铰接塔平台非线性动力特性研究
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摘要
铰接塔平台为适应水深100m～500m的顺应式平台,可用于单点系泊、装载终端、控制塔、海底安装等生产环节。铰接塔平台承受非线性海洋载荷作用,其摇摆运动特性和运动稳定的研究对于确保平台的安全具有重要的意义。本文应用非线性动力学理论和方法,研究了铰接塔平台复杂海洋环境作用下的非线性运动特性和运动稳定性,以及单点系泊铰接塔的运动特性及系泊张力的冲击特性。本文的工作和主要结论如下:
(1)全面综述了铰接塔平台动力响应研究的进展和现状,确定了本文的研究方向和关键研究问题。
(2)建立了铰接塔塔柱刚性运动动力学模型,应用Morison公式计算水动力,依据拉格朗日原理建立了铰接塔平台强非线性运动方程。分别考虑波浪作用和波流联合作用,研究了铰接塔平台超谐共振、亚谐共振、组合共振、混沌运动。研究表明,波浪作用下平台系统将出现2、3、5…倍超谐共振, 2 vm +vn组合共振,波流联合作用下平台系统将出现2、3、4、5…倍超谐共振,v m + vn组合共振、2 vm + vn组合共振,在特定波浪载荷下平台系统会发生1/2亚谐共振,在高频率波浪激励下平台系统会出现混沌运动。
(3)考虑铰接塔塔柱弹性变形,应用Morison公式计算平台瞬时位置的水动力,建立了铰接塔平台摇摆运动的偏微分方程,采用伽辽金方法求解波流作用下非线性动力响应。研究表明,波浪激励下铰接塔二阶模态将发生2、3、4倍超谐共振,弹性铰接塔在波浪作用下振动是不对称的。使用有限元方法计算了不同环境载荷条件下铰接塔的动力学响应,验证了本文简化模型的合理性。
(4)提出用增量谐波平衡法,结合Floquet理论和增量弧长法研究铰接塔-油轮分段线性恢复刚度,平方阻尼系统的方法。研究了系统超谐、亚谐、谐波、组合谐波解,幅频特性和分岔、混沌特性。用最大Lyapunov 指数图验证了三种由倍周期分岔导致的混沌运动,用内插胞映射的方法验证了混沌运动和稳定周期解共存以及系统运动对初值极度敏感的非线性动力学现象。
(5)基于铰接塔-油轮系统运动响应,分析了该系统系泊力变化特性。研究表明,在相同状态参数条件下系泊力具有多种不同的响应,当系统受到扰动,系统运动的形式极易发生突变,系泊力响应在不同响应中跳跃,由此可以初步解释铰接塔-油轮系统系泊力的突变机理。
The articulated tower platform(ATP) is the a kind of compliant platform designed for water depths of 100 to 500 m, it is used as single point mooring or as loading terminals, control towers, seabed installation facilities. The ATP suffers from nonlinear ocean environment loads, the study of swing motion and motion stability of ATP is very important for safety of the platform. In this paper, the theory and method of nonlinear dynamics are adopted to study the nonlinear dynamic behavior and motion stability of the ATP under complicated environment loads. In addition, the motion behavior of single point mooring articulated tower and the impact property of mooring force are studied. The main work and the conclusions obtained are as following:
(1) The research development and current state of the nonlinear dynamics of ATP at abroad and home is summarized synthetically. The study direction and key problems of the thesis are presented.
(2) The dynamics model of rigid body of ATP is put forward considering non-elastic tower, the hydrodynamic force is evaluated for the ATP by using Morison formula, the highly nonlinear motion equation is established according to Lagrange principle. Sup-harmonic resonance, sub-harmonic resonance, combination-harmonic resonance and chaos motion of ATP were investigated using numerical method with considering wave loads and combined wave-current loads respectively. The results show that the sup-harmonic resonance of orders 2,3,5…and 2 v m +vn combination-harmonic resonance will occur under wave loads; sup-harmonic resonance of orders 2,3,4,5…, v m + vn and 2 v m +vn combination-harmonic resonance will occur under combined wave-current loads; sub-harmonic resonance of order 1/2 was found in certain case under wave loads; Chaotic motion was found under high frequency wave excitation.
(3) Considering elastic deformation of the tower, the hydrodynamic force is evaluated at the instantaneous position of ATP using Morison formula, the partial differential equation of swing motion is established. The nonlinear response of ATP is calculated by Galerkin’s method. The results show that the second mode of ATP arises sup-harmonic resonance of orders 2,3,4 , and the unsymmetry of the motion of ATP is found under wave excitation. The dynamical response of ATP and the moment variety of tower are analyzed under different environmental loads by the finite element method, and the simplified model is validated in this paper.
(4) The calculation methods using incremental harmonic balance method, Floquet theory and incremental arc length method are presented for nonlinear dynamical response of ATP-Tanker considering piecewise linear restoring force and square damping. Sup-harmonic, sub-harmonic, harmonic, combination harmonic response, amplitude frequency characteristic and the behaviors of bifurcation and chaos are studied. Three kinds of chaotic motion caused by period doubling bifurcation in this system are verified by the max Lyapunov exponent diagram. The coexistence of chaos and stable period motion, and the sensibility of system motion depending on the initial value are explained by the interpolated cell mapping method.
(5) Based on the response of ATP-Tanker system, the varying characteristic of mooring force is studied. It is found that there are many different mooring force responses for the same set of system parameters, the abrupt change occurs easily in the motion of the system when it is perturbed and the mooring force response jumps among the several different values. This can primarily explain the mechanism of the abrupt change of the mooring force for the ATP-Tanker system.
(1)全面综述了铰接塔平台动力响应研究的进展和现状,确定了本文的研究方向和关键研究问题。
(2)建立了铰接塔塔柱刚性运动动力学模型,应用Morison公式计算水动力,依据拉格朗日原理建立了铰接塔平台强非线性运动方程。分别考虑波浪作用和波流联合作用,研究了铰接塔平台超谐共振、亚谐共振、组合共振、混沌运动。研究表明,波浪作用下平台系统将出现2、3、5…倍超谐共振, 2 vm +vn组合共振,波流联合作用下平台系统将出现2、3、4、5…倍超谐共振,v m + vn组合共振、2 vm + vn组合共振,在特定波浪载荷下平台系统会发生1/2亚谐共振,在高频率波浪激励下平台系统会出现混沌运动。
(3)考虑铰接塔塔柱弹性变形,应用Morison公式计算平台瞬时位置的水动力,建立了铰接塔平台摇摆运动的偏微分方程,采用伽辽金方法求解波流作用下非线性动力响应。研究表明,波浪激励下铰接塔二阶模态将发生2、3、4倍超谐共振,弹性铰接塔在波浪作用下振动是不对称的。使用有限元方法计算了不同环境载荷条件下铰接塔的动力学响应,验证了本文简化模型的合理性。
(4)提出用增量谐波平衡法,结合Floquet理论和增量弧长法研究铰接塔-油轮分段线性恢复刚度,平方阻尼系统的方法。研究了系统超谐、亚谐、谐波、组合谐波解,幅频特性和分岔、混沌特性。用最大Lyapunov 指数图验证了三种由倍周期分岔导致的混沌运动,用内插胞映射的方法验证了混沌运动和稳定周期解共存以及系统运动对初值极度敏感的非线性动力学现象。
(5)基于铰接塔-油轮系统运动响应,分析了该系统系泊力变化特性。研究表明,在相同状态参数条件下系泊力具有多种不同的响应,当系统受到扰动,系统运动的形式极易发生突变,系泊力响应在不同响应中跳跃,由此可以初步解释铰接塔-油轮系统系泊力的突变机理。
The articulated tower platform(ATP) is the a kind of compliant platform designed for water depths of 100 to 500 m, it is used as single point mooring or as loading terminals, control towers, seabed installation facilities. The ATP suffers from nonlinear ocean environment loads, the study of swing motion and motion stability of ATP is very important for safety of the platform. In this paper, the theory and method of nonlinear dynamics are adopted to study the nonlinear dynamic behavior and motion stability of the ATP under complicated environment loads. In addition, the motion behavior of single point mooring articulated tower and the impact property of mooring force are studied. The main work and the conclusions obtained are as following:
(1) The research development and current state of the nonlinear dynamics of ATP at abroad and home is summarized synthetically. The study direction and key problems of the thesis are presented.
(2) The dynamics model of rigid body of ATP is put forward considering non-elastic tower, the hydrodynamic force is evaluated for the ATP by using Morison formula, the highly nonlinear motion equation is established according to Lagrange principle. Sup-harmonic resonance, sub-harmonic resonance, combination-harmonic resonance and chaos motion of ATP were investigated using numerical method with considering wave loads and combined wave-current loads respectively. The results show that the sup-harmonic resonance of orders 2,3,5…and 2 v m +vn combination-harmonic resonance will occur under wave loads; sup-harmonic resonance of orders 2,3,4,5…, v m + vn and 2 v m +vn combination-harmonic resonance will occur under combined wave-current loads; sub-harmonic resonance of order 1/2 was found in certain case under wave loads; Chaotic motion was found under high frequency wave excitation.
(3) Considering elastic deformation of the tower, the hydrodynamic force is evaluated at the instantaneous position of ATP using Morison formula, the partial differential equation of swing motion is established. The nonlinear response of ATP is calculated by Galerkin’s method. The results show that the second mode of ATP arises sup-harmonic resonance of orders 2,3,4 , and the unsymmetry of the motion of ATP is found under wave excitation. The dynamical response of ATP and the moment variety of tower are analyzed under different environmental loads by the finite element method, and the simplified model is validated in this paper.
(4) The calculation methods using incremental harmonic balance method, Floquet theory and incremental arc length method are presented for nonlinear dynamical response of ATP-Tanker considering piecewise linear restoring force and square damping. Sup-harmonic, sub-harmonic, harmonic, combination harmonic response, amplitude frequency characteristic and the behaviors of bifurcation and chaos are studied. Three kinds of chaotic motion caused by period doubling bifurcation in this system are verified by the max Lyapunov exponent diagram. The coexistence of chaos and stable period motion, and the sensibility of system motion depending on the initial value are explained by the interpolated cell mapping method.
(5) Based on the response of ATP-Tanker system, the varying characteristic of mooring force is studied. It is found that there are many different mooring force responses for the same set of system parameters, the abrupt change occurs easily in the motion of the system when it is perturbed and the mooring force response jumps among the several different values. This can primarily explain the mechanism of the abrupt change of the mooring force for the ATP-Tanker system.
引文
[1] 吴应湘,李华,曾晓辉等,深海采油平台发展现状和设计中的关键问题,中国造船工程学会学报,2002,43 卷增刊:80-87
[2] 安国亭,卢佩琼编著,海洋石油开发工艺与设备,天津大学出版社,2001
[3] 徐继祖,李维扬等编著,海洋工程结构动力分析,天津大学出版社,1992
[4] 虞维明,周岳银,陈秀妹编著,海洋平台的建造与维修,海洋出版社, 1992
[5] G.E.Burnd,G.C.D’Amorim, Buoyant tower for phase one development of Garoupa field. In 9th Annual Offshore Technology conference, 1977,177-184
[6] D.L.Hays, M.McSwiggan, R.Vilain, Operation of an articulated oil loading column at the Beryl field in the North Sea, In 11th Annual Offshore Technology Conference, 1979,1805-1815
[7] J.R.Smith, The application of a concrete articulating buoyant column to offshore drilling and production systems,In Proceedings Symposium on New Technology for Exploration and Exploitation of Oil and Gas Resources,1979,377-389
[8] J.R.Smith , R.S.Taylor,The development of articulated buoyant system column system as an aid to economic offshore production,In European Offshore Petroleum Conference and Exhibition, 1980,545-557
[9] H.G.Butt, J.Salewski,A large-scale test with the concrete articulated tower conat in the vicinity of the research platform ‘nordsee’,In International Conference on Marine Science and Ocean Engineering, 1980,31-47
[10] A.Naess,Loads and motions of an articulated loading platform with moored tanker. In 12th Annual Offshore Technology Conference,1980,409-417
[11] S.Fjeld,S.Flogel,Deepwater platforms problem areas,In European Offshore Petroleum Conference and Exhibition,1980,223-234
[12] Anonymous,BHP brings Challis on stream with world’s largest SALRAM. Ocean Industry, 1990,53-56
[13] J.R.Morison, M.P.O’Brien, J.W.Johnson, S.A.Schaaf,The force by surface waves on piles,Petroleum Transactions, AIME, 1950,189:149-154
[14] C.L.Kirk,R.K. Jain,Response of articulated tower to waves and current ,The 9th Annual Offshore Technology Conference, 1977,545-552
[15] S.Chakrabarti ,D.C.Cotter,Analysis of a tower-tanker system. In Proceedings of the 10th Annual Offshore Technology Conference, 1978,1301-1310
[16] O.A. Olsen, A.Braathen, A.E.Loken,Slow and high frequency motions and loadsof articulated single point mooring system for large tanker,Norwegian Marine Research, 1978,14-28
[17] S.K.Chakrabarti,D.C.Cotter,Motion analysis of articulated tower, Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 1979,105:281-292
[18] S.K.Chakrabarti,D.C.Cotter,Transverse motion of articulated tower,Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 1980,107:65-77
[19] C.H.Kim,P.A.Luh,Motions and loads of an articulated loading platform in waves. IEEE, Ocean Engineering, 1981,956-961
[20] R.K.Jain,C.L.Kirk,Dynamic response of a double articulated offshore loading structure to non-collinear waves and current,Journal of Energy Resources Technology, 1981,103:41-47
[21] P.K. Muhuri,A.S.Gupta,Stochastic stability of tethered buoyant platforms,Ocean Engineering, 1983,10(6): 174-479
[22] J.M.T. Thompson, A.R. Bokaian,R.Ghaffai,Stochastic and chaotic motions of compliant offshore structures and articulated mooring towers, Journal of Energy Resources Technology, 1984,106:191-198
[23] T.E,Schellin,T.Koch,Calculated response of an articulated tower in eaves comparison with mode tests, In Proceedings of the For the Offshore Mechanics and Arctic Engineering Symposium, ASME, 1985, 1
[24] J.Chantrel,P.Maro,Subharmonic response of articulated loading platform,In Proceedings of the 6th Conference on Offshore Mechanics and Arctic Engineering, 1987, 35-45
[25] K.Jain,T.K.Datta,Stochastic response of articulated towers,In Deep Offshore Technology 4th International Conference and Exhibit, 1987,191-208
[26] T.K.Datta,A.K.Jain,Response of articulated tower platforms to random wind and wave forces,Computers and Structures, 1990,34(1): 137-144
[27] C.Y. Liaw. Bifurcations of subharmonics and chaotic motions of articulated towers. Engineering Strtuctures,1988,117-124
[28] S.Y.Hanna, D.I.Karsen, J.Y.Yeung,Dynamic response of a compliant tower with multiple articulations,In Seventh International Conference on Offshore Mechanics ana Arctic Engineering, 1988, 257-269
[29] I.H. Hevacioglu,A.Incecik,Dynamic analysis of coupled articulated tower and floating production system,In The 7th International Conference on Offshore Mechanics and Arctic Engineering, 1988, 279-287
[30] C.W.Liaw,N.J.Shankar, K.S.Chua, Large motion analysi of compliant structures using Euler parameters, Ocean Engineering, 1989,16(6): 545-557
[31] J.W. Leonard,R.A.Young,Coupled response of compliant offshore platforms.Engineering Structures, 1985,7:74-84
[32] L.N.Virgin, S.R.Bishop,Catchment regions of multiple dynamic responses in nonlinear problems of offshore mechanics,Journal of Offshore Mechanics and Arctic Engineering, 1990,127-133
[33] A.K.Jain,T.K.Datta,Nonlinear behavior of articulated tower platforms to random sea,Journal of Engineering for Industry, 1991,113:238-240
[34] H.S.Choi,J.Y.K.Lou,Nonlinear behavior of an articulated offshore loading platform, Applied Ocean Research, 1991,12(2):63-74
[35] O.Gottlieb, C.S. Yim, R.T.Hudspeth,Analysis of nonlinear response of an articulated tower. International Journal of Offshore and Polar Engineering, 1992,2(1): 61-66
[36] C.Y.Liaw, N.J.Shankar,K.S.Chua,Subharmonic motions and wave force structure interaction,Marine Structures, 1992,5:281-295
[37] L.E. Borgman,Ocean wave simulation for engineering design,Journal of the Waterways and Harbors Division, ASCE,1969, 95:557-583
[38] Z.Ran, M.H.Kim,Response of articulated loading platform in irregular waves, Journal of water, port, coastal, and ocean engineering, 1995,121(6):286-293
[39] R.Adrezin,P.Bar-Avi,H.Benaroya,Dynamic response of compliant offshore structures review, Journal of Aerospace engineering, 1996,9(4):114-131
[40] P.Bar-Avi,H.Benaroya,Response of a two degrees of freedom articulated tower to different environmental conditions, 1996,31(5):717-741
[41] P.Bar-Avi,H.Benaroya,Non-linear dynamics of an articulated tower in the ocean, Journal of Sound and Vibration,1996, 190(1):77-103
[42] P.Bar-Avi,H.Benaroya,Stochastic response a two DOF articulated tower, Journal of Sound and Vibration, 1997,32(4):639-655
[43] Suhail Ahmad, N. Islam,Wind-induced response of double articulated towers, Journal of Wind Engineering and Industrial Serodynamics, 1997,67-68:937
[44] A.Rgahothama, S.Narayanan,Bifurcation and chaos of an articulated loading platform with piecewise non-linear stiffness using the incremental harmonic balance method,Ocean Engineering, 2000,27:1087-1107
[45] Dadi.GSRR, Ahmad.S, Majage.ASA,Behaviour of an articulated tower due to wave diffraction,Chennai, International Conference in Ocean Engineering – 2001, India: Indian Institute of Technology Madras,Ocean Engineering,2002.
[46] Ahmed S,Islam N,Response of multi hinged articulated tower in waves,Chennai, International Conference in Ocean Engineering – 2001, India: Indian Institute of Technology Madras,Ocean Engineering,2002
[47] Islam,Ahmad,Random wave response of double pendulum articulated offshoretower,Proceedings of the 22nd International Conference on Offshore Mechanics and Arctic Engineering, Mexico: American Society of Mechanical Engineers, 2003, 489-498.
[48] Shashikala A.P,Route to chaos of an articulated offshore loading platform. Proceedings of the 23rd International Conference on Offshore Mechanics and Arctic Engineering, Canada: American Society of Mechanical Engineers, 2004, 309-319.
[49] K.F.Haverty,J.F.McNamara,B.Moran,Finite dynamics motions of articulated offshore loading towers, In Proceedings of the Internation Conference of Marine Research Ship Technology and Ocean Engineering,1982
[50] J.F. McNamara,M.Lane,Practicle modeling for articulated risers and loading columns,Journal of Energy Resources Technology, 1984, 444-450
[51] G. Sebastiani,R.Brandi,F.D.Lena,A.Nista,Highly compliant column for tanker mooring and oil production in 1000m wate depth, In The 16th Annual Offshore Technology Conference, 1984,379-388
[52] C.Ganapathy, B.P. Smantaray,K.Nagamani,Truss type articulated towers for deep water applications,In The 9th International Conference on Offshore Mechanics and Artic Engineering, 1990,477-484
[53] K.M.Mathisen,P.G. Bergan,Large displacement analysis of submerged multibody systems,Engineering Computations, 1992,9:609-634
[54] L.L Seller,J.M.Niedzwecki,Response characteristics of multi-articulated offshore towers,Ocean Engineeing, 1992,19(1): 1-20
[55] Nagamani.K, Baskar.K, Dynamic response of two-hinged articulated tower,Int. Conf. in Ocean Engineering COE '96, Madras (India): ALLIED, CHENNAI (INDIA) ,1996,17-20.
[56] K.Nagamanj, C,Ganapath,Finite element analysis of nonlinear dynamic response of articulated tower, Computers & Structures, 1996,59(2):213-223
[57] K.Nagamani,C.Ganapathy,The dynamic response of a three-leg articulated tower. Ocean Engineering ,2000,27:1455-1471
[58] S. M. Han,H. Benaroya, Non-linear coupled transverse and axial vibration of a compliant structure, part 1: Formulation and free vibration, Journal of Sound and Vibration, 2000,237(5):837-873
[59] S. M. Han,H. Benaroya,Non-linear coupled transverse and axial non-linear coupled transverse and axial vibration of a compliant structure, part 2: Forced vibration, Journal of Sound and Vibration, 2000,237(5):875-900
[60] Seon Mi Han ,Haym Benaroya ,Comparison of linear and nonlinear response of a compliant tower to random wave forces,Chaos,Solitons and Fractals 2002,14:269-291
[61] Seon Mi Han ,Haym Benaroya, A parametric study of a flexible ocean tower, Chaos, Solitons and Fractals,2002,14:183-201
[62] S.N.Kuchnicki,H.Benaroya,Coupled transverse and axial motion of a compliant structure in response to vortex-shedding loads, Journal of Sound and Vibration 2002,257(5): 903-929
[63] S. M. Han, H. Benaroya, Vibration of a compliant tower in three-dimensions, Journal of Sound and Vibration ,2002, 250(4):675-709
[64] K.Yoshida, H.Suzuki,N.Ona,Control of dynamic response of tower-like offshore structures in waves,In 7th International Conference on Offshore Mechanics and Arctic Engineering, ASME,1988,1:249-255
[65] K.Yoshida, H.Suzuki,S.Mishima,Response control of articulated compliant tower,In Eighth International Conference on Offshore Mechanics and Arctic Engineering, 1989,199-207
[66] B.P.Suneja, T.K.Datta, Open-Close loop active control of articulated leg platform, Journal of engineering mechanics,1998, 124(7):734-740
[67] B.P.Suneja, T.K.Datta, Active control of ALP with improved performance function, Ocean Engineering, 1998,25(10):817-835
[68] Suneja B.P. , Datta T.K,Performance of linear and nonlinear control laws for the response reduction of ALP, International Shipbuilding Progress, 2000, 47(451):325-347
[69] N.Haritos, Modelling the Inertial Force Dominant Response of Compliant Offshore Structures in 3-Dimensional Wave Environments,Environment International. 1995,21(5): 637-645
[70] P.Bar-Avi,H.Benaroya, Nonlinear Dynamics of an articulated tower in the ocean. Journal of Sound and vibration ,1996,190(1): 77-103
[71] J.F.威尔逊编著,海洋结构动力学,扬国金等译,石油工业出版社,1991
[72] 竺艳蓉编著,海洋工程波浪力学,天津大学出版社,1991
[73] J.盖斯维特著,海洋环境与建筑物设计,吴宜倜译,海洋出版社,1992
[74] M. Issacson, Wave and current forces on a fixed offshore structure, Canadian Journal of Civil Engineering ,1989,15.937-947
[75] 李玉成编著,波浪对海上建筑物的作用,海洋出版社,1990
[76] Sparpkaya T,Vortex-induced oscillation : a selective review. Journal of Applied Mechanics,1979,46(2):241-248.
[77] Iwan W D,The vortex-induced oscillation of non-uniform structural system, Journal of Sound and Vibration, 1981,79(2):291-301
[78] Lyons G J, Petal M H. A prediction technique for vortex induced transverse response of marine riser and tethers,Journal of Sound and Vibration, 1986,111(3):467-487
[79] Bokaian A,Lock-in prediction of marine riser and tethers,Journal of sound Vibration, 1994,175(5):607-623
[80] 丁学成,结构动力学,北京:高等教育出版社,1992
[81] 铁摩辛柯,D.H.杨,工程中的振动问题, 北京:人民铁道出版社,1978
[82] 唐友刚,高等结构动力分析,天津:天津大学出版社,2002
[83] 曹树谦,张文德,振动结构模态分析,天津:天津大学出版社,2001
[84] Dong Yanqiu, Xie Geng, Lou J Y K. Stability of vortex-induced oscillations of tension leg platform tethers,Ocean Engineering, 1992,19(6):551-571
[85] 马驰,董艳秋,海洋平台张力腿在两种边界条件下的涡激非线性振动的比较研究,船舶力学,2000,4(1):56-65
[86] 马驰,董艳秋,胡志敏,TLP 张力腿涡激非线性响应,天津大学学报,2000,33(6):401-706
[87] 董艳秋,波流联合作用下海洋平台张力腿的涡激非线性振动,海洋学报, 1994,13(3):121-129
[88] 董艳秋,深水海洋平台张力腿涡激非线性振动响应研究,中国造船,1992,3:80-92
[89] 董艳秋,张力腿平台涡激非线性振动稳定性研究,中国造船,1992,4:85-97
[90] TANG You-gang, Gu Jia-yang, ZUO Jian-li, MIN Jian-qin. Nonlinear dynamics response of casing pipe under combined wave-current[J],Applied Mathematics and Mechanics, 2005, 26( 8): 1040-1046.
[91] 董艳秋,深海采油平台波浪载荷及响应,天津:天津大学出版社,2005
[92] ANSYS Element Reference, 10 th Edition, SAS IP, Inc.
[93] 王勖成,邵敏,有限单元法基本原理和树值分析方法,北京:清华大学出版社,1997
[94] ANSYS Theory Reference, 9 th Edition, SAS IP, Inc.
[95] Friedmann P., Hammond C.E., Woo T.H., Efficient numerical treatment of periodic systems with application to stability problems, International Journal of Numerical Methods in Engineering ,1977,11:1117–1136
[96] S. Maezawa, H. Kumano, Y. Minakuchi, Forced vibrations in an unsymmetric piecewise-linear system excited by general periodic force functions, Bulletin of the JSME ,1980,23:68–75
[97] S.L. Lau, Y.K. Cheung, Amplitude incremental variational principle for nonlinear vibration of elastic system,American Society of Mechanical Engineers,Journal of Applied Mechanics ,1981,48:959–964
[98] Cheung Y.K., Lau S.L., Incremental time–space finite strip method for nonlinear structural vibrations, Earthquake Engineering and Structural Dynamics, 1982, 10:239–253
[99] Lau S.L., Cheung Y.K., Wu S.Y., A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems, Journal of Applied Mechanics ,1982, 49:849–853
[100] Lau S.L., Cheung Y.K., Wu S.Y., Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems, Journal of Applied Mechanics ,1983,50:871–876
[101] C. Pierre, A.A. Ferri, E.H. Dowell, Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method, Journal of Applied Mechanics , 1985, 52:958–964
[102] Y.S. Choi, S.T. Noah, Forced periodic vibration of unsymmetric piecewise -linear systems, Journal of Sound and Vibration,1988,121:117–126
[103] Wong C.W., Zhang W.S., Lau S.L., Periodic forced vibration of unsymmetrical piecewise linear systems by incremental harmonic balance method, Journal of Sound and Vibration,1991, 149(1):91–105
[104] Choi H.S., Lou J.Y.K., Nonlinear behavior and chaotic motions of an sdf with piecewise nonlinear stiffness, International Journal of Nonlinear Mechanics ,1991,26 (5):461–473
[105] Lau S.L., Zhang S., Nonlinear vibrations of piecewise linear systems by incremental harmonic balance method, Journal of Applied Mechanics,1992, 59:153–163
[106] S.L. Lau, S.W. Yuen, Solution diagram of nonlinear dynamic systems by the IHB method, Journal of Sound and Vibration, 1993,167:303–316
[107] Leung A.Y.T., Chui S.K., Nonlinear vibration of coupled Duffing’s oscillator by an improved incremental harmonic balance method, Journal of Sound and Vibration ,1995,181 (4):619–633
[108] S. Narayanan, P. Sekar, A frequency domain based numeric-analytical method for non-linear dynamical systems, Journal of Sound and Vibration ,1998, 211:409–424
[109] A. Cardona, A. Lerusse, M. Gerardin, Fast Fourier nonlinear vibration analysis, Computational Mechanics ,1998,22:128–142
[110] A. Raghothama, S. Narayanan, Bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method, Journal of Sound and Vibration ,1999,226: 469–492
[111] H. Wolf, M. Stegilc, The influence of neglecting small harmonic terms on estimation of dynamical stability of the response of non-linear oscillators, Computational Mechanics , 1999,24:230–237.
[112] A. Raghothama, S. Narayanan, Bifurcation and chaos in escape equation model by incremental harmonic balancing, Chaos Solitons & Fractals , 2000,11:1349–1363
[113] Q. Cao, L. Xu, K. Djidjeli, Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise linearity, Chaos Solitons & Fractals, 2001, 12:1917–1927
[114] K.Y. Szea, S.H. Chen, J.L. Huang, The incremental harmonic balance method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration ,2005,281:611–626
[115] 陆启韶,常微分方程定性方法和分岔,北京:北京航空航天大学出版社,1989
[116] A.Y.T. Leung, Nonlinear natural vibration analysis of beams by selective coefficient increment, Computational Mechanics, 1989,5:73-80
[117] Yushu Chen, Andrew Y. T. Leung, Bifurcation and Chaos in Engineering, London : Springer, 1998
[118] J.L. Hudson, H.C. Killory, Chaos in a four-variable piecewise-linear system of differential equations, IEEE Transactions on Circuits and Systems ,1998,35:902–908
[119] 张伟,杨绍普等,非线性系统的周期振动和分岔(非线性力学丛书),北京:科学出版社,2002
[120] 闻邦春,李以农,韩清凯,非线性振动理论中的解析方法及工程应用,沈阳:东北大学出版社,2001
[121] Hubertus F., Von Bremen, Firdaus E.Udwadia, Wlodek Proskurowski, An efficient QR based method for the computation Lyuapunov exponents, Physics D,1997,101:1-16
[122] Gyorgy Bbarna, Ichiro Tsuds, A new method for computing Lyapunov exponents, Physics Letters A, 1993,175:421-427
[123] Wolf A., Swift J.B., Swinney H.L., Vastano J.A., Determining Lyapunov exponents from a time series, Physica D ,1988,16:285–317
[124] Tongue B.H., Gu K., Interpolated cell mapping of dynamical systems, Journal of Applied Mechanics ,1988,55:461–466.
[2] 安国亭,卢佩琼编著,海洋石油开发工艺与设备,天津大学出版社,2001
[3] 徐继祖,李维扬等编著,海洋工程结构动力分析,天津大学出版社,1992
[4] 虞维明,周岳银,陈秀妹编著,海洋平台的建造与维修,海洋出版社, 1992
[5] G.E.Burnd,G.C.D’Amorim, Buoyant tower for phase one development of Garoupa field. In 9th Annual Offshore Technology conference, 1977,177-184
[6] D.L.Hays, M.McSwiggan, R.Vilain, Operation of an articulated oil loading column at the Beryl field in the North Sea, In 11th Annual Offshore Technology Conference, 1979,1805-1815
[7] J.R.Smith, The application of a concrete articulating buoyant column to offshore drilling and production systems,In Proceedings Symposium on New Technology for Exploration and Exploitation of Oil and Gas Resources,1979,377-389
[8] J.R.Smith , R.S.Taylor,The development of articulated buoyant system column system as an aid to economic offshore production,In European Offshore Petroleum Conference and Exhibition, 1980,545-557
[9] H.G.Butt, J.Salewski,A large-scale test with the concrete articulated tower conat in the vicinity of the research platform ‘nordsee’,In International Conference on Marine Science and Ocean Engineering, 1980,31-47
[10] A.Naess,Loads and motions of an articulated loading platform with moored tanker. In 12th Annual Offshore Technology Conference,1980,409-417
[11] S.Fjeld,S.Flogel,Deepwater platforms problem areas,In European Offshore Petroleum Conference and Exhibition,1980,223-234
[12] Anonymous,BHP brings Challis on stream with world’s largest SALRAM. Ocean Industry, 1990,53-56
[13] J.R.Morison, M.P.O’Brien, J.W.Johnson, S.A.Schaaf,The force by surface waves on piles,Petroleum Transactions, AIME, 1950,189:149-154
[14] C.L.Kirk,R.K. Jain,Response of articulated tower to waves and current ,The 9th Annual Offshore Technology Conference, 1977,545-552
[15] S.Chakrabarti ,D.C.Cotter,Analysis of a tower-tanker system. In Proceedings of the 10th Annual Offshore Technology Conference, 1978,1301-1310
[16] O.A. Olsen, A.Braathen, A.E.Loken,Slow and high frequency motions and loadsof articulated single point mooring system for large tanker,Norwegian Marine Research, 1978,14-28
[17] S.K.Chakrabarti,D.C.Cotter,Motion analysis of articulated tower, Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 1979,105:281-292
[18] S.K.Chakrabarti,D.C.Cotter,Transverse motion of articulated tower,Journal of the Waterway, Port, Coastal and Ocean Division, ASCE, 1980,107:65-77
[19] C.H.Kim,P.A.Luh,Motions and loads of an articulated loading platform in waves. IEEE, Ocean Engineering, 1981,956-961
[20] R.K.Jain,C.L.Kirk,Dynamic response of a double articulated offshore loading structure to non-collinear waves and current,Journal of Energy Resources Technology, 1981,103:41-47
[21] P.K. Muhuri,A.S.Gupta,Stochastic stability of tethered buoyant platforms,Ocean Engineering, 1983,10(6): 174-479
[22] J.M.T. Thompson, A.R. Bokaian,R.Ghaffai,Stochastic and chaotic motions of compliant offshore structures and articulated mooring towers, Journal of Energy Resources Technology, 1984,106:191-198
[23] T.E,Schellin,T.Koch,Calculated response of an articulated tower in eaves comparison with mode tests, In Proceedings of the For the Offshore Mechanics and Arctic Engineering Symposium, ASME, 1985, 1
[24] J.Chantrel,P.Maro,Subharmonic response of articulated loading platform,In Proceedings of the 6th Conference on Offshore Mechanics and Arctic Engineering, 1987, 35-45
[25] K.Jain,T.K.Datta,Stochastic response of articulated towers,In Deep Offshore Technology 4th International Conference and Exhibit, 1987,191-208
[26] T.K.Datta,A.K.Jain,Response of articulated tower platforms to random wind and wave forces,Computers and Structures, 1990,34(1): 137-144
[27] C.Y. Liaw. Bifurcations of subharmonics and chaotic motions of articulated towers. Engineering Strtuctures,1988,117-124
[28] S.Y.Hanna, D.I.Karsen, J.Y.Yeung,Dynamic response of a compliant tower with multiple articulations,In Seventh International Conference on Offshore Mechanics ana Arctic Engineering, 1988, 257-269
[29] I.H. Hevacioglu,A.Incecik,Dynamic analysis of coupled articulated tower and floating production system,In The 7th International Conference on Offshore Mechanics and Arctic Engineering, 1988, 279-287
[30] C.W.Liaw,N.J.Shankar, K.S.Chua, Large motion analysi of compliant structures using Euler parameters, Ocean Engineering, 1989,16(6): 545-557
[31] J.W. Leonard,R.A.Young,Coupled response of compliant offshore platforms.Engineering Structures, 1985,7:74-84
[32] L.N.Virgin, S.R.Bishop,Catchment regions of multiple dynamic responses in nonlinear problems of offshore mechanics,Journal of Offshore Mechanics and Arctic Engineering, 1990,127-133
[33] A.K.Jain,T.K.Datta,Nonlinear behavior of articulated tower platforms to random sea,Journal of Engineering for Industry, 1991,113:238-240
[34] H.S.Choi,J.Y.K.Lou,Nonlinear behavior of an articulated offshore loading platform, Applied Ocean Research, 1991,12(2):63-74
[35] O.Gottlieb, C.S. Yim, R.T.Hudspeth,Analysis of nonlinear response of an articulated tower. International Journal of Offshore and Polar Engineering, 1992,2(1): 61-66
[36] C.Y.Liaw, N.J.Shankar,K.S.Chua,Subharmonic motions and wave force structure interaction,Marine Structures, 1992,5:281-295
[37] L.E. Borgman,Ocean wave simulation for engineering design,Journal of the Waterways and Harbors Division, ASCE,1969, 95:557-583
[38] Z.Ran, M.H.Kim,Response of articulated loading platform in irregular waves, Journal of water, port, coastal, and ocean engineering, 1995,121(6):286-293
[39] R.Adrezin,P.Bar-Avi,H.Benaroya,Dynamic response of compliant offshore structures review, Journal of Aerospace engineering, 1996,9(4):114-131
[40] P.Bar-Avi,H.Benaroya,Response of a two degrees of freedom articulated tower to different environmental conditions, 1996,31(5):717-741
[41] P.Bar-Avi,H.Benaroya,Non-linear dynamics of an articulated tower in the ocean, Journal of Sound and Vibration,1996, 190(1):77-103
[42] P.Bar-Avi,H.Benaroya,Stochastic response a two DOF articulated tower, Journal of Sound and Vibration, 1997,32(4):639-655
[43] Suhail Ahmad, N. Islam,Wind-induced response of double articulated towers, Journal of Wind Engineering and Industrial Serodynamics, 1997,67-68:937
[44] A.Rgahothama, S.Narayanan,Bifurcation and chaos of an articulated loading platform with piecewise non-linear stiffness using the incremental harmonic balance method,Ocean Engineering, 2000,27:1087-1107
[45] Dadi.GSRR, Ahmad.S, Majage.ASA,Behaviour of an articulated tower due to wave diffraction,Chennai, International Conference in Ocean Engineering – 2001, India: Indian Institute of Technology Madras,Ocean Engineering,2002.
[46] Ahmed S,Islam N,Response of multi hinged articulated tower in waves,Chennai, International Conference in Ocean Engineering – 2001, India: Indian Institute of Technology Madras,Ocean Engineering,2002
[47] Islam,Ahmad,Random wave response of double pendulum articulated offshoretower,Proceedings of the 22nd International Conference on Offshore Mechanics and Arctic Engineering, Mexico: American Society of Mechanical Engineers, 2003, 489-498.
[48] Shashikala A.P,Route to chaos of an articulated offshore loading platform. Proceedings of the 23rd International Conference on Offshore Mechanics and Arctic Engineering, Canada: American Society of Mechanical Engineers, 2004, 309-319.
[49] K.F.Haverty,J.F.McNamara,B.Moran,Finite dynamics motions of articulated offshore loading towers, In Proceedings of the Internation Conference of Marine Research Ship Technology and Ocean Engineering,1982
[50] J.F. McNamara,M.Lane,Practicle modeling for articulated risers and loading columns,Journal of Energy Resources Technology, 1984, 444-450
[51] G. Sebastiani,R.Brandi,F.D.Lena,A.Nista,Highly compliant column for tanker mooring and oil production in 1000m wate depth, In The 16th Annual Offshore Technology Conference, 1984,379-388
[52] C.Ganapathy, B.P. Smantaray,K.Nagamani,Truss type articulated towers for deep water applications,In The 9th International Conference on Offshore Mechanics and Artic Engineering, 1990,477-484
[53] K.M.Mathisen,P.G. Bergan,Large displacement analysis of submerged multibody systems,Engineering Computations, 1992,9:609-634
[54] L.L Seller,J.M.Niedzwecki,Response characteristics of multi-articulated offshore towers,Ocean Engineeing, 1992,19(1): 1-20
[55] Nagamani.K, Baskar.K, Dynamic response of two-hinged articulated tower,Int. Conf. in Ocean Engineering COE '96, Madras (India): ALLIED, CHENNAI (INDIA) ,1996,17-20.
[56] K.Nagamanj, C,Ganapath,Finite element analysis of nonlinear dynamic response of articulated tower, Computers & Structures, 1996,59(2):213-223
[57] K.Nagamani,C.Ganapathy,The dynamic response of a three-leg articulated tower. Ocean Engineering ,2000,27:1455-1471
[58] S. M. Han,H. Benaroya, Non-linear coupled transverse and axial vibration of a compliant structure, part 1: Formulation and free vibration, Journal of Sound and Vibration, 2000,237(5):837-873
[59] S. M. Han,H. Benaroya,Non-linear coupled transverse and axial non-linear coupled transverse and axial vibration of a compliant structure, part 2: Forced vibration, Journal of Sound and Vibration, 2000,237(5):875-900
[60] Seon Mi Han ,Haym Benaroya ,Comparison of linear and nonlinear response of a compliant tower to random wave forces,Chaos,Solitons and Fractals 2002,14:269-291
[61] Seon Mi Han ,Haym Benaroya, A parametric study of a flexible ocean tower, Chaos, Solitons and Fractals,2002,14:183-201
[62] S.N.Kuchnicki,H.Benaroya,Coupled transverse and axial motion of a compliant structure in response to vortex-shedding loads, Journal of Sound and Vibration 2002,257(5): 903-929
[63] S. M. Han, H. Benaroya, Vibration of a compliant tower in three-dimensions, Journal of Sound and Vibration ,2002, 250(4):675-709
[64] K.Yoshida, H.Suzuki,N.Ona,Control of dynamic response of tower-like offshore structures in waves,In 7th International Conference on Offshore Mechanics and Arctic Engineering, ASME,1988,1:249-255
[65] K.Yoshida, H.Suzuki,S.Mishima,Response control of articulated compliant tower,In Eighth International Conference on Offshore Mechanics and Arctic Engineering, 1989,199-207
[66] B.P.Suneja, T.K.Datta, Open-Close loop active control of articulated leg platform, Journal of engineering mechanics,1998, 124(7):734-740
[67] B.P.Suneja, T.K.Datta, Active control of ALP with improved performance function, Ocean Engineering, 1998,25(10):817-835
[68] Suneja B.P. , Datta T.K,Performance of linear and nonlinear control laws for the response reduction of ALP, International Shipbuilding Progress, 2000, 47(451):325-347
[69] N.Haritos, Modelling the Inertial Force Dominant Response of Compliant Offshore Structures in 3-Dimensional Wave Environments,Environment International. 1995,21(5): 637-645
[70] P.Bar-Avi,H.Benaroya, Nonlinear Dynamics of an articulated tower in the ocean. Journal of Sound and vibration ,1996,190(1): 77-103
[71] J.F.威尔逊编著,海洋结构动力学,扬国金等译,石油工业出版社,1991
[72] 竺艳蓉编著,海洋工程波浪力学,天津大学出版社,1991
[73] J.盖斯维特著,海洋环境与建筑物设计,吴宜倜译,海洋出版社,1992
[74] M. Issacson, Wave and current forces on a fixed offshore structure, Canadian Journal of Civil Engineering ,1989,15.937-947
[75] 李玉成编著,波浪对海上建筑物的作用,海洋出版社,1990
[76] Sparpkaya T,Vortex-induced oscillation : a selective review. Journal of Applied Mechanics,1979,46(2):241-248.
[77] Iwan W D,The vortex-induced oscillation of non-uniform structural system, Journal of Sound and Vibration, 1981,79(2):291-301
[78] Lyons G J, Petal M H. A prediction technique for vortex induced transverse response of marine riser and tethers,Journal of Sound and Vibration, 1986,111(3):467-487
[79] Bokaian A,Lock-in prediction of marine riser and tethers,Journal of sound Vibration, 1994,175(5):607-623
[80] 丁学成,结构动力学,北京:高等教育出版社,1992
[81] 铁摩辛柯,D.H.杨,工程中的振动问题, 北京:人民铁道出版社,1978
[82] 唐友刚,高等结构动力分析,天津:天津大学出版社,2002
[83] 曹树谦,张文德,振动结构模态分析,天津:天津大学出版社,2001
[84] Dong Yanqiu, Xie Geng, Lou J Y K. Stability of vortex-induced oscillations of tension leg platform tethers,Ocean Engineering, 1992,19(6):551-571
[85] 马驰,董艳秋,海洋平台张力腿在两种边界条件下的涡激非线性振动的比较研究,船舶力学,2000,4(1):56-65
[86] 马驰,董艳秋,胡志敏,TLP 张力腿涡激非线性响应,天津大学学报,2000,33(6):401-706
[87] 董艳秋,波流联合作用下海洋平台张力腿的涡激非线性振动,海洋学报, 1994,13(3):121-129
[88] 董艳秋,深水海洋平台张力腿涡激非线性振动响应研究,中国造船,1992,3:80-92
[89] 董艳秋,张力腿平台涡激非线性振动稳定性研究,中国造船,1992,4:85-97
[90] TANG You-gang, Gu Jia-yang, ZUO Jian-li, MIN Jian-qin. Nonlinear dynamics response of casing pipe under combined wave-current[J],Applied Mathematics and Mechanics, 2005, 26( 8): 1040-1046.
[91] 董艳秋,深海采油平台波浪载荷及响应,天津:天津大学出版社,2005
[92] ANSYS Element Reference, 10 th Edition, SAS IP, Inc.
[93] 王勖成,邵敏,有限单元法基本原理和树值分析方法,北京:清华大学出版社,1997
[94] ANSYS Theory Reference, 9 th Edition, SAS IP, Inc.
[95] Friedmann P., Hammond C.E., Woo T.H., Efficient numerical treatment of periodic systems with application to stability problems, International Journal of Numerical Methods in Engineering ,1977,11:1117–1136
[96] S. Maezawa, H. Kumano, Y. Minakuchi, Forced vibrations in an unsymmetric piecewise-linear system excited by general periodic force functions, Bulletin of the JSME ,1980,23:68–75
[97] S.L. Lau, Y.K. Cheung, Amplitude incremental variational principle for nonlinear vibration of elastic system,American Society of Mechanical Engineers,Journal of Applied Mechanics ,1981,48:959–964
[98] Cheung Y.K., Lau S.L., Incremental time–space finite strip method for nonlinear structural vibrations, Earthquake Engineering and Structural Dynamics, 1982, 10:239–253
[99] Lau S.L., Cheung Y.K., Wu S.Y., A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems, Journal of Applied Mechanics ,1982, 49:849–853
[100] Lau S.L., Cheung Y.K., Wu S.Y., Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems, Journal of Applied Mechanics ,1983,50:871–876
[101] C. Pierre, A.A. Ferri, E.H. Dowell, Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method, Journal of Applied Mechanics , 1985, 52:958–964
[102] Y.S. Choi, S.T. Noah, Forced periodic vibration of unsymmetric piecewise -linear systems, Journal of Sound and Vibration,1988,121:117–126
[103] Wong C.W., Zhang W.S., Lau S.L., Periodic forced vibration of unsymmetrical piecewise linear systems by incremental harmonic balance method, Journal of Sound and Vibration,1991, 149(1):91–105
[104] Choi H.S., Lou J.Y.K., Nonlinear behavior and chaotic motions of an sdf with piecewise nonlinear stiffness, International Journal of Nonlinear Mechanics ,1991,26 (5):461–473
[105] Lau S.L., Zhang S., Nonlinear vibrations of piecewise linear systems by incremental harmonic balance method, Journal of Applied Mechanics,1992, 59:153–163
[106] S.L. Lau, S.W. Yuen, Solution diagram of nonlinear dynamic systems by the IHB method, Journal of Sound and Vibration, 1993,167:303–316
[107] Leung A.Y.T., Chui S.K., Nonlinear vibration of coupled Duffing’s oscillator by an improved incremental harmonic balance method, Journal of Sound and Vibration ,1995,181 (4):619–633
[108] S. Narayanan, P. Sekar, A frequency domain based numeric-analytical method for non-linear dynamical systems, Journal of Sound and Vibration ,1998, 211:409–424
[109] A. Cardona, A. Lerusse, M. Gerardin, Fast Fourier nonlinear vibration analysis, Computational Mechanics ,1998,22:128–142
[110] A. Raghothama, S. Narayanan, Bifurcation and chaos in geared rotor bearing system by incremental harmonic balance method, Journal of Sound and Vibration ,1999,226: 469–492
[111] H. Wolf, M. Stegilc, The influence of neglecting small harmonic terms on estimation of dynamical stability of the response of non-linear oscillators, Computational Mechanics , 1999,24:230–237.
[112] A. Raghothama, S. Narayanan, Bifurcation and chaos in escape equation model by incremental harmonic balancing, Chaos Solitons & Fractals , 2000,11:1349–1363
[113] Q. Cao, L. Xu, K. Djidjeli, Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise linearity, Chaos Solitons & Fractals, 2001, 12:1917–1927
[114] K.Y. Szea, S.H. Chen, J.L. Huang, The incremental harmonic balance method for nonlinear vibration of axially moving beams. Journal of Sound and Vibration ,2005,281:611–626
[115] 陆启韶,常微分方程定性方法和分岔,北京:北京航空航天大学出版社,1989
[116] A.Y.T. Leung, Nonlinear natural vibration analysis of beams by selective coefficient increment, Computational Mechanics, 1989,5:73-80
[117] Yushu Chen, Andrew Y. T. Leung, Bifurcation and Chaos in Engineering, London : Springer, 1998
[118] J.L. Hudson, H.C. Killory, Chaos in a four-variable piecewise-linear system of differential equations, IEEE Transactions on Circuits and Systems ,1998,35:902–908
[119] 张伟,杨绍普等,非线性系统的周期振动和分岔(非线性力学丛书),北京:科学出版社,2002
[120] 闻邦春,李以农,韩清凯,非线性振动理论中的解析方法及工程应用,沈阳:东北大学出版社,2001
[121] Hubertus F., Von Bremen, Firdaus E.Udwadia, Wlodek Proskurowski, An efficient QR based method for the computation Lyuapunov exponents, Physics D,1997,101:1-16
[122] Gyorgy Bbarna, Ichiro Tsuds, A new method for computing Lyapunov exponents, Physics Letters A, 1993,175:421-427
[123] Wolf A., Swift J.B., Swinney H.L., Vastano J.A., Determining Lyapunov exponents from a time series, Physica D ,1988,16:285–317
[124] Tongue B.H., Gu K., Interpolated cell mapping of dynamical systems, Journal of Applied Mechanics ,1988,55:461–466.