| 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 rotorsystem undergoing multiple nonlinear stimulations; stabilities analyses of large.scale systems; the nonlinear dynamical balancing technique for high.speed rotors. Themain 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. Thepresent 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 analyticalform 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 isshowed 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 therotors 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 and4.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) andwithout it (static.undetermined). A 48DOF rotor model is proposed for studying IIactual large.scale steam turbine generators. We also discuss the rotor modelingproblems 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 itscorresponding real ODE are proved. The stabilities and convergences of pluralNewton 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 theunbalance masses of the rotors, the influences of the joint stiffness an...
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