Parametric resonance of printing viscoelastic film with time-varying tension and velocity coupling based on the variable amplitude method

In high-precision roll-to-roll coating processes, periodic tension perturbations occur at both ends of the tensioned film due to factors such as roll eccentricity and diameter variations. Simultaneously, the film undergoes variable-velocity linear motion. According to the fundamental principles of Newtonian mechanics, tension perturbations inevitably induce acceleration responses. Based on the dynamic static equilibrium method and D’Alembert theory, the coupling equation between time-varying tension and velocity are explored, and then the nonlinear vibration control equation of printing motion viscoelastic film under time-varying tension and velocity coupling was established. The Galerkin method is applied to discretize and the variable amplitude method is employed to solve the amplitude expression of the parametric resonance response of the film system, and the Jacobi matrix is used to carry out the stability of the system’s stationary solution. The ratio of the tension variation frequency to the intrinsic frequency of the system is investigated for its influence on the vibration amplitude of the film, and MATLAB programming is used to calculate and analyze the influence of the parameters such as the aspect ratio, initial velocity, tension variation coefficient and viscoelastic coefficient on the parametric resonance of the film system.