Jeffcott Rotor Research Articles

Bifurcations of Limit Cycles in Rotating Shafts Mounted On Partial Arc and Lemon Bore Journal Bearings in Elastic Pedestals

Abstract The paper investigates the bifurcations encountered in a simple rotor dynamic system interacting with nonlinear impedance forces, generated by the supporting journal bearings of realistic profile geometry. Bearing configurations of finite arc length and of finite width, as implemented in standard design of turbomachinery have been selected, namely the cylindrical partial arc and the elliptical (lemon) bore profile. The way in which the key design parameters influence the stability of elastic or rigid Jeffcott rotor is discussed. In the scope of this study, the following bearing design parameters are considered: arc length, length to diameter ratio, geometric preload and offset, and properties of the supporting pedestal by codimension-two studies. The bearing model is coupled to a six degree of freedom shaft-disc-pedestal model with nonlinear forces calculated from the journal kinematics, bearing design and operating conditions by numerical evaluation of the Reynolds equation for laminar, isothermal flow on a two dimensional mesh. An autonomous system of differential equations is implemented. Stability of fixed points and of limit cycles for this system is evaluated applying numerical continuation. The results confirm that minor variations in journal bearing design and pedestal properties have the potential to render substantial changes in the quality of stability and the bifurcation set of the rotor dynamic system. Specific bearing profiles render significant increment of instability threshold speed while at the same time supercritical Hopf bifurcations can be shifted to subcritical with resulting instability envelopes to be generated at speeds lower than the threshold speed.

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.

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.

Noise-assisted response steering for a rotor–stator system

In the present article, torsional vibrations and whirling motions of an extended Jeffcott rotor–stator system confined within a fixed stator are numerically explored for both forward and backward whirling motions. The governing equations are constructed in a generalized polar coordinate system for describing non-constant whirling speeds and impact motions. Torsional vibrations and whirling motions are examined by using the derived reduced-order equations. It is found that both forward and backward whirling motions with contact have different, but constant steady state whirling speeds. Furthermore, numerical studies have been undertaken to understand the influence of noise on the system dynamics. The trigonometric representation of non-Gaussian excitation as a harmonic excitation with a random phase modulation is modified by introducing a new state variable. The Euler–Maruyama simulation scheme is used for numerical integration. It is observed that for a low value of friction coefficient, with a sufficient level of noise introduction in the drive speed, the system dynamics for the cases of forward and backward whirling with contact can be significantly influenced. The present studies point to potential beneficial uses of noise modulated rotor drive speeds for controlling rotor–stator contact interactions.

Analysis of Ship Propulsion Shafting Vibration Using Coupled Torsional-Bending Model

The paper describes a coupled torsional-bending model of ship propulsion system vibrations. The developed model is based on the modified model of the Jeffcott rotor. In order to test the model and determine the coupled torsional-bending vibrations, several cases were analyzed. First, a reference case corresponding to a fully axisymmetric ship propulsion system is set up. Then, the influence of the constant radial force in the vertical direction at the position of the stern bearing was analyzed, such as the conditions of navigation of the ship at calm sea under partial or fully loaded hull. Finally, the case of sailing on rough sea is analyzed, when the propeller racing occurs due to the stern lifting out of the sea. For simplicity, the harmonic law of amplitude changing of the vertical radial force on the stern tube bearing as well as of the propeller load was adopted. Based on the results of numerical analysis, it was found that the proposed model well describes the case of coupling of torsional and bending vibrations of the propulsion system.

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.

Analytical Study of Nonlinear Vibration in a Rub-Impact Jeffcott Rotor

The purpose of this work is to explore the nonlinear vibration of a rub-impact Jeffcott rotor. In the first stage, the motion is not affected by the friction force, but in the second stage, the motion is influenced by the normal force and the friction force. The governing equations of the rotor of this model are derived in this paper. In consequence, there appears a difference between the two stages. We establish an approximate analytical solution for nonlinear vibrations corresponding to two stages with the mention of the location of jumps. The obtained results are compared with the numerical integration results. The steady-state response and the stability of the solutions are analytically determined for the two stages. The stability of a full annular rub solution is studied with the help of the Routh–Hurwitz criterion. Effects of different parameters of the system, the saddle-node bifurcation (turning points) and the Hopf bifurcation are presented. The main contribution lies in the analytical approximation solution based on the Optimal Auxiliary Functions Method.

Bifurcation analysis of rotor/bearing system using third-order journal bearing stiffness and damping coefficients

Hydrodynamic journal bearings are used in many applications which involve high speeds and loads. However, they are susceptible to oil whirl instability, which may cause bearing failure. In this work, a flexible Jeffcott rotor supported by two identical journal bearings is used to investigate the stability and bifurcations of rotor bearing system. Since a closed form for the finite bearing forces is not exist, nonlinear bearing stiffness and damping coefficients are used to represent the bearing forces. The bearing forces are approximated to the third order using Taylor expansion, and infinitesimal perturbation method is used to evaluate the nonlinear bearing coefficients. The mesh sensitivity on the bearing coefficients is investigated. Then, the equations of motion based on bearing coefficients are used to investigate the dynamics and stability of the rotor-bearing system. The effect of rotor stiffness ratio and applied load on the Hopf bifurcation stability and limit cycle continuation of the system are investigated. The results of this work show that evaluating the bearing forces using Taylor’s expansion up to the third-order bearing coefficients can be used to profoundly investigate the rich dynamics of rotor-bearing systems.

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.

Impact of non-synchronous whirl on post-resonance backward whirl in vertical cracked rotors

Certain heavy-duty industrial machinery uses vertical rotating shafts. Such rotating systems typically undergo transient operation phases to reach their nominal speeds; recurrent passage through resonance generally occurs during startup and coast down operations. The Non-synchronous whirl between the shaft rotational speed and its whirling in the whirl orbit in the vicinity of the resonance zones, significantly affecting the breathing mechanism of a propagating transverse crack. Therefore, the appearance of Non-synchronous whirl during the passage through resonance rotational speed activates the crack breathing mechanism. To verify this observation, Jeffcott rotor and finite element models with induced breathing cracks are considered here for geometrically symmetrical and asymmetrical vertical rotor configurations. The mathematical models associated with these systems yield linear time-varying equations of motion incorporating the angular acceleration rate. The appearance of a Non-synchronous is found activating the crack breathing mechanism, which immediately excites the post-resonance backward whirl even with small crack depths in the rotor system. Accordingly, the excitation of post-resonance backward whirl can be considered as a strong indicator of an early crack propagation phase in vertical rotors.

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.

A forced response-based method to track instability of rotating systems

The paper is focused on the stability computation of a rotating system affected by parametric excitation, by using a frequency based computational method. The test case is a Jeffcott rotor excited by the unbalance of the disk and by the parametric excitation due to varying compliance of the rolling bearing elements of the supports. The stability of a system is typically studied using Floquet's theory based on direct time integration, or less often using Harmonic Balance Method (HBM) and Hill's Method, based on computations in frequency domain. In this paper all these methods are applied and produce cross-confirming results. However, it should be noted that these methods require separate, computationally demanding analyses, which are usually not applied to systems with a large number of degrees of freedom. Standard analyses of rotating systems typically entail the computation of Frequency Response Curves using HBM. In the paper it will be shown that the standard application of HBM cannot track the presence of instability. This deficiency is especially dangerous as it may lead the designer to overlook potentially dangerous speed ranges. The paper proposes an alternative method to track the presence of instability while computing the frequency response of the system here named Jacobian Based Approach (JBA). Its results are validated against standard stability analysis techniques.

Nonlinear forced vibration analysis of the composite shaft-disk system combined the reduced-order model with the IHB method

In this paper, the internal resonance phenomena of a composite shaft-disk system with multi-degrees-of-freedom are analyzed. The force caused by the unbalanced mass of the disk is considered as an external excitation force. The shaft is simply supported. Shear deformation and gyroscopic effects are considered. The strain–displacement relationship of the shaft element is expressed using the Timoshenko beam theory. Each node has 5 degrees of freedom. SHBT (simplified homogenized beam theory) is applied to calculate the stiffness of the composite shaft. WQEM (weak form quadrature element method) is used to construct the element matrices, and the system matrices are established using the element matrix assembly rule of the FEM (finite element method). The reduced-order model is applied to reduce the calculation time. IHB (incremental harmonic balance) method is utilized to solve the nonlinear equations of motion of the composite shaft-disk system. The nonlinear vibration characteristics of the Jeffcott rotor are analyzed using the proposed method and compared with the results of previous researches, and the results are very similar. Based on these considerations, the nonlinear vibration phenomena of the composite shaft-disk system with multi-degrees-of-freedom are considered at the several resonance points.

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.

Identification of rotor-stator rub and dependence of dry whip boundary on rotor parameters

Rotor rub is one of the most critical and catastrophic event that occurs in gas turbine aircraft engines and other rotating machineries. Rotor rubbing introduces non-linearity in the system which makes the study of such phenomena difficult. The present study models a Jeffcott rotor along with a stator. This paper aims to develop a rotor rub model with Smoothening Function (SF) which can detect the onset of rotor rub efficiently whilst revealing the properties of Instantaneous Frequencies (IF) of rubbing rotor. First, the Adaptive Chirp Mode Decomposition Method (ACMD) has been used to detect the rub and establish the properties of IF. Next, the benefits of smoothening function in the rotor rub model have been discussed in terms of computation time. Further, the nonlinear dynamical behavior of the model has been studied using Poincare maps and Bifurcation diagram. The results reveal that IF of rotor possesses oscillating property which can be observed using ACMD method and can be used to detect the onset of rub. Moreover, use of SF effectively reduces the computation time by a factor of 11.6 when compared with classic rub model. The rotor orbits, Poincare Maps and Bifurcation diagram clearly shows that the rotor exhibits properties such as full annular rub, partial rub along with quasi-periodic motion. The presence of backward whirl using full spectrum has been observed and boundary for dry whip has been identified. Stability analysis using Floquet Theory reveals that the dry whip makes the rotor unstable. Dry whip boundary’s dependence on the parameters such as rotor stiffness ratio, damping ratio and friction coefficient has been discusses in detail.

Nonlinear vibration mechanism of elastic beam in large overall rotation with rotor eccentricity effect

The eccentric effect of rotor has seriously affected on the stability and precision of the mechanism at high speed for light and flexible mechanism in industrial application. The analysis of nonlinear vibration mechanism must be completed to realize precision and stable control of elastic beam in a large overall rotation. Firstly, the slender flexible rod is assumed as Euler-Bernoulli beam. We apply the deformation theory of elastic beam to establish its kinematic constraint relationship. Its dynamic model is built with Galerkin method and Hamilton principle. Then the Lagrange method is used to establish the rotor dynamics model based on the Jeffcott rotor. The coupling part between the motor and the elastic beam is assumed to be rigidly connected, and the dynamic model of the motor-elastic beam system can be obtained. Lastly, we analyze the nonlinear vibration mechanism of the elastic beam with rotor eccentricity effect through applying the fourth-order Runge-Kutta method and the superposition principle of traveling wave. It provides an important theoretical basis for making an effective vibration suppression strategy for elastic beam.

Local Sensitivity Analysis of Steady-State Response of Rotors with Rub-Impact to Parameters of Rubbing Interfaces

Local sensitivity analysis, which describes the relative importance of specific design parameters to the response of systems, is crucial for investigating dominant factors in optimal design. In this paper, local sensitivity analysis of the response of rotors with rub-impact to parameters of rubbing interfaces is carried out. The steady-state motion of the rotor is evaluated by a harmonic balance method and the sensitivity coefficients for every rotation speed over the speed range are derived analytically. Two classical models, including the Duffing oscillator and the gap model, are utilized to validate the accuracy and capability of the adopted methods and high accuracy is shown. Numerical investigations of sensitivities of steady-state response of rotors to parameters of rubbing interfaces are then carried out, based on a lumped Jeffcott rotor and a finite element model respectively. Conclusions are drawn that the response of rotors subjected to rubbing problems is more sensitive to initial clearance than other parameters of the applied friction model. With increase of initial gap, the response of rotors becomes more sensitive and the range of region subjected to rub-impact forces shrinks until the separation of rotor and stator.