ}]}^{2} - z \cdot [\sin^{2} (φ) \cdot \frac{\partial^{2} w}{\partial r^{2}}, + \frac{\cos^{2} (φ)}{r} \cdot \frac{\partial w}{\partial r} + \frac{\cos^{2} (φ)}{r^{2}} \cdot \frac{\partial^{2} w}{\partial φ^{2}}], \\ {\bar{γ}}_{r φ} = \frac{1}{r} \cdot \frac{\partial u}{\partial φ} + \frac{\partial v}{\partial r} - \frac{v}{r} + \frac{\partial w}{\partial r} \cdot (\frac{1}{r} \cdot \frac{\partial w}{\partial φ}) - z \cdot [\frac{2 \cos (2 φ)}{r} \cdot \frac{\partial^{2} w}{\partial r \partial φ} - \sin (2 φ) \frac{\partial w}{\partial r} - \frac{2 \cos (2 φ)}{r^{2}} \cdot \frac{\partial w}{\partial φ}], \end{cases}

(1)

where w is the vibration displacement; u and v are the midplane displacements along the X- and Y-axes, respectively;

{\bar{ε}}_{r}

⁠,

{\bar{ε}}_{φ}

⁠, and

{\bar{γ}}_{r φ}

are the polar coordinate strains along the hoop and radial directions, respectively; and z is the horizontal coordinate point.

The transformation of Eq. is as follows:

{\bar{ε}}_{r} - \frac{\partial (r \cdot {\bar{ε}}_{φ})}{\partial r} - 0.5 \cdot {(\frac{\partial w}{\partial r})}^{2} = 0 .

(2)

The equilibrium differential equation for the motion of the flexible graphene electronic membrane in polar coordinates is expressed as

\{\begin{cases} \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{cases}

(3)

Omitting the physical force, for symmetric problems, there are f_r = f_φ = 0 and τ_rφ = 0, and the internal forces are δσ_r = F_Tr and δσ_φ = F_Tφ.

The compatible equation obtained by substituting Eq. into Eq. is as follows:

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

(4)

Since the bending stiffness of the membrane is very small and can be ignored, only the middle plane strain energy is considered when establishing the deformation potential energy,

\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}

(5)

where A remains constant, with φ_A and φ_B expressed in radians.

The equation describing the electron membrane in polar coordinates is presented as

\{\begin{cases} F_{T r} = \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{cases}

(6)

Substituting Eq. into Eq. gives the following results:

U_{ε_{2}} = \frac{π E δ}{1 - v^{2}} \int ({\bar{ε}}_{r}^{2} + {\bar{ε}}_{φ}^{2} + 2 v \cdot {\bar{ε}}_{r} \cdot {\bar{ε}}_{φ}) \cdot r d r .

(7)

Substituting geometric equation into Eq. yields

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

(8)

The total kinetic energy is

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

(9)

where Ω_f is the area of the electron membrane, w is the lateral deflection, and v is Poisson’s ratio. Using the stationary value method to solve the above energy equation with a boundary, another functional form can be obtained,

δ Π = δ \int_{t_{3}}^{t_{4}} (T - U_{p} + Π_{Γ}) d t = 0 .

(10)

The variation in the density of the membrane is illustrated in Fig. 2, where α denotes the density coefficient and $ρ (r)$ represents the density function.

FIG. 2.

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Density variations in the moving graphene electronic membrane.

The density variation function

ρ (r)

of the membrane in motion is given by

\{\begin{cases} \frac{ρ_{0}}{b - a} \cdot [(b - a) + 2 α (r - a)], a \leq r \leq \frac{b + a}{2}, \\ \frac{ρ_{0}}{b - a} \cdot [- a + α (- 2 r + b)], \frac{b + a}{2} \leq r \leq b . \end{cases}

(11)

To normalize the piecewise density function, it can be expressed as

ρ_{0} (1 + α) - 2 α ρ_{0} \cdot |\frac{r - a}{b - a} - \frac{1}{2}| .

(12)

The energy functional of Eqs. and is simplified by using Green’s equation,

\begin{gathered} \iint_{Ω} (\frac{\partial X}{\partial r} + \frac{\partial Y}{\partial φ}) d x d y = \oint [X \cos (n, r) + Y \cos (n, φ)] d s . \end{gathered}

(13)

In Eq. (13), (n, r) and (n, φ) represent the angle between the outer normal n of the edge of the moving flexible graphene electronic membrane and the r-axis and φ -axis, respectively.

Using the D’Alembert principle to substitute Eqs. – into Eqs. and , the nonlinear vibration equations for the moving membrane with varying densities are as follows:

\{\begin{cases} - ρ (r) \cdot δ \cdot \{\frac{\partial^{2} w}{\partial t^{2}} + 2 U_{1} \cdot \cos (φ) \cdot \frac{\partial^{2} w}{\partial r \partial t} - 2 U_{1} \cdot \frac{\sin (φ)}{r} \cdot \frac{\partial^{2} w}{\partial φ \partial t} \\ + U_{1}^{2} \cdot [\cos^{2} (φ) \cdot \frac{\partial^{2} w}{\partial r^{2}} - \frac{\sin (2 φ)}{r} \cdot \frac{\partial^{2} w}{\partial r \partial φ} + \frac{\sin^{2} (φ)}{r} \\ \cdot \frac{\partial w}{\partial r} + \frac{\sin (2 φ)}{r^{2}} \cdot \frac{\partial w}{\partial φ} + \frac{\sin^{2} (φ)}{r^{2}} \cdot \frac{\partial^{2} w}{\partial φ^{2}}]\} + D_{0} \cdot \frac{1}{r} \\ \cdot \frac{d}{d r} \{r \cdot \frac{d [\frac{1}{r} \cdot \frac{d (r \cdot \frac{d w}{d r})}{d r}]}{d r}\} - \frac{1}{r} \cdot \frac{d (r \cdot F_{T r} \cdot \frac{d w}{d r})}{d r} + ϑ \cdot \frac{\partial w}{\partial t} \\ = \bar{p} \cdot \cos (ω t), r^{2} \cdot \frac{\partial^{2} F_{T r}}{\partial r^{2}} + 3 r \cdot \frac{\partial F_{T r}}{\partial r} + 0.5 E δ \cdot {(\frac{\partial w}{\partial r})}^{2} = 0 . \end{cases}

(14)

Introducing the dimensionless quantity into Eq. gives the following:

\begin{gathered} ς = \frac{r}{a}, χ = \frac{a}{b}, \bar{w} = \frac{w}{δ}, Ω = ω \cdot \sqrt{\frac{ρ_{0} \cdot a^{4}}{E \cdot δ^{3}}}, τ = t \cdot \sqrt{\frac{E \cdot δ^{3}}{ρ_{0} \cdot a^{4}}}, \\ c = U_{1} \cdot \sqrt{\frac{ρ_{0} \cdot a^{2}}{E \cdot δ^{3}}}, ℘ = ϑ \cdot \sqrt{\frac{a^{4}}{ρ_{0} \cdot E \cdot δ^{3}}}, f = \frac{ρ_{0} \cdot b^{2}}{E \cdot δ^{3}} \cdot F_{T r}, \\ p = \bar{p} \cdot \frac{a^{4}}{E \cdot δ^{4}} . \end{gathered}

(15)

Incorporating the dimensionlessness into Eq. (14)

In Eq. , when D₀ = 0, its dimensionless form can be expressed as follows:

\begin{align} [(1 + α) - 2 α |\frac{ς - 1}{\frac{1}{χ} - 1} - \frac{1}{2}|] \cdot (\frac{\partial^{2} \bar{w}}{\partial τ^{2}} + 2 c \cdot \frac{\partial^{2} \bar{w}}{\partial ς \partial τ} + c^{2} \cdot \frac{\partial^{2} \bar{w}}{\partial ς^{2}}) - χ^{2} \cdot f \\ \cdot \frac{d^{2} \bar{w}}{d ς^{2}} - χ^{2} \cdot f \cdot \frac{d \bar{w}}{d ς} \cdot \frac{1}{ς} - χ^{2} \cdot \frac{d \bar{w}}{d ς} \cdot \frac{d f}{d ς} + ℘ \cdot \frac{\partial \bar{w}}{\partial τ} = p \cdot \cos (Ω t), \end{align}

(16a)

ς^{2} \cdot \frac{\partial^{2} f}{\partial ς^{2}} + 3 ς \cdot \frac{\partial f}{\partial ς} + \frac{1}{2 \cdot r^{2}} \cdot {(\frac{d \bar{w}}{d ς})}^{2} = 0,

(16b)

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} .

(16c)

The boundary conditions for the dimensionless nonlinear vibration of an axially transported membrane are articulated as

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

(17)

A. Bubnov–Galerkin method and mode assumption

The nonlinear large deflection test function and internal force function are assumed to satisfy the boundary value and compatibility equations, respectively, and are expressed as follows:

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

(18)

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

(19)

where

\bar{w} (ς, φ, τ)

represents the lateral large deflection test function;

f (ς, φ, τ)

represents the internal force function; and C_i(t) is a function of time.

Combining Eqs. , , and , the Bubnov–Galerkin algorithm is utilized to obtain the following:

\begin{align} \iint_{S} \{[(1 + α) - 2 α |\frac{ς - 1}{\frac{1}{χ} - 1} - \frac{1}{2}|] \cdot (\frac{\partial^{2} \bar{w}}{\partial τ^{2}} + 2 c \cdot \frac{\partial^{2} \bar{w}}{\partial ς \partial τ} + c^{2} \cdot \frac{\partial^{2} \bar{w}}{\partial ς^{2}}) \\ - χ^{2} \cdot f \cdot \frac{d^{2} \bar{w}}{d ς^{2}} - χ^{2} \cdot f \cdot \frac{d \bar{w}}{d ς} \cdot \frac{1}{ς} - χ^{2} \cdot \frac{d \bar{w}}{d ς} \cdot \frac{d f}{d ς} + ℘ \cdot \frac{\partial \bar{w}}{\partial τ}\} \\ \cdot W (ς, φ) d s = \iint_{s} p \cos (Ω τ) W (ς, φ) d s . \end{align}

(20)

Simplifying Eq. gives the nonlinear forced vibration equation for a moving membrane with varying densities as follows:

m \cdot {C_{i}}^{''} + μ \cdot {C_{i}}^{'} + K \cdot C_{i} - Θ \cdot {C_{i}}^{3} = F \cdot \cos (Ω τ),

(21)

where the coefficient of Eq. is expressed as follows:

\{\begin{cases} m = \sum_{i = 1}^{n} \iint_{s} [(1 + α) - 2 α |\frac{ς - 1}{\frac{1}{χ} - 1} - \frac{1}{2}|] \cdot W_{i}^{2} (ς, φ) d s, \\ μ = [(1 + α) - 2 α |\frac{ς - 1}{\frac{1}{χ} - 1} - \frac{1}{2}|] \cdot \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} ℘ \cdot W_{i}^{2} (ς, φ) d s, \\ K = \sum_{i = 1}^{n} \iint_{s} c^{2} \cdot [(1 + α) - 2 α |\frac{ς - 1}{\frac{1}{χ} - 1} - \frac{1}{2}|] \cdot \frac{\partial^{2} W_{i}}{\partial ς^{2}} \cdot W_{i} (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, \\ F = \sum_{i = 1}^{n} \iint_{s} p \cdot W_{i} (ς, φ) d s . \end{cases}

(22)

B. Nonlinear parameter vibration

Transforming the nonlinear vibration equation for the moving membrane with variable densities results in the following:

\ddot{I} + 2 ξ \dot{I} + B I + ψ I^{3} = F_{1} \cos (Ω τ),

(23)

where I is the time function, τ is dimensionless time, ξ is the damping ratio, ψ is the cubic stiffness coefficient, F₁ represents the external excitation force, Ω denotes the angular frequency, and B indicates the stiffness.

Using the multiscale algorithm, a small parameter, ɛ (0 < ɛ ≪ 1), and a detuning parameter σ are introduced to describe the degree of closeness to the natural frequency of the system, and the parameters of Eq. are processed as follows:

Ω = \frac{ω}{ω_{0}} = 1 + ε \cdot σ, ξ = ε \cdot \bar{ξ}, ψ = ε \cdot \bar{ψ}, F_{1} = ε \cdot \bar{F_{1}},

(24)

where ω₀ is the natural frequency of the system; then, Eq. can be changed to

\ddot{I} + 2 ε \bar{ξ} \cdot \dot{I} + B \cdot I + ε \bar{ψ} \cdot I^{3} = ε \bar{F_{1}} \cos ((1 + ε \cdot σ) \cdot τ) .

(25)

The scale factors T_n = ɛⁿ τ and

D_{n} = \frac{\partial}{\partial T_{n}}

are introduced and defined as follows:

\{\begin{cases} \frac{\partial}{\partial τ} = D_{0} + ε \cdot D_{1} + \dots, \\ \frac{\partial^{2}}{\partial τ^{2}} = D_{0}^{2} + 2 ε D_{0} D_{1} + \dots . \end{cases}

(26)

Supposing the solution to Eq. and substituting Eq. into Eq. yields the following:

\begin{gathered} (D_{0}^{2} + 2 ε D_{0} D_{1}) \cdot (u_{0} (T_{0}, T_{1}) + ε \cdot u_{1} (T_{0}, T_{1})) + 2 ε \bar{ξ} \cdot (D_{0} + ε D_{1}) \\ \cdot (u_{0} (T_{0}, T_{1}) + ε \cdot u_{1} (T_{0}, T_{1})) + B \cdot (u_{0} (T_{0}, T_{1}) + ε \cdot u_{1} (T_{0}, T_{1})) \\ + ε \cdot \bar{ψ} \cdot {(u_{0} (T_{0}, T_{1}) + ε \cdot u_{1} (T_{0}, T_{1}))}^{3} = ε \cdot \bar{F_{1}} \cos (T_{0} + σ T_{1}) . \end{gathered}

(27)

Assuming that the coefficients of ɛ⁰ and ɛ¹ are zero and ignoring the higher-order terms of ɛ, Eq. can be obtained as follows:

\{\begin{cases} D_{0}^{2} u_{0} (T_{0}, T_{1}) + B \cdot u_{0} (T_{0}, T_{1}) = 0, \\ D_{1}^{2} u_{1} (T_{0}, T_{1}) + B \cdot u_{1} (T_{0}, T_{1}) = - 2 D_{0} D_{1} \cdot u_{0} (T_{0}, T_{1}), \\ - 2 \bar{ξ} D_{0} \cdot u_{0} (T_{0}, T_{1}) - \bar{ψ} \cdot u_{0}^{3} (T_{0}, T_{1}) + \bar{F_{1}} \cos (T_{0} + σ T_{1}) . \end{cases}

(28)

The general solution

u_{0} (T_{0}, T_{1})

of Eq. is as follows:

u_{0} = [A (T_{1}) \cdot \exp (i \cdot T_{0}) + A^{*} (T_{1}) \cdot \exp (i \cdot T_{0})] \cdot B,

(29)

where A is a complex number and A* is the conjugate of A. Substituting Eq. into Eq. and eliminating the long term (cc), Eq. becomes

\begin{gathered} 2 i B \cdot A^{'} (T_{1}) + 2 i B \bar{ξ} \cdot A (T_{1}) + 3 B^{3} A^{2} (T_{1}) \cdot A^{*} (T_{1}) \\ \bar{ψ} - \frac{\bar{F_{1}}}{2} \exp (i σ T_{1}) + c c = 0 . \end{gathered}

(30)

Expressing A(T₁) as a series gives

\{\begin{cases} A (T_{1}) = \frac{1}{2} a (T_{1}) \cdot \exp [i \cdot β (T_{1})], \\ A^{'} (T_{1}) = \frac{1}{2} \{a^{'} (T_{1}) \cdot \exp [i \cdot β (T_{1})] + a (T_{1}) \\ \cdot \exp [i \cdot β (T_{1})] \cdot i \cdot β^{'} (T_{1})\} . \end{cases}

(31)

Substituting Eq. into Eq. gives

\begin{gathered} i B a^{'} - a B β^{'} + i B \bar{ξ} a + \frac{3 B^{3}}{8} \bar{ψ} \cdot a^{3} \\ - \frac{1}{2} \cdot \bar{F_{1}} [\cos (σ T_{1} - β) + i \sin (σ T_{1} - β)] = 0 . \end{gathered}

(32)

The real and imaginary parts of Eq. are

a^{'} = - \bar{ξ} a + \frac{1}{2} \cdot \bar{F_{1}} \cdot \frac{1}{B} \sin (σ T_{1} - β),

(33a)

a β^{'} = \frac{3 B^{2}}{8} \bar{ψ} a^{3} - \frac{1}{2} \cdot \bar{F_{1}} \cdot \frac{1}{B} \cos (σ T_{1} - β) .

(33b)

In these equations, a(T₁) and β(T₁) are real numbers, introducing the following relation form:

ϕ = σ T_{1} - β (T_{1}),

(34a)

ϕ^{'} = σ - β^{'} (T_{1}) .

(34b)

Equation is turned into an autonomous system, Eq. is substituted into Eq. , and then, Eq. is used to obtain the following:

a^{'} = - \bar{ξ} a + \frac{1}{2} \bar{F_{1}} \cdot \frac{1}{B} \sin (ϕ),

(35a)

a ϕ^{'} = σ a - \frac{3 B^{2}}{8} \bar{ψ} a^{3} + \frac{1}{2} \bar{F_{1}} \cdot \frac{1}{B} \cos (ϕ) .

(35b)

Considering the stable operation of the system, a′ = 0 and ϕ′ = 0; then, Eq. can be further simplified as

\bar{ξ} a = + \frac{1}{2} \bar{F_{1}} \cdot \frac{1}{B} \sin (ϕ),

(36a)

- σ a + \frac{3 B^{2}}{8} \bar{ψ} a^{3} = + \frac{1}{2} \bar{F_{1}} \cdot \frac{1}{B} \cos (ϕ) .

(36b)

Here, Eq. needs to be squared and superimposed on each other to obtain

{(\bar{ξ} a)}^{2} + {(σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2})}^{2} \cdot a^{2} = \frac{1}{4} \cdot {\bar{F_{1}}}^{2} \cdot \frac{1}{B^{2}} .

(37)

Using Eq. to solve Eq. , the frequency response equation of Ω with a is

{(\bar{ξ} a)}^{2} + {(σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2})}^{2} \cdot a^{2} = \frac{1}{4} \cdot {\bar{F_{1}}}^{2} \cdot \frac{1}{B^{2}} .

(38)

According to Eq. , for a certain amplitude, it corresponds to two frequencies, Ω_1,2. However, these two frequencies may be unstable. Therefore, it is necessary to analyze the stability of the system dynamics equation. Consider the following relationships:

\{\begin{cases} {\dot{X}}_{1} = f_{1} (x_{1}, x_{2}), \\ {\dot{X}}_{2} = f_{2} (x_{1}, x_{2}) . \end{cases}

(39)

According to Eq. , the relationship of the Jacobian matrix can be expressed as follows:

[\begin{matrix} \frac{\partial f_{1}}{\partial x_{1}} & \frac{\partial f_{1}}{\partial x_{2}} \\ \frac{\partial f_{2}}{\partial x_{1}} & \frac{\partial f_{2}}{\partial x_{2}} \end{matrix}] .

(40)

Equation is the control equation of the moving graphene electron membrane with different densities, and its state variables are a and ϕ. Using Eq. instead of Eq. , the following results can be obtained:

\{\begin{cases} f_{1} (a, ϕ) = - \bar{ξ} a + \frac{1}{2} \bar{F_{1}} \cdot \frac{1}{B} \sin (ϕ), \\ f_{2} (a, ϕ) = σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2} + \frac{1}{2} \bar{F_{1}} \cdot \frac{1}{B a} \cos (ϕ) . \end{cases}

(41)

Therefore, according to Eq. , the Jacobian matrix of the system is

J = [\begin{matrix} - \bar{ξ} & - a (σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2}) \\ \frac{1}{a} (σ - \frac{9 B^{2}}{8} \bar{ψ} a^{2}) & - \bar{ξ} \end{matrix}] .

(42)

The stable operation of the system depends on the eigenvalues of the Jacobian matrix, and the characteristic equation of Eq. is

|\begin{matrix} - \bar{ξ} - λ & - a (σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2}) \\ \frac{1}{a} (σ - \frac{9 B^{2}}{8} \bar{ψ} a^{2}) & - \bar{ξ} - λ \end{matrix}| .

(43)

Expanding the determinant of Eq. gives

λ^{2} + 2 \bar{ξ} λ + {\bar{ξ}}^{2} + (σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2}) (σ - \frac{9 B^{2}}{8} \bar{ψ} a^{2}) = 0 .

(44)

Therefore, the eigenvalue of the system is

λ_{1, 2} = - \bar{ξ} \pm \sqrt{- (σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2}) (σ - \frac{9 B^{2}}{8} \bar{ψ} a^{2})} .

(45)

According to the Routh criterion, the stable motion of the system should satisfy the following:

(σ - \frac{3 B^{2}}{8} \bar{ψ} a^{2}) (σ - \frac{9 B^{2}}{8} \bar{ψ} a^{2}) + {\bar{ξ}}^{2} > 0,

(46a)

σ = \frac{Ω - 1}{ε} .

(46b)

For a damped system, the sum of the eigenvalues of the Jacobian matrix needs to be negative, and then, it can be determined that the eigenvalue contains at least one negative number; then, the fixed point (a, ϕ) is a stable node.

III. RESULT ANALYSIS AND NUMERICAL COMPARISON

The nonlinear amplitude–frequency response shown in Eq. (38) and Routh’s stability criterion inequality shown in Eq. (46) for a flexible moving graphene electronic membrane include the dimensionless external excitation force, dimensionless velocity, ratio of the inner and outer diameters, and dimensionless damping parameter. A multiscale algorithm and fourth-order Runge–Kutta technique are used for numerical computation, and the nonlinear primary resonance properties and chaos are analyzed.

A. The feasibility of analyzing parametric vibration by the multiscale algorithm with a variable density coefficient

With the basic parameter set, the dimensionless velocity (c = 0.3), Poisson’s ratio (v = 0.15), internal to external diameter ratio (χ = 0.5), frequency (Ω = 1), initial value (⁠ $[0.01, 0]$ ⁠), external excitation (F₁ = 1), dimensionless damping (ξ = 0.1), and density coefficients (α) are 0.1 and 0.5. The nonlinear amplitude–frequency response curves (a − Ω) and global bifurcation graphs of the moving flexible graphene electron membrane were acquired by using the fourth-order Runge–Kutta technique and the multiscale algorithm, as shown in Figs. 3 and 4.

FIG. 3.

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Nonlinear amplitude–frequency response charts of the moving flexible graphene electron membrane. (a) Global bifurcation graph obtained by the fourth-order Runge–Kutta technique with a density coefficient of 0.1. (b) Global bifurcation graph obtained by the multiscale algorithm with a density coefficient of 0.1. (c) Global bifurcation graph obtained by the fourth-order Runge–Kutta technique with a density coefficient of 0.5. (d) Global bifurcation graph obtained by the multiscale algorithm with a density coefficient of 0.5.

FIG. 4.

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Bifurcation diagrams for nonlinear parametric dynamics in a moving graphene electron membrane. (a) Global bifurcation graphs with a density coefficient of 0.1. (b) Global bifurcation graphs with a density coefficient of 0.5.

Figure 3 shows the numerical results of the nonlinear amplitude–frequency response based on the multiscale algorithm and fourth-order Runge–Kutta technique. It can be seen that (1) as the density coefficient increases, the curvature of the a − Ω curve decreases, which illustrates that the increase in the density coefficient can reduce the strength of the nonlinear term of the electron membrane. At the same time, the nonlinear term can reduce the amplitude of the amplitude–frequency response. However, the excitation frequency ratio Ω will increase correspondingly. (2) The solution to the nonlinear amplitude–frequency response of the moving flexible graphene electronic membrane has multiple value regions and contains the distinction between the stable solution and the unstable solution, and the amplitude–frequency response has a jumping phenomenon. (3) When the density coefficient is 0.1, the a − Ω curve obtained by the fourth-order Runge–Kutta technique exhibits unstable vibration near the maximum value and then quickly jumps to a smaller amplitude region. When the external excitation force is constant, the multiscale algorithm is adopted. As the frequency ratio Ω increases, the path of amplitude a is 1 → 2 → 3 → 4 → 5, and 3 is the jump point. In this process, the amplitude–frequency curve (a − Ω) has multiple solutions. In contrast, the frequency ratio Ω changes from large to small values, the path of amplitude a is 5 → 4 → 6 → 2 → 1, and 2 is the jump point. When the density coefficient is 0.5, the multiscale algorithm is used for calculations, and the path for a density coefficient of 0.5 is the same as the path for a density coefficient of 0.1. During this process, the amplitude–frequency response curve obtained by the multiscale algorithm will not change, but, with the fourth-order Runge–Kutta technique, when the maximum value is exceeded, it will diverge. Figure 4 presents the overall bifurcation diagram of the inverse frequency sweep of the moving flexible graphene electronic membrane when the density coefficients are 0.1 and 0.5. Compared with Fig. 3, it can be proven that the nonlinear parametric vibration curve obtained by the multiscale algorithm conforms to the global bifurcation graphs. In addition, the a − Ω curve exhibits a jumping phenomenon, and the jumping threshold is different when the density coefficient is different.

When addressing the nonlinear parametric vibration issue, the multiscale algorithm exhibits greater stability compared to the fourth-order Runge–Kutta method.

B. The influence of the density coefficient on nonlinear primary resonance and chaos

When altering only one fundamental parameter, the others remain constant: the dimensionless speed as c = 0.3, Poisson’s ratio set to v = 0.15, the internal to external diameter ratio as χ = 0.5, the external motivation as F₁ = 1, the dimensionless damping coefficient as ξ = 0.1, the detuning parameter as ɛσ = 0, the frequency as Ω = 1, and the original number as $[0.01, 0]$ ⁠.

On the basis of the characteristic curve in Fig. 5, we obtain the following:

The nonlinear primary resonance amplitude response of the system is more sensitive to the external excitation force. The larger the external excitation force is, the larger the amplitude of the primary resonance. Under different external excitation forces, as the density coefficient increases, the primary resonance amplitude increases slowly. When the density coefficient reaches a certain value, inflection spots will appear (the inflection point of F₁ = 1 is α = 0.74, the inflection point of F₁ = 3 is α = 0.56, and the inflection point of F₁ = 5 is α = 0.8). At this time, if the density coefficient increases, the primary resonance of the system is at a stable amplitude value. The results indicate that reducing the external disturbance excitation force and choosing a certain density coefficient can control the nonlinear primary resonance amplitude value of the flexible graphene electronic membrane in motion.
The primary resonance amplitude a of the system depends on the axial dimensionless velocity. As the dimensionless velocity increases, the system will exhibit the phenomenon in which the amplitude decreases, and it will jump repeatedly within a certain density coefficient range. The system solution is a multivalued solution, with at most three solutions and at least one solution. When c = 0, as the density coefficient increases, the main resonance amplitude of the system gradually increases. There is a corner at α = 0.37, and the amplitude of the main resonance does not increase until α = 0.53. The system exhibits a bifurcation jump, divergence, and instability.

FIG. 5.

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Density coefficient and amplitude characteristic curve of the moving membrane. (a) Various external excitation forces (ξ = 0.1, c = 0.3). (b) Various dimensionless speeds (ξ = 0.1, F₁ = 1). (c) Various detuning parameters (c = 0.3, ξ = 0.1, F₁ = 1). (d) Various damping speeds (c = 0.3, F₁ = 1).

When c = 1, the primary resonance amplitude of the system increases with increasing density coefficient, but the primary resonance amplitude is always a single value that is stable and slowly increasing. When c = 1.5, the increase in the density coefficient has little effect on the amplitude of the primary resonance. As the density coefficient continues to increase until α = 0.12 and α = 0.57, static bifurcation occurs, and the amplitude of the primary resonance sharply increases and decreases, respectively.

The global bifurcation graphs of the density coefficient and displacement in Fig. 6 show that when the system is within the range of the nonlinear primary resonance density coefficients (0, 0.8), the electron membrane is in a chaotic movement. To further study the chaotic process, the density coefficients α = 0 and α = 0.4 and the inflection point α = 0.74 of the nonlinear primary resonance were selected, and the Poincaré map, time history diagram, phase portrait, and power spectrum were obtained for these three points. The results are shown in Figs. 7–9. When parameter α = 0 is set, three fixed points appear on the Poincaré map [Fig. 7(b)], the phase portrait [Fig. 7(c)] forms a closed-loop, and the power spectrum [Fig. 7(d)] displays a discontinuous spectrum, indicating a periodic motion in the flexible graphene electron membrane. Conversely, with parameter α = 0.4, the Poincaré map [Fig. 8(b)] shows a dense distribution of points, the phase portrait [Fig. 8(c)] turns into a fragmented curve, and the power spectrum [Fig. 8(d)] becomes continuous, suggesting a chaotic movement of the membrane. When the density coefficient is at the inflection point α = 0.74 of the nonlinear primary resonance, the Poincaré map [Fig. 9(b)] is distributed with unclosed dense points, the trajectory of phase portrait curve [Fig. 9(c)] is an irregular unbroken chart, the power spectrum [Fig. 9(d)] is a discretized spike, and the membrane is in a state of chaos. In conclusion, the system has experienced a path changing from periodic to chaotic.

FIG. 6.

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Global bifurcation graphs of the displacement and density coefficient.

FIG. 7.

Nonlinear vibration characteristic graphs when the density coefficient is 0. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.3, F1 = 1, α = 0, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graphs when the density coefficient is 0. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.3, F₁ = 1, α = 0, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

FIG. 8.

Nonlinear vibration characteristic graphs when the density coefficient is 0.05. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.3, F1 = 1, α = 0.4, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graphs when the density coefficient is 0.05. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.3, F₁ = 1, α = 0.4, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

FIG. 9.

Nonlinear vibration characteristic graphs when the density coefficient is 0.74. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.3, F1 = 1, α = 0.74, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graphs when the density coefficient is 0.74. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.3, F₁ = 1, α = 0.74, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

C. Influence of dimensionless velocity on nonlinear primary resonance and chaos

Consider a flexible graphene electron membrane with the following physical parameters: damping coefficient of 0.1, Poisson’s ratio of 0.15, dimensionless angular frequency of 1, external driving force of 1, internal to external diameter ratio of 0.5, density coefficient of 0.1, and initial values labeled as $[0.01, 0]$ ⁠. Under these conditions, the characteristic curve of the membrane is depicted in Fig. 10.

FIG. 10.

Dimensionless velocity and characteristic amplitude curve of the moving membrane. (a) Different density coefficients (ξ = 0.1, F1 = 1). (b) Different external excitation forces (ξ = 0.1, α = 0.1). (c) Different detuning parameters (ξ = 0.1, α = 0.1, F1 = 1). (d) Different damping ratios (α = 0.1, F1 = 1).

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Dimensionless velocity and characteristic amplitude curve of the moving membrane. (a) Different density coefficients (ξ = 0.1, F₁ = 1). (b) Different external excitation forces (ξ = 0.1, α = 0.1). (c) Different detuning parameters (ξ = 0.1, α = 0.1, F₁ = 1). (d) Different damping ratios (α = 0.1, F₁ = 1).

It can be concluded from Fig. 10 that (1) the nonlinear primary resonance amplitude is symmetrical to c = 0. When αρ is a certain value, the system will exhibit static bifurcation; the smaller the αρ is, the more static bifurcations there are. As the density coefficient increases, the primary resonance area of the system becomes narrower, while the amplitude of the primary resonance increases. When the dimensionless velocity increases, the system jumps, and the amplitude of the main resonance undergoes a sudden change. When αρ = 0.1, the dimensionless speed increases within the range of $[0, 1.2]$ ⁠, and the amplitude of the primary resonance is in the slowly increasing area. When the dimensionless velocity changes within the range of $[1.2, 1.4]$ ⁠, the system will exhibit a multivalued area, and the primary resonance amplitude value will jump within this range. When c = 1.4, saddle knot static bifurcation occurs for the first time, and when c = 1.7, saddle knot static bifurcation occurs for the second time. Since the curve of the nonlinear primary amplitude value a is symmetrical about c = 0, the negative velocity changes in the same way as the positive velocity. When αρ = 3, as the dimensionless speed increases, the system will undergo static bifurcation at c = 1.3. (2) With an increasing external excitation force, the primary resonance amplitude also increases (⁠ $a_{, F_{1} = 1} < a_{, F_{1} = 3} < a_{, F_{1} = 5}$ ⁠). As the dimensionless velocity increases, the primary resonance amplitude of the nonlinear motion of the flexible graphene electronic membrane will experience a single value area → bifurcation → a multivalue jumping area → a flat area → a multivalue jumping area → bifurcation → a zone with rapidly decreasing amplitude. (3) As the dimensionless velocity increases, the nonlinear motion of the membrane is influenced by the detuning parameters. The larger the detuning parameter is, the more frequent the multi-value jump phenomenon appears in the dimensionless velocity range.

As shown in Fig. 11, increasing the dimensionless velocity causes the flexible graphene electronic membrane to alternate between cycles and chaos, indicating a transition from periodic to chaotic nonlinear vibration. At 0 ≤ c < 0.45, irregularly distributed global bifurcation points make the membrane unstable. At 0.45 ≤ c < 0.74, the bifurcation graph shows regular points, resulting in stable motion. As dimensionless speed rises, chaotic states widen. At 0.74 ≤ c < 1.3, the bifurcation graph thickens, pushing the membrane into chaos. If the dimensionless velocity continues to increase to 1.3 ≤ c < 2, the membrane returns to a periodic and stable state.

FIG. 11.

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Bifurcation graphs of dimensionless velocity and displacement.

The dynamic behavior of the flexible graphene electronic membrane transitions from chaotic to periodic states, including period-doubling and intermittent chaos. To thoroughly explore the chaotic characteristics, analyses focus on c = 0 and c = 0.7, and the nonlinear primary resonance saddle node of the static bifurcation point c = 1.4, as illustrated in Figs. 12–14. At state c = 0, the Poincaré map shows dense spots, the phase chart displays broken lines, and the power spectrum is continuous, indicating a chaotic movement of the membrane. At state c = 0.7, the membrane exhibits a steady period-doubling behavior. At state c = 1.4, it operates in a quintuple cycle mode.

FIG. 12.

Nonlinear vibration characteristic graphs when the velocity is 0. v1 = 0.15, Ω = 1, χ = 0.5, c = 0, F1 = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graphs when the velocity is 0. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0, F₁ = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

FIG. 13.

Nonlinear vibration characteristic graphs when the velocity is 0.7. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.7, F1 = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graphs when the velocity is 0.7. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.7, F₁ = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

FIG. 14.

Nonlinear vibration characteristic graphs when the velocity is 1.4. v1 = 0.15, Ω = 1, χ = 0.5, c =1.4, F1 = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graphs when the velocity is 1.4. v₁ = 0.15, Ω = 1, χ = 0.5, c =1.4, F₁ = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

D. Influence of damping ratio on nonlinear primary resonance and chaos

Figures 15–19 use the damping ratio as a parameter to acquire the feature diagrams of the moving flexible graphene electron membrane with different damping ratios.

FIG. 15.

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Damping ratio and characteristic amplitude curve of the moving flexible graphene electron membrane. (a) Different density coefficients (c = 0.3, α = 0.1, F₁ = 1). (b) Different dimensionless velocities (ξ = 0.3, α = 0.1, F₁ = 1). (c) Different external excitation forces (ξ = 0.3, α = 0.1, c = 0.3). (d) Different detuning parameters (ξ = 0.3, α = 0.1, c = 0.3, F₁ = 1).

FIG. 16.

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Bifurcation graphs of damping ratio and displacement.

FIG. 17.

Nonlinear vibration characteristic graph when the damping is 0.1. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.3, F1 = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graph when the damping is 0.1. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.3, F₁ = 1, α = 0.1, ξ = 0.1. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

FIG. 18.

Nonlinear vibration characteristic graph when the damping is 0.4. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.3, F1 = 1, α = 0.1, ξ = 0.4. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graph when the damping is 0.4. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.3, F₁ = 1, α = 0.1, ξ = 0.4. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

FIG. 19.

Nonlinear vibration characteristic graph when the damping is 0.9. v1 = 0.15, Ω = 1, χ = 0.5, c = 0.3, F1 = 1, α = 0.1, ξ = 0.9. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

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Nonlinear vibration characteristic graph when the damping is 0.9. v₁ = 0.15, Ω = 1, χ = 0.5, c = 0.3, F₁ = 1, α = 0.1, ξ = 0.9. (a) Time schedule diagram; (b) Poincaré map; (c) phase portrait; (d) power spectrum.

Figure 15 shows that the change in the damping ratio has little effect on the amplitude of the nonlinear primary resonance. (1) When the damping ratio increases, the greater the density coefficient of the system is, the greater the primary resonance amplitude value (a_,αρ=0.1 < a_,αρ=3 < a_,αρ=5); that is, reducing the density coefficient can reduce the nonlinear primary resonance. (2) The value of the nonlinear primary resonance is proportional to the dimensionless velocity (a_,c=1.5 < a_c=0 < a_,c=1). (3) The nonlinear primary resonance amplitude value increases with the increasing external exciting force (⁠ $a_{, F_{1} = 1} < a_{, F_{1} = 3} < a_{, F_{1} = 5}$ ⁠). When ɛσ = 3, the nonlinear primary resonance amplitude value exhibits a multivalue jump phenomenon. This means that within a certain range of damping ratios, appropriately changing the detuning parameters can effectively reduce the value of the nonlinear primary resonance amplitude and improve the stability of the system.

According to Figs. 16–19, when 0 < c < 0.28, the global bifurcation points are irregularly distributed, indicating that the electron membrane is in a divergent and unstable state. When 0.28 ≤ c < 1, the bifurcation graph is distributed with ordered spots, displaying that the electronic membrane moving within the dimensionless velocity range will be in periodic motion. When in state ξ = 0.1, the Poincaré map shows dense dots, the phase-plane portrait appears as a virtual broken line, and the power spectrum is continuous, indicating chaotic motion under this damping condition. In state ξ = 0.4, the Poincaré map shows two fixed points, the phase-plane portrait forms a closed loop, and the power spectrum displays discrete peaks, indicating a two-period cycle. In state ξ = 0.9, the Poincaré map shifts to three fixed points, the phase-plane portrait turns into a closed curve, and the power spectrum continues to show discrete peaks, signifying a transition to three-period motion. In short, when the mobile flexible graphene electronic membrane is transmitted at a dimensionless velocity of 0.3, as the damping coefficient increases, the motion of the system will gradually change from single-period motion to multiperiod motion.

IV. CONCLUSIONS

This paper investigates the nonlinear primary resonance and chaotic characteristics of an axially moving flexible graphene electronic membrane. By employing the displacement function and Bubnov–Galerkin method to separate space–time variables in the vibration equation, the nonlinear vibration differential equation was addressed using the multiscale technique. This study investigates the impact of key parameters—density coefficient, velocity, damping, excitation force, and detuning—on nonlinear primary resonance and chaos.

From the amplitude–frequency curve, it can be concluded that the moving electron membrane has the characteristics of a rigid spring. As the density coefficient increases, the nonlinearity of the flexible electronic membrane will become weaker, and the amplitude effect of the amplitude–frequency response will decrease; however, the excitation frequency ratio will increase accordingly.
Stable and unstable solutions to the nonlinear amplitude–frequency response problem, multivalue regions, and amplitude jump processes of the moving electron membrane were acquired. At the same time, it was proven that in nonlinear primary resonance analysis, the application of the multiscale method is more accurate and effective than the fourth-order Runge–Kutta technique.
Regarding the density coefficient as a variable, when the density coefficient reaches a certain value, an inflection point will appear (the inflection point of F₁ = 1 is α = 0.74, the inflection point of F₁ = 3 is α = 0.56, and the inflection point of F₁ = 5 is α = 0.8). Continuing to increase the density coefficient, the resonance amplitude will not change significantly, and the primary resonance of the system exhibits a stable amplitude, indicating that selecting a certain density coefficient can control the nonlinear primary amplitude of the moving electron membrane. The increase in the damping ratio has little effect on the relationship curve between the density coefficient and primary resonance amplitude, basically maintaining the same trend, and the curve is relatively close. The density coefficient of the system is within the range of (0, 0.8), and the membrane is in a chaotic movement. This means that in the case of a variable density coefficient, the primary resonance value and chaos of the moving electron membrane cannot be controlled at the same time.
Regarding the velocity as a variable, the nonlinear primary resonance amplitude a is symmetrical about c = 0. When αρ is a certain value, the system will exhibit saddle knot static bifurcations. The smaller the αρ is, the more static bifurcations there are. When the dimensionless velocity increases, the system will jump. The primary resonance amplitude first rises rapidly, then drops suddenly and undergoes a process of changing from stable to unstable, which occurs frequently as the velocity increases.
Regarding the damping ratio as a variable: for different density coefficients, dimensionless velocities, external excitation forces, and detuning parameters, the nonlinear main amplitude of the curve of the motion of membrane will not become obvious with increasing damping ratio, which means that increasing the damping ratio cannot effectively reduce the nonlinear primary resonance amplitude of the system. However, appropriately increasing the damping ratio can reduce the possibility of chaotic movement and increase the stability of the system.

ACKNOWLEDGMENTS

This work was supported in part by the National Natural Science Foundation of China (Grant No. 52075435), the Basic Public Welfare Research Program of Zhejiang Province, China (Grant No. LGG21F010002), the Education Department of Zhejiang Province (Grant No. Y202455090), and Huzhou College (Grant No. 2024HXKM05).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Shudi Ying: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Jimei Wu: Conceptualization (equal); Data curation (equal). Zhiduan Cai: Formal analysis (lead). Yuling Wang: Investigation (lead).

DATA AVAILABILITY

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

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© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Chaotic dynamics of flexible graphene electronic membranes with variable density in motion

I. INTRODUCTION

II. DYNAMICAL MODEL

A. Bubnov–Galerkin method and mode assumption

B. Nonlinear parameter vibration

III. RESULT ANALYSIS AND NUMERICAL COMPARISON

A. The feasibility of analyzing parametric vibration by the multiscale algorithm with a variable density coefficient

B. The influence of the density coefficient on nonlinear primary resonance and chaos

C. Influence of dimensionless velocity on nonlinear primary resonance and chaos

D. Influence of damping ratio on nonlinear primary resonance and chaos

IV. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Chaotic dynamics of flexible graphene electronic membranes with variable density in motion Open Access

I. INTRODUCTION

II. DYNAMICAL MODEL

A. Bubnov–Galerkin method and mode assumption

B. Nonlinear parameter vibration

III. RESULT ANALYSIS AND NUMERICAL COMPARISON

A. The feasibility of analyzing parametric vibration by the multiscale algorithm with a variable density coefficient

B. The influence of the density coefficient on nonlinear primary resonance and chaos

C. Influence of dimensionless velocity on nonlinear primary resonance and chaos

D. Influence of damping ratio on nonlinear primary resonance and chaos

IV. CONCLUSIONS

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

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Related Content

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Chaotic dynamics of flexible graphene electronic membranes with variable density in motion