Nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support

Roll-to-roll manufacturing is the primary means of flexible electronics to facilitate scale-up production, and the web as the printed substrate is guided by the intermediate roller in this process. This paper aims to investigate the nonlinear superharmonic resonance and chaotic motion of a moving web under an intermediate nonlinear support. This support is modeled as a nonlinear elastic spring with linear and nonlinear stiffness, and the equation of motion of the web attached with nonlinear elastic support is derived from the D’Alembert principle and von Karman theory. The resulting equation is reduced to the two-degree ordinary differential equations via the Galerkin truncation, the superharmonic resonance responses of the web system are obtained by the multi-scale method, and the bifurcation analysis and stability are analyzed by the Runge-Kutta numerical method. The results indicate that the linear and nonlinear stiffnesses have a significant effect on amplitude-frequency responses and chaotic motion. This study provides an exploration of vibration behaviors of the web in flexible manufacturing, thereby laying the foundation for the improvement of fabrication productivity

Chaotic dynamics of fractional viscoelastic PET membranes subjected to combined harmonic and variable axial loads

Nonlinear chaotic vibrations of fractional viscoelastic PET (polyethylene terephthalate) membranes subjected to combined harmonic and variable axial loads is investigated in this paper. Axial tension variations arise from the machine disturbances of the processing line of roll-to-roll manufacturing. The viscoelasticity of PET membrane is characterized by the fractional Kelvin-Voigt model. Based on the Hamilton principle, the equation of motion of the membrane is established with the consideration of geometric nonlinearity, and the Galerkin procedure is employed to discretize the resulting governing equation. For the solution, the finite difference method is utilized in conjunction with the Caputo-type fractional derivative to reliably estimate the nonlinear response of fractional viscoelastic PET membrane. The reliability of this numerical strategy is proved by the available results of the fractional system and comparison examples. The influence of system parameters on chaotic behaviors is described by the bifurcation diagram and the detailed responses at the set bifurcation parameters. The fractional model together with the analysis provides a fundamental framework for the control of viscoelastic substrates in flexible manufacturing