Jeffcott Rotor Research Articles

Stability of steady-state solutions of Jeffcott rotor with varying rotational speed

AbstractThe behavior of flexible rotating systems with varying rotational speeds is essential in engineering applications. Analysis methods that consider linear dynamic models and many existing nonlinear analysis approaches assume constant rotational speed. These approaches are unsuited to study the dynamic interaction between driving torque and whirling motion in this class of applications. In this paper, an analysis of the stability and control of a Jeffcott rotor under varying operational conditions is presented. A nonlinear dynamic model of the system is formulated to enable a detailed stability and parametric analysis. A proportional-integral (PI) torque command is employed to achieve a steady-state rotational speed. Assuming constant lateral control effort, system equilibrium points and their stability characteristics as functions of the system’s parameters are analyzed. A control law that minimizes the lateral effort is derived. A feedback proportional lateral control strategy is introduced to enhance the system’s region of stability, particularly in the supercritical speed range. Finally, a simulation study is conducted to validate the analytical findings. Simulation results demonstrate the effectiveness of the proposed approach for defining stable operating conditions and improving system performance.

Resonance and attraction domain analysis of asymmetric duffing systems with fractional damping in two degrees of freedom

This article establishes a two degree of freedom asymmetric Duffing system model with fractional damping through the study of the Jeffcott rotor system with horizontal support, and analyzes the dynamic behavior of its resonance and attraction domain structures. The second-order approximate solution of the system equation is obtained using a multiscale method, and the slow flow modulation equations of both the amplitudes and phases are extracted. The comparison between approximate and numerical solutions has shown their high consistency and the Routh Hurwitz criterion is utilized to study the stability of the system. Finally, we explore the effect of fractional damping on the vibration behavior of the system using amplitude frequency curves, attractive watersheds, and time trajectory plots, including both negative and positive values of the nonlinear stiffness coefficient. The analysis shows that if the coefficient and order of the fractional damping term are reasonably selected, the horizontal and vertical displacements of the system vibration can be effectively controlled. The importance of this work lies in its application in rotor vibration, which has significant implications for the study of the system under investigation.

Oscillatory systems with two degrees of freedom and van der Pol coupling: Analytical approach

In this paper steady‐state vibrations of the two‐degrees‐of‐freedom oscillatory systems with van der Pol coupling are investigated. The model is a system of two differential equations with weak nonlinearity. A new solving procedure based on D′Alembert's method and the method of time‐variable amplitude and phase is developed. The main advantage of the method in comparison to others is that it gives the solution of the system of two coupled weak nonlinear equations in the form that is simple to analyze, as it has the same form as the solution of the corresponding system of linear equations. In the paper two types of systems are considered: one, a two‐mass system with two degrees of freedom, and second, the one‐mass system with two degrees of freedom. The torsional vibrations of a two‐mass system and vibrations of a Jeffcott rotor with two‐degrees‐of‐freedom are analyzed. Analytically obtained results are numerically tested. It is obtained that the difference between analytic and numeric results is small and almost negligible. As the accuracy of the analytic solution is high, it is suitable for application in technics and engineering. Conclusions about steady‐state self‐sustainable oscillators, orbital, and limit cycle motions are given.

On the operational similarities of bladed rotor vibrations with casing contacts

AbstractRotor-to-stator rubbing in rotating machinery, resulting from tight clearances, introduces complex dynamics that can potentially lead to high vibrations and machine failure. Historically, the rubbing models were addressed using cylinder-to-cylinder contacts; however, recent attention has shifted towards examining blade-tip contact in turbines, which affects the systems dynamics and efficiency. This study investigates the impact of the variations in blade number on bladed rotor systems, emphasizing on the types of motion that occur as function of the operational speed in the sub-critical range. A simplified bladed rotor model has been developed, using a Jeffcott rotor with blades represented as damped elastic pendulums. The equations of motion are derived and numerical simulations are performed to explore the system’s behaviour with varying blade numbers (3, 5, 7, and 10) in order to analyse displacements, contact forces and bifurcation diagrams as function of the rotating speed. Results reveal distinct regions: periodic motion (I and III) and chaotic motion (II and IV) appear alternatively in the bifurcation diagram, with the chaotic regions occurring at specific fractions of the natural frequency and the number of blades. The study concludes that chaotic motions are associated with larger displacements and higher contact forces, and the vibrational behaviour becomes less hazardous as the number of blades increases. In addition, the appearance of periodic and chaotic motions occur in the same regions by scaling the rotating speed with the number of blades and natural frequency of the system. From an operational perspective, this dynamic investigation offers valuable insights into the severity of blade rubbing in industrial systems. It can guide the implementation of mitigation solutions to prevent worst-case failure scenarios and help to perform adjustments to either operational or design parameters.

Modelling, analysis, and identification in a cracked and unbalanced Jeffcott rotor supported on foil bearings

Modelling, analysis, and identification in a cracked and unbalanced Jeffcott rotor supported on foil bearings

A Novel Hybrid Approach to the Diagnosis of Simultaneous Imbalance and Shaft Bowing Faults in a Jeffcott Rotor-Bearing System

Ensuring optimal performance and reliability in rotor-bearing systems is crucial for industrial applications. Imbalances and shaft bowing in these systems can lead to decreased efficiency and increased vibrations. The early detection and mitigation of a rotor’s faults are essential, and model-based fault identification has gained much attention in the manufacturing industry over the years. Over the past two decades, however, the development of fault diagnosis rules with data-driven and artificial intelligence (AI) methods has become a trend, and in the foreseeable future the combination of AI with big data will become mainstream. Nevertheless, the critical role of rotating machinery in manufacturing introduces a challenge, as often insufficient fault data are available. This limitation renders the establishment of diagnostic rules using data-driven methods and AI technologies impractical. In light of these challenges, this study proposes a novel hybrid approach that combines a physical model with machine learning (ML) techniques for the diagnosis of multi-faults (imbalances and shaft bowing are demonstrated) in a Jeffcott rotor. To overcome the lack of real-world labeled fault datasets, a physics-based Jeffcott rotor model is first derived and then used to generate abundant fault datasets for ML. Subsequently, simulated data are employed for the training of an artificial neural network (ANN), enabling the network to learn from and analyze the vast array of generated data. The results prove that a well-trained feed-forward neural network (FNN) can accurately isolate and diagnose imbalance and shaft bowing faults using the simulated and real data from the Jeffcott rotor experiment. These physics-based and ML approaches prove effective particularly for multi-faults, offering new possibilities for advanced rotor system monitoring and maintenance strategies in industrial applications.

Nonlinear Dynamic Analysis for a Jeffcott Rotor Considering Varying Connection Stiffness of Bolted Joint

Bolted joints have been widely used in rotor assemblies of aircraft engines. A typical bolt joint is composed of two metal parts connected mechanically with pre-tensioned bolts. During operations, the bolting pretensions can vary with the changing contacting relationship between the parts, which makes the connection stiffness of the joint nonlinear on the macroscopic scale. The fundamental problems of the dynamical analysis for bolted joints are (1) to identify the transition of the change of the connection stiffness and properly model the overall connection stiffness and (2) to investigate the influence of such variable stiffness on the rotor vibration. In this paper, an analytical model is proposed considering the effect of bolted joints. The bolt of the joint is simplified as a two-node spring-like connection element considering different axial tension and compression stiffness, non-uniform preload of bolts and the contact interface friction described based on Coulomb friction. The dynamic equations of the rotor system are established based on the lumped mass modeling and the proposed connection element, and numerically solved through the Runge–Kutta method. The results indicate that the rotor system exhibits a variety of states, such as period-1 motion, multi-period motion, quasi-periodic motion and chaotic motion. The effects of friction coefficient and preload of the bolts on the nonlinear dynamic behaviors of the bolted rotor system are demonstrated in detail.

Physics-Informed Neural Networks for the Condition Monitoring of Rotating Shafts.

Condition monitoring of rotating shafts is essential for ensuring the reliability and optimal performance of machinery in diverse industries. In this context, as industrial systems become increasingly complex, the need for efficient data processing techniques is paramount. Deep learning has emerged as a dominant approach due to its capacity to capture intricate data patterns and relationships. However, a prevalent challenge lies in the black-box nature of many deep learning algorithms, which often operate without adhering to the underlying physical characteristics intrinsic to the studied phenomena. To address this limitation and enhance the fusion of data-driven methodologies with the fundamental physics of the system under study, this paper leverages physics-informed neural networks (PINNs). Specifically, a simple but realistic numerical case study of an extended Jeffcott rotor model, encompassing damping effects and anisotropic supports for a more comprehensive modelling, is considered. PINNs are used for the estimation of five parameters that characterize the health state of the system. These parameters encompass the radial and angular position of the static unbalance due to the disk installed on the shaft, the stiffness along the principal axes of elasticity, and the non-rotating damping coefficient. The estimation is conducted solely by exploiting the displacement signals from the centre of the disk and, to showcase the efficacy and precision provided by this novel methodology, various scenarios involving different constant rotational speeds are examined. Additionally, the impact of noisy input data is also taken into account within the analysis and the performance is compared to that of traditional optimization algorithms used for parameters estimation.

Fault Severity Estimation in Cracked Shafts by Integration of Phase Space Topology and Convolutional Neural Network

With the rapid advancement of industrial systems and the unavoidable complications and interconnectedness in systems, diagnostics of industrial machinery are achieving paramount importance. Accurate estimation of health condition of industrial machinery becomes more challenging due to the inherent nonlinearity, complexity, and uncertainty of the observations. Nonlinear dynamic analysis has proven to be a powerful tool for providing information about the health condition of a system that can be used for diagnostic applications. The current study particularly focuses on crack depth estimation using phase space analysis. Phase space provides a topological representation of the dynamics of the system and is highly informative about the health condition. The information suitable for diagnostics is employed by Convolutional Neural Networks, which are known to be powerful in extracting spatial information from maps. The proposed diagnostic method is evaluated on a Jeffcott rotor model with transverse crack in the rotating shaft to estimate the severity of the fault from the phase space topology as a case study.

Finite element analyses of rotor/bearing system using second-order journal bearings stiffness and damping coefficients

The rotor-bearing system is a crucial component of rotating machinery, such as turbines, pumps, compressors, and turbogenerators, which are widely used in various advanced engineering fields. This work presented a methodology for studying the behavior of an elastic Jeffcott rotor supported by two similar fluid film journal bearings. A finite element model using consistent matrix formulation was employed to simulate the shaft, including the external load, with four degrees of freedom per node. The small perturbation method was used to evaluate the second-order bearing coefficients of a journal bearing of finite length. These coefficients were further integrated with the finite element model to evaluate the dynamic response of the flexible rotors. Moreover, the system equations of motion were presented in dimensional form. The results of the second-order bearing coefficients analysis agreed with nonlinear analysis when the speed was less than the threshold speed, while there was a pronounced difference in second-order analysis when the speed was above the threshold speed. The behavior of the rotor-bearing system was studied using dynamic response and orbit diagrams, revealing that changes in rotational speed significantly affected the rotor’s stability.

An experimental and finite element investigation for determining critical speeds of a cracked jeffcott rotor model

Rotating machines like pumps, compressors, generators, and steam and gas turbines are widely used in Industry 4.0. The researchers have gained more attention by modeling a cracked Jeffcott rotor system through a finite element approach. The present work consists of theoretical and finite element analysis of the Jeffcott rotor model with consideration of different crack depths, crack locations, and shaft material. Finite element results and their experimental verification obtained by FFT analysis show that increasing crack depth decreases the first critical speed of the system. Also, an increasing number of cracks results in decreases in the system’s first critical speed. The presence of a crack changes the stiffness of the shaft which results in, a decreasing the natural frequency. The present study approach provides an improved platform for crack diagnostic information to the vibration condition monitoring community for using Ansys software effectively as a finite element tool for performing rotordynamic analysis in Industry 4.0.

Experimental Detection of Lubrication Characteristics in a Solid–Liquid Two-Phase Flow Supported Bearing-Rotor System Using Vibration Monitoring Techniques

This work aims to investigate the friction and vibration behaviors of a liquid–solid two-phase flow supported bearing-rotor system for the early monitoring and assessing of oil contamination resulting from externally ingested particles. Based on a Jeffcott rotor test rig, the lubricating oil in journal bearing was mixed with micron-scale silica particles of different sizes and concentrations. The vibration monitoring techniques utilizing eddy current, photoelectric, and piezoelectric sensors were conducted to detect the rotor displacement amplitude and bearing acceleration online. According to the quantitative evaluation, the vibration frequency spectrum, shaft trajectory, and the bearing surface characteristics were analyzed and classified at different conditions of particle parameters. Furthermore, the lubricating mechanism at bearing interface which possesses a strong interactive coupling effect with rotor vibration was deduced and illustrated. The results showed that although the sizes of ingested micron particles were less than the minimum oil-film thickness, they usually exhibited detrimental effects on the flow stability and fluid support. As the particle size or concentration increased, the interaction between particles and bearing surfaces led to the burrs in time-domain waveform, reverse displacements in rotor orbit, higher frequency harmonics in vibration spectrum, and the scratches on the bearing surface. The occurrence of continuous three-body friction could induce misalignment, blockage, oil starvation, and rub-impact fault. Furthermore, the detectability of particle contamination and damage microstructure of bearing surface was described for reference.

The Influence of Coordinate Systems on the Stability Analysis of Lateral–Torsional Coupled Vibration

Stability analysis of lateral–torsional coupled vibration is obligatory for rotating machinery, such as aero-engines. However, the state-of-the-art method may lead to stability misjudgment under different coordinate systems. The cause of this misjudgment has not yet been well explored. The purpose of this paper is to clarify the error source of the stability analysis in a more comprehensive manner. A vertical Jeffcott rotor model including torsion vibration is built, and the Lagrange approach is applied to establish the motion equations. The coordinate transformation matrix is used to transfer the motion equations into the rotating coordinate system, making the coefficients of the motion equation constants. The differences in the unstable speed regions in the two coordinate systems are captured. The limitations of the Floquet theory and Hill’s determinant analysis in the stability estimation of the lateral–torsional coupled vibration are explained. It is found that, for Hill’s method, increasing the number of the harmonic truncation cannot correct the misjudgment, and the matrix truncation is the fundamental error source. The above research provides more accurate theoretical support for the analysis of the lateral–torsional coupling instability of rotors.

Experimental investigation on a Jeffcott rotor with combined coupling misalignment using time-frequency analysis

Abstract Rotating machinery, such as turbo-jet engines, operate at a high rotational speed and passes through critical zones. The dynamic response of high-speed machines is critical for long-term stability and functioning. In this work, a fast and effective method for detecting coupling misalignment utilising time-frequency analysis (TFA) based on both the adaptive noise added complete ensemble empirical mode decomposition and wavelet-based denoising is presented. This novel and innovative method detect the coupling misalignment feature via the amplitude modulation aspect in the envelope analysis of the fault-containing intrinsic mode function. The Hilbert spectrum analysis provides spontaneous frequency and spectral energy in the time-frequency domain. The experiments were performed for various rotor accelerations and combined parallel and angular coupling misalignments using a laboratory test rig. The suggested approach gives excellent denoising efficiency and can improve misalignment identification accuracy. Additionally, it may be highly helpful for machinery that starts and stops often.

A data driven method applied on nonlinear vibration of flexible rotor supported on centralized squeeze film damper

This paper discusses the application of Dynamic Mode Decomposition (DMD) on vibration response of flexible rotor supported by a Centralized Squeeze Film Damper (CSFD). The mathematics behind DMD and its applications to rotor vibration data such as linearization, reconstruction has been established. Vibration data from the flexible rotor system supported on CSFD exhibits both linear and nonlinear behavior. Further, DMD has been applied on periodic and quasi-periodic vibration data of CSFD for prediction and forecasting. DMD is observed to have successfully linearized a nonlinear system with the linear Henkel Matrix. The reconstruction and forecasting of time series data from rotor has been achieved with a maximum error of 0.2% even for nonlinear vibration data. Further, White Gaussian Noise has been intermixed with CSFD data to demonstrate the noise reduction capability of DMD method for nonlinear systems. Lastly, the application of DMD algorithm on experimental data from Jeffcott rotor based test setup has been shown.

Structural design optimization of a high-speed rotor system for mechanical vibration mitigation

This work proposes a design optimization approach for a high-speed rotor system, commonly used in a wide range of industrial applications, in order to address the issue with the flexural flexibility of a rotor system which can cause significant vibration and noise problems due to shaft whirl. For this purpose, the numerical modeling of a flexible rotor system was used to carry out the optimization process considering the effects of the mass and transverse natural frequency to the optimized design. Torque and geometrical sizing were also incorporated as the constraints in the optimization. The rotor system was modeled as a multi-disc Jeffcott rotor system that allows a rapid design optimization process. A case study on a high-speed rotor was carried out to demonstrate the effectiveness of the developed optimization approach, yielding the reduction of the rotor mass by at least 45.6%, while the fundamental transverse natural frequency has been increased by 2.91% as the result.

Investigation of analytical solution of super harmonic resonance of rotor system in permanent magnet synchronous motors considering mixed eccentricity

The unbalanced magnetic pull results in the nonlinearity of the system for the permanent magnet synchronous motor rotor having mixed eccentricity. The mass unbalance and static eccentricity form multi-frequency excitation, which results in the coupling effect of forward and backward whirling motions on the steady-state of super harmonic resonance. The analysis is difficult due to the phases of complex amplitudes of two motions which is no longer time invariant compared with the main resonance. To solve this problem, an analysis method of complex amplitudes is presented in this paper. The rotor is considered a Jeffcott rotor, and multi-scale method is applied to this system. A mathematical analysis method of the complex amplitude phases is employed for their time variant characteristics. The results show that both phases are linear time variant and their variation rates are both equal in magnitude to the difference between twice the excitation frequency and natural frequency of the generating system. The characteristics lead to the frequency shift of the two motions by the difference value from the natural frequency of the generating system. The numerical analysis reveals that the phases are independent of the initial condition and hence the complex amplitudes. The analysis solves two problems: the analytical solution of the rotor system carrying phases information can be expressed, which is in good agreement with the numerical result, and the solution construction is analyzed easily, which indicates the contribution of every component of the analytical solution. The phase characteristics analysis is the prerequisite for the frequency characteristics which can be easily carried out, and the coupling mechanism of the two motions is explained.

Rotordynamic analysis of a complex high-speed rotor

High-speed turbomachinery such as cryogenic turboexpander, turbocharger, turbo-compressor, and dental drill machines are prone to rigorous vibration and instability. The rotors of such high-speed turbomachinery cannot be analyzed by the fundamental Jeffcott rotor model. This is because, under the conditions of high-speed operation, the flexible rotor experiences many factors such as gyroscopic effect, shear deformation, axial torque, internal friction and viscous damping. These factors can influence catastrophic subsynchronous whirl in the system. Turbomachinery rotors have complex geometries and involve discs, turbine, compressor and bearings. In addition, complex turbomachinery rotors operate in high-speed conditions where tangential forces come in to picture, which further influences the behavior of a rotor and tends to destabilize the turbomachinery. These instabilities can be addressed by the design of a suitable rotor and its supporting bearings. In the present work, a complex turboexpander rotor is designed, which is supported on gas foil bearings, and its modal behavior is studied under high-speed environment. The rotor is small-sized and incorporates a disc, a turbine, a compressor and a pair of gas foil bearings with support structures. Finite element modeling is carried out, and the bearing dynamic coefficients are considered while modeling. The dynamics of the rotor is studied by estimating critical speeds, mode shapes and unbalance response.

Mass-eccentricity nonlinear evolution mechanism of combined rotor in fretting slip process

This paper studied the mass-eccentricity nonlinear evolution in fretting slip process. Firstly, a mechanical analysis of combined Jeffcott rotor under rotating condition is carried out, and the mass-eccentricity evolution laws in sticking, partial sliding, and fully sliding stages were preliminarily obtained. Subsequently, based on the measured preload values of the combine rotor, the nonlinear evolution process of mass-eccentricity was reproduced by numerical simulation. Finally, the effects of the preload distribution and rotation speed on mass-eccentricity evolution are systematically discussed. The results show that: the initial mass-eccentricity is closely related to the preload distribution, and the accuracy of mass-eccentricity evolution model (in fully sliding stage) is: R-square= 0.997. The research content provides strong support for the study of connection structure dynamics.

On the rub-impact force, bifurcations analysis, and vibrations control of a nonlinear rotor system controlled by magnetic actuator integrated with PIRC-control algorithm

This article presents the Proportional Integral Resonant Controller (PIRC-controller) as a novel control strategy to suppress the lateral vibrations and eliminate nonlinear bifurcation characteristics of a vertically supported rotor system. The proposed control algorithm is incorporated into the rotor system via an eight-pole electromagnetic actuator. The control strategy is designed such that the control law (PIRC-controller) is employed to generate eight different control currents depending on the air-gap size between the rotor and the electromagnetic poles. Then, the generated electrical currents are utilized to energize the magnetic actuator to apply controllable electromagnetic attractive forces to suppress the undesired lateral vibrations of the considered rotor system. According to the suggested control strategy, the whole system can be represented as a mathematical model using classical mechanics' principle and electromagnetic theory, in which, the rub-impact force between the rotor and the stator is included in the derived model. Then, the obtained discrete dynamical model is analyzed using perturbation techniques and validated numerically through bifurcation diagrams, frequency spectrums, Poincare maps, time responses, and steady-state whirling orbit. The obtained results illustrate that the proposed control algorithm can mitigate the nonlinear vibration and eliminate the catastrophic bifurcations of the rotor system when the control gains are designed optimally. In addition, the system dynamics are analyzed when the rub-impact occurrence between the rotor and the pole housing is unavoidable. The acquired results revealed that the system may perform periodic-1, periodic-n, or quasiperiodic motion with one of two oscillation modes depending on both the impact stiffness coefficient and the dynamic friction coefficient.Article HighlightsNonlinearity dominates the uncontrolled rotor response, where it suffers from the jump phenomenon and multiple solutions.The proposed controller forces the Jeffcott rotor to respond as a linear system with small oscillation amplitudes.The rotor oscillates with full-annular-rub or partial-rub-impact mode when rub-impact occurs between the rotor and stator.