高维复杂转子系统非线性动力学的若干现代问题研究
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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
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
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