On the limit velocity and buckling phenomena of axially moving orthotropic membranes and plates

Abstract

a b s t r a c tIn this paper, we consider the static stability problems of axially moving orthotropic membranes and plates. The study is motivated by paper production processes, as paper has a fiber structure which can be described as orthotropic on the macroscopic level. The moving web is modeled as an axially moving orthotropic plate. The original dynamic plate problem is reduced to a two-dimensional spectral problem for static stability analysis, and solved using analytical techniques. As a result, the minimal eigenvalue and the corresponding buckling mode are found. It is observed that the buckling mode has a shape localized in the regions close to the free boundaries. The localization effect is demonstrated with the help of numerical examples. It is seen that the in-plane shear modulus affects the strength of this phenomenon. The behavior of the solution is investigated analytically. It is shown that the eigenvalues of the cross-sectional spectral problem are nonnegative. The analytical approach allows for a fast solver, which can then be used for applications such as statistical uncertainty and sensitivity analysis, real-time parameter space exploration, and finding optimal values for design parameters.

On reliability of systems with moving material subjected to fracture and instability

“…When the values of T 0 , c T and a T are set, the volatility parameter s T is obtained from Eq. (7). For the crack length process, when c ξ and a ξ are given, the shape parameter δ ξ is obtained from Eq.…”

“…Studies of moving elastic strings were continued by Sack [2] and Archibald and Emslie [3], who were also the first published authors on axially moving materials in English. Later studies of axially moving materials concern, e.g., stability of traveling two-dimensional elastic [4,5], orthotropic [6,7] and viscoelastic [8,9] plates. Extensive literature reviews on the studies of axially moving materials can be found in [5,7,9].…”

Vibrations of a continuous web on elastic supports

“…In our previous studies (see, e.g., [3][4][5][6][7][8], we have considered mathematical modelling of the mechanical aspects of the paper making process, using analytical and numerical approaches. The processing of paper, steel, fabric or rubber, and looping systems such as band saws and timing belts, all belong to the class of systems known as axially moving materials.…”

The origin of in-plane stresses in axially moving orthotropic continua

“…It is also known that the lack of web tension will result in a loss of stability of the moving web, which from the application A C C E P T E D M A N U S C R I P T viewpoint, disturbs the required smooth advancing of the web (see e.g. [4,3]). On the other hand, high tension may cause web breaks, which deteriorates production efficiency (see e.g.…”