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Article(id=1241784307379934030, tenantId=1146029695717560320, journalId=1241715181215068175, issueId=1241784304989175988, articleNumber=null, orderNo=null, doi=10.6052/1672-6553-2025-042, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1740412800000, receivedDateStr=2025-02-25, revisedDate=1745337600000, revisedDateStr=2025-04-23, acceptedDate=null, acceptedDateStr=null, onlineDate=1773994050115, onlineDateStr=2026-03-20, pubDate=null, pubDateStr=null, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773994050115, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773994050115, creator=13701087609, updateTime=1773994050115, updator=13701087609, issue=Issue{id=1241784304989175988, tenantId=1146029695717560320, journalId=1241715181215068175, year='2025', volume='23', issue='10', pageStart='1', pageEnd='96', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773994049546, creator=13701087609, updateTime=1773994239413, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241785101412012589, tenantId=1146029695717560320, journalId=1241715181215068175, issueId=1241784304989175988, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241785101412012590, tenantId=1146029695717560320, journalId=1241715181215068175, issueId=1241784304989175988, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=18, endPage=25, ext={EN=ArticleExt(id=1241784308143297362, articleId=1241784307379934030, tenantId=1146029695717560320, journalId=1241715181215068175, language=EN, title=Pedestrian-Induced Lateral Sway Analysis of Flexible Suspension Footbridges under Geometric Nonlinearity, columnId=1241784307421872311, journalTitle=Journal of Dynamics and Control, columnName=Research Articles, runingTitle=null, highlight=null, articleAbstract=

To investigate the influence of geometric nonlinearity on human-induced vibrations of flexible suspension bridges. A nonlinear finite-element model of a flexible pedestrian suspension bridge is established based on an engineering background, and validated using measured data. On this basis, nonlinear transient vibration analysis of the suspension bridge, considering geometric nonlinearity, is conducted. The analysis reveals that the structural displacement response time histories and time-frequency characteristics under different main cable sag-to-span ratios and excitation amplitudes, as well as the response-excitation amplitude curves. The results further show that single-frequency excitation at low-order vertical modes can induce high-order frequency vibrations at 1：2 and 1：3 ratios. When the ratio of vertical to horizontal natural frequencies is close to 2：1, a certain level of vertical excitation on the main girder can cause lateral sway of the structure. Increasing the main cable sag-to-span ratio can effectively suppress vertical and lateral coupling vibrations. As the vertical excitation level increases, the sway amplitude exhibits a sudden jump and significant increase at a critical excitation level. Under pedestrian-induced excitation, the flexible suspension bridge exhibits significant geometric nonlinear vibration characteristics.

, correspAuthors=Liang Xu, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Junbao Yang, Liang Xu, Houjun Kang, Yi Hui, Jianbing Chen), CN=ArticleExt(id=1241784313667195811, articleId=1241784307379934030, tenantId=1146029695717560320, journalId=1241715181215068175, language=CN, title=几何非线性影响下柔性悬索桥人致晃动分析, columnId=1241784307690307771, journalTitle=动力学与控制学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

为了研究结构几何非线性对柔性悬索桥人致振动的影响情况，以某柔性人行悬索桥为工程背景建立其非线性有限元模型，并基于实测结果进行验证；据此开展悬索桥考虑几何非线性的非线性振动瞬态分析，得到了不同主缆垂跨比和激励幅值影响下的结构位移响应时程和时频，以及响应—激励幅值曲线.结果表明，在低阶竖向模态的单频激励会引起1：2和1：3的高阶频率振动；当竖向和横向自振频率比接近2：1时，一定水平的主梁竖向激励会引起结构的横向晃动；增大主缆垂跨比能有效抑制竖向和侧向耦合振动的发生；随着竖向激励水平的提高，晃动幅值会在某个临界激励处突发性跳跃和显著增大；在行人强迫激励下柔性悬索桥表现出了显著的几何非线性振动特性.

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† E-mail:
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language=EN, label=Fig.7, caption=The first three lateral and vertical vibration mode shapes of the Ramello pedestrian suspension bridge：（a） Lateral modes；（b） Vertical modes, figureFileSmall=WfbN9sIHtVSgV0VEHRp9GQ==, figureFileBig=RZYOuaa1IWwfaq2AoxtjMQ==, tableContent=null), ArticleFig(id=1241784322252935291, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=图7, caption=Ramello人行悬索桥的前3阶侧向和竖向振动模态形状：（a）侧向模态；（b）竖向模态, figureFileSmall=WfbN9sIHtVSgV0VEHRp9GQ==, figureFileBig=RZYOuaa1IWwfaq2AoxtjMQ==, tableContent=null), ArticleFig(id=1241784322374570113, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Fig.8, caption=Time-history and corresponding time-frequency spectrum of mid-span displacement response of the main girder under Case 1：（a） Vertical displacement response at mid-span；（b） Lateral displacement response at mid-span, figureFileSmall=1JZPtAnshiAPWE8pBZmj/Q==, 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articleId=1241784307379934030, language=CN, label=图9, caption=对应工况1条件下的主缆响应时程图：（a）1/2跨竖向；（b）1/2跨侧向；（c）1/4跨竖向；（d）1/4跨侧向, figureFileSmall=YiZAVvfYOG3c+ldTTyHd+g==, figureFileBig=/KlPXHE6t7YIdYuI2YbZuQ==, tableContent=null), ArticleFig(id=1241784322970161296, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Fig.10, caption=Time-history and corresponding time-frequency spectrum of mid-span displacement response of the main girder under Case 2：（a） Vertical direction；（b） Lateral direction, figureFileSmall=E7OB8vPLN9Ozze2Q8Clp+g==, figureFileBig=JoNk3eLzsqXJlM+9VuK7oQ==, tableContent=null), ArticleFig(id=1241784324454944918, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=图10, caption=对应工况2条件下的主梁跨中位移响应时程图和时频谱：（a）竖向；（b）侧向, figureFileSmall=E7OB8vPLN9Ozze2Q8Clp+g==, figureFileBig=JoNk3eLzsqXJlM+9VuK7oQ==, tableContent=null), ArticleFig(id=1241784324580774042, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Fig.11, caption=Time-history and corresponding time-frequency spectrum of mid-span displacement response of the main girder under Case 3：（a） Vertical direction；（b） Lateral direction, figureFileSmall=bH5kxMTkiWwFFmqtXCcSbQ==, figureFileBig=ZmQsY3N2NJ20uCWpExfXAw==, tableContent=null), ArticleFig(id=1241784324664660126, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=图11, caption=对应工况3条件下的主梁跨中位移响应时程图和时频谱：（a）竖向；（b）侧向, figureFileSmall=bH5kxMTkiWwFFmqtXCcSbQ==, figureFileBig=ZmQsY3N2NJ20uCWpExfXAw==, tableContent=null), ArticleFig(id=1241784324815655077, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Fig.12, caption=Time-history spectrum of mid-span displacement response of the main girder under varying sag-to-span ratios of main cables：（a） Vertical direction；（b） Lateral direction, figureFileSmall=OL638VHP/oNHjcKmheVyJA==, figureFileBig=R5T8jv8OPhOIjxFUUAIlOQ==, tableContent=null), ArticleFig(id=1241784324912124073, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=图12, caption=不同主缆垂跨比影响下的主梁跨中位移时程图：（a）竖向；（b）侧向, figureFileSmall=OL638VHP/oNHjcKmheVyJA==, figureFileBig=R5T8jv8OPhOIjxFUUAIlOQ==, tableContent=null), ArticleFig(id=1241784325033758893, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Fig.13, caption=Variation of the main girder's response with the amplitude of external excitation：（a） At mid-span；（b） At quarter-span, figureFileSmall=+llawC1PkvlAM37gX25vTw==, figureFileBig=5F1YNLL0g9JHIbri9zaLqQ==, tableContent=null), ArticleFig(id=1241784325134422193, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=图13, caption=主梁响应随外激励幅值的变化情况：（a）跨中；（b）1/4跨, figureFileSmall=+llawC1PkvlAM37gX25vTw==, figureFileBig=5F1YNLL0g9JHIbri9zaLqQ==, tableContent=null), ArticleFig(id=1241784325226696885, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Table 1, caption=

Geometric and mechanic parameter of Ramello suspension footbridge[13]

, figureFileSmall=null, figureFileBig=null, tableContent=
ComponentParameterUnitValue
Actual bridge Finite element model
Area（Ab）m2 1.127×10-3×2 2.438×10-2
Vertical moment of inertia（Iz）m4 2.118×10-6 2.118×10-6
Longitudinal beam Lateral moment of inertia（Iy）m4 1.786×10-3 1.786×10-3
Elastic modulus（Eb） GPa 210 210
Density（ρb）kg/m3 7850 7850
Area（Ag）m2 1.434×10-3 1.434×10-5
Transverse beam Elastic modulus（Eb）GPa 210 2.1×105
Density（ρg）kg/m3 7850 1.0×10-6
Area（Ah）m2 7.29×10-4 7.29×10-4
Hanger Elastic modulus（Eh）GPa 210 2.1×105
Density（ρh）kg/m3 7850 1.0×10-6
Area（Ac）m2 1.26×10-3×3 2.84×10-3
Cable Elastic modulus（Ec）GPa 160 160
Density（ρc）kg/m378507850
), ArticleFig(id=1241784325365108921, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=表1, caption=

Ramello人行悬索桥的几何和材料参数[13]

, figureFileSmall=null, figureFileBig=null, tableContent=
ComponentParameterUnitValue
Actual bridge Finite element model
Area（Ab）m2 1.127×10-3×2 2.438×10-2
Vertical moment of inertia（Iz）m4 2.118×10-6 2.118×10-6
Longitudinal beam Lateral moment of inertia（Iy）m4 1.786×10-3 1.786×10-3
Elastic modulus（Eb） GPa 210 210
Density（ρb）kg/m3 7850 7850
Area（Ag）m2 1.434×10-3 1.434×10-5
Transverse beam Elastic modulus（Eb）GPa 210 2.1×105
Density（ρg）kg/m3 7850 1.0×10-6
Area（Ah）m2 7.29×10-4 7.29×10-4
Hanger Elastic modulus（Eh）GPa 210 2.1×105
Density（ρh）kg/m3 7850 1.0×10-6
Area（Ac）m2 1.26×10-3×3 2.84×10-3
Cable Elastic modulus（Ec）GPa 160 160
Density（ρc）kg/m378507850
), ArticleFig(id=1241784325461577916, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Table 2, caption=

Modal results of Ramello suspension footbridge：comparison between finite element model and field measured

, figureFileSmall=null, figureFileBig=null, tableContent=
ModeMode shapeModel frequency/HzMeasured frequency/HzΔf（%）MACModeMode shapeModel frequency/HzMeasured frequency/HzΔf（%）MAC
1 1st LS 0.315///11 3rd VA 1.301 1.345 3.34 0.928
2 1st VA 0.431///12 VS 3rd 1.505 1.531 1.71 0.886
3 1st VS 0.606 0.612 1.03 0.987 13 4th LS 1.589///
4 2nd VA 0.855 0.894 4.42 0.998 14LS 5th 1.591///
5 2nd LS 0.913///15 4th VA 1.768 1.803 1.95 0.986
6 3rd LS 0.972///16VS 4th 1.947 1.980 1.66 0.983
7 1st LA 1.047///17 4th LA 2.021///
8 2nd VS 1.052 1.089 3.44 0.973 18LA 5th 2.021///
9 2nd LA 1.141///19 6th VS 2.160///
103rd LA1.216///205th VA2.2642.3112.02/
), ArticleFig(id=1241784325566435518, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=表2, caption=

Ramello人行悬索桥的模态分析结果：有限元与实测值对比

, figureFileSmall=null, figureFileBig=null, tableContent=
ModeMode shapeModel frequency/HzMeasured frequency/HzΔf（%）MACModeMode shapeModel frequency/HzMeasured frequency/HzΔf（%）MAC
1 1st LS 0.315///11 3rd VA 1.301 1.345 3.34 0.928
2 1st VA 0.431///12 VS 3rd 1.505 1.531 1.71 0.886
3 1st VS 0.606 0.612 1.03 0.987 13 4th LS 1.589///
4 2nd VA 0.855 0.894 4.42 0.998 14LS 5th 1.591///
5 2nd LS 0.913///15 4th VA 1.768 1.803 1.95 0.986
6 3rd LS 0.972///16VS 4th 1.947 1.980 1.66 0.983
7 1st LA 1.047///17 4th LA 2.021///
8 2nd VS 1.052 1.089 3.44 0.973 18LA 5th 2.021///
9 2nd LA 1.141///19 6th VS 2.160///
103rd LA1.216///205th VA2.2642.3112.02/
), ArticleFig(id=1241784325671293122, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=EN, label=Table 3, caption=

Loading cases

, figureFileSmall=null, figureFileBig=null, tableContent=
Load casek1/mFrequency/Hzk2/（N/m2）Af /（N/m）
1 4.53 0.606 275 1247
2 4.53 1.947 50 227
34.530.9723061386
), ArticleFig(id=1241784325822288068, tenantId=1146029695717560320, journalId=1241715181215068175, articleId=1241784307379934030, language=CN, label=表3, caption=

荷载工况

, figureFileSmall=null, figureFileBig=null, tableContent=
Load casek1/mFrequency/Hzk2/（N/m2）Af /（N/m）
1 4.53 0.606 275 1247
2 4.53 1.947 50 227
34.530.9723061386
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几何非线性影响下柔性悬索桥人致晃动分析
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杨军宝 1 , 徐亮 1, 2, † , 康厚军 3 , 回忆 4 , 陈建兵 1
动力学与控制学报 | 研究论文 2025,23(10): 18-25
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动力学与控制学报 | 研究论文 2025, 23(10): 18-25
几何非线性影响下柔性悬索桥人致晃动分析
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杨军宝1, 徐亮1, 2, † , 康厚军3, 回忆4, 陈建兵1
作者信息
  • 1.苏州科技大学　土木工程学院，苏州　250009
  • 2.重庆交通大学　山区桥梁及隧道工程国家重点实验室，重庆　402247
  • 3.广西大学　土木建筑工程学院，南宁　530004
  • 4.重庆大学　土木工程学院，重庆　400044

通讯作者:

Pedestrian-Induced Lateral Sway Analysis of Flexible Suspension Footbridges under Geometric Nonlinearity
Junbao Yang1, Liang Xu1, 2, † , Houjun Kang3, Yi Hui4, Jianbing Chen1
Affiliations
  • 1.School of Civil Engineering, Suzhou University of Science and Technology, Suzhou　250009, China
  • 2.State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing　402247, China
  • 3.School of Civil Engineering and Architecture, Guangxi University, Nanning　530004, China
  • 4.School of Civil Engineering, Chongqing University, Chongqing　400044, China
doi: 10.6052/1672-6553-2025-042
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摘要
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为了研究结构几何非线性对柔性悬索桥人致振动的影响情况，以某柔性人行悬索桥为工程背景建立其非线性有限元模型，并基于实测结果进行验证；据此开展悬索桥考虑几何非线性的非线性振动瞬态分析，得到了不同主缆垂跨比和激励幅值影响下的结构位移响应时程和时频，以及响应—激励幅值曲线.结果表明，在低阶竖向模态的单频激励会引起1：2和1：3的高阶频率振动；当竖向和横向自振频率比接近2：1时，一定水平的主梁竖向激励会引起结构的横向晃动；增大主缆垂跨比能有效抑制竖向和侧向耦合振动的发生；随着竖向激励水平的提高，晃动幅值会在某个临界激励处突发性跳跃和显著增大；在行人强迫激励下柔性悬索桥表现出了显著的几何非线性振动特性.

柔性悬索桥  /  人致振动  /  非线性动力学  /  晃动失稳

To investigate the influence of geometric nonlinearity on human-induced vibrations of flexible suspension bridges. A nonlinear finite-element model of a flexible pedestrian suspension bridge is established based on an engineering background, and validated using measured data. On this basis, nonlinear transient vibration analysis of the suspension bridge, considering geometric nonlinearity, is conducted. The analysis reveals that the structural displacement response time histories and time-frequency characteristics under different main cable sag-to-span ratios and excitation amplitudes, as well as the response-excitation amplitude curves. The results further show that single-frequency excitation at low-order vertical modes can induce high-order frequency vibrations at 1：2 and 1：3 ratios. When the ratio of vertical to horizontal natural frequencies is close to 2：1, a certain level of vertical excitation on the main girder can cause lateral sway of the structure. Increasing the main cable sag-to-span ratio can effectively suppress vertical and lateral coupling vibrations. As the vertical excitation level increases, the sway amplitude exhibits a sudden jump and significant increase at a critical excitation level. Under pedestrian-induced excitation, the flexible suspension bridge exhibits significant geometric nonlinear vibration characteristics.

flexible suspension bridge  /  pedestrian-induced vibration  /  nonlinear dynamics  /  swaying instability
杨军宝, 徐亮, 康厚军, 回忆, 陈建兵. 几何非线性影响下柔性悬索桥人致晃动分析. 动力学与控制学报, 2025 , 23 (10) : 18 -25 . DOI: 10.6052/1672-6553-2025-042
Junbao Yang, Liang Xu, Houjun Kang, Yi Hui, Jianbing Chen. Pedestrian-Induced Lateral Sway Analysis of Flexible Suspension Footbridges under Geometric Nonlinearity[J]. Journal of Dynamics and Control, 2025 , 23 (10) : 18 -25 . DOI: 10.6052/1672-6553-2025-042
正文
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引言
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柔性悬索桥凭借跨越能力强、施工便捷及造型美观等优势，成为城市与山区景观桥的首选[1].但其轻柔低频特性导致结构自振频率接近行人步频，引发大幅共振甚至失稳垮塌，如2022年印度古吉拉特邦悬索桥坍塌（致死141人）、2021年和2024年分别在我国河南中牟县和新疆伊犁自治州相继发生柔性悬索桥人致倾覆等事故，均凸显人致振动研究的紧迫性[2].
人致大幅振动是所有人行桥面临的共性难题.行人周期性力与桥梁固有频率匹配时极易诱发振动，典型案例为2000年伦敦千禧桥开放首日因行人同步行走引发的发散式晃动，该现象促使全球学者聚焦人致振动机理研究[3].现有机理学说主要包括三派：（1）直接共振理论认为侧向激励频率与桥梁固有频率共振导致失稳[4]，但无法解释临界人数后的振幅突变；（2）负阻尼理论提出行人群体形成负阻尼抵消结构正阻尼[5]，但未能阐明竖向与横向频率2：1的关联；（3）几何非线性理论指出柔性桥梁的模态耦合效应是核心机制，当竖向自振频率为横向的2倍时，竖向能量通过内共振触发侧向振幅跃变[6-8]，该理论通过弹簧摆模型复现了临界阈值下的失稳现象，成为解释人行桥失稳的主流框架.
既有研究多基于试验与理论分析，试验虽能获取真实响应数据，但成本高且条件受限；理论模型依赖简化假设，与实际偏差较大[9].相比之下，有限元方法兼具普适性与经济性，可规避试验限制并减少假设误差，已广泛应用于人行桥振动研究[10]，但现有成果多局限于线性分析及普通桥型[11].针对显著非线性特性的悬索桥[12]，其人致跨模态失稳特性仍不清楚.
为此，本研究以实际柔性悬索桥为背景，建立有限元模型并进行试验验证，针对考虑几何非线性的人致悬索桥跨模态振动特性进行研究，为柔性悬索桥的防振减灾设计提供了有益参考.
1　人行悬索桥系统仿真
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1.1　工程背景
本文以意大利La Spezia乡村的Ramello人行悬索桥为研究对象（图1），该桥为单跨结构，跨长90 m、宽2.28 m，因结构轻盈对荷载敏感，曾发生显著人致振动.其主梁体系包含：纵向1 m间距钢横梁、横向1.78 m间距工字钢纵梁及木质桥面板（纵梁支撑于吊杆连接的横梁，桥面板置于纵梁上），整体由双钢绞线主缆（单缆截面3.77×10-3 m2，垂度7.05 m）通过吊杆悬吊.主缆两端分别锚固于距左塔7.9 m、右塔7.2 m处，单侧设89根间距1 m的吊杆[几何构型见图1(b)].主缆、吊杆和横/纵梁均为钢材，其弹性模量210 GPa、密度7850 kg/m3（见表1）；木板密度700 kg/m3，结构阻尼比为0.5%，其余参数详见文献[12].
1.2　建模过程
基于ANSYS建立Ramello人行悬索桥有限元模型，几何与材料参数参照表1.模型主梁采用鱼骨模型模拟，其中横梁视为刚性梁（仅绕纵轴转动）；双纵梁等效为单根，桥面板及横梁质量等效至纵梁.单元划分：纵梁离散为450个Beam4梁单元，单主缆由90个仅受拉的Link10单元构成，吊杆采用Link180单元（通过实常数定义拉压状态）.吊杆与主缆/主梁采用主从节点约束（X/Y/Z平动耦合），主缆和纵梁两端铰接（平动固定，转动自由），模型构型见图2.主缆内力通过迭代找形确定（流程见图3），跨中位移为控制条件，其误差阈值ε≤1×10-6，确保主梁水平且主缆呈二次抛物线；经找形后的主缆张力（Nc）、最终构型和目标构型误差分别如图4和5所示；吊杆初始内力设为0，经找形迭代后在静力平衡状态下的各吊杆内部张力（Nh）如图6所示.
1.3　动力特性与模型验证分析
基于有限元模型的模态分析获得了其前20阶模态（见表2），主要表现为竖向与侧向振动，其中LS/LA、VS/VA分别代表横向和竖向对称/反对称模态，1st~3rd表示阶数（如1st LS为第1阶横向对称振动）.模型频率与Elyas Bayat等实测数据对比显示误差Δf范围为1.03%~4.42%，考虑实测试验离散性，该误差处于合理接受范围.
图7呈现了前3阶竖向和侧向的模态形状，从图中可以看出，竖向振动模态均为整体模态，而侧向振动模态存在局部模态和整体模态之分，局部模态以主缆的振动为主导，整体模态为主梁和主缆共同主导，这与实测结果一致.进一步采用模态置信准则（MAC）对模态形状进行对比验证，当MAC值大于0.8时即表明模态形状吻合良好.对本模型振型与实测结果计算，所得MAC值见表2，结果显示所有模态中最小MAC值为0.886，其他MAC值皆大于0.9.上述对比结果显示，本文模型的模态频率和形状皆与实测值相近，表明本文有限元模型能够真实反映Ramello人行悬索桥的模态特性.
2　非线性振动响应分析
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2.1　加载工况
开展人致强迫振动分析时考虑最不利情况，即将行人沿纵向满布全桥，且所有行人对桥梁进行同步激励并与之同频共振.中国《城市人行天桥与人行地道技术规范》（CJJ 69-95）[14]、瑞典规范Bro2004[15]、法国人行桥指南[16]在内的多国规范指出，当行人与桥梁共振时，行人荷载的1阶谐波荷载分量占主导作用，此时行人竖向和横向作用均可简化为简谐荷载.以瑞典规范Bro2004为例，对于高密度人群分布的情况，行人荷载可视为沿跨径均布的脉动正弦波动载荷F（t）：
式中，Af=k1×k2，，其中B为桥面宽度，L为桥梁净跨（k1参考第1节的几何参数可计算得4.53）；k2为人群荷载密度，fs为步行频率，t为步行时间.
阻尼采用瑞利阻尼模型C=αM+βK，其中α=2ω1ω2ξ/（ω1+ω2），β=2ω1ω2ξ/（ω1+ω2），α和β为瑞利阻尼系数.参考Elyas Bayat等人的实测结果，阻尼比ξ取0.5%[12]，ω1和ω2分别取结构的第1和第3阶频率，计算得α和β分别为0.013029和0.00173，在ANSYS中采用完全法中的Newmark法并打开大变形开关进行瞬态分析，时间步长取0.01 s，子荷载步取20步.
由式（1）可知，人群激励引起桥梁共振时，人群荷载可视为单频简谐激励.本次分析选取如表3所示的三种工况，工况1以第1阶竖向对称振动自振频率0.606 Hz（1stVS）作为行人激励频率fs，人群密度k2取300 N/m2，在主梁竖向施加沿跨径均布的简谐荷载，激励幅值Af为1247 N/m；工况2以第4阶竖向对称振动自振频率1.947 Hz（4thVS）作为行人激励频率fs，激励幅值Af为227 N/m；工况3以第4阶竖向对称振动自振频率的一半（0.972 Hz）作为行人激励频率fs，激励幅值Af为1386 N/m.上述工况的几何和材料参数参考第1节.
2.2　结果分析
图8为工况1条件下的主梁位移响应时程图，从图中可以看出，主梁在竖向激励下除产生竖向振动[图8(a)]外，还出现侧向晃动[图8(b)]，其中跨中竖向振幅接近0.17 m，显著大于侧向振幅3.4×10-3 m.时频谱显示，竖向位移以0.61 Hz（1st VS）单频简谐振动为主，侧向出现0.30 Hz响应频率，即竖向模态（0.61 Hz）与侧向模态（0.30 Hz）间发生2：1内共振.这表明当外激励超过某阈值时，竖向振动能量通过耦合关系转移至侧向引发内共振.
图9为工况1条件下主缆的位移时程响应.主缆竖向响应与主梁基本一致，但侧向响应存在一定独立性，因吊杆对竖向相对运动约束强，而对横向约束较弱，导致主梁更易发生横向晃动.
图10为工况2条件下主梁的位移时程响应及其时频谱.激励主梁第4阶竖向对称模态频率（4th VS：1.947 Hz）时，横向振幅为0.047 m，约为竖向振幅（0.0046 m）的10倍.时程响应显示，悬索桥侧向与竖向自由度间的非线性耦合关系使较小竖向外激励（Af=227 N/m）也能引发显著横向晃动.结论表明，悬索桥的几何非线性引发竖向-横向模态耦合效应：低荷载下仅表现为竖向振动，当荷载超过临界值时，侧向振幅突变并出现幅值跳跃（跨中与四分点响应反向变化），此类非线性失稳机制对行人安全构成显著威胁.
图11给出了工况3条件下主梁的位移时程响应及其时频谱.该图显示，工况3的激励条件同样出现了竖向与侧向自由度之间的内共振现象，其中被间接激励的横向振幅为0.15 m，约为被直接激励的竖向振幅（0.02 m）的7倍.
为研究主缆垂跨比（d/l）对跨模态竖向和侧向的影响，分别将原模型的垂跨比增大和减小5%，重新建立有限元模型（d/l=0.0744、0.0783、0.0823），取工况2所对应的激励幅值和频率进行时程分析，主梁跨中位移的竖向和侧向时程结果如图12所示.由图可知，主缆的垂跨比降低5%（d/l=0.0744），竖向振动幅值变为0.0044 m，相比原垂跨比（d/l=0.0783）降低约6%，侧向振动的幅值变为0.046 m，相比原垂跨比降低约2%；主缆的垂跨比增大5%（d/l=0.0823），竖向振动幅值变为0.0059 m，相比原垂跨比增大约26%，侧向振动的幅值随时间逐渐衰减为0，此时结构只发生竖向强迫振动.垂跨比的改变使得结构的动力特性发生变化，结构可能不再满足发生2：1跨模态耦合振动的条件，因此通过合理调整结构的垂跨比能有效避免竖向和侧向耦合振动的发生.
为研究激励幅值Af对人行悬索桥晃动的影响，在相同激励频率、加载方式和初始条件下（工况1），逐步增加激励幅值，得到主梁竖向和侧向跨中和四分点的稳态响应幅值如图13所示.为清晰地展示低外激励幅值部分的变化，在横坐标0~1200 N/m范围进行了压缩.参数分析表明，竖向振动幅值随外激励幅值增加呈现复杂变化规律，可分为三个阶段：
阶段一，外激励幅值Af从0增至945 N/m时，竖向振动幅值初期呈线性变化，系统处于线性共振状态，仅存在竖向单频振动；后期因几何非线性逐渐显现，变化趋势转为非线性；
阶段二，外激励幅值Af在945 N/m至1189 N/m范围内进一步增加，当激励幅值达到临界值时，系统出现鞍结分叉，原有的稳定解消失，出现非稳定解，系统侧向振动幅值跳跃到较大的稳定解，导致侧向振幅突然增大，竖向振动动能累计到一定程度后则激发侧向振动.图13(a)局部放大显示，主梁竖向振幅在外激励幅值为1189 N/m时达局部最大值0.17 m，此时横向振幅为3.96×10-3m，相对较小可忽略.然而，外激励幅值微小增加会引发侧向振幅约20倍的突然跳跃，且跨中竖向振幅骤减，四分点位置竖向振幅骤增；
阶段三，跳跃后，竖向和横向振幅变化趋势趋于平缓，振动能量因非线性和耦合关系进入饱和状态.
综上所述，竖向振动幅值随外激励幅值Af增加呈现阶段性变化特征，从线性增长到临界区域的复杂变化（受非线性解和初始条件影响），再到高负载下的饱和状态，反映了系统从线性到非线性的转变过程.外激励幅值通过与固有频率相互作用，显著改变系统振动状态.
该类桥梁具有显著的几何非线性动力特性.低水平竖向激励下仅表现为竖向振动，但随着行人负载增加，竖向振动与横向晃动耦合，导致横向振动发展并改变竖向振幅变化趋势，可能引发显著幅值跳跃，影响行人行走稳定性及结构安全，严重威胁行人安全.
3　结论
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本文以意大利Ramello人行悬索桥为工程背景建立其非线性有限元模型，然后针对与Ramello人行悬索桥类似的固有频率落在行人荷载频率范围内容易引起结构大幅振动的柔性悬索桥开展了三种工况的人致振动分析，并得到了如下主要结论：
（1）柔性悬索桥在行人竖向激励下表现出显著几何非线性特征，当竖向与横向自振频率比接近2：1时，竖向振动激励（如0.61 Hz）可通过内共振机制激发侧向模态（如0.30 Hz），导致竖向能量向侧向转移.荷载水平超过临界阈值（如工况1的945 N/m）时，侧向振幅突增20倍以上，呈现典型的幅值跳跃失稳现象，揭示了索-梁耦合体系的非线性模态相互作用本质；
（2）即使竖向激励幅值较低（如工况2的227 N/m），仍可能因非线性耦合引发显著侧向晃动（横向振幅达竖向的10倍）.激励频率接近竖向自振频率时，主梁侧向振幅随荷载增加呈三阶段变化：初始线性增长→临界区突变（能量转移）→高荷载饱和，揭示了行人密度与振幅突变的强相关性；
（3）悬索桥设计中需重点关注竖向-横向模态频率的整数比关系（如2：1），并预判临界激励阈值和几何非线性导致的幅值跳跃效应，避免因行人同步激励触发耦合振动；
（4）通过合理调整主缆垂跨比等结构参数，使得系统失去竖向和侧向的耦合振动条件，从而实现对侧向振动的有效控制.
研究结果为柔性悬索桥的稳定性评估与振动控制提供了非线性动力学理论支撑.
基金
收起
  • 山区桥梁及隧道工程国家重点实验（重庆交通大学）开放研究基金资助项目(SKLBT-2319)
  • 江苏省高等学校自然科学研究项目(24KJB410005)
  • 国家自然科学基金资助项目(52078087)
文献
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参考文献 引证文献
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2025年第23卷第10期
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文章信息
doi: 10.6052/1672-6553-2025-042
  • 接收时间：2025-02-25
  • 首发时间：2026-03-20
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  • 收稿日期：2025-02-25
  • 修回日期：2025-04-23
基金
State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University(SKLBT-2319)
山区桥梁及隧道工程国家重点实验（重庆交通大学）开放研究基金资助项目(SKLBT-2319)
Natural Science Foundation of the Jiangsu Higher Education Institutions of China(24KJB410005)
江苏省高等学校自然科学研究项目(24KJB410005)
National Natural Science Foundation of China(52078087)
国家自然科学基金资助项目(52078087)
作者信息
    1.苏州科技大学　土木工程学院，苏州　250009
    2.重庆交通大学　山区桥梁及隧道工程国家重点实验室，重庆　402247
    3.广西大学　土木建筑工程学院，南宁　530004
    4.重庆大学　土木工程学院，重庆　400044

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2种不同金属材料的力学参数
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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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