Abstract：A three degree-of-freedom continuous dynamic model for an iced transmission line 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. 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, providing theoretical support to practical engineering.
霍冰 1,2， 刘习军 1,2，张素侠 1,2，刘鹏 1,2. 覆冰导线非线性舞动系统的奇异性和混沌分析[J]. 振动与冲击, 2015, 34(13): 36-41.
HUO Bing1,2 LIU Xi-jun 1,2 ZHANG Su-xia 1,2 LIU Peng1,2 . Singularity and chaos of nonlinear galloping for an iced transmission line. JOURNAL OF VIBRATION AND SHOCK, 2015, 34(13): 36-41.
[1] 王少华, 蒋兴良, 孙才新. 输电线路导线舞动的国内外研究现状[J]. 高电压技术, 2005, 31(10):14-17.
WANG Shao-hua, JIANG Xing-liang, SUN Cai-xin. Study status of conductor galloping on transmission line [J]. High Voltage Engineering, 2005, 31(10):14-17.
[2] Denhartog J P. Transmission line vibration due to sleet [J]. Electrical Engineering, 1932, 51(6): 413-413.
[3] Nigol O, Buchan P G. Conductor galloping part I: Den Hartog mechanism [J]. Power Apparatus and Systems, IEEE Transactions on, 1981, 100(2): 699-707.
[4] Nigol O, Buchan P G. Conductor galloping part I: Torsional mechanism [J]. Power Apparatus and Systems, IEEE Transactions on, 1981, 100(2): 708-720.
[5] Yu P, Shah A H, Popplewell N. Inertially coupled galloping of iced conductors [J]. Journal of Applied Mechanics, 1992, 59(1), 140-145.
[6] Luongo A, Zulli D, Piccardo G. Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables [J]. Journal of Sound and Vibration, 2008, 315 (2008): 375-393.
[7] Lee C L, Perkins N C. Nonlinear oscillations of suspended cables containing a two-to-one internal resonance [J]. Nonlinear Dynamics, 1992, 3:465-490.
[8] Benedettini F, Rega G, Alaggio R. Non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions [J]. Journal of Sound and Vibration, 1995, 182(5): 775-798.
[9] Nayfeh A H, Arafat H N, Chin C M, et al. Multimode interactions in suspended cables [J]. Journal of Vibration and Control, 2002, 8(3): 337-387.
[10] 李黎, 陈元坤, 夏正春, 等. 覆冰导线舞动的非线性数值仿真研究[J]. 振动与冲击, 2011, 30(8): 107-111.
LI Li, CHEN Yuan-kun, XIA Zheng-chun, et al. Nonlinear numerical simulation study of iced conductor galloping [J]. Journal of Vibration and Shock, 2011, 30(8): 107-111.
[11] 赵莉, 严波, 蔡萌琦, 等. 输电塔线体系中覆冰导线舞动数值模拟研究[J]. 振动与冲击, 2013, 32(18):113-120.
ZHAO Li, YAN Bo, CAI Meng-qi, et al. Numerical simulation for galloping of iced conductors in a transmission tower-line system [J]. Journal of Vibration and Shock, 2013, 32(18):113-120.
[12] Liu F H, Zhang Q C, Wang W. Analysis of hysteretic strongly nonlinearity for quad iced bundle conductors [J]. Chinese Physics Letters, 2010, 27(3): 034703.
[13] Qin Z H, Chen Y S, Zhan X, et al. Research on the galloping and anti-galloping of the transmission line [J]. International Journal of Bifurcation and Chaos, 2012, 22 (2): 1250038.
[14] Liu X J, Huo B, Zhang S X. Nonlinear dynamic analysis on the rain-wind-induced vibration of cable considering the equilibrium position of rivulet [J]. Abstract and Applied Analysis, 2013, 2013: 927632(10).
[15] 李欣业, 张华彪, 高仕赵, 等. 三自由度模型覆冰输电导线舞动的数值仿真分析[J]. 河北工业大学学报, 2010, 39(3): 1-5.
LI Xin-ye, ZHANG Hua-biao, GAO Shi-zhao, et al. Numerical analysis of galloping of iced power transmission lines [J]. Journal of Hebei University of Technology, 2010, 39(3): 1-5.
[16] 侯磊, 陈予恕. 输电线路导线舞动中的混沌运动研究[J]. 振动工程学报, 2014, 27(1): 75-83
HOU Lei, CHEN Yu-shu. Study on chaos in galloping of the transmission line [J]. Journal of Vibration Engineering, 2014, 27(1): 75-83.
[17] 熊蕊, 刘向东. 含PID控制器的迟滞非线性控制系统的主共振及奇异性[J]. 振动与冲击, 2014, 33(8): 72-77.
XIONG Rui, LIU Xiang-dong. Principal resonance and singularity of a hysteretic nonlinear control system with a PID controller [J]. Journal of Vibration and Shock, 2014, 33(8): 72-77.
[18] 王晓东, 陈予恕. 一类电力系统的分岔和奇异性分析[J]. 振动与冲击, 2014, 33(4): 1-6.
WANG Xiao-dong, CHEN Yu-shu. Bifurcation and singularity analysis for a class of power systems [J]. Journal of Vibration and Shock, 2014, 33(4): 1-6.
[19] 孟遂民, 孔伟. 架空输电线路设计[M]. 北京: 中国电力出版社, 2007.
MENG Sui-min and KONG Wei. Design of Overhead Transmission Line [M]. Beijing: China Electric Power Press, 2007.
[20] Zhang Q, Popplewell N, Shah A H. Galloping of bundle conductor [J]. Journal of Sound and Vibration, 2000, 234(1): 115-134.
[21] McComber P, Paradis, A. A cable galloping model for thin ice accretions [J]. Atmospheric Research, 1998, 46(1): 13-25.
[22] 蔡君艳, 刘习军, 张素侠. 覆冰四分裂导线舞动近似解析解分析[J]. 工程力学, 2013, 30(5): 305-310.
CAI Jun-yan, LIU Xi-jun, ZHANG Su-xia. Analysis of approximate analytical solution on galloping of iced quad bundle conductors [J]. Engineering Mechanics, 2013, 30(5): 305-310.
[23] 陈予恕. 非线性振动[M]. 北京: 高等教育出版社, 2002.
CHEN Yu-shu. Nonlinear Vibration [M]. Beijing: Higher Education Press, 2002.
[24] 陈予恕. 非线性振动系统的分岔和混沌理论[M]. 北京: 高等教育出版社, 1993.
CHEN Yu-shu. Bifurcation and Chaos in Nonlinear Vibration [M]. Beijing: Higher Education Press, 1993.
