Rotordynamic System Research Articles

Application of active magnetic bearings in flexible rotordynamic systems – A state-of-the-art review

Abstract In this paper a critical review of literature on applications of Active Magnetic Bearings (AMBs) systems in flexible rotordynamic systems have been presented. AMBs find various applications in rotating machinery; however, this paper mainly focuses on works in vibration suppression and associated with the condition monitoring using AMBs. It briefly introduces reader to the AMB working principle, provides details of various hardware components of a typical rotor-AMB test rig, and presents a background of traditional methods of vibration suppression in flexible rotors and the condition monitoring. It then moves on to summarize the basic features of AMB integrated flexible rotor test rigs available in literature with necessary instrumentation and its main objectives. A couple of lookup tables provide summary of important information of test rigs in papers within the scope of this article. Finally, future directions in AMB research within the paper’s scope have been suggested.

Assessment of accuracy of modal parameters of a rotor dynamic system using different types of finite elements

The main objective of this work is to check the compatibility of Finite element method (FEM) to determine the modal parameters of the rotor system and to compare the results with analytical method. Current work is an attempt to implement different finite element (FE) models for the estimation of modal parameters. Two different order elements from Beam, Axisymmetric and Solid FE models are considered. The analysis is carried out on the rotor system and the accuracy of the results and the computation time of the FE models are estimated. The FE results are compared with the analytical method. It is observed that FE model can be used to simulate even at higher order frequencies, which is a limitation of analytical model. The analysis shows that the FE results closely match with analytical method and also the study summarizes that proper choosing of FE helps in developing rotor models in both a desirable accuracy and efficiency. Finally, a Campbell diagram is generated using the analytical method and FEM. The critical speeds are also calculated from the Campbell diagram. The closeness of the results between the two methods is checked.

Shooting/continuation based bifurcation analysis of large order nonlinear rotordynamic systems

This study introduces an improved numerical algorithm that is capable of analyzing nonlinear vibrations and bifurcations of general, finite, large-order rotordynamic systems supported on nonlinear bearings. An industrial rotor generally consists of several sections and stages, but numerical shooting/continuation method has been applied to a simple Jeffcott type rotor instead of complex models due to the computational burden of the numerical procedure; it becomes significant when the rotor combined with nonlinear finite bearing models. Here, some mathematical/computational techniques such as a deflation algorithm and the parallel computing are suggested for acceleration along with the conventional treatment of model reduction scheme. An eight-stage compressor rotor supported by two identical five-pad tilting pad journal bearings (TPJB) is selected as a mechanical model to test the numerical incorporation of the algorithms. The rotor beam is modelled with 35 nodes, 140 DOF based on Euler beam theory, and the fluid reaction forces from the two TPJB are calculated using simplex, triangular type finite meshes on the pads. In the numerical procedure, the shooting/continuation combined with the acceleration schemes identifies the solution curves of periodic responses and determines their stability. The orbital motions of coexistent responses are obtained from the solution manifolds.

Use of Bond Graph Technique for Rotor Dynamic System Modelling and Comparative Analysis of Vibration Frequency using Fast Fourier Transform

Use of Bond Graph Technique for Rotor Dynamic System Modelling and Comparative Analysis of Vibration Frequency using Fast Fourier Transform

Viscoelastic Dampers for Rotors: Modeling and Validation at Component and System Level

The aim of this paper is to propose a methodology for the modelling, testing, and the parameter identification of viscoelastic supports for rotating machines at the component and system level. Although the use of this type of dampers is common in rotordynamic applications, the strict dependence on the working frequency of the material parameters makes their behavior hard to predict and recommends grounding the design on experimental data of the characteristics. A dedicated test rig is adopted to characterize the supports and validate the modelling approach at component level. A parameter identification procedure is carried out from the experimental results to extract the mechanical properties of the supports. To this end, a Genetic Algorithm (GA) is adopted to search the most fitting values of damping and stiffness of the reference model. The choice of GA is motivated by the need of adopting a technique that can be easily implemented on industrial control units, being that the dedicated machine is supposed to be used in production lines for testing procedures. The results are obtained in terms of complex stiffness as a function of the frequency and are then translated in classical stiffness and damping components that are commonly used in rotordynamics. Afterwards, they are introduced in the model of a turbo-molecular pump adopted as a case study to evaluate the correctness of the proposed methodology at system level at stand-still and with the pump in rotation. The proposed approach is effective, showing a good match between the numerical model and the experimental results even with a relatively low order model of the viscoelastic supports and on a complex rotordynamic system.

Mitigation of nonlinear oscillations of a Jeffcott rotor System with an optimized damper and nonlinear energy sink

Contact occurrence between disk and stator as result of undesirable vibrations produced by eccentricity of the disk is one of the most destructive and common phenomena in rotor dynamics systems. In this work, utilizing tuned mass damper (TMD) and nonlinear energy sink (NES) are suggested as a solution for preventing contact occurrence. The mass and angular position of absorbers determine their efficiency for resisting the eccentricity force produced by the disk, and their stiffness and damping coefficients determine the displacement scope of the absorber. In order to efficiently design absorbers, optimization is proposed. In this suggested optimization process, complex averaging method is used in order for deriving the equations of motion of the system in presence of dynamic absorbers at the steady state condition. Afterwards, for determining trustworthiness of each absorber’s performance, system’s behavior is studied for different values of its parameters such as rotational speed, stiffness, clearance and eccentricity in presence of each absorber. From the obtained results, it can be perceived that TMD and NES are as efficient as possible and they have exactly the same positive influence on the system’s vibrations. The reliability of the proposed optimization process can be determined by the results.

Rotordynamic Energy Expressions for General Anisotropic Finite Element Systems

It is often desirable to identify the critical components that are active in a particular mode shape or an operational deflected shape (ODS) in a complex rotordynamic system with multiple rotating groups and bearings. The energy distributions can help identify the critical components of a rotor bearing system that may be modified to match the design requirements. Although the energy expressions have been studied by researchers in the past under specific limited conditions, these expressions require computing the displacements and velocities of all degrees-of-freedom (DOFs) over one full cycle. They do not address the overall time dependency of the energies and energy distributions, and their effect on the interpretation of a mode shape or an ODS. Moreover, a detailed finite element formulation of these energy expressions including the effects of anisotropy, skew-symmetric stiffness, viscous and structural damping have not been identified by the authors in the open literature. In this article, a detailed account of orbit characteristics and planarity for isotropic and anisotropic systems is presented. The effect of orbit characteristics on the energy expressions is then discussed. An elegant approach to obtaining time-dependent kinetic and strain energies of a mode shape or an ODS directly from the structural matrices and complex eigenvectors/displacement vectors is presented. The expressions for energy contributed per cycle by various types of damping and the destabilizing skew-symmetric stiffness that can be obtained in a similar way are also shown. The conditions under which the energies and energy distributions are time-invariant are discussed. An alternative set of energy expressions for isotropic systems with the DOFs reduced by half is also presented.

Control strategies for DC motors driving rotor dynamic systems through resonance

Rotor dynamic systems require considerably higher power/torque to accelerate through the structural resonance. However, most sources of mechanical power are non-ideal, i.e., they can only provide a limited amount of power. If there is insufficient power to overcome the resonance then the rotor speed may get caught at resonance and the persistent high vibrations can damage the machine. Various proposed solutions to this problem deal with modifications to the mechanical structure and active/semi-active control of structural parameters. This article proposes modification to the prime mover so that peak available power is delivered exactly at the structural resonance frequency. The limited power/non-ideal prime mover considered in this article is a direct current (DC) motor and the structural resonance happens due to forcing from an eccentric rotor disk and vibrations of a flexible weakly damped foundation. Various control strategies to modify the torque-speed characteristics of permanent magnet, shunt and series wound DC motors to promote escape through resonance are considered. Also, the characteristic curves for rotor/motor speed versus the DC supply voltage are obtained for the considered DC motor types from which the unattainable steady angular speeds and the speed jumps due to Sommerfeld effect are computed. Transient simulations are performed using bond graph models for this multi-energy domain (here, electro-mechanical) system. It is shown that a switched control permitting to switch between shunt and series DC motor configurations gives better regulation over the power delivery at the resonant frequency as well as super-critical operating speeds in the neighborhood of structural resonance.

Sommerfeld effect in a two-disk rotor dynamic system at various unbalance conditions

Near resonance behavior of a shaft-rotor system driven by a motor with limited power gives rise to the Sommerfeld effect wherein the rotor speed gets stuck till a specific level of excitation power is reached. In this paper, a semi-analytical method is used to observe the Sommerfeld effect of a two-disk rotor system driven through a direct current motor. The effect of relative unbalance position on vibration amplitude of a two disk rotor-shaft system is studied in detail. We observe the change in critical power input for passage through resonance by changing the unbalance positions on the disks. We also investigate the case where two close resonance frequencies interfere with the jump phenomenon associated with the Sommerfeld effect. Prediction of critical power input in complex rotor system is highly essential. Else the system may be destroyed due to large vibration amplitudes generated during operations. In this article, drive and rotor dynamics are modeled together by using multi-energy domain bond graph approach and theoretically obtained steady-state characteristics are validated through transient simulations.

Dynamic characteristics of rub-impact on rotor system with cylindrical shell

Cylindrical shells are widely used as rotating components in turbo-machines. Radial rub-impact contact tends to occur between the rotating shells and the stator due to the small gap and high speed of rotation. In this paper, a new model is developed to simulate the rub-impact contact force between the rotating shell and stator in the rotor system, where the effects of the geometrical parameters and deformation angular position of the shell on the contact stiffness are taken into account. The curvature variation of the shell subjected to rub-impact contact is defined using the transformation of the Fredholm Equation of the second kind, which is expressed as a function of the shell thickness and radius, the radial rubbing loads and the contact region term. Then, the formulation for the shell deformation is derived, and the relation among the contact region, deformation and forces is obtained. The rub-impact contact model is validated by comparing with the traditional piecewise-smooth linear-elastic contact model and the finite element analysis results. The proposed rub-impact contact model is introduced into the dynamic model of the rotor system with a cylindrical shell using the Sanders shell theory. Dynamic characteristics of the rotor system under two types of rub impact are investigated by analyzing the waveform, frequency spectrum, orbits, the rub-impact force distribution and motion stability. The effects of cylindrical shell on the dynamics of rotor system are evaluated. The proposed rub-impact contact model is capable of simulating the rub impact behaviors of complex rotor systems involving cylindrical shells more accurately.

Comparing the floquet stability of open and breathing fatigue cracks in an overhung rotordynamic system

Rotor cracks represent an uncommon but serious threat to rotating machines and must be detected early to avoid catastrophic machine failure. An important aspect of analyzing rotor cracks is understanding their influence on the rotor stability. It is well-known that the extent of rotor instability versus shaft speed is exacerbated by deeper cracks. Consequently, crack propagation can eventually result in an unstable response even if the shaft speed remains constant. Most previous investigations of crack-induced rotor instability concern simple Jeffcott rotors. This work advances the state-of-the-art by (a) providing a novel inertial-frame model of an overhung rotor, and (b) assessing the stability of the cracked overhung rotor using Floquet stability analysis. The rotor Floquet stability analysis is performed for both an open crack and a breathing crack, and conclusions are drawn regarding the importance of appropriately selecting the crack model. The rotor stability is analyzed versus crack depth, external viscous damping ratio, and rotor inertia. In general, this work concludes that the onset of instability occurs at lower shaft speeds for thick rotors, lower viscous damping ratios, and deeper cracks. In addition, when comparing commensurate cracks, the breathing crack is shown to induce more regions of instability than the open crack, though the open crack generally predicts an unstable response for shallower cracks than the breathing crack. Keywords: rotordynamics, stability, rotor cracks.

Health Monitoring and Vibratory Fault Prediction of Rotating Machinery

The major rotating machines such as large centrifugal or axial flow compressor, gas turbine and aero-engine are in the value chains of high-end and the core aspects of the industry factories, regarded as important embodiments of the national core competence in industry and high-technology development. The health monitoring and fault diagnosis and prediction, belonging to the technologies of prognosis and health management (PHM) are widely focused in recent years and developing constantly. The principles of health monitoring and vibratory fault prediction of rotating machinery are introduced in this paper. The dynamics of rotor system and structures are introduced, and the vibration problems of the rotating machine or structures are interpreted. The diagnosis and prediction of vibration faults happening on these machines commonly are given with examples of bearing faults of a turbine test-rig. At last, some important research tasks in future are prompted.

Effects of thermo hydrodynamic (THD) floating ring bearing model on rotordynamic bifurcation

This paper introduces a numerical scheme for simulating instabilities of a nonlinear rotordynamic system including thermal effects in the fluid film bearings. The method utilizes shooting/arc-length continuation, and simultaneous, finite element based solutions of the variable viscosity Reynolds equation and the energy equation. This provides a means to investigate the effects of the thermo-hydrodynamic THD model on bifurcations and nonlinear rotordynamic stability. A “Jeffcott” type rigid rotor is modeled as supported on double-layered fluid film, floating ring bearings (FRB). The FRB are known to produce highly nonlinear forces as functions of relative and absolute internal displacements and velocities. Both autonomous (free vibration) and non-autonomous (mass unbalanced excitation) cases and algorithms are presented. The computational workload and execution time required for determining coexisting periodic solutions is significantly reduced by employing deflation and parallel computing methods. The THD model nonlinear responses and bifurcation diagrams are compared with isoviscous model results for various lubricant supply temperatures. The autonomous case, THD model orbit sizes and onset of Hopf and saddle–node bifurcations for coexisting steady state responses, may have significant differences relative to the isothermal model results. The onset of Hopf bifurcation is strongly dependent on thermal conditions, and the saddle–node bifurcation points are significantly shifted compared to the isothermal model. This tends to increase the likelihood of bifurcation from a machine operators standpoint. In the non-autonomous case, large unbalance forces create sub-synchronous and quasi-periodic responses at low spin speeds. The responses stability and onset of bifurcations of these responses are highly reliant on the lubricant supply temperature.

The effect of blade vibration on the nonlinear characteristics of rotor–bearing system supported by nonlinear suspension

The influence of blade vibration on the nonlinear characteristics of rotor–bearing system is non-ignorable in estimating system performance. The extensive studies simplify the rotor system as lumped mass points. The influence of shaft’s bending and shear and the flexibility are usually ignored. The present paper is aim to analyze the nonlinear dynamic behavior of a continuum model. The continuum model of flexible blade–rotor–bearing coupling system is established, simplifying the shaft as Timoshenko beam. The Lagrange method is utilized to derive the differential equation of motion of system. Then, the nonlinear equations of coupling system are numerically solved using the Newmark-\(\upbeta \) method. The results obtained through the proposed model are compared with the rotor–bearing system without the blades. The effect of several parameters such as rotational speed, the damping coefficient and the length of blade on the nonlinear dynamics of rotor system have been investigated. Inclusive of the analysis methods of bifurcation diagram, three-dimensional spectral plots, time-base analysis, Poincare maps and spectral plots are used to analyze the behavior of the coupling system under different operating conditions, which exhibits rich dynamic behavior of the system.

Vibration analysis of a multi-fault transient rotor passing through sub-critical resonances

Various faults may simultaneously occur in a rotor dynamic system. It is crucial to detect and monitor these faults to ensure the safe operation of machinery. Usually, vibration response is utilized for fault diagnosis; however, the identification process becomes more challenging when multiple faults exhibit similar symptoms in the vibration response. In this paper, the vibrational transient response of a cracked rotor is analyzed in the presence of unbalance and coupling misalignment. The study focuses on investigating the possibility, advantages and limitations of using the sub-critical startup response in solving the fault identification problem. Numerical simulations using finite element modeling and experimental investigations are carried out for different individual and multi-fault conditions. Various factors are taken into consideration, including sub-critical limit speed, crack parameters, unbalance parameters and misalignment severity. Numerical and experimental investigations showed that the sub-critical transient response is suitable for the fault detection and diagnosis of the simultaneously existing faults studied. Compared to the critical response, the sub-critical response is more informative and each fault shows a unique feature that can be extracted without the need for sophisticated post signal processing. The study provides useful insights on the interaction between the various fault effects on the vibratory response and presents a simple method for condition monitoring of industrial equipment.

Rolling element bearing fault diagnosis under slow speed operation using wavelet de-noising

Abstract Rolling element bearing is widely used in rotating machines and any bearing defect therein can severely affect the overall machine performance. Vibration monitoring is often used to detect bearing faults. However, some rotor dynamic systems operate at very slow speed such as the slewing bearing used in dumpers, rolling mills, and cranes etc. At slow operating speed, the fault related signal features get smeared/masked due to slip, vibration of other machine parts, noise and disturbances. Thus, it becomes very difficult to identify the bearing characteristic frequencies (BCFs) from the vibration signature. In this paper, a novel diagnosis scheme based on envelope analysis and wavelet de-noising with sigmoid function based thresholding is used to extract the fault related symptoms from noisy vibration signatures of defective ball bearings operating at slow speed. The vibration signal is assumed to be composed of a deterministic part representing large scale features, a stochastic part and noise components. After sigmoid function based thresholding of the wavelet coefficients, a Bayesian estimator is used to obtain an approximation of the large scale features in the signal. The uncorrelated noise component in the signal is removed whereas the high-frequency structural ringing of the bearing induced due to impacts with the faults is retained in the large scale features. The envelope spectrum of the large scale features is used for fault diagnosis. The developed diagnosis scheme is tested using the experimental data collected from a machine fault simulator system.

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.

Impedance Synthesis Based Vibration Analysis of Geared Transmission System

The severity of gear noise response depends on the sensitivity of geared rotor system dynamics to the transmission error. As gearbox design trending towards lighter weight and lower noise, the influence of housing compliance on system dynamic characteristics cannot be ignored. In this study, a gear-shaft-bearing-housing coupled impedance model is proposed to account for the effect of housing compliance on the vibration of geared transmission system. This proposed dynamic model offers convenient modeling, efficient computing, and ability to combine computed parameters with experimental ones. The numerical simulations on system dynamic characteristics are performed for both a rigid housing configuration and a flexible one. Natural frequencies, dynamic mesh forces, and dynamic bearing reaction loads are computed, and the housing compliance contribution on system dynamic characteristics is analyzed. Results show that increasing housing compliance will decrease the system natural frequencies and will affect the dynamic bearing reaction loads significantly but have very little influence on the dynamic mesh force. Also, the analysis shows that bearing stiffness has significant influence on the degree of housing contribution on system dynamic characteristics.

The Thermohydrodynamic Analysis of Sliding Bearing High-Speed Motorized Spindle by Rotor Dynamic

This is paper presents thermohydrodynamic characteristics of high speed motorized spindle sliding bearing rotor system. The dynamic characteristic of the oil film bearing is affected by temperature increment, thereby affecting the high-speed spindle rotor system dynamics. This study applied the hydrodynamic lubrication theory, the influence of temperature on the viscosity of lubricating oil, associated with the bearing stiffness, oil film damping and other performance parameters, is considered in generalized Reynolds equation of oil film bearing. The theoretical model of the sliding bearing rotor system is established by using the transfer matrix method to analyze the dynamic characteristic and verified by experiments. The results show the high temperature environment in the motorized spindle and the friction of the bearing lead to oil temperature rise and viscosity reduction, which influences the bearing capacity, stiffness and damping, hence impact on the critical speeds and modal shapes of the sliding bearing rotor system.

Rotor Vibration under the Coupled Effects of Mass Unbalance and Asymmetric Bearings

Large unbalance in rotor-dynamic systems is typically responsible for high energy vibrations and the consequent decrease in machine life. This paper presents an analytical model developed using Lagrangian mechanics and partial differential equations (PDEs) for the purpose of early fault-detection in rotor-bearing systems. The model was validated through a Fortran based program, RDA99 developed by Adams (2010), by successfully quantifying the single-peak unbalance response of the simple 8 DOF and 12 DOF rotor-bearing mass stations over two cases. Case I uses bearings with symmetric stiffness and damping matrix. The critical speed for Case I occurred at 1690 rpm and orbital shapes of each mass station was found to be circular with forward-whirl orbits. In Case II asymmetrical bearing stiffness and damping coefficient matrices demonstrate an anisotropic system. Critical speed occurred at 1655 rpm and rotor, bearing and pedestal orbits were seen to be elliptical and changing with shaft speed. Both cases demonstrated a significant shaft bending contribution to the disk displacement.