基于弹性细杆理论的深海立管和系泊线动力学模型研究
摘要
弹性细杆问题有着广泛的实际背景,立管、系泊线、钻杆、电缆、绳索等等都可将弹性细杆作为其力学模型。在海洋开发急速发展的今天,促使弹性细杆力学发展的一个新领域就是深海立管和系泊系统。
本文以弹性细杆理论为基础推导了适用于深海立管和系泊线的运动微分方程。在推导过程中,本文考虑了多种类型立管和系泊线常见的布置形态,如悬垂状态、张紧状态等,使用有限元法在统一的整体坐标系中模拟了三维空间中的立管和系泊线。通过多单元类型的设置模拟了实际工程中存在的多类型材料组成的系泊线,并保证了结构截面的平滑过渡。本文推导的以深海中的立管和系泊线为对象基于弹性细杆理论的运动微分方程解决了悬链线模型不能处理的动态刚度问题,处理了梁模型不能模拟的深海管线动力响应问题,以及结构极端细长性引起的小应变大位移的几何非线性问题。在对结构的边界条件的处理中,根据实际工程情况,本文给出了弹簧阻尼约束和海底的模拟等处理方法。通过弹簧阻尼约束,可以方便地设定边界条件坐标系,在坐标系中可以定义节点约束如线约束、弹簧约束和阻尼约束等。本文采用一层线性弹簧模拟平坦的海底实现海底条件对管线运动的影响。
基于以上理论,本文使用Fortran90语言编制了深海立管和系泊线的静力和动力分析程序,用于求解结构在动载荷作用下的运动规律和应力变化情况。由于本文采用时域分析法,莫里森公式中的非线性项等内容均得到了保留,不需要进行线性化处理,保证了载荷更接近实际情况。程序中特别注重了输入文件的格式,以期静力分析和动力分析共用一个文件,差别仅在几个参数的修改。经过实践应用证明,弹性细杆理论是解决深海立管和系泊线问题的有效理论。
Elastic rod problem has extensive practical background, for example, riser, drilling line, mooring line, fiber, cable, slender stems of plants and DNA, etc. Elastic rod can be as their mechanical model. With the rapid development of marine technology, a new application field for elastic rod theory is the deepwater riser and mooring engineering.
The elastic rod theory described here is designed to support static and dynamic analysis for riser and mooring line. In global coordinate, arbitrary layout in three-dimensional space can be simulated for many types of risers and mooring lines, such as suspension and taut condition. Different element types are available in one line simulation in order to get close to the actual structure. Different sections of the line are ensured smooth transition. The unified differential motion equations can be used for risers and mooring lines, including terms of bending stiffness and inside fluid. Elastic rod theory can be used to solve problem of dynamical stiffness for deepwater risers and mooring lines. At the same time, dynamic response analysis can be performed by elastic rod theory together with geometric nonlinearity problem. Finite difference method and finite element method are all available to solve the motion equations. Finite element method is used to get the unknowns. The spring and damping system is included to simulate boundary condition for the actual engineering. In boundary coordinate, the node can be set line constraint when the rotation of the node is free. For catenary mooring lines and steel catenary riser, linear spring is used to simulate flat seabed.
With the support of elastic rod theory, a program is made with Fortran90 to get structure response with the varying loads. The main application of the program is to perform static analysis and dynamic analysis. Time domain method is effective and accurate to solve relative motion of lines and the nonlinear terms in Morison equation. The characteristics of large displacement and finite deformation for riser are also considered to deal with nonlinear stiffness matrix. The same format input file is used for convenience. The elastic rod theory is proved effectively to solve problems of mooring lines and risers.
本文以弹性细杆理论为基础推导了适用于深海立管和系泊线的运动微分方程。在推导过程中,本文考虑了多种类型立管和系泊线常见的布置形态,如悬垂状态、张紧状态等,使用有限元法在统一的整体坐标系中模拟了三维空间中的立管和系泊线。通过多单元类型的设置模拟了实际工程中存在的多类型材料组成的系泊线,并保证了结构截面的平滑过渡。本文推导的以深海中的立管和系泊线为对象基于弹性细杆理论的运动微分方程解决了悬链线模型不能处理的动态刚度问题,处理了梁模型不能模拟的深海管线动力响应问题,以及结构极端细长性引起的小应变大位移的几何非线性问题。在对结构的边界条件的处理中,根据实际工程情况,本文给出了弹簧阻尼约束和海底的模拟等处理方法。通过弹簧阻尼约束,可以方便地设定边界条件坐标系,在坐标系中可以定义节点约束如线约束、弹簧约束和阻尼约束等。本文采用一层线性弹簧模拟平坦的海底实现海底条件对管线运动的影响。
基于以上理论,本文使用Fortran90语言编制了深海立管和系泊线的静力和动力分析程序,用于求解结构在动载荷作用下的运动规律和应力变化情况。由于本文采用时域分析法,莫里森公式中的非线性项等内容均得到了保留,不需要进行线性化处理,保证了载荷更接近实际情况。程序中特别注重了输入文件的格式,以期静力分析和动力分析共用一个文件,差别仅在几个参数的修改。经过实践应用证明,弹性细杆理论是解决深海立管和系泊线问题的有效理论。
Elastic rod problem has extensive practical background, for example, riser, drilling line, mooring line, fiber, cable, slender stems of plants and DNA, etc. Elastic rod can be as their mechanical model. With the rapid development of marine technology, a new application field for elastic rod theory is the deepwater riser and mooring engineering.
The elastic rod theory described here is designed to support static and dynamic analysis for riser and mooring line. In global coordinate, arbitrary layout in three-dimensional space can be simulated for many types of risers and mooring lines, such as suspension and taut condition. Different element types are available in one line simulation in order to get close to the actual structure. Different sections of the line are ensured smooth transition. The unified differential motion equations can be used for risers and mooring lines, including terms of bending stiffness and inside fluid. Elastic rod theory can be used to solve problem of dynamical stiffness for deepwater risers and mooring lines. At the same time, dynamic response analysis can be performed by elastic rod theory together with geometric nonlinearity problem. Finite difference method and finite element method are all available to solve the motion equations. Finite element method is used to get the unknowns. The spring and damping system is included to simulate boundary condition for the actual engineering. In boundary coordinate, the node can be set line constraint when the rotation of the node is free. For catenary mooring lines and steel catenary riser, linear spring is used to simulate flat seabed.
With the support of elastic rod theory, a program is made with Fortran90 to get structure response with the varying loads. The main application of the program is to perform static analysis and dynamic analysis. Time domain method is effective and accurate to solve relative motion of lines and the nonlinear terms in Morison equation. The characteristics of large displacement and finite deformation for riser are also considered to deal with nonlinear stiffness matrix. The same format input file is used for convenience. The elastic rod theory is proved effectively to solve problems of mooring lines and risers.
引文
[1]Garrett D L. Dynamic analysis of slender rods. Journal of Energy Resources Technology, Transaction of ASME,1982,104:302-307P
[2]Pauling J R, Webster W C. A Consistent Large-Amplitude Analysis of the Coupled Response of a TLP and Tension System.1986
[3]Kimb A T A M. Coupled-dynamic analysis of floating structures with polyester mooring lines. Ocean Engineering,2008,35(7):7-9P
[4]Seyed M H P A. Review of flexible riser modelling and analysis techniques. Engineering Structures,1999,17(4):293-304P
[5]Chen X, Zhang J. Dynamic Analysis of Mooring Lines by Using Three Different Methods. NORWAY:2001
[6]Chen X, Zhang J, Iranib P J A M. Dynamic analysis of mooring lines with inserted springs. Applied Ocean Research,2002,23(5)
[7]Garrett D L. Coupled analysis of floating production systems. Ocean Engineering, 2004,32(7):13-14P,15-17P,14-15P,16-17P
[8]Cohen H. A nonlinear theory of elastic directed curves. Int. J. Engng Sci., 1966,(4):511-524P
[9]Zajac E E. Stability of two planar loop elasticas. J. Appl. Mech.,1962
[10]Antman S S. The thory of rods, New York:Springer,1972
[11]Antman S S. Nonlinear Problems of Elasticity, New York:Springer New York,1995
[12]Knap R H. Helical wire stresses in bent cables. Trans. ASME J. Offshore Mech. And Artic Engng,1988,110:55-66P
[13]Coyne J. Analysis of the for,ulation and elimination of loops in twisted cable. IEEE J. of Oceanic Engen,1990,15(2):72-83P
[14]Seemann W. Deformation of an elastic helix in contact with a rigid cylinder. Archive of Applied Mechanics,1996,67:117-139P
[15]Van Der H G H M. The spatial complexity of localized buckling in rods with noncircular cross section. SIAM J. Appl. math.,1998,59(1):198-221P
[16]Van Der H G H M. Lock-on to tape-like behavor in the torsional buckling of anisotropoic rods. Physica,1998, D112:201-224.
[17]张雄,王天舒.计算动力学.北京：清华大学出版社,2007
[18]聂武,刘玉秋.海洋工程结构动力分析.哈尔滨：哈尔滨工程大学出版社,2002
[19]唐友刚.高等结构动力学.天津大学出版社,2002
[20]仇伟德.机械振动.石油大学出版社,2001
[21]Kirchhoff G. Uber das Gleichgewicht und die Bewegung eines unendlich dunnen elastischen stabes. J. Rein Angew. Math.,1859,56:285-313P
[22]Clebsch A. Theorie der Elasticitat Fester Korper, Leipzig:Teubner,1862
[23]Love A E H. A treatise on mathematical theory of elasticity, New York: Dover,1927
[24]Timoshenko S P. History of strength of materials, New York:McGraw-Hill, 1953
[25]Timoshenko S P. Strength of materials, New York:Van Nostrand Company, 1957
[26]Timoshenko S P. Theory of elastic stability.2nd ed., New York:McGraw-Hill,1961
[27]Ericksen J L, Truesdell C. Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal.,1958,1:293-323P
[28]Reissner R. On a one-dimensional, large-displacement, finite-strain beam-theory. Stud. Appl. Math.,1973,52:87-95P
[29]Ilyukhin A A. Spatial problems of the nonlinear throry of elastic rods, Kiev: Naukova Dumka,1979
[30]Frisch-Fay R. Flxible bars, London:Butterworth,1962
[31]Simo J C, Marsden J E, Krishnaparasad P S. The Hamiltonian structure of nonlinear elasticity:the material and convective representation of solids,rods and plates. Arch. Rat. Mech. Anal.,1988,104:125-183P
[32]Tucker W R, Wang C. On the Effective Control of Torsional Vibrations in Drilling Systems.1999,224(1):101 - 122P
[33]Ma W, Webster W C. An Analytical Approach to Cable Dynamics Theory and User Manual:SEA GRANT PROJECT R/OE-26,1994
[34]Ran Z. Coupled dynamic analysis of floating structures in waves and currents:[Ph.D学位论文].Texas A&M University.,2000
[35]Arcandra. Hull/mooring/riser coupled dynamic analysis of a deepwater floating platform with polyester lines:[Ph.D.学位论文].Texas A&M University. 2001
[36]Tucker R W, Wang C. Torsional Vibration Control and Cosserat Dynamics of a Drill-Rig Assembly. Meccanica,2003,38(1):145-161P
[37]刘延柱.弹性细杆的非线性力学——DNA力学模型的理论基础.清华大学出版社,2006
[38]刘延柱.曲线的运动学.力学与实践.2004
[39]刘延柱,薛纭.受轴向力和扭矩作用的直杆的平衡稳定性.力学与实践.2005,27(1)：64-65页
[40]薛纭,张毅.超细长弹性杆的力学模型及其边界条件.应用科学学报.2007,25(3)：306-310页
[41]王兴波,陈希祥,黎丽梅.任意参数形式下的Frenet公式.湖南理工学院学报(自然科学版).2005
[42]哈尔滨工业大学理论力学教研组.理论力学.北京：高等教育出版社,1997
[43]范钦珊.材料力学.北京：清华大学出版社出版,2004
[44]Randall R E. Elements of Ocean Engineering,上海：上海交通大学出版社,2002
[45]张亮,李云波.流体力学.哈尔滨：哈尔滨工程大学出版社,2001
[46]王勖成,邵敏.有限单元法基本原理和数值方法.北京：清华大学出版社,1997
[47]聂武,孙丽萍.船舶计算结构力学.哈尔滨：哈尔滨工程大学出版社,2003
[48]谈梅兰,王鑫伟.一种有效的分析任意空间形状曲杆单元的位移函数. 工程力学.2004,21(3)：134-137页
[49]王国荣,俞耀明,徐兆亮.数值分析：王国荣,俞耀明,徐兆亮.机械工业出版社,2005
[50]鞠衍清,张风雷.基于MATLAB的单摆周期近似解的比较.大学物理.2007,26(3)：6-9页
[51]鞠衍清,张风雷.单摆运动周期近似公式的数值推导及修正.大学物理.2007,26(5)：15-17页
[52]罗勇锋,龚志强,李水.大幅度摆动单摆的高精度周期公式研究.科技咨询导报.2007,(28)：143页
[53]张燕林,李建国.单摆周期几种近似公式比较.河南教育学院学报：自然科学版.2005,(2)
[2]Pauling J R, Webster W C. A Consistent Large-Amplitude Analysis of the Coupled Response of a TLP and Tension System.1986
[3]Kimb A T A M. Coupled-dynamic analysis of floating structures with polyester mooring lines. Ocean Engineering,2008,35(7):7-9P
[4]Seyed M H P A. Review of flexible riser modelling and analysis techniques. Engineering Structures,1999,17(4):293-304P
[5]Chen X, Zhang J. Dynamic Analysis of Mooring Lines by Using Three Different Methods. NORWAY:2001
[6]Chen X, Zhang J, Iranib P J A M. Dynamic analysis of mooring lines with inserted springs. Applied Ocean Research,2002,23(5)
[7]Garrett D L. Coupled analysis of floating production systems. Ocean Engineering, 2004,32(7):13-14P,15-17P,14-15P,16-17P
[8]Cohen H. A nonlinear theory of elastic directed curves. Int. J. Engng Sci., 1966,(4):511-524P
[9]Zajac E E. Stability of two planar loop elasticas. J. Appl. Mech.,1962
[10]Antman S S. The thory of rods, New York:Springer,1972
[11]Antman S S. Nonlinear Problems of Elasticity, New York:Springer New York,1995
[12]Knap R H. Helical wire stresses in bent cables. Trans. ASME J. Offshore Mech. And Artic Engng,1988,110:55-66P
[13]Coyne J. Analysis of the for,ulation and elimination of loops in twisted cable. IEEE J. of Oceanic Engen,1990,15(2):72-83P
[14]Seemann W. Deformation of an elastic helix in contact with a rigid cylinder. Archive of Applied Mechanics,1996,67:117-139P
[15]Van Der H G H M. The spatial complexity of localized buckling in rods with noncircular cross section. SIAM J. Appl. math.,1998,59(1):198-221P
[16]Van Der H G H M. Lock-on to tape-like behavor in the torsional buckling of anisotropoic rods. Physica,1998, D112:201-224.
[17]张雄,王天舒.计算动力学.北京：清华大学出版社,2007
[18]聂武,刘玉秋.海洋工程结构动力分析.哈尔滨：哈尔滨工程大学出版社,2002
[19]唐友刚.高等结构动力学.天津大学出版社,2002
[20]仇伟德.机械振动.石油大学出版社,2001
[21]Kirchhoff G. Uber das Gleichgewicht und die Bewegung eines unendlich dunnen elastischen stabes. J. Rein Angew. Math.,1859,56:285-313P
[22]Clebsch A. Theorie der Elasticitat Fester Korper, Leipzig:Teubner,1862
[23]Love A E H. A treatise on mathematical theory of elasticity, New York: Dover,1927
[24]Timoshenko S P. History of strength of materials, New York:McGraw-Hill, 1953
[25]Timoshenko S P. Strength of materials, New York:Van Nostrand Company, 1957
[26]Timoshenko S P. Theory of elastic stability.2nd ed., New York:McGraw-Hill,1961
[27]Ericksen J L, Truesdell C. Exact theory of stress and strain in rods and shells. Arch. Rational Mech. Anal.,1958,1:293-323P
[28]Reissner R. On a one-dimensional, large-displacement, finite-strain beam-theory. Stud. Appl. Math.,1973,52:87-95P
[29]Ilyukhin A A. Spatial problems of the nonlinear throry of elastic rods, Kiev: Naukova Dumka,1979
[30]Frisch-Fay R. Flxible bars, London:Butterworth,1962
[31]Simo J C, Marsden J E, Krishnaparasad P S. The Hamiltonian structure of nonlinear elasticity:the material and convective representation of solids,rods and plates. Arch. Rat. Mech. Anal.,1988,104:125-183P
[32]Tucker W R, Wang C. On the Effective Control of Torsional Vibrations in Drilling Systems.1999,224(1):101 - 122P
[33]Ma W, Webster W C. An Analytical Approach to Cable Dynamics Theory and User Manual:SEA GRANT PROJECT R/OE-26,1994
[34]Ran Z. Coupled dynamic analysis of floating structures in waves and currents:[Ph.D学位论文].Texas A&M University.,2000
[35]Arcandra. Hull/mooring/riser coupled dynamic analysis of a deepwater floating platform with polyester lines:[Ph.D.学位论文].Texas A&M University. 2001
[36]Tucker R W, Wang C. Torsional Vibration Control and Cosserat Dynamics of a Drill-Rig Assembly. Meccanica,2003,38(1):145-161P
[37]刘延柱.弹性细杆的非线性力学——DNA力学模型的理论基础.清华大学出版社,2006
[38]刘延柱.曲线的运动学.力学与实践.2004
[39]刘延柱,薛纭.受轴向力和扭矩作用的直杆的平衡稳定性.力学与实践.2005,27(1)：64-65页
[40]薛纭,张毅.超细长弹性杆的力学模型及其边界条件.应用科学学报.2007,25(3)：306-310页
[41]王兴波,陈希祥,黎丽梅.任意参数形式下的Frenet公式.湖南理工学院学报(自然科学版).2005
[42]哈尔滨工业大学理论力学教研组.理论力学.北京：高等教育出版社,1997
[43]范钦珊.材料力学.北京：清华大学出版社出版,2004
[44]Randall R E. Elements of Ocean Engineering,上海：上海交通大学出版社,2002
[45]张亮,李云波.流体力学.哈尔滨：哈尔滨工程大学出版社,2001
[46]王勖成,邵敏.有限单元法基本原理和数值方法.北京：清华大学出版社,1997
[47]聂武,孙丽萍.船舶计算结构力学.哈尔滨：哈尔滨工程大学出版社,2003
[48]谈梅兰,王鑫伟.一种有效的分析任意空间形状曲杆单元的位移函数. 工程力学.2004,21(3)：134-137页
[49]王国荣,俞耀明,徐兆亮.数值分析：王国荣,俞耀明,徐兆亮.机械工业出版社,2005
[50]鞠衍清,张风雷.基于MATLAB的单摆周期近似解的比较.大学物理.2007,26(3)：6-9页
[51]鞠衍清,张风雷.单摆运动周期近似公式的数值推导及修正.大学物理.2007,26(5)：15-17页
[52]罗勇锋,龚志强,李水.大幅度摆动单摆的高精度周期公式研究.科技咨询导报.2007,(28)：143页
[53]张燕林,李建国.单摆周期几种近似公式比较.河南教育学院学报：自然科学版.2005,(2)