Jeffcott Rotor Model Research Articles

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.

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.

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.

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.

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.

Interrogating the Lead-Up to a Critical Speed in Rotordynamics

Abstract This paper presents a new approach to predicting an incipient critical speed in a rotating shaft. Based on the classical governing equations of motion for an eccentric mass on a flexible shaft (the Jeffcott rotor model), the approach is centered on examining the behavior of small perturbations or random disturbances to infer the approach of a critical speed (resonance). Such disturbances, that may be based on intentional probing, or simply the result of naturally occurring fluctuations, cause small transients. It is the changing nature of these transients (as characterized by their associated eigenvalues) that is used to assess the proximity to a critical speed. In this paper, the material developed is based on analysis, but generating the data from simulations or experiments will be the next step. The approach is a kind of stress-test, conceptually not dissimilar to structural health monitoring and damage detection but here directed toward the lead-up to resonance.

Model and experimental verification of a four degrees-of-freedom rotor considering combined eccentricity and electromagnetic effects

In this paper, the combined eccentricity includes the mass eccentricity, static radial eccentricity and angular eccentricity in the rotor of a permanent magnet synchronous motor, and their electromagnetic effects are considered. Approximate analytical expressions of the unbalanced magnetic pull and the additional transverse electromagnetic moment are developed for the combined eccentricity. Moreover, a four degrees-of-freedom rotor system dynamics model is established that describes the influences of these electromagnetic forces and the moments on the rotor with combined eccentricity, when it is not mounted at the midspan of the rotor shaft. The comparisons of the degenerate cases of the four degrees-of-freedom model and Jeffcott rotor model with the radial and angle eccentricities are employed, and the simulation results are in good agreement with one another; further, a critical speed analysis is performed. For experimental validation, the operating mode of the rotor is selected carefully. The rotor of the permanent magnet synchronous motor in an electric vehicle is more compact in the axial direction and hence is considered to be a rigid rotor for which the critical speed is above the operating range of the angular velocity; however, half of the critical speed could be located in this range. Thus, super harmonic resonance occurs, which is discussed based on the combined eccentricity model, and the dynamic response is verified by experiments in which the results agree well in terms of the trajectory shape of the rotor compared with the results of the dynamics model.

An improved NOFRFs-based fault feature extraction method and its application to quantitative diagnosis in rotor rub-impact

In this paper, the Jeffcott rotor model has been set up to extract and describe the nonlinear characteristics caused by the local rub-impact. During a mechanical fault diagnosis, the detection methods frequently lead to misjudgment only based on the output behaviors. Hence, nonlinear output frequency response functions (NOFRFs) have gradually developed. Like the linear transfer function, NOFRFs can reflect the relationship between the spectrums and the nonlinear system parameters. However, in the fault diagnosis of the rotor system, the current fault feature extraction methods based on NOFRFs can describe the damage severity tend qualitatively, but diagnostic results cannot be guaranteed by only analyzing the vibration response under a single rotational speed. In the present study, a definite amplification coefficient αn is clarified to keep the same magnification ratio of the high-order NOFRFs. It achieves the purpose that describes the amplified fault characteristics quantitatively. Furthermore, a novel fault diagnosis criterion—weighted contribution ergodic index (TR(n)), is proposed to consider system responses in a wide range of harmonic excitations, so a more comprehensive system characteristic can be obtained to make the diagnosis results more reliable. Simulation results have demonstrated that the new fault diagnosis method can enhance the sensitivity and stability to identify the rub-impact fault. Simultaneously, the experiments based on the rotor bench have verified the superiority of the new method in practical application.

Negative potential energy content analysis in cracked rotors whirl response

Appearance of transverse cracks in rotor systems mainly affects their stiffness content. The stability of such systems at steady-state running is usually analyzed by using the Floquet’s theory. Accordingly, the instability zones of rotational speeds are dominated by negative stiffness content in the whirl response in the vicinity of critical rotational speeds. Consequently, an effective stiffness measure is introduced here to analyze the effect of the crack and the unbalance force vector orientation on the intensity of negative potential and stiffness content in the whirl response. The effective stiffness expression is obtained from the direct integration of the equations of motion of the considered cracked rotor system. The proposed effective stiffness measure is applied for steady-state and transient operations using the Jeffcott rotor model with open and breathing crack models. The intensity of negative potential and stiffness content in the numerical and experimental whirl responses is found to be critically depending on the propagation level of the crack and the unbalance force vector orientation. Therefore, this can be proposed as a crack detection tool in cracked rotor systems that either exhibit recurrent passage through the critical rotational speeds or steady-state running.

Effect of Coupled Torsional and Transverse Vibrations of the Marine Propulsion Shaft System

In this study, the coupled torsional–transverse vibration of a propeller shaft system owing to the misalignment caused by the shaft rotation was investigated. The proposed numerical model is based on the modified version of the Jeffcott rotor model. The equation of motion describing the harmonic vibrations of the system was obtained using the Euler–Lagrange equations for the associated energy functional. Experiments considering different rotation speeds and axial loads acting on the propulsion shaft system were performed to verify the numerical model. The effects of system parameters such as shaft length and diameter, stiffness and damping coefficients, and cross-section eccentricity were also studied. The cross-section eccentricity increased the displacement response, yet coupled vibrations were not initially observed. With the increase in the eccentricity, the interaction between two vibration modes became apparent, and the agreement between numerical predictions and experimental measurements improved. Given the results, the modified version of the Jeffcott rotor model can represent the coupled torsional–transverse vibration of propulsion shaft systems.

Natural Frequency Analysis of a Jeffcott Rotor Having Stepped Shaft

The Rotating shafts are mechanical elements used to transmit power or motion. A shaft with a step or steps is widely used instead of a shaft with a fixed (non-variable) diameter when operating at high speeds. The aim of this research is to study the effect of the step amount and its location in the shaft on the natural frequencies of the Jeffcott rotor model. Analytical methods are used to find an approximate formulation to obtain the natural frequencies of the Jeffcott rotor model neglecting the shaft mass. Lagrange equations are used to develop dynamic equations assuming elastic shaft with steps carrying a disk. The finite element method by using ANSYS is used to validate and compare the results obtained in the analytical method. The results obtained analytically and numerically were compatible and in good agreement. In addition, some parameters such as the step amount and the stepped shaft length are changed to check its effects on the natural frequencies. the results showed that the natural frequencies increase with an increase in the amount and length of the stepped part, while they decrease the closer the disc position to the center.

Natural Frequency Analysis of a Jeffcott Rotor Having Stepped Shaft

The Rotating shafts are mechanical elements used to transmit power or motion. A shaft with a step or steps is widely used instead of a shaft with a fixed (non-variable) diameter when operating at high speeds. The aim of this research is to study the effect of the step amount and its location in the shaft on the natural frequencies of the Jeffcott rotor model. Analytical methods are used to find an approximate formulation to obtain the natural frequencies of the Jeffcott rotor model neglecting the shaft mass. Lagrange equations are used to develop dynamic equations assuming elastic shaft with steps carrying a disk. The finite element method by using ANSYS is used to validate and compare the results obtained in the analytical method. The results obtained analytically and numerically were compatible and in good agreement. In addition, some parameters such as the step amount and the stepped shaft length are changed to check its effects on the natural frequencies. the results showed that the natural frequencies increase with an increase in the amount and length of the stepped part, while they decrease the closer the disc position to the center.

Control of a smart electro-magnetic actuator journal integrated bearing to a common equilibrium position: A simulation study

The integration of plain journal bearings (JBs) and active magnetic bearings (AMBs) has previously been introduced as an innovative bearing concept in rotating machinery, known as smart electro-magnetic actuator journal integrated bearings (SEMAJIBs). The integrated bearing system tends to exploit the advantages and eliminate the deficiencies of each individual bearing technology. However, the integration of the two bearing technologies introduces new design and control challenges, such as operating the two bearings at a common equilibrium position. The mismatch between the equilibrium positions of the two bearings consumes both bearing load capacities leading to a load sharing problem. A novel experimentally validated control scheme has been adopted in this study to eliminate this problem and to ensure the availability of the integrated bearing load capacities. The scheme introduces a low frequency periodic biasing that enables the controller to detect non-zero static force in the AMB by sensing rotor motion at the bias carrier frequency. Consequently, the AMB controller can modify its magnetic center to match the JB equilibrium position resulting in zero static force in the AMB. Additionally, the control scheme permits the conventional control of AMBs to overcome JBs instabilities. The effectiveness of the control scheme has been demonstrated through numerical simulations using a Jeffcott rotor model supported on a pair of SEMAJIBs, for both stable and unstable operating regimes of the rotor.

Analysis of Vibration Characteristics of Rotor Eccentricity for Generator in a Mobile Power Station

Due to machining accuracy, installation error and other factors, the rotor has various eccentricity, which will cause adverse effects on the operation of generator of certain mobile power station and even lead to all systems failure. In order to investigate the generator rotor’s vibration characteristics, taking a military mobile power station generator as the research object, a Jeffcott rotor model is established which considering rotor eccentricity with cubic term in this paper. It is consistent with the Duffing equation in any transverse motion equation. It is simplified to Duffing equation for solving. The analytical solution is solved by the Runge-Kutta method. By studying the effects of various types mechanical parameters on the transverse vibration, the rotor system’s main resonance is studied: the stiffness has extremely few effect on the amplitude change, however, the stability of the system changes greatly with them. While the amplitude only changes and the mass eccentricity reaches a certain value, the rotor system is prone to rub. Resonance amplitude, the response and phase of the system are all affected by the damping, therefore, damper is one of the most effective ways to suppress vibration of generator.

Vibration Characteristics of PMSM Grinding Motorized Spindle

This paper presents electromechanical vibration characteristics in parallel misalignment of a shaft by analyzing the vibration of PMSM grinding motorized spindle. The unbalanced magnetic pull (UMP) of the PMSM grinding motorized spindle is established as electromechanical model based on the Maxwell stress tensor, which is used to the spindle to cause the electromechanical vibration and calculated under different eccentricities through FEA (Finite Element Analysis). It increases with increasing eccentricity under no load and on load and is maximum value when mixing eccentricity. Applying a Jeffcott rotor model to analyse the dynamic characteristics of the spindle with UMP excitation and mass imbalance. The number of pole-pair is 6 and the number of slots is 36 for the PMSM grinding motorized spindle, the vibrations of 1fr, 2fr and 3fr are generated and the limit cycle is more complicated at 4500rpm. Simulated results agree well with the experimental results, and it indicates the electromechanical vibration with the shaft eccentricity and the UMP is obviously on the shaft.

Design and Experimentation of New Balancing Technique Based on the Jeffcott Rotor Model

Unbalance is one of the most common causes of machinery vibrations, present in rotating machines. The influence coefficient method of balancing used for in-situ balancing requires vibration data (Phase angles and vibration values) of many trial runs which may result in a lengthy balancing process. The vibration data can be measured with the help state-of-the-art four channel Adash 4400 VA4 Pro FFT analyzer platform along with data acquisition system and electronic instrumentation. It has been identified in the literature that it is possible to balance the rotor without the use of trial weight. This paper presents a new systematic approach of balancing method based on the Jeffcott rotor dynamic model. The mathematical model on Jeffcott rotor model is presented here. The proposed method requires vibration data of two test runs at different speeds without trial weight. For experimental verification of proposed method, a test rig has been developed. Systematic experiments done on the Jeffcott rotor model at different rotating speed are described in this paper. The experimental results shows the success of proposed balancing method. The importance of identifying type of rotor unbalance (static, couple, quasi-static, or dynamic) and the method to detect type of unbalance present in the rotor is presented.

H∞ and H2 optimization procedures for optimal design of support parameters of a flexible rotor

Rotating machinery support design with the aim of reducing the synchronous unbalance response has significant importance regarding the various applications of these machineries. In this paper, we present H∞ and H2 methods for calculating the optimum support flexibility and damping of flexible rotors to minimize vibrational amplitudes in the vicinity of the rotor first critical speed. First, the governing equations for the Jeffcott rotor model mounted on flexible supports are derived and the optimal parameters for the supports have been analytically achieved using H∞ and H2 optimization procedures. The approach method of the tuned damper support system is similar to that designed for dynamic vibration absorber optimization. The H∞ optimum parameters such as tuning frequency and damping ratios are expressed based on fixed-point theory to minimize the rotor amplitudes, as well as, the H2 optimum parameters to minimize the mean square motion of flexible rotor as analytical formulas. It is proven by numerical simulations that the system optimization design can effectively improve the synchronous unbalance response. Comparison of two optimization procedure showed the vibrational amplitudes with H2 optimization was smaller than H∞ one.

Comparative study of cracked rotor responses for simply supported end conditions and fluid film bearings

In this paper, a comparative study of cracked rotor responses for simply supported end conditions and fluid film bearings is presented. A 2 degree of freedom (dof) Jeffcott rotor model is used. A cracked rotor is represented by switching crack model. Fluid film bearings is modelled by short bearing approach given by Ocvirk theory. Stiffness damping coefficients of the bearings are incorporated in the rotor equation of motion. The equation of motion is solved by using RungeKutta numerical integration method. Cracked rotor responses are analysed by using Fast Fourier Transform (FFT) and orbit plots. It is found that for simply supported end conditions, cracked rotor shows 1X,2X,3X,4X and 5X frequency components to be significant in horizontal direction. For fluid film bearings, cracked rotor shows 1X,2X,3X,5X and 7X frequency components to be significant in horizontal direction

Rub impact response analysis of Jeffcott rotor for simply supported end conditions and fluid film bearings: a comparative study

In this paper, a comparative study on rub-impact responses of rotor for simply supported end conditions and fluid film bearings is presented. A 2 degree of freedom (dof) Jeffcott rotor model is used. Fluid film bearing is modelled using short bearing approach given by Ocvirk theory. Stiffness damping coefficients of the bearings are incorporated in the rotor equation of motion. The equation of motion is solved by using Runge-Kutta numerical integration method. Rub-impact responses are analysed by using Fast Fourier Transform (FFT) and orbit plots. It is found that with compare to simply supported end conditions 6X, 7X, 8X and 9X frequency components are absent in the Y direction response at operating speed ω = 1.3ωn for the rotor supported on fluid film bearings. In addition, 2X and 3X frequency components are present in the Z direction response at operating speed ω = 1.3ωn.