偏心索风致振动的稳定性分析
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摘要
索结构以其质量轻、承载力大、安装简便等优点广泛运用于实际工程中,如悬索桥、膜结构的穹顶、输电线等。冬季雨雪的作用使得悬索在覆冰后产生偏心的现象,在风作用下振动形式更为复杂,并可能造成灾难性的后果。08年春季我国南方大规模的冻雨现象使得输电线路被毁坏造成大规模停电,给国家造成了巨大的损失,因此对于偏心索结构风致振动的振动形式和稳定性的研究以及其各种非线性动力学行为的分析是有必要的。
本文考虑了覆冰索面内、面外和转动三个方向的振动形式,并且考虑到了在偏心效应下的耦合振动。将风荷载表示为攻角的非线性函数,通过Hamilton原理导出动力学方程。采用Galerkin法将控制方程离散化,并且用数值解验证了稳定性条件。在给定的Hopf分岔设计点附近,采用近似解析方法确定了稳定域的边界形状,节省了一定的计算工作量。主要工作包括：
(1)建立了偏心索在风致力作用下三个自由度方向上的耦合振动的模型,将风荷载表示为攻角的非线性函数,通过Hamilton原理导出动力学方程。采用Galerkin法将控制方程离散化为常微分方程。
(2)分析所建立方程的非线性动力学行为,根据Routh-Hurwitz判据得到索平衡构形在参数空间内的稳定域,确定了发生Hopf分岔的临界风速,对于得到的稳定域给出了数值解的验证,并通过与以往文献中类似问题的两个自由度模型比较说明了本文建立的三个自由度模型的优点。
(3)结合本文提出的模型建立了适合本问题的索单元的有限元离散方法。推导得到了索单元的刚度阵和质量阵。结合有限元程序对偏心索风致振动的问题得到了有限元的解法。
Suspend cables are wildly used in cable bridges, membrane structures and transmission lines for its low weights, high strength and easy to construct. In the winter when the cable was covered by ice caused by the rain and snow, the eccentricity makes the in-plane and out-planes vibration couple with the torsion vibration. So the motions of the cable become much more complicated, and it maybe leads to some terrible results. In the spring of 2008 the ice-covered transmission lines fractured in the south of China which made lots of economic loss and large-scale power-failure. So it's necessary to investigate the nonlinear dynamics and the stable region of the eccentricity cables loaded by wind.
The wind-induced vibration is considered to be the in-plane, out-plane and the torsion vibrations. The wind load is considered as a nonlinear function of the angle of attack, and Hamilton principal is used to obtain the governing partial differential equation of motion. Galerkin approach is adopted to reduce the governing partial differential equation to an approximate system with few numbers of DOFs. The stability conditions can be verified with numerical solutions. At the given point of Hopf bifurcation an approximation is used to get the stability region saving lots of time and workload. The research work can be summarized as follows:
(1) A 3-degree-of-freedom eccentricity cable loaded by wind model is established. The wind load is considered as a nonlinear function of the angle of attack, and Hamilton principal was used to obtain the governing partial differential equation of motion. The Galerkin approach is adopted to reduce the governing partial differential equation to differential equations.
(2) Analysis the differential equations, explicit conditions are provided for loss of stability based on the Routh-Hurwitz criterion while for generation of stable limit cycles Hopf bifurcation. The stability conditions can be verified with numerical solutions. Compare with the 2-degree-of-freedom model established in other papers; obtain the advantages of the model in the research.
(3) According to the 3-degree-of-freedom model established in the paper, a cable element is built for the finite element discrimination. The mess matrix and the stiffness matrix are derived. A finite element program is used to solve the wind induced vibration of the eccentricity cable.
本文考虑了覆冰索面内、面外和转动三个方向的振动形式,并且考虑到了在偏心效应下的耦合振动。将风荷载表示为攻角的非线性函数,通过Hamilton原理导出动力学方程。采用Galerkin法将控制方程离散化,并且用数值解验证了稳定性条件。在给定的Hopf分岔设计点附近,采用近似解析方法确定了稳定域的边界形状,节省了一定的计算工作量。主要工作包括：
(1)建立了偏心索在风致力作用下三个自由度方向上的耦合振动的模型,将风荷载表示为攻角的非线性函数,通过Hamilton原理导出动力学方程。采用Galerkin法将控制方程离散化为常微分方程。
(2)分析所建立方程的非线性动力学行为,根据Routh-Hurwitz判据得到索平衡构形在参数空间内的稳定域,确定了发生Hopf分岔的临界风速,对于得到的稳定域给出了数值解的验证,并通过与以往文献中类似问题的两个自由度模型比较说明了本文建立的三个自由度模型的优点。
(3)结合本文提出的模型建立了适合本问题的索单元的有限元离散方法。推导得到了索单元的刚度阵和质量阵。结合有限元程序对偏心索风致振动的问题得到了有限元的解法。
Suspend cables are wildly used in cable bridges, membrane structures and transmission lines for its low weights, high strength and easy to construct. In the winter when the cable was covered by ice caused by the rain and snow, the eccentricity makes the in-plane and out-planes vibration couple with the torsion vibration. So the motions of the cable become much more complicated, and it maybe leads to some terrible results. In the spring of 2008 the ice-covered transmission lines fractured in the south of China which made lots of economic loss and large-scale power-failure. So it's necessary to investigate the nonlinear dynamics and the stable region of the eccentricity cables loaded by wind.
The wind-induced vibration is considered to be the in-plane, out-plane and the torsion vibrations. The wind load is considered as a nonlinear function of the angle of attack, and Hamilton principal is used to obtain the governing partial differential equation of motion. Galerkin approach is adopted to reduce the governing partial differential equation to an approximate system with few numbers of DOFs. The stability conditions can be verified with numerical solutions. At the given point of Hopf bifurcation an approximation is used to get the stability region saving lots of time and workload. The research work can be summarized as follows:
(1) A 3-degree-of-freedom eccentricity cable loaded by wind model is established. The wind load is considered as a nonlinear function of the angle of attack, and Hamilton principal was used to obtain the governing partial differential equation of motion. The Galerkin approach is adopted to reduce the governing partial differential equation to differential equations.
(2) Analysis the differential equations, explicit conditions are provided for loss of stability based on the Routh-Hurwitz criterion while for generation of stable limit cycles Hopf bifurcation. The stability conditions can be verified with numerical solutions. Compare with the 2-degree-of-freedom model established in other papers; obtain the advantages of the model in the research.
(3) According to the 3-degree-of-freedom model established in the paper, a cable element is built for the finite element discrimination. The mess matrix and the stiffness matrix are derived. A finite element program is used to solve the wind induced vibration of the eccentricity cable.
引文
[1]BILLAH K. Y and SCANIAN R. H. Resonance, Tacoma Narrow bridge failure and undergraduate physics textbooks. [J]. American Association of Physics Teachers 1991 Am. J. phys.59 (2). February
[2]DIMAROGONAS A O. The origins of vibration theory. [J]. Journal of Sound and Vibration,1990,140:181-189
[3]NAFEH A.H and BALACHANDRAN B. Applied nonlinear dynamics analytical computational and experimental methods. [J] New York:Wiley,1995
[4]席德勋,席沁编著.非线性物理学[M].南京：南京大学出版社,2007.
[5]刘延柱,陈立群编著.非线性振动[M].北京：高等教育出版社,2001.
[6]THOMPSON J.M.T and STEWART H. B. Nonlinear dynamics and chaos[M]. John Wiley and Sons,Ltd,England.2002
[7]COOK N.J. The designer's guide to wind loading of building structures [M]. London: Butterworth,1985.
[8]黄本才汪从军编著.结构抗风分析原理及应用[M]上海：同济大学出版社,2008
[9]SOCKEL H. Wind excited vibration of structures [M], Wien:Springer-Verlag,1994
[10]MURAKAMI S. Current status and future trends in computational wind engineering [J]. Journal of Wind Engineering and Industrial Aerodynamics,1997,67&68,3-34.
[11]CHEN Y., Wu J. Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network [J]. Physica D,1999,134:185-199.
[12]RAGHOTHAMA A., NARAYANAN S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay [J]. Nonlinear Dynamics,2002,27:341-365.
[13]DEN HARTOG J. P., Transmission Line Vibration Due to Sleet Trans [J]. AIEE,1932, Vol.51, part 4, pp.1074-1086
[14]BLEVINS R. D. and IWAN W. D. The Galloping Response of a Two-Degree-of-Freedom System, [J].ASME Journal of Applied Mechenics,1974,Vol.41, pp.1113-1118.
[15]DESAI Y M., POPPLEWELL N., SHAH A H, and CHAN J K., Static and Dynamic Behavior of Mechanics Components Associated with Electrical Transmission Lines-Ⅲ:Part A-Theoretical Perspective, [J], Shock and Vib. Dig,1989, Vol.21, pp.3-8
[16]RAO G. V. and Iyengar R. Internal resonance and non-linear response of a cable under periodic excitation [J], Journal of Sound and Vibration,1991 149,25-41
[17]NAYFEH A H., ARAFAT H N. Multimode Interactions in Suspended Cable [J] Journal of Vibration and Countrol,8:337-387,2002
[18]YU P., POPPLEWELL N. and SHAH A H. A Geometrical Approach Assessing Instability Trends for Galloping [J] Journal of Applied Mechanics 1991 Vol.58 784-791
[19]LEE C. L. and PERKINS N. C. Three-dimensional oscillations of suspended cables involving simultaneous internal resonances [J], Proceedings of ASME Winter Annual Meeting 1992, AMD-144.59-67
[20]YU P., POPPLEWELL N. And SHAH A. H. Initially Coupled Galloping of Iced Conductors [J] Journal of Sound and Vibration 1992 Vol.59 140-145
[21]BENEDETTINI F., REGA G. and ALAGGIO R. non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions, [J] Journal of Sound and Vibration 1995 182(5),775-798
[22]ARAFAT H. N., NAYFHE A. H. Non-linear responses of suspended cables to primary resonance excitations, [J], Journal of Sound and Vibration 266 (2003) 325-354
[23]SRINIL N., REGA G., CHUCHEEPSAKUL S., Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cable, [J] Journal of Sound and Vibration 269 (2004) 823-852
[24]BERLIOZ A., LAMARQUE C. H. Nonlinear Vibration of an inclined cable [J] Journal of Vibration and Acoustics 2005 Vol.127 315-323
[25]DESAI Y M., YU P., POPPLEWELL N. and SHAH A. H. Finite element modeling of transmission line galloping [J] Computers and Structures Vol.57 No.3 407-420.1995
[26]顾明,马文勇.两种典型覆冰导线气动力特性及稳定性分析[J]同济大学学报,2009.
[27]马文勇,顾明,全涌,黄鹏覆冰导线任意方向驰振分析方法[J]同济大学学报(自然科学版)2010 Vol.38No.1
[28]MUKHOPADYAY V. and DUGUNDJ I. Addendum-wind-excited vibrations of a square section cantilever beam in smooth flow. [J]Journal of Sound and Vibration 56(2),309-311.
[29]RICHARDSON J. A. S. The time-line method for assessing galloping exposure. [C] IEEE Paper 82 WM 083-4, IEEE, New York, N.Y.1982
[30]ERICSSON L. E. Limit amplitude of galloping cables[C]. AlAA 83-0132, Amer. Inst. Aeronaut, and Astronautics Meeting, Jan.1983
[31]EGBERT R. I. Estimation of maximum amplitudes of conductor galloping by describing function analysis[C]. IEEE 85 WM 202-7, IEEE, New York, N. Y.1985
[32]GATTULLI V. and LEPIDI M. Nonlinear interactions in the planar dynamics of cable-stayed beam[J]. International Journal of Solids and Structures 40 (2003) 4729-4748
[33]GATTULLI V. and MARTINELLI L. Nonlinear oscillations of cables under harmonic loading using analytical and finite element models[J]. Computer methods in applied mechanics and engineering.193 (2004) 69-85
[34]MATSUMTOT M., ATTOH T., KITAZAWZ M. Response characteristics of rain-wind induced vibration of stay-cables of cable-stayed bridges. [J] Journal of Wind Engineering and Industrial Aerodynamics,1995,57:323-333.
[35]PEI U. and NAHRATH N. Modeling of rain-wind induced vibrations. [J] Wind and Structures, 2003,6(1):41-52
[36]DONEA J. A Taylor-Galerkin method for convective transport problems. [M] John Wiley & Sons, Ltd.1984
[37]JACK H. Theory of Functional Differential Equation [M]. New York:Spring-Verlag,1977.
[38]刘延柱,陈文良,陈立群编著.振动力学[M].北京：高等教育出版社,1998.
[39]GUCKENHEIMER J. and HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. [M].New York:springer-Verlag,1983.
[40]VIALAR T. Complex and chaotic nonlinear dynamics:advances in economics and finance, mathematics and statistics. [M]. Berlin Heidelberg:New York:springer-Verlag,2009
[41]GUCKANHEIMER J., HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields [M].New York:Springer-Verlag,1983
[42]GUCKENHEIMER J.and HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York:Springer-Verlag,1983.
[43]KUZNETSOV Y A. Elements of applied bifurcation theory. New York:Springer-Verlag,2004
[44]HIKAMI Y.,SHIRAISHI N. Rain-wind induced vibrations of cables stayed bridges. [J]. Journal of wind Engineering and Industrial Aerodynamics,29(1-3):409-418,1988.
[45]BURTON D. CAO D Q., et al. On the stability of stay cables under light wind and rain conditions. [J]. Journal of Sound and Vibration,279(1-2):89-117,2005.
[46]SALMON R. Practical use of Hamilton's principle. [J], Fluid Mech vol.132.pp.431-444.1983
[47]徐赵东,马乐为编著.结构动力学.[M].北京：高等教育出版社,2007
[48]刘延柱陈立群编著.非线性动力学.[M].上海：上海交通大学出版社,2000
[49]LEE C. L., PERKINS C. N. Nonlinear oscillations of suspended cables containing a two-to-one internal resonance. [J].Nonlinear Dynamic,3(6):465-490,1992
[50]秦朝红陈予恕李军,1：1内共振悬索的二维奇异性分析[J]. Applied Mathematics and Mechanics, Vo.l 31, No.2, Feb.15,2010
[51]SEYRANIAN A P., MAILYBAEV A A. Multiparameter Stability Theory with Mechanical Applications [M] World Scientific Publishing Co. Pet. Ltd.2003
[52]KEVORKIAN J., JULIAN D., Multiple scale and singular perturbation methods [M]. New York: springer-Verlag,1996.
[53]GHOSH S., LEE K.,MOORTHY S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method. [J]. International Journal of Solids and Structures Volume 32, Issue 1, January 1995, Pages 27-62
[54]Wang Yuefang, Lu Lefeng, Huang Lihua. Analysis and design optimization of axially moving structure with stability constraint under wind excitations. [J]. Advances in Structural Engineering,2007,10(6):655-661
[55]吕乐丰,王跃方,刘迎曦。横向风荷载作用下轴向行进弦线自激振动和稳定性分析,工程力学,2008,25(2)： 40-45
[2]DIMAROGONAS A O. The origins of vibration theory. [J]. Journal of Sound and Vibration,1990,140:181-189
[3]NAFEH A.H and BALACHANDRAN B. Applied nonlinear dynamics analytical computational and experimental methods. [J] New York:Wiley,1995
[4]席德勋,席沁编著.非线性物理学[M].南京：南京大学出版社,2007.
[5]刘延柱,陈立群编著.非线性振动[M].北京：高等教育出版社,2001.
[6]THOMPSON J.M.T and STEWART H. B. Nonlinear dynamics and chaos[M]. John Wiley and Sons,Ltd,England.2002
[7]COOK N.J. The designer's guide to wind loading of building structures [M]. London: Butterworth,1985.
[8]黄本才汪从军编著.结构抗风分析原理及应用[M]上海：同济大学出版社,2008
[9]SOCKEL H. Wind excited vibration of structures [M], Wien:Springer-Verlag,1994
[10]MURAKAMI S. Current status and future trends in computational wind engineering [J]. Journal of Wind Engineering and Industrial Aerodynamics,1997,67&68,3-34.
[11]CHEN Y., Wu J. Minimal instability and unstable set of a phase-locked periodic orbit in a delayed neural network [J]. Physica D,1999,134:185-199.
[12]RAGHOTHAMA A., NARAYANAN S. Periodic response and chaos in nonlinear systems with parametric excitation and time delay [J]. Nonlinear Dynamics,2002,27:341-365.
[13]DEN HARTOG J. P., Transmission Line Vibration Due to Sleet Trans [J]. AIEE,1932, Vol.51, part 4, pp.1074-1086
[14]BLEVINS R. D. and IWAN W. D. The Galloping Response of a Two-Degree-of-Freedom System, [J].ASME Journal of Applied Mechenics,1974,Vol.41, pp.1113-1118.
[15]DESAI Y M., POPPLEWELL N., SHAH A H, and CHAN J K., Static and Dynamic Behavior of Mechanics Components Associated with Electrical Transmission Lines-Ⅲ:Part A-Theoretical Perspective, [J], Shock and Vib. Dig,1989, Vol.21, pp.3-8
[16]RAO G. V. and Iyengar R. Internal resonance and non-linear response of a cable under periodic excitation [J], Journal of Sound and Vibration,1991 149,25-41
[17]NAYFEH A H., ARAFAT H N. Multimode Interactions in Suspended Cable [J] Journal of Vibration and Countrol,8:337-387,2002
[18]YU P., POPPLEWELL N. and SHAH A H. A Geometrical Approach Assessing Instability Trends for Galloping [J] Journal of Applied Mechanics 1991 Vol.58 784-791
[19]LEE C. L. and PERKINS N. C. Three-dimensional oscillations of suspended cables involving simultaneous internal resonances [J], Proceedings of ASME Winter Annual Meeting 1992, AMD-144.59-67
[20]YU P., POPPLEWELL N. And SHAH A. H. Initially Coupled Galloping of Iced Conductors [J] Journal of Sound and Vibration 1992 Vol.59 140-145
[21]BENEDETTINI F., REGA G. and ALAGGIO R. non-linear oscillations of a four-degree-of-freedom model of a suspended cable under multiple internal resonance conditions, [J] Journal of Sound and Vibration 1995 182(5),775-798
[22]ARAFAT H. N., NAYFHE A. H. Non-linear responses of suspended cables to primary resonance excitations, [J], Journal of Sound and Vibration 266 (2003) 325-354
[23]SRINIL N., REGA G., CHUCHEEPSAKUL S., Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cable, [J] Journal of Sound and Vibration 269 (2004) 823-852
[24]BERLIOZ A., LAMARQUE C. H. Nonlinear Vibration of an inclined cable [J] Journal of Vibration and Acoustics 2005 Vol.127 315-323
[25]DESAI Y M., YU P., POPPLEWELL N. and SHAH A. H. Finite element modeling of transmission line galloping [J] Computers and Structures Vol.57 No.3 407-420.1995
[26]顾明,马文勇.两种典型覆冰导线气动力特性及稳定性分析[J]同济大学学报,2009.
[27]马文勇,顾明,全涌,黄鹏覆冰导线任意方向驰振分析方法[J]同济大学学报(自然科学版)2010 Vol.38No.1
[28]MUKHOPADYAY V. and DUGUNDJ I. Addendum-wind-excited vibrations of a square section cantilever beam in smooth flow. [J]Journal of Sound and Vibration 56(2),309-311.
[29]RICHARDSON J. A. S. The time-line method for assessing galloping exposure. [C] IEEE Paper 82 WM 083-4, IEEE, New York, N.Y.1982
[30]ERICSSON L. E. Limit amplitude of galloping cables[C]. AlAA 83-0132, Amer. Inst. Aeronaut, and Astronautics Meeting, Jan.1983
[31]EGBERT R. I. Estimation of maximum amplitudes of conductor galloping by describing function analysis[C]. IEEE 85 WM 202-7, IEEE, New York, N. Y.1985
[32]GATTULLI V. and LEPIDI M. Nonlinear interactions in the planar dynamics of cable-stayed beam[J]. International Journal of Solids and Structures 40 (2003) 4729-4748
[33]GATTULLI V. and MARTINELLI L. Nonlinear oscillations of cables under harmonic loading using analytical and finite element models[J]. Computer methods in applied mechanics and engineering.193 (2004) 69-85
[34]MATSUMTOT M., ATTOH T., KITAZAWZ M. Response characteristics of rain-wind induced vibration of stay-cables of cable-stayed bridges. [J] Journal of Wind Engineering and Industrial Aerodynamics,1995,57:323-333.
[35]PEI U. and NAHRATH N. Modeling of rain-wind induced vibrations. [J] Wind and Structures, 2003,6(1):41-52
[36]DONEA J. A Taylor-Galerkin method for convective transport problems. [M] John Wiley & Sons, Ltd.1984
[37]JACK H. Theory of Functional Differential Equation [M]. New York:Spring-Verlag,1977.
[38]刘延柱,陈文良,陈立群编著.振动力学[M].北京：高等教育出版社,1998.
[39]GUCKENHEIMER J. and HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. [M].New York:springer-Verlag,1983.
[40]VIALAR T. Complex and chaotic nonlinear dynamics:advances in economics and finance, mathematics and statistics. [M]. Berlin Heidelberg:New York:springer-Verlag,2009
[41]GUCKANHEIMER J., HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields [M].New York:Springer-Verlag,1983
[42]GUCKENHEIMER J.and HOLMES P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York:Springer-Verlag,1983.
[43]KUZNETSOV Y A. Elements of applied bifurcation theory. New York:Springer-Verlag,2004
[44]HIKAMI Y.,SHIRAISHI N. Rain-wind induced vibrations of cables stayed bridges. [J]. Journal of wind Engineering and Industrial Aerodynamics,29(1-3):409-418,1988.
[45]BURTON D. CAO D Q., et al. On the stability of stay cables under light wind and rain conditions. [J]. Journal of Sound and Vibration,279(1-2):89-117,2005.
[46]SALMON R. Practical use of Hamilton's principle. [J], Fluid Mech vol.132.pp.431-444.1983
[47]徐赵东,马乐为编著.结构动力学.[M].北京：高等教育出版社,2007
[48]刘延柱陈立群编著.非线性动力学.[M].上海：上海交通大学出版社,2000
[49]LEE C. L., PERKINS C. N. Nonlinear oscillations of suspended cables containing a two-to-one internal resonance. [J].Nonlinear Dynamic,3(6):465-490,1992
[50]秦朝红陈予恕李军,1：1内共振悬索的二维奇异性分析[J]. Applied Mathematics and Mechanics, Vo.l 31, No.2, Feb.15,2010
[51]SEYRANIAN A P., MAILYBAEV A A. Multiparameter Stability Theory with Mechanical Applications [M] World Scientific Publishing Co. Pet. Ltd.2003
[52]KEVORKIAN J., JULIAN D., Multiple scale and singular perturbation methods [M]. New York: springer-Verlag,1996.
[53]GHOSH S., LEE K.,MOORTHY S. Multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method. [J]. International Journal of Solids and Structures Volume 32, Issue 1, January 1995, Pages 27-62
[54]Wang Yuefang, Lu Lefeng, Huang Lihua. Analysis and design optimization of axially moving structure with stability constraint under wind excitations. [J]. Advances in Structural Engineering,2007,10(6):655-661
[55]吕乐丰,王跃方,刘迎曦。横向风荷载作用下轴向行进弦线自激振动和稳定性分析,工程力学,2008,25(2)： 40-45