Jointing of thin elastic rods and generalized Kirchhoff conditions at nodes

Abstract

The paper presents various models of thin elastic curved rods and their joints. Such rods and their connections are important in applied research and have a long history. Nevertheless, the questions of mathematical research in the description of the procedure of transition from a multidimensional model to a one-dimensional one are not fully investigated. This paper presents the results when a thin object is pulled together into a one-dimensional object. In this case, various difficulties arise. For example, a more detailed consideration is required by the problem of describing the gluing conditions at the vertices of the limiting neck. It turns out that the type of specific bonding conditions depends on various physical premises. On the other hand, the unique solvability in various functional spaces depends on the selected gluing conditions. The first part of the paper describes various categories of bonding conditions. In the second part of the work, various examples of the construction of thin elastic curved rods and their joints will be given, when the ultimate task is supplemented by certain gluing conditions. In the conclusion of the second part, a numerical calculation of the natural frequencies of free vibrations of the joints of elastic thin curved rods is presented.

Keywords

• elastic curved thin rod
• curved rod
• kirchhoff's condition in knots
• theory of elasticity
• equation of elasticity theory
• model of rod connection