Jeffcott Rotor System Research Articles

Bifurcation Analysis of a Transversely Cracked Nonlinear Jeffcott Rotor System at Different Resonance Cases

This work focuses on the dynamical behaviour and bifurcations of a vertically supported Jeffcott rotor system having a transverse crack and nonlinear stiffness characteristics at the primary, sub-harmonic, and super-harmonic resonance cases. The nonlinear restoring force due to the bearing-clearance, the crack breathing, the disc eccentricity, and the orientation angle between the crack and imbalance direction are considered in the system model. The equations governing the system motion are derived and solved analytically by applying the Multiple Scales Perturbation Technique (MSPT). The slow-flow modulating equations are obtained and the spinning speed response curve is plotted. The whirling orbit and amplitude spectrum are constructed in the three considered resonance cases. The acquired results provide a better understanding of the main reasons of the super- and sub-harmonic resonance excitations. In additions, we concluded that the suitable resonance case that can be used for early detections of the cracks in the rotating shafts is the sub-harmonic resonance case. Finally, the obtained results are confirmed numerically and compared with the work published in the literature

On vibration behavior and motion bifurcation of a nonlinear asymmetric rotating shaft

This article investigates the nonlinear vibrations of asymmetric vertically supported Jeffcott rotor system. Asymmetry in both linear and nonlinear stiffness coefficients of the rotating shaft is considered. The disk eccentricity and its orientation angle are included in the system model. Asymptotic analysis is sought to obtain an analytical approximate solution for the considered system model in the primary resonance case. Bifurcation diagrams for the different system parameters are obtained to explore the system steady-state lateral vibrations. The main acquired results revealed that (1) the symmetric system can oscillate by one of three stable forward whirling amplitudes at the same rotational speed depending on the initial position of the rotating disk. (2) Asymmetry in the linear stiffness coefficient does not affect the symmetry of the whirling motion, but it may change the system natural frequency. (3) Asymmetry in the nonlinear stiffness coefficient is responsible for both asymmetrical and backward whirling motions. All obtained analytical results have been verified via solving the system original equations numerically, where the analytical and numerical results are in excellent agreement.

Nonlinear dynamic behavior and stability of a rotor/seal system with the dynamic vibration absorber

The dynamic vibration absorbers have been applied to attenuate the critical or unbalanced vibration but may create the fluid-induced vibration instability in the rotor/seal system. The major purpose of this study is devoted to the effects of the dynamic vibration absorber on the nonlinear dynamic behavior and stability of the fluid-induced vibration in the rotor/seal system. The dynamic vibration absorber is attached on the shaft in the perpendicular directions. The model of the rotor/seal-dynamic vibration absorber system is established as the modified Jeffcott rotor system, and Muszynska nonlinear seal force is applied. The numerical method is used for the dynamic behavior analysis. The effects of the natural frequency and damping ratio of the dynamic vibration absorber on the dynamic behavior are discussed. The stability of the rotor/seal-dynamic vibration absorber system is judged by the eigenvalue theory. The variations of the instability threshold with the parameters of the dynamic vibration absorber are obtained. The results show that the instability threshold and instability vibration frequency are changed by the dynamic vibration absorber. The parameters of the dynamic vibration absorber must be selected carefully to avoid reducing the instability threshold and causing the instability vibration to occur in advance when the dynamic vibration absorber is applied to attenuate the critical or unbalanced vibration of the rotor/seal system.

Vibration mitigation of a nonlinear rotor system with linear and nonlinear vibration absorbers

Steady state dynamics of a horizontally supported Jeffcott rotor system under nonlinear restoring forces has been studied in this work. Vibration reduction of the rotor system using linear tuned mass dampers (TMD), nonlinear energy sinks (NES), and combined energy sinks (TMD-NES) was studied. In addition, the analysis on the effects of different parameters on the dynamics of the rotor system has been conducted. In this study, semi-analytical modified complex averaging technique (MCXA), numerical arc-length continuation method, and ODE integration method have been used to solve the governing equations. The obtained results showed, all three types of absorbers including TMD, NES, and combined TMD-NES sufficiently good performance in vibration reduction of primary rotor system. In a closer view, it shows that TMD-NES, TMD, and NES have respectively better performance in vibration reduction. On the other hand, combined TMD-NES, NES, and TMD give respectively broader frequency range of stability. Finally, it's concluded that in spite of good efficiency in vibration reduction for the design condition, all types of absorbers should be modified to be more robust against changes in parameters of primary system.

Nonlinear dynamic analysis using the complex nonlinear modes for a rotor system with an additional constraint due to rub-impact

Abstract A rub-impact can bring the impact load and friction force onto the rotor, and in the meantime, it can also produce the additional effect of changing the stiffness of the rotor system. Past analysis and numerical simulation work related to rub-impact problems primarily focus on the complicated nonlinear vibration responses caused by the rubbing force. However, the influence of the additional effect on the rotor’s modal characteristics is rarely studied. The present study investigates the modal characteristic of the rotor system with the additional constraint and how it affects the rotor’s responses. The governing equation for a modified Jeffcott rotor system with a rub-impact is established first. Next, the complex nonlinear modes concept is introduced, and the corresponding solution method is derived. Finally, the modal characteristics are analyzed in detail, such as the modal frequency, the modal damping, the stability and the interaction between the nonlinear mode motion and rotor’s responses. The results show that the rubbing rotor system possesses both forward whirl mode motion and backward whirl mode motion. The magnitudes of the modal frequencies for both the forward whirl and backward whirl increase with an increase in the amplitude of the mode motion (modal amplitude). Nevertheless, the magnitudes are limited to an interval range, which can be approximately determined through the linear rotor without rub-impact and the coupled linear rotor/stator system. Differing from the forward whirl mode motion, the backward whirl mode motion can be unstable, since its modal damping may be less than zero in certain cases. Moreover, the instability of the backward whirl mode motion is only the primary physical mechanism for the partial rub transmitting into dry whip.

Bifurcations of periodic motion of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft

This article investigates the periodic motion bifurcations of a horizontally supported nonlinear Jeffcott rotor system having transversely cracked shaft. The nonlinear spring characteristics due to Hertz contact force and bearing clearance, disc weight, disc eccentricity, breathing of the shaft crack, and angle between the crack and imbalance directions are included in the system model. A mathematical model governing the cracked system lateral vibrations is derived and then analyzed utilizing asymptotic analysis in the primary resonance case. Effects of disc eccentricity, creak depth, and angle between the crack and imbalance directions on the system response curves are studied. The analysis revealed that at a small crack depth, the system executes both forward and backward whirling motions at a specific range of the disc spinning speed, while the backward whirling orbits disappear as the crack depth increases. In addition, at zero disc eccentricity, the cracked system does not oscillate unless the system linear stiffness coefficient is reduced by about 11% as a result of shaft crack. Moreover, there is a spinning speed range of the rotating shaft at which two stable periodic solution attractors appear beside the trivial solution one when the linear stiffness coefficient of the system is reduced to 20% or more. The obtained analytical results are confirmed numerically that showed a very good agreement with the numerical ones. Finally, the acquired results are compared with the work published in the literature.

Time-delayed positive-position and velocity fefe edback controller to suppress the lateral vibrations in nonlinear Jeffcott-rotor system

Time-delayed positive-position and velocity fefe edback controller to suppress the lateral vibrations in nonlinear Jeffcott-rotor system

ENHANCING PD-CONTROLLER EFFICIENCY VIA TIMEDELAYS TO SUPPRESS NONLINEAR SYSTEM OSCILLATIONS

Within this work, the conventional PD-controller is proposed to suppress the lateral oscillations of anonlinear Jeffcott-rotor system. Time-delays in the control loop are included in the system model. The slow-flowmodulating equations governing the whole system dynamics are obtained. Influence of the controller gains andtime-delays on response curves are explored. The analyses showed that the controller efficiency has beenenhanced at specific values of time-delays; while the system may lose its stability at the other values. Thephenomenon of enhancing the controller efficiency was explained and accordingly the optimal values of the loopdelayshave been reported.

A general electromagnetic excitation model for electrical machines considering the magnetic saturation and rub impact

The electromagnetic vibration of electrical machines with an eccentric rotor has been extensively investigated. However, magnetic saturation was often neglected. Moreover, the rub impact between the rotor and stator is inevitable when the amplitude of the rotor vibration exceeds the air-gap. This paper aims to propose a general electromagnetic excitation model for electrical machines. First, a general model which takes the magnetic saturation and rub impact into consideration is proposed and validated by the finite element method and reference. The dynamic equations of a Jeffcott rotor system with electromagnetic excitation and mass imbalance are presented. Then, the effects of pole-pair number and rubbing parameters on vibration amplitude are studied and approaches restraining the amplitude are put forward. Finally, the influences of mass eccentricity, resultant magnetomotive force (MMF), stiffness coefficient, damping coefficient, contact stiffness and friction coefficient on the stability of the rotor system are investigated through the Floquet theory, respectively. The amplitude jumping phenomenon is observed in a synchronous generator for different pole-pair numbers. The changes of design parameters can alter the stability states of the rotor system and the range of parameter values forms the zone of stability, which lays helpful suggestions for the design and application of the electrical machines.

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

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

Active magnetic bearing-based tuned controller to suppress lateral vibrations of a nonlinear Jeffcott rotor system

Active magnetic bearing system is an up-to-date technology that supports rotors without physical contacts and facilitates the vibration control in rotating machinery. Within this research, a tuned positive position feedback controller is proposed to control the lateral vibrations in a Jeffcott rotor system having cubic and quadratic nonlinearities. The controller is integrated into the system via four electromagnetic poles that act as actuators. The nonlinearity due to the electromagnetic coupling is included in the system model. A second-order approximate solution to the system governing equations is sought by utilizing asymptotic analyses. Bifurcation behaviours are investigated for both the system and controller parameters. The influence of the air-gap size, bias current, disc eccentricity, feedback gain, and control gain on the vibration amplitudes has been explored. The analytical results approved that the proposed controller can reduce the vibration amplitudes close to zero at any spinning speed even at large disc eccentricity. Then, numerical confirmations for the acquired analytical results have been performed that illustrated an excellent agreement with the analytical ones. Finally, a comparison with already published articles is included.

Influences of time-delays on the performance of a controller based on the saturation phenomenon

Abstract The nonlinear vibrations control of a vertically supported Jeffcott-rotor system is investigated within this article. A modified version of the controllers based on the saturation phenomenon is connected to the system via quadratic coupling nonlinearity. Time-delays in the control loop are included in the considered model to explore their influences on the controller performance and stability chart. Asymptotic analyses are sought to obtain an approximate solution to the time-delayed differential equations simulating the whole system dynamics. Bifurcation control is conducted by means of bifurcation-diagrams. The analyses illustrated that the proposed controller can suppress the system vibrations effectively, but the system may lose its stability via Hopf bifurcations if time-delays exceed a specific value. The limits of the time-delays at which the system solution remains stable have been determined. Then, the acquired results are simulated numerically by utilizing the ODE45 and DDE23 Matlab solvers. The nature of the system motion is analyzed by constructing Poincare-map, frequency-spectrum, and whirling-orbits. The numerical results showed that the system has chaotic and quasi-periodic motions in addition to the periodic ones. Finally, the optimal control parameters are reported and a comparison with recently published articles is included.

Utilizing time-delays to quench the nonlinear vibrations of a two-degree-of-freedom system

Although, time-delays are considered an undesirable phenomenon in the active control system, this article shows how to utilize time-delays to mitigate the oscillations of a two-degree-of-freedom nonlinear model simulating a horizontally supported Jeffcott-rotor system. The multiple scales method is conducted to obtain a second-order asymptotic solution to the system governing equations. The slow-flow modulating equations of both the amplitudes and phases are extracted. The conditions that make the time-delayed controller works as a damper or exciter are clarified. The effects of the control gains and time-delays on the system stability are investigated. The analyses illustrated that both negative and positive displacement feedback control can mitigate the system vibrations to an excellent level if the time-delays are chosen within their optimal values. A method for selecting the optimal values of the time-delays is included. Then, the acquired analytical results are approved by solving the system equations numerically. The analytical and numerical results confirmed that the vibration peak has been reduced by about 80% without time-delays, and by 95% with the optimal values of the time-delays. Finally, a comparison with recently published articles concerning time-delayed feedback control is included.

Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system

Time-delay is an unavoidable phenomenon in active control systems. Measuring of the system states, processing of the measured signals, executing the control laws, conditioning and enforcing the control actions are the main reasons of time-delayed systems. This paper studies the vibration control of a horizontally suspended Jeffcott-rotor system having cubic and quadratic nonlinearities via time-delayed position-velocity controller. The intervals of the time-delays (τ1 and τ2) at which the system response is stable has been studied. The τ1−τ2 plane is constructed to illustrate the area at which the system solutions are stable. The influences of the controller gains on the stable-solutions area in τ1−τ2 plane are explored. The analysis revealed that the time-delay increases the vibration amplitudes and can destabilize the system solution in the case of negative position feedback control, while at positive position feedback control it improves the vibration suppression performance. The time-delays mechanism in stabilizing and destabilizing the dynamical systems is explained. Then, we proposed a simple and concrete method to determine the optimal value for time-delays that can improve the vibrations suppression efficiency. The acquired analytical results are confirmed numerically and the optimal working conditions of the system are concluded. Finally, a comparison with the papers that published previously is included.

Nonlinear vibration control of a horizontally supported Jeffcott-rotor system

A positive position feedback (PPF) controller is proposed to control the nonlinear vibrations of a horizontally supported Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The controller is coupled to the system with 1:1 internal resonance. A second order approximate solution to the system governing equations is constructed by applying the multiple scales perturbation technique (MSPT). The bifurcation analyses of the Jeffcott-rotor system before and after control are conducted. The effects of the different controller parameters on the system frequency–response curves are investigated. Optimum working conditions of the controlled system are extracted to be used in the design of such systems. Numerical simulations demonstrated a good agreement with the approximate results that obtained by MSPT. A comparison is provided with already published work.

비선형 진동계 정규모드의 수치적 계산 연구

Nonlinear normal modes(NNMs) is a branch of periodic solution of nonlinear dynamic systems. Determination of stable periodic solution is very important in many engineering applications since the stable periodic solution can be an attractor of such nonlinear systems. Periodic solutions of nonlinear system are usually calculated by perturbation methods and numerical methods. In this study, numerical method is used in order to calculate the NNMs. Iteration of the solution is presented by multiple shooting method and continuation of solution is presented by pseudo-arclength continuation method. The stability of the NNMs is analyzed using Floquet multipliers, and bifurcation points are calculated using indirect method. Proposed analyses are applied to two nonlinear numerical models. In the first numerical model nonlinear spring-mass system is analyzed. In the second numerical model Jeffcott rotor system which has unstable equilibria is analyzed. Numerical simulation results show that the multiple shooting method can be applied to self excited system as well as the typical nonlinear system with stable equilibria.

On the nonlinear oscillations of a horizontally supported Jeffcott rotor with a nonlinear restoring force

This paper deals with the vibration analysis of a horizontally supported Jeffcott rotor system. Both nonlinear restoring force and the rotor weight are considered in the system modeling. The model shows a small difference between the natural frequencies of the vertical and horizontal mode. The multiple scales perturbation technique is utilized to obtain a second-order approximate solution at the simultaneous resonance case. The bifurcation analyses are conducted. The stability of the obtained solution is investigated by applying Lyapunov first method. The influences of all the parameters on the system behavior are explored. The Effect of both the negative and positive values of the nonlinear stiffness coefficient is studied. At the large rotor eccentricity, the analysis revealed the following: (1) the existence of three different stable solutions in an interval of the rotational speed. (2) The disk exposed to two consecutive jumps if its speed crossed the resonant speed. (3) For a soft spring, localized and nonlocalized oscillation in both the horizontal and vertical mode occurs. (4) For a hard spring, nonlocalized oscillation occurs in the two directions in addition to the localized motion in the vertical direction only (5) The system is very sensitive to initial conditions. Then, numerical simulations are performed to confirm the accuracy of the approximate results. It is found that the predictions from the analytical solutions are in a good agreement with the numerical simulations. Finally, a comparison with previously published work is included.

Nonlinear PD-controller to suppress the nonlinear oscillations of horizontally supported Jeffcott-rotor system

This paper investigates the vibration control of a horizontally suspended Jeffcott-rotor system. A nonlinear restoring force and the rotor weight are considered in the system model. The system frequency (angular speed) -response curve is plotted at different values of the rotor eccentricity. The analysis illustrated that the system has a high oscillation amplitude and exhibits some nonlinear behaviors before control. A Proportional-Derivative (PD)-controller is integrated into the system via two pairs of electromagnetic magnetic poles. The nonlinearity due to the electromagnetic coupling is considered in the system model. A second-order approximate solution is obtained by utilizing multiple scales perturbation method. The bifurcation analyses of the controlled system are conducted. The results showed the high efficiency of the controller to mitigate the nonlinear vibrations of the considered system. Numerical simulations are carried out to validate the accuracy of the analytical results. The numerical results confirmed the excellent agreement with the analytical solutions. Then, the optimal working conditions of the system are concluded. Finally, a comparative study with previously published work is reported.

Unbalance and resonance elimination with active bearings on a Jeffcott Rotor

In this contribution we have proven theoretically and practically that active bearings are able to eliminate both bearing forces and the resonance of a Jeffcott Rotor system. Active bearings can displace a rotor such that its center of mass always stays in the rotational center. The proposed collocated controller is able to keep this state at any rotational speed, leading to an elimination of bearing forces and resonances. We analytically demonstrated that the closed-loop system is always stable, even without knowledge of the rotor's properties. The generalization of the proposed control approach for force-free operation either using displacement or force actuators enables its use for all kinds of active bearings. Moreover, the control approach allows a real time estimation of the rotor's eccentricity. The low parameter count and the unproblematic stability behavior qualify the controller for many applications.

Investigation of dynamic characteristics of a rotor system with surface coatings

A Jeffcott rotor system with surface coatings capable of describing the mechanical vibration resulting from unbalance and rub-impact is formulated in this article. A contact force model proposed recently to describe the impact force between the disc and casing with coatings is employed to do the dynamic analysis for the rotor system with rubbing fault. Due to the variation of penetration, the contact force model is correspondingly modified. Meanwhile, the Coulomb friction model is applied to simulate the friction characteristics. Then, the case study of rub-impact with surface coatings is simulated by the Runge-Kutta method, in which a linear interpolation method is adopted to predict the rubbing instant. Moreover, the dynamic characteristics of the rotor system with surface coatings are analyzed in terms of bifurcation plot, waveform, whirl orbit, Poincaré map and spectrum plot. And the effects of the hardness of surface coatings on the response are investigated as well. Finally, compared with the classical models, the modified contact force model is shown to be more suitable to solve the rub-impact of aero-engine with surface coatings.