斜拉索拱考虑拱受外激励作用下1∶1内共振分析
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摘要
研究在拱受外激励作用下斜拉索拱结构中索拱之间1∶1内共振问题.当拱的某阶频率接近索的某阶频率时,可导致索拱之间出现1∶1内共振,利用已建立的斜拉索拱非线性动力学耦合面内运动微分方程,采用Galerkin方法把斜拉索拱的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到拱主共振情况下的平均方程,研究在拱受到外激励作用下拱振动对索振动产生的影响,同时对斜拉索拱内共振时的稳定、分叉及混沌情况进行了分析.结果表明:拱受到外激励产生共振后,通过索拱之间的内共振容易激发对柔性索的振动,导致索出现较大的幅值.能量在索拱之间相互传递,原本静止的索也可能出现共振,共振频域区间内索拱振动将出现跳跃、分叉及混沌等复杂的非线性动力学行为.
The internal resonance of cable-stayed arch under external excitation on arch with one-to-one internal resonance was investigated.The frequency of the cables and the frequency of the arch were studied,resulting in a one-to-one internal resonance,The coupling nonlinear dynamic equations of the cable-stayed arch derived by using Hamilton principle were used.First,the Galerkin method was used to discrete the nonlinear equation of planar motion.Then,the method of multiple scales was applied to perturb the discrete equations of motion,and the averaged equation under the primary resonances of arch was obtained.The impact of the vibration of the arch over the vibration of cables under external excitation on the arch was studied,and the equilibrium solution,the period solution and chaotic solution of averaging equations were examined.Numerical calculation results have shown that,after the resonance of the arch appears under the external excitation,the resonance of cables is easily excited through the internal resonance between cables and arch,resulting in a larger amplitude of cables.The energy is transferred between the cables and the arch,and the cable-stayed arch show complicated nonlinearity dynamic behavior,such as leap,bifurcation and chaos within the resonance frequency region.
The internal resonance of cable-stayed arch under external excitation on arch with one-to-one internal resonance was investigated.The frequency of the cables and the frequency of the arch were studied,resulting in a one-to-one internal resonance,The coupling nonlinear dynamic equations of the cable-stayed arch derived by using Hamilton principle were used.First,the Galerkin method was used to discrete the nonlinear equation of planar motion.Then,the method of multiple scales was applied to perturb the discrete equations of motion,and the averaged equation under the primary resonances of arch was obtained.The impact of the vibration of the arch over the vibration of cables under external excitation on the arch was studied,and the equilibrium solution,the period solution and chaotic solution of averaging equations were examined.Numerical calculation results have shown that,after the resonance of the arch appears under the external excitation,the resonance of cables is easily excited through the internal resonance between cables and arch,resulting in a larger amplitude of cables.The energy is transferred between the cables and the arch,and the cable-stayed arch show complicated nonlinearity dynamic behavior,such as leap,bifurcation and chaos within the resonance frequency region.
引文
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[3]刘开国.索-拱体系的静力与动力特性分析[J].建筑钢结构进展,2003,5(4):49-51.LIU Kai-guo.Static and dynamic characteristic analysis for ca-ble-arch system[J].Progress in Steel Building Structures,2003,5(4):49-51.(In Chinese)
[4]陈宝春,孙潮.石潭溪大桥施工受力分析[J].中国公路学报,1998,11(4):51-56.CHEN Bao-chun,SUN Chao.Mechanic analysis of the con-struction phase of Shitanxi Bridge[J].China Journal of High-way and Transport,1998,11(4):51-56.(In Chinese)
[5]剧锦三,郭彦林.索-拱结构的平面内稳定性研究[J].建筑结构学报,2001,22(2):84-87.JU Jin-san,GUO Yan-lin.Instability behavior in the plane ofcable-arch structure[J].Journal of Building Structures,2001,22(2):84-87.(In Chinese)
[6]丁建国.索-拱结构受跨中集中力作用时的稳定性分析[J].南京理工大学学报,2003,27(2):214-217.DING Jian-guo.The analysis of stability on cable-arch struc-ture acted by a concentrated load at the middle point of a span[J].Journal of Nanjing University of Science and Technology,2003,27(2):214-217.(In Chinese)
[7]赵跃宇,康厚军,王连华,等.索-拱结构面内稳定性研究[J].湖南大学学报:自然科学版,2006,33(3):1-5.ZHAO Yue-yu,KANG Hou-jun,WANG Lian-hua,et al.In-stability behavior in the plane of cable-arch structure[J].Jour-nal of Hunan University:Natural Sciences,2006,33(3):1-5.(In Chinese)
[8]吕建根.大跨度索拱组合体系非线性静动力性能研究[D].长沙:湖南大学土木工程学院,2007.LV Jian-gen.The research on nonlinear static-dynamic me-chanics performance of long-span cable-stayed arch system[D].Changsha:College of Civil Engineering,Hunan Universi-ty,2007.(In Chinese)
[9]NAYFEH A H.Perturbation methods[M].New York:Wi-ley-Interscience,1981.
[10]NAYFEH A H,BALACHANDRAN B.Applied nonlineardynamics[M].New York:Wiley-Interscience,1994.
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