摘要
大型旋转机械是现代工业社会中的关键设备,而众多非线性激励因素又是
大型旋转机械中的关键因素。近二十余年来,非线性转子动力学一直是国际学
术界和工程界关心的前沿和热点。然而,非线性转子动力学在理论、方法和应
用方面尚有许多问题待解决,如①综合非线性因素作用下转子系统的非线性动
力学机理;②大系统稳定裕度的计算方法;③转子大系统异常故障振动的非线
性治理方法等问题。本文针对这三个问题取得了以下具体研究成果:
(1) 从八个方面综述了现代非线性转子动力学的研究现状和存在问题。
(2) 建立了一个4DOF单跨弹性转子在非线性油膜力、非线性内阻力和非线
性弹性力作用下的非线性动力学模型,提出了求周期解的数值计算方法,以及
计算周期解周期数及分岔点的算法。发现当转子通过油膜失稳进入倍周期运动
时,由于非线性内阻力作用,还会发生概周期分岔形式的二次分岔,产生约1/6
倍工频的低频运动。
(3) 提出了转子大系统的理论建模准则,考虑非线性油膜力、联轴节刚度及
标高等因素的影响,建立了一个16DOF两跨四盘四支撑不平衡弹性转子模型,
并提出了大型汽轮发电机组全轴系的48DOF理论模型,建立了考虑陀螺效应时
的复数形式的非线性动力学模型。
(4) 针对求复数ODE 周期解及稳定性分析,讨论了复数打靶法和Floquet
理论的应用问题,证明了复数ODE与实数ODE 对应Jacobian和单值矩阵的关
系。将复数打靶法应用于16DOF两跨转子非线性动力学模型,对轴系中特有的
非线性动力学现象进行了分析,发现了轴系失稳时存在的双低频现象。研究了
轴系不平衡量、联轴节刚度及轴承标高变化对轴系失稳特性的影响。
(5) 对求解高维非线性动力学系统的一种半解析半数值方法——坐标平面
投影法(CPP)进行了初步探讨,通过对Lorenz和Rossler混沌系统的奇点稳定
性分析发现,应用CPP可以从降维系统的奇点性态得到整体系统的部分性态,
从理论上证明了降维系统奇点集与原系统奇点集的关系,发现这两种混沌系统
中的奇怪吸引子为“鞍—结—焦”型奇怪吸引子。
(6) 考虑非线性油膜力的影响,提出了高速转子动平衡的非线性传递函数
法。实验证明,该方法能够较准确地得到加重质量的幅值和相位,且简单易行。
Large.scale rotary machines are key equipments in the modern society, while
the nonlinear stimulations are the key factors in large.scale rotary machines. In the
late over 20 years, nonlinear rotor dynamics is theadvancing front and the hotspot in
the international academe andengineering area. However, there are many problems
on theory, method as well as application in nonlinear dynamics that have not been
solved up to day. For example, the nonlinear dynamical mechanism of the rotor
system undergoing multiple nonlinear stimulations; stabilities analyses of large.scale
systems; the nonlinear dynamical balancing technique for high.speed rotors. The
main results obtained in this paper are as following:
(1) The latest achievements on modern rotordynamics are summarized into eight
parts, laying emphases on the nonlinear dynamical problems in every part. The
present situation and remaining problems on rotordynamics are discussed in detail.
The future research direction about this subject is pointed out.
(2) A 4DOF nonlinear symmetric rotor model with single.span, single disk is
established in the second part of the paper, considering the linear external damping,
nonlinear oil film forces, nonlinear internal damping and the nonlinear stiffness of the
rotor. The internal damping forces and the nonlinear elastic forces in the analytical
form are deduced from the Kelvin.Voigt viscoelastic model. A smart method basing
on the concept of period number is proposed for evaluating periodic solutions of the
nonlinear system and determining the periodic.doubling bifurcation points. It is
showed by the results that, the rotor has different dynamics with respect to present
results obtained by other researchers while the rotor is subjected to internal damping,
nonlinear oil film forces and nonlinear elastic forces simultaneously. With increase of
the rotary speed the rotor, quasi.periodic movement caused by internal damping will
occur. This is the second bifurcation of the rotor, leading to low frequency movement
at about 1/6 rotary speed.
(3) In the third part of the paper, we focus on how to establish a rotor model with
multi.span and multi.disk including main linear and nonlinear stimulations on the
rotors system. The rules for establishing this kind of rotors model are put forward at
the beginning of this part. Then a 16DOF rotors system with 2.span, 4.disk and
4.support is modeled in the form of plural ODE, considering the unbalance of mass,
external damping, nonlinear oil film forces as well as elevation of the bearings. The
equivalent stiffness of the shaft is deduced by FEM while only considering transverse
vibrations of the rotors. The equivalent loads of the bearings are also obtained in two
cases, with the influence of the elevation of the supports (static.determined) and
without it (static.undetermined). A 48DOF rotor model is proposed for studying
II
actual large.scale steam turbine generators. We also discuss the rotor modeling
problems considering the gyroscopic effects.
(4) For the purpose of evaluating high.dimensional plural ODE and analyzing
the stabilities of its solutions, a new method named plural shooting is discussed in the
fourth part of the paper. Combining Floquet theory, this method can help us to obtain
the stabilities of solutions of high.dimensional plural ODE fleetly. The relationships
of the Jacobians and monodromy matrices between the plural ODE and its
corresponding real ODE are proved. The stabilities and convergences of plural
Newton shooting and Broyden rank 1 plural shooting are compared. By this method,
the 16DOF rotors model established in the last part is studied, especially focusing on
some special phenomena of the rotors system, for example, the influences of the
unbalance masses of the rotors, the influences of the joint stiffness an
引文
[1] Nelson HD, Rotordynamic modeling and analysis procedures: a review, JSME
Inter. J. Series C, 1998, 41(1):1.12
[2] Baxter NL, Case Study of Rotor in Stability in the Utility Industry, Public
Service, USA, 1984
[3] 金宗武,张宁,赵凯等,应用非线性振动理论诊治大型汽轮发电机组的现
场故障,非线性动力学学报,1998, 5(S):32.35
[4] 金宗武,刘泉,张宁等,天津北郊变电站160MVAR调相机猝发振动的诊
断分析,天津电力技术,1995, 4:5.10
[5] 陈予恕,田家玉,金宗武等,非线性动力学理论与大型火电机组振动故障
综合治理技术,中国机械工程,1999, 10(9): 1063.1068
[6] 张文,转子动力学基础,北京:科学出版社,1990
[7] 阮跃,国产200MW机组常见故障的诊断和预防,汽轮机技术,1997,39(4):
200.208
[8] 伍行健,史铁林,叶能安等,汽轮发电机组故障的振动特征与自动识别,
华中理工大学学报,1993, 21(1):105.110
[9] 能源部,机电部,国产200MW机组低频振动分量普查工作报告,1990
[10]姚福生,半速涡动与油膜振荡,发电设备,1990, 1: 3.9
[11]Rankine WJ McQ, On the centrifugal force of rotating shaft, The Engineer,
London, 1869, 27: 249
[12]闻邦椿,顾家柳,夏松波,王正主编,高等转子动力学——理论、技术与
应用,北京:机械工业出版社,2000
[13]F.ppl A, Das problem der Laval.schen turbinewelle, Civilingenieur, 1895, 41:
248.250
[14]Jeffcott HH, The lateral vibration of loaded shafts in the neighborhood of a
whirling speed.the effect of want of balance, Phil. Mag., 1919, 6(37): 304.314
[15]Usher AP, A history of Mechanical Inventions, Dover Publications, 1929
[16]Holmes AG, Ettles CMM and Mayes IW, A periodic behaviour of a rigid shaft
in short journal bearings, Int. J. Numer Methods Engng. 1978, 12(4): 695.702
[17]陈予恕,孟泉.非线性转子—轴承系统的分岔.振动工程学报,1996,9
(3):266-275
[18]Adiletta, G., Guido, A. R., and Rossi, C., .Chaotic motions of a rigid rotor in
short journal bearings., Nonlinear Dynamics 10, 1996, 251-269
[19]Adiletta, G., Guido, A. R., and Rossi, C., .Nonlinear Dynamics of arigid
unbalanced rotor in journal bearings. Part I: Theoretical analysis., Nonlinear
111
天津大学博士论文
Dynamics 14, 1997, 57-87
[20]Adiletta, G., Guido, A. R., and Rossi, C., .Nonlinear Dynamics of a rigid
unbalanced rotor in journal bearings. Part II: Experimental analysis., Nonlinear
Dynamics 14, 1997, 157-189
[21]Chu F, Zhang Z, Periodic, quasi-periodic and chaotic vibrations of a rub-impact
rotor system supported on oil film. Int. J. Eng. Science 1997, 35(10-11):
963-973,
[22]丁千,陈予恕,弹性转子—滑动轴承系统稳定性分析,应用力学学报,2000,
17(3): 111.116
[23]Cao Shuqian, Chen Yushu. Bifurcation of Unbalanced Flexible Rotor with
th
Nonlinear Oil Film Forces and Internal Damping. Proc. of the 5 ICVE. China
Aviation Industry Press, 2002.9, (ISBN 7-80183-045-8). 2002 (Nanjing),
387-392
[24]郑惠萍,滑动轴承不平衡转子系统非线性动力学稳定性及其稳定裕度的研
究,[博士学位论文],天津:天津大学,2000
[25]袁小阳,朱均,多圆盘转子—滑动轴承系统自激振动的计算与分析,西安
交通大学学报,1997, 31(8):80.86
[26]曹树谦,陈予恕.多跨不平衡轴系的非线性动力学建模.非线性动力学学
报,2002,9(1,2):26-32
[27]Holzer H, Die derechnung die drehschwingungen, Julius Springer, Berlin, 1921
[28]顾家柳等,转子动力学,北京:国防工业出版社,1985
[29]Archer JS, Consistent matrix formulations for structural analysis using finite
element techniques, AIAA Journal, 1965, 3: 1910.1918
[30]Newkirk BL, Shaft whipping, General Electric Review, 1924, 27: 169.178
[31]孟光,转子动力学研究的回顾与展望,振动工程学报,2002, 15(1): 1.9
[32]虞烈,刘恒,谢友柏,轴承—转子动力学,中国机械工程,1999, 10(11):
1290.1295
[33]Newkirk BL, Journal bearing instability: A review, Inst. Mech. Conf. on Lubr.
And Wear. 1957
[34]Lund JW, Review of the concept of dynamic coefficients for fluid journal
bearings, J. of Tribology, 1987, 109: 37.41
[35]Lund JW, Spring and damping coefficients for the tilting pad journal bearing,
Trans. ASLE, 1964, 7;342.352
[36]Lund JW, Calculation of stiffness and damping properties of gas bearing,
ASME, J. of Lubr. Tech., 1968, 90: 793.803
[37]Glienicke J, Experimental investigation of the stiffness and damping
coefficients of turbine bearings and their application to instability prediction, J.
Bearings for Reciprocating and Turbomachinery, IME Symposium in
112
参考文献
Mottingham, 1966, 122.135
[38]钟一谔,何衍宗,王正.转子动力学.北京:清华大学出版社,1987
[39]Someya T, Journal.bearing data book, Springer, Tokyo, 1989
[40]Smith DN, Journal bearing dynamic characteristics.effect of inertia of
lubricant, I. Mech. E. Conv. On Lab. & Wear, 1965, 37.44
[41]Bannister RH, A theoretical and experimental investigation illustrating the
influence of non.linearity and misalignment on eight oil film force coefficients,
IME, 1976
[42]张直明,朱钧,滑动轴承的液体动力润滑理论,北京:高等教育出版社,
1986
[43]孟泉,大型高速转子油膜振荡失稳机理非线性分析:[博士学位论文].天
津:天津大学,1995
[44]Zhang W, Zhang HS, Xu XF, Study of general nonlinear formula of oil.film
force acting on a journal with unsteady motion, Asia.Pacific Vibration
Conference, Korea, 1997: 9.13
[45]张文,崔升,徐小峰等,动载轴承非稳态油膜力的一般数学模型,In 王大
钧,曲广吉主编,工程力学进展,北京:北京大学出版社,1998:158.167
[46]张慧生,张文,裘祖干等,有限长平轴承非稳妥油膜承载力导数的计算方
法及其性质,复旦学报(自然科学版),1998, 37(5): 584.594
[47]张文,郑铁生,马建敏等,油膜轴承瞬态非线性油膜力的力学建模及其表
达式,自然科学进展,2002, 12(3):255.260
[48]孟志强,非线性转子—轴承系统动力学中几个关键问题的研究,[博士学位
论文],西安:西安交通大学,2002
[49]朱均等,非线性油膜力数据库及拟合表达式输出程序使用说明书,西安交
通大学润滑理论与轴承研究所,2000
[50]王文,张直明,油叶型轴承非线性油膜力数据库,上海工业大学学报,1993,
14(4):299-305
[51]Childs DW, The space shuttle main engine high.pressure fuel turbopump rotor
dynamic instability problem, Trans. ASME, J. of Engineering for power, 1978,
100: 48.57
[52]Black HF, Effects of hydraulic forces in annular pressure seals on the vibration
of centrifugal pump rotors, J. Mech. Eng. Sci., 1969, 11(2): 206.213
[53]Kostyuk AG, A theoretical analysis of the aerodynamic forces in the labyrinth
glands of turbomachines, Teploenergetica, 1972, 19(11): 31.34
[54]Iwatsubo T et al, Flow induced force and flow pattern of labyrinth seal, NASA,
CP2250, 1982, 202.222
[55]Childs DW, Scharrer JK, Experimental rotor dynamic coefficient results for
113
天津大学博士论文
teeth.on.rotor and teeth.on.stator labyrinth gas seals, Gas Turbines and
Powers, 1986, 108: 599.604
[56]鲁周勋,谢友柏,邱大谋,迷宫气体密封转子动力学特性分析,动力工程,
1994,14(1):1.6
[57]Bently DE, Muszynska A, Perturbation tests of bearing/seal for evaluation of
dynamic coefficients, Rotor Dynamic Instability, ASME, AMD, 1983: 55
[58]Muszynska A, Whirl and whip.rotor/bearing stability problems, JSV, 1986,
110(3): 443.462
[59]Muszynska A, Bently DE, Frequency.swept rotating input perturbation
techniques and identification of the fluid force models in rotor/bearing/seal
systems and fluid handling machines, JSV, 1990, 143(1): 103.124
[60]Tam, LT, Przekwas AJ, Muszynska A, et al, Numerical and analytical study of
fluid dynamic forces in seals and bearings, Trans. ASME, J. of Vibration,
Acoustics, Stress and Reliability in Design, 1998, 110: 315.325
[61]Thomas HJ, Unstable oscillations of turbine rotors due to steam leakage in the
sealing glands and the buckets, Bulletin Scientifique, AJM, 1958, 71:
1039.1064
[62]Alford JS,Protecting turbomachinery from self.excited rotorwhirl, Trans.
ASME, J. of Engineering for Power, 1965, 335.344
[63]Urlichs K, Clearance flow.generated transverse of forces at the rotors thermal
turbomachines, Dissertation, Technical University of Munich, 1975, (English
translation on NASA TM.77292, 1983)
[64]Wohlrab R, Experimental determination of gap flow.conditioned force at
turbine stages and their effect on running stability of simple rotors, 1975,
(English translation on NASA TM.77293, 1983)
[65]Vance JM, Landadio FJ, Experimental measurement of Alford.s force in axial
flow trubomachinery, Trans. ASME, J. of Engineering for Gas Turbines and
Power, 1984, 106: 585.590
[66]柴山,张耀明,曲庆文等,汽轮机间隙气流激振分析,中国工程科学,
2001,3(4): 69.72
[67]柴山,马浩,曲庆文等,大型旋转机械气流激振力研究综述,山东工程学
院学报,1998, 12(4): 1.13
[68]杨建刚,朱天云,高伟,汽流激振对轴系稳定性的影响分析,中国电机工
程学报,1998, 18(1): 9.11
[69]柴山,张耀明,马浩等,汽轮机调节级的气流激振力分析,应用数学和力
学,2001, 22(7): 706.712
[70]Beaty RF, Differentiating rotor response due to radial rubbing, Trans. ASME, J.
of Vibration, Acoustics, Stress, Reliability in Design, April, 1985, 151.160
114
参考文献
[71]Choy FK, Padovan J, Nonlinear transient analysis of rotor.casting rub events,
JSV, 1987, 113(3): 529.545
[72]岳国金,晏励堂,李其汉,转子碰摩的特征分析,航空学报,1990, 11(10):
B499.B502
[73]Ehrich FF, Some observations of chaotic vibration phenomena in high speed
rotor dynamics, Trans. ASME, J. of Vibration, Acoustics, Stress, Reliability in
Design, 1991, 113: 50.57
[74]Ehrich FF, Spontaneous sidebanding in high speed rotor dynamics,Trans.
ASME, J. of Vibration, Acoustics, Stress, Reliability in Design, 1992, 114:
418.505
[75]Goldman P, Muszynska A, Chaotic behaviour of rotor/stator systems with rubs,
Trans. ASME, J. of Engineering for Gas Turbines and Power, 1994, 116:
692.701
[76]褚福磊,冯寇平,张正松,碰摩转子系统中的阵发性及混沌现象,航空动
力学报,1996, 11(3): 261.264
[77]Chu F, Zhang Z, Bifurcation and chaos in rub.impact Jeffcott rotor system,
JSV, 1998, 210(1): 1.18
[78]丁千,陈予恕,转子碰摩运动的非稳态分析,航空动力学报,2000, 15(2):
191.195
[79]孙政策,徐健学,龚璞林,转子系统碰摩行为的研究,振动工程学报,2000,
13(3): 474.480
[80]闫民,陈予恕,曹树谦,转子系统非线性动力学DEM 建模研究,力学学
报,2001, 33(3): 390.402
[81]Kimball A, Internal friction theory of shaft whipping, General Electric Review,
1924, 17: 244
rd
[82]Timoshenko S, Vibration problems in engineering, 3 ed., D. Van Nostrand Co.,
New York, 227.232, 1961
[83]Ehrich FF, Shaft whirl induced by rotor internal damping, J. of App. Mech.,
1964, 31: 279.282
[84]Tondl A, Some problems of rotor dynamics, Publishing House of the
Czechoslovak Academy of Sciences, Prague, 1965, 17.69
[85]Gunter EJJ, The influence of internal friction on the stability of high.speed
rotor, J. of Engineering for Industry, Nov. 1967, 683.688
[86]Torby J, The effect of structural damping upon the whirling of rotors, J. of App.
Mech., 1979, 46: 469.470
[87]Hendrick SL, The effect of ciscoelasticity on the vibration of a rotor, J. of App.
Mech., 1986, 53: 412.416
[88]Crandall SH, The influences of material creep on rotor dynamics, Proc. of the
115
天津大学博士论文
Inter. Conf. on Vib. Prob. in Eng., 1986, 43.50
[89]张文,粘弹转轴的动力失稳,复旦学报(自然科学版),1984, 43(4): 425.433
[90]Zhang W, Dynamic stability of the rotating shaft made of Boltzmann
viscoelastic solid, J. of App. Mech., 1986, 53: 424.429
[91]Zhang W, A general theory for viscoelastic rotor dynamics, Rotating
th
Mechinery Dynamics, 11 Biennial Conference on Mechanical Vibration and
Noise, Boston, Sep. 1987, 183.189
[92]Zhang W, et al, Creep instability of high spinning viscoelastic rod, Creep in
th
Structure, 4 IUTAM Symposium, Springer.Verlag, 1990
[93]Horner GC, Pilkey WD, The Riccati transfer matrix method, J. of Mech.
Design Trans, ASME, 1978, 100(4): 297.302
[94]Prohl MA, A general method for calculating critical speeds of flexible rotors, J.
of Appl. Mech. Trans, ASME, 1945, 12(3): 142.148
[95]Myklestad NO, A new method for calculating natural modes of uncoupled
bending vibration of airplane wings and other types of beams, J. of Aero. Sci.,
1944, 11: 153.162
[96]王正,Riccati传递矩阵法的奇点及其消除方法,振动与冲击,1987,2: 74.78
[97]曹树谦,丁千,陈予恕等.具有滑动轴承的稳态转子系统有限元建模分
析.汽轮机技术,1999,41(6):347-350,354
[98]Yamamoto T, On the critical speeds of a shaft, Memoirs of the Faculty of
Engineering, Nagoya University, 1954, 6, 106.174
[99]Yamamoto T, Ota H, On unstable vibrations of a shaft carrying an
unsymmetrical rotor, J. of Applied Mechanics, 1964, 31: 515.522
[100]Yamamoto T, Ishida Y, Theoretical discussion on vibration of the rotating shaft
with nonlinear spring characteristics, Ingenier.Archv, 1977, Vol.46, Heft.2,
125.135
[101]Pinkus O, Sternlicht B, Theory of Hydrodynamic Lubrication, McGraw.Hill
Book Company, New York, 1961
[102]Lund JW, Nielsen HB, Instability threshold of an unbalanced, rigid rotor in
short journal bearings, IME Conference, Vibration in rotating machinery, 1980,
91.95
st
[103]Yamauchi S, The nonlinear vibration of flexible rotors, 1 report, development
of a new analysis technique, Proc. of the JSME, Series C, 1983, 49(446):
1862.1868
[104]Gardner M,Myers C,Savage M.Analysis of limit-cycle response in fluid-film
journal bearings using the method of multiple scales.TheQuarterly Journal of
Mechanics and Applied Mathematics,1985,38:27~45
[105]Muszynska A, Stability of whirl and whip in rotor/bearing systems, JSV, 1988,
127(1): 49.64
116
参考文献
[106]Nataraj C, Nelson HD, Periodic solutions in rotor dynamic systems with
nonlinear supports: a general approach, J. of Vibration, Stress, and Reliability
in Design, Trans. ASME, 1989, 111: 187.193
[107]Jean AN, Nelson HD, Periodic response investigation of large order non.linear
rotordynamic systems using Collocation, JSV, 1990, 143(2): 473.489
[108]Adams ML,Abu-Mahfouz IA, Exploratory research on chaos concepts as
diagnostic tools for assessing rotating machinery vibration signatures.
Proceedings of IFToMM Fourth International Conference on Rotor Dynamics,
Chicago.USA,1994.29~39
[109]Ishida Y.Nonlinear vibration and chaos in rotordynamics.JSME International
Journal,Series C: Dynamics, Control, Robotics, Design and Manufacturing,
1994,37(2):237.245
[110]Kim YB, Noah ST, Bifurcation analysis for a modified Jeffcott rotor with
bearing clearance, Nonlinear Dynamics, 1990, 1:221.241
[111]Ehrich FF.Observations of subcritical superharmonic and chaotic response in
rotordynamics. J. of Vibration, Acoustics, Stress, Reliability in Design, 1992,
114:93.100
[112]Choi SK,Noah ST.Mode-locking and chaos in a Jeffcott rotor with bearing
clearances.J. of Applied Mechanics,1994,61:131.138
[113]Gonsalves DH, Neilson RD, Barr ADS, A study of response of a
discontinuously nonlinear rotor system, Nonlinear Dynamics, 1995, 7: 451.70
[114]Ganesan R.Dynamic responseand stability of a rotor-support system with
non-symmetric bearing clearances.Mech Mach Theory,1996,31(6):781.798
[115]Karpenko EV, Wiercigroch M, Cartmell M, Regular and chaotic dynamics of a
discontinuously nonlinear rotor system, Chaos, Solitons & Fractals, 2002, 13:
1231.1242
[116]Shaw J and Shaw SW. Instabilities and bifurcations in a rotating shaft. JSV,
1989; 132(2): 227-244
[117]Shaw J and Shaw SW. Non-linear resonance of an unbalanced rotating shaft
with internal damping. JSV, 1991; 147(3): 435-451
[118]Dimarogonas AD,Papadopoulos CA.Vibration of cracked shaft in bending.
JSV,1983,91(4):583~593
[119]Mayes IW,Davies WGR.Analysis of the response of a multi-rotor-bearing
system containing a transverse crack in a rotor.Journal of Vibration,Acoustics,
Stress,and Reliability in Design,1984,106:139~145
[120]Nelson HD,Nataraj C.The dynamic of a rotor system with a cracked shaft.
Journal of Vibration,Acoustics,Stress,and Reliability in Design,1986,
108:189~196
[121]Gasch R.A survey of the dynamic behavior of a simple rotating shaft with a
117
天津大学博士论文
transverse crack.JSV,1993,162:313~332
[122]Tsai TC,Wang YZ.The vibration of a multi-crack rotor.Int.J. of Mechanical
Science,1997,39(9):1037~1053
[123]Sekhar AS, Dey JK, Effects of cracks on rotor system instability, Mechanism
and Machine Theory,2000, 35: 1657.1674
[124]孟光,薜中擎,带挤压油膜阻尼器的柔性转子非线性相应的Duffing特性
分析,航空动力学报,1989,4(2):173.178
[125]Golubitsky M, Schaeffer DG, Singularities and Groups in Bifurcation Theory,
Vol. 1, New York: Springer.Verlag, 1985
[126]陆启韶,分岔与奇异性,上海:上海科技教育出版社,1995
[127]Chen YS, Langford WF, The subharmonic bifurcation solution of nonlinear
Mathieu.s equation and Euler dynamically buckling problem, Acta Mech
Sinica, 1988, 4(4): 350.362
[128]Yu P, Huseyin K, Parametrically excited non.linear systems: a comparison of
certain methods, Int. J. Non.linear Mechanics, 1998, 33(6): 967.978
[129]Chen YS and Leung AYT. Bifurcation and Chaos in Engineering. London:
Springer.Verlag, 1998
[130]陈予恕,唐云等,非线性动力学中的现代分析方法,北京:科学出版社,
1992
[131]陈予恕,非线性振动系统的分叉和混沌理论,北京:高等教育出版社,1993
[132]孟泉,陈予恕,非线性转子—轴承系统油膜失稳新机理研究,非线性动力
学学报,1995, 2(3): 189.197
[133]孟泉,陈予恕,非线性转子—轴承系统1/2亚谐共振全局分岔研究,非线
性动力学学报,1996,3(2):137.144
[134]陈予恕,丁千,非线性转子动力学的稳定性和分岔,非线性动力学学报,
1996, 3(1):13-22
[135]陈予恕,丁千,非线性转子—密封系统低频失稳机理研究:平衡系统的
Hopf分岔,非线性动力学学报,1996,3(3):197-205
[136]丁千,陈予恕,非线性转子—密封系统低频失稳机理研究:不平衡系统的
亚谐共振,非线性动力学学报,1996,3(4):310-320
[137]陈予恕,丁千,非线性转子—密封系统的稳定性和Hopf分叉,振动工程
学报,1997,10(3):368-374
[138]丁千,陈予恕,非线性转子—密封系统的亚谐共振失稳机理研究,振动工
程学报,1997,10(4):404-412
[139]张宇,陈予恕,转子—轴承—基础非线性动力学研究,振动工程学报,1998,
11(1):24-30
118
参考文献
[140]陈予恕,丁千等,非线性转子的低频振动失稳机理分析,应用力学学报,
1998,15(1):113-117
[141]Chen YS, Ding Q, C.L method and its application to subharmonic vibration of
th
nonlinear rotor systems, Proc. of the 1 Int. Conf. On the Integration of
Dynamics, Monitoring and Control, DYMAC99, Starr, 1999, 369.375
[142]陈予恕,丁千,C-L方法及其在工程非线性动力学问题中的应用,应用数
学和力学,2001,22(2):127-134
[143]褚福磊,张正松,碰摩转子系统的混沌特性,清华大学学报:自科版,1996,
36(7):52-57
[144]褚福磊,方泽南,带有支座松动故障的转子--轴承系统的混沌特性,清华
大学学报:自科版,1998,38(4):60-63
[145]褚福磊,唐云,碰摩转子系统的稳定性,清华大学学报:自科版,2000,
40(4):119-123
[146]陆启韶,张思进,王士敏,转子—弹性机壳系统碰摩的分段光滑模型分析,
振动工程学报,2000, 13(2):178.187
[147]张思进,陆启韶,碰摩转子系统的非光滑分析,力学学报,2000, 32(1):
59.69
[148]张彦梅、陆启韶,张思进,一种非稳态油膜力模型下转子系统的碰摩分岔
分析,振动工程学报,2002, 15(1): 68.73
[149]李松涛,许庆余等,迷宫密封转子系统非线性动力稳定性的研究,应用力
学学报,2002,19(2):27-30
[150]柴山,张耀明,转子偏心引起的气流激振力分析,机械工程学报,2000,
36(4):34-37
[151]郑吉兵,孟光,考虑非线性涡动时裂纹转子的分叉与混沌特性,振动工程
学报,1997,10(2):190-197
[152]郑吉兵,孟光,一种确定非线性裂纹转子解的形式的新方法,力学学报,
1998,30(1):51-57
[153]王立平,杜润生,带有轴承间隙的裂纹转子分叉与混沌特性,振动工程学
报,2000,13(2):241-246
[154]杨积东,徐培民等,裂纹转子分岔、混沌行为研究,固体力学学报,2002,
23(1):115-119
[155]李晓峰,史铁林等,带有涡动的裂纹转子故障特征研究,应用数学和力学,
2002,23(6):643-65
[156]刘荣强,夏松波,汪光明,轴承标高对多跨轴系振动及稳定性的影响,哈
尔滨工业大学学报,1995, 27(1): 127.131
119
天津大学博士论文
[157]李立,郑铁生,齿轮-转子-滑动轴承系统时变非线性动力特性研究,应用
力学学报,1995,12(1):15-24
[158]袁小阳,朱均,不平衡转子—滑动轴承系统稳定性的非线性研究,振动与
冲击,1996,15(1):71-76
[159]夏南,孟光,非线性系统周期强迫不平衡响应的稳定性分析,力学学报,
2001,33(1):128-133
[160]郭丹,陈绍汀,非线性动力系统全局分析的变胞胞映射法与转子/轴承系统
的全局稳定性,应用力学学报,1996,13(4):8-19
[161]牛玉清,徐鉴,多圆盘转子系统的周期运动及其稳定性分析,振动工程学
报,2000,13(1):30-36
[162]刘恒,刘恭忍,非线性动力系统多重周期解的伪不动点追踪法,力学学报,
1999,31(2):222-229
[163]张卫,朱均,转子—轴承系统稳定性的试验研究,机械工程学报,1997, 33
(6):53.57
[164]张卫,朱均,转子—轴承系统的稳定裕度,机械工程学报,1995, 31(2):
57.62
[165]郑惠萍,陈予恕,滑动轴承转子系统抗扰动稳定裕度的研究,汽轮机技术,
2001,43(1):35-37
[166]郑惠萍,陈予恕,非自治系统用瞬态响应求周期解稳定度的方法,应用力
学学报,2002,19(2):75-77
[167]郑惠萍,薛禹胜,陈予恕,基于轨迹的非线性转子系统参数稳定裕度的计
算,机械强度,2002, 24(2):185.189
[168]Klaus F, Muti.plane balancing of elastic rotors.fundamental theories and
practical application, Schenck Maschi.Nenfabrik GMBH DARMSTADT
Germany, 1956
[169]Bishop RED, Gladwell GML, The vibration and balancing of an unbalanced
flexible rotor, J. of Mech. Eng. Sci., 1, 1959
[170]Kellenberger W, Should a flexible rotor be balanced in N or N+2 planes? Trans.
ASME, J. of Engineering for Industry, 1972, 548.560
[171]张延安,刘细求,曹树谦,高速转子动平衡中的灵敏度分析及应用,非线
性动力学学报,2001,8(4):296-302
[172]邱海,屈梁生等,神经网络在转子动平衡中应用的几个关键问题,机械工
程学报,2001, 37(1): 88.91,112
[173]Foiles WC, Allaire PE, Gunter EJ, Min-max optimum flexible rotor balancing
compared to weighted least squares, IMECHE CONFERENCE
TRANSACTIONS, 2000 (6): 141-148
120
参考文献
[174]Rajalingham C, Bhat RB, Rakheja S, Automatic balancing of flexible vertical
rotors using a guided ball, Int. J. Mech. Sci. 1998, 40(9): 825.834
[175]刘永光,夏松波,焦映厚等,非线性转子系统自动平衡方法研究,哈尔滨
工业大学学报,1999, 31(3):77.83
[176]Sperling L, Ryzhik B, LinzCh, etal, Simulation of two-plane automatic
balancing of a rigid rotor, Mathematics and Computers in Simulation,2002, 58:
351.365
[177]Edwards S, Lees AW, Fault diagnosis of rotating machinery, Shock and
Vibration Digest, 1998, 30(1): 4.13
[178]Earrar CR, Duffey TA, Vibration.based damage detection in rotating
machinery, Key Engineering Materials, 1999, 167: 224.235
th
[179]Randall RB, Some developments in machine condition monitoring, In: Proc. 5
Inter. Conf. On Rotor Dynamics, IFToMM, Germany, 1998, 83.95
[180]张彦梅,陆启韶等,基于小波包分解的一种非稳态油膜力轴承-转子系统的
碰摩故障诊断,河南师范大学学报(自然科学版),2002,30(3):28-31
[181]彭志科,何永勇等,小波尺度谱在振动信号分析中的应用研究,机械工程
学报,2002,38(3):122-126
[182]陈耀式,汪乐宇,基于组合式神经网络的转子系统状态预测模型,中国电
机工程学报,2001,21(1):30-34,39
[183]Chen WJ, Rajan M, Rajan SD, et al, The optimal design of squeeze film
dampers for flexible rotor systems, J. of Mechanisms, Transmissions and
Automation in Design, 1988, 110(2): 166.174
[184]EI.Shafei A, Unbalance response of a Jeffcott rotor incorporationg long
squeeze fiml dampers, J. of Vibration and Acoustics, 1991, 113(1): 85.94
th
[185]Ulbrich H, Active vibration control of rotors, In: proc. of 5 Inter. Conf. On
Rotor Dynamics, IFToMM, Germany, 1998, 16.31
[186]刘迎澍,黄天,磁悬浮轴承研究综述,机械工程学报,2000,36(11):5.9
[187]Palazzolo AB, Piezoelctric pushers for active vibrationcontrol of rotating
machinery, Trans.ASME J.of Vibration,Acoustics,Stress, Reliability in
Design,1989, 111:298.305
[188]Straub FK, Merkley DJ, Design of a smart material actuator for rotor control,
Smart Materials and Structures, 1997, 6(3): 223.234
[189]Bormann J, Ulbrich H, Isolation of vibration to avoid dynamic interactions
rd
between a telescope and its foundation by active control, In: Proc. 3 Inter.
Conf. On Motion and Vibration Control, Chiba, Japan, 1996
[190]Santos IF, Design and evaluation of two types of active tilting pad journal
bearings, IUTAM Symposium on the Active Control of Vibration, England,
1994
121
天津大学博士论文
[191]Mu C, Gu JL, Vibration control and stability analysis of an active squeeze film
damper bearing and rotor system, In: Proc. of the Inter. Conf. On HBRSD,
Xi.an, 1990
[192]Yao GZ, Meng G, The vibration control of a rotor system by disk type
electro.rheological damper, JSV, 1999, 219(1): 175.188
[193]Kicinski J, Drozdowski R, The nonlinear analysis of the effect of support
construction properties on the dynamic properties of multi.support rotor
systems, JSV, 1997, 206(4): 523.539
[194]Bucciarell LL, On the instability of rotating shafts due to internal damping, J.
of Applied Mechanics, 1982, 49: 425.428
[195]Hendricks SL, The effects of viscoelasticity on the vibration of a rotor, J. of
Applied Mechanics, 1986, 53: 412.416
[196]丁千,材料内阻、密封力激励下转子系统失稳机理的非线性分析,[博士学
位论文],天津:天津大学,1997
[197]Thomson WT, Theory of Vibration with Applications, Englewood Cliffs, N.J.,
Prentice-Hall, 1972(汤姆逊著,振动理论及其应用,胡宗武译,北京:煤
炭工业出版社,1980)
[198]曹树谦,张文德,萧龙翔,振动结构模态分析—理论、实验与应用,天津:
天津大学出版社,2001年
[199]徐小峰,张文.一种非稳态油膜力模型下刚性转子的分岔和混沌特性.振
动工程学报,2000;13(2):247-252
[200]Hanselman D and Littlefield B, 精通Matlab5: 综合辅导与指南,李人厚等
译,西安:西安交通大学出版社,2001
[201]黄文虎,武新华,焦映厚,夏松波等,非线性转子动力学研究综述,振动
工程学报,2000, 13(4):497.509
[202]黄文虎.振动与冲击手册(第一卷).北京:国防工业出版社,1988
[203]Cveticanin L, Resonant vibrations of nonlinear rotors, Mech. Mach. Theory,
1995, 30(4): 581.588
[204]Chen AH, Zhong J, Mechanism analysis and mathematical description of
mechanical abrupt faults, Chinese J. of Mechanical: Engineering, 1999, 12(1):
55.57
[205]Chen YS, Sun HJ, Complex inner product averaging method for calculating
normal form of ODE, Appl. Math. Mech., 2001, 22(12): 1368.1374
[206]Cveticanin L.Vibrations of a textile machine rotor.JSV,1984,97(2):181.187
[207]Cveticanin L.The oscillations of a textile machine rotor on which the textile is
wound up.Mech. Mach. Theory,1991,26(3):253.260
[208]Ishida Y,Ikeda T,Yamamoto T.Effects of nonlinear spring characteristics on
122
参考文献
the dynamic unstable region of an unsymmetrical rotor.Bulletin of JSME,
1986,29(247):200.207
[209]Choi YS,Noah ST.Nonlinear steady-state response of a rotor-support system.
Journal of Vibration,Acoustics,Stress,and Reliability in Design,1987,
109:255~261
[210]Shiau TN,Jean AN.Prediction of periodic response of flexible mechanical
systems with nonlinear characteristics.Journal of Vibration and Acoustics,
1990,112:501~507
[211]Kang Y,Shih YP,Lee AC.Investigation on the steady-state responses of
asymmetric rotors. Journal of Vibration and Acoustics,1992,114:194.208
[212]Cveticanin L, Normal modes of vibration for continuous rotors with slow time
variable mass, Mech. Mach. Theory, 1997, 32(7): 881.891
[213]Moreira M, Antunes J, Pina H, Nonlinear analysis of the orbital motions of
immersed rotors using a spectral / Galerkin approach,Communications in
Nonlinear Scienceand Numerical Simulation, 2002, 7: 123.137
[214]陈予恕,非线性振动,天津:天津科技出版社,1983
[215]Nayfeh AH, Mook DT, Nonlinear oscillations, New York: Wiley, 1979
[216]胡海岩,应用非线性动力学,北京:航空工业出版社,2000
[217]Ling FH, Wu XX, Fast Galerkin method and its application to determine
periodic solutions of nonlinear oscillators, Int. J. of Non. Mech., 1987, 22(2):
89.98
[218]Ji JC, Yu L, Leung AYT, Bifurcaion behavior of a rotor supported by active
magnetic bearings, JSV, 2000, 235(1): 133.151
[219]李欣业,陈予恕,吴志强等,多自由度内共振系统非线性模态的分岔特性,
力学学报,2002, 34(3): 401.407
[220]李欣业,多自由度内共振系统的非线性模态及其分岔,[博士学位论文],
天津:天津大学,2000
[221]Adams ML, Nonlinear dynamics of flexible multi.bearing rotors, JSV, 1980,
71(1): 129.144
[222]Hashimoto H, Wada S, Sumitomo M, The effects of fluid inertia forces on the
dynamic behaviour of shortjournal bearings insuper laminar flow regime,
Trans. ASME, J. of Tribology, 1988, 110: 539.547
[223]Genta G, et al, Condition for noncircular whirling of nonlinear isotropic rotors,
Nonl. Dyn. 1993, 4:153.181
[224]Brown RD, et al, Chaos in the unbalance response of journal bearings, Nonl.
Dyn., 1994, 5:421.432
[225]Choi YS, Noah ST, Response and stability analysis of piecewise.linear
oscillators uner multi.forcing frequencies, Nonl. Dyn., 1992, 3: 105.121
123
天津大学博士论文
[226]刘恒,虞烈,谢友柏等,非线性周期非自治系统的Poincare型胞映射方法
及其应用,西安交通大学学报可,1998, 32(4):97.101
[227]刘恒,虞烈,谢友柏等,非线性不平衡轴承转子系统全局特性及其稳定性
准则的研究,机械工程学报,1999, 35(2):61.65
[228]刘恒,虞烈,谢友柏等,Poincare型胞映射分析方法及其应用,力学学报,
1999, 31(1):91.99
[229]Sundararajan P, Noah ST, San Andres LA, Fluid inertia effects on the nonlinear
response of a squeeze.film supported rigid rotor system, Proc. of IFToMM, 4th
Inter. Conf. On Rotor Dynamics, Chicago, IK, 1994, 333.340
[230]Sundararajan P, Noah ST, Dynamics of forced nonlinear systems using
shooting arc length continuation method.application to rotor systems, Trans.
ASME, J. of Vibration and Acoustics, 1995
[231]Hwang JL, Shiau TN, An application of the generalized polynomial expansion
method to nonlinear rotor bearing systems, J. of Vibrations and Acoustics,
Trans. ASME, 1991, 113: 299.308
[232]Padovan J, et al, Nonlinear transient finite element analysis of
rotor.bearing.stator systems, Computer and Structures, 1984, 18(4): 629.639
[233]McLean LJ, Hahn EJ, Unbalance behavior of squeeze film damped multi.mass
flexible rotor bearing systems, J. of Lubrication Technology, Trans. ASME,
1983, 105: 22.28
[234]郑铁生,复杂非线性转子—轴承系统动力特性数值分析,力学学报,2001,
33(3):377.389
[235]Nelson HD, et al, Nonlinear analysis of rotor.bearing systems using
component mode synthesis, Trans. ASME, J. of Engineering for Power, 1983,
105: 606.614
[236]HollHJ, Belyaev AK, Irschik H, A numerical algorithm fornonlinear dynamic
problems based on BEM, Engineering Analysis with Boundary Elements, 1999,
23: 503.513
[237]凌复华,非线性振动系统周期解的数值分析,应用数学和力学,1983, 4(4):
489.504
[238]凌复华,非线性振动系统周期运动及其稳定性的数值研究,力学进展,1986,
16(1)
[239]Morrison DD, Riley JD, Zancanaro JF, Multiple shooting method for two point
boundary value problems, Comm. ACM, 1962, 5: 613.614
[240]Stoer J, Bulirsch R, Introduction to Numerical Analysis, Springer.Verlag, New
York, 1980
[241]Hsu CS, A theory of cell-to-cell mapping dynamical system.J. Appl. Mech.,
1980, 47, 931.939
124
参考文献
[242]徐健学,高维非线性动力系统全局分析—胞胞映射法应用,应用数学和力
学,1985, 6(11):953.962
[243]Tongue BH, On the global analysis of nonlinear systems through interpolated
cell mapping, Physica, D, 1987, 28: 401.408
[244]Hsu CS, Cell.to.cell mapping: a method of global analysis for nonlinear
systems, Springer.Verlag, New York, 1987
[245]Hsu CS, Global analysis by cell mapping, Inter. J. Bifurcation and Chaos, 1992,
2(4): 727.771
[246]Levitas J, Weller T and Singer J, Poincare-like simple cellmapping for
nonlinear dynamical systems. JSV, 1994, 176, 641.662
[247]Hong L, Xu JX, Crises and chaotic transients studied by the generalized cell
mapping digraph method, Physics Letters A,1999, 262: 361.375
[248]Hsu CS, An unraveling algorithm for global analysis of dynamicalsystems, an
application of cell-to-cell mapping. J. Appl. Mech.,1980, 47: 940.948.
[249]Liu H, Yu L, Xie YB, et al, The continuous Poincare-like cellmapping method
and itsapplication to nonlineardynamics analysis of a bearing.rotor system,
Tribology International, 1998, 31(7): 369.375,
[250]李庆扬,常微分方程数值解法(刚性问题与边值问题),北京:高等教育
出版社,1991.9
[251]Goodman TR, Lance GN, The numerical integration of two.point boundary
value problems, Math. Tables and other Aids Comput. 1956, 10: 82.86
[252]陆启韶,常微分方程的定性方法和分叉,北京:北京航空航天大学出版社,
1989
[253]Roose D, Lust K, Champneys, et al, A Newton.Picard shooting method for
computing periodic solutions of large.scale dynamical systems, Chaos,
Solitons & Fractals, 1995, 5(10): 1913.1925
[254]李德信,徐健学,求解非线性系统周期轨道及其周期的一种方法,机械强
度,2002,24(1):35.38
[255]周纪卿,朱因远,非线性振动,西安:西安交通大学出版社,1998.9
[256]武新宇,射频集成电路周期稳态快速模拟算法的研究,微电子学,2002,
32(3):161-164
[257]《数学手册》编写组,数学手册,北京:高等教育出版社,1979
[258]Kron G, Diakoptics, Macdonald, London, 1973
[259]王慕秋,稳定性理论中方程组的分解问题,科学记录,1960, 4(1):1.5
[260]王慕秋,稳定性参数区域之扩大,数学学报,1975, 18(2)
[261]秦元勋,王慕秋,王联,运动稳定性理论与应用,北京:科学出版社,1981
[262]Baily FN, The application of Lypunov.s second method to interconnected
125
天津大学博士论文
systems, J. SIAM, Control Series A, 1966, 3(3): 443.462
[263]刘永清,黄启宇,钱祥征等,几类方向性微分方程组的稳定性及稳定域的
估计,第一届全国微分方程学术会议论文,1961年8月
[264]疏松桂,范鸣世,直流同步随动系统的拆喷分析出,数学学报,1961, 11
(1)
[265]刘永清,李雅普诺夫函数的分解问题,自动化学报,1965,3(3):176.182
[266]刘永清,宋中昆,大型动力系统的理论与应用—分解、结构与稳定性(卷
1),广州:华南工学院出版社,1988
[267]Xue Y, Van Cutsem Th, Ribbens.Pavella M, Criterion for transient stability
assessment of large.scale electric power systems, In: Proc. of IMACS/IFAC
Symp. On Modeling and Simulation. Lille, 1986
[268]Dahl OGC, Electric Power Circuits, Vol. II: Power System Stability, New York:
McGraw.Hill, Inc., 1938
[269]Skilling HH, Yamakawa MH, A Graphical solution of transient stability,
Electrical Eng. , 1940
[270]Xue Y, Van Cutsem Th, Ribbens.Pavella M, A simple direct method for fast
transient stablitity assessment of large power systems, IEEE Trans. On Power
Systems, PWRS, 1988, 3(2)
[271]Xue Y, Extended equal area criterion: foundations and applications (Invited
Paper), In: The 4th Symp. Of Specialists in Electric Operational and Expansion
Planning, Brasil, 1994
[272]薛禹胜,运动稳定性量化理论——非自治非线性多刚体系统的稳定性分
析,南京:江苏科学技术出版社,1999
[273]檀斌,薛禹胜,马尔金系统的稳定性分析,非线性动力学学报,2001, 8(2):
93.100
[274]檀斌,薛禹胜,质点弹簧系统稳定性的量化分析,控制理论与应用,2001,
18(5):709-713
[275]廖浩辉,唐云,电力系统暂态稳定性的数学表述,非线性动力学学报,2001,
8(4): 317.327
[276]邹云,邱志鹏,薛禹胜,非自治动力学系统的同步稳定性,应用数学学报,
2001, 24(1): 155.157
[277]薛禹胜,周海强,单弹簧哈密顿系统在两个不同周期力作用下的动态行为
分析,非线性动力学学报,2001,8(4):311-318
[278]薛禹胜,非线性动力学的轨线分析方法,非线性动力学学报,2002, 9(1.2):
6.18
[279]曹树谦,陈予恕,薛禹胜,坐标平面投影法及其在Lorenz混沌系统中的应
用,非线性动力学学报,2002, 9(4)
126
参考文献
[280]暑痤脲饴? 务篪羼蜮钼囗梃 桢滂眈蜮屙眍耱栾疱溴朦眍泐鲨觌?
祉钽铎屦睇踵怛铐铎耔耱屐, 啮趑 尊噔?, 1965, 1(6): 736.741
[281]李炳熙,一个三阶非线性微分方程的周期解,应用数学学报,1977, 4: 39.48
[282]李炳熙,高维动力系统的周期轨道:理论和应用,上海:上海科学技术出
版社,1984
[283]Sparrow C, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors,
Springer.Verlag, New York, 1982
[284]Guckenheimer J, Holmes P, Nonlinear oscillations, dynamical systems, and
bifurcations of vector fields, New York: Springer-Verlag, 1983
[285]Schrier G van der, Maas LRM, The diffusionless Lorenz equations; Shil.nikov
bifurcations andreduction to an explicit map, Physica D, 2000, 141: 19.36
[286]Brunnet LG, Chaté H, Phase coherence in chaotic oscillatory media, Physica A,
1998, 257: 347.356
[287]Phillipson PE, Schuster P, Bifurcation dynamics of three dimensional systems,
Internat. J. Chaos and Bifurcation, 1999, TBI Preprint No. pks 99.005
[288]Everett LJ, Two.plane balancing of rotor system without phase response
measurements, ASME J. of Vib. Acou., Stress and Reliability in Res., 1987,
109: 162.167
[289]Bishop RED, Some experiments of the balancing of small flexible rotor, Part
I.theory, J. Mech. Eng. Sci., 1968, 5(1)
[290]Goodman TP, A least squares method for computing balancing correction, J. of
Eng. Indu. Trans. ASME Series B, 1964, 86(3): 273.279
[291]Foiles WC, Allaire PE, Gunter, EJ, Review: rotor balancing, Shock and
Vibration, 1998, 5(5,6): 325.336
[292]Parkinson AG, Schneider H, Balancing of flexible rotors.some considerations
on modal convergence, In: Proc. of 5th Inter. Conf. On Rotor Dyn. IFToMM,
Germany, 1998, 653.663
[293]Rao JS, A note on Jeffcott warped rotor, Machanism and Machine Theory,
2001, 36: 563.575
[294]曹树谦,陈予恕,丁千等,高速转子动平衡的传递函数法,机械强度,2002,
24(4):500.504
[295]Federn K, Multi.plane balancing of elastic rotors, fundamental theories and
practical application, Carl Schenck Maschinenfabrik, Gernamy, 1956
[296]Bishop RED, Parkinson AG., On the Use of Balancing Machines for Flexible
Rotors, ASMEVibr. Conf., l. 73(1971).
[297]Kellenberge W, Should a Flexible Rotor Be Balanced in N or N+2 Planes,
ASME Vibr. Conf.,1, 55(1971).
[298]Tessarzik JM, Badglev RH, Flexible Rotor Balancing By the Exact
Point-Speed Influence Coefficient Method, ASME Vibr. Conf., 1. 91(197l).
127
天津大学博士论文
[299]Lund JW, Tonneson I, Analysis and Experiments on Multi-Plane Balancing of
a Flexible Rotor, ASME Vibr. Conf., 1, 74(1971).
[300]Giers A, Recbnergestutzes Auswucbten elestischer Rotoren, ATM Archiv fur
technisches Messen 1974.
[301]Darlow MS, Smalley AJ, and Parkinson. A. G., A Unified Approach to
Flexible Rotor Balancing: Outline and Experimental Verification, Vibrations in
Rotating Machinery, 2nd Inter. Conf., C340/80(1981)
[302]张文德,刘习军,萧龙翔,用互谱密度函数估算两个同频振动信号的相位
差,力学与实践(已录用)