Thermal coupling model of a traveling flexible printing electronical membrane subjected to nonlinear electrostatical force

A flexible printing electronical membrane is an electron equipment made by precisely spraying conductive metal ink such as silver on a soft membrane substrate. With its advantages of light weight and flexibility, it can adapt to changing working environments and is widely used in aerospace, wearable electronics and other fields. Nevertheless, during the manufacturing preparation of roll-to-roll printing membranes, the high-speed movement of printing electronical membranes under tension is affected by the impact of hot air from the drying oven and the electrostatic interference generated by friction in transmission, which restricts the overprint accuracy and preparation velocity of flexible electronical membranes. To address this issue, the nonlinear forced vibrational characteristics of a traveling flexible printing electronical membrane on temperature coupling subjected to nonlinear electrostatic force were investigated. The roll-to-roll printed intelligent RFID electron membrane is the research target. On the basis of the energy approach and the heat conduction equation considering the effect of deformation, the nonlinear vibrational equations of an axially traveling flexible printing electronical membrane coupled with temperature under the function of nonlinear electrostatical excitation force were derived. The Bubnov–Galerkin algorithm was applied to discretize the vibration partial differential equations; by making full use of the quartic Runge–Kutta numerical algorithm to calculate the approximate solution of equations, the phase portraits, Poincaré maps, time history diagrams, power spectra, and bifurcation plots of the nonlinear vibrations of the traveling printing electronical membrane were used to explore the effects of movement velocities, electrostatical field, and thermal coupling coefficients. The findings obtained the stable working domain and the divergence instability domain of the traveling flexible printing electronic membrane, which provided a theory fundamental for enhancing the stable craft of a printing electronical membrane.