| Citation: |
Peng Liu, Anqi Zhou, Bing Huo, Xijun Liu. LIMIT CYCLE BIFURCATIONS IN THE IN-PLANE GALLOPING OF ICED TRANSMISSION LINE*[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1355-1374. doi: 10.11948/20190202
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LIMIT CYCLE BIFURCATIONS IN THE IN-PLANE GALLOPING OF ICED TRANSMISSION LINE*
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Abstract
In this paper, we establish a mathematical model to describe in-plane galloping of iced transmission line with geometrical and aerodynamical nonlinearities using Hamilton principle. After Galerkin Discretization, we get a two-dimensional ordinary differential equations system, further, a near Hamiltonian system is obtained by rescaling. By calculating the coefficients of the first order Melnikov function or the Abelian integral of the near-Hamiltonian system, the number of limit cycles and their locations are obtained. We demonstrate that this model can have at least 3 limit cycles in some wind velocity. Moreover, some numerical simulations are conducted to verify the theoretical results.-
Keywords:
- Galloping /
- limit cycle /
- bifurcation /
- Melnikov function.
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References
[1] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations, 250, Springer Science & Business Media, New York, 2012. [2] J. Den Hartog, Transmission line vibration due to sleet, Transactions of the American Institute of Electrical Engineers, 1932, 4(51), 1074–1076. [3] M. Han, H. Yan, J. Yang and C. Lhotka, On the number of limit cycles of some liénard systems, Canadian Applied Mathematics Quarterly, 2009, 17(1), 61–83. [4] M. Han and P. Yu, Normal forms, Melnikov functions and bifurcations of limit cycles, 181, Springer Science & Business Media, London, 2012. [5] D. Hilbert, Mathematical problems, Bulletin of the American Mathematical Society, 1902, 8, 437–479. doi: 10.1090/S0002-9904-1902-00923-3 [6] L. Hou and Y. Chen, Study on chaos in galloping of the transmission line, Journal of Vibration Engineering, 2014, 27(1), 75–83. [7] Y. Hou and M. Han, Melnikov functions for planar near-hamiltonian systems and hopf bifurcations, Journal of Shanghai Normal University(Natural Science), 2006, 35(1), 1–10. [8] B. Liu, K. Zhu, X. Li and X. Zhan, Hysteresis phenomenon in the galloping of the d-shape iced conductor, Mathematical Problems in Engineering, 2013, 784239. [9] B. Liu, K. Zhu, X. Sun et al., A contrast on conductor galloping amplitude calculated by three mathematical models with different dofs, Shock and Vibration, 2014, 2014, 781304. [10] F. Liu, Q. Zhang and Y. Tan, Analysis of high codimensional bifurcation and chaos for the quad bundle conductor's galloping, Chinese Physics Letters, 2010, 27(4), 044702. doi: 10.1088/0256-307X/27/4/044702 [11] X. Liu and B. Huo, Nonlinear vibration and multimodal interaction analysis of transmission line with thin ice accretions, International Journal of Applied Mechanics, 2015, 7(01), 1550007. doi: 10.1142/S1758825115400074 [12] Y. Liu and Y. Chen, Global bifurcations for a generalized codimension-4 duffing-van der pol equation, Journal of Vibration and Shock, 2011, 30(1), 69–72. [13] A. Luongo, D. Zulli and G. Piccardo, A linear curved-beam model for the analysis of galloping in suspended cables, Journal of Mechanics of Materials and Structures, 2007, 2(4), 675–694. doi: 10.2140/jomms.2007.2.675 [14] A. Luongo, D. Zulli and G. Piccardo, Analytical and numerical approaches to nonlinear galloping of internally resonant suspended cables, Journal of Sound and Vibration, 2008, 315(3), 375–393. doi: 10.1016/j.jsv.2008.03.067 [15] P. McComber and A. Paradis, A cable galloping model for thin ice accretions, Atmospheric Research, 1998, 46(1), 13–25. [16] S. Meng and W. Kong, Design of Overhead Transmission Line, China Electric Power Press, Beijing, 2007. [17] O. Nigol and P. Buchan, Conductor galloping part i-den hartog mechanism, IEEE transactions on power apparatus and systems, 1981, 2(PAS-100), 699–707. [18] O. Nigol and P. Buchan, Conductor galloping-part ii torsional mechanism, IEEE Transactions on Power Apparatus and Systems, 1981, 2(PAS-100), 708–720. [19] Z. Qin, Y. Chen, X. Zhan et al., Research on the galloping and anti-galloping of the transmission line, International Journal of Bifurcation and Chaos, 2012, 22(02), 1250038. doi: 10.1142/S0218127412500381 [20] X. Sun, P. Yu and Q. Bin, Global existence and uniqueness of periodic waves in a population model with density-dependent migrations and allee effect, International Journal of Bifurcation and Chaos, 2017, 27(12), 1750192. doi: 10.1142/S0218127417501929 [21] Z. Yan, Z. Yan, Z. Li and T. Tan, Nonlinear galloping of internally resonant iced transmission lines considering eccentricity, Journal of Sound and Vibration, 2012, 331(15), 3599–3616. doi: 10.1016/j.jsv.2012.03.011 [22] P. Yu, A. Shah and N. Popplewell, Inertially coupled galloping of iced conductors, Journal of applied mechanics, 1992, 59(1), 140–145. doi: 10.1115/1.2899419 [23] Q. Zhang, N. Popplewell and A. Shah, Galloping of bundle conductor, Journal of Sound and Vibration, 2000, 234(1), 115–134. -
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Peng Liu, Anqi Zhou, Bing Huo, Xijun Liu. LIMIT CYCLE BIFURCATIONS IN THE IN-PLANE GALLOPING OF ICED TRANSMISSION LINE*[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1355-1374. doi: 10.11948/20190202
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- Figure 1. The theoretical model of transmission line: (a) configuration, and (b) schematic diagram of dynamic displacement
- Figure 2. The schematic of velocity vector and the aerodynamics forces acting on the conductor section
- Figure 3. The phase portrait of system (2.23)
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Figure 4. Simulation of system (2.22) for
$ U = 3.9 $ , with initial value$ x(0) = 0.08 $ ,$ y(0) = 0 $ , converging to a stable limit cycle: (a) time evolution; and (b) phase portrait -
Figure 5. Simulation of system (2.22) for
$ U = 3.9 $ , with initial value$ x(0) = 2.95 $ ,$ y(0) = 0 $ , converging to a stable limit cycle: (a) time evolution; and (b) phase portrait -
Figure 6. Steady-state time history of system (2.17) for
$ U = 3.9 $ -
Figure 7. Simulation of system (2.22) for
$ U = 19.55 $ , with initial value$ x(0) = -0.62 $ ,$ y(0) = 0 $ , converging to a stable limit cycle: (a) time evolution; and (b) phase portrait -
Figure 8. Steady-state time history of system (2.17) for
$ U = 19.55 $