Juha Jeronen

Mechanics of moving materials

1 Introduction 2 Travelling strings, beams, panels, membranes and plates 3 Stability analysis 4 Non-homogeneous tension profile 5 Travelling panels made of viscoelastic material 6 Travelling panels interacting with external flow 7 Fracture and fatigue of travelling plates 8 Some optimization problems References Index

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. The linearized 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 analyzed for plates with small flexural rigidity.

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.

Vibrations of a continuous web on elastic supports

ABSTRACT We consider an infinite, homogenous linearly elastic beam resting on a system of linearly elastic supports, as an idealized model for a paper web in the middle of a cylinder-based dryer section. We obtain closed-form analytical expressions for the eigenfrequencies and the eigenmodes. The frequencies increase as the support rigidity is increased. Each frequency is bounded from above by the solution with absolutely rigid supports, and from below by the solution in the limit of vanishing support rigidity. Thus in a real system, the natural frequencies will be lower than predicted by commonly used models with rigid supports.

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 eigenfunction are determined. It is seen that even at very low free-stream fluid velocities, the buckling shape may become antisymmetric in the cross direction.

On Bifurcation Analysis of Implicitly Given Functionals in the Theory of Elastic Stability

In this paper, we analyze the stability and bifurcation of elastic systems using a general scheme developed for problems with implicitly given functionals. An asymptotic property for the behaviour of the natural frequency curves in the small vicinity of each bifurcation point is obtained for the considered class of systems. Two examples are given. First is the stability analysis of an axially moving elastic panel, with no external applied tension, performing transverse vibrations. The second is the free vibration problem of a stationary compressed panel. The approach is applicable to a class of problems in mechanics, for example in elasticity, aeroelasticity and axially moving materials (such as paper making or band saw blades).

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-analytical solution, where the fluid flow is solved analytically in terms of the panel displacement function, and then strongly coupled into the partial differential equation describing the panel displacement. The panel displacement, accounting also for the fluid–structure interaction, can then be solved numerically from a single integrodifferential equation. In the first section of this chapter, we will set up and solve the problem of axial potential flow obstructed by the travelling panel. In the second section, we will use the results to solve the fluid–structure interaction problem, and give so me numerical examples.

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.

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-plane (transverse) vibrations of the material only, as this is the most relevant aspect of the physics from the viewpoint of dynamical stability, which will be the focus of later chapters. We will look at both one-dimensional and two-dimensional models, and consider variants with and without bending rigidity. The in-plane tension fields, which affect the out-of-plane behaviour, will be considered at the end of the chapter.

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 the fracture (longevity of the plate) as a function of in-plane tension and other problem parameters are used for formulation and we will study the case as an optimization problem. Our target is to find the optimal in-plane tension to maximize the performance function of paper production. This problem is solved analytically and the obtained results are presented as formulae and numerical tables.