Problems of dynamics and stability of a moving web, travelling between two rollers at a constant velocity, are studied using analytical approaches. Transverse vibrations of the web are described by a partial differential equation small inhomogeneities in the tension may have a large effect on the divergence forms.
PrefaceMechanics of moving materials is currently attracting considerable attention.Interest in research in the eld has grown in connection with the rapid development of process industry and precision machinery, especially in the eld of paper making.Specialized areas of investigation include dynamics and mechanical stability of travelling elastic materials, failure and reliability analysis, including modern fracture mechanics, and runnability optimization problems taking into account factors of uncertainty and incomplete data.Problems of mechanics of moving materials have not only practical but also theoretical importance. The investigation into new types of mathematical problems is interesting in itself. It is noteworthy that there are some nonconservative stability problems, and runnability optimization problems under fracture and stability constraints with uncertainties in positioning and sizes of initial defects (cracks), for which there are no systematic techniques of investigation. The most appropriate approach to tackle the complex problem varies depending on the case.This monograph is devoted to the exposition of new ways of formulating and solving problems of mechanics of moving materials. We present some research results concerning dynamics of travelling elastic strings, membranes, panels and plates. We study mechanical stability of axially moving elastic panels, accounting for the interaction between the structural element and its environment, such as axial potential ow. Most of the attention in this book is devoted to out-of-plane dynamics and stability analysis for isotropic and orthotropic travelling elastic and viscoelastic materials, with and without uid-structure interaction, using analytical and numerical approaches. Also such topics as fracture and fatigue are discussed in the context of moving materials. The last part of the book deals with some runnability optimization problems with physical constraints arising from the stability and fatigue analyses including uncertainties in the parameters. The approach taken in this monograph is to proceed analytically as far as is reasonable, and only then nish the investigation numerically.In this book, we oer a systematic and careful development of many aspects of mechanics of travelling materials, particularly for panels and plates.Some of the presented results are new, and some have appeared only in specialized journals or in conference proceedings. Some aspects of the theory presented here, such as the semi-analytical treatment of the uidstructure interaction problem of a travelling panel, studies of spectral problems with free boundary singularities, and optimization of problem parameters under crack growth and instability constraints have not been considered before to any extent.Important new results relate to optimization of runnability with a longevity constraint. Damage accumulation is modelled using the theory of fatigue crack growth, with the travelling material element subjected to cyclic load-3 ing. Uncertainties must be accounted for, beca...
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.
We study safe conditions and process effectiveness of systems of moving materials from the viewpoint of failures including fracture and loss of stability. The web is modelled as a thin elastic plate made of brittle material, travelling between a system of supports at a constant velocity, and subjected to bending, in-plane tension and small initial cracks. We study crack growth under cyclic in-plane tension and transverse buckling of the web analytically. We seek optimal in-plane tension that maximizes a performance vector function consisting of the number of cycles before fracture, the critical velocity and process effectiveness. The present way of applying optimization in the studies of fracture and stability is new and affords an analytical tool for process analysis techniques.
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