考虑行人随机性的人行桥人致横向振动稳定性分析
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摘要
2000年英国千禧桥大幅横向振动事件的发生,揭示了在人行桥的人致横向振动中,存在着振动发散不稳定现象:人群数目的小额增加会导致有限振幅的桥梁振动突变为大幅桥梁振动。这个现象已被证实为是由人-桥的相互作用引起,但其中机理仍未得到明确的解释。众多学者包括目前已有的规范所提出的模型,大部分基于确定性的思路或者单个试验观测结果,忽略了行人荷载中明显存在的随机性。行人荷载实质是一个复杂的窄带随机过程,包含行人不同步伐间的作用随机性以及不同行人之间的作用随机性,这些随机性导致的行人荷载将明显不同于按确定性考虑的情况。因此,该文提出了考虑人桥相互作用以及行人横向荷载随机性的非线性随机模型,将行人横向荷载划分为两部分:与桥梁振动无关的横向荷载以及与桥梁振动(振幅)相关的荷载,并考虑了单个行人不同步伐间的荷载随机性造成的激励过程窄带性。基于随机平均法推导得到的It?方程,进行了随机P-分叉分析和随机D-分叉分析,并以此得到人行桥横向振动发散失稳临界状态。最后利用所提方法对伦敦千禧桥北边跨进行了算例分析,验证了该文所提方法的有效性,并得到了一些有益的结论。
The infamous incident of large lateral vibration on the London Millennium Bridge in 2000 reveals that divergence of instability exists in pedestrian-induced vibration:a small increase in the number of pedestrians will cause the amplitude of vibration of the footbridge to become divergent.Such divergence of an instability phenomenon has been proven to be caused by the interaction between the pedestrian and the footbridge,although the mechanisms still have not been clearly explored.Most existing models are based on deterministic methods or results of a single test,ignoring the obvious randomness in the pedestrian load.In fact,pedestrian load is a complex stochastic process characteristized by narrow-band,including the large intra-subject and inter-subject randomness,which cause the real pedestrian load to be significantly different from that of a deterministic case.A nonlinear stochastic model for footbridges considering the pedestrian-bridge interaction and the randomness among the pedestrian lateral load was proposed.The pedestrian lateral load was considered as a narrow-band excitation process caused by the intra-subject variability, and was divided into a static load independent of footbridge vibration and vibration amplitude–dependent load.Based on the It? equations derived by the stochastic averaging method, the stochastic P-bifurcation and stochastic D-bifurcation were analyzed to gain the critical condition for triggering a large lateral vibration of footbridge.Finally, through the case study of the northern span of the London Millennium Bridge, the availability of the proposed method was confirmed and meaningful conclusions were obtained.
The infamous incident of large lateral vibration on the London Millennium Bridge in 2000 reveals that divergence of instability exists in pedestrian-induced vibration:a small increase in the number of pedestrians will cause the amplitude of vibration of the footbridge to become divergent.Such divergence of an instability phenomenon has been proven to be caused by the interaction between the pedestrian and the footbridge,although the mechanisms still have not been clearly explored.Most existing models are based on deterministic methods or results of a single test,ignoring the obvious randomness in the pedestrian load.In fact,pedestrian load is a complex stochastic process characteristized by narrow-band,including the large intra-subject and inter-subject randomness,which cause the real pedestrian load to be significantly different from that of a deterministic case.A nonlinear stochastic model for footbridges considering the pedestrian-bridge interaction and the randomness among the pedestrian lateral load was proposed.The pedestrian lateral load was considered as a narrow-band excitation process caused by the intra-subject variability, and was divided into a static load independent of footbridge vibration and vibration amplitude–dependent load.Based on the It? equations derived by the stochastic averaging method, the stochastic P-bifurcation and stochastic D-bifurcation were analyzed to gain the critical condition for triggering a large lateral vibration of footbridge.Finally, through the case study of the northern span of the London Millennium Bridge, the availability of the proposed method was confirmed and meaningful conclusions were obtained.
引文
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[5]Nakamura S.Model for lateral excitation of footbridges by synchronous walking[J].Journal of Structural Engineering,2004,130(1):32-37.
[6]Piccardo G,Tubino F.Parametric resonance of flexible footbridges under crowd-induced lateral excitation[J].Journal of Sound&Vibration,2008,311(1-2):353-371.
[7]Ingólfsson E T,Georgakis C T.A stochastic load model for pedestrian-induced lateral forces on footbridges[J].Engineering Structures,2011,33(12):3454-3470.
[8]Ingólfsson E T,Georgakis C T,Ricciardelli F,et al.Experimental identification of pedestrian-induced lateral forces on footbridges[J].Journal of Sound&Vibration,2012,330(6):1265-1284.
[9]Roberts T M.Lateral pedestrian excitation of footbridges[J].Journal of Bridge Engineering,2005,10(1):107-112.
[10]Newland D.Pedestrian excitation of bridges[J].Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science,2004,218(5):477-492.
[11]Erlicher S,Trovato A,Argoul P.A modified hybrid Van der Pol/Rayleigh model for the lateral pedestrian force on a periodically moving floor[J].Mechanical Systems&Signal Processing,2013,41(1):485-501.
[12]Kumar P,Kumar A,Erlicher S.A nonlinear oscillator model to generate lateral walking force on a rigid flat surface[J].International Journal of Structural Stability&Dynamics,2017,18(2):1850020-1―1850020-31.
[13]Macdonald J H G.Lateral excitation of bridges by balancing pedestrians[J].Proceedings of the Royal Society A Mathematical Physical&Engineering Sciences,2009,465(2104):1055-1073.
[14]Bocian M,Macdonald J H G,Burn J F.Biomechanically inspired modelling of pedestrian-induced forces on laterally oscillating structures[J].Journal of Sound&Vibration,2012,331(16):3914-3929.
[15]Carroll S P,Owen J S,Hussein M F M.Reproduction of lateral ground reaction forces from visual marker data and analysis of balance response while walking on a laterally oscillating deck[J].Engineering Structures,2013,49(2):1034-1047.
[16]Carroll S P,Owen J S,Hussein M F M.Experimental identification of the lateral human-structure interaction mechanism and assessment of the inverted-pendulum biomechanical model[J].Journal of Sound&Vibration,2014,333(22):5865-5884.
[17]Ricciardelli F,Pizzimenti A D.Lateral walking-induced forces on footbridges[J].Journal of Bridge Engineering,2007,12(6):677-688.
[18]Bocian M,Macdonald J H G,Burn J F,et al.Experimental identification of the behaviour of and lateral forces from freely-walking pedestrians on laterally oscillating structures in a virtual reality environment[J].Engineering Structures,2015,105(2015):62-76.
[19]Brownjohn J M W,Pavic A,Omenzetter P.A spectral density approach for modelling continuous vertical forces on pedestrian structures due to walking[J].Canadian Journal of Civil Engineering,2004,31(1):65-77.
[20]?ivanovi?S,Pavi?A,Reynolds P.Probability-based prediction of multi-mode vibration response to walking excitation[J].Engineering Structures,2007,29(6):942-954.
[21]李泉,樊健生,聂建国.双向行走激励条件下人行桥的随机振动问题研究[J].工程力学,2010,27(7):74-81.Li Quan,Fan Jiansheng,Nie Jianguo.Random vibration of footbridge generated by crowds’two-way walking[J].Engineering Mechanics,2010,27(7):74-81.(in Chinese)
[22]朱前坤,李宏男,杜永峰,等.不同行走步速下人行桥振动舒适度定量化评估[J].工程力学,2016,33(10):97-104.Zhu Qiankun,Li Hongnan,Du Yongfeng,et al.Quantitative evaluation of vibration serviceability of pedestrian brigde under different walking speed[J].Engineering Mechanics,2016,33(10):97-104.(in Chinese)
[23]Racic V,Brownjohn J.Mathematical modelling of random narrow band lateral excitation of footbridges due to pedestrians walking[J].Computers&Structures,2011,90/91(1):116-130.
[24]Racic V,Brownjohn J M W.Stochastic model of near-periodic vertical loads due to humans walking[J].Advanced Engineering Informatics,2011,25(2):259-275.
[25]Fernando Jimenez-Alonso J,Saez A,Caetano E,et al.Vertical crowd-structure interaction model to analyze the change of the modal properties of a footbridge[J].Journal Of Bridge Engineering,2016,21(8):C4015004-1-C4015004-19.
[26]Venuti F,Racic V,Corbetta A.Modelling framework for dynamic interaction between multiple pedestrians and vertical vibrations of footbridges[J].Journal of Sound&Vibration,2016,379(2016):245-263.
[27]李慧乐,夏禾.基于车桥耦合随机振动分析的钢桥疲劳可靠度评估[J].工程力学,2017,34(2):69-77.Li Huile,Xia He.Fatigue reliability evaluation of steel bridges based on coupling random vibration analysis of train and bridge[J].Engineering Mechanics,2017,34(2):69-77.(in Chinese)
[28]Ricciardelli F,Mafrici M,Ingólfsson E T.Lateral pedestrian-induced vibrations of footbridges:characteristics of walking forces[J].Journal of Bridge Engineering,2014,19(9):1265-1266.
[29]Ricciardelli F,Demartino C.Design of footbridges against pedestrian-induced vibrations[J].Journal Of Bridge Engineering,2016,21(8):C4015003-1-C4015003-13.
[30]袁旭斌.人行桥人致振动特性研究[D].上海:同济大学,2006.Yuan Xubin.Research on pedestrian-induced vibration of footbridge[D].Shanghai:Tongji University,2006.(in Chinese)
[31]Venuti F,Bruno L.Crowd-structure interaction in lively footbridges under synchronous lateral excitation:Aliterature review[J].Physics of Life Reviews,2009,6(3):176-206.
[32]Bruno L,Venuti F.Crowd-structure interaction in footbridges:Modelling,application to a real case-study and sensitivity analyses[J].Journal of Sound&Vibration,2009,323(1/2):475-493.
[33]Shinozuka M,Deodatis G.Simulation of stochastic processes by spectral representation[J].Applied Mechanics Reviews,1991,44(4):191-204.
[34]Jia B,Yu X,Yan Q,et al.Nonlinear stochastic analysis for lateral vibration of footbridge under pedestrian narrowband excitation[J].Mathematical Problems in Engineering,2017,2017(22):1-12.
[35]朱位秋.随机振动[M].北京:科学出版社,1992.Zhu Weiqiu.Random vibration[M].Beijing:Science Press,1992.(in Chinese)
[36]Rong H,Wang X,Xu W,et al.Subharmonic response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random excitation[J].Physical Review E,2009,80(2):026604-1-026604-8.
[37]冯长水.窄带随机激励下时滞反馈控制的强非线性系统随机动力学研究[D].杭州:浙江大学,2008.Feng Changshui.Stochastic dynamics of strongly nonlinear systems with time-delayed feedback control under narrow-band random excitations[D].Hangzhou:Zhejiang University,2008.(in Chinese)
[38]杨志安,赵开元.有界窄带激励下微梁系统主共振[J].工程力学,2017,34(增刊1):19-25.Yang Zhian,Zhao Kaiyuan.Primary response of micro-beam system subjected to narrow-band random excitation[J].Engineering Mechanics,2017,34(Suppl1):19-25.(in Chinese)
[39]Rong H,Xu W,Wang X,et al.Principal response of van der pol-duffing oscillator under combined deterministic and random parametric excitation[J].Applied Mathematics and Mechanics,2002,23(3):299-310.
[2]陈政清,刘光栋.人行桥的人致振动理论与动力设计[J].工程力学,2009,26(增刊2):148-159.Chen Zhengqing,Liu Guangdong.Pedestrian-induced vibration theory and dynamic design of footbridges[J].Engineering Mechanics,2009,26(Suppl 2):148-159.(in Chinese)
[3]Fujino Y,Pacheco B M,Nakamura S-I,et al.Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge[J].Earthquake Engineering&Structural Dynamics,1993,22(9):741-758.
[4]Fujino Y,Siringoringo D M.A conceptual review of pedestrian-induced lateral vibration and crowd synchronization problem on footbridges[J].Journal of Bridge Engineering,2016,21(8):C4015001-1―C4015001-12.
[5]Nakamura S.Model for lateral excitation of footbridges by synchronous walking[J].Journal of Structural Engineering,2004,130(1):32-37.
[6]Piccardo G,Tubino F.Parametric resonance of flexible footbridges under crowd-induced lateral excitation[J].Journal of Sound&Vibration,2008,311(1-2):353-371.
[7]Ingólfsson E T,Georgakis C T.A stochastic load model for pedestrian-induced lateral forces on footbridges[J].Engineering Structures,2011,33(12):3454-3470.
[8]Ingólfsson E T,Georgakis C T,Ricciardelli F,et al.Experimental identification of pedestrian-induced lateral forces on footbridges[J].Journal of Sound&Vibration,2012,330(6):1265-1284.
[9]Roberts T M.Lateral pedestrian excitation of footbridges[J].Journal of Bridge Engineering,2005,10(1):107-112.
[10]Newland D.Pedestrian excitation of bridges[J].Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science,2004,218(5):477-492.
[11]Erlicher S,Trovato A,Argoul P.A modified hybrid Van der Pol/Rayleigh model for the lateral pedestrian force on a periodically moving floor[J].Mechanical Systems&Signal Processing,2013,41(1):485-501.
[12]Kumar P,Kumar A,Erlicher S.A nonlinear oscillator model to generate lateral walking force on a rigid flat surface[J].International Journal of Structural Stability&Dynamics,2017,18(2):1850020-1―1850020-31.
[13]Macdonald J H G.Lateral excitation of bridges by balancing pedestrians[J].Proceedings of the Royal Society A Mathematical Physical&Engineering Sciences,2009,465(2104):1055-1073.
[14]Bocian M,Macdonald J H G,Burn J F.Biomechanically inspired modelling of pedestrian-induced forces on laterally oscillating structures[J].Journal of Sound&Vibration,2012,331(16):3914-3929.
[15]Carroll S P,Owen J S,Hussein M F M.Reproduction of lateral ground reaction forces from visual marker data and analysis of balance response while walking on a laterally oscillating deck[J].Engineering Structures,2013,49(2):1034-1047.
[16]Carroll S P,Owen J S,Hussein M F M.Experimental identification of the lateral human-structure interaction mechanism and assessment of the inverted-pendulum biomechanical model[J].Journal of Sound&Vibration,2014,333(22):5865-5884.
[17]Ricciardelli F,Pizzimenti A D.Lateral walking-induced forces on footbridges[J].Journal of Bridge Engineering,2007,12(6):677-688.
[18]Bocian M,Macdonald J H G,Burn J F,et al.Experimental identification of the behaviour of and lateral forces from freely-walking pedestrians on laterally oscillating structures in a virtual reality environment[J].Engineering Structures,2015,105(2015):62-76.
[19]Brownjohn J M W,Pavic A,Omenzetter P.A spectral density approach for modelling continuous vertical forces on pedestrian structures due to walking[J].Canadian Journal of Civil Engineering,2004,31(1):65-77.
[20]?ivanovi?S,Pavi?A,Reynolds P.Probability-based prediction of multi-mode vibration response to walking excitation[J].Engineering Structures,2007,29(6):942-954.
[21]李泉,樊健生,聂建国.双向行走激励条件下人行桥的随机振动问题研究[J].工程力学,2010,27(7):74-81.Li Quan,Fan Jiansheng,Nie Jianguo.Random vibration of footbridge generated by crowds’two-way walking[J].Engineering Mechanics,2010,27(7):74-81.(in Chinese)
[22]朱前坤,李宏男,杜永峰,等.不同行走步速下人行桥振动舒适度定量化评估[J].工程力学,2016,33(10):97-104.Zhu Qiankun,Li Hongnan,Du Yongfeng,et al.Quantitative evaluation of vibration serviceability of pedestrian brigde under different walking speed[J].Engineering Mechanics,2016,33(10):97-104.(in Chinese)
[23]Racic V,Brownjohn J.Mathematical modelling of random narrow band lateral excitation of footbridges due to pedestrians walking[J].Computers&Structures,2011,90/91(1):116-130.
[24]Racic V,Brownjohn J M W.Stochastic model of near-periodic vertical loads due to humans walking[J].Advanced Engineering Informatics,2011,25(2):259-275.
[25]Fernando Jimenez-Alonso J,Saez A,Caetano E,et al.Vertical crowd-structure interaction model to analyze the change of the modal properties of a footbridge[J].Journal Of Bridge Engineering,2016,21(8):C4015004-1-C4015004-19.
[26]Venuti F,Racic V,Corbetta A.Modelling framework for dynamic interaction between multiple pedestrians and vertical vibrations of footbridges[J].Journal of Sound&Vibration,2016,379(2016):245-263.
[27]李慧乐,夏禾.基于车桥耦合随机振动分析的钢桥疲劳可靠度评估[J].工程力学,2017,34(2):69-77.Li Huile,Xia He.Fatigue reliability evaluation of steel bridges based on coupling random vibration analysis of train and bridge[J].Engineering Mechanics,2017,34(2):69-77.(in Chinese)
[28]Ricciardelli F,Mafrici M,Ingólfsson E T.Lateral pedestrian-induced vibrations of footbridges:characteristics of walking forces[J].Journal of Bridge Engineering,2014,19(9):1265-1266.
[29]Ricciardelli F,Demartino C.Design of footbridges against pedestrian-induced vibrations[J].Journal Of Bridge Engineering,2016,21(8):C4015003-1-C4015003-13.
[30]袁旭斌.人行桥人致振动特性研究[D].上海:同济大学,2006.Yuan Xubin.Research on pedestrian-induced vibration of footbridge[D].Shanghai:Tongji University,2006.(in Chinese)
[31]Venuti F,Bruno L.Crowd-structure interaction in lively footbridges under synchronous lateral excitation:Aliterature review[J].Physics of Life Reviews,2009,6(3):176-206.
[32]Bruno L,Venuti F.Crowd-structure interaction in footbridges:Modelling,application to a real case-study and sensitivity analyses[J].Journal of Sound&Vibration,2009,323(1/2):475-493.
[33]Shinozuka M,Deodatis G.Simulation of stochastic processes by spectral representation[J].Applied Mechanics Reviews,1991,44(4):191-204.
[34]Jia B,Yu X,Yan Q,et al.Nonlinear stochastic analysis for lateral vibration of footbridge under pedestrian narrowband excitation[J].Mathematical Problems in Engineering,2017,2017(22):1-12.
[35]朱位秋.随机振动[M].北京:科学出版社,1992.Zhu Weiqiu.Random vibration[M].Beijing:Science Press,1992.(in Chinese)
[36]Rong H,Wang X,Xu W,et al.Subharmonic response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random excitation[J].Physical Review E,2009,80(2):026604-1-026604-8.
[37]冯长水.窄带随机激励下时滞反馈控制的强非线性系统随机动力学研究[D].杭州:浙江大学,2008.Feng Changshui.Stochastic dynamics of strongly nonlinear systems with time-delayed feedback control under narrow-band random excitations[D].Hangzhou:Zhejiang University,2008.(in Chinese)
[38]杨志安,赵开元.有界窄带激励下微梁系统主共振[J].工程力学,2017,34(增刊1):19-25.Yang Zhian,Zhao Kaiyuan.Primary response of micro-beam system subjected to narrow-band random excitation[J].Engineering Mechanics,2017,34(Suppl1):19-25.(in Chinese)
[39]Rong H,Xu W,Wang X,et al.Principal response of van der pol-duffing oscillator under combined deterministic and random parametric excitation[J].Applied Mathematics and Mechanics,2002,23(3):299-310.