Galloping Model And Analytical Method Of Iced Transmission Line

The galloping of the iced transmission line will cause the great damage to the whole society. In order to better understand the theory of galloping of iced transmission line, a three-degree-of-freedom galloping model based on curved beam theory has been proposed. Also, the Multiple Scale Perturbation Method (MSM) is introduced to analyze bifurcation and stability of the iced transmission line.Firstly,based on curved beam theory proposed by Zhu and converted curvilinear coordinates raised by Mason, three-dimension Lagrange strain tensor of iced transmission line is established. Then, according to Hamilton principle, a four-degree-of-freedom (tangential, nominal, bi-nominal and torsional) galloping model considering iced eccentricity is formulated. Moreover, due to the property of transmission line, a reduced three-degree-of-freedom (nominal, bi-nominal and torsional) galloping model is obtained in view of bending, rotation and eccentricity of cross section. Furthermore, the initial rotational angle is introduced into the model of attack angle in addition to the rotational displacement, nominal velocity and bi-nominal velocity. Finally, According to Galerkin method, three-degree-of-freedom discrete galloping model and aerodynamic discrete model are obtained.Secondly, in order to test the accuracy of three-degree-of-freedom galloping model based on curved beam theory, three galloping models based on cable structure are established, such as one-degree-of-freedom (vertical) , two-degree-of-freedom (vertical and horizontal) and three-degree-of-freedom (vertical, horizontal and rotational) galloping models. Moreover, one classical galloping example (cross section of D shape) is chosen to analyze the forgoing four galloping models. Comparing vertical displacements of the four galloping models and the experimental data, it can be found that the accuracy of the three-degree-of-freedom galloping model based on curved beam theory is higher than the other three galloping models based on cable structure. Furthermore, the reasons for the difference between the four galloping models and experimental data are discussed.Finally, the three-degree-of-freedom galloping model based on curved beam theory and MSM are introduced to analyze the 1:1 and 2:1 resonant cases. In order to establish the model suitable for MSM, a two-degree-of-freedom (nominal and bi-nominal) galloping model is reduced from the three-degree-of-freedom galloping model due to the property of iced transmission line. In view of the above reduced two-degree-of-freedom galloping model, 1:1 and 2:1 Reduced Amplitude Modulation Equations (RAME) are formulated with the method of MSM to analyze the bifurcations of 1:1 and 2:1 resonant cases. Moreover, the Jacobian matrix of 1:1 and 2:1 RAME are established to analyze the stability of the forgoing bifurcations of 1:1 and 2:1 resonant cases except the special branch (one-mode (nominal) galloping), in which case the Jacobian matrix of 2:1 RAME becomes singular. In order to solve this problem, a mixed polar-Cartesian form of 2:1 RAME is adopted. Furthermore, according to the experimental data recently completed in Chinese Aerodynamic Research and Development Center, the drag and lift coefficients are fitted in the form of cubic equations. In view of the above fitting data, the bifurcation and stability of the 1:1 and 2:1 resonant cases considering the eccentricity and without considering the eccentricity are discussed. The results turn out that the iced eccentricity plays an important role on bifurcation and stability. Finally, in the 2:1 resonant case without considering the eccentricity, one special phenomenon (multiple stabilities) is found. Also, from the analysis of numerical integration of RAME and reduced model in time history, it is proved that this phenomenon actually appears as the wind speed is in the wind speed range of multiple stabilities, which is obtained by analyses of bifurcation and stability of RAME.