Statics and dynamics of curved rods based on Bernoulli hypotheses and relations for a rectilinear rod

Abstract

Method for calculating the statics and dynamics of curved rods, based on the equations for a rectilinear rod, is described and justified in detail. Bernoulli's hypotheses and the variational method are applied. The main advantage and special feature of these formulas is that the simplest formulas that are valid for rectilinear rods are used for the calculations of curved rods. These formulas do not contain parameters characterizing the curvatures of the longitudinal axis of the rod. This feature is an essential factor in the calculation of curved rods, where the information about their longitudinal axis is given discretely, since no special methods of approximation of discretely given data are required, which enable to obtain information about the radius-vector of the rod longitudinal axis and its derivatives with the required high accuracy. Solutions of test static and dynamic problems are provided. Bending of a rod with a longitudinal axis in the form of a circle, a naturally twisted rod, and a spring fluctuation are considered. Comparison of the calculation results with the data published in the literature illustrates the reliability and high accuracy of the solutions obtained.