Whirl Vibration Research Articles

Analytical Determination of Stick–Slip Whirling Vibrations and Bifurcations in Rotating Machinery

In rotating machinery, aerodynamic forces and oil film forces often lead to cross-coupling stiffness. This paper is aimed at studying the stick-slip whirling vibrations induced by the piecewise smooth rotor/stator frictions in a modified flexible rotor subjected to cross-coupling stiffness. Governing equations determining the sliding region and boundaries of piecewise discontinuous friction are defined. This analytical study was conducted to discuss the complex vibrations and bifurcations. Various types of sliding motions (continuous pure rolling, continuous crossing, and grazing–sliding) were observed in this research. Further, as for discussing the impacts of the parameters on nonlinear sliding vibrations, a parametric study was conducted. The obtained results reveal that with an increase in the cross-coupling stiffness coefficient, continuous pure rolling occurs earlier, and the disk vibration time around the contact regime becomes shorter. For studying the self-excited backward whirling vibration of stick–slip nonlinear motions, analytical formulations are established. Detailed vibration amplitude and frequency studies of friction-induced backward whirling vibrations were carried out. Numerical simulations were performed to compare them with the analytical solutions and to validate the results as well. The proposed theory and results provide fresh perspectives for predicting friction-induced whirlings and creating proper designs for turbo machinery.

Numerical and experimental investigation on the whirling vibration characteristic of ship propulsion shafting under multi-source uncertainty

Conventional numerical models struggle to accurately capture the whirling vibration characteristics of ship propulsion shafting due to the influence of multi-source uncertainties and gyroscopic effects. Hence, this study develops nonparametric modelling to incorporate multi-source uncertainties into the dynamic equations for shafting. Based on random matrix theory, these uncertainties are represented by random matrices for mass, damping, gyroscopic moment, and stiffness. The whirling frequency under different uncertainties is analysed, revealing that stiffness significantly influences on whirling frequency, followed by mass. Conversely, the gyroscopic moment has the most negligible impact. The range of the first four-order whirling frequencies is estimated accurately using confidence intervals. Additionally, the bearing vibration response is calculated by considering gravity, unbalanced force, and misalignment force. The findings suggest that the time-domain waveform fluctuates within a defined range when considering uncertainty, and unbalance and misalignment are exacerbated, leading to poor stability. Finally, a vibration signal acquisition experiment and modal testing are conducted using a test bench. The experimental results further confirm the effectiveness of nonparametric modelling that considers multi-source uncertainties and gyroscopic effects as more accurate.

Research on the excitation force and vortex dynamics characteristics of pump-jet propulsor induced by shafting whirling vibration: Non-uniform blade tip clearance

The propulsion shafting whirling vibration causes non-uniform dynamic changes in the rotor tip clearance, which directly have a significant influence on the excitation force and vortex dynamic characteristics of the pump-jet propulsor. In the current study, based on improved delay detached eddy simulation, the influence of non-uniform blade tip clearance on the excitation force and vortex dynamics characteristics of the pump-jet propulsor is studied under design conditions. The results show that the application of propulsion shafting whirling vibration induces significant changes in the excitation force of the pump-jet propulsor. The rotor blades modulate the excitation forces of the stator blades and duct. The transverse and vertical excitation forces are more significant than the longitudinal excitation force. The magnitude change in the circular orbit shows a linear relationship with the excitation force magnitude. The characteristic frequency of the transverse and vertical excitation forces of each component is the shaft rotation frequency. In contrast, the characteristic frequency of the longitudinal excitation force is twice the shaft rotation frequency. In the elliptical orbit, the excitation force of each component is compressed or stretched in the time domain, and the dominant frequency is shifted in the frequency domain; there is no longer a linear relationship between the vibration magnitude change and the excitation force magnitude. Furthermore, an energy generation mechanism in the wake field of the pump-jet propulsor induces vortex frequency due to the whirling vibration of the propulsion shafting system.

Propeller hydrodynamic excitation influenced by shafting whirling vibration considering different nonuniform inflow conditions: A numerical study

The hydrodynamic excitation generated by a propeller rotating in nonuniform inflow causes the propulsion shafting to vibrate, and this vibration feeds back to the propeller hydrodynamics, giving rise to two-way coupling in the propeller–shafting system. Investigated here are unsteady propeller hydrodynamic excitation influenced by shafting whirling vibration considering different nonuniform inflow conditions. This is done using the delayed detached eddy simulation and a method that can simulate the two-way coupling between the propeller hydrodynamics and shafting rotor dynamics. First, the simulation method was introduced, and the method as well as the propeller hydrodynamic model are validated. Second, numerical simulations are reported considering two-way coupling with different nonuniform inflows. The results show that, the two-way coupling has an influence on the amplitude and phase of the excitation, while the frequency remains still. In terms of degree of freedom, the radial excitation amplitude increases by 31.65% after considering the two-way coupling. The coupling causes spatial and temporal unevenness of the propeller inflow velocity and angle of attack, accompanied by a larger fluctuation magnitude, both result in the variation of amplitude and phase in the hydrodynamic excitation and rotor dynamic response. Different nonuniform inflow influences the radial hydrodynamic excitation and the shafting orbit.

Influence of torsional stick-slip vibration on whirl behavior in drill string system

Torsional and lateral vibrations, which are important causes of drill string failure, are harmful vibration modes that can occur during drilling. As its form of expression, stick-slip and whirl significantly limit the drilling efficiency and the bottom hole assembly (BHA) performance. Prediction of the correlation between these vibration modes has an important significance in adjusting basic parameters to avoid the effects of stick-slip and whirl vibration in the drilling process. In this work, in order to provide a general understanding of the vibration mechanism, a coupled torsional stick-slip and whirl model with 6 degrees of freedom (6-DOF) was proposed, and numerical simulations were conducted. The Lagrange method was used to derive the model, and the stabilizer-borehole wall interaction, drill collar-borehole wall interaction, and bit-rock interaction were additionally considered. Torsional vibrations were modeled with respect to the entire drill string system, while the lateral vibration was restricted to the BHA part. The results showed a correlation between torsional and lateral vibration modes when stick-slip vibration occurs and disappears. These correlations were performed by analyzing the time domain, frequency domain, and correlation coefficients. The relationship between torsional and lateral vibration modes was verified by performing a qualitative comparison with field experiments.

Numerical study on unsteady propeller exciting characteristics considering shafting whirling vibration effect

In this paper, the unsteady exciting characteristics of a marine propeller influenced by the shafting whirling vibration effect are numerically studied by employing the dynamic sliding multi-region method and a Reynolds-averaged Navier–Stokes equation model. The propeller and shafting system inevitably vibrates during operation, where the multiple exciting forces can cause the shafting to rotate and whirl around different center lines at the same time, which is usually ignored in former studies on propeller exciting characteristics. To represent the complex whirling motion of the propeller, it is defined as a combination of revolution around its theoretical center line and rotation around its real center line. Firstly, a model for predicting the unsteady propeller exciting characteristics under different whirling motions is established, and an experiment of propeller transient forces is designed to validate the numerical model. Secondly, the influence mechanism of different whirling motions on the propeller exciting characteristics and propeller velocity field fluctuation are studied. Due to periodic fluctuation of the inflow velocity influenced by whirling, the propeller exciting fluctuates periodically. The linear relationship between propeller exciting and whirling amplitudes is found in the CFD and experimental results, which is caused by the linear increase of inflow velocity fluctuation amplitudes.

Linear stability and nonlinear rotor responses of the gas foil bearing with multiple sliding beams

This article reports the stability and dynamic characteristics of a passive gas foil bearing (MSGFB) in which multiple sliding beams provide the elastic support and Coulomb damping. The linear and nonlinear dynamic models of rotor-MSGFB system are established, both considering the complex contact constraints inside foil structures. The instability mechanism of rotor-foil bearing system is revealed from the equivalent dynamic stiffness and the critical whirl speed. The half speed whirl and the frequency locked vibrations are both observed in the simulations, indicating the nonlinearities of gas film and foil structure respectively. The results also show that larger rotor preload of MSGFB help suppress the half speed whirl vibrations and assist improving high speed rotor stability especially under the friction effect.

Morton effect prediction with validation using a CFD based CNN for pad inlet temperatures

The Morton Effect (ME) is a thermal-fluid–structure interaction instability occurring in rotating machinery supported by hydrodynamic journal bearings. The mechanism of ME consists of a bowed rotor or mass imbalance induced shaft synchronous whirl vibration in the bearing, which causes local, asymmetric heating of the journal, which causes shaft bending, potentially leading to increasing vibration. This study presents an original ME simulation approach that includes a CFD (Computational Fluid Dynamics) bearing groove model, embedded in a deep learning algorithm for computational efficiency and non-expert usage. The groove model provides a 2D oil temperature distribution at the leading edge of the bearing pads, yielding a more accurate journal axisymmetric temperature distribution which is the source of the ME. The paper provides validation of the approach by test result correlation, and illustrates the effects of parameter variation and configuration variation, by examining various oil injection types. The approach may be used for correcting ME occurring in existing machinery, or for designing machinery to avoid the ME.

Effects of Ship Propulsion Shafting Alignment on Whirling Vibration and Bearing Temperature Response

Ship’s propulsion shafting is one of the main sources of ship vibration and noise. The shafting, whirling vibrations, and alignment are important factors that affect the comfort, stability, and reliability during a ship’s navigation. However, the mechanism of the interacting of the both factors is not fully revealed. In this paper, the effect of shafting alignment on whirling vibration and the bearing temperature response is studied by experiment. The test scheme is designed reasonably according to the theoretical analysis. The results show that the horizontal component of the shafting whirling vibration can be effectively reduced by adjusting the shafting alignment state while the vertical component is not. The shafting axis balancing position (SABP) slightly moves upward in high speed, which should be considered in the dynamic alignment design of the shafting, especially for the high-speed shafting. Little ABSB (the angle between the shafting centre line and the No. 1 bearing centre line) is beneficial to the stable operation of shafting, while appropriately increasing the ABSB and bearing load is beneficial to reducing the shafting whirling vibration. By balancing the relationship between bearing load and ABSB, the performance of whirling vibration and bearing temperature response can be optimized.

Active vibration control of an elastic rotor by using its deformation as controlled variable

An elastic gyroscopic rotor system is modelled using the finite element method. The rotor is supported with two piezoelectric actuators which allow a dynamic manipulation of the rotational axis. The piezos are modelled as displacement actuators causing a base excitation. It is known from the literature that actively reducing the displacements of the rotor in the self-centring area leads to increasing bearing forces. We propose an approach which theoretically allows the use of the rotor displacements for active control without significantly increasing the bearing forces, even in the self-centring area. We use the relative displacements of the rotor for control which are proportional to the deformation of the rotor. The relative displacements are the difference between the absolute displacements, which describe the rotor centre, and a virtual active rotor axis which is a linear interpolation of the theoretically load-free actuator displacements of the piezos. A combination of integral force feedback and the least-mean-squares algorithm is chosen for control which takes full advantage of complex time signals. The analytic optimum of this control shows resonances which match with the resonances of the unsupported rotor if only the bearing forces are minimized. We refer to them as force-free resonances. While minimizing the absolute displacements in addition to the bearing forces only eliminates the first resonance, minimizing the relative displacements in addition eliminates all resonances. Using an additional weighting of the controller part prevents the force-free resonances and eliminates the difference between using the absolute and relative displacements. This is confirmed by experiments on the test-rig. However, there is no additional resonance when using the absolute displacement in addition to the bearing forces. Using the relative displacements allow for a higher reduction of the bearing forces caused by unbalance excitation in comparison to the absolute displacements when no weighting of the controller output is used. The test-rig results also show that it is not possible to eliminate the first conventional resonance peak which comprise forward and backward whirl vibrations by only controlling the bearing forces in the first bearing plane. The additional consideration of the absolute or relative displacements is able to eliminate these resonances with just a low increase of the bearing forces after the second resonance.

The Characteristics of Rotordynamic Forces generated by Mechanical Seals

Evaluating rotordynamic reaction force caused by whirl excitation is very effective technique in order to investigate the dynamic properties of the mechanical seal. Especially, it is easy to understand the stability of the rotating component by investigating rotordynamic tangential force. In this paper, the stability of the mechanical seal was discussed by measuring the rotordynamic tangential force depending on the whirl frequency. Judging from the result, the mechanical seal used for this study has stable performance that the tangential force always acts against whirling direction at 1,000 rpm of rotational speed and up to 30 Hz of whirl frequency. Furthermore, whirl frequency ratio and effective damping were calculated using cross-coupled stiffness and direct damping which were obtained by solving quaternary linear equations after whirl excitation test. Whirl frequency ratio was always bellow 1.0, and effective damping was also always positive at all frequencies. Those results indicate that the mechanical seal definitely generate the force to stabilize this whirl vibration.

Marine Propulsion Shafting: A Study of Whirling Vibrations

Whirling vibration is an important part of the calculations of the design of a marine shaft. In fact, all classification societies require a propulsion shafting whirling vibration calculation giving the range of critical speeds, i.e., free whirling vibration calculation. However, whirling vibration is a source of fatigue failure of the bracket and aft stern tube bearings, destruction of high-speed shafts with universal joints, noise, and hull vibrations. There are numerous uncertainties in the calculation of whirling vibration, namely, in the shafting system modeling and in the determination of excitement and damping forces. Moreover, whirling vibration calculation mathematics is much more complex than torsional or axial calculations. The marine propulsion shaft can be studied as a selfsustained vibration system, which can be modeled using the Van der Pol equation. In this document, a new way to solve the Van der pol equation is presented. The proposed method, based on a variational approach without local minima extra to the solution, converges for whatever initial point and parameter in the Van der Pol equation.

Calculation of the Dynamic Characteristics of Ship’s Aft Stern Tube Bearing Considering Journal Deflection

Abstract Dynamic properties are vital for the working reliability of aft stern tube bearings. However, the determination of such properties currently involves several simplifications and assumptions. To obtain its dynamic characteristics accurately, the aft stern tube bearing was divided into several bearing segments. The oil film reaction force was considered in the calculation of shaft alignment, and the journal deflection and actual oil film thickness were obtained accordingly. Subsequently, the perturbed Reynolds equation was solved using the finite difference method when the dynamic characteristics of journal bearings with finite width were evaluated. Then, a calculation program was developed and verified by comparing with the results of other studies. Finally, the dynamic characteristics were calculated under different revolutions. The results showed that the stiffness at the vertical direction of the aft stern tube bearing was several times that of the horizontal direction and varied with the revolutions of the shafting system. These findings can provide the foundation for the precise calculation of the journal trajectory under dynamic conditions, as well as for the evaluation of the oil film thickness. Moreover, the results led to favorable conditions for the accurate calculation of the shafting whirling vibration.

Simulation, Test, and Mitigation of ½× Forward Whirl Following Rotor Drop Onto Auxiliary Bearings

Abstract 1/2× forward whirl repeatedly occurred after a test rotor spinning at 5800 rpm was dropped onto ball bearing type auxiliary bearings (AB), utilized as a backup for magnetic bearings (MB). The measured contact forces that occurred between the rotor and the AB during the ½× subsynchronous vibration were about thirteen times larger than the static reaction force. The vibration frequency coincided with the rotor-support system natural frequency with the rotor at rest on the AB, an occurred at ½ of the rotor spin speed when dropped. The test rig provided measurements of rotor-bearing contact force, rotor orbit (vibrations), and rotational speed during rotor drop events. A simulation model was also developed and demonstrated that parametric excitation in the form of a Mathieu Hill model replicated the measured ½× forward whirl vibrations. The simulation model included a nonlinear, elastic-thermal coupled, ball bearing type AB model. The transient model successfully predicted the ½× vibration when the rotor was passing 5800 RPM as well, and the simulation results quantitatively agreed well with the test results in the frequency domain. Several approaches for mitigating the 1/2× forward whirl were presented such as adding an elastomer O-ring or waviness spring in the AB support system. Measurements confirmed that adding AB dampers effectively mitigated the ½ subsynchronous forward whirl and significantly reduced the contact forces.

Effects of static and imbalance loads on nonlinear response of rigid rotor supported on gas foil bearings

Abstract This paper presents an experimental and numerical investigation into the effects of static and imbalance loads on the nonlinear rotordynamic performance of a rigid rotor supported on gas foil bearings (GFBs). The interesting phenomenon that static loads can lead to high sub-synchronous motions in rotor-GFB systems is reported for the first time in this paper. The effects of static and imbalance loads on coast-down tests, synchronous response amplitudes, peak response frequency, appearance of sub-synchronous vibrations and time extent of coast-down responses are measured and discussed. A rotordynamic model based on FEM and unsteady Reynolds equation is derived to predict the nonlinear response of the system. The foil structure and gas film in-series are considered to derive the unsteady bearing force. The predictions are consistent with the test data. The relationships between static or imbalance loads and nonlinear response, especially the vibrations in sub-synchronous frequencies, are discussed. In the numerical simulations, as the static and imbalance loads increase, the vibration component changes from mainly synchronous to sub-synchronous and then back to synchronous vibrations. When the static and imbalance loads change, the natural frequency of the rotor-GFB system could be shifted into or away from the threshold of the whirl vibration with a typical factor, λ = 0.5, and lead to serious sub-synchronous vibrations. The experimental data and theoretical predictions help with the engineering design and implementation of GFBs in high-performance micro turbomachineries.

Effect of Unbalance Force Vector Orientation on the Whirl Response of Cracked Rotors

The combined effect of a crack with unbalanced force vector orientation in cracked rotor-bearing-disk systems on the values and locations of critical whirl amplitudes is numerically and experimentally investigated here for starting up operations. The time-periodic equations of motion of the cracked system are formulated according to the finite element (FE) time-varying stiffness matrix. The whirl response during the passage through the critical whirl speed zone is obtained via numerical simulation for different angles of the unbalance force vector. It is found that the variations in the angle of unbalance force vector with respect to the crack opening direction significantly alters the peak values of the critical whirl amplitudes and their corresponding critical whirl speeds. Consequently, the critical speeds of the cracked rotor are found to be either shifted to higher or lower values depending on the unbalance force vector orientation. In addition, the peak whirl amplitudes are found to exhibit significant elevation in some zones of unbalance force angles whereas significant reduction is observed in the remaining zones compared with the crack-free case. One of the important findings is that there exists a specific value of the unbalance force angle at which the critical whirl vibration is nearly eliminated in the cracked system compared with the crack-free case. These all significant numerical and experimental observations can be employed for crack damage detection in rotor systems.

Development of optimal controller design method to compensate for vibrations caused by unbalanced force in rotor system based on Nyquist diagram

In this paper, an optimal controller design method is proposed to compensate for vibrations caused by unbalanced force in the rotor system. The vibrations caused by unbalanced force are the major root cause of excessive whirling vibration in the rotor system, and it is important to compensate for the vibration to maintain its stable operation. The proposed design method can optimize a performance of the controller based on the vector locus of open loop characteristics on the Nyquist diagram. To verifiy the effectiveness, the proposed design method was employed for three typical active vibration control methods. The experimental results show that the proposed method can design the optimal parameters to compensate for the whirling vibrations of the rotor system.

Stochastic resonance energy harvesting for a rotating shaft subject to random and periodic vibrations: influence of potential function asymmetry and frequency sweep

Stochastic resonance is referred to as a physical phenomenon that is manifest in nonlinear systems whereby a weak periodic signal can be significantly amplified with the aid of inherent noise or vice versa. In this paper, stochastic resonance is considered to harvest energy from two typical vibrations in rotating shafts: random whirl vibration and periodic stick-slip vibration. Stick-slip vibrations impose a constant offset in centrifugal force and distort the potential function of the harvester, leading to potential function asymmetry. A numerical analysis based on a finite element method was conducted to investigate stochastic resonance with potential function asymmetry. Simulation results revealed that a harvester with symmetric potential function generates seven times higher power than that with asymmetric potential function. Furthermore, a frequency-sweep analysis also showed that stochastic resonance has hysteretic behavior, resulting in frequency difference between up-sweep and down-sweep excitations. An electromagnetic energy harvesting system was constructed to experimentally verify the numerical analysis. In contrast to traditional stochastic resonance harvesters, the proposed harvester uses magnetic force to compensate the offset in the centrifugal force. System identification was performed to obtain the parameters needed in the numerical analysis. With the identified parameters, the numerical simulations showed good agreement with the experiment results with around 10% error, which verified the effect of potential function asymmetry and frequency sweep excitation condition on stochastic resonance. Finally, attributed to compensating the centrifugal force offset, the proposed harvester generated nearly three times more open-circuit output voltage than its traditional counterpart.

A Finite Element Method With Full Bit-Force Modeling to Analyze Drillstring Vibration

Drillstring vibration is detrimental to drilling operations. It is crucial to understand the underlying mechanisms to circumvent these vibrations and to help improve drilling performance. This paper presents a six degrees-of-freedom (DOF) finite element method (FEM) model to characterize the drillstring dynamics. In addition, a comprehensive bit-force model is developed and included as a boundary condition to the model, corresponding to the vibrations in axial, lateral, and torsional directions. This bit-force model considers the bottom hole assembly (BHA) eccentricity, mud damping, bit–rock interaction, and their coupling mechanisms. Simulation results have shown good agreement with field observations and experimental data in the literature. The utility of this modeling framework is demonstrated in the paper through case studies for normal operation, stick–slip vibration, and whirl vibration.

Externally pressurized gas journal bearing with slot restrictors arranged in the axial direction

In this study, a novel rectangular slot restriction-type externally pressurized gas journal bearing was developed for use in high-speed rotating machinery, such as medical devices and industrial machines. The proposed bearing has several rectangular slot restrictors arranged in the inner surface of the bearing. To measure the bearing characteristics, a model was developed for the numerical calculation of the pressure distribution in the bearing clearance and the static characteristics of the bearing. The proposed bearing, which consists of two parts, was designed and can be manufactured using appropriate techniques. In this study, a prototype bearing with eight slots in its surface was manufactured as a test piece for fundamental tests. The diameter and length of the test bearing are 30 and 40mm, respectively. The roundness of the bearing was measured using a three-dimensional coordinate measuring machine, and the results were used in the analysis. The pressure distribution and static characteristics obtained experimentally were found to be in good agreement with the calculated values. In the rotational tests, the rotor speed exceeded 380Hz (22,800rpm), and whirl vibration did not occur. During testing, the maximum rotor vibration amplitude was 0.002mm, corresponding to an eccentricity ratio of 0.3.