Periodic motions and limit cycles of linear cable galloping
Tóm tắt
In this paper, analytical galloping dynamics of linear cables with the uniform airflow of the wind is discussed through a two-degree-of-freedom nonlinear oscillator. The nonlinearity in the two-degree-of-freedom oscillator is only from aero-dynamic forces caused by the uniform airflow. The analytical solutions of periodic motions for linear cable galloping are discussed from the analytical method with the finite Fourier series. The corresponding stability and bifurcation of the periodic motions of the linear cable galloping are determined through the two-degree-of-freedom nonlinear system. Frequency-amplitude analysis of periodic motions and limit cycles of linear cable galloping are presented. Numerical illustrations of trajectories and amplitude spectrums are given for galloping motions of linear cables. From such analytical solutions, galloping phenomenon in flow-induced vibration can be further understood. The galloping dynamics of linear cables is similar to the dynamics of the van der Pol oscillator.
Tài liệu tham khảo
den Hartog JP (1932) Transmission line vibration due to sleet. AIEE Trans 4(51):1074–1086
Parkinson GV (1989) Phenomena and modeling of flow-induced vibrations of buff bodies. Prog Aerosp Sci 26:169–224
Blevins RD (1977) Flow-induced vibration. Van Nostrand Reinhold Co., New York
Blevins RD(1974) Flow induced vibration. Ph.D. thesis California Institute of Technology, Pasadena
Nigol O, Buchan PG (1981) Conductor galloping 2: torsional mechanism. IEEE Trans 100(2):708–720
Richardson AS Jr (1981) Dynamic analysis of lightly iced galloping in two degrees of freedom. Proc IEEE C 128(4):211–218
Richardson AS Jr (1988) Bluff body aerodynamics. ASCE J Struct Eng 112(7):1723–1726
Edwards AT, Madeyski A (1956) Progress report on the investigation of galloping of transmission line conductors. AIEE Trans 75:666–686
Blevins RD, Iwan WD (1974) The galloping response of a two-degree-of-freedom system. ASME J Appl Mech 96(3):1113–1118
Desai YM, Shah AH, Popplewell N (1990) Galloping analysis for two degree-of-freedom oscillator. J Eng Mech 116(12):2583–2602
Yu P, Shah AH, Popplewell N (1993) Three-degree-of-freedom model for galloping (Parts I and II). J Eng Mech 119:2405–2448
Slate JE (1959) Aeroelastic instability of a structural angle section. Ph.D. thesis University of British Columbia, Vancouver
Nayfeh AH (1981) Introduction to perturbation techniques. Wiley, New York
Luo ACJ (2012) Continuous dynamical systems. Higher Education Press/L&H Scientific Publishing, Beijing/Glen Carbon
Luo ACJ (2014) Toward analytical chaos in nonlinear systems. Wiley, New York
Luo ACJ, Huang JZ (2012) Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance. J Vib Control 18:1661–1671
Luo ACJ, Huang JZ (2012) Analytical dynamics of period-m flows and chaos in nonlinear Systems. Int J Bifurc Chaos 22(4):Article No. 1250093
Luo ACJ, Huang JZ (2012) Analytical routes of period-1 motions to chaos in a periodically forced Duffing oscillator with a twin-well potential. J Appl Nonlinear Dyn 1:73–108
Luo ACJ, Huang JZ (2012) Unstable and stable period-m motions in a twin-well potential Duffing oscillator. Discontin Nonlinear Complex 1:113–145
Luo ACJ, Yu B (2013) Analytical solutions for stable and unstable period-1 motions in a periodically forced oscillator with quadratic nonlinearity. ASME J Vib Acoust 135(3):Article No:034503
Luo ACJ, Yu B (2013) Period-m motions and bifurcation trees in a periodically excited, quadratic nonlinear oscillator. Discontin Nonlinear Complex 2(3):263–288
Luo ACJ, Yu B (2015) Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. J Vib Control 21(5):896–906
Luo ACJ, Laken AB (2013) Analytical solutions for period-m motions in a periodically forced van del Pol oscillator. Int J Dyn Control 1(2):99–155
Cartwright ML, Littlewood JE (1945) On nonlinear differential equations of the second order I. The equation \(\ddot{y}-k(1-y^{2}){\dot{y}}+y=b\lambda k\cos (\lambda t+\alpha ), k\) large. J Lond Math Soc 20:180–189
Levinson N (1949) A second order differential equation with singular solutions. Ann Math Second Ser 50(1):127–153
Luo ACJ, Yu B (2015) Bifurcation trees of period-1 motions to chaos in a two-degree-of-freedom, nonlinear oscillator. Int J Bifurc Chaos 25:Article No.1550179
Luo ACJ, Yu B (2016) Analytical period-1 motions to chaos in a two-degree-of-freedom oscillator with a hardening nonlinear spring. Int J Dyn Control (in press)
Caetano EDS (2007) Cable vibrations in cable-stayed bridges, International Association for Bridge and Structural Engineering, Zurich, Switzerland