大型转子-基础-地基系统的非线性动力分析

针对实际工程中的大型机组，在线性理论分析基础上，引入转子系统的非线性油膜力项，采用子结构模态综合法，形成一个比较接近实际大型汽轮发电机组的包括陀螺转子-非稳态非线性油膜转承-弹性基础-地基系统的非线性系统计算模型。通过对系统方程进行分块直接积分求解，得到了不同位置的轴承在不同转速和不同转子偏心量下引起的系统非线性动力学现象，为大机组的非线性分析和改进提供较完善的理论分析和计算的基础。     

考虑进油压力的滑动轴承非线性油膜力数据库

通过对ReyTlolds方程的非线性变换，提出了考虑进油压力边界条件时径向滑动轴承非线性油膜力数据库的建库方法，扩展了油膜力数据库计算方法的应用，通过引入2个有限数据域的新变量对转子轴心速度项和进油压力边界条件进行有限化处理，得到了更符合实际工况的连续性油膜力数据库及计算模型，同有限元法对比分析了非线性油膜力数据库的适用性．结果表明，非线性油膜力数据库模型的精度较高，所建立的非线性油膜力计算模型可用于对转子系统瞬态运动进行简便和快捷的分析．     

Nonlinear dynamic analysis of a rub-impact rotor supported by oil film bearings

A general model of a rub-impact rotor system is set up and supported by oil film journal bearings. The Jacobian matrix of the system response is used to calculate the Floquet multipliers, and the stability of periodic response is determined via the Floquet theory. The nonlinear dynamic characteristics of the system are investigated when the rotating speed and damping ratio is used as control parameter. The analysis methods are inclusive of bifurcation diagrams, Poincaré maps, phase plane portraits, power spectrums, and vibration responses of the rotor center and bearing center. The analysis reveals a complex dynamic behavior comprising periodic, multi-periodic, chaotic, and quasi-periodic response. The modeling results thus obtained by using the proposed method will contribute to understanding and controlling of the nonlinear dynamic behaviors of the rotor-bearing system.     

Planetary gearbox with localised bearings and gears faults: simulation and time/frequency analysis

Planets bearings of planetary gear sets exhibit high rate of failure; detection of these faults which may result in catastrophic breakdowns have always been challenging. The objective of this paper is to investigate the planetary gears vibration properties in healthy and faulty conditions. To seek this goal a previously proposed lumped parameter model (LPM) of planetary gear trains is integrated with a more comprehensive bearing model. This modified LPM includes time varying gear mesh and bearing stiffness and also nonlinear bearing stiffness due to the assumption of Hertzian contact between the rollers/balls and races. The proposed model is completely general and accepts any inner/outer race bearing defect location and profile in addition to its original capacity of modelling cracks and spalls of gears; therefore, various combinations of gears and bearing defects are also applicable. The model is exploited to attain the dynamic response of the system in order to identify and analyze localized faults signatures for inner and outer races as well as rolling elements of planets bearings. Moreover, bearing defect frequencies of inner/outer race and ball/roller and also their sidebands are discussed thoroughly. Finally, frequency response of the system for different sizes of planets bearing faults are compared and statistical diagnostic algorithms are tested to investigate faults presence and growth.     

Nonlinear dynamic analysis of a rotor system with fixed-tilting-pad self-acting gas-lubricated bearings support

Based on the nonlinear theory, the unbalanced responses of a fixed-tilting-pad gas-lubricated journal bearing-rigid rotor system are investigated. A?time-dependent mathematical model is established to describe the pressure distribution of gas-lubricated journal bearing with nonlinearity. The rigid rotor supported by a fixed-tilting-pad self-acting gas-lubricated journal bearing is modeled. The differential transformation method has been employed to solve the time-dependent gas-lubricated Reynolds equation, and the dynamic motion equation has been solved by the direct integral method. The unbalanced responses of the rotor system supported by fixed-tilting-pad gas-lubricated journal bearings are analyzed by the orbit diagram, Poincaré map, time series, and spectrum diagram. The numerical results reveal periodic, period-3, and quasiperiodic motions of nonlinear behaviors of the system. Finally, the effects of pivot ratio and preload coefficient on the nonlinear dynamic characteristics of the fixed-tilting-pad self-acting gas-lubricated journal bearing-rotor system are investigated.     

复杂机车振动环境下牵引电机轴承服役寿命评估

牵引电机是铁路机车的动力源, 其关键零部件(支承轴承等)的服役性能直接影响机车传动系统的稳定性和可靠性. 对于重载机车, 传统轴承服役寿命评估方法主要基于定载荷工况, 难以准确评估轨道不平顺等复杂外部激励作用下电机轴承的服役寿命. 因此, 本文根据车辆-轨道耦合动力学理论, 考虑轨道车辆运行过程中的轮轨相互作用和齿轮啮合作用, 建立了具有牵引动力传动系统的机车-轨道耦合动力学模型; 采用线性损伤累积准则和ISO 281标准计算方法, 评估了复杂机车振动环境下牵引电机轴承的服役寿命. 结果表明, 在轨道随机不平顺激励下, 机车轮轨垂向力、齿轮啮合力、牵引电机内部转子离心力、不平衡磁拉力等明显增大; 在复杂机车振动环境中, 电机轴承内部滚子-滚道相互作用加剧, 传动端与非传动端轴承的疲劳寿命缩短; 随着线路状态的不断恶化和机车运行速度的提高, 牵引电机轴承的预测寿命里程不断减小; 由于传动端轴承承受较大的外部动态载荷, 传动端轴承的服役寿命明显低于非传动端轴承. 本文提出的评估方法可为机车牵引电机轴承的设计、选型和寿命评估提供理论指导.      

Parameter variation and data mining of oil-film bearings: a stochastic study on the Reynolds’s equation of lubrication

The modeling and simulation process of oil-film bearing dynamics constitutes a rather essential task integrated in the workflow of various mechanical products. Specifically, in the turbo charger industry, the correct capture and understanding of the associated nonlinear rotating dynamics is of utmost importance, since the system’s efficiency and lifetime span depends on it. The root cause of the nonlinear rotordynamic effects is the oil-film concentrated in the rotor’s journal bearings. Its behavior is highly coupled with both the system’s geometric and dynamic configuration. The dynamics of the oil-film are described by the well-known Navier–Stokes equation, which under a series of assumptions and simplifications results to the, so-called, Reynolds equation. In this paper, the Reynolds equation is numerically solved based on a finite difference scheme and several parameter variation studies are conducted in an effort to pinpoint the most influential parameters—journal bearing geometric dimensions, oil-film properties and rotor-velocity-driven inputs—with respect to designated responses—friction, oil-film pressure force, minimum oil-film thickness and boundary oil-flow—all of which are regarded as important in terms of the aforementioned system’s efficiency and lifetime span. Based on multivariate analysis algorithms, correlation outcomes and global sensitivity results are presented. In an effort to capture possible nonlinear phenomena, which might not be possible via linear data mining tools, the Spearman rank-order coefficient and self-organizing maps methodology are applied.     

Effects of axial preload of ball bearing on the nonlinear dynamic characteristics of a rotor-bearing system

This research studies the effects of axial preload on nonlinear dynamic characteristics of a flexible rotor supported by angular contact ball bearings. A dynamic model of ball bearings is improved for modeling a five-degree-of-freedom rotor bearing system. The predicted results are in good agreement with prior experimental data, thus validating the proposed model. With or without considering unbalanced forces, the Floquet theory is employed to investigate the bifurcation and stability of system periodic solution. With the aid of Poincarè maps and frequency response, the unstable motion of system is analyzed in detail. Results show that the effects of axial preload applied to ball bearings on system dynamic characteristics are significant. The unstable periodic solution of a balanced rotor bearing system can be avoided when the applied axial preload is sufficient. The bifurcation margins of an unbalanced rotor bearing system enhance markedly as the axial preload increases and relates to system resonance speed.     

3D Dynamic Modelling of Spatial Geared Systems

In this paper, a nonlinearly dynamic model for a spatial geared systemwith intersecting-axes, which consists of bevel gears, bearings, andshafts, is proposed based on a specific finite element theory. A newtype of spatial gear element which is consistent with the specific finiteelement theory is developed and completely describes all thedeformations that exist for the spatial motions of a pair of bevel gears,the time-variant meshing stiffness, and various types of gear errors inmanufacturing and assembly. The 3D motions of the spatial geared systemin axial, lateral, and torsional directions are coupled in the model. Theproposed approach has been coded into a software system and a dynamicanalysis for the spatial geared system is carried out. The nonlinearinfluences of axial, lateral and torsional stiffnesses of the shaft onthe vibration of the spatial geared system are especially investigated.The lateral stiffness changes the resonance peak frequencies of thespatial geared system and the torsional stiffness greatly affects thesize of the dynamic load and vibration amplitude.     

非定常短轴承油膜力公式的变分修正

本文采用了变分方法对非定常短轴承的油膜压力分布公式进行了修正，既保留了短轴承公式的简洁形式，又使其适用于大长径比轴承。得出了具有足够精度、适合轴颈大扰动情况下的有限长圆柱轴承非定常油膜力的解析公式。与差分充零算法相比，短轴承公式的结果在轴承长径比为0．6时，误差已经超过百分之二百，而本方法计算结果的误差小于百分之五。因此采用本方法既提高了短轴承油膜力公式的计算精度，又保持了油膜力公式的简洁形式，不失为进行转子-轴承系统非线性动力分析的一种有效方法。     

Study on catastrophic mechanism for rotor drop transient vibration following magnetic bearing failure

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Nonlinear dynamic characteristic of a face gear drive with effect of modification

Face gear drive is one of the main directions of research for aeronautical transmission for its advantages, but the vibration induced gear noise and dynamic load are rarely involved by researchers. The present work examines the complex, nonlinear dynamic behavior of a 6DOF face gear drive system combining with time varying stiffness, backlash, time varying arm of meshing force and supporting stiffness. The mesh pattern of the face gear drive system is analyzed when the modification strategy is applied and the effect of modification on the dynamics response, the time varying arm of meshing force based on the TCA is deduced. The dynamic responses of the face gear drive system show rich nonlinear phenomena. Nonlinear jumps, chaotic motions, period doubling bifurcation and multiple coexisting stable solutions are detected but different from the spur and bevel gear dynamics, which don’t occur near primary and higher harmonic resonance.     

Steady-state behaviour of flexible rotordynamic systems with oil journal bearings

This paper deals with the long term behaviour of flexible rotor systems, which are supported by nonlinear bearings. A system consisting of a rotor and a shaft which is supported by one oil journal bearing is investigated numerically. The shaft is modelled using finite elements and reduced using a component mode synthesis method. The bearings are modelled using the finite-length bearing theory. Branches of periodic solutions are calculated for three models of the system with an unbalance at the rotor. Also self-excited oscillations are calculated for the three models if no mass unbalance is present. The results show that a mass unbalance can stabilize rotor oscillations.     

Bifurcation and chaos analysis of multistage planetary gear train

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.     

Influence of periodic excitation on self-sustained vibrations of one disk rotors in arbitrary length journals bearings

Interaction of forced and self-sustained vibrations of one disk rotor is described by nonlinear finite-degree-of-freedom dynamical system. The shaft of the rotor is supported by two journal bearings. The combination of the shooting technique and the continuation algorithm is used to study the rotor periodic vibrations. The Floquet multipliers are calculated to analyze periodic vibrations stability. The results of periodic motions analysis are shown on the frequency response. The quasi-periodic motions are investigated. These nonlinear vibrations coexist with the periodic forced vibrations.     

Coupled lateral-torsional-axial vibrations of a helical gear-rotor-bearing system

Considering the axial and radial loads, a math- ematical model of angular contact ball bearing is deduced with Hertz contact theory. With the coupling effects of lateral, torsional and axial vibrations taken into account, a lumped-parameter nonlinear dynamic model of helical gearrotor-bearing system （HGRBS） is established to obtain the transmission system dynamic response to the changes of dif- ferent parameters. The vibration differential equations of the drive system are derived through the Lagrange equation, which considers the kinetic and potential energies, the dis- sipative function and the internal/external excitation. Based on the Runge-Kutta numerical method, the dynamics of the HGRBS is investigated, which describes vibration properties of HGRBS more comprehensively. The results show that the vibration amplitudes have obvious fluctuation, and the frequency multiplication and random frequency components become increasingly obvious with changing rotational speed and eccentricity at gear and bearing positions. Axial vibration of the HGRBS also has some fluctuations. The bearing has self-variable stiffness frequency, which should be avoided in engineering design. In addition, the bearing clearance needs little attention due to its slightly discernible effect on vibration response. It is suggested that a careful examination should be made in modelling the nonlinear dynamic behavior of a helical gear-rotor-bearing system.     

Nonlinear analysis of a rub-impact rotor-bearing system with initial permanent rotor bow

A general model of a rub-impact rotor-bearing system with initial permanent bow is set up and the corresponding governing motion equation is given. The nonlinear oil-film forces from the journal bearing are obtained under the short bearing theory. The rubbing model is assumed to consist of the radial elastic impact and the tangential Coulomb type of friction. Through numerical calculation, rotating speeds, initial permanent bow lengths and phase angles between the mass eccentricity direction and the rotor permanent bow direction are used as control parameters to investigate their effect on the rub-impact rotor-bearing system with the help of bifurcation diagrams, Lyapunov exponents, Poincaré maps, frequency spectrums and orbit maps. Complicated motions, such as periodic, quasi-periodic even chaotic vibrations, are observed. Under the influence of the initial permanent bow, different routes to chaos are found and the speed when the rub happens is changed greatly. Corresponding results can be used to diagnose the rub-impact fault in this kind of rotor systems and this study may contribute to a further understanding of the nonlinear dynamics of such a rub-impact rotor-bearing system with initial permanent bow.     

受电磁激励的连续转子-轴承系统非线性振动与分岔

研究了受横向不平衡电磁激励的转子．轴承系统的非线性振动响应。首先将转子．轴承系统简化为带有质量不平衡并受横向激励的连续梁,由于短轴承的油膜力和电磁力的共同激励,系统振动具有强非线性特性。用Galerkin方法把偏微分控制方程离散为常微分方程组,采用四阶Runge—Kutta法对该系统进行数值仿真研究。其次比较了转轴分别在电磁力、油膜力单独作用和两种力共同作用下的振动特性,研究表明电磁力和油膜力对转子系统的非线性振动和分岔有着不同的贡献：油膜力的存在抑制了拟周期运动的发生,延长了稳定运行区域;电磁力拉长了拟周期发生的区域,降低了转子系统发生突发性破坏的风险。最后给出了系统响应随转速、电磁参数、油膜粘度等控制参数变化的分岔图,表明：系统在两个方向的运动随控制参数的变化趋势基本相同,经历了周期、倍周期、拟周期等非线性运动交替出现的过程;且油膜粘度的增大有利于转子系统的安全运行。     

Preload effect on nonlinear dynamic behavior of a rigid rotor supported by noncircular gas-lubricated journal bearing systems

This paper presents the effect of preload, as one of the design parameters, on nonlinear dynamic behavior of a rigid rotor supported by gas-lubricated noncircular journal bearings. A finite element method has been employed to solve the Reynolds equation in static and dynamical states and the dynamical equations are solved using the Runge–Kutta method. To analyze the behavior of the rotor center in horizontal and vertical directions under different operating conditions, dynamic trajectory, power spectra, Poincare maps, and bifurcation diagrams are used. Results of this study reveal how the complex dynamic behavior of two types of noncircular bearing systems comprising periodic, KT-periodic, and quasi-periodic responses of the rotor center varies with changes in preload value.     

弹性箔片动压径向气体轴承动特性的实验研究

建立了新型弹性箔片动压气体轴承性能测试实验台，在静载60N、转速9000r／min条件下进行轴承性能测试试验，通过中心插值法获得轴承中心的扰动速度和加速度，采用时域最小二乘法计算不同激振频率下的8个轴承动态线性刚度和阻尼系数，考察了激振频率对轴承动特性系数的影响．结果表明：轴颈中心扰动轨迹为椭圆形且振幅随动态激振频率增加而增大；刚度和阻尼系数的直接项随动态激振频率增加而减小；交叉刚度和阻尼相对激振频率的变化较小，且其值小于直接项；当激振频率与轴颈转动频率相同时，工频成分的影响较大，轴心扰动幅值明显增大，刚度和阻尼的直接项小于其它频率时的值．