Nonlinear Parametric Vibration and Chaotic Behaviors of an Axially Accelerating Moving Membrane

Abstract

Nonlinear vibration characteristics of a moving membrane with variable velocity have been examined. The velocity is presumed as harmonic change that takes place over uniform average speed, and the nonlinear vibration equation of the axially moving membrane is inferred according to the D’Alembert principle and the von Kármán nonlinear thin plate theory. The Galerkin method is employed for discretizing the vibration partial differential equations. However, the solutions concerning to differential equations are determined through the 4th order Runge–Kutta technique. The results of mean velocity, velocity variation amplitude, and aspect ratio on nonlinear vibration of moving membranes are emphasized. The phase-plane diagrams, time histories, bifurcation graphs, and Poincaré maps are obtained; besides that, the stability regions and chaotic regions of membranes are also obtained. This paper gives a theoretical foundation for enhancing the dynamic behavior and stability of moving membranes.