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

ＩｎｔｒｏｄｕｃｔｉｏｎＳｉｎｃｅｔｈｅｅｎｄｏｆ 1 980ｓ,ａｓａｈｉｇｈａｎｄｎｅｗｔｅｃｈｎｏｌｏｇｙｆｏｒａｃｔｉｖｅｖｉｂｒａｔｉｏｎｃｏｎｔｒｏｌ,ａｃｔｉｖｅｍａｇｎｅｔｉｃｂｅａｒｉｎｇｓ (ＡＭＢｓ)ｈａｖｅｂｅｅｎａｐｐｌｉｅｄｉｎｈｉｇｈｐｅｒｆｏｒｍａｎｃｅｃｅｎｔｒｉｆｕｇａｌｃｏｍｐｒｅｓｓｏｒｓａｎｄｐｕｍｐｓｉｎｖｏｌｖｅｄｉｎｌａｒｇｅｔｙｐｅｐｏｗｅｒｓｔａｔｉｏｎｓ,ｊｅｔ＿ｐｌａｎｅｓ,ｓｐａｃｅｃｒａｆｔｓａｎｄｐｅｔｒｏｌｅｕｍｏｉｌｐｉｐｉｎｇｓｙｓｔｅｍｓ,…     

水轮发电机转子偏心引起的非线性电磁振动

由于机械和电磁相互耦合,水轮发电机的电磁振动具有强非线性特征。根据不平衡磁拉力与转子偏心的非线性函数关系,通过简化的各向同性的单圆盘转子系统,建立了水轮发电机转子电磁振动的非线性系统。利用非线性振动理论的多尺度方法,从理论上分析了该系统强迫振动的稳态响应,进而研究了水轮发电机转子偏心引起的电磁振动。研究发现不平衡磁拉力使系统的涡动频率下降,使运动的中心发生变化;并且会引起两倍转频的振动。最后用模型转子仿真试验的结果验证了这些理论分析的结论。     

Bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings

Auxiliary bearings are utilized in practical installations of magnetically suspended rotating machines with the main functions to provide support to the rotating machines during their non-operational period, and to protect the magnetic bearings and the rotating assembly from being damaged due to power loss during operation. The relatively small clearances of these bearings, which are typically half of that of the magnetic bearings, may at time cause contact between the rotor and these bearings to occur even during normal operation of the rotating machines. The work presented herein examines the bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, which occurs during contact between the rotor and the auxiliary bearings. Numerical results revealed the occurrence of period-doubling bifurcation resulting in vibrations of period-2, -4, -8, -16 and -32, as well as quasi-periodic and chaotic vibrations. The results further showed that for a relatively small rotor imbalance magnitude, which is within the prescribed level of certain classes of practical rotating machinery, such nonlinear dynamical phenomena would not have been discovered had the auxiliary bearings forces been omitted in the model of the rotor-bearing system. As these bearings are essential elements in practical installations of magnetically supported rotating machines, failure to include them in the rotor-bearing model may result in incorrect prediction of the rotor’s vibration response.     

Theoretical rotor dynamic analysis of two-pole induction motors regarding excitation due to static rotor eccentricity

The paper shows a theoretical rotor dynamic analysis of two-pole induction motors regarding excitation due to static rotor eccentricity, which causes an oscillating magnetic force in the air gap of the machine. The hereby caused rotor vibrations are mathematically described, showing the orbit movements of the rotor centre, the shaft journals and the bearing housings. The influence of the rotor speed as well as influence of the direction of the magnetic force on the vibration behavior is investigated, based on a numerical example. On the one hand the aim of the paper is to derive the mathematical coherences—based on a simplified rotor dynamic model—between rotor dynamics, electromagnetic and the oil film characteristics of the sleeve bearings for two-pole induction machines, considering electromagnetic excitation due to a static rotor eccentricity. On the other hand the aim is to derive a method for calculating the maximum height of the shaft vibrations—as a worst case—caused by a static rotor eccentricity. Therefore the paper shall prepare the basis for implementing this method into more detailed numerical programs, e.g., finite element programs.     

Fully suspended, five-axis, three-magnetic-bearing dynamic spin rig with forced excitation

A significant advancement in the dynamic spin rig (DSR), i.e., the five-axis, three-magnetic-bearing DSR, is used to perform vibration tests of turbomachinery blades and components under rotating and non-rotating conditions in a vacuum. The rig has three magnetic bearings as its critical components: two heteropolar radial active magnetic bearings and a magnetic thrust bearing. The bearing configuration allows full vertical rotor magnetic suspension along with a feedforward control feature, which enables the excitation of various modes of vibration in the bladed disk test articles. The theoretical, mechanical, electrical, and electronic aspects of the rig are discussed. Also presented are the forced-excitation results of a fully levitated, rotating and non-rotating, unbladed rotor and a fully levitated, rotating and non-rotating, bladed rotor in which a pair of blades were arranged 180° apart from each other. These tests include the “bounce” mode excitation of the rotor in which the rotor was excited at the blade natural frequency of 144 Hz. The rotor natural mode frequency of 355 Hz was discerned from the plot of acceleration versus frequency. For non-rotating blades, a blade-tip excitation amplitude of approximately 100 g A⁻¹ was achieved at the first-bending critical (≈144 Hz) and at the first-torsional and second-bending blade modes. A blade-tip displacement of 1.778×10⁻³m (70 mils) was achieved at the first-bending critical by exciting the blades at a forced-excitation phase angle of 90° relative to the vertical plane containing the blades while simultaneously rotating the shaft at 3000 rpm.     

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.     

Sommerfeld effect in a gyroscopic overhung rotor-disk system

Deflection of a rotor-disk at the free end of a flexible overhung rotor-shaft causes rotation about diametral axis and consequently leads to a strong gyroscopic coupling in a spinning overhung rotor system. When the rotor is spun up about its axis, the unbalance in the rotor-disk causes transverse and rotational vibrations to increase as the spin speed approaches the critical speed of the rotor. These transverse and rotational vibrations dissipate a lot of energy, and if the rotor is driven through a non-ideal drive, i.e., a motor which can supply a limited amount of power, then the entire motor power may be spent to account for the energy dissipation. As a result, the rotor speed may get stuck in resonance at the critical speed or jump through the critical speed to a much higher speed with lower transverse and rotational vibration levels. These symptoms, normally referred to as the Sommerfeld effect, occur due to the intrinsic energetic coupling between the drive and the driven systems and are important design considerations for development of various rotating machinery with flexible rotor-shafts or supports (bearings). Sommerfeld effect in a strongly gyroscopic rotor dynamic system is studied in this article. The dynamics of an overhung rotor system near the regimes of Sommerfeld effect is studied by using a discrete and a continuous shaft-rotor model coupled with the model of the non-ideal motor drive. The models are developed using multi-energy domain modeling approach in bond graph model form. A steady-state analysis of power transfer mechanism is used to postulate the ideal characteristics of Sommerfeld effect in the neighborhood of the critical speed, and thereafter, full transient analysis is performed with aid of the bond graph model-generated coupled equations of motion to validate the postulated characteristics of the Sommerfeld effect.     

Nonlinear oscillation of active magnetic bearing–rotor systems with a time-delayed proportional–derivative controller

Nonlinear Dynamics - A time-delayed proportional–derivative controller is applied to suppress the nonlinear vibration of a rigid rotor suspended by the active magnetic bearing (AMB) subjected...     

Effects of flexible support stiffness on the nonlinear dynamic characteristics and stability of a turbopump rotor system

Using a flexible support is an efficient approach to solving the subsynchronous problems in a turbopump. In this paper, nonlinear rotordynamic analysis of a liquid fuel turbopump with a flexible support is presented using a dynamic modeling including two key destabilizing factors, nonlinear hydrodynamic forces induced by seals and internal rotor damping. The methodology of the partitioned direct integration method (PDIM) is described for reducing the computational efforts efficiently. Combining the PDIM and the shooting method, a nonlinear stability analysis of the rotor system is performed effectively. The numerical results, which are in good agreement with test data, indicate that the effects of flexible support stiffness k on the dynamic characteristics and stability of the rotor system are significant. The first critical speed of the rotor system rises as a nonlinear function of k markedly. The second critical speed varies slightly and approximates a linear variation as k increases. The onset speed of instability of the rotor system rises initially and then reduces as k increases. The effect of seal nonlinearities at low k is contrary to that at high k and the effect of seal length on the system stability is more significant than that of seal radius. The results explain the nature of the subsynchronous motion of a turbopump rotor system with flexible support and can be used in the design and operation of a liquid fuel turbopump rotor system to eliminate its rotordynamic problem.     

Nonsynchronous vibrations of rotor system induced by oil-block inside the rotating drum

Lubrication oil in a rotor system guarantees the rotating components working smoothly and protects the system from being damaged due to friction. A volume of lubrication oil, however, sometimes leaks into the inner cavity of shaft and drums of rotor system and forms an oil-block during rotating operation. The oil-block usually induces abnormal vibration of the rotating machine, which is often observed in practical cases, such as in aero-engine. The work in this paper studies the nonsynchronous vibration (NSV) induced by an oil-block in a rotating drum of a Jeffcott rotor system, which consists of a shaft, a drum and two supporting isotropic bearings. The additional effect due to an oil-block rotating on the inner wall of the drum is included into rotor system differential equations considering the Coriolis acceleration and friction interaction between the oil-block and the drum. Numerical simulations are carried out under two rotating speeds conditions: a lower one and a higher one than the first critical rotor speed, which are defined as rigid rotor case and flexible rotor case. Numerical results states the transverse vibrations by bifurcation diagrams, shaft center trajectories, frequency spectra and Poincare diagrams, which reveal multi-periodic, quasi-periodic and other complex motions due to the existing of oil-block. The internal friction coefficient and mass of the oil-block are found to have a significant effect on the generation and development of NSV. As the oil-block case is very common in practice, the investigation of NSV caused by oil-block in rotor system would benefit the understanding of complex phenomena and contribute to fault detection and diagnosis of rotating machine.     

Bifurcation of limit cycles in fluid film bearings

Rotors supported by journal bearings may become unstable due to self-excited vibrations when a critical rotor speed is exceeded. Linearised analysis is usually used to determine the stability boundaries. Non-linear bifurcation theory or numerical integration is required to predict stable or unstable periodic oscillations close to the critical speed. In this paper, a dynamic model of a short journal bearing is used to analyse the bifurcation of the steady state equilibrium point of the journal centre. Numerical continuation is applied to determine stable or unstable limit cycles bifurcating from the equilibrium point at the critical speed. Under certain working conditions, limit cycles themselves are shown to disappear beyond a certain rotor speed and to exhibit a fold bifurcation giving birth to unstable limit cycles surrounding the stable supercritical limit cycles. Numerical integration of the system of equations is used to support the results obtained by numerical continuation. Numerical simulation permitted a partial validation of the analytical investigation.     

多油楔轴承支承的转子系统在临界转速与失稳转速之间的半频振动

本文分析了多油楔轴承支承的单质量对称转子系统的非线性振动和稳定特性,研究了临界转速与失稳转速之间半频振动产生原因及其危害性.本文提出的计算方法和计算结果可供设计、计算和应用时参考.     

Dual-mode global stabilization of high-order saturated integrator chains: LMI-based MPC combined with a nested saturated feedback

Nonlinear Dynamics - In the active magnetic bearings (AMBs) supported rotating machinery, touchdown bearings are considered as safety devices to support the rotor in the deficiency of...     

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.     

Vibration suppression of a time-varying stiffness AMB bearing to multi-parametric excitations via time delay controller

The applications of active magnetic bearings are growing in industry due to its amazing advantages in reducing friction losses. In this research, the vibration of a two-degree-of-freedom rotor, active magnetic bearings system is suppressed via a nonlinear time delay controller at the confirmed worst resonance case. The selected resonance case is the simultaneous primary and sub-harmonic resonance case. The main aim of this paper was to study the effects of the nonlinear, time delay controller on the behavior of the vibrating system. The multiple time scale perturbation technique is applied to obtain an approximate solution to the second-order approximation. The steady-state solution is obtained around the worst resonance case. The stability of the system is studied applying both frequency response equations and phase-plane method. The worst resonance case is confirmed applying numerical technique. The effects of the different parameters on the steady-state response of the vibrating system are investigated. The obtained approximate solution is validated numerically. Some recommendations are given regarding the design of such system. At the end of the work, a comparison is made with the available published work.     

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。     

Detection of a slant crack in a rotor bearing system during shut-down

Abstract

The transient response of a slant-cracked rotor system during shut-down has been analyzed while it is decelerating through the critical speed. Vibration response has been simulated by using finite element method (FEM) for flexural vibrations. An unbalance force and a harmonically-varying torque is applied on the rotor system. Subharmonic frequency components at an interval frequency corresponding to torsional frequency were found in the frequency domain (Fast Fourier Transform (FFT) plot). These peaks are centered on the critical speed of the rotor system when a slant crack is present and can be used to detect the crack. The fundamental mode shapes of the rotor bearing systems are wavelet transformed to identify the crack location. A peak is observed at the crack location in the spatial variation of the wavelet-transformed fundamental mode shape. A few suggestions for the detection of slant cracks in the rotor during shut-down are described.     

Subharmonic resonance of a symmetric ball bearing–rotor system

The performance of a ball bearing–rotor system is often limited by the occurrence of subharmonic resonance with considerable vibration and noise. In order to comprehend the inherent mechanism and the feature of the subharmonic resonance, a symmetrical rotor system supported by ball bearings is studied with numerical analysis and experiment in this paper. A 6DOF rotordynamic model which includes the non-linearity of ball bearings, Hertzian contact forces and bearing internal clearance, and the bending vibration of rotor is presented and an experimental rig is offered for the research of the subharmonic resonance of the ball bearing–rotor system. The dynamic response is investigated with the aid of orbit and amplitude spectrum, and the non-linear system stability is analyzed using the Floquet theory. All of the predicted results coincide well with the experimental data to validate the proposed model. Numerical and experimental results show that the resonance frequency is provoked when the speed is in the vicinity of twice synchroresonance frequency, while the rotor system loses stability through a period-doubling bifurcation and a period-2 motion i.e. subharmonic resonance occurs. It is found that the occurrence of subharmonic resonance is due to the together influence of the non-linear factors, Hertzian contact forces and internal clearance of ball bearings. The effect of unbalance load on subharmonic resonance of the rotor system is minor, which is different from that of the sliding bearing–rotor system. However, the moment of couple has an impact influence on the subharmonic resonances of the ball bearing–rotor system. The numerical and experimental results indicate that the subharmonic resonance caused by ball bearings is a noticeable issue in the optimum design and failure diagnosis of a high-speed rotary machinery.     

Active magnetic bearing based force measurement using the multi-point technique

Active magnetic bearings (AMBs) are currently used in a number of commercial and research applications. AMBs allow for higher rotational speeds than rolling element bearings while eliminating the need for lubrication. In addition, they also may be used concurrently as support bearings and non-invasive force sensors by monitoring bearing flux-density through the addition of Hall Effect probes or through measurement of bearing currents applied to magnetic circuit models of the bearing. Many current-based approaches are limited because of model assumptions due to unknowns in the final geometry/assembly of the bearing itself resulting in unknown flux distribution. This paper addresses a new strategy for AMB current-based force measurement, the Multi-Point Method, which uses a system identification approach in order to improve the accuracy of force measurements in field conditions with no additional hardware. The Multi-Point Method described in this paper is a system identification approach to force measurement where both gap and forces are determined through the use of perturbation current application, recording of resulting states, and the use of an on-line algorithm. The Multi-Point Method has been shown to be capable of providing force measurements that are more accurate and more precise over a range of vertical rotor displacements (mimicking misalignment, assembly variations, and thermal growth effects on field gaps) than conventional single point current-based measurements that rely on assumed gaps in static tests. This paper presents the results of a proof-of-concept static test in a non-rotating two bearing rotor. Initial static experimental results demonstrate that the multi-point force predictions are within 1.03% of the known rotor force on average, with an average standard deviation of 0.83%. Comparatively, the conventional single point force predictions were within 5.76% of the known force with an average standard deviation of 6.17%. Dynamic effects due to eddy currents and hysteresis effects also degrade current-based force model accuracy and will be addressed in future work. The goal of this study is to demonstrate the importance of accounting for geometric parameters through a system identification approach to improve the fundamental current-based model accuracy.     

Lateral vibrations of soft mounted converter-fed induction motors caused by dynamic air gap torques: a mathematical analysis of orbital movements

This paper shows that dynamic air gap torques in converter-fed induction motors may not only cause torsional vibrations in the drive train, but may also cause lateral vibrations of the motor itself—including bending vibrations of the rotor shaft—, if the motor is mounted on a soft foundation. Based on a simplified analytical model, the mathematical correlation between rotor-stator interaction, oil film characteristics of sleeve bearings, the influence of a soft foundation—coupling angular and lateral displacement of the stator—and excitation from dynamic air gap torques is shown. In addition to the absolute orbits of the rotor, stator, shaft journals and bearing housings, also the relative orbits between rotor and stator and between shaft journals and bearing housings are mathematically described. Further, the paper shows the influence of the oil film coefficients of the sleeve bearings on the elliptical shape of the orbits. Therefore, the paper presents a possibility to optimize the whole system—converter, motor and foundation—concerning lateral vibrations, caused by dynamic air gap torques.