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RESEARCH PRODUCT

Travelling Strings, Beams, Panels, Membranes and Plates

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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.