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bifurcation AND UNIVERSAL UNFOLDING FOR A ROTATING SHAFT WITH UNSYMMETRICAL STIFFNESS

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Acta Mechanica Sinica 2002年第2期18卷 181-187页

作者：陈芳启吴志强陈予恕 Department of Mathematics Tianjin UniversityTianjin 300072China Liuhui Center for Applied MathematicsNankai University & Tianjin University Department of Mechanics Tianjin UniversityTianjin 300072China

The 1/2 subharmonic resonance bifurcation and universal unfoldingare studied for a rotating shaft with unsymmetrical *** bifurcation behav-ior of the response amplitude with respect to the detuning parameter was studied forthis class of problems by Xiao et ***,it is highly hnportant to research thebifurcation behavior of the response amplitude with respect to the unsymmetry ofstiffness for this ***,by means of the singularity theory,the bifurcationand universal unfolding of amplitude with respect to the unsymmetrical stiffness pa-rameter are *** results indicate that it is a high codimensional bifurcationproblem with codimension 5,and the universal unfolding is *** the mechan-ical background,we study four forms of two parameter unfoldings contained in theuniversal *** transition sets in the parameter plane and the bifurcationdiagrams are *** results obtained in this paper show rich bifurcation phe-nomena and provide some guidance for the analysis and design of dynamic bucklingexperiments of this class of system,especially,for the choice of system parameters.

关键词： shaft with unsymmetrical stiffness 1/2 subharmonic resonance bifurcation universal unfolding

bifurcation and chaos characteristics of hysteresis vibration system of giant magnetostrictive actuator

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Chinese Physics B 2020年第2期29卷 165-176页

作者： Hong-Bo Yan Hong Gao Gao-Wei Yang Hong-Bo Hao Yu Niu Pei Liu College of Mechanical Engineering Inner Mongolia University of Science&TechnologyBaotou 014010China Baotou Research Institute of Rare Earths State Key Laboratory of Bayan Obo Rare Earth Resource Researches and Comprehensive UtilizationBaotou 014030China

Chaotic motion and quasi-periodic motion are two common forms of instability in the giant magnetostrictive actuator(GMA).Therefore,in the present study we intend to investigate the influences of the system damping coefficient,system stiffness coefficient,disc spring cubic stiffness factor,and the excitation force and frequency on the output stability and the hysteresis vibration of the *** this regard,the nonlinear piezomagnetic equation,Jiles-Atherton hysteresis model,quadratic domain rotation model,and the GMA structural dynamics are used to establish the mathematical model of the hysteresis vibration system of the ***,the multi-scale method and the singularity theory are used to determine the eo-dimensional two-bifurcation characteristics of the ***,the output response of the system is simulated to determine the variation range of each parameter when chaos is ***,the fourth-order Runge-Kutta method is used to obtain the time domain waveform,phase portrait and Poincare mapping diagrams of the ***,the obtained three graphs are *** obtained results show that when the system output is stable,the variation range of each parameter can be ***,the stability interval of system damping coefficient,system stiffness coefficient,and the coefficient of the cubic stiffness term of the disc spring are ***,the stability interval of the exciting force and the excitation frequency are determined.

关键词： giant magnetostrictive actuator(GMA) nonlinear hysteresis bifurcation chaos

bifurcation and chaos study on transverse-torsional coupled 2K-H planetary gear train with multiple clearances

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Journal of Central South University 2016年第1期23卷 86-101页

作者：盛冬平朱如鹏靳广虎陆凤霞鲍和云 College of Mechanical and Electrical Engineering Nanjing University of Aeronautics and Astronautics

A new non-linear transverse-torsional coupled model was proposed for 2K-H planetary gear train, and gear's geometric eccentricity error, comprehensive transmission error, time-varying meshing stiffness, sun-planet and planet-ring gear pair's backlashes and sun gear's bearing clearance were taken into consideration. The solution of differential governing equation of motion was solved by applying variable step-size Runge-Kutta numerical integration method. The system motion state was investigated systematically and qualitatively, and exhibited diverse characteristics of bifurcation and chaos as well as non-linear behavior under different bifurcation parameters including meshing frequency, sun-planet backlash, planet-ring backlash and sun gear's bearing clearance. Analysis results show that the increasing damping could suppress the region of chaotic motion and improve the system's stability significantly. The route of crisis to chaotic motion was observed under the bifurcation parameter of meshing frequency. However, the routes of period doubling and crisis to chaos were identified under the bifurcation parameter of sun-planet backlash; besides, several different types of routes to chaos were observed and coexisted under the bifurcation parameter of planet-ring backlash including period doubling, Hopf bifurcation, 3T-periodic channel and crisis. Additionally, planet-ring backlash generated a strong coupling effect to system's non-linear behavior while the sun gear's bearing clearance produced weak coupling effect. Finally, quasi-periodic motion could be found under all above–mentioned bifurcation parameters and closely associated with the 3T-periodic motion.

关键词： planetary gear train bifurcation chaos transverse-torsional coupling backlash bearing clearance

bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

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Applied Mathematics and Mechanics(English Edition) 2017年第9期38卷 1233-1246页

作者： Kuan LU Yushu CHEN Lei HOU School of Astronautics Harbin Institute of TechnologyHarbin 150001China College of Engineering The University of IowaIowa CityIA 52242U.S.A. School of Energy Science and Engineering Harbin Institute of TechnologyHarbin 150001China

A method for seeking main bifurcation parameters of a class of nonlinear dynamical systems is proposed. The method is based on the effects of parametric varia- tion of dynamical systems on eigenvalues of the Frechet matrix. The singularity theory is used to study the engineering unfolding （EU） and the universal unfolding （UU） of an arch structure model, respectively. Unfolding parameters of EU are combination of concerned physical parameters in actual engineering, and equivalence of unfolding parameters and physical parameters is verified. Transient sets and bifurcation behaviors of EU and UU are compared to illustrate that EU can reflect main bifurcation characteristics of non- linear systems in engineering. The results improve the understanding and the scope of applicability of EU in actual engineering systems when UU is difficult to be obtained.

关键词： bifurcation singularity engineering unfolding (EU) universal unfolding(UU) nonlinear dynamics

bifurcation and chaos of a 4-side fixed rectangular thin plate in electromagnetic and mechanical fields

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Journal of Zhejiang University-Science A(Applied Physics & Engineering) 2009年第1期10卷 62-71页

作者： Wei-guo ZHU Xiang-zhong BAI School of Civil Engineering and Mechanics Yanshan University Qinhuangdao 066004 China Department of Transportation Engineering Huaiyin Institute of Technology Huai 'an 223001 China

We studied the problem of bifurcation and chaos in a 4-side fixed rectangular thin plate in electromagnetic and me-chanical *** on the basic nonlinear electro-magneto-elastic motion equations for a rectangular thin plate and the ex-pressions of electromagnetic forces,the vibration equations are derived for the mechanical loading in a steady transverse magnetic *** the Melnikov function method,the criteria are obtained for chaos motion to exist as demonstrated by the Smale horseshoe *** vibration equations are solved numerically by a fourth-order Runge-Kutta *** bifurcation dia-gram,Lyapunov exponent diagram,displacement wave diagram,phase diagram and Poincare section diagram are obtained.

关键词： Rectangular thin plate Electromagnetic-mechanical coupling Melnikov function method Runge-Kutta method bifurcation Chaos

bifurcation and Chaos Analysis of Nonlinear Rotor System with Axial-grooved Gas-lubricated Journal Bearing Support

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Chinese Journal of Mechanical Engineering 2014年第2期27卷 358-368页

作者： ZHANG Yongfang HEI Di Lü Yanjun WANG Quandai MüLLER Norbert School of Printing and Packaging Engineering Xi'an University of Technology State Key Laboratory of Mechanical Transmission Chongqing University College of Engineering Michigan State University School of Mechanical and Precision Instrument Engineering Xi'an University of Technology Department of Mechnical and Electrical Engineering Shaanxi Railway Institute

Axial-grooved gas-lubricated journal bearings have been widely applied to precision instrument due to their high accuracy, low friction, low noise and high stability. The rotor system with axial-grooved gas-lubricated journal bearing support is a typical nonlinear dynamic system. The nonlinear analysis measures have to be adopted to analyze the behaviors of the axial-grooved gas-lubricated journal bearing-rotor nonlinear system as the linear analysis measures fail. The bifurcation and chaos of nonlinear rotor system with three axial-grooved gas-lubricated journal bearing support are investigated by nonlinear dynamics theory. A time-dependent mathematical model is established to describe the pressure distribution in the axial-grooved compressible gas-lubricated journal bearing. The time-dependent compressible gas-lubricated Reynolds equation is solved by the differential transformation method. The gyroscopic effect of the rotor supported by gas-lubricated journal bearing with three axial grooves is taken into consideration in the model of the system, and the dynamic equation of motion is calculated by the modified Wilson-0-based method. To analyze the unbalanced responses of the rotor system supported by finite length gas-lubricated journal bearings, such as bifurcation and chaos, the bifurcation diagram, the orbit diagram, the Poincar6 map, the time series and the frequency spectrum are employed. The numerical results reveal that the nonlinear gas film forces have a significant influence on the stability of rotor system and there are the rich nonlinear phenomena, such as the periodic, period-doubling, quasi-periodic, period-4 and chaotic motion, and so on. The proposed models and numerical results can provide a theoretical direction to the design of axial-grooved gas-lubricated journal bearing-rotor system.

关键词： axial-grooved gas journal bearing differential transformation method nonlinear bifurcation chaos

bifurcation analysis of aero-engine's rotor system under constant maneuver load

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Applied Mathematics and Mechanics(English Edition) 2015年第11期36卷 1417-1426页

作者： Lei HOU Yushu CHEN School of Astronautics Harbin Institute of Technology

When an aircraft is hovering or doing a dive-hike flight at a fixed speed, a constant additional inertial force will be induced to the rotor system of the aero-engine, which can be called a constant maneuver load. Take hovering as an example. A Jeffcott rotor system with a biased rotor and several nonlinear elastic supports is modeled, and the vibration characteristics of the rotor system under a constant maneuver load are analytically studied. By using the multiple-scale method, the differential equations of the system are solved, and the bifurcation equations are obtained. Then, the bifurcations of the system are analyzed by using the singularity theory for the two variables. In the EG-plane, where E refers to the eccentricity of the rotor and G represents the constant maneuver load, two hysteresis point sets and one double limit point set are obtained. The bifurcation diagrams are also plotted. It is indicated that the resonance regions of the two variables will shift to the right when the aircraft is maneuvering. Furthermore, the movement along the horizontal direction is faster than that along the vertical direction. Thus, the different overlapping modes of the two resonance regions will bring about different bifurcation modes due to the nonlinear coupling effects. This result lays a theoretical foundation for controlling the stability of the aero-engine＇s rotor system under a maneuver load.

关键词： constant maneuver load aero-engine Jeffcott rotor system two-variablesingularity bifurcation

bifurcation and buckling analysis of a unilaterally confined self-rotating cantilever beam

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Acta Mechanica Sinica 2006年第2期22卷 177-184页

作者： Shifu Xiao Bin Chen Min Yang Southwest Institute of Structural Mechanics EO. Box 919-401 Mianyang 621900 China Department of Mechanics & Engineering Science PKU Beijing 100871 China

A nonlinear dynamic model of a simple nonholonomic system comprising a self-rotating cantilever beam subjected to a unilateral locked or unlocked constraint is established by employing the general Hamilton＇s Variational Principle. The critical values, at which the trivial equilibrium loses its stability or the unilateral constraint is activated or a saddle-node bifurcation occurs, and the equilibria are investigated by approximately analytical and numerical methods. The results indicate that both the buckled equilibria and the bifurcation mode of the beam are different depending on whether the distance of the clearance of unilateral constraint equals zero or not and whether the unilateral constraint is locked or not. The unidirectional snap-through phenomenon （i.e. catastrophe phenomenon） is destined to occur in the system no matter whether the constraint is lockable or not. The saddle-node bifurcation can occur only on the condition that the unilateral constraint is lockable and its clearance is nonzero. The results obtained by two methods are consistent.

关键词： Non-holonomic system bifurcation Buckling Assumed modes method Shooting method

bifurcation analysis of polynomial models for longitudinal motion at high angle of attack

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Chinese Journal of Aeronautics 2013年第1期26卷 151-160页

作者： Shi Zhongke Fan Li School of Automation Northwestern Polytechnical University

To investigate the longitudinal motion stability of aircraft maneuvers conveniently, a new stability analysis approach is presented in this paper. Based on describing longitudinal aerodynamics at high angle-of-attack （a ＆lt; 50 ） motion by polynomials, a union structure of two-order differential equation is suggested. By means of nonlinear theory and method, analytical and global bifurcation analyses of the polynomial differential systems are provided for the study of the nonlinear phenomena of high angle-of-attack flight. Applying the theories of bifurcations, many kinds of bifurcations, such as equilibrium, Hopf, homoclinic （heteroclinic） orbit and double limit cycle bifurcations are discussed and the existence conditions for these bifurcations as well as formulas for calculating bifurcation curves are derived. The bifurcation curves divide the parameter plane into several regions; moreover, the complete bifurcation diagrams and phase portraits in different regions are obtained. Finally, our conclusions are applied to analyzing the stability and bifurcations of a practical example of a high angle-of-attack flight as well as the effects of elevator deflection on the asymptotic stability regions of equilibrium. The model and analytical methods presented in this paper can be used to study the nonlinear flight dynamic of longitudinal stall at high angle of attack.

关键词： bifurcation High angle of attack Longitudinal motion Polynomials Stability