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

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.