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Wind excited vibrations generated by the vortex shedding are very common in high-voltage overhead transmission lines. Although such vibrations are barely perceptible due to their low amplitudes (less than a conductor diameter), controlling them, however, is extremely important since they may lead to conductor fatigue. Mathematical models are therefore necessary for the computation of these vibrations, not only to evaluate the risk of potential damage to the transmission line but also for studying the efficiency of the damping measures. For single conductor transmission lines, the so-called energy-balance method gives fairly good results for the estimation of vibration amplitudes. However, the problem becomes more involved for the conductor bundles attached with different damping devices. A modified form of the energy-balance method is thus needed. This thesis presents a mathematical-mechanical model for modeling the vibrations of the conductor bundles with many spacer dampers and Stockbridge dampers, by considering subconductors as continuous systems. External damping devices are incorporated into the model by means of their complex impedance matrices. The presented model results in a smaller system matrix in comparison to what is obtained while modeling the conductor as a discrete system. This modeling procedure yields a complex non-polynomial transcendental eigen- value problem (TEVP). Solving such a TEVP is simple for smaller systems, e.g., for single conductor transmission lines. However, for a comparatively bigger system like the one in case of conductor bundles, it is a formidable task to obtain all the system eigenvalues in a certain frequency domain. The first goal of the presented thesis is to find e±cient numerical methods for obtaining the solutions of TEVPs. Different numerical techniques are discussed and their results are compared. Newton's approach for the solution of the eigenvalue problem is found to be an e±cient solution technique for obtaining the complex eigenvalues of a TEVP of the current type. After obtaining the complex eigenvalues and eigenmodes of the system, an energy balance principle is presented in order to obtain the actual vibration amplitudes in the subconductors. In energy balance the energy input from the wind is equated with the energy loss due to external damping and the conductor's self damping. Wind power input is normally obtained using data from the wind tunnel experiments or from the experiments carried out with transmission lines in the field, for laminar and turbulent wind speeds. Wind power input data is only available for single oscillating rods/cylinders, as obtaining such data via experiments for different configurations of multi-cylinders, is a difficult task. The second goal of the presented work is to numerically obtain the wind power input for the oscillating cylinders in a tandem arrangement. A finite-volume approach is used for the solution of the Navier-Stokes equations. Moving grids are used to incorporate the movements of the cylinders. Firstly, accuracy and feasibility of the numerical results are verified by solving the flow around a single cylinder, and comparing the obtained results with the available experimental data. For similar setups as used in the experiments from different researchers, numerical wind power input for a single oscillating cylinder is obtained. Good agreement with the experimental results is found. The numerical approach is subsequently further extended to obtain the wind power input for two oscillating cylinders in a tandem arrangement. A considerable difference between the wind power inputs for the downstream cylinder in a cylinder-tandem and for a single oscillating cylinder is observed. The newly obtained wind power input is utilized for the energy balancing in a bundled conductor benchmark problem, and results are discussed. |
