Whirl Orbit Research Articles

Numerical and experimental analysis on the whirl orbit with inner loop in cracked rotor system

Cracks in rotor systems are generated because of internal as well as external heavy loading conditions. The external loading conditions could be attributed to thermal gradients (such as in turbomachinery) or other environmental conditions that result in cracks developments. The internal loading conditions can be attributed to cyclic fatigue loading during continuous and recurrent shaft's running which is the most common cause for cracks propagation. The presence of cracks in rotor systems induces nonlinearities into the system which affects its whirl response. Many published literature on cracked rotors have indicated that when the rotational speed is at sub-critical resonance of order 1/n (1/2, 1/3,1/4, etc.…) of the critical rotational speed, the fundamental whirl orbit takes patterns including inner loop/loops of order (n-1) or outer loop/loops of order (n+1) depending on the critical whirling direction. The present study investigates the strong impact of the magnitude of unbalance force on whirling pattern with inner loop in a cracked rotor. Numerical analysis is carried out on the mathematical model of the considered cracked rotor. The unbalance force magnitude is observed to be the main factor affecting the inner loop pattern in the whirl orbit rather than the ½ of the critical resonance speed. Accordingly, the appearance of inner loop in shaft's whirl orbit due to crack propagation in the rotor system is found here to be present before and after passing through resonance speed. This inner loop whirl orbit pattern is not restricted to be associated with ½ of the critical rotational speed as believed in plenty of literature. This kind of whirling is observed to be strongly depending on the unbalance force magnitude rather than the subcritical resonance speed. The experimental results also validate the numerical simulation findings. Therefore, the whirl orbits of the cracked rotor can be categorized into pre-resonance whirl orbit (Pr-WO), resonance whirl orbit (R-WO) and post-resonance whirl orbit (Po-WO) based on the magnitude of the unbalance force. The resultant numerical and experimental findings on the whirl orbit with inner loop can provide a robust insight into the cracked rotor whirl behavior.

Study on Rotor-Bearing System Vibration of Downhole Turbine Generator under Drill-String Excitation

Downhole turbine generators (DHTG) installed within drill-string are susceptible to internal and external excitation during the drilling process, causing significant dynamic loads on bearings, and thereby reducing the bearing’s service life. In this study, a finite element model of an unbalanced rotor-bearing system (RBS) of DHTG with multi-frequency excitations, based on the Lagrangian motion differential equation, is established. The responses of the RBS under different drill-string excitations in terms of time-domain response, whirl orbit, and spectrum are analyzed. For a constant rotor speed, lateral harmonic translational and lateral oscillation both transform the whirl orbit to quasi-periodic, while axial rotation only changes the response amplitude. Changing the duration of pulse excitation leads to different response forms. Then, the dynamic characteristics of the RBS supported by a squeeze film damper (SFD) are investigated. The results indicate that SFD effectively reduces the displacement response amplitude and bearing force near the critical speed. As the axial rotation angular velocity of the drill-string increases, the first critical speed and displacement response decrease, while the variation of lateral oscillation frequency and amplitude has limited impact on them. The established model provides a means for analyzing the dynamic characteristics of DHTG’s RBS under drill-string excitations during the design stage.

Dynamic analysis of slant cracked rotor system considering nonlinear oil film force.

In this paper, considering the combined effects of nonlinear oil film forces and cracks on the rotor-bearing system, the differential equations of motion with 4 degrees of freedom are established by Lagrangian method. Then, the Lundgren-Kutta method is used to solve them and the results of the model are compared with the experimental data. The study demonstrate that the cracked rotor-bearing system is relatively stable at subcritical speeds, mostly in the period-1 motion. But near 1/3 of the critical speed, there is an inner loop in its whirl orbit and a significant increase in the 2x frequency component. When the system speed rises to the region near 1/2 of the critical speed, though the bifurcation motion and a relatively high 2x frequency can be observed, there are no other reliable fault characteristics. The study proves that the rotor crack fault diagnosis method based on the whirl orbits is convincing for slant cracked rotors.

Analysis of the Turbulent Lubrication of a Textured Hydrodynamic Journal Bearing

In order to investigate the turbulent lubrication performance of a textured hydrodynamic journal bearing (THJB), a model of turbulent lubrication was established in this paper. The variations in the Reynolds number, oil film thickness, oil film pressure, bearing capacity, attitude angle, and side leakage flow with structural and working parameters were studied, and the axis whirl orbit was further analyzed. The results show that turbulent lubrication is suitable for the actual operating conditions of THJBs. The Reynolds number decreases with the eccentricity ratio in the pressure-bearing zone but increases with rotational speeds, whereas the variation in the maximum oil film pressure increases and the minimum oil film thickness decreases with the eccentricity ratio under various Reynolds numbers. The bearing capacity decreases with the dimple diameter, depth, oil film thickness, and clearance ratio but increases with the length/diameter ratio and dimple spacing. As the eccentricity ratio increases, the attitude angle decreases, but the side leakage flow increases. In addition, the system tends to be unstable as the rotational speed and length/diameter ratio increase, and the friction and wear on the surface are three-body friction. This work not only helps in analyzing the characteristics of a THJB under actual operating conditions but also provides support for research on the simulation of THJB’s lubrication mechanism of THJB via computational fluid dynamics.

Experimental Study of Coupled Torsional and Lateral Vibration of Vertical Rotor-to-Stator Contact in an Inviscid Fluid

Diagnosis of faults in a rotor system operating in a fluid is a complex task in the field of rotating machinery. In an ideal scenario, a forced shutdown due to rotor-stator contact failure would necessitate the replacement of the rotor or stator. However, factors such as time constraints, economic considerations, and the aging of infrastructure make it imprudent to abruptly shut down machinery that can still be safe to operate. The purpose of this paper is to present an experimental study that validates the theoretical results of the dynamic behavior and friction detection using the wavelet synchrosqueezing transformation (WSST) method for recurrent rotor-stator contacts in a fluid environment, as presented in a previous study. The investigation focused on the analysis of whirl orbits, shaft deflection, and fluctuation frequency during passage through critical speeds. The WSST method was used to decompose the dynamic responses of the rotor in the supercritical speed zone into several supercomponents. The variation of the high-frequency component was studied based on the fluctuation of the instantaneous frequency (IF) technique. Additionally, the fast Fourier transform (FFT) method, in conjunction with the WSST technique, was used to calculate the variation in the amplitude of high-order frequencies in the vibration signal spectrum. The experimental study revealed that the split in resonance caused by rubbing effects is reduced when the rotor and stator interact with an inviscid fluid. However, despite the effects of elasticity and fluid boundaries generating self-excitation at low frequencies and uneven motion due to stator clearance, the experimental results were consistent with the theoretical analysis, demonstrating the effectiveness of the contact detection method based on WSST.

Dynamic Behavior Analysis of a Rotating Shaft with an Elliptical Breathing Surface Crack

PurposeIn this paper, dynamic behavior of a rotor system with an elliptical breathing crack that simulates the real shape of the crack front is investigated.MethodsA finite element model of the cracked rotor system is developed. The crack breathing mechanism is modelled based on an improved breathing model which considers the inclination of the neutral axis of the cracked element cross-section during shaft rotation. Harmonic balance method is used to solve the equations of motion of the rotor system for steady-state response characteristics. The effect of some parameters such as crack depth, crack shape factor and the spinning speed is investigated.Results and conclusionsThe results show that the unique whirl orbits behavior during passage through the subcritical speeds serve as a key indicator of crack presence in the shaft. The effects of the crack front curvature and the breathing model are revealed. The value of shape factor affects the whirl orbit characteristics such as size of the inner or outer loops and the amount by which the orbits rotate while crossing the subcritical speeds. The presented model considering the real crack front shape may contribute towards improved modelling of cracked rotors and better interpretation of measured vibration response.

A novel method for cage whirl motion capture of high-precision bearing inspired by U-Net

To solve the problem of cage whirl motion capture and evaluation, this paper developed an efficient non-contact measurement method based on semantic segmentation technology. An encoder–decoder network whose backbone is U-Net is constructed by introducing residual learning and attention mechanism for cage motion state segmentation. A random move augmentation strategy is used to simulate the random movement of cage mass center. The network is trained with 1368 high-speed cage rotational images using the augmentation strategy. Additionally, 150 images are validation set, and 5000 images under different operating conditions are test set. A trained network is applied to the cage whirl motion capture under different operating conditions by matching the suitable parameters during the training phase. The results show that our method effectively predicts the trend of cage whirl motion, with the predicted cage whirl orbit used for the accurate analysis of cage rotational stability.

Investigation on cage motions in high speed angular contact ball bearing with clearance fit between the outer ring and housing

As one of the bearing vibration sources, cage motions were often investigated for distinguishing the bearing performance. However, when the ball bearing works under complex conditions, there may be clearance fit between outer ring and housing, cage motions are affected seriously, which is less considered in previous studies. Thus, a dynamic model for the ball bearing with clearance fit was proposed, and effects of clearance fit on the cage motions were studied. The constructed model is verified by the experiment in the literature. In the obtained results, the whirl orbit, poincare map, cage velocity deviation ratio were given for defining the whirl stability of cage motions. The cage whirl stability declined with the enlargement of clearance fit and initial contact angle. Whereas, the cage whirl stability enhanced with the increasing guiding clearance. The above results may offer some advice for the design and using of angular contact ball bearing.

Investigation on the Cage Whirl State of Cylindrical Roller Bearings under High Speed and Light Load

This paper explores the cage whirl state under different operating conditions based on the cylindrical roller bearing dynamics model. The stability and whirl characteristics of the cage whirl are evaluated by considering the rotation and force of the cage and roller. The results demonstrate that the cage whirl state is divided into three types according to the cage whirl orbit: disordered whirl, local periodic whirl and regular circular whirl. With disordered whirl, the whirl speed and whirl radius of the cage are unpredictable, and the force of the cage changes frequently. With local periodic whirl, the rotation speed, whirl speed and the force on the cage change periodically or stay constant. In the case of a regular circular whirl, the force on the cage should be considered to evaluate the stability of the cage whirl. In addition, the collision force between the roller and cage determines whether the cage whirl speed is stable or not, and the bearing slipping has no obvious effect on cage whirl stability.

Simplified Improvement of Higher Order Extended Bulk Flow Analysis Specialized for Derivation of Rotordynamic Coefficients of the Annular Plain Seal With Large Static Eccentricity

Abstract Rotordynamic (RD) fluid force of turbomachinery is known as one of the causes of the shaft vibration problem. Bulk flow analysis is the analytical method of RD fluid force, and it is widely used in design stage. Conventional bulk flow analysis has been basically carried out under the assumption of concentric circular whirl orbit. While, the actual turbomachinery is often operated with static eccentricity due to the effect of static load and so on. However, there are not many studies that have investigated the case with large static eccentricity, in particular, the case more than half the seal clearance. In this paper, the linear RD coefficient for the plain annular seal is focused on and examined for the range that static eccentricity is up to 65% of seal clearance. In this range of static eccentricity, it requires the nonlinear analysis extended to the fifth order and largely increases the number of equations to be solved. This paper proposed simplification of higher order extended bulk flow (SHEBF) analysis, and as a result, the number of analytical equations to be solved is significantly reduced. The validity of this method is confirmed both numerically and experimentally.

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.

Dynamic Response Analysis of Tilting Pad Journal Bearing Considering Fluid-Structure Interaction

The transient hydrodynamic lubrication model of tilting pad journal bearings (TPJBs) was established by the computational fluid dynamics (CFD) method and the self-developed dynamic grid program. The fluid-structure interaction between the flow field and the rotor motion, the pads rotations was realized. The feasibility of the model is proved by comparing with the experimental data. The dynamic response of TPJBs under the various unbalance, the loading modes and the rotating speeds was studied. The dynamic response of TPJBs is further analyzed through a research of the relationships among the shaft whirl orbits, transient force acting on the shaft, rotation angles of the pads and transient oil film force of the pads. With the increase of unbalance, the whirl orbits expand and whirl orbits centers rise continuously. The whirl orbits and orbit center attitude angles of TPJBs are smaller than those of fixed-pad journal bearings. Compare with the load between pads, the whirl orbits are smaller and whirl orbits centers drop slightly under the load on pads. With the increase of rotating speed, the whirl orbits expand nonlinearly, whirl orbit center rises nonlinearly. The transient force acting on the shaft, the rotation angles of the pads and the transient oil film force of the pads change periodically, and the period and frequency of these changes are the same as that of the shaft rotation. The maximum force acting on the shaft appear before the maximum shaft center position (the vertexes of the whirl orbit).

Numerical and experimental study on dynamic characteristics of tilting-pad journal bearings considering pivot stiffness in a vertical rotor system

Purpose The purpose of this paper is to study the influence of pivot stiffness on the dynamic characteristics of tilting-pad journal bearings (TPJBs) and the stability of the bearing-rotor system. Design/methodology/approach A theoretical numerical model is established, and the influences of pivot stiffness on TPJBs and a bearing-rotor system are analyzed. Then, two kinds of pivot structures with different stiffness are designed and the vibration characteristics are tested on the vertical rotor bearing test bench. Findings The pivot stiffness has an obvious effect on the dynamic characteristics of the TPJBs and the stability of the bearing-rotor system. As a result of appropriate pivot stiffness, the critical speed and the vibration amplification factor can be reduced, the logarithmic decay rate and the stability of the rotor system can be effectively increased. While the journal whirl orbit is smoother and the rubbing is obviously reduced when the bearings have flexible pivots. Originality/value The influence of pivot stiffness on TPJBs and a vertical rotor-bearing system is studied by theoretical and experimental methods.

Orbit Decomposition Method for Rotordynamic Coefficients Identification of Annular Seals

The elliptical orbit whirl model is widely used to identify the frequency-dependent rotordynamic coefficients of annular seals. The existing solution technique of an elliptical orbit whirl model is the transient computational fluid dynamics (CFD) method. Its computational time is very long. For rapid computation, this paper proposes the orbit decomposition method. The elliptical whirl orbit is decomposed into the forward and backward circular whirl orbits. Under small perturbation circumstances, the fluid-induced forces of the elliptical orbit model can be obtained by the linear superposition of the fluid-induced forces arising from the two decomposed circular orbit models. Due to that the fluid-induced forces of circular orbit, the model can be calculated with the steady CFD method, and the transient computations can be replaced with steady ones when calculating the elliptical orbit whirl model. The computational time is significantly reduced. To validate the present method, its rotordynamic results are compared with those of the transient CFD method and experimental data. Comparisons show that the present method can accurately calculate the rotordynamic coefficients. Elliptical orbit parameter analysis reveals that the present method is valid when the whirl amplitude is less than 20% of seal clearance. The effect of ellipticity on rotordynamic coefficients can be ignored.

Experimental Study on Cage Dynamic Characteristics of Angular Contact Ball Bearing in Acceleration and Deceleration Process

During acceleration and deceleration processes of a bearing, the kinetic relationship of collision and friction between the cage and the rolling element is complex, which has an important influence on the dynamic characteristics and stability of the rolling bearing. In this study, experimental investigation of the dynamic motions of a cage under various external radial loads was performed during acceleration and deceleration for an angular contact ball bearing with a constant axial load of 100 N. In a bearing test rig, cage motion in the axial and radial directions is measured via laser sensors; the bearing inner ring speed and the cage speed are measured by using eddy current and photoelectric speed sensors, respectively. The results reveal that acceleration and deceleration of the bearing significantly affect the dynamic characteristics of the cage. During the acceleration process, there is a critical speed at which the cage forms a circular whirl orbit. When the inner ring speed is lower than the critical speed, the mass center of the cage swings along the guide land but enters a circular whirl orbit once the critical speed is exceeded. Further, the greater the acceleration and radial load, the higher the critical speed. Similar results are revealed during the deceleration process. In addition, the cage slip ratio is proportional to the acceleration. Conversely, the cage slip ratio is inversely proportional to the radial load; however, the increasing radial load has minimal effect on cage slip rate when the radial load is greater than 100 N.

Investigation of the whirling motion and rub/impact occurrence in a 16-pole rotor active magnetic bearings system with constant stiffness

In this paper, the weight of the rotor is considered in a 16-pole rotor active magnetic bearing system with constant stiffness. The equations of motion are derived to show the asymmetry between the rotor’s horizontal and vertical displacements. Accordingly, the rotor may exhibit forward, backward, intermediate, or hybrid whirls. The possibility to overcome the backward whirl and to symmetrize the rotor’s motion again is discussed. Also, the rotor may rub/impact with the stator legs depending on the values of the adopted system parameters. The multiple-scales method is utilized to extract the approximate solutions of the studied model and to analyze its nonlinear dynamics and the aforementioned whirls. The discussion is enhanced by different analytical plots such as the rotor’s responses to its eccentricity $$f$$ and rotation speed $$\varOmega $$ . Numerical validation is carried out to demonstrate how these analytical plots describe precisely the nonlinear dynamical behavior of the whole system. Finally, whirl orbit maps are plotted to simulate the real-life motion of the rotor at different whirls and at rub/impact occurrence.

Effect of flight/structural parameters and operating conditions on dynamic behavior of a squeeze-film damped rotor system during diving–climbing maneuver

During aircraft maneuvering flights, engine's rotor-bearing systems are subjected to parametric excitations and additional inertial forces, which may cause severe vibration and abnormal operation. Based on Lagrange's principle combined with finite element modeling, the differential equations of motion for a squeeze film damped rotor-bearing system mounted on an aircraft in maneuvering flight are derived. Using Newmark–Hilber–Hughes–Taylor integration method, dynamic characteristics of the nonlinear rotor system under maneuvering flight are investigated. The factors are considered, involving mass unbalance, oil–film force, gravity, parametric excitations and additional inertial forces, and instantaneous static eccentricity of journal induced by maneuvering loads. The effects of forward velocity, radius of curvature, rotating speed, mass unbalance, oil–film clearance, and elastic support stiffness on transient responses of rotor system are discussed during diving–climbing maneuver. The results indicate that when the aircraft performs a diving–climbing maneuver in the vertical plane, the journal deviates from the center of oil–film outer ring, and the excursion direction of whirl orbit is determined by centrifugal acceleration and additional gyroscopic moment. The journal whirls asynchronously around the instantaneous static eccentricity and its magnitude is related to the maneuvering loads and the supporting stiffness. Increasing forward velocity or decreasing pitching radius, the rotor vibration will enter earlier into or withdraw later from the relatively large eccentricity. Rotating near critical speeds or excessive mass unbalances should be prevented during maneuvering flights. For large maneuver, the oil–film radial clearance needs to be enlarged properly to avoid hard contact between journal and outer ring. In addition, the stiffness of elastic support needs to be appropriately determined for damping performance. Overall, it provides a flexible approach with good expandability to predict dynamic characteristics of on-board squeeze-film damped rotor system during maneuvering flights in the design process.

Dynamic modeling and nonlinear analysis of a rotor system supported by squeeze film damper with variable static eccentricity under aircraft turning maneuver

A high-speed rotating rotor system mounted on an aircraft is inevitably induced by parametric excitations, inertial forces and gyroscopic moments caused by maneuvering flights. With the increase of maneuverability, this effect becomes more significant, which may lead to severe vibrations and even abnormal operation. Using Lagrange's principle, equations of motion for a squeeze film damped rotor system relative to a maneuvering aircraft are derived, considering additional inertial forces and variable static eccentricity of journal, of which the mechanism and computation procedure under turning maneuver are analyzed and proposed. The effects of flight status, structural parameters, and operating conditions on transient responses are discussed, including forward velocity, turning radius, rotating speed, mass unbalance, and elastic support stiffness. The results indicate that when an aircraft negotiates a turn, the offset direction of whirl orbit is determined by centrifugal acceleration and gyroscopic moment. The journal whirls around variable static eccentricity, of which the magnitude is related to maneuvering loads and supporting stiffness. Increasing forward velocity or decreasing turning radius, the rotor vibration will enter earlier into or withdraw later from the relatively large eccentricity. During maneuvering flight, rotating near critical speeds and excessive mass unbalances should be prevented. The elastic support stiffness has a great impact on vibration behavior of rotor system in maneuvering flights. Using finite element modeling combined with mechanism analysis, dynamic characteristics of rotor system supported by squeeze film damper (SFD) under aircraft maneuver can be predicted, which provides theoretical and technical support for dynamic design of engine's rotor system mounted on aircraft with high maneuverability and agility.

Modelling and analysis of a cracked rotor: a review of the literature and its implications

Timely detection of fatigue cracks is necessary to avoid catastrophic failure of rotating machines which can lead to economical losses and accidental risk. This paper presents an exhaustive literature survey on theoretical and experimental vibration analysis of the rotating shaft in the presence of crack. Various non-destructive methods adopted by several researchers for crack detection in rotating machinery have been discussed. The vibration-based crack detection methods such as vibration-based diagnostics methods and vibration-based signal processing techniques have been broadly categorized along with their advantages and disadvantages. In general, various methodologies such as breathing mechanism, finite-element method, Hilbert–Huang transform, artificial intelligence techniques, wavelet transform and wavelet finite-element transform have been applied to investigate the presence of crack in the rotating shaft. The parameters such as natural frequencies, $$1{\times }$$, $$2{\times }$$ and $$3{\times }$$ harmonic components of dynamic response, critical speeds and whirl orbits have been significantly influenced due to the presence of crack in the rotating shaft. Several studies have been carried out to study variations in these parameters. Still, there is a need of more reliable and accurate modelling approach to detect variations in these parameters. In this paper, an attempt has been made to deliver all the modelling approaches implemented by the various researchers to detect crack within the rotating shaft. The modelling approaches are categorized based on the methodologies adopted by the various researchers to detect crack. Moreover, the critical observations made from the proposed modelling approaches are summarized and presented.

Crack characteristic analysis of multi-fault rotor system based on whirl orbits

In this paper, the nonlinear crack characteristics of different multi-fault rotor system are analyzed for crack detection based on whirl orbits. The nonlinear crack response can bring instability of the rotating shaft. The identification of the nonlinear characteristics is very important in the crack detection. Considering the interaction of rub-impact forces, time-varying stiffness and nonlinear oil-film forces, the nonlinear dynamic models are established. The numerical simulations put emphasis on the evolution of whirl orbits by studying orbit morphological characteristics near subcritical speed for a rotor system with crack. The bifurcation diagrams, time series and frequency spectra are used to analyze the nonlinear coupled behaviors of the multi-fault rotor system. The typical features and rules of whirl orbits are found in the rotor system with crack. The morphological characteristics of whirl orbit present the number and orientation changes of inner loops during the subcritical speed region, even though multi-fault exist in the rotor system with crack. The experimental whirl orbits of a rotor system with crack during passage through the 1/2 subcritical speeds agree with the theoretical analysis. The orbit morphological characteristics will offer a method for crack detection in multi-fault rotor system.