a b s t r a c tProblems of stability of an axially moving elastic band travelling at constant velocity between two supports and experiencing small transverse vibrations are considered in a 2D formulation. The model of a thin elastic plate subjected to bending and tension is used to describe the bending moment and the distribution of membrane forces. The stability of the plate is investigated with the help of an analytical approach. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence (buckling). Then the static forms of instability are investigated, and critical regimes are studied as functions of geometric and mechanical problem parameters. It is shown that in the limit of a narrow strip, the 2D formulation reduces to the classical 1D model. In the limit of a wide band, there is a small but finite discrepancy between the results given by the 1D model and the full 2D formulation, where the discrepancy depends on the Poisson ratio of the material. Finally, the results are illustrated via numerical examples, and it is observed that the transverse displacement becomes localised in the vicinity of free boundaries.

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...

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.

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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.

The out-of-plane dynamic response of a moving plate, travelling between two rollers at a constant velocity, is studied, taking into account the mutual interaction between the vibrating plate and the surrounding, axially flowing ideal fluid. Transverse displacement of the plate (assumed cylindrical), is described by an integrodifferential equation that includes a local inertia term, Coriolis and centrifugal forces, the aerodynamic reaction of the external medium, the vertical projection of membrane tension, the bending resistance, and external perturbation forces. In the two-dimensional model thus set up, the aerodynamic reaction is found analytically as a functional of the cylindrical displacement, using the techniques of complex analysis. The resulting integro-differential problem is discretized in space with the Fourier-Galerkin method, and integrated in time with the diagonalization method. Examples are computed with physical parameters corresponding to air and some paper materials. The effects of the surrounding fluid on the critical velocity and first natural frequency are investigated, for stationary air, for an air mass moving with the plate, and for some arbitrary axial fluid velocities. The obtained results are applicable for both an ideal membrane and a plate with nonzero bending rigidity.