Nonlinear vibration and stability of a moving printing web with variable density based on the method of multiple scales

An axially moving printing web with variable density in a printing process causes a geometric nonlinear vibration, and a nonlinear vibration system is established using the von Karman nonlinear plate theory and the D’Alembert principle. The time and displacement variables are separated using the Galerkin method. The ordinary differential equation of a web is solved using the method of multiple scales. The amplitude–frequency response equation of a moving web is obtained. The time histories, phase–plane portraits, and amplitude–frequency curves of the system are obtained by numerical calculations. The influence of different dimensionless speeds and variable density coefficients on the nonlinear vibration characteristics of the printing web is analyzed. The results show that the overprinting accuracy can be ensured by making a reasonable choice of web speed in the stable region.