Jeffcott Rotor Research Articles

Identification and separation of coupled vibration sources in multi-rotor gas turbines under time-varying speed conditions

Vibration sources in multi-rotor gas turbines under time-varying speed conditions are strongly coupled in the time–frequency domain, which prevents further mining of dynamic information from measured signals. Hence, a novel scheme of vibration source identification and separation is proposed to obtain individual source information. Firstly, the order and resonance components are identified by order analysis, and precise order numbers are further determined by computed order tracking. Then, in four typical vibration source coupling scenarios, the performances of Vold-Kalman filter (VKF), several typical adaptive mode decomposition methods and derivative methods are comparatively studied using Jeffcott rotor simulation system. Finally, the optimal source separation based on VKF and variational mode decomposition/extraction (VMD/VME) is determined. Furthermore, the settings of VKF filter bandwidth are studied quantitatively, and the effects of VMD and VME penalty factors are analyzed. In conclusion, the proposed scheme effectively identifies and accurately separates vibration sources from the test bed and gas turbine datasets, which provides important evidences for vibration monitoring and control as well as vibration transmission analysis of gas turbines.

A novel method for rotor fault diagnosis based on deep transfer learning with simulated samples

For solving the problem of data without failure samples, a novel transfer unsupervised learning method called classifier constrained domain adaptation network (CCDAN) is proposed for extracting transfer characteristics from simulated samples by theory model for experimental rotor fault diagnosis. Firstly, a dynamical model of Jeffcott rotor with crack fault is established and the achieved vibration responses of the system are used as generated simulation samples. Then, the multilayer convolutional network and multiple-kernel maximum mean discrepancy (MK-MMD) are adopted to extract similar characteristics between source and target domains. Finally, two independent classifiers are designed to constrain the features which have fallen near the decision boundary in network learning, which could effectively increase the accuracy and promote the fault identifying generalization. The proposed CCDAN method is verified through two unsupervised transfer tasks of crack rotor fault datasets. The comprehensive experiment analyses conclude the proposed method can learn the transferable characteristics of rotor crack fault diagnosis and its classification accuracy is superior to the existing intelligent transfer network of fault diagnosis.

Nonlinear vibration response of a helicopter supercritical tail rotor drive shaft equipped with a dry friction damper

PurposeThe supercritical design of tail rotor drive shaft has attracted more attention in helicopter design due to its high power–weight ratio and low maintenance cost. However, there exists excessive vibration when the shaft passes through the critical frequency. Dry friction damper is the equipment applied to the drive shaft to suppress the excessive vibration. In order to figure out the damping mechanism of the dry friction damper and improve the damping efficiency, the dynamic model of the shaft/damper system is established based on the Jeffcott rotor model.Design/methodology/approachThe typical frequency response of the system is studied through bifurcation diagrams, amplitude-frequency characteristic curves and waterfall frequency response spectrum. The typical transient responses under frequency sweeps are also obtained.FindingsThe results show that the response of the system changes from periodic no-rub motion to quasi-periodic rub-impact motion, and then to synchronous full annular rub-impact, and finally, back to periodic no-rub motion. The slip of the rub-impact ring improves the stability of the system. Besides, the effects of the system parameters including critical dry friction force, rub-impact friction coefficient, initial clearance on the stability and the vibration damping capacity are studied. It is observed that the stability changes significantly varying the three parameters respectively. The vibration damping capacity is mainly affected by the critical dry friction force and the initial clearance.Originality/valuePresented results provide guidance for the design of the dry friction damper.

Stability and Analysis of Vibrations Bifurcation Based on Dynamic Modeling of a Solar Titan 130 Gas Turbine

Abstract This work is interested in studying the vibrations’ stability in gas turbines, compared to a dynamic model of the Jeffcott rotor, the parameters of the gas turbine will be estimated for the purpose of developing a mathematical model which points out the behavior of the turbine rotor in virtue of the data of actual vibration measurements. The addition of the geometric nonlinearities of the vibrations unveils a supercritical Hopf bifurcation, the unstable modes are repulsed towards a stable limit cycle of small radius. The vibration measurements and analyzes were performed on a gas turbine Solar TITAN 130 type mounted in a gas compression station in Djelfa, Algeria. The turbine was in the operating mode when the vibration parameters were read. Bifurcation monitoring bounds the specific operating ranges, extracts efficiently the dynamic behavior core of the studied turbine and consequently affords a new approach to efficiently dimension the turbine rotor.

Dynamics of a non-linear Jeffcott rotor in supercritical regime

This work is concerned with the prediction and simulation of limit cycles of a nonlinear Jeffcott rotor. Geometrical nonlinearity (related to large flexural displacements) in both stiffness and internal damping is considered. Classical instabilities due to linear internal damping occurring in the supercritical range are modified by nonlinear effects and lead to oscillating limit cycles at a radius whose value is found analytically as a function of the rotating speed. If the rotor is additionally excited by unbalance, the response spectrum also contains the excitation frequency, and the bifurcation leads to quasi-periodic limit cycles. The dynamics of this system is studied numerically via the harmonic balance method, path continuation and bifurcation tracking. In particular, an original procedure for switching to a branch of quasi-periodic solutions is proposed. To preserve the integrity of the machine post bifurcation, one then considers the possibility of constraining the transverse displacement of the shaft. Such possible rotor–stator interaction generates three-frequency quasi-periodic oscillations. Frequency response curves show that the quasi-periodic solutions created by the bouncing rotor are mainly driven by the friction coefficient.

Adaptive modeling of vibrations and structural fatigue for analyzing crack propagation in a rotating system

This study is concerned with adaptive modeling of vibrations and structural fatigue in a rotating system with a propagating crack. The system consists of a Jeffcott rotor with a transverse surface crack at mid-span. It undergoes low and high cycle loadings due to an unbalance force. The high cycle loading (HCL) and the consequent high cycle fatigue (HCF) can accelerate crack growth, structural degradation and occurrence of mechanical failure. Therefore, accurate modeling is critical for lifetime prediction and prognostics requirements. There is a significant difference between the time constants of the rotor’s dynamic behavior and that of crack growth. Hence, in most of the prior studies, it has been assumed that the crack depth is constant, which implies that the system undergoes low cycle loading (LCL). In some studies that do focus on HCL, the loading scenario is assumed to be a summation of LCL steps with constant length. This study shows that assuming constant and pre-defined length for these steps can significantly affect the accuracy of the model leading to inaccurate lifetime prediction and poor prognostics in general. Hence, in this study, a novel approach is proposed in which the LCL step length is variable and adaptive. Considering different relationships proposed for modeling the fatigue crack growth, fluctuation of stress intensity factor (SIF) is the most important factor for driving the crack growth. Hence, an index is defined based on the rate of SIF fluctuation. Critical thresholds of this index are set at rapid and slow crack growth conditions. The level of proximity to these thresholds determines each LCL step length, which is hence determined adaptively. The equations governing the rotor motion and crack growth are formulated using Newton’s second law of motion and Paris–Erdogan’s fatigue crack growth law, respectively. The equations are solved numerically using the Runge–Kutta method. It is shown that this novel adaptive approach makes the modeling process realistic and computationally efficient while providing high accuracy. The results of this study could be of significance in decision making processes that would require lifetime prediction and prognostics of crack growth.

Experimental investigation for the dynamic mechanical properties of a vertical anisotropically supported rotor

The dynamic mechanical properties of rotors with anisotropic supporting stiffness have been intensively studied in recent decades. However, the transient unbalance response and the whirl motion when passing through backward critical speeds have not been adequately performed. In addition, there is little published experimental data on the influence of the support stiffness anisotropy. This paper attempts to investigate the steady-state and transient unbalance response of the rotor with anisotropic supports. Response characteristics are summarized based on a modified Jeffcott model first. Next, A vertical Jeffcott rotor test rig with anisotropic elastic supports is established, and the stiffness property of different supports is measured. The steady-state unbalance response and the transient response when accelerating are measured accurately and investigated thoroughly. The influence of stiffness anisotropy and the sensor mounting inclination are investigated experimentally, and whirl modes are observed and explained. The results reveal that due to the stiffness anisotropy in the supports, the amplitude and the number of peaks of the steady-state unbalance response in the x and y directions are different, and backward whirl motions can be aroused both in the steady-state and transient unbalance response. The diversity of critical speeds and range of backward whirl regions reduce for the decreasing of stiffness anisotropy. Then we found that the number of peaks and the amplitude of transient unbalance response are relevant to the sensor mounting inclination. Moreover, the experimental results confirm the appearance of straight-line orbits and the alternations of the whirl modes when passing through critical speeds, and the mechanism are specified using the beat vibration theory as well.

Design of a Herringbone-Grooved Bearing for Application in an Electrically Driven Air Compressor

A turbo compressor was investigated to ensure the operational reliability of the charging of fuel cell systems. This study investigated air-lubricated herringbone bearings to support the high-speed rotating shaft. For reliable operation of the rotor bearing system, stable operation in the whole speed range (up to 120 krpm), as well as low lift-off speed, is an important issue. Some publications containing guidelines for an optimized design in terms of stability and lift-off behavior date back to the 1970s, with some simplifying assumptions (such as narrow groove theory and small eccentricity analysis). Many publications have addressed the calculations, as well as the optimization of herringbone-grooved bearings; however, general design guidelines are still missing in the view of the authors. Although the investigations related to bearings for the support of a lightweight rotor for a special compressor of a fuel cell unit, this study could also indicate favorable bearing designs for other high-speed applications. Here, the compressible Reynolds equation was solved in the whole solution domain using a conservative finite difference scheme, and the corresponding bearing characteristics were determined. In a perturbation analysis, the linearized dynamic coefficients of the herringbone bearing are calculated. To compare the suitability and performance of the various herringbone-grooved bearing designs, especially at high speed, the simple model of a Jeffcott rotor airborne with two identical herringbone-grooved journal bearings (HGJBs) was used. The geometrical parameters of the HGJBs were varied, and their effects on bearing characteristics and stability were evaluated. Recommendations concerning favorable geometrical bearing parameters for a sufficiently high stability threshold speed and reasonable low lift-off speed were the result of the parameter study.

Combination resonances of rotor systems with asymmetric residual preloads in bolted joints

For highly-loaded rotor systems of aero-engines, residual preloads in the same row of bolts are generally asymmetric after multiple operational cycles. The asymmetry in residual bolt preloads could lead to time-varying stiffness of rotors, which become parametrically excited systems. This paper investigates the combination resonances of bolted rotor systems with asymmetric preloads under multi-frequency excitation. A new breathing model induced by asymmetric bolt preloads is proposed, which takes the rotor whirling position into account, instead of using the assumption of weight dominance. Theoretical studies are conducted on a Jeffcott rotor with asymmetric bolt preloads under two-frequency excitation, whose approximate analytical solution is obtained through the Harmonic Balance Method. Results show that combination tones arise in the vibration spectra, whose frequencies are the combinations of two excitation frequencies. The amplitude of a combination tone reaches the maximum value when its frequency is in the vicinity of natural frequencies, which correspond to the combination resonance. To extend these results to practical rotors with more complex structures, the finite element modeling method of bolted joints with asymmetric preloads is presented based on the Euler-Bernoulli beam theory. Experimental validations are performed on a practical aero-engine dual rotor system with an inter-shaft bearing. Good agreement is shown in the comparison between numerical and experimental results, which indicate that combination resonances of practical dual rotor systems could occur in the vicinity of dual-rotor coupled vibration mode.

Vertical Transient Response Analysis of a Cracked Jeffcott Rotor Based on Improved Empirical Mode Decomposition

The crack-induced changes in the vertical transient response of a rotating shaft–disc system, Jeffcott rotor, are investigated for transverse crack detection. The crack is considered as a breathing crack. A novel breathing function is proposed, in which the partially open–closed crack breathing behavior is interpolated between the fully open and closed crack behaviors. The breathing crack excites superharmonic response components of the transient as well as the subharmonic components. A Hilbert–Huang transform based on an improved empirical mode decomposition algorithm is subsequently formulated to evaluate the time–frequency representation of the vertical transient response of the rotor to detect the crack. The results show that the proposed breathing function can effectively reduce the computational effort without sacrificing the accuracy of the crack breathing behavior in the presence of small cracks. It is shown that time–frequency representations based on an improved empirical mode decomposition algorithm can lead to the detection of smaller cracks compared with those based on the empirical mode decomposition algorithm.

Enhancing vibration mitigation in a Jeffcott rotor with active magnetic bearings through parametric excitation

In previous studies of linear rotary systems with active magnetic bearings, parametric excitation was introduced as an open-loop control strategy. The parametric excitation was realized by a periodic, in-phase variation of the bearing stiffness. At the difference between two of the eigenfrequencies of the system, a stabilizing effect, called anti-resonance, was found numerically and validated in experiments. In this work, preliminary results of further exploration of the parametric excitation are shared. A Jeffcott rotor with two active magnetic bearings and a disk is investigated. Using Floquet theory, a deeper insight into the dynamic behavior of the system is obtained. Aiming at a further increase of stability, a phase difference between excitation terms is introduced.

Instability threshold analysis and optimization of a rotor system considering the Alford force

Under certain circumstances, the Alford force might induce instability of the rotor system. The mechanism and influence factors of the Alford force induced instability were analyzed with the Jeffcott rotor. Then a method for calculating the instability threshold was proposed. Finally, the optimization model of a high-speed micro-turbojet engine was established to optimize the instability threshold. The results show that the anisotropy of support stiffness and the damping ratio can both improve the threshold of instability. The instability threshold can be greatly improved by optimizing bearing stiffness and diameter of the rotor system.

Coupled bending torsional vibrations of viscoelastic rotors with fractional damper

The behavior of a Jeffcott rotor with lateral-torsional coupling is investigated in the presence of internal and external damping and eccentricity. The governing equations are derived based on the Lagrange method. Also, the Laplace method and linearization is used to solve the governing equations for free vibrations analysis. For a rotor with unbalance, the instability occurs when the real part of eigenvalues has positive values, and at the same time, it is the intersection point between the lines of natural frequencies. The instability speed increases with increasing the external damping, yet dependent on the internal damping and unbalance. Also, it is demonstrated that the rotor critical speed is affected by the order of fractional derivatives.

Analysis of subharmonic and quasi-periodic vibrations of a Jeffcott rotor supported on a squeeze-film damper by the IHB method

In rotors supported on a squeeze-film damper (SFD) with static eccentricity, subharmonic and quasi-periodic vibrations are generated by rotor unbalance. In this paper, subharmonic and quasi-periodic vibrations in a Jeffcott rotor supported on an SFD are analyzed using the modified incremental harmonic balance (IHB) method. Oil film forces in the damper are calculated using π film theory and short bearing approximation. These forces are first calculated in the time domain and then transformed into the frequency domain by Fourier transformation. The oil film forces in the frequency domain are multiplied by a transformation matrix to match the matrices used in the classical IHB method. To apply the continuation technique, the Jacobian matrix of residuals is calculated analytically. The stability analysis of the calculated solutions is analyzed using Floquet theory. Using this theory, saddle-node, symmetry breaking, period-doubling, and secondary Hopf bifurcations are detected. The phase plane, Poincaré sections, and time history at these bifurcation points are calculated using the Runge–Kutta method.

The influence of the preload on the nonlinear dynamic performance of the rod-fastened Jeffcott rotor system

For a rod-fastened Jeffcott rotor (RFJR) system, this paper establishes a nonlinear dynamic model considering contact interface, preload, unbalanced mass, nonlinear oil film force, and gives the dynamic equation of the rotor system. Then the fourth-order Runge–Kutta method is applied to study the nonlinear dynamics and bifurcation characteristics of the RFJR system. The bifurcation diagram, time-domain vibration waveform, frequency spectrum, phase trajectory, and Poincare section are applied to investigate the influence of the preload on the nonlinear dynamic performance of the rotor system. The results show that the preload affects the instability speed of the synchronous periodic solution of the RFJR system, the rotor system’s first critical speed, and the rotor system’s vibration amplitude passing the first critical speed. Whether the preload of the rotor is saturated or not has a very different effect on the vibration characteristics of the RFJR system. Assuming the fastening force of the tie rods is uneven, it will make the combined rotor have a certain initial bending deformation. At this time, the detuning rate of the tie rod will affect the rotor system’s nonlinear dynamic performance. In addition, the phase angle between the initial bending deformation and the mass eccentricity of the rotor also has a certain impact on the nonlinear dynamic characteristics of the combined rotor. The results obtained in this paper will provide a reference for understanding the nonlinear dynamics of the combined rotor and further research.

Static and Dynamic Characteristics of Journal Bearings Under Uncertainty: A Non-Probabilistic Perspective

Abstract This paper provides a comprehensive evaluation of crucial performances of hydrodynamic journal bearings subject to inherent uncertainties from a non-probabilistic perspective. Uncertainties can arise for various reasons such as the evolving service environment, manufacturing errors and assembling imperfections. An efficient non-intrusive frame named the Chebyshev interval finite difference method is proposed to solve the interval Reynolds equations, which allows accurate predictions of variabilities of performance parameters. The uncertain geometrical properties, lubricant viscosity and external load are considered and their effects on the static and dynamic characteristics of the uncertain journal bearing are intensively discussed in different situations. New insightful results are found in various case studies and numerical validations of the proposed method are carried out. Moreover, the sensitivity analysis was carried out and the application of the proposed frame to a Jeffcott rotor was demonstrated. The interval uncertainties are found to have severe effects on the hydrodynamic journal bearings, causing potential critical influences in the corresponding rotating systems. Results and methods in the current study provide references to robust assessment and design of journal bearings, and further guide the investigations on rotor-journal bearing systems.

Dynamic characteristics analysis of a vertical Jeffcott rotor-brush seal system

ABSTRACT Turbomachinery is used widely in the energy and power industries. The configuration of the rotor system is one of the major concerns in rotating machinery. This paper describes the development of a mathematical dynamic model of the vertical Jeffcott rotor–brush seal system considering bristle interference. The effects of the main structural and operating parameters, such as the rotational speed, rotor mass, installation distance of the brush seal, and system damping, on the nonlinear dynamic behavior of the vertical system are analyzed. The bifurcation points, stability, and vibration responses of the system are investigated using axis orbits, time histories, Poincaré maps, spectrum diagrams, bifurcation diagrams, and largest Lyapunov exponents. The results indicate that the amplitude of a vertical rotor system is smaller than that of a horizontal rotor system. The amplitude decreases as the installation distance and damping level increase. Reducing the rotor mass would enhances the stability of a rotor system.

Investigation of vibration characteristics of a Jeffcott rotor system under the influence of nonlinear restoring force, hydrodynamic effect, and gyroscopic effect

Investigation of vibration characteristics of a Jeffcott rotor system under the influence of nonlinear restoring force, hydrodynamic effect, and gyroscopic effect

Approximate Analytical Solutions to Nonlinear Oscillations of Horizontally Supported Jeffcott Rotor

The present paper focuses on nonlinear oscillations of a horizontally supported Jeffcott rotor. An approximate solution to the system of governing equations having quadratic and cubic nonlinearities is obtained in two cases of practical interest: simultaneous and internal resonance. The Optimal Auxiliary Functions Method is employed in this study, and each governing differential equation is reduced to two linear differential equations using the so-called auxiliary functions involving a moderate number of convergence-control parameters. Explicit analytical solutions are obtained for the first time in the literature for the considered practical cases. Numerical validations proved the high accuracy of the proposed analytical solutions, which may be used further in the study of stability and in the design process of some highly performant devices.

Analysis of nonlinear vibration and stability of Jeffcott rotor supported on squeeze-film damper by IHB method

In this paper, a methodology to analyze the nonlinear vibration of a Jeffcott rotor supported on a squeeze-film damper (SFD) with centering springs, which is widely used in high-speed rotating machines, is proposed using the incremental harmonic balance (IHB) method. In this paper, the IHB method is modified to analyze the dynamic behavior of rotor systems supported on fluid-film bearings, such as SFDs. The processing of the mass matrix, stiffness matrix, and linear force matrix proceeds in exactly the same way as in classical IHB. The nonlinear force generated in the oil film of the SFD is calculated using the alternating frequency/time method and the transformation matrix and incorporated into the computational processing of the classical IHB method. This calculation method is first proposed in this paper. Solutions computed using the proposed method are compared with solutions computed using numerical integration. The results are very close. The stability of the calculated solutions is determined using the Floquet theory. Based on this, frequency–response curves according to the change in various parameters are constructed. The proposed method can be effectively used to analyze the nonlinear vibration characteristics of rotor systems supported on fluid-film bearings, such as SFDs.