Nonlinear vibration of fractional-order PET films with air resistance and non-uniform temperature distribution

The chaotic vibration analysis of fractional-order viscoelastic polyethylene terephthalate (PET) films under air resistance and non-uniform temperature fields is investigated. During the printing production process, moving viscoelastic PET films in the oven hot-air system are not only subjected to non-uniform temperature distributions and hot air, but they are also influenced by the friction forces generated by the airflow field on the film surface, causing the printed PET film to vibrate and affecting print quality. Based on the von Karman theory of large-deflection thin plates and Hamilton's principle, a fractional-order nonlinear vibration differential equation is established for the motion of PET films in a non-uniform temperature distribution field during roll-to-roll printing processes. The finite difference method and discrete Caputo-type fractional derivatives are applied to obtain the nonlinear vibration response of the fractional-order viscoelastic PET film under non-uniform temperature distribution. Through programming and calculation using MATLAB, bifurcation diagrams, phase trajectory diagrams, time history diagrams, phase diagrams, and Poincaré section diagrams are obtained to analyze the effects of air resistance, fractional-order degree, and non-uniform temperature on the nonlinear vibration of the film.