一般认为当激励频率不变时悬索响应相位为恒定值，而实际上非线性效应使相位随时间呈周期变化，且其特性与激励密切相关。为研究在分布和端部两种典型激励下悬索瞬时相频特性的异同，首先，利用伽辽金法将两种激励单独作用下的运动控制方程离散为常微分方程，采用多尺度法进行摄动求解。其次，通过数值算例，分别算得两种激励下不同Irvine参数λ2（以及对应垂跨比）和激励频率Ω时悬索的响应。最后，利用Hilbert变换分别得到响应和激励的瞬时相位，进而研究两者相位差及其幅值在λ2-Ω平面内的变化规律。研究表明：无论那种激励下，系统高阶近似解中的漂移项和二倍频项将使响应相位随时间呈周期变化。两种激励下悬索频响方程的右侧项存在差异，使响应幅值a不同，进而可以通过高阶解中的漂移项和二倍频项影响相频特性。分布和端部激励下响应-激励的瞬时相位差幅值pmax均会在λ2≈3.0且Ω≈1.12为中心的局部范围内突然增大并呈反对称分布。但是，前者突变范围为狭长带域，而后者为点域，且前者量值明显大于后者。
Abstract
It is generally believed that the response phase of the cables is constant when the excitation frequency is fixed, but in fact, the nonlinear effect makes the phase change periodically with time, and its characteristics are closely related to the excitations. The similarities and differences of the instantaneous phase-frequency characteristics of suspended cables under external and end excitations are studied. Firstly, the governing equations of motion under the two kinds of excitations are discretized into ordinary differential equations by using Galerkin method, and the Multiple Scales Method (MSM) is used to solve the equations. Secondly, the responses of the cables with different Irvine parameters λ2 (and the corresponding sag-span ratio) and excitation frequency Ω under two kinds of excitations are calculated through numerical examples, respectively. Finally, the instantaneous phases of the responses and the excitations are obtained by Hilbert transform to study the variations of their phase difference and amplitude in the λ2-Ω plane. It is shown that the drift term and the doubling-frequency term in the high-order approximate solution of the system make the response phase vary periodically with time under any kind of excitation. The right term of the frequency response equations of the suspended cables under the two excitations are different, which makes the response amplitude different, and then the phase-frequency characteristics can be affected by the drift term and the doubling-frequency term in the high-order solution. The amplitudes pmax of the instantaneous phase difference between the response and excitation under both external and end excitation suddenly increases in a local range which shows an anti-symmetric distribution centered at λ2 ≈ 3.0 and Ω ≈ 1.12. However, the value of the former is a long and narrow band, while the latter is a point field. Besides, the value of the former is significantly larger than that of the latter.
Key words: Suspended cable; Instantaneous phase-frequency characteristics; External excitation; End excitation; Instantaneous phase difference
关键词
悬索 /
瞬时相频特性 /
分布激励 /
端部激励 /
瞬时相位差
{{custom_keyword}} /
Key words
Suspended cable /
Instantaneous phase-frequency characteristics /
External excitation /
End excitation /
Instantaneous phase difference
{{custom_keyword}} /
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Irvine H M. Cable Structures[J]. Canadian Journal of Civil Engineering, 1988, 9(4):129-142.
[2] 孙测世, 周水兴. 不同端部激励下斜拉索的空间耦合振动[J]. 力学季刊, 2017, 038(003):545-551.
SUN Ceshi, ZHOU Shuixing. Spatial coupling vibrations of a stay cable subjected to different support excitations[J]. Chinese Quarterly of Mechanics, 2017, 038(003):545-551.
[3] Savor Z, Radic J, Hrelja G. Cable vibrations at Dubrovnik bridge[J]. Bridge Structures, 2006,2(2): 97-106.
[4] 符旭晨，周岱，吴筑海. 斜拉索的风振与减振[J]. 振动与冲击, 2005, 23(3)：29-32.
FU Xuchen, ZHOU Dai, WU Zhuhai. Study on wind induced vibration and vibration reduction of stayed cables[J]. Journal of Vibration and Shock, 2005, 23(3)：29-32.
[5] Ni Y Q, Wang X Y, Chen Z Q, et al. Field observations of rain-wind-induced cable vibration in cable-stayed Dongting Lake Bridge [J]. Journal of Wind Engineering and industrial Aerodynamics, 2007, 95(5): 303-328.
[6] Matsumoto M, Shiraishi N, Shirato H. Rain-wind induced vibration of cables of cable-stayed bridges[J]. Journal of Wind Engineering and industrial Aerodynamics, 1992, 43: 2011-2022.
[7] 李国豪. 桥梁结构稳定与振动(修订版) [M]. 北京：中国铁道出版社，1996
LI Guohao. Stability and Vibration of Bridge Structures[M]. Peking: China Railway Publishing House. 1996.
[8] 王 涛，沈锐利. 基于动力非线性有限元法的索－梁相关振动研究[J]. 振动与冲击, 2014, 34(5)：159-167.
WANG Tao, SHEN Ruili. Cable-beam Vibration study with nonlinear dynamic FEM[J]. Journal of Vibration and Shock, 2014, 34(5): 159-167.
[9] Rega G, Alaggio R, Benedettini F. Experimental investigation of the nonlinear response of a hanging cable. Part P: Local Analysis [J]. Nonlinear Dynamics, 1997, 14: 89-117.
[10] Zhao Y, Huang C, Chen L, et al. Nonlinear vibration behaviors of suspended cables under two-frequency excitation with temperature effects[J]. Journal of Sound and Vibration. 2018, 416: 279-294.
[11] Wang L H, Zhao Y Y. Nonlinear interactions and chaotic dynamics of suspended cables with three-to-one internal resonances[J]. International Journal of Solids and Structures,2006,43(25): 7800-7819.
[12] Rega G, Alaggio R, Benedettini F. Experimental investigation of the nonlinear response of a hanging cable. Part I: Local Analysis [J]. Nonlinear Dynamics, 1997, 14(2): 89-117.
[13] Bossens F, Preumont A. Active tendon control of cable-stayed bridges: a large-scale demonstration[J]. Earthquake Engineering & Structural Dynamics, 2001, 30: 961-979.
[14] Baicu C F, Rahn C D, Dawson D M. Exponentially stabilizing boundary control of string-mass systems[J]. Journal of Vibration and Control, 1999, 5(3): 491-502.
[15] Zhao Y, Huang C, Chen L, et al. Nonlinear vibration behaviors of suspended cables under two-frequency excitation with temperature effects[J]. Journal of Sound and Vibration. 2018, 416: 279-294.
[16] Kim H K, Kim K T, Lee H, et al. Performance of unpretensioned wind stabilizing cables in the construction of a cable-stayed bridge[J]. Journal of Bridge Engineering, 2013, 18(8): 722-734.
[17] 邓正科, 孙测世, 杨汝东. 不同索力斜拉索的主共振瞬时相频特性[J]. 应用数学和力学. 2021, 42(11): 1126-1135.
DENG Zhengke, SUN Ceshi, YANG Rudong. Transient primary resonance phase-frequency characteristics of stay cables with different tensions[J]. Applied Mathematics and Mechanics. 2021, 42(11): 1126-1135.
[18] Wang, L H, Zhao, Y Y. Large amplitude motion mechanism and non-planar vibration character of stay cables subject to the support motions [J]. Journal of Sound and Vibration, 2009, 327(1-2): 121-133.
[19] Zhao Y B, Sun C S, Wang Z Q, et al. Analytical solutions for resonant response of suspended cables subjected to external excitation[J]. Nonlinear Dynamics, 2014, 78(2): 1017-1032.
[20] Sun C S, Zhao Y B, Wang Z Q, et al. Effects of longitudinal girder vibration on nonlinear cable responses in cable-stayed bridges [J]. European Journal of Environmental and Civil Engineering. 2017. 01. 94-107.
[21] Wu Q, Takahashi K, Nakamura S. Formulae for frequencies and modes of in-plane vibrations of small-sag inclined cables[J]. Journal of Sound and Vibration, 2005, 279(3-5): 1155-1169.
[22] Arafat H.N, Nayfeh A.H. Non-linear responses of suspended cables to primary resonance excitations [J]. Journal of Sound and Vibration, 2003, 266(2): 325-354.
[23] Sun C S, Zhou X K, Zhou S X. Nonlinear responses of suspended cable under phase-differed multiple support excitations [J]. Nonlinear DynamicsDyn, 2021, 104(2): 1097-1116.
[24] 李金海, 李世兵, 张松林. 考虑斜拉索刚度, 垂度, 阻尼的非线性运动方程研究[J]. 防灾减灾工程学报, 2010, 2(1): 222-225.
LI Jinhai, LI Shibing, ZHANG Songlin. Study on nonlinear equations of motion of stay cables considering stiffness, sag and damping[J]. Journal of Disaster Prevention and Mitigation Engineering, 2010, 2(1): 222-225.
{{custom_fnGroup.title_cn}}
脚注
{{custom_fn.content}}
PDF(1285 KB)
