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DOI 10.1007/s12206-020-1011-9

Journal of Mechanical Science and Technology 34 (11) 2020 Original Article DOI 10.1007/s12206-020-1011-9 Keywords: · Bifurcation and chaos · Oblique supports · Traveling membrane · Fourth-order Runge-Kutta method · External excitation

Correspondence to: Jimei Wu [email protected]

Citation: Shao, M., Qing, J., Wu, J. (2020). Bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation. Journal of Mechanical Science and Technology 34 (11) (2020) 4513~4523. http://doi.org/10.1007/s12206-020-1011-9

Received May 5th, 2020 Revised

August 9th, 2020

Bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation Mingyue Shao, Jiajuan Qing and Jimei Wu Faculty of Printing, Packing and Digital Media Engineering, Xi’an University of Technology, Xi’an 710048, China

Abstract

This paper analyzes bifurcation and chaos of the traveling membrane on oblique supports subjected to external excitation. Through coordinate transformation, the Von Karman nonlinear plate theory was employed to derive the non-linear governing equations of membrane on oblique supports in terms of axial movement in the oblique coordinate system. The boundary conditions were also given in the oblique coordinate system. The approximate solution for the governing equations was found via the Galerkin method as well as the fourthorder Runge-Kutta numerical computing method. The nonlinear dynamic techniques including Lyapunov exponents diagrams, bifurcation plots, Poincare maps, phase trajectories, and time histories were introduced to analyze the impacts of the angles of oblique supports, aspect ratios and traveling velocities on dynamics responses and the various forms of vibration regarding the membrane system.

Accepted August 13th, 2020 † Recommended by Editor No-cheol Park

© The Korean Society of Mechanical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2020

1. Introduction The membrane structure is a common material in engineering components, aerospace devices, and wearable electronics, etc. In printing engineering, the oblique rollers are often used to change the direction of membrane movement and support the traveling membrane during the roll-to-roll transmission process. As a typically flexible material, a membrane has the characteristics of light weight, small stiffness and large deformation after the forces have acted upon it. The traveling membrane on oblique supports subjected to external excitation will appear nonlinear dynamic behaviors such as bifurcation and chaos, which leads to defects such as surface scratches, wrinkling and deformation. These phenomena can seriously affect the processing accuracy and efficiency of the membrane products. In addition, the printing quality of the end products is susceptible to degradation based on the transverse displacement of the membrane. Nonlinear dynamic behaviors of axially moving systems on oblique supports have fascinated researchers for decades. Indeed there is some lit