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 a
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Travelling Strings, Beams, Panels, Membranes and Plates
Abstract 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.
2.1 Out-of-Plane Vibrations In the following, we will introduce the four most common linear models for small outof-plane vibrations of a travelling thin elastic material. These are the travelling string, panel, membrane and plate. The string and the panel are simple one-dimensional models, while the membrane and plate models are two-dimensional, accounting for variations in the displacement along both in-plane axes. The panel and the plate resist bending, while the string and the membrane can support only tensile loads. For the membrane and plate, both isotropic and orthotropic variants will be considered. In later chapters, we will examine dynamical stability predictions from these models. For the plate model, in this book we focus on rectangular plates with SFSF boundary conditions, where two opposite edges are simply supported (S), and the other two edges are free of tractions (F). For the panel model, we will use the simply supported boundary conditions. For an analysis of the travelling plate in the case of SSSS N. Banichuk et al., Mechanics of Moving Materials, Solid Mechanics and Its Applications 207, DOI: 10.1007/978-3-319-01745-7_2, © Springer International Publishing Switzerland 2014
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2 Travelling Strings, Beams, Panels, Membranes and Plates
boundary conditions, i.e., simply supported on all sides, see Luo and Hamidzadeh (2004) and Marynowski (2008).
2.1.1 Travelling Strings The simplest way to model a moving material experiencing out-of-plane vibrations is the equation of the travelling string (also known as the threadline equation). The one-dimensional string representation has been used in many fundamental studies and it provides a basis for more advanced analysis. Moreover, it turns out that the behaviour predicted by two-dimensional models of moving materials, when the strip of material is long and narrow, reduces to the
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