Multi-scale Method Galloping Analysis of the Iced Transmission Line Based on Curved-beam Model and Considering Elastic Boundary Condition2

Firstly, in order to cover the vacancy in the theoretical analysis of the icedtransmission line and deeply clarify the galloping mechanism, a3-DOF gallopingcurved-beam model of the iced transmission line considering stiffness of insulatorstrings and adjacent spans as boundary conditions is formulated based on thecurved-beam theory with three-dimensional Lagrange tensor considering boundarycondition, Hamiton principle and Galerkin method This model is mainly developedfrom the three-degree-of–freedom(vertical, horizontal and torsional) iced transmissionline galloping model based on curved beam theory and considering section eccentric,Secondly, in order to prove the correctness of the above cured-beam model takingboundary condition into account, the program code of galloping model based oncurved-beam and cable model developed by Mathematica is used to solve gallopingequations of the iced transmission line. Take the classical D-shaped transmission linecross section as an example, the accuracy of the above model is validated.The resultsindicate that the3-DOF galloping curved-beam model considering the bending stiffnessis more superior than the cable theory model. Taking accout for the elastic boundaryconditions, the model is in best agreement with experimental results, which has higheraccuracy than the models based on cable theory among all models. In addition, the moredegrees of freedom of the galloping models have, the smaller the deviation betweengalloping model amplitude and experimental results will be. Subsequently, According tothe experimental data of crescent moon shaped cross section, the galloping rule of theiced transmission line is preliminarily investigated. Compared with the anchoredconstraint on both ends, vertical and horizontal galloping displacement amplitudesdiminish and rotational displacement amplitude raise accordingly after consideringelastic boundary condition. Afterward, an example of the crescent moon shaped crosssection show that the impacts of elastic boundary condition aiming at gallopingamplitude, critical wind velocity and model function are pinpoint. The Resultdemonstrates that with the increase of the boundary spring stiffness, vertical andhorizontal displacement amplitudes raise and rotational displacement amplitudediminish independently. Both the critical wind velocity and supercritical wind speed forgalloping increase. Simultaneously, the range of the wind speed for galloping becomeslarger. The boundary stiffness has a slight effect on horizontal and rotational modefunctions and has significant influence on vertical and longitudinal mode functions. Finally, a two-degree-of-freedom (vertical and horizontal) galloping model isreduced from the three-degree-of-freedom galloping model due to the frequencycharacteristic of rotational direction, then using the multi-scale method in the nonlinearvibration to deduce the1:1and2:1internal resonance simplified amplitude equationrespectively. Afterward, through the resonance amplitude equation, this paper attachesimportance to analyze a variety of the bifurcation in both cases. Meanwhile, thegalloping performances (bifurcation and stability) have been ascertained under the fixedboundary condition and elastic boundary condition in the1:1and2:1internal resonancecases considering the eccentric of cross-section or not considering the eccentric.Analysis shows that under the elastic boundary condition, whether considering theeccentric or not, the galloping law are roughly in line with those in fixed boundarycondition, but the corresponding galloping amplitudes are smaller than those in fixedboundary condition. Likewise, under the elastic boundary condition, compared with thefixed boundary condition, Both the critical wind velocity and supercritical wind speedfor galloping increase. But it is worth noticing that the increase of the supercritical windvelocity is greater than that of the critical wind speed. This conclusion is the same as thegalloping law obtained by the numerical solution and the influence law of elasticboundary condition aiming at the critical wind velocity and supercritical wind speedunder the identical example in the fourth chapter of this paper. In the further analysis, inorder to identify the effect of the axial model function on the model established in thispaper, the galloping performances between the model used in this paper and the modelset up by Yan zhimiao under the anchored boundary condition is contrasted. The Resultsillustrate that the critical wind velocity and supercritical wind speed in the model of thispaper increase slightly, and meanwhile the vertical and horizontal galloping amplitudesdecrease, compared with the model established by Yan zhimiao under the case of1:1internal resonance (low sag),. Under the case of2:1internal resonance (larger sag), theeffect of the model function on the galloping bifurcation and stability cannot be ignored,especially in the eccentric case. The critical wind velocity and supercritical wind speedin the model of this paper increase more than the model proposed by Yan zhimiao. Theeccentric of the cross-section exerts large impact on the galloping bifurcation andstability.