Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string

Abstract

In the numerous works trying to describe the free vibration of stretched elastic strings, phenomena due to non-linear geometrical modifications in a three-dimensional large amplitude motion are separated from those due to the string's stiffness, in which case torsional vibrations cannot be taken into account. In order to fill this gap with a more complete model and to specify its field of validity the problem set up again, starting from three-dimensional elasticity for finite displacements with small strains of a slender homogeneous elastic cylindrical rod under initial stress. The original geometrical construction which gives the cross-section's orientation during the motion is detailed. With the symmetry of the problem preserved, it provides a description of simple physical meaning and in particular the torsion is used as a parameter. The goal of this study being the determination of the deformed shape of the central line, the three spatial variables are reduced to one through expressions for the resultant forces and moments on the cross-sections, leading to macroscopic equations of motion which are simplified in the case of uncoupled transverse and longitudinal modes. For the first time, the equations of motion introduce a coupling torsion term and include existing string and beam models, which can thus be considered as particular cases. The assumptions and generalizations are discussed and the influences of the different terms are described. While this work is based on the limits imposed by material linearity, it provides a useful basic tool for the study of the transverse vibrations of slender elastic beams, including the commonly avoided torsional phenomena.