Dynamics and stability of turbocharger rotors

The paper discusses the bifurcation and stability behavior of (automotive) turbochargers with full-floating ring bearings. Turbocharger rotors exhibit a highly nonlinear behavior due to the nonlinearities introduced by the floating ring bearings. A flexible multibody model of the rotor/bearing system is presented. Numerical run-up simulations are compared with corresponding test rig measurements. The nonlinear oscillation effects are thoroughly investigated by means of simulated and measured rotor vibrations. The influence of various system parameters on the bifurcation behavior of the rotor/bearing system is analyzed. The article examines rotors supported in full-floating ring bearings with plain circular bearing geometry in the inner and outer oil gap. By recapitulating the well-known oil whirl and oil whip phenomena for single and double oil film bearings, the paper gives an overview on the fundamental dynamic effects occurring in turbocharger systems.     

The effects of unbalance on oil whirl

The nonlinear behavior of an unbalanced rotor supported in a fluid film bearing is analyzed. A simplified two dimensional model is adopted which uses the long-bearing approximation with a -film to account for cavitation. This model has been thoroughly studied by Myers [1] in the balanced case, where it is shown that the whirl instability is the result of a Hopf bifurcation. The implications of imbalance are studied in this paper. This leads to the study of a periodically perturbed Hopf bifurcation. It is shown that the dynamics in this situation can, especially under certain nonlinear resonance conditions, have an extremely complicated dependence on the system parameters and the rotor speed. Complete bifurcation diagrams are presented for a particular rotor model which demonstrate this dependence.     

多种非线性力作用下不平衡弹性转子的分岔特性

研究了4自由度不平衡弹性转子在非线性油膜力、非线性内阻力和非线性弹性力联合作用下的动力学特性。结果表明，当只有非线性油膜力作用时，转子只存在由于油膜失稳而导致的倍周期分岔。而当非线性油膜力与非线性内阻力共同作用时，在油膜失稳后，转子产生低频振动。转速继续增加，还会诱发内阻失稳，产生概周期运动。在倍周期分岔中，存在分岔激变现象。本文发现的由于油膜涡动而导致的内阻失稳(概周期运动)是一种未见报道的转子失稳模式(组合失稳)，它与油膜失稳(倍周期运动)一起可作为转子故障诊断的典型失稳模式。     

非稳态非线性油膜力作用下柔性轴裂纹转子的动力特性分析

分析了在动载轴承非稳态非线性油膜力作用下，具有横向裂纹柔性轴Jeffcott转子在非线性涡动影响下的动力特性。通过数值计算表明，在油膜失稳转速前，随着裂纹轴刚度变化比的增大，系统在低转速区域内具有丰富的非线性动力行为，出现倍周期分叉及混沌现象，涡动振幅随转速升高而减小，直到非稳态非线性油膜失稳，在无裂纹转子油膜临界失稳点处发现了类Hopf分叉现象，系统运动由平衡变为拟周期运动；裂纹转子在油膜临界失稳时的系统运动亦为拟周期运动，裂纹转子轴刚度变化对油膜失稳点及油膜失稳之后转子的运动影响不大，转子系统作拟周期运动。     

Numerical and experimental analysis of the first-and second-mode instability in a rotor-bearing system

Aiming at the oil film instability of the sliding bearing at high speeds, a rotor test rig is built to study the non-linear dynamic behaviours caused by the first- and second-mode instability. A lumped mass model (LMM) of the rotor system considering the gyroscopic effect is established, in which the graphite self-lubricating bearing and the sliding bearing are simulated by a spring–damping model and a nonlinear oil film force model based on the assumption of short bearings, respectively. Moreover, a finite element model is also established to verify the validity of the LMM. The researches focus on the effects of two loading conditions (the first- and second-mode imbalance excitation) on the onset of instability and nonlinear responses of the rotor-bearing system by using the amplitude–frequency response, spectrum cascade, vibration waveform, orbit, and Poincaré map. Finally, experiments are carried out on the test rig. Simulation and experiment all show that oil film instability can excite complicated combination frequency components about the rotating frequency and the first-/second-mode whirl/whip frequency.     

高速涡轮增压器轴承-转子系统不平衡响应及稳定性分析

汽车涡轮增压器广泛采用浮环轴承支承的小型轻质转子系统,以实现100 000~300 000 r/min的工作转速,提高发动机功率和动力性能,并降低燃油消耗和排放. 在此超高速工况下,动压油膜的强非线性作用和转子固有的不平衡效应使该系统呈现出复杂的动力学现象,其中油膜涡动、振荡、跳跃、倍周期分岔和混沌等非线性动力学行为对增压器的健康运转意义重大,因而备受关注. 本文作者从摩擦学动力学耦合的角度出发,基于流体动压轴承润滑理论和有限差分法计算非稳态油膜压力,结合达朗贝尔原理和传递矩阵法建立了转子离散化动力学方程,提出了一种由双油膜浮环支承的涡轮增压器转子系统动力学模型,并从转子轨迹、轴承偏心率、频谱响应、庞加莱映射和分岔特性等方面比较分析,描述了该非线性轴承-转子系统的不平衡效应及油膜失稳特征. 结果表明:转子一般在相对低速下作稳定的单周期不平衡振动,在高转速下其被油膜失稳引起的次同步涡动所抑制,但不平衡量的增加可阻碍转子以拟周期运动通向混沌运动的路径;适当不平衡补偿下,由于内、外油膜间交互的非线性刚度和阻尼作用,在油膜失稳区间之间的中高速区会出现适合增压器健康运转的稳定区间.      

Oil whirl of rotors: A numerical method of prediction

Summary A numerical precedure is suggested for the forecast of the onset speed for oil whirl in shafts rotating on lubricated plain bearings. The procedure uses matrix manipulations of the type recently proposed in a generalization of the Myklestadt-Holzer method for the prediction of amplitudes of vibration in unbalanced shafts.

---

Sommario Si suggerisce un procedimento numerico che consente di prevedere l'insorgere in un asse ruotante (anche su più cuscini) di vibrazioni autoeccitate nelle quali il veicolo del trasferimento dell'energia dal moto rotatorio a quello vibratorio è il lubrificante nei cuscini (oil whirl). Il procedimento si basa su sviluppi matriciali del tipo di quelli usati per la previsione di ampiezze di vibrazione in assi sbilanciati.

---

     

Nonlinear Dynamics of a Rigid Unbalanced Rotor in Journal Bearings. Part I: Theoretical Analysis

The dynamic behaviour of a rigid rotor supported on plain journal bearings was studied, focusing particular attention on its nonlinear aspects. Under the hypothesis that the motion of the rotor mass center is plane, the rotor has five Lagrangian co-ordinates which are represented by the co-ordinates of the mass center and the three angular co-ordinates needed to express the rotor's rotation with respect to its center of mass. In such conditions, the system is characterised not only by the nonlinearity of the bearings but also by the nonlinearity due to the trigonometric functions of the three assigned angular co-ordinates. However, if two angular co-ordinates have values that are generally quite small because of the small radial clearances in the bearings, the system is de facto linear in these angular co-ordinates. Moreover, if the third angular co-ordinate is assumed to be cyclic [18], the number of degrees of freedom in the system is reduced to four and nonlinearity depends solely on the presence of the journal bearings, whose reactions were predicted with the -film, short bearing model. After writing the equations of motion in this way and determining a numerical routine for a Runge–Kutta integration the most significant aspects of the dynamics of a symmetrical rotor were studied, in the presence of either pure static or pure couple unbalance and also when both types of unbalance were present. Two categories of rotors, whose motion is prevailingly a cylindrical whirl or a conical whirl, were put under investigation.     

Oil whirl in porous metal bearings

Summary Peculiar vibrations of journal bearings, tipical of balanced rigid rotors, are examined in order to elucidate the phenomenon in phisical terms and supply design diagrams for limiting its harmful effects in porous metal bearings.

---

Sommario Viene esaminato un particolare fenomeno vibratorio della coppia cinematica pernocuscinetto, caratieristico dei rotori rigidi e perfettamente bilanciati. Con tale studio si è inteso chiarire fisicamente il fenomeno, e soprattutto fornire, relativamente ai cuscinetti porosi, dei diagrammi alti, in sede di progetto, a limitare gli effetti nocivi da esso derirabili.

---

     

A note on internal damping induced self-excited vibration in a rotor by considering source loading of a DC motor drive

It is well known that rotors become unstable beyond a certain threshold spinning speed due to the non-conservative circulatory forces, which arise out of rotating internal damping. In this note, it is shown that if the source loading of the non-ideal drive is considered then this instability manifests itself as a constant rotor spinning speed and a constant amplitude whirl orbit about an unstable equilibrium. A DC motor drive is considered and the corresponding steady-state spinning frequency and whirl orbit amplitude are analytically derived as functions of the drive and the rotor system parameters.     

轴承内部润滑油分布及回流的试验观察与研究

润滑是轴承良好工作性能和长寿命的重要保证,而球-滚道接触区外润滑油层的分布直接决定了接触区内润滑油膜的形成. 本文作者搭建了基于轴承模型的试验台,将轴承外圈替换成玻璃环,从而利用光学方法实现对钢球与外圈滚道形成的接触区及其附近润滑油层分布的直接观测和测量. 本文作者重点探究了供油量、速度等工况参数和润滑油黏度等物化特性对接触区附近润滑油分布和回流的影响规律,结果表明气穴变化与回流特征均受供油量、速度和润滑油黏度影响.      

Analytical bifurcation analysis of a rotor supported by floating ring bearings

Like with other types of fluid bearings, rotors supported by floating ring bearings may become unstable with increasing speed of rotation due to self-excited vibrations. In order to study the effects of the nonlinear bearing forces, within this contribution a perfectly balanced symmetric rotor is considered which is supported by two identical floating ring bearings. Here, the bearing forces are modeled by applying the short bearing theory for both fluid films. A linear stability analysis about the static equilibrium position of the rotor shows that for a critical revolution speed the real part of an eigenvalue pair changes its sign. By means of a center manifold reduction it is shown that this destabilization of the steady state is due to a Hopf-bifurcation. Furthermore, the type of this bifurcation is determined as well as the existence and stability of limit-cycles. Notably it is found that depending on the parameters of the floating ring bearing subcritical as well as supercritical bifurcations may occur. Additionally, the analytical results obtained from the center manifold reduction are compared to numerical results by a continuation method. In conclusion, the influences of bearing design parameters on the stability and on the limit-cycles are discussed.     

基于荧光法的滚动轴承内部润滑油层分布研究

润滑油在轴承内的分布及其变化规律对轴承的润滑性能有显著影响. 在本文中搭建了滚动轴承模拟试验台,基于激光诱导荧光方法实现了滚动轴承内钢球-外圈接触区附近润滑油分布的观察与测量,获得了润滑油供给油层分布的三维形貌图,研究了不同供油量和转速对轴承内部供给油层分布的影响规律. 试验结果表明充分润滑条件下相邻钢球-外圈接触区供给油池之间会形成相互连接的油带;在高速情况下,钢球-外圈接触区供给油层厚度受前一个接触区尾部空穴影响而减小;供油量的增加会增大表观油池,但并不意味着入口有效供油层的增加.      

轴同期涡动和轴承座激振对轴承油膜稳定性的影响

在自行设计的轴承试验台上研究了轴颈同期涡动和轴承座激振对油膜层流失稳的影响 .结果表明 :随着轴颈偏心的增大 ,涡动倾向增强 ,流体出现层流失稳 ;当涡动倾向较弱时 ,影响油膜层流失稳的主要因素是轴承激振的强度以及低频激振 ,而高频激振对其影响较小 ;但当涡动倾向较强时 ,轴承激振的强度以及激振频率对油膜层流的失稳几乎没有影响     

Flexible shaft whirl induced by dry friction and its stability

This paper is devoted to the study of the whirling phenomena of flexible rotors due to dry friction. The mechanical model used here is a two-degree-of-freedom system in which the rubbing plane is not coincident with the rotating plane of the lumped mass. The characteristic equation of whirl speeds is derived and the whirling modes are obtained. The dynamic stability of each admissible whirling motion is also discussed. The results show that the whirl speeds are always higher than the critical speed of the shaft.     

Chaotic motions of a rigid rotor in short journal bearings

In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, , the modified Sommerfeld number relating to the lubricated bearings, and , the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (=0.05), whereas higher values (>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance . The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.Nomenclature C radial clearance of bearing (m) - D diameter of bearing (m) - e dimensional eccentricity of journal (m) - e _s value of e corresponding to the static position of the journal - E dimensional static unbalance of rotor (m) - f _x, f _y =F _x/(P), F _y/(P): non-dimensional components of fluid film force - F _x, F _y dimensional components of fluid film force (N) - g acceleration of gravity (m/s²) - L axial length of bearing (m) - m % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabg2da9maalaaabaGaeqyYdC3aaWbaaSqa% beaacaaIYaaaaaGcbaGaeqyYdC3aa0baaSqaaGabciaa-bdaaeaaca% WFYaaaaaaakiabg2da9maalaaabaGaeqyYdC3aaWbaaSqabeaacaaI% YaaaaOGaam4qaaqaaiabeo8aZjaadEgaaaaaaa!4C14!\[ = \frac{{\omega ^2 }}{{\omega _0^2 }} = \frac{{\omega ^2 C}}{{\sigma g}}\]: half non-dimensional mass of rotor - M half mass of rotor (kg) - n angular speed of rotor (in r.p.m.=60/2) - t time     

On the wrinkling of a pre-stressed annular thin film in tension

Asymptotic properties of the neutral stability curves for a linear boundary eigenvalue problem which models the wrinkling instability of an annular thin film in tension are considered. The film is subjected to imposed radial displacement fields on its inner and outer boundaries and, when these loads are sufficiently large, the film is susceptible to wrinkling. The critical values at which this onset occurs are dictated by the solution of a fourth-order ordinary differential eigensystem whose eigenvalue λ is a function of μ(?1), a quantity inversely proportional to the non-dimensional bending stiffness of the film, and n, the number of half-waves of the wrinkling pattern that sets in around the annular domain. Previously, Coman and Haughton [2006. Localised wrinkling instabilities in radially stretched annular thin films. Acta Mech. 185, 179-200] employed the compound matrix method together with a WKB technique to characterise the form of λ(μ,n) which essentially is related to a turning point in a reduced eigenproblem. The asymptotic analysis conducted therein pertained to the case when this turning point was not too close to the inner edge of the annulus. However, in the thin film limit μ→∞, the wrinkling load and the preferred instability mode are given by a modified eigenvalue problem that involves a turning point asymptotically close to the inner rim. Here WKB and boundary-layer asymptotic methods are used to examine these issues and comparisons with direct numerical simulations made.     

预紧方式对弹流润滑下角接触球轴承内部力学特性的影响

油膜厚度预测在评估弹流润滑(EHL)下角接触球轴承的性能和耐久性方面发挥着重要的作用. 耦合拟静力学理论和自旋下椭圆接触弹流模型,以干接触角接触球轴承拟静力学分析方法为基础,建立了定压和定位预紧方式下考虑弹流润滑和钢球自旋运动的角接触球轴承的拟静力学分析模型. 采用快速傅里叶变换(FFT)计算椭圆接触的弹性变形,运用Gauss-Seidel迭代方法求解Reynolds方程,得到自旋弹流模型的完全数值解,将其代入轴承拟静力学模型中迭代,得到轴承内部接触载荷、三维接触压力及三维膜厚分布. 对采用不同预紧方式的SKF7210型角接触球轴承进行分析,结果表明:富油润滑下,当轴承转速从0增大到15 000 r/min时,定压预紧时内圈轴向位移减小17.83%,而定位预紧时内圈承受的轴向载荷增大23.17%;定压预紧方式下球与内外滚道间膜厚均略大于定位预紧. 此外,不同预紧方式下,外圈上的中心膜厚大于内圈10%. 与干接触相比,定压下考虑弹流润滑内圈上接触载荷略大0.64%.