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Article(id=1243226199703535865, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, articleNumber=null, orderNo=null, doi=10.7511/jslx20240618001, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1718640000000, receivedDateStr=2024-06-18, revisedDate=1722873600000, revisedDateStr=2024-08-06, acceptedDate=null, acceptedDateStr=null, onlineDate=1774337824035, onlineDateStr=2026-03-24, pubDate=1761580800000, pubDateStr=2025-10-28, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774337824035, onlineIssueDateStr=2026-03-24, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774337824035, creator=13701087609, updateTime=1774337824035, updator=13701087609, issue=Issue{id=1243226190786441246, tenantId=1146029695717560320, journalId=1242798230522609684, year='2025', volume='42', issue='5', pageStart='699', pageEnd='888', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1774337821909, creator=13701087609, updateTime=1774338282025, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1243228120724128564, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1243228120724128565, tenantId=1146029695717560320, journalId=1242798230522609684, issueId=1243226190786441246, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=722, endPage=728, ext={EN=ArticleExt(id=1243226200861163784, articleId=1243226199703535865, tenantId=1146029695717560320, journalId=1242798230522609684, language=EN, title=In plane free vibration analysis of cable-beam composite structure with cable damage, columnId=1243226193193971746, journalTitle=Chinese Journal of Computational Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

In order to investigate the effect of cable damage on in-plane free vibration characteristics of cable-beam composite structures, three dimensionless parameters of cable damage intensity, extent and position are introduced in this paper to establish an in-plane dynamic model of single-cable cantilever beam composite structures with cable damage. The eigenvalue problem of in-plane free vibration of a single cable-cantilever beam model is solved by the method of separation of variables. At the same time, the finite element models under undamaged and damaged conditions of the cable are established for verification, and the results are in good agreement with the theoretical results. The results show that the frequency of the combined structure will decrease obviously only when the sag or damage intensity and extent of the cable are large. When the damage intensity and extent of the cable increase to a certain value, 1∶1 in-mode resonance phenomena tend to appear for high order frequencies. The asymmetric initial configuration caused by damage can increase the frequency of the combined structure, and some mixed modes of the combined structure change to local modes, while some local modes change to mixed modes.

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为了探究拉索损伤对索-梁组合结构的面内自由振动特性的影响，引入拉索损伤程度、范围和位置的三个无量纲参数，建立拉索损伤下的单索-悬臂梁组合结构面内动力学模型。应用分离变量法，确定单索-悬臂梁模型的面内自由振动特征值问题。同时，建立拉索无损和有损下的有限元模型进行对比验证，所得结果与本文理论结果吻合良好。研究表明，拉索损伤一般在拉索垂度或损伤程度、范围较大的情况下，才会对组合结构频率有明显减小；当拉索损伤程度和损伤范围增大到一定值后，高阶频率之间容易出现1∶1模态内共振现象；损伤导致的拉索非对称初始构型可导致组合结构的频率增大，组合结构部分混合模态转变为局部模态，部分局部模态转变为混合模态。

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丛云跃*（1991-），男，博士，助理教授（E-mail：）.

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丛云跃*（1991-），男，博士，助理教授（E-mail：）.

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丛云跃*（1991-），男，博士，助理教授（E-mail：）.

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Study on nonlinear coupled vibrations of damaged suspended cables with symmetry-breaking[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 471-481. 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First 10 frequencies of the single-cable cantilever beam model

, figureFileSmall=null, figureFileBig=null, tableContent=
 前10阶频率
1st2nd3rd4th5th6th7th8th9th10th 
无损理论值0.20400.61101.04351.62512.06743.10093.16744.13465.16765.2294
ANSYS0.20450.61041.03301.62042.06613.09853.15474.13315.16615.2006
相对误差/%0.48780.09831.00600.29010.06290.07740.40260.15000.02900.5538
有损理论值0.19140.60470.95661.62291.88742.83123.16593.77444.71795.2281
ANSYS0.19270.60420.94291.61821.88572.82863.15293.77234.71625.1982
相对误差/%0.67460.08281.45300.29040.09020.09190.41230.81030.05570.5752
), ArticleFig(id=1243244117283684359, tenantId=1146029695717560320, journalId=1242798230522609684, articleId=1243226199703535865, language=CN, label=表1, caption=

单索-悬臂梁模型的前10阶频率

, figureFileSmall=null, figureFileBig=null, tableContent=
 前10阶频率
1st2nd3rd4th5th6th7th8th9th10th 
无损理论值0.20400.61101.04351.62512.06743.10093.16744.13465.16765.2294
ANSYS0.20450.61041.03301.62042.06613.09853.15474.13315.16615.2006
相对误差/%0.48780.09831.00600.29010.06290.07740.40260.15000.02900.5538
有损理论值0.19140.60470.95661.62291.88742.83123.16593.77444.71795.2281
ANSYS0.19270.60420.94291.61821.88572.82863.15293.77234.71625.1982
相对误差/%0.67460.08281.45300.29040.09020.09190.41230.81030.05570.5752
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考虑拉索损伤的索-梁组合结构面内自由振动分析
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吴俊谕 1 , 丛云跃 1, 2 , 康厚军 1, 2 , 郭铁丁 1, 2 , 苏潇阳 1, 2
计算力学学报 | 研究论文 2025,42(5): 722-728
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计算力学学报 | 研究论文 2025, 42(5): 722-728
考虑拉索损伤的索-梁组合结构面内自由振动分析
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吴俊谕1, 丛云跃1, 2 , 康厚军1, 2, 郭铁丁1, 2, 苏潇阳1, 2
作者信息
  • 1.广西大学　土木建筑工程学院，南宁　530004
  • 2.广西大学　工程力学研究中心，南宁　530004

    • 丛云跃*（1991-），男，博士，助理教授（E-mail：）.

In plane free vibration analysis of cable-beam composite structure with cable damage
Junyu WU1, Yunyue CONG1, 2 , Houjun KANG1, 2, Tieding GUO1, 2, Xiaoyang SU1, 2
Affiliations
  • 1.School of Civil Engineering, Guangxi University, Nanning 530004, China
  • 2.Engineering Mechanics Research Center, Guangxi University, Nanning 530004, China
出版时间: 2025-10-28 doi: 10.7511/jslx20240618001
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摘要
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为了探究拉索损伤对索-梁组合结构的面内自由振动特性的影响，引入拉索损伤程度、范围和位置的三个无量纲参数，建立拉索损伤下的单索-悬臂梁组合结构面内动力学模型。应用分离变量法，确定单索-悬臂梁模型的面内自由振动特征值问题。同时，建立拉索无损和有损下的有限元模型进行对比验证，所得结果与本文理论结果吻合良好。研究表明，拉索损伤一般在拉索垂度或损伤程度、范围较大的情况下，才会对组合结构频率有明显减小；当拉索损伤程度和损伤范围增大到一定值后，高阶频率之间容易出现1∶1模态内共振现象；损伤导致的拉索非对称初始构型可导致组合结构的频率增大，组合结构部分混合模态转变为局部模态，部分局部模态转变为混合模态。

索梁组合结构  /  拉索损伤  /  结构动力学  /  自由振动  /  内共振

In order to investigate the effect of cable damage on in-plane free vibration characteristics of cable-beam composite structures, three dimensionless parameters of cable damage intensity, extent and position are introduced in this paper to establish an in-plane dynamic model of single-cable cantilever beam composite structures with cable damage. The eigenvalue problem of in-plane free vibration of a single cable-cantilever beam model is solved by the method of separation of variables. At the same time, the finite element models under undamaged and damaged conditions of the cable are established for verification, and the results are in good agreement with the theoretical results. The results show that the frequency of the combined structure will decrease obviously only when the sag or damage intensity and extent of the cable are large. When the damage intensity and extent of the cable increase to a certain value, 1∶1 in-mode resonance phenomena tend to appear for high order frequencies. The asymmetric initial configuration caused by damage can increase the frequency of the combined structure, and some mixed modes of the combined structure change to local modes, while some local modes change to mixed modes.

cable-beam composite structure  /  cable damage  /  structural dynamics  /  free vibration  /  internal resonance
吴俊谕, 丛云跃, 康厚军, 郭铁丁, 苏潇阳. 考虑拉索损伤的索-梁组合结构面内自由振动分析. 计算力学学报, 2025 , 42 (5) : 722 -728 . DOI: 10.7511/jslx20240618001
Junyu WU, Yunyue CONG, Houjun KANG, Tieding GUO, Xiaoyang SU. In plane free vibration analysis of cable-beam composite structure with cable damage[J]. Chinese Journal of Computational Mechanics, 2025 , 42 (5) : 722 -728 . DOI: 10.7511/jslx20240618001
正文
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1　引言
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大跨桥梁是一个国家的经济命脉，桥梁工程对经济发展有着巨大的推动作用，正因如此，大跨桥梁的安全和耐久性越来越受到重视[1]。拉索在强风、地震极端环境下容易发生大幅振动而造成疲劳损伤，导致结构动力学特性变化，受到广泛关注[2]。
目前，损伤拉索的力学特性研究已得到国内外相关学者的重视。Jiang等[3]对腐蚀环境下的拉索进行疲劳分析，发现腐蚀和疲劳会极大缩减拉索寿命。Sun等[4]建立拉索在腐蚀状态下的力学模型，重点分析腐蚀损伤对索力、垂度和频率的影响。Bouaanani[5]采用有限差分法，研究损伤位置和损伤范围对水平悬索模态的影响。Xu等[6]研究拉索断丝对拉索的静力学和动力学特性的影响，研究表明断丝损伤位置显著影响拉索的垂度。王立彬等[7]推导损伤拉索的索力和线形公式，分析数值方法、经典的等效弹性模量公式两种方法计算等效弹性模量的误差。刘庆宽等[8]研究斜拉索表面损伤对风致振动特性的影响，研究表明在不利风向下，表面的划痕、裂缝和凹痕等损伤会使拉索的振幅增大。Liu等[9]通过实验研究发现桥梁钢索在腐蚀、微动和疲劳振动等作用下，会加速钢丝索的腐蚀，使钢丝索的开裂加快，导致钢丝索寿命降低。文献[10，11]推导出损伤拉索的非线性运动方程，研究损伤对索主共振响应和面内模态共振响应的影响。诸骏[12]以单索悬臂梁体系为研究对象，使用回传射线矩阵和同伦分析法研究拉索损伤对组合体系的线性及非线性动力特性影响，但未分析损伤程度、范围等参数对索-梁组合结构的动力学影响。已有研究大多为单根悬索损伤的动力学研究。为此，本文建立拉索损伤下的索-悬臂梁组合结构面内自由振动模型，应用变量分离法并建立相应的有限元模型，详细探究拉索损伤程度、范围和损伤导致的拉索非对称构型对组合结构固有特性的影响。
2　索-梁组合结构动力学模型
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2.1　基本构型及假设
图1给出受损拉索构型及考虑拉索损伤的索-梁组合结构的静态构型。建立图1（b）中坐标系xc-oc-yc来描述拉索的运功，坐标系xb-ob-yb来描述梁的运动。vb、ub和vc、uc分别为梁和拉索的横向、纵向运动。θ为拉索的倾角。la、lc分别表示梁和拉索的跨长。EcAc为索的初始轴向刚度，EcAd为索出现损伤后的轴向刚度。拉索损伤为损伤部位的整体截面削减，其中Ad为拉索损伤部位的截面面积。假设损伤不会引起材料损失[13]，即保持拉索的单位长度质量不变。损伤拉索在无应力状态下的构形如图1（a）所示，定义索损伤区域为[a1，a2]，其中a1、a2分别为损伤位置起始点和终点的坐标，L0为索无应力状态下的初始索长。引入如下的参数定义拉索损伤对其初始索力和垂度的影响。
其中Hd（H）和bd（b）分别为有损伤（无损）时的拉索初始索力在水平方向上的分量以及初始垂度。χ2、κ2分别称为水平张力折减系数和垂度增大系数。由索力和垂度的关系可知χ2=1/κ2。
使用三个无量纲参数来定义索的损伤程度、范围和位置[14]：η为损伤程度，δ为损伤范围，α0为损伤位置。
用分段函数来描述损伤在索上的分布情况
对于张紧的拉索，损伤位置的变化对张力损失的影响很小，张力损失仅取决于损伤的程度和范围[19]。故有
为建立约化模型，引入下面的假设：
（1）在索的局部坐标系中，索的垂跨比很小（b/lc<1/10），斜拉索的静态构型可由二次抛物线表示。
（2）式（5）中拉索初始构型具有对称性。损伤会导致其初始构型为非对称，此时采用样条曲线拟合受损悬索的初始构形[15]。得到以下分段函数。
（3）忽略索的抗弯、抗扭、抗剪刚度和轴向惯性力的影响；忽略梁的剪切应变、轴向转动、扭转。
2.2　运动方程及边界条件
基于上述假设，应用Hamilton原理，并引入以下无量纲变量，即可得到索和梁的面内运动方程。
索和梁的运动方程为
其中j=0时，n=0；j=1、2、3时，n=1。mc、Ac、cc、H、Ec、yj分别为斜拉索的单位长度质量、截面面积、阻尼系数、初始索力在水平方向上的分量、弹性模量及拉索的静态构型。mb、cb、Ib、Eb分别为梁的单位长度质量、阻尼系数、截面惯性矩、弹性模量。（∙）、（′）分别表示对时间和坐标的导数。
组合结构应满足的几何边界、连续条件为
3　面内特征值问题
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忽略高阶项以及索的平均动应变e（τ）中积分号内的平方非线性项，得到无阻尼自由振动方程为
应用分离变量法，令
其中i表示虚数单位。将式（19）代入索、梁的线性振动方程，得
同理，得到线性平均动应变为
得到相应的边界和连续条件为
易知，方程（20，21）的解可表示为
将式（27，29）代入线性平均动应变式（22，24），再联合式（26）的边界和连续条件。可得到组合结构的特征方程为
其中，当n分别取0或1时，X矩阵为
式中T0、T1分别为6阶、10阶的系数矩阵。特征方程（30）的系数行列式是关于组合结构面内自振频率ω的函数，使系数行列式的值为零，可得到在拉索损伤下组合结构的各阶频率值。
4　算例分析
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4.1　特征值问题
拉索和梁的物理参数如下，斜拉索单位长度质量mc=10.4 kg/m，弹性模量Ec=210 Gpa，长度lc=150 m，截面面积Ac=6.273×10－3 m2，水平倾角θ=30o，初始索力H=1 MN；梁单位长度质量mb=4.4×104 kg/m，弹性模量Eb=34.5 Gpa，截面面积Ab=16.3 m2，截面惯性矩Ib=9.8 m4。为验证本文理论和方法的正确性，利用有限元软件ANSYS建立相同参数下的单索-悬臂梁的组合结构模型，索使用LINK1单元，梁使用Beam3单元。表1为拉索具有对称初始构型时，应用本文损伤理论方法和有限元方法计算得到的组合结构在拉索无损和有损两种状况下的前10阶频率值。其中，损伤程度η=0.5，损伤范围δ=0.2。从表1频率数据可以看出，相对误差最大仅为1.430%，两种方法的结果吻合良好。
4.2　参数分析
图2和图3给出拉索无损和有损且拉索具有对称初始构型下组合结构的前5阶频率随拉索垂度的变化曲线。图4给出考虑损伤导致的拉索非对称初始构型后的组合结构前5阶频率随拉索垂度的变化曲线。这里均取损伤程度η=0.5，损伤范围δ=0.2。从图2可以看出，随着拉索垂度的增加，结构的第1阶和第2阶频率基本不变，结构的其他各阶频率只是在一定的垂度变化区间内相应减小，这说明增大拉索的垂度只在特定范围导致其某一阶频率减小，而对其他各阶频率没影响[3]。同时看到，结构的相邻两阶频率随着拉索垂度的变化出现相互靠近又分离的现象，即veering现象。此时结构的两阶频率接近，将导致1∶1模态内共振，振动能量可在两阶模态间传递，动力学行为复杂，导致振动控制难度增加。图3与图2相比，在拉索垂度较小的情况下，损伤一般不会使结构频率产生明显改变，而在垂度较大时对结构频率产生影响但仍不显著。若垂度不变，拉索损伤使结构的各阶频率减小，高阶频率（第4阶和第5阶）之间的veering现象在更小垂度时出现。从图4可以看到，考虑拉索损伤导致的非对称初始构型，结构频率随拉索垂度变化明显，在给定的参数区间内，出现更多veering现象，如图4所示第2阶和第3阶、第3阶和第4阶分别出现新的veering现象。
随后，考虑拉索的对称初始构型，图5和图6给出不同损伤范围和损伤程度对结构的前6阶频率影响曲线。考虑拉索的非对称初始构型，图7和图8给出拉索的不同损伤范围和损伤程度对结构的前7阶频率影响曲线。从图5和图6中看到，结构各阶频率随着拉索损伤程度的增加而减小，损伤程度越大，结构频率减小愈显著。损伤范围增大，如图6所示，结构频率降低愈明显，且第4阶和第5阶频率出现新的veering现象。图5和图6分别给出veering前后组合结构的第4阶和第5阶模态，可以看出，发生veering现象后结构的第4阶和第5阶模态发生了交换。值得注意的是，第3、5、6阶频率变化明显而第1、2、4阶频率变化不明显。这是因为组合结构的第1、2、4阶为混合模态，表现为索和梁共同参与的振动，拉索的损伤对整个结构的振动影响较小。第3、5、6阶为局部模态，此时组合结构只有拉索在振动，损伤对拉索的振动影响显著。
从图7和图8看到，考虑损伤导致的拉索非对称初始构型会使系统的各阶频率增大。这是由于拉索的非对称初始构型（见公式（6））使运动方程中的线性平均动应变增大，从而提高了结构的刚度。同时，随着拉索损伤程度和范围的增大，系统的高阶频率之间出现更多新的veering现象，如图8所示。同时观察图7和图8中随着拉索的损伤程度和损伤范围的增加，第2、4、5、7阶频率显著减小，第1、3、6阶频率几乎不变。同样是此时组合结构的第1、3、6阶模态表现为混合模态，拉索的损伤对整个结构的振动影响较小；第2、4、5、7阶模态表现为局部模态，拉索的损伤对整个结构的振动影响较大。相比于图5和图6，组合结构的第2、3、4、6阶模态发生转变，即混合模态转变为局部模态，局部模态转变为混合模态。
5　结论
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为探究拉索损伤对索组合结构动力学特性的影响，本文建立拉索损伤下的单索-悬臂梁组合结构面内振动动力学模型。应用变量分离法，求解单索-悬臂梁模型的面内自由振动特征值问题。同时，建立相应的有限元模型，有限元所得结果与本文理论吻合良好。然后，在理论模型推导的基础上，围绕垂度、损伤程度和范围等参数开展系统的参数分析。探究在参数变化下系统固有特性的变化规律。研究表明拉索损伤对索组合结构的固有特性产生显著影响，特定参数下可导致多阶模态共振，主要结论如下。
（1）拉索损伤一般在拉索垂度或损伤程度、范围较大的情况下，才会对组合结构频率有明显的影响。当拉索损伤程度和损伤范围增大到一定值后，高阶频率之间容易出现veering现象。
（2）考虑损伤导致的拉索非对称初始构型，组合结构的频率增大，veering区间内存在模态交换现象。随着拉索垂度、损伤程度和范围的增大，更多相邻模态之间出现新的veering现象。
综上，拉索损伤初期对索-悬臂梁组合结构的动力学特性影响较小，达到一定程度后会显著改变组合结构的动力特性，组合结构易发生模态共振。实际工程中需要定期做好大跨桥梁拉索的健康监测。
基金
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  • 国家自然科学基金(12202109; 12372006; 12372007)
  • 广西科技基地与人才专项-青年创新人才科研专项(桂科AD22035930)
文献
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Lepidi M, Gattulli V, Vestroni F. Static and dynamic response of elastic suspended cables with damage[J]. International Journal of Solids and Structures, 2007, 44(25-26): 8194-8212.
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赵珧冰, 郑攀攀, 陈林聪, 等. 受损悬索对称性破缺下非线性耦合振动研究[J]. 力学学报, 2022, 54(2): 471-481.
(ZHAO Yaobing, ZHENG Panpan, CHEN Lincong, et al. Study on nonlinear coupled vibrations of damaged suspended cables with symmetry-breaking[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 471-481. (in Chinese))
2025年第42卷第5期
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doi: 10.7511/jslx20240618001
  • 接收时间：2024-06-18
  • 首发时间：2026-03-24
  • 出版时间：2025-10-28
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  • 收稿日期：2024-06-18
  • 修回日期：2024-08-06
基金
国家自然科学基金(12202109; 12372006; 12372007)
广西科技基地与人才专项-青年创新人才科研专项(桂科AD22035930)
作者信息
    1.广西大学　土木建筑工程学院，南宁　530004
    2.广西大学　工程力学研究中心，南宁　530004
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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