Multi-modal Galloping For A Slender-long Structure With Non-circular Cross Section Subjected To Wind Disturbance

A type of a slender long structure is widely applied in the practical engineering including transmission line and inclined cable, which is characterized by small stiffness, low mass and light damping. And the longitudinal dimension of the structure is much larger than that of the transversal one. The structure can bear tension only but not resist compression and bending moment. For the standing exposure to the natural environment, the circular cross section of the structure can be transformed non-circular one in sleety weather, and the large vibration is susceptible to be stimulated under the wind disturbance. The galloping of the iced transmission line can lead to wire jumper, tripping operation and tower collapse. The vibration of the inclined cable induced by the wind and rain will result in the fatigue damage of the bridge, causing negative effects to the traffic safety. The theoretical analysis, numerical simulation and experiment verification are carried on the nonlinear galloping of slender-long structure with nearly elliptic non-circular cross section subjected to mean wind. Galloping mechanism and abundant nonlinear dynamics are studied. The present study has main features concluded as follows:A three degree-of-freedom continuous dynamic model for slender-long structure with nearly elliptic non-circular cross section is proposed for describing the coupling of in-plane, out-of-plane and torsional vibrations, which is built on the basis of Hamilton principle with the consideration of geometric and aerodynamic nonlinearities. Galerkin procedure is applied to spatially disperse the partial differential governing equations. Together with average method, the bifurcation equation is derived from the average equations. The relevance of the bifurcated, unfolding and physical parameters is established, in which the bifurcated and unfolding parameters are separated and decoupled. Transition sets and their corresponding regions of original physical parameters are then made on the bifurcation equation by employing the singularity theory. The topological structures of bifurcated curves in different regions are presented, where saddle nodes and jumping phenomenon are found in certain regions. Since only the first quartile in the bifurcated curves does make sense in the practical engineering, the transition set is translated into restrained one based on the restrained bifurcation theory. Numerical procedures are then implemented in the stable and jumping regions, respectively. The bifurcated diagrams obtained by numerical calculations are consistent with those derived by theoretical analysis, where periodic and chaotic solutions are observed.The nonlinear dynamic behavior of the non-circular cross section structure, which is shaped by the rivulet moving on the section, is investigated with the consideration of the equilibrium position of the moving rivulet. The partial differential governing equations of three-degree-of-freedom for the structure are established, which are proposed for describing the nonlinear interactions among the inplane, out-of-plane vibration and the oscillation of the moving rivulet. The Galerkin method is also applied to discretize the partial differential governing equations. The approximately analytic solution is obtained by using the method of averaging. The unique correspondence between the wind and the equilibrium position of the rivulet is ascertained. The presence of rivulet at certain positions on the surface of cable is then proved to be one of the trigger for galloping. The nonlinear dynamic phenomena of the structure are then studied by varying the parameters including mean wind velocity, Coulomb damping force, damping ratio, the span length, and the initial tension of the structure on the model. Jumping phenomenon is observed twice successively in certain wind velocities, raising and finishing the restricted galloping.The slender-long structure with nearly elliptic non-circular cross section subjected to wind disturbance is investigated under harmonic axial excitation. On the basis of Hamilton principle, a two degree-of-freedom dynamic model along with boundary condition are established for describing the coupling of in-plane and axial vibrations. The reduced model is derived by eliminating the axial vibration and Galerkin method is then applied to spatially disperse the partial differential equation. The galloping of straight tower-transmission line with ice accretions is taken as an instance for calculating. The basic range of the excited force is determined through the calculation of dynamic tension initiated by the adjacent span. Numerical procedures are implemented to analyze the influences of excited frequency and force on the system. Abundant motion patterns of period-doubling, almost period and chaos are observed when the excited frequency is approximately equal to the natural frequency of the iced transmission line. The numerical verification is carried out on the model without axial excitation to further prove that galloping of adjacent span can lead to obvious increase of amplitude, decline of critical wind velocity and instability of iced transmission line galloping.Multiscale method is then employed to obtained the amplitude-frequency response equation of 1/2 subharmonic resonance in the parametric excited system of the slender-long structure with nearly elliptic non-circular cross section, which is subjected to both wind disturbance and harmonic axial excitation. Take the galloping of straight tower-transmission line with ice accretions as an instance. The amplitude-frequency response curves with varying topological structures are then depicted under different parameters, which are performed to be softening-spring characteristic and one or two saddle nodes are observed. Analytical and numerical verifications are carried out on the model without axial excitation to further prove the instabilities of multi-solutions and jumping phenomenon caused by the parametric excitation. The transition sets and bifurcated curves are derived by using Singular theory. The regions corresponding to the amplitude-frequency response curves are found out, and nonlinear parametric controller is designed according to different topological structure to eliminate the saddle nodes. The simplest formula of the system and the universal unfolding of the 2-codimension degenerated bifurcation are deduced by applying Normal Form transformation. And the Pitchfork and Hopf bifurcated point sets are then got through the stability analysis of the equilibrium points. Melnikov method is applied to solve the heteroclinic bifurcation set. The point sets obtained partition the parametric plane into four regions, in which the number of equilibrium points and vector fields in the neighbourhood of the equilibrium points are varied.The multimodal galloping is carried out on the slender-long structure with nearly elliptic non-circular cross section. The coupling model of the first four in-plane modes and the first torsional mode are established(Model 1). The transformation of modal galloping with the continuous variations of parameters and modal interactions are analyzed. Eigenvalue analysis is applied on linear system to determine the switch of different modal galloping in plane wind-span(U-l), where Hopf bifurcation occurs and mono-modal, bi-modal and multi-modal galloping are observed. Various numerical procedures are implemented to capture the outstanding nonlinear dynamic features of every regional galloping in plane U-l. Higher modes are inclined to be stimulated by higher wind velocity and longer span. Internal resonance is observed and investigated to interpret the phenomenon of modal transition which is also analyzed by accounting for the influence of initial conditions on galloping with either symmetric or anti-symmetric modes. The model are then established on the coupling of the first four in-plane and torsional modes(Model 2). Through the eigenvalue analysis on the Model 2, the first four in-plane modes are all subjected to two Hopf bifurcations successively as the mean wind velocity increasing, performing restricted galloping feature. The lower mass per length of the structure has lower critical wind velocity, whose higher modes are more easily to be stimulated. Model 1 and Model 2 are then connected by discussing the impact of the torsional damping ratio on the galloping modes. Larger torsional damping ratio will cause the increase of the critical wind velocity corresponding to the second Hopf bifurcated point. Model 2 is identical to the Model 1 when the torsional damping ratio is infinite.The wind tunnel especially for continuous slender-long structure is established. And the experimental verification of multi-modal galloping is carried out on the continuous slender-long structure with nearly elliptic non-circular cross section. Laser sensor is adopted to measure the displacement at different spans. The experimental contents include the contrast test between slender-long structures with circular cross section and non-circular one, the galloping mode variations with respect to wind velocities, the influences of damping ratio and cross sectional area on the galloping mode. The experimental results implied that the instability of the aerodynamic force caused by the non-circular cross section is the most essential reason for the galloping; the galloping mode are performed to be restrict feature as the wind velocity increases, with the decrease of the previous mode and the rise of the next mode, where mono-modal, bi-modal and multi-modal galloping are observed; higher modes are susceptible to be stimulated by the larger wind velocity and lower mass per length, which are all coincident with the theoretical results qualitatively.