垂索-辅助索系统的建模与固有特性

孙测世; 焦德望; 赵碧航; 谭超

doi:10.6052/j.issn.1000-4750.2022.02.0136

垂索-辅助索系统的建模与固有特性

doi: 10.6052/j.issn.1000-4750.2022.02.0136

重庆交通大学土木工程学院，重庆 400074

基金项目: 国家自然科学基金项目(51808085)

详细信息

作者简介:
焦德望(1997−)，男，山东聊城人，硕士生，主要从事桥梁非线性动力学研究(E-mail: jiaodw126@163.com)

赵碧航(1996−)，男，贵州遵义人，硕士生，主要从事桥梁非线性动力学研究(E-mail: 1050126836@qq.com)

谭　超(1995−)，男，重庆人，硕士生，主要从事桥梁非线性动力学研究(E-mail: 13075423708@163.com)

通讯作者:
孙测世(1985−)，男，湖南益阳人，副教授，博士，硕导，主要从事桥梁非线性动力学研究(E-mail: suncs@hnu.edu.cn)

中图分类号: U441+.3
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- 被引次数: 0
出版历程
- 收稿日期: 2022-02-09
- 修回日期: 2022-06-02
- 网络出版日期: 2022-06-10
- 刊出日期: 2023-11-25

MODELING AND INHERENT CHARACTERISTICS OF SAGGED-CABLE-CROSSTIE SYSTEM

College of Civil Engineering, Chongqing Jiaotong University, Chongqing 400074, China

摘要

摘要: 辅助索被认为是长索振动控制中一种富有潜力的手段。为研究安装辅助索后系统的固有特性及其随关键参数的变化规律，建立了由N根垂索和M道辅助索组成的索网系统的模型，得到其无量纲运动方程，引入边界条件、连续条件和平衡条件进行求解。将模型退化到双索-辅助索系统，求得其无量纲频率方程。采用数值分析研究了Irvine参数λ²、辅助索刚度和位置、波速比η等关键参数对系统频率和模态的影响。研究表明，当两索参数相同时，垂度仅影响反相频率，故同相振动和反相振动的ω-λ²曲线分别表现为“穿越”(cross-over)现象和“转向”(veering)现象。当λ²为某特定值时，任意辅助索刚度下的一阶反相振动频率均等于一阶同相频率，从而频率曲线均通过第一个“穿越”点。两索参数相同时，增大辅助索刚度仅提高系统反相振动频率，但其增幅不超过1。两索波速不同时，系统所有频率均随波速差异增大而发生往高阶的“跳阶”(jumping)现象，且频率阶次越高“跳阶”次数越多。
- 垂索 /
- 辅助索 /
- 同相振动 /
- 反相振动 /
- 固有特性
Abstract: Crosstie is a promising mean for vibration mitigation of long inclined cables. In order to study the inherent characteristics of the system with crosstie and its variation with key parameters, a model for the sagged-cable-crosstie system composed of N vertical cables and of M crossties was established. The dimensionless equation of motion was obtained and then solved by introducing boundary, continuity, and equilibrium conditions. The general model was reduced to a double-cable-crosstie system, and its dimensionless frequency equation was obtained. The effects of key parameters on the frequencies and modes of the system were studied by numerical analysis, such as Irvine parameter λ², stiffness and location of crossties, and wave velocity ratio η. The results show that the sag only affects the out-of-phase frequency when the parameters of the two cables are the same, so the ω-λ² curves of in-phase vibration and out-of-phase vibration show cross-over phenomenon and veering phenomenon, respectively. When λ² is a certain value, the first order out-of-phase vibration frequency under any crosstie stiffnesses is equal to the first order in-phase frequency, thusly the frequency curves pass through the first cross-over point. If the parameters of the two sagged cables are the same, increasing the stiffness of the crosstie only increases the out-of-phase vibration frequency of the system, but the increase is not more than 1. If the wave velocities of the two sagged cables are different, all the frequencies of the system will be “jumping” to higher orders with the increase of the wave velocity difference, and the higher the frequency order, the more the number of “jumping”.
- sagged-cable /
- crosstie /
- in-phase vibration /
- out-of-phase vibration /
- inherent characteristic

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图 1 垂索-辅助索系统

Figure 1. The sagged-cable-crosstie system

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图 2 ω-$\lambda_j^2 $曲线

Figure 2. Curves of ω-$\lambda_j^2 $

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图 3 第一个“穿越”点附近频率曲线和典型模态图

Figure 3. Frequency curves and typical modes near the first cross-over point

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图 4 ω的三维曲面图

Figure 4. The 3-D surfaces of ω

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图 5 典型模态

Figure 5. Typical modes

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图 6 ω-η曲线图及典型模态图

Figure 6. Curves of ω-η and typical modes

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表 1 模态频率对比

Table 1. Comparison of frequencies

频率阶次	$\lambda_j^2=2 $			$\lambda_j^2=0 $			对应模态
频率阶次	本文/Hz	文献[29]/Hz	ANSYS/Hz	本文/Hz	文献[12]/Hz	ANSYS/Hz	对应模态
1	1.4630	1.463	1.4630	1.3562	1.3562	1.3562	一阶同相
2	1.8696	1.870	1.8696	1.8082	1.8082	1.8085	一阶反相
3	2.7124	2.712	2.7124	2.7124	2.7124	2.7133	二阶同相
4	3.6165	3.617	3.6167	3.6165	3.6165	3.6181	二阶反相
5	4.0727	−	4.0731	4.0685	4.0685	4.0705	三阶同相
6	5.4341	−	5.4348	5.4724	5.4247	5.4278	三阶反相
7	5.4247	−	5.4255	5.4247	5.4247	5.4278	四阶同相
8	5.4247	−	5.4255	5.4247	5.4247	5.4278	四阶反相
9	6.7818	−	6.7831	6.7809	6.7809	6.7852	五阶同相
10	7.2330	−	7.2344	7.2330	7.2330	7.2377	五阶反相
注：“−”表示文献[29]未给出该数据。

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表 2 拉索参数^[33]

Table 2. Cable parameters

索号	索长/m	索力/kN	单位长度质量/(kg·m⁻¹)
1	154.08	3831	70.1
2	139.70	3351	70.1
3	125.78	3204	65.2
4	112.28	2732	52.9
5	99.38	2394	52.9

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表 3 模态频率对比

Table 3. Comparison of frequencies

频率阶次	本文/Hz	文献[33]/Hz	ANSYS/Hz
1	1.0168	1.02	1.0112
2	1.7543	1.80	1.7441
3	1.9618	1.98	1.9532
4	2.0568	2.08	2.0507
5	2.1127	2.12	2.1081
6	2.1322	2.15	2.1328
7	2.1718	2.18	2.1726
8	2.1831	2.19	2.1832
9	2.2044	2.22	2.2046
10	2.2758	2.28	2.2789

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参考文献(38)

[1]	康厚军, 赵跃宇, 王涛. 阻尼对斜拉索大幅振动的影响[J]. 湖南大学学报(自然科学版), 2008, 35(12): 1 − 6. KANG Houjun, ZHAO Yueyu, WANG Tao. Influence of damping on large amplitude vibration of stay cables [J]. Journal of Hunan University (Natural Science Edition), 2008, 35(12): 1 − 6. (in Chinese)
[2]	孙一飞, 刘庆宽, 李震, 等. 非标准圆斜拉索风致振动时频特征研究[J]. 工程力学, 2021, 38(增刊): 52 − 57. doi: 10.6052/j.issn.1000-4750.2020.05.S009 SUN Yifei, LIU Qingkuan, LI Zhen, et al. Study on time-frequency characteristics of wind-induced vibration of non-standard circular stay cables [J]. Engineering Mechanics, 2021, 38(Suppl): 52 − 57. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.05.S009
[3]	郑云飞, 刘庆宽, 战启芳, 等. 螺旋线参数对斜拉索气动特性影响的试验研究[J]. 工程力学, 2020, 37(增刊): 301 − 306. doi: 10.6052/j.issn.1000-4750.2019.04.S057 ZHENG Yunfei, LIU Qingkuan, ZHAN Qifang, et al. Experimental study on the influence of helix parameters on the aerodynamic characteristics of stay cables [J]. Engineering Mechanics, 2020, 37(Suppl): 301 − 306. (in Chinese) doi: 10.6052/j.issn.1000-4750.2019.04.S057
[4]	王慧萍, 孙利民, 胡晓伦. 基于可变摩擦阻尼力的斜拉索半主动控制算法[J]. 工程力学, 2015, 32(11): 94 − 99. doi: 10.6052/j.issn.1000-4750.2014.04.0336 WANG Huiping, SUN Limin, HU Xiaolun. Semi active control algorithm of stay cable based on variable friction damping force [J]. Engineering Mechanics, 2015, 32(11): 94 − 99. (in Chinese) doi: 10.6052/j.issn.1000-4750.2014.04.0336
[5]	刘菁, 梁栋, 杨柳, 等. 齿轮齿条式惯质粘滞阻尼器的拉索减振机理研究[J]. 工程力学, 2023, 40(1): 155 − 167. doi: 10.6052/j.issn.1000-4750.2021.08.0597 LIU Jing, LIANG Dong, YANG Liu, et al. Study on cable damping mechanism of rack and pinion inertial viscous damper [J]. Engineering Mechanics, 2023, 40(1): 155 − 167. (in Chinese) doi: 10.6052/j.issn.1000-4750.2021.08.0597
[6]	孙利民, 狄方殿, 陈林, 等. 考虑垂度影响的拉索-双粘滞阻尼器系统振动分析[J]. 工程力学, 2022, 39(8): 49 − 60. doi: 10.6052/j.issn.1000-4750.2021.04.0262 SUN Limin, DI Fangdian, CHEN Lin, et al. Vibration analysis of cable double viscous damper system considering sag effect [J]. Engineering Mechanics, 2022, 39(8): 49 − 60. (in Chinese) doi: 10.6052/j.issn.1000-4750.2021.04.0262
[7]	刘菁, 梁栋. 垂度拉索-惯质阻尼器体系的减振分析[J]. 振动与冲击, 2021, 40(16): 29 − 38. LIU Jing, LIANG Dong. Vibration reduction analysis of inertia damper system with deflection cable [J]. Vibration and Shock, 2021, 40(16): 29 − 38. (in Chinese)
[8]	KRENK S, NIELSEN S. Vibrations of a shallow cable with a viscous damper [J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 2002, 458(2018): 339 − 357
[9]	HE X, CAI C, WANG Z, et al. Experimental verification of the effectiveness of elastic cross-ties in suppressing wake-induced vibrations of staggered stay cables [J]. Engineering Structures, 2018, 167: 151 − 165. doi: 10.1016/j.engstruct.2018.04.033
[10]	CARACOGLIA L, JONES N P. In-plane dynamic behavior of cable networks. Part 1: Formulation and basic solutions [J]. Journal of Sound and Vibration, 2005, 279(3/4/5): 969 − 991. doi: 10.1016/j.jsv.2003.11.058
[11]	CARACOGLIA L, JONES N P. In-plane dynamic behavior of cable networks. Part 2: Prototype prediction and validation [J]. Journal of Sound and Vibration, 2005, 279(3/4/5): 993 − 1014. doi: 10.1016/j.jsv.2003.11.059
[12]	AHMAD J, CHENG S. Effect of cross-link stiffness on the in-plane free vibration behaviour of a two-cable network [J]. Engineering Structures, 2013, 52: 570 − 580. doi: 10.1016/j.engstruct.2013.03.018
[13]	AHMAD J, CHENG S. Analytical study on in-plane free vibration of a cable network with straight alignment rigid cross-ties [J]. Journal of Vibration and Control, 2013, 21(7): 1299 − 1320.
[14]	AHMAD J, CHENG S, GHRIB F. An analytical approach to evaluate damping property of orthogonal cable networks [J]. Engineering Structures, 2014, 75(15): 225 − 236.
[15]	GIACCU G F, CARACOGLIA L. Effects of modeling nonlinearity in cross-ties on the dynamics of simplified in-plane cable networks [J]. Structural Control and Health Monitoring, 2012, 19: 348 − 369. doi: 10.1002/stc.435
[16]	GIACCU G F, CARACOGLIA L. Generalized power-law stiffness model for nonlinear dynamics of in-plane cable networks [J]. Journal of Sound and Vibration, 2013, 332(8): 1961 − 1981. doi: 10.1016/j.jsv.2012.12.006
[17]	GIACCU G F, CARACOGLIA L, BARBIELLINI B. Modeling “unilateral” response in the cross-ties of a cable network: Deterministic vibration [J]. Journal of Sound and Vibration, 2014, 333(19): 4427 − 4443. doi: 10.1016/j.jsv.2014.04.030
[18]	HE X, PENG Y, JING H, et al. Parametric studies on the dynamic properties of stay cables interconnected with uniformly distributed cross-ties [J]. Advances in Structural Engineering, 2019, 22(4): 882 − 892. doi: 10.1177/1369433218786544
[19]	CHEN W, ZHANG Z, ZHEN X, et al. Effect of bending stiffness on the in-plane free vibration characteristics of a cable network [J]. Journal of Mechanical Science and Technology, 2020, 34(11): 4439 − 4463. doi: 10.1007/s12206-020-1006-6
[20]	SUN L, HONG D, CHEN L. Cables interconnected with tuned inerter damper for vibration mitigation [J]. Engineering Structures, 2017, 151(15): 57 − 67.
[21]	ZHOU P, LIU M, XIAO H G, et al. Feasibility of using a negative stiffness damper to two interconnected stay cables for damping enhancement [J]. International Journal of Structural Stability and Dynamics, 2019, 19(1): 1 − 22.
[22]	ZHOU H, YANG H, PENG Y, et al. Damping and frequency of twin cables with a crosslink and a viscous damper [J]. Smart Structures and Systems, 2019, 23(6): 669 − 682.
[23]	ZHOU H, WU Y, LI L, et al. Free vibrations of a two-cable network inter-supported by cross-links extended to ground [J]. Smart Structures and Systems, 2019, 23(6): 653 − 667.
[24]	周海俊, 陈朝骏, 杨夏, 等. 形状记忆合金辅助索-简化索网模型振动特性的数值模拟及试验研究. 重庆交通大学学报(自然科学版) [J]. 2014, 33(1): 8 − 11. ZHOU Haijun, CHEN Chaojun, YANG Xia, et al. Numerical simulation and experimental study on vibration characteristics of shape memory alloy assisted cable-simplified cable net model [J]. Journal of Chongqing Jiaotong University (Natural Science Edition), 2014, 33 (1): 8 − 11. (in Chinese)
[25]	YANG X, ZHOU H J, YANG X, et al. Shape memory alloy strands as cross-ties: Fatigue behavior and model-cable net tests [J]. Engineering Structures, 2021, 245: 1 − 13.
[26]	SHI F, ZHOU Y, OZBULUT O E, et al. Development and experimental validation of anchorage systems for shape memory alloy cables [J]. Engineering Structures, 2021, 228: 1 − 13.
[27]	SUN L, HONG D, CHEN L. In‐plane free vibrations of shallow cables with cross-ties [J]. Structural Control and Health Monitoring, 2019, 26(10): 1 − 20.
[28]	CHEN L, XU Y, SUN L. A component mode synthesis method for reduced-order modeling of cable networks in cable-stayed bridges [J]. Journal of Sound and Vibration, 2020, 491(20): 1 − 24.
[29]	AHMAD J. In-plane linear dynamic behavior and impact of key system parameters of low-sagged cable network [J]. Journal of Sound and Vibration, 2021, 14(18): 1 − 20.
[30]	DI F, CHEN L, SUN L. Free vibrations of hybrid cable networks with external dampers and pretensioned cross-ties [J]. Structural Control and Health Monitoring, 2021, 156: 1 − 24.
[31]	DI F, SUN L, CHEN L. In‐plane dynamic behaviors of two-cable networks with a pretensioned cross-tie [J]. Structural Control and Health Monitoring. 2021, 2755: 1 − 24.
[32]	IRVINE H M. Cable structures [M]. New York: The Massachusetts Institute of Technology Press, 1981.
[33]	AHMAD J, CHENG S, GHRIB F. Impact of cross-tie design on the in-plane stiffness and local mode formation of cable networks on cable-stayed bridges [J]. Journal of Sound and Vibration, 2016, 363: 141 − 155. doi: 10.1016/j.jsv.2015.09.052
[34]	GATTULLI V, LEPIDI M. Localization and veering in the dynamics of cable-stayed bridges [J]. Computers and Structures, 2007, 85(21/22): 1661 − 1678.
[35]	周海俊, 朱亚峰, 杨夏, 等. 拉索-弹簧系统的振动特性研究 [J]. 振动工程学报, 2012, 25(5): 522 − 526. ZHOU Haijun, ZHU Yafeng, YANG Xia, et al. Study on vibration characteristics of cable spring system [J]. Journal of Vibration Engineering, 2012, 25 (5): 522 − 526. (in Chinese)
[36]	禹见达, 张湘琦, 于浩, 等. 多索复合阻尼索 [P]. 湖南: CN206408524U, 2017-08-15. YU Jianda, ZHANG Xiangqi, YU Hao, et al. Multi-cable composite damping cable [P]. Hunan: CN 206408524U, 2017-08-15. (in Chinese)
[37]	禹见达, 唐伊人, 张湘琦, 等. 复合阻尼索设计及减振性能试验研究 [J]. 振动工程学报, 2018, 31(4): 591 − 598. YU Jianda, TANG Yiren, ZHANG Xiangqi, et al. Design of composite damping cable and experimental study on its vibration reduction performance [J]. Journal of Vibration Engineering, 2018, 31 (4): 591 − 598. (in Chinese)
[38]	YU J, DUAN Z, ZHANG X, et al. Wind-induced vibration control of high-rise structures using compound damping cables [J]. Shock and Vibration, 2021, 2021: 1 − 9.