温度场中悬索受多频激励组合联合共振响应研究
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摘要
基于温度变化对拉索张拉力和垂度的影响,利用Hamilton变分原理,引入拟静态假设,推导温度场中受多频激励下悬索的非线性运动微分方程。利用Galerkin法得到离散后的无穷维方程,并考虑一阶正对称模态,利用多尺度法求解系统发生组合联合共振时的幅频响应方程组,并判断稳态解的稳定性。考虑四组垂跨比及四种温度变化工况,通过数值算例探究悬索组合联合共振的响应特性及其受温度变化影响。研究结果表明:多频激励时系统同时展现出组合共振和超谐波共振响应的特性;此时稳态解个数、共振区间、响应幅值及其相位等均会发生改变;温度变化会使得组合共振和超谐波共振发生定性和定量的变化,从而导致联合共振响应亦发生明显的定性和定量的改变;组合联合共振响应受温度变化的影响与悬索的垂跨比和温度变化幅度密切相关;为了更好地区分系统受多频激励下的的稳态解,可以通过研究解的相位来分辨。
Here, considering effects of temperature on cable's tension force and sag, nonlinear equations of motion of a suspended cable under multi-frequency excitation were derived by using Hamilton's principle and the quasi-static hypothesis. Galerkin procedure was adopted to discretize nonlinear dynamic equations. When the system's combined joint resonances happened, the amplitude-frequency response equations were solved using the multi-scale method considering the first order symmetric mode, and the stability of their steady-state solutions was judged. Considering 4 sag-to-span ratios and 4 temperature variation cases, through numerical examples the response characteristics of the suspended cable system's combined joint resonances and the effects of temperature were studied. The results showed that the system under multi-frequency excitation exhibits the characteristics of combined resonances and super-harmonic resonances simultaneously; number of steady-state solutions, interval of resonances, amplitude and phase of responses all vary; temperature variation can make combined resonances and super-harmonic ones have qualitative and quantitative changes; temperature variation affects the system's combined joint resonance responses, they are closely related to the cable's sag-to-span ratio and temperature change amplitude; studying phase of steady-state solutions under multi-frequency excitation can distinguish these solutions better.
Here, considering effects of temperature on cable's tension force and sag, nonlinear equations of motion of a suspended cable under multi-frequency excitation were derived by using Hamilton's principle and the quasi-static hypothesis. Galerkin procedure was adopted to discretize nonlinear dynamic equations. When the system's combined joint resonances happened, the amplitude-frequency response equations were solved using the multi-scale method considering the first order symmetric mode, and the stability of their steady-state solutions was judged. Considering 4 sag-to-span ratios and 4 temperature variation cases, through numerical examples the response characteristics of the suspended cable system's combined joint resonances and the effects of temperature were studied. The results showed that the system under multi-frequency excitation exhibits the characteristics of combined resonances and super-harmonic resonances simultaneously; number of steady-state solutions, interval of resonances, amplitude and phase of responses all vary; temperature variation can make combined resonances and super-harmonic ones have qualitative and quantitative changes; temperature variation affects the system's combined joint resonance responses, they are closely related to the cable's sag-to-span ratio and temperature change amplitude; studying phase of steady-state solutions under multi-frequency excitation can distinguish these solutions better.
引文
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[14] 康厚军,郭铁丁,赵跃宇,等.大跨度斜拉桥非线性振动模型与理论研究进展[J].力学学报,2016,48(3):519-535.KANG Houjun,GUO Tieding,ZHAO Yueyu,et al.Review on nonlinear vibration and modeling of large span cable-stayed bridge[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):519-535.
[15] REGA G.Nonlinear vibrations of suspended cables,Part II:deterministic phenomena[J].Applied Mechanics Reviews,2004,57:479-514.
[16] XIA Y,CHEN B,WENG S,et al.Temperature effect on the vibration properties of civil structures:a literature review and case studies[J].Journal of Civil Structural Health Monitoring,2012(2):29-46.
[17] 李顺龙,李惠,欧进萍,等.考虑温度和风速影响的桥梁结构模态参数分析[J].土木工程学报,2009(4):100-106.LI Shunlong,LI Hui,OU Jinping,et al.identification of modal parameters of bridges considering temperature and wind effects[J].China Civil Engineering Journal,2009(4):100-106.
[18] 闵志华,孙利民,淡丹辉.影响斜拉桥模态参数变化的环境因素分析[J].振动与冲击,2009,28(10):99-105.MIN Zhihua,SUN Limin,DAN Danhui.Effect analysis of environmental factors on structural modal parameters of a cable-stayed bridge[J].Journal of Vibration and Shock,2009,28(10):99-105.
[19] 邓扬,丁幼亮,李爱群.环境条件影响下悬索桥模态频率变异性的定量评价[J].振动与冲击,2011,30(8):230-236.DENG Yang,DING Youliang,LI Aiqun.Quantitative evaluation of variability in modal frequencies of a suspension bridge under environment conditions[J].Journal of Vibration and Shock,2011,30(8):230-236.
[20] 赵珧冰,孙测世.温度效应对端部激励斜拉索共振响应影响[J].计算力学学报,2017,34(5):644-649.ZHAO Yaobing,SUN Ceshi.Temperature effects on the resonance responses of stay cables under supported excitation[J].Chinese Journal of Computational Mechanics,2017,34(5):644-649.
[21] 赵珧冰,彭剑.温度变化对悬索主共振响应影响分析[J].振动与冲击,2017,36(15):240-244.ZHAO Yaobing,PENG Jian.Effects of temperature changes on primary resonances of suspended cables[J].Journal of Vibration and Shock,2017,36(15):240-244.
[22] TREYSSEDE F.Free vibrations of cables under thermal stress[J].Journal of Sound and Vibration,2009,327(1):1-8.
[23] LEPIDI M,GATTULLI V.Static and dynamic response of elastic suspended cables with thermal effects[J].International Journal of Solids and Structures,2012,49(9):1103-1116.
[24] 赵珧冰,黄超辉,金波.悬索面内次谐波共振受温度效应影响研究[J].振动与冲击,2018,37(6):1-6.ZHAO Yaobing,HUANG Chaohui,JIN Bo.Investigation of temperature effects on the in-plane sub-harmonic resonances of suspended cables[J].Journal of Vibration and Shock,2018,37(6):1-6.
[25] ZHAO Y B,PENG J,ZHAO Y Y,et al.Effects of temperature variations on nonlinear planar free and forced oscillations at primary resonance of suspended cables[J].Nonlinear Dynamics,2017,89:2815-2827.
[26] PLAUT R H,HAQUANG N,MOOK D T.Simultaneous resonances in non-linear structural vibrations under two-frequency excitation[J].Journal of Sound and Vibration,1986,106 (3):361-376.附录 A 附录 B
[2] NAYFEH A H,MOOK D T.Nonlinear oscillations[M].JohnWiley Sons,Inc,1979.
[3] 王祖尧,丁虎,陈立群.多频激励磁悬浮能量采集[J].动力学与控制学报,2017,15(2):125-130.WANG Zuyao,DING Hu,CHEN liqun.Energy harvesting of magnetic levitationunder multi-frequency excitations[J].Journal of Dynamics and Control,2017,15(2):125-130.
[4] 楼京俊,何其伟,朱石坚.多频激励软弹簧型Duffing系统中的混沌[J].应用数学和力学,2004,25(12):1299-1304.LOU Jingjun,HE Qiwei,ZHU Shijian.Chaos in the softening duffing system under multi-frequency periodic forces[J].Applied Mathematics and Mechanics,2004,25(12):1299-1304.
[5] YANG S P,NAYFEH A H,MOOK D T.Combination resonances in the response of the Duffing oscillator to a three-frequency excitation[J].Acta Mechanica,1998,131 (3):235-245.
[6] 杨德森,董雷,时洁,等.多频激励Duffing系统振动状态研究[J].振动与冲击,2011,30(12):19-21.YANG Desen,DONG Lei,SHI jie,et al.Duffing system vibration behavior under multi-frequency excitation[J].Journal of Vibration and Shock,2011,30(12):19-21.
[7] 李韶华,张雪锋,杨绍普.多频激励下van der Pol系统主参数-组合共振[J].振动、测试与诊断,2003,23(3):188-191.LI Shaohua,ZHANG Xuefeng,YANG Shaopu.Principal parametric-combination resonance in a multi-frequency excited van der pol ’s oscillator[J].Journal of Vibration Measurement and Diagnosis,2003,23(3):188-191.
[8] 赵志宏,张明,杨绍普,等.多频激励Duffing-van der Pol 系统振动状态研究[J].振动与冲击,2013,32(19):76-79.ZHAO Zhihong,ZHANG Ming,YANG Shaopu,et al.A Duffing-van der Pol system’s vibration behavior under multi-frequency excitation[J].Journal of Vibration and Shock,2013,32(19):76-79.
[9] 姜源,申永军,温少芳,等.分数阶达芬振子的超谐与亚谐联合共振[J].力学学报,2017,49(5):1008-1019.JIANG Yuan,SHEN Yongjun,WEN Shaofang,et al.Super-harmonic and sub-harmonic simultaneous resonances of fractional-order Duffing oscillator[J].Chinese Journal of Theoretical and Applied Mechanics,2017,49(5):1008-1019.
[10] 赵跃宇,赵珧冰,马建军,等.Winkler地基上有限长梁联合共振分析[J].地震工程与工程振动,2012,32(2):160-166.ZHAO Yueyu,ZHAO Yaobing,MA Jianjun,et al.Simultaneous resonance analysis of finite-length beams on Winkler foundation[J].Journal of Earthquake Engineering and Engineering Vibration,2012,32(2):160-166.
[11] 胡宇达,吕书锋,李佰洲.叠层板在两项简谐激励作用下的组合共振分析[J].工程力学,2010,27(6):40-44.HU Yuda,Lü Shufeng,LI Baizhou.Nonlinear combination resonances of orthotropic laminated plate under two-term harmonic excitation[J].Engineering Mechanics,2010,27(6):40-44.
[12] SAHOO B,PANDA L N,POHIT G.Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam[J].Nonlinear Dynamics,2015,82 (4):1721-1742.
[13] 黄建亮,陈树辉.外激励力作用下的轴向运动梁非线性振动的联合共振[J].振动工程学报,2011,24 (5):455-460.HUANG Jianliang,CHEN Shuhui.Combination resonance of nonl inear forced vibrat ion of an axially moving beam[J].Journal of Vibration Engineering,2011,24 (5):455-460.
[14] 康厚军,郭铁丁,赵跃宇,等.大跨度斜拉桥非线性振动模型与理论研究进展[J].力学学报,2016,48(3):519-535.KANG Houjun,GUO Tieding,ZHAO Yueyu,et al.Review on nonlinear vibration and modeling of large span cable-stayed bridge[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(3):519-535.
[15] REGA G.Nonlinear vibrations of suspended cables,Part II:deterministic phenomena[J].Applied Mechanics Reviews,2004,57:479-514.
[16] XIA Y,CHEN B,WENG S,et al.Temperature effect on the vibration properties of civil structures:a literature review and case studies[J].Journal of Civil Structural Health Monitoring,2012(2):29-46.
[17] 李顺龙,李惠,欧进萍,等.考虑温度和风速影响的桥梁结构模态参数分析[J].土木工程学报,2009(4):100-106.LI Shunlong,LI Hui,OU Jinping,et al.identification of modal parameters of bridges considering temperature and wind effects[J].China Civil Engineering Journal,2009(4):100-106.
[18] 闵志华,孙利民,淡丹辉.影响斜拉桥模态参数变化的环境因素分析[J].振动与冲击,2009,28(10):99-105.MIN Zhihua,SUN Limin,DAN Danhui.Effect analysis of environmental factors on structural modal parameters of a cable-stayed bridge[J].Journal of Vibration and Shock,2009,28(10):99-105.
[19] 邓扬,丁幼亮,李爱群.环境条件影响下悬索桥模态频率变异性的定量评价[J].振动与冲击,2011,30(8):230-236.DENG Yang,DING Youliang,LI Aiqun.Quantitative evaluation of variability in modal frequencies of a suspension bridge under environment conditions[J].Journal of Vibration and Shock,2011,30(8):230-236.
[20] 赵珧冰,孙测世.温度效应对端部激励斜拉索共振响应影响[J].计算力学学报,2017,34(5):644-649.ZHAO Yaobing,SUN Ceshi.Temperature effects on the resonance responses of stay cables under supported excitation[J].Chinese Journal of Computational Mechanics,2017,34(5):644-649.
[21] 赵珧冰,彭剑.温度变化对悬索主共振响应影响分析[J].振动与冲击,2017,36(15):240-244.ZHAO Yaobing,PENG Jian.Effects of temperature changes on primary resonances of suspended cables[J].Journal of Vibration and Shock,2017,36(15):240-244.
[22] TREYSSEDE F.Free vibrations of cables under thermal stress[J].Journal of Sound and Vibration,2009,327(1):1-8.
[23] LEPIDI M,GATTULLI V.Static and dynamic response of elastic suspended cables with thermal effects[J].International Journal of Solids and Structures,2012,49(9):1103-1116.
[24] 赵珧冰,黄超辉,金波.悬索面内次谐波共振受温度效应影响研究[J].振动与冲击,2018,37(6):1-6.ZHAO Yaobing,HUANG Chaohui,JIN Bo.Investigation of temperature effects on the in-plane sub-harmonic resonances of suspended cables[J].Journal of Vibration and Shock,2018,37(6):1-6.
[25] ZHAO Y B,PENG J,ZHAO Y Y,et al.Effects of temperature variations on nonlinear planar free and forced oscillations at primary resonance of suspended cables[J].Nonlinear Dynamics,2017,89:2815-2827.
[26] PLAUT R H,HAQUANG N,MOOK D T.Simultaneous resonances in non-linear structural vibrations under two-frequency excitation[J].Journal of Sound and Vibration,1986,106 (3):361-376.附录 A 附录 B
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