Jeffcott Rotor System Research Articles

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.

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.

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.

Periodic response analysis of a Jeffcott-rotor system under modified saturation-based control

In this paper, two modified nonlinear saturation-based controllers and negative velocity feedback controllers are integrated to suppress the horizontal and vertical vibrations of a horizontally supported Jeffcott-rotor system at primary resonance excitation and the presence of 1:1 and 1:2 internal resonances. The second order approximations and the amplitude equations are obtained by applying the integral equation method to analyze the nonlinear behavior of this model. The stability of the steady-state solutions is determined based on the Floquet theory. The necessity of adding a negative velocity feedback to the main system is stated. The effects of different control parameters on the frequency-response curves and the force-response curves are investigated. Time histories of the whole system are included to show the response with and without control. It is shown that the main systems steady-state amplitude under the proposed control is reduced by about 99.8% and the negative velocity feedback can suppress the transient vibrations and prevent the main system having the large amplitude vibration. Finally, numerical simulations are provided to verify the effectiveness of the integral equation method.

Non-Linear Interactions of Jeffcott-Rotor System Controlled by a Radial PD-Control Algorithm and Eight-Pole Magnetic Bearings Actuator

Within this work, the radial Proportional Derivative (PD-) controller along with the eight-poles electro-magnetic actuator are introduced as a novel control strategy to suppress the lateral oscillations of a non-linear Jeffcott-rotor system. The proposed control strategy has been designed such that each pole of the magnetic actuator generates an attractive magnetic force proportional to the radial displacement and radial velocity of the rotating shaft in the direction of that pole. According to the proposed control mechanism, the mathematical model that governs the non-linear interactions between the Jeffcott system and the magnetic actuator has been established. Then, an analytical solution for the obtained non-linear dynamic model has been derived using perturbation analysis. Based on the extracted analytical solution, the motion bifurcation of the Jeffcott system has been investigated before and after control via plotting the different response curves. The obtained results illustrate that the uncontrolled Jeffcott-rotor behaves like a hard-spring duffing oscillator and responds with bi-stable periodic oscillation when the rotor angular speed is higher than the system’s natural frequency. It is alsomfound that the system, before control, can exhibit stable symmetric motion with high vibration amplitudes in both the horizontal and vertical directions, regardless of the eccentricity magnitude. In addition, the acquired results demonstrate that the introduced control technique can eliminate catastrophic bifurcation behaviors and undesired vibration of the system when the control parameters are designed properly. However, it is reported that the improper design of the controller gains may destabilize the Jeffcott system and force it to perform either chaotic or quasi-periodic motions depending on the magnitudes of both the shaft eccentricity and the control parameters. Finally, to validate the accuracy of the obtained results, numerical simulations for all response curves have been introduced which have been in excellent agreement with the analytical investigations.

Applications of the integral equation method on nonlinear multi degree of freedom vibration systems

This paper compares the accuracy of the integral equation method and the multiple scales method in solving the amplitude equation of multi degree of freedom nonlinear vibration systems. We consider three examples: a two-degree-of-freedom stainless-steel beam system controlled by a saturation controller, a three-degree-of-freedom rotating compressor blade model and a four-degree-of-freedom horizontally supported Jeffcott-rotor system controlled by a PPF controller. The amplitude equations are obtained by applying the integral equation method and the method of multiple scales. The stability analysis is achieved based on the Floquet theory together with Routh–Hurwitz criterion. Furthermore, we modified the iterative procedure of the integral equation method to make the analytic approximate solution more accurate. Finally, the analyses show that in most cases, the analytical solutions obtained by the integral equation method are more excellent agreement with the numerical solutions than the most commonly used method of multiple scales. Therefore, the integral equation method is worth popularizing to obtain the approximate analytical solutions of the multi-degree-of-freedom nonlinear vibration system.

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.

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

Vibration features of rotor unbalance and rub-impact compound fault

To meet the requirements of improving work efficiency, the rotating machinery represented by gas turbines reduces the gap between rotor and stator, making rub-impact occur frequently. However, a few researchers combined with the current gas turbine condition monitoring concluded that it could diagnose the rub-impact fault. In this study, the vibration features that can help diagnose the compound fault of rotor unbalance and rub-impact are researched. Taking the Jeffcott rotor system as the research object, the mathematical model is constructed considering the coupling of radial and torsion, and the Runge-Kutta method is used to solve the differential equation. Through the analysis of the model results, it is concluded that the compound fault can be diagnosed by the vibration features-reduction of the fundamental frequency amplitude. Finally, the correctness of the model is verified by experiments. At the same time, the vibration features of the rotor and stator are compared to show the consistency of vibration features, which shows that the proposed features can diagnose this compound fault.

Dynamic analysis of Jeffcott rotor under uncertainty based on Chebyshev convex method

Rotor system is often affected by various uncertain factors during work, which may affect the normal working process of the rotor system. In this paper, aiming at the uncertainty of the Jeffcott rotor system, a non-probabilistic convex model is constructed to describe the uncertain parameters. By introducing the non-embedded Chebyshev expansion function, the Chebyshev Convex Method (CCM) is proposed. Then, the CCM is employed to calculate the rotor response when the mass and bearing stiffness are respectively uncertain parameters. Meanwhile, by changing the mass of the rotor disc and the bearing support position on the Bently RK4 rotor rig, the Jeffcott rotor with uncertain parameters is simulated. The numerical and experimental results show that the proposed method is valid and effective for quantifying the uncertain parameters of the Jeffcott rotor system. In addition, the Jeffcott rotor response with multiple uncertain parameters is discussed in detail. The results show that compared with the parameters related to the material properties of the rotor such as mass and density, when the parameters related to the bearing support such as stiffness and damping are uncertain, it will have a greater impact on the response of the rotor system.

Periodic, Quasi-Periodic, and Chaotic Motions to Diagnose a Crack on a Horizontally Supported Nonlinear Rotor System

This work aims to diagnose the crack size of a nonlinear rotating shaft system based on the qualitative change of the system oscillatory characteristics. The considered system is modeled as a two-degree-of-freedom horizontally supported nonlinear Jeffcott rotor system. The influence of the crack size on the system whirling motion for the primary, superharmonic, and subharmonic resonance cases are investigated utilizing the bifurcation diagram, Poincaré map, frequency spectrum, and whirling orbit. The obtained numerical results revealed that the cracked system whirling motion is subjected to a continuous qualitative change as the crack size increases for the superharmonic resonance case, where the system can exhibit period-1, period-2, quasi-periodic, period-3, period-doubling, chaotic, and period-2 motions, sequentially. In addition, an asymmetry is observed in the system whirling orbit due to both the shaft weight and shaft crack. Moreover, it is found that the disk eccentricity does not affect the nature of these motions. Accordingly, we illustrated a simple method to diagnose the existence of such a crack and to quantify its size via monitoring the system lateral vibrations at the superharmonic resonance. Finally, all the obtained numerical results are concluded and a comparison with already published work is included.

Passive vibration reduction of a squeeze film damper for a rotor system with fit looseness between outer ring and housing

Clearances between bearing outer ring and sleeve can generally be maintained to provide a margin for the thermal expansion of the bearings. However, temperature variation, improper assembly and long-term vibration can enlarge the clearances and accelerate mechanical wear, leading to what is known as the fit looseness fault. Therefore, it is important to study a fit looseness fault model and investigate how to control the vibration coming from the fit looseness fault. In this paper, a Jeffcott rotor system with three disks was modeled as a single unit. A fit looseness model was applied in the whole rotor model to study the contact problems and response characteristics using a numerical integration method. Then, a squeeze film damper model was applied to assess the vibration reduction effects on the whole rotor system with the fit looseness fault. By comparing the results of the fit looseness fault without squeeze film damper and with squeeze film damper, it is found that the squeeze film damper can reduce nonlinear vibration responses effectively generated by the fit looseness fault for the nonlinear contact. This research work contributes to understanding the mechanism of fit looseness fault and controlling strong nonlinear vibration responses due to the fit clearances.

Nonlinear stochastic dynamics of a rub-impact rotor system with probabilistic uncertainties

This paper presents a stochastic model for performing the uncertainty and sensitivity analysis of a Jeffcott rotor system with fixed-point rub-impact and multiple uncertain parameters. A probabilistic nonlinear formulation is developed based on the combination of the harmonic balance method and an alternate frequency/time procedure (HB-AFT) in stochastic form. The non-intrusive generalized polynomial chaos expansion (gPCE) with unknown deterministic coefficients is employed to represent the propagation of uncertainties on rotor dynamics. In conjunction with the path continuation scheme and the Floquet theory, the developed model enables one to expediently evaluate the uncertainty bounds and probability density functions (PDFs) on periodic solution branches and associated stabilities. A global sensitivity analysis is then carried out by evaluating Sobol’s indices from gPCE to quantitatively ascertain the relative influence of different stochastic parameters on vibrational behaviors and conditions for the occurrence of rub-impact. The efficiency of the proposed algorithm for nonlinear stochastic dynamics of rub-impact rotors is validated with Monte Carlo simulation. Parametric studies are finally carried out to investigate the effects of multiple random parameters on the probabilistic variability in nonlinear responses of rub-impact rotors, which reveals the necessary to consider input uncertainties in analyses and designs to ensure the sustainable system performance.

Research on the Dynamic Characteristics of Magnetic Bearing-rotor System with Auxiliary Bearing

Non-linear factors such as magnetic force and rub-impact force are comprehensively considered on the basis of the classic Jeffcott rotor system model, and the dynamic model of the magnetic bearing-rotor system is established. The dimensionless motion equation is obtained by introducing dimensionless parameters. The pattern types and distribution regions of adjacent fundamental periodic motions of the system are obtained based on multi-objective and multi-parameter co-simulation analysis, and the dynamic behaviour is analysed through three-dimensional bifurcation diagram, two-parameter bifurcation diagram, single-parameter bifurcation diagram and axis locus diagram. The reasonable matching range of parameters of system and speed ratio is obtained in this paper, which optimizes the structural parameter design of the rotor system and determines its reasonable and stable working interval.

Thermohydrodynamic Characteristics and Stability Analysis for a Journal Hybrid Floating Ring Bearing Within Laminar and Turbulent Mixed Flow Regime

Abstract There exist laminar and turbulent flow in the inner and outer fluid film of the hybrid floating ring bearing at high speed. The finite element method (FEM) and finite difference method (FDM) are used to solve the thermohydrodynamic (THD) lubrication model within the laminar and turbulent mixed flow regime which governs the pressure, temperature, viscosity, and Reynolds number of the two-layer fluid films together for a type of the journal hybrid floating ring bearing with deep/shallow pockets. The static and dynamic characteristics are analyzed including the load capacity, the friction moment, the volume flowrate, the stiffness, and damping coefficients with consideration of the mixed flow and thermal effect based on the floating ring balance. The unitized kinematic model of the shaft and floating ring is proposed to deduce the instability criteria and calculate the threshold rotational speed of a rigid Jeffcott rotor system supported on two hybrid floating ring bearings by the Routh–Hurwitz method. The results present that there are non-uniform pressure and temperature profile within the fluid film field, and the laminar flow and turbulent flow appear at certain positions of film through theory and experiment, separately. The mixed flow effect promotes the bearing load capacity and the friction moment at high speed and eccentricity, and the system threshold speed drops within the mixed flow regime. Therefore, the thermal effect and mixed flow existing in the fluid film should be considered into the lubrication analysis for the floating ring bearing.

Improved incremental transfer matrix method for nonlinear rotor-bearing system

Rotor system supported by nonlinear bearing such as squeeze film damper (SFD) is widely used in practice owing to its wide range of damping capacity and simplicity in structure. In this paper, an improved and effective Incremental transfer matrix method (ITMM) is first presented by combining ITMM and fast Fourier transform (FFT). Afterwards this method is applied to calculate the dynamic characteristics of a Jeffcott rotor system with SFD. The convergence difficulties incurred caused by strong nonlinearities of SFD has been dealt by adopting a control factor. It is found that for the more general boundary problems where the boundary conditions are not at input and output ends of a chain system, the supplementary equation is necessarily added. Additionally, the Floquet theory is used to analyze the stability and bifurcation type of the obtained periodic solution. The semi-analytical results, including the periodic solutions of the system, the bifurcation points and their types, are in good agreement with the numerical method. Furthermore, the involution mechanism of the quasi-periodic and chaotic motions near the first-order translational mode and the second order bending mode of this system is also clarified by this method with the aid of Floquet theory.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two Neimark bifurcations (NBs) of period-1 motions, one NB of period-2 motions, and four NBs of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling (PD) bifurcations exist for the period-1, period-2, and period-4 motions. The unstable PD bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable PD bifurcation is the third order source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

On the steady-state forward and backward whirling motion of asymmetric nonlinear rotor system

Nonlinear lateral vibrations and the corresponding whirling motions of asymmetric horizontally supported rotor system are investigated within this article. The rotating shaft is modeled as a two-degree-of-freedom nonlinear Jeffcott-rotor system. The nonlinear restoring force of the rotating shaft, asymmetry in both the linear and nonlinear stiffness coefficients, the disk weight, the disk eccentricity, and the eccentricity orientation angle are included in the studied model. The asymptotic analysis is utilized to derive the autonomous amplitude-phase modulating equations that govern the system lateral vibrations in the horizontal and vertical directions. Bifurcation diagrams of both the system vibration amplitudes and the corresponding phase angles are obtained. The main acquired results revealed that the existence of the asymmetry in the rotating shaft stiffness coefficients widens the spinning speed interval at which the system may have more than one stable solution. In addition, increasing the linear asymmetrical stiffness coefficient can eliminate the system backward whirling motion. Finally, numerical confirmations for the obtained analytical results are performed that illustrated their accuracy in the prediction of the system vibration amplitudes and the whirling direction whether forward, backward, or along a straight line.

On bifurcation trees of period-1 to period-2 motions in a nonlinear Jeffcott rotor system

In this paper, periodic motions in a nonlinear rotor system are predicted through an implicit mapping method. The bifurcation trees of different period-1 to period-2 motion are presented, and there are many segments for different period-1 and period-2 motions. Harmonic frequency-amplitude characteristics of periodic motions are discussed for analysis and design of nonlinear rotor systems. Numerical simulations of periodic motions are completed for comparison of predicted and numerical results. The predictions of periodic motions in the nonlinear rotor system can provide guides for practical measurements of rotor systems, and one can have a better understanding of nonlinear rotor dynamics compared to the traditional perturbation analysis. Such studies presented in this paper just throw out some guides and ideas to predict complex motions in nonlinear rotor systems. The authors hope one can follow what presented in this paper to do further studies in nonlinear rotor dynamical systems and to improve design and control of nonlinear rotor systems.

Nonlinear Vibration Analysis of Generator Rotor of a Military General Mobile Power Plant

The generator is the core component of the military general mobile electric power plant. In general, the rotor of generator has different degrees of eccentricity which affect the operation and even cause system failure. To explore the vibration characteristics of the rotor, the rotor of generator is taken as research object. A Jeffcott rotor system with cubic terms is established and reduced to Duffing equation. The multi-scale method is used to solve the first approximate solution of the forced vibration. The amplitude-frequency, phase-frequency response equation and the resonance conditions are obtained. The influence of different mechanical parameters on the lateral vibration and the main resonance of the system are studied: eccentricity and the stiffness have little effect on the amplitude variation, but have a great influence on the stability region; the main resonance frequency and the excitation change the magnitude of the system common amplitude value; the damping not only has a great influence on the common amplitude value, but also changes the response and phase.