Vibration characteristics of a hydraulic generator unit rotor system with parallel misalignment and rub-impact

The object of this research aims at the hydraulic generator unit rotor system. According to fault problems of the generator rotor local rubbing caused by the parallel misalignment and mass eccentricity, a dynamic model for the rotor system coupled with misalignment and rub-impact is established. The dynamic behaviors of this system are investigated using numerical integral method, as the parallel misalignment, mass eccentricity and bearing stiffness vary. The nonlinear dynamic responses of the generator rotor and turbine rotor with coupling faults are analyzed by means of bifurcation diagrams, Poincaré maps, axis orbits, time histories and amplitude spectrum diagrams. Various nonlinear phenomena in the system, such as periodic, three-periodic and quasi-periodic motions, are studied with the change of the parallel misalignment. The results reveal that vibration characteristics of the rotor system with coupling faults are extremely complex and there are some low frequencies with large amplitude in the 0.3–0.4× components. As the increase in mass eccentricity, the interval of nonperiodic motions will be continuously moved forward. It suggests that the reduction in mass eccentricity or increase in bearing stiffness could preclude nonlinear vibration. These might provide some important theory references for safety operating and exact identification of the faults in rotating machinery.     

非线性碰摩力对碰摩转子分叉与混沌行为的影响

研究了具有非线性碰摩力的转子局部碰摩的分叉与混沌运动,利用计算机仿真对某发动机转子的碰摩故障进行了数值模拟,讨论了转子系统参数的变化对转子混池运动状态的影响,并与碰摩实验结果进行了比较,发现了具有非线性碰摩力的转子局部碰摩转子系统的各种多周期运动和混沌运动及其演变过程。     

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.     

航空发动机整机耦合动力学模型及振动分析

面向航空发动机整机振动, 建立了航空发动机转子-滚动轴承-机匣耦合动力学模型. 该模型具有如下特点: (1)考虑转子、滚动轴承及机匣之间的耦合作用; (2)考虑了实际航空发动机的弹性支承及挤压油膜阻尼效应; (3)将转子考虑为等截面自由欧拉梁模型, 运用模态截断法进行分析; (4)考虑了滚动轴承间隙、非线性赫兹接触力以及变柔性VC(Varyingcompliance)振动; (5)考虑了转子与机匣之间的碰摩故障. 运用数值积分方法研究了航空发动机的整机振动规律, 包括: 滚动轴承VC振动分析、弹性支承刚度对耦合系统临界转速的影响、转轴模态截断阶数NM对系统响应的影响分析、挤压油膜阻尼器参数对系统响应的影响分析、突加不平衡的瞬态响应分析以及转静碰摩故障特性分析等.      

转子与定子碰摩的非线性动力学研究

转子与定子碰摩的动力学行为及其与系统参数关系对旋转机械设计和安全运行至关重要. 转子/定子碰摩系统是一类多参数控制的高维非光滑强非线性系统, 其动力学行为非常复杂. 本文主要从动力学与控制的角度对过去半个世纪有关转子与定子碰摩的研究成果进行归纳和总结, 其目的在于帮助读者尽可能全面系统地了解该问题的研究现状, 提炼尚待解决的问题, 以求推动转子与定子碰摩研究的进一步深入, 并为解决实际问题提供帮助. 本文将首先从碰摩局部模型和碰摩系统模型两个方面对已有的转子/定子碰摩系统的建模进行了简单的划分和归类. 其次, 以典型碰摩非线性响应为主线, 分别介绍了有关同频碰摩响应、谐波周期碰摩响应、准周期局部碰摩响应、干摩擦自激反向全周涡动响应、碰摩的全局响应行为以及碰摩响应的分岔与混沌方面的研究成果. 接下来讨论了在主被动抑制碰摩振动方面取得的结果. 最后, 给出结论并提出有待进一步研究的问题.     

Nonlinear dynamic analysis for coupled vehicle-bridge system with harmonic excitation

In this paper, a multi-degree-of-freedom lumped parameter coupled vehicle-bridge dynamic model is proposed considering the nonlinearities of suspension and tire stiffness/damping and the nonlinear foundation of bridge. In terms of modelling, the continuous expressions of the kinetic energy, potential energy and the dissipation function are constructed. The dynamic equations of the coupled vehicle-bridge system (CVBS) are derived and discretized using Galerkin’s scheme, which yield a set of second-order nonlinear ordinary differential equations with coupled terms. The numerical simulations are conducted by using the Newmark-β integration method to perform a parametric study of the effects on excitation amplitude, suspension stiffness and position relation. The bifurcation diagram, 3-D frequency spectrum and largest Lyapunov exponent are demonstrated in order to better understand the vibration properties and interaction between the vehicle and bridge with the key system parameters. It can be found that the nonlinear dynamic characteristics such as parametric resonance, jump phenomena, periodic, quasi-periodic and chaotic motions are strongly attributed to the interaction between vehicle and bridge. Significantly, under the combined internal and external excitations, the vibration amplitudes of the CVBS have a certain degree of dependence on the external excitation. Suspension stiffness could lead to complex dynamics such as the higher-order bifurcations increase and the chaotic regions broaden. The increasing of distance could effectively control the nonlinear vibration of CVBS. The application of the proposed nonlinear coupled vehicle-bridge model would bring higher computational accuracy and make it possible to design the vehicle and bridge simultaneously.     

Dynamic response of a rub-impact rotor system under axial thrust

A model of a rigid rotor system under axial thrust with rotor-to-stator is developed based on the classic impact theory and is analyzed by the Lagrangian dynamics. The rubbing condition is modeled using the elastic impact-contact idealization, which consists of normal and tangential forces at the rotor-to-stator contact point. Mass eccentricity and rotating speed are used as control parameters to simulate the response of rotor system. The motions of periodic, quasi-periodic and chaotic are found in the rotor system response. Mass eccentricity plays an important role in creating chaotic phenomena.     

考虑定子质量和碰摩面刚度的转子/定子系统碰摩响应的全局动力学特性研究

对考虑定子质量和碰撞面刚度的四自由度转子/定子模型的全局响应特性进行了研究。首先,通过解析方法确定了无碰摩响应的边界,然后,求解了同频全周碰摩解并进行了稳定性分析得到了同频全周碰摩响应的区域。在此基础上,利用非线性动力学分析的数值方法,确定了准周期局部碰摩与反向涡动失稳的边界,由此得出了参数平面内的不同碰摩响应的稳定区域图。进而研究了定子与转子质量比和碰撞面刚度对碰摩全局响应特性的影响,得到了不同碰摩响应共存及随系统参数变化的典型全局响应分岔图。     

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.     

Response and bifurcation analysis of a MDOF rotor system with a strong nonlinearity

A new HB (Harmonic Balance)/AFT (Alternating Frequency Time) method is further developed to obtain synchronous and subsynchronous whirling response of nonlinear MDOF rotor systems. Using the HBM, the nonlinear differential equations of a rotor system can be transformed to algebraic equations with unknown harmonic coefficients. A technique is applied to reduce the algebraic equations to only those of the nonlinear coordinates. Stability analysis of the periodic solutions is performed via perturbation of the solutions. To further reduce the computational time for the stability analysis, the reduced system parameters (mass, damping, and stiffness) are calculated in terms of the already known harmonic coefficients. For illustration, a simple MDOF rotor system with a piecewise-linear bearing clearance is used to demonstrate the accuracy of the calculated steady-state solutions and their bifurcation boundaries. Employing ideas from modern dynamics theory, the example MDOF nonlinear rotor system is shown to exhibit subsynchronous, quasi-periodic and chaotic whirling motions.     

碰摩裂纹转子轴承系统的周期运动稳定性及实验研究

根据碰摩裂纹耦合故障转子轴承系统的非线性动力学方程，利用求解非线性非自治系统周期解的延拓打靶法，研究了系统周期运动的稳定性。研究发现，小偏心量下系统周期运动发生Hopf分岔，大偏心量下系统周期运动发生倍周期分岔，偏心量的加大使周期解的稳定性明显降低；系统碰摩间隙变小，碰摩影响了油膜涡动的形成，使失稳转速有所提高；裂纹深度的加大降低了系统周期运动的稳定性。本文的研究为转子轴承系统的安全稳定运行提供了理论参考。     

Nonlinear behavior analysis of geared rotor bearing system featuring confluence transmission

In this paper, the nonlinear vibration characteristics of geared rotor bearing system and the interactions among gears, shafts, and plain journal bearings were studied. First, with the consideration of backlash, transmission error, time-varying mesh stiffness, and layout parameters, the dynamic model of geared rotor bearing system featuring confluence transmission was proposed. The nonlinear oil-film forces were computed with the Reynolds equation for finite-length journal bearings. Second, the responses of meshing vibration and bearing vibration were discussed. The numerical results revealed that the system exhibited a diverse range of periodic, sub-harmonic, and chaotic behaviors. Under different ranges of rolling frequency, the system got into chaos state through different roads. Moreover, in lower frequency, meshing vibration showed coexist of different periodic motions. Lastly, couplings of nonlinear oil-film force and nonlinear gear mesh force were discussed through a range of rolling frequencies. Gear-bearing dynamic interactions were demonstrated through the analysis of dynamic gear loads and dynamic bearing loads, and the coupling effect behaved different when rolling frequency changed.     

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.     

非线性转子-机匣系统的分岔行为研究

建立了一类非线性转子-机匣系统的碰摩模型.应用数值分析的方法对其进行研究,得到了不同参数变化下系统响应随转速变化的分岔图,分析了系统参数变化对分岔过程的影响,并作出了在相应参数状态和特定转速下的Poincare截面图,揭示系统参数变化对非线性碰摩转子-机匣系统分岔特性的影响.     

Dynamics of a Rigid Unbalanced Rotor with Nonlinear Elastic Restoring Forces. Part I: Theoretical Analysis

In a previous paper, the dynamic behaviour of a Jeffcott rotor was studied in the presence of pure static unbalance and nonlinear elastic restoring forces. The present paper extends the analysis to a rigid rotor with an axial length such as to make the transverse moment of inertia greater than the axial one. As in the previous investigation, the elastic restoring forces are assumed to be nonlinear and the effects of couple unbalance are also included but, unlike the Jeffcott rotor, the system exhibits six degrees-of-freedom. The Lagrangian coordinates were fixed so as to coincide with the three coordinates of the centre of mass of the rotor and the three angular coordinates needed in order to express the rotor's rotations with respect to a reference frame having its origin in the centre of mass. The precession motions of such a rotor turn out to be cylindrical at low angular speeds and exhibit a conical aspect when operating at higher speeds. The motion equations of the rotor were written with reference to a system that was subsequently adopted for the experimental analysis. The particular feature of this system was the use of a steel wire (piano wire) for the rotor shaft, suitably constrained and with the possibility of regulating the tension of the wire itself, in order to increase or reduce the nonlinear character of the system. The numerical analysis performed with integration of the motion equations made it possible to point out that chaotic solutions were manifested only when the tension in the wire was given the lowest values – i.e. when the system was strongly nonlinear – in the presence of considerable damping and rotor unbalance values that were so high as to lose any practical significance. Under conditions commonly shared by analogous real systems characterised by poor damping, where the contribution to nonlinearity is almost entirely due to elastic restoring forces, the analysis pointed out that precession motions may be manifested with a periodic character, whether synchronous or not, or a quasi-periodic character, but in no case is the solution chaotic.     

Rotordynamic model for electromagnetic excitation caused by an eccentric and angular rotor core in an induction motor

The paper shows a rotordynamic model for electromagnetic excitation caused by an eccentric and angular rotor core in an induction motor. It is shown that an eccentric rotor core leads to an electromagnetic force and an angular rotor core to an electromagnetic moment, which both force the rotor to vibrate. For these two kinds of magnetic unbalance, a rotordynamic model was developed considering the influence of the oil film stiffness and damping of the sleeve bearings, the stiffness of the end-shields and bearing housings, the stiffness of the rotor, the electromagnetic stiffness—radial and angular electromagnetic stiffness—the mass moment of inertia and the gyroscopic effect of the rotor. With this model, the absolute orbits of the shaft centre, the shaft journals and the bearing housings can be calculated, as well as the relative orbits between the shaft journals and the bearing housings. Additionally, the bearing housing velocities can also be computed. In addition to the mathematical derivation of the model, also a numerical example is shown for clarification. The aim of the paper is, on the one hand, to show the mathematical coherences—based on an analytical model—between rotordynamics and the electromagnetics for an induction motor with an eccentric and angular rotor core and, on the other hand, to derive a calculation method for evaluating the vibration sensitivity regarding these two different kinds of magnetic unbalance.     

The numerical modeling of rotor–stator rubbing in rotating machinery: a comprehensive review

Nonlinear Dynamics - The rotor–stator rubbing in rotating machinery generated as a consequence of rotor imbalance, shaft misalignment, and casing deformation is a potential threat to the...     

异步进动转子与限位器碰摩试验

利用限位器来限制储能飞轮实验转子的大幅度低频异步进动,设计了转子与限位器碰摩试验装置,研究转子的碰摩振动。分析了转子内表面碰摩力对转子运动的影响。转子与内置式限位器发生稳定的局部碰摩时,转子低频进动幅值不再增加,转子自转速度保持不变。碰摩转子的强迫振动在时域及频域都表现出了复杂性,碰摩冲击作为宽频激励,能够激励出转子-支承系统的第二模态正向进动。     

密封环挠性安装形式对干气密封动态追随性的影响

基于气体润滑理论,并通过小扰动法建立了螺旋槽干气密封微扰膜压控制方程,在高速高压条件下获得了气膜动态特性系数;基于动力学相关知识,在考虑转轴轴向振动的情况下,利用气膜轴向动态刚度和阻尼系数分别求解了静环挠性安装、动环挠性安装和两环均挠性安装的干气密封挠性环运动方程.在不同轴向激励振幅、激励频率、挠性环质量、弹簧刚度和辅助密封圈阻尼下分别研究了三种典型结构干气密封动态追随性并进行了对比分析.结果表明:当轴向激励频率较高或挠性环质量较大时,静环挠性安装干气密封在刚受到外界激励时膜厚突变相对严重,动态追随性较差;在轴向激励频率较低且挠性环质量较小时,静环挠性安装干气密封相比动环挠性安装干气密封表现出更好的动态追随性;在三种密封环挠性安装形式中,两环均挠性安装干气密封动态追随性最好,且具有绝对优势.     

Periodic and Chaotic Motions of a Rotor-Active Magnetic Bearing with Quadratic and Cubic Terms and Time-Varying Stiffness

In this paper, we use the asymptotic perturbation method to investigate nonlinear oscillations and chaotic dynamics in a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. Because of considering the weight of the rotor, the formulation on the electromagnetic force resultants includes the quadratic and cubic nonlinearities. The resulting dimensionless equations of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions are a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found that there exist period-3, period-4, period-6, period-7, period-8, quasiperiodic and chaotic modulated amplitude oscillations in the rotor-AMB system with the time-varying stiffness. It is seen from the numerical results that there are the phenomena of the multiple solutions and the soft-spring type and the hardening-spring type in nonlinear frequency-response curves for the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller is considered to be a controlling force which can control the chaotic response in the rotor-AMB system to a period n motion.