Numerical aspects of determination of natural frequencies of a power transmission line cable equipped with in-line fittings

In the analysis of Aeolian vibration response of electric power transmission lines, the modified energy balance method (MEBM) is often used. The first and crucial step in applying the MEBM is an accurate determination of the natural frequencies and modes of the system, or in other words, the eigenvalues and eigenvectors of the system matrix. In this paper an efficient numerical procedure that searches for frequency parameters s that make the system matrix J(s) of transmission line cable singular is considered. A realistic case where the cable is equipped with in-line fittings such as Stockbridge vibration dampers and aircraft warning spheres is taken into account. It is shown that the rank of the considered system matrix J(s) is either full, or the full rank minus one. From a numerical point of view, this is an important property of the system matrix since multiple eigenvalues without a full set of eigenvectors, i.e., with Jordan blocks of order greater than 1, are very sensitive to small perturbations. The developed numerical procedure, which is easily parallelizable, consists of the hybrid minimization method paired with the Singular Value Decomposition (SVD) for the detection of the singularity of the matrix. Presented numerical examples illustrate advantages of the numerical procedure proposed.