Spatial Vibrations of Power Transmission Conductors with Ice Deposits

The problem on free spatial vibrations of an overhead transmission line conductor with an asymmetric mass distribution over the cross-section that is caused by ice deposits, which impart an asymmetric shape to the cross-section, is considered. As a result, an eccentricity between the centers of torsional stiffness and mass in the cross-section is formed; and a dynamic relation of vertical, torsional, and “pendulum” vibrations develops with the conductor leaving the sagging plane. The conductor is modeled as a flexible heavy elastic rod that resists only stretching and torsion. The case of a slightly sagging conductor, when the tension and curvature of its centerline can be considered constant within the span, is investigated. It is also considered that the elasticity of the ice casing is small compared to the elasticity of the conductor. The mathematical model considering the interaction of longitudinal, torsional, and transverse waves polarized in the vertical and horizontal planes is analyzed. The ratios of phase velocities of all types of waves are analyzed and a group of particular subsystems determining partial vibrations is identified. Partial and natural frequencies and vibration modes of the conductor are investigated. Analytical solutions for the problem on determining the spectrum of natural frequencies and spatial vibration modes are obtained. The effect of an ice casing on the spectrum of conductor vibrations is investigated. A dependence of the wave number of torsional vibrations on the frequency is found. Such a dependence is determined not only by the elastic-inertial, but also by the gravitational factor, which is strongly manifested for conductors in long spans, especially subjected to galloping. This circumstance is essential for the analysis of the phenomenon of galloping from the standpoint of linking the occurrence of galloping with the convergence of the frequencies of torsional and transverse modes during conductor icing. It is shown that the ratio of these frequencies causing the self-oscillatory process is considerably complicated.