Travelling Strings, Beams, Panels, Membranes and Plates

In this chapter, we will introduce in a general manner some of the most common models for axially travelling materials, which will be used in the rest of the book. We will introduce the linear models of travelling strings, panels, and plates. It will be assumed that the material is thin, i.e. its planar dimensions are much larger than its thickness. We will work in the small displacement regime, that is, with linear models approximating the behaviour of the system near the trivial equilibrium. As is well known in the theory of elasticity, this approximation allows for a decoupling of the in-plane and out-of-plane components in the dynamics of the system. We will concentrate on small out-of-…

On modelling and stability of axially moving viscoelastic materials

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

In 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 demon…

Optimization and analysis of processes with moving materials subjected to fatigue fracture and instability

We study systems of traveling continuum modeling the web as a thin elastic plate of brittle material, traveling 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. peerReviewed

Dynamic Analysis for Axially Moving Viscoelastic Poynting–Thomson Beams

This paper is concerned with dynamic characteristics of axially moving beams with the standard linear solid type material viscoelasticity. We consider the Poynting–Thomson version of the standard linear solid model and present the dynamic equations for the axially moving viscoelastic beam assuming that out-of-plane displacements are small. Characteristic behaviour of the beam is investigated by a classical dynamic analysis, i.e., we find the eigenvalues with respect to the beam velocity. With the help of this analysis, we determine the type of instability and detect how the behaviour of the beam changes from stable to unstable.

Dynamic analysis for axially moving viscoelastic panels

In this study, stability and dynamic behaviour of axially moving viscoelastic panels are investigated with the help of the classical modal analysis. We use the flat panel theory combined with the Kelvin–Voigt viscoelastic constitutive model, and we include the material derivative in the viscoelastic relations. Complex eigenvalues for the moving viscoelastic panel are studied with respect to the panel velocity, and the corresponding eigenfunctions are found using central finite differences. The governing equation for the transverse displacement of the panel is of fifth order in space, and thus five boundary conditions are set for the problem. The fifth condition is derived and set at the in-…

Travelling Panels Interacting with External Flow

This chapter is devoted to the analysis of the travelling panel, submerged in axially flowing fluid. In order to accurately model the dynamics and stability of a lightweight moving material, the interaction between the material and the surrounding air must be taken into account somehow. The light weight of the material leads to the inertial contribution of the surrounding air to the acceleration of the material becoming significant. In the small displacement regime, the geometry of the vibrating panel is approximately flat, and hence flow separation is unlikely. We will use the model of potential flow for the fluid. The approach described in this chapter allows for an efficient semi-analyti…

On static instability and estimates for critical velocities of axially moving orthotropic plates under inhomogeneous tension

In this study, models for axially moving orthotropic plates are investigated analytically. Linearised Kirchhoff plate theory is used, and the energy forms of steady-state models are considered. With the help of the energy forms,ho- mogeneous and inhomogeneous tension profiles, some limits for the critical web velocity are found analytically. A numerical example is given about effects of the shear moduls in the case of an inhomogenous tenstion profile. nonPeerReviewed

Travelling Panels Made of Viscoelastic Material

In this chapter, our focus is to analyse the behaviour of moving panels using viscoelastic materials. As the reader will have noticed, all the models discussed in previous chapters have concerned the case of a purely elastic material. The deformation of an elastic material depends only on the applied forces; it has no explicit time dependence. Paper, however, is a more complicated material: it is viscoelastic. In addition to elastic properties, it has also time-dependent viscous properties, which cause the phenomena of creep and relaxation (see, e.g., Alava and Niskanen 2006). One of the simplest models for a viscoelastic solid is the Kelvin–Voigt model, which consists of a linear spring an…

Stochastic analysis of the critical velocity of an axially moving cracked elastic plate

In this study, a probabilistic analysis of the critical velocity for an axially moving cracked elastic and isotropic plate is presented. Axially moving materials are commonly used in modelling of manufacturing processes, like paper making and plastic forming. In such systems, the most serious threats to runnability are instability and material fracture, and finding the critical value of velocity is essential for efficiency. In this paper, a formula for the critical velocity is derived under constraints for the probabilities of instability and fracture. The significance of randomness in different model parameters is investigated for parameter ranges typical of paper material and paper machin…

Some Optimization Problems

In this chapter, the problems of safety analysis and optimization of a moving elastic plate travelling between two rollers at a constant axial velocity are considered. We will use a model of a thin elastic plate subjected to bending and in-plane tension (distributed membrane forces). We will study transverse buckling (divergence) of the plate and its brittle and fatigue fracture caused by fatigue crack growth under cyclic in-plane tension (loading). Our aim is to find the safe ranges of velocities of an axially moving plate analytically under the constraints of longevity and stability. In the end of this chapter, the expressions for critical buckling velocity and the number of cycles before…

Estimates for Divergence Velocities of Axially Moving Orthotropic Thin Plates

Some models for axially moving orthotropic thin plates are investigated analytically via methods of complex analysis to derive estimates for critical plate velocities. Linearised Kirchhoff plate theory is used, and the energy forms of steady-state models are considered with homogeneous and inhomogeneous tension profiles in the cross direction of the plate. With the help of the energy forms, some limits for the divergence velocity of the plate are found analytically. In numerical examples, the derived lower limits for the divergence velocity are analysed for plates with small flexural rigidity. peerReviewed

Theoretical study on travelling web dynamics and instability under non-homogeneous tension

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 that includes the centrifugal force, in-plane tension, elastic reaction and nonstationary inertial terms. The model of a thin elastic plate subjected to bending and non-homogeneous 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 studies of small out-of-plane vibrations. The influence of linearly distributed in-plane tension on the characteristics of the web vibration…

Non-Homogeneous Tension Profile

In this chapter, we will look at the influence of a skewed tension profile on the divergence instability of a travelling, thin elastic plate. The travelling plate is subjected to axial tension at the supports, but the tension distribution along the supports is not uniform. For the nonuniformity, we will use a linear distribution. First, we will perform a dynamic analysis of small time-harmonic vibrations, after which we will concentrate on the divergence instability problem. We will see that a small inhomogeneity in the applied tension may have a large effect on the divergence modes, and that inhomogeneity in the tension profile may significantly decrease the critical velocity of the plate.

Navier-Stokes equations

Fracture and Fatigue of Travelling Plates

In this chapter, problems of fracture and stability of a moving plate, travelling in a system of rollers at a constant velocity, are studied. It is known that in the manufacturing process, there may occur many kinds of defects in the paper web, such as edge cracks and blister and fiber cuts. Our aim is to tackle this problem and analyse theoretically how the defects change the behaviour. We will use the model of a thin elastic plate made of brittle material. A plate with initial cracks is studied, subjected to constant tension and cyclic tension. As a result, we will show how to find safe parameter ranges of transport velocities and in-plane tensions when fracture, stability and constraints…

On displacement-velocity coupling and the origin of in-plane stress in orthotropic moving continua

In this paper, we address the problem of the origin of in-plane stresses in continuous, two-dimensional high-speed webs. In the case of thin, slender webs, a typical modeling approach is the application of a stationary in-plane model, without considering the effects of in-plane velocity field. However, for high-speed webs this approach is insufficient, because it neglects the coupling between the total material velocity and the deformation experienced by the material. By using a mixed Lagrange–Euler approach in model derivation, the solid continuum problem can be transformed to solid a continuum flow problem. Mass conservation in the flow problem, and the behaviour of free edges in the two-dime…

Tietotekniikan kandidaattiseminaarin kehityspolkuja 2012-2015

Jyväskylän yliopiston Informaatioteknologian tiedekunnan Tietotekniikan laitoksella selvitettiin kandidaatintutkielmien tehokkaamman edistymisen mahdollisuuksia kandidaattiseminaaria kehittämällä ja ohjausresursseja keskittämällä. Aiemman toteutustavan analysoinnin ja tutkimusongelman muotoilun pohjalta suunniteltiin ja toteutettiin useita tutkimus- ja kehitystoimenpiteitä, joiden tuloksena syntyi vaiheistamiseen ja tiedeviestinnän entistä saumattomampaan integrointiin perustuva KandiX-malli. Tässä raportissa kuvataan mallin iteratiivista kehittämistä ja vuosien 2012-2015 aikana saatuja tuloksia, joiden mukaan kandidaatintutkielman työstämiseen kuluva aika on lyhentynyt muutamaan kuukauteen…

Mechanics of Moving Materials

Stability of moving viscoelastic panels interacting with surrounding fluid

We study a model describing the out-of-plane vibrations of an axially moving viscoelastic panel submerged in flowing fluid. The panel is assumed to travel at a constant velocity between two fixed supports, and it is modeled as a flat panel made of viscoelastic Kelvin-Voigt material. The fluid flow is modeled with the help of the added mass coefficients. The resulting dynamic equation is a partial differential equation of fifth order in space. Five boundary conditions are set for the studied problem. The behavior of the panel is analyzed with the help of its eigenvalues (eigenfrequencies). These characteristics are studied with respect to the velocity of the panel. In our study, we have incl…

Stability of a Tensioned Axially Moving Plate Subjected to Cross-Direction Potential Flow

We analyze the stability of an axially moving Kirchhoff plate, subjected to an axial potential flow perpendicular to the direction of motion. The dimensionality of the problem is reduced by considering a cross-directional cross-section of the plate, approximating the axial response with the solution of the corresponding problem of a moving plate in vacuum. The flow component is handled via a Green’s function solution. The stability of the cross-section is investigated via the classical Euler type static linear stability analysis method. The resulting eigenvalue problem is solved numerically using Hermite type finite elements. As a result, the critical velocity and the corresponding eigenfun…

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

In this paper, we address the problem of the origin of in-plane stresses in continuous, two-dimensional high-speed webs. In the case of thin, slender webs, a typical modeling approach is the application of a stationary in-plane model, without considering the effects of the in-plane velocity field. However, for high-speed webs this approach is insufficient, because it neglects the coupling between the total material velocity and the deformation experienced by the material. By using a mixed Lagrange–Euler approach in model derivation, the solid continuum problem can be transformed into a solid continuum flow problem. Mass conservation in the flow problem, and the behaviour of free edges in th…

Static instability analysis of an elastic band travelling in the gravitational field

Static instability analysis is performed for an axially moving elastic band, which is travelling at a constant velocity in a uniform gravitational field between two supports. The buckling of the band is investigated with the help of admitting small transverse deflections. The model of a thin elastic beam (panel) subjected to bending, centrifugal forces and nonhomogeneous tension (including a gravitational term) is used. Buckling analysis and estimation of the critical velocities of elastic instability are based on variational principles and variational inequalities. As a result, explicit formulas for upper and lower limits for critical velocities are found. It is shown analytically that a c…