Ball Bearing Rotor System Research Articles

Dynamic modeling and reliability analysis of the ball bearing rotating accuracy considering the uncertain bearing-shaft interference fit

The rotating accuracy of the rolling bearing is closely related to its precision class and the actual fitting parameters. However, there are few quantitative evaluation models to describe their relationships. In this paper, considering the effect of the uncertain but bounded interference amounts between the bearing inner ring and the shaft, the Chebyshev interval analysis method is utilized and a rotating accuracy dynamic reliability model is presented based on the two degrees of freedom (2-DOF) rotor-ball bearing system. Firstly, the relationships between the bearing radial clearance and bearing-shaft interference amounts are investigated based on the shaft-bearing interference fit model. Thus, in terms of Chebyshev interval analysis method, the 2-DOF rotor-ball bearing system with uncertain bearing-shaft interference amounts is established. Then the rotating accuracy dynamic reliability model of the ball bearing is presented based on the Chebyshev interval theory. The effects of the average values and deviation coefficients of the initial interference amounts on the actual interference amounts, radial clearance and the resultant rotating accuracy dynamic reliability of the ball bearing are analyzed. The results show that the uncertain bearing-shaft interference amounts have a significant effect on the rotating accuracy dynamic reliability of the ball bearing and reasonable design could improve the rotating accuracy of the ball bearing. The results calculated by the present study are compared to the Monte Carlo simulation (MCS), which show the validity of the study.

Effects of interference fit variation on dynamic characteristics of full ceramic ball bearing rotor system

Bearing fit is one of the key aspects in determining the performance of FCBB (full ceramic ball bearing), which affects FCBB internal contact condition and support stiffness. In this paper, the interference fit value variations of FCBB induced by the centrifugal displacement and thermal deformation is calculated, the mapping relationship between interference fit and internal clearance is established, and the influence of preload and interference fit value on the dynamic characteristics of FCBB is discussed in detail. Finally, the vibration experiment of FCBB is verified by ABLT-9 bearing life testing machine. The research shows that interference fit value is the main factor affecting internal clearance. The increase of internal clearance will reduce the bearing radial stiffness and increase the non-uniformity of load distribution, while the preload will weaken the influence of internal clearance on radial stiffness and load distribution. The results provide a theoretical basis for improving the FCBB’s service performance and optimizing their assembly parameters.

A modified IHB method for nonlinear dynamic and thermal coupling analysis of rotor-bearing systems

This paper presents a modified incremental harmonic balance (IHB) method to analyze the nonlinear dynamic and thermal coupling characteristics of rotor-bearing systems after thermal balance. The IHB method is modified by tensor contraction and fast Fourier transform (FFT) to efficiently calculate the residual vector and Jacobian matrix. Nonlinear dynamic and thermal full coupling is implemented through the nonlinear part of the Jacobian matrix. The presented method achieves the simultaneous and efficient calculation of the motion and heat balance equations in the frequency domain according to the improvement measures. The effectiveness of the presented method is tested using the vibration response and temperature variation analysis for the coupled thermo-mechanical model of a rotor-ball bearing system. Furthermore, the performance comparisons show the good consistency of results between the presented method and the step-by-step (S-S) method, which is a partitioned iterative solution method for dealing with fully coupled problems in the time domain. The presented method has greater computational efficiency, more accurate temperature prediction, and better recognition of nonlinear phenomena in the system rather than the S-S method. Consequently, the presented method has a broad application prospect in solving fully coupled thermo-mechanical problems of more complex and higher dimensional rotor-bearing systems.

Research on the vibration characteristics of the ball bearing considering the uncertain bearing-shaft interference fit

Due to the exist of the inevitable dimensional tolerance and assembling errors in the ball bearing, the interference amounts between the inner ring and the connected shaft are uncertain but bounded, which could directly affect the dynamics performance of the rotor-ball bearing system. Whereas, most of the works on the field of dynamics in a ball bearing have considered the varying compliance (VC) vibration as deterministic vibration and there are few quantitative studies considering the uncertain factors. Therefore, in this paper, an uncertain ball bearing-rotor dynamic model is presented based on the Chebyshev interval method. Firstly, the uncertain interference amounts between the bearing inner ring and the connected shaft are modeled by the Chebyshev interval method. Then the radial clearance and actual interference amounts of ball bearing are derived by the bearing-shaft interference fit model. The time-dependent ball-raceway contact probability is formulated and the influence of the average and deviation coefficient of the initial interference amounts on the vibration behaviors of the rotor under various rotational speeds are investigated based on the double degrees of freedom (2-DOF) ball bearing system. The numerical cases show that the amounts of the bearing-shaft interference fit are uncertain and could be affected by rotational speed, which have an obvious influence on the dynamic performance of the rotor-ball bearing system. Reasonable design of the interference amounts could reduce the dispersion of the displacement and time-varying stiffness. The numerical results obtained by the present study are compared to the Monte Carlo simulation (MCS), which show the validity of the study.

Nonlinear dynamics and thermal bidirectional coupling characteristics of a rotor-ball bearing system

This paper focuses on the nonlinear dynamics and thermal bidirectional coupling characteristics of a rotor-ball bearing system. The motion equations of the rotor system are formulated by the Lagrange method, where the radial clearance of the ball bearing affected by thermal characteristics is considered. The heat transfer model of the ball bearing is established by combining the lumped parameter with the full thermal network, in which the frictional heat generation induced by the dynamic load of the ball bearing is considered. The heat transfer model is coupled with the motion equations of the rotor system via the mechanical model of the ball bearing. The dynamics and thermal bidirectional coupling model is solved by the step-by-step iterative solving procedure to detect the interaction between the vibration response of the rotor and the temperature variation of the ball bearing. The results show that the thermal expansion of the ball bearing results in the dynamical variation of the effective radial clearance, which makes a significant influence on nonlinear dynamics and thermal characteristics of the rotor-ball bearing system. On the other hand, the nonlinear dynamic response of the rotor leads to the nonlinear thermal characteristics of the ball bearing through the influence of the dynamic load. It is observed that there are more complicated jump phenomena and motion patterns in the vibration response and temperature variation compared to the case without thermal effects. Moreover, the jump phenomena affected by the initial radial clearance, rotor eccentricity, and lubricant viscosity are analyzed in detail. The results obtained in this paper can provide more insight into the bidirectional coupling mechanism of the nonlinear dynamics and thermal characteristics.

Investigation on the misalignment of full ceramic ball bearing rotor system caused by varying gap in wide temperature ranges

The temperature-varying gap between the full ceramic ball bearings and the steel pedestal in wide temperature ranges leads to misalignment of the bearing rotor system, and there is a significant impact on the vibration of the rotor system. In this paper, the dynamic model considering the temperature-varying gap and time-varying contact between the full ceramic ball bearing and the steel pedestal in wide temperature ranges is proposed. The rigid body elements and the inner rings are coupled in the model, and the influence of the temperature-varying gap is analyzed. Results show that the increased gap at high temperatures brings about the looseness of the full ceramic ball bearing, the misalignment occurs between the axes of the two full ceramic ball bearings due to the looseness. The interaction between the full ceramic ball bearing and the steel pedestal becomes more significant, which causes the vibration of the rotor system to rise. Appropriately increase the interference between the full ceramic ball bearing and the steel pedestal helps reduce the gap at high temperature, thus avoiding the excessive vibration caused by misalignment and improving the rotation accuracy of the rotor system. The research results provide a reference basis for the use of full ceramic ball bearings in wide temperature ranges.

Digital twin-driven machine learning: ball bearings fault severity classification

Machine learning algorithms (MLAs) are increasingly being used as effective techniques for processing vibration signals obtained from complex industrial machineries. Previous applications of automatic fault detection algorithms in the diagnosis of rotating machines were mainly based on historical operating data sets, limiting the diagnostic reliability to devices with an extended operating history. Moreover, physically collected data are often restricted by the conditions of acquisition and the specific elements for which they were recorded. Digital twin (DT) provides a powerful tool able to generate a huge amount of training data for MLAs. However, the DT model must be accurate enough to substitute the experiments. This work aims to escape the experience requirement by using a simulation-driven MLA based on the multifactorial analysis of fault indicators associated with a DT. To achieve this approach, a numerical model of a rotor-ball bearing system is developed. The latter is updated according to a parameter update scheme based on a comparison between the relevant features of the experimentally measured signals and the signals simulated by the model. These features are chosen as the selected input parameters of the MLA classifier. The results show that after updating, the developed DT has provided a reliable diagnostic with an adaptive degradation analysis, which makes the simulated data suitable for the construction of a machine learning predictive model. Two common MLAs, (multi-kernel support vector machine) and (k nearest neighbor’s algorithm), were trained using the simulated data and validated later against experimental datasets.

Study on the Dynamic Problems of Double‐Disk Rotor System Supported by Deep Groove Ball Bearing

Aiming at the analysis of the dynamic characteristics of the rotor system supported by deep groove ball bearings, the dynamic model of the double‐disk rotor system supported by deep groove ball bearings was established. In this paper, the nonlinear finite element method is used combined with the structural characteristics of deep groove ball bearings. Based on the nonlinear Hertz contact theory, the mechanical model of deep groove ball bearings is obtained. The excitation response results of the rotor system nodes are solved by using the Newmark‐β numerical solution method combined with the Newton–Raphson iterative method. The vibration characteristics of the rotor system supported by deep groove ball bearings are studied deeply. In addition, the effect of varying compliance vibration (VC vibration) caused by the change in bearing support stiffness on the dynamics of the system is considered. The time domain and frequency domain characteristics of the rotor system at different speeds, as well as the influence of bearing clearance and bearing inner ring’s acceleration on the dynamics of the rotor system are analyzed. The research shows that the VC vibration of the bearing has a great influence on the motion of the rotor system when the rotational speed is low. Moreover, reasonable control of bearing clearance can reduce the mutual impact between the bearing rolling element and the inner or outer rings of the bearing and reduce the influence of unstable bearing motion on the vibration characteristics of the rotor system. The results can provide theoretical basis for the subsequent study of the nonlinear vibration characteristics of the deep groove ball bearing rotor system.

Vibration Analysis of Flexible Rotor with Angular Contact Ball Bearings Using a General Bearing Stiffness Model

The vibration analysis of flexible rotor systems supported by angular contact ball bearings is presented. Vibration analysis of rotor-ball bearing systems has often been performed via simplification of supporting bearings as linear springs with constant stiffness. In this study, an improved model of rotor-ball bearing systems was proposed. It utilizes a general bearing model based on response and time-dependent bearing characteristics. The system equations of motion were established using the finite-element method and numerically solved using the Newmark-β method. The method was used to recalculate the bearing stiffness matrices at every interval of numerical integration as a function of the instantaneous bearing displacements using a separated five-degrees-of-freedom bearing model. The method was verified via comparison with experimental data available in the literature. The extended simulations were conducted to investigate the unbalanced responses of a rotor-ball bearing system using the proposed and conventional methods. Numerical results showed a meaningful discrepancy between the vibrational responses obtained by the proposed model using the response and timedependent bearing stiffness model and the traditional constant-stiffness model.

The Varying Compliance Resonance in a Ball Bearing Rotor System Affected by Different Ball Numbers and Rotor Eccentricities

The varying compliance (VC) vibration directly reflects the oscillation intensity of a rolling bearing, and it can be fully revealed in the VC resonance. Moreover, we define the bearing vibration intensity as the bearing vibration information in this paper. Besides the rolling element number of the bearing, the rotor eccentricity is also an inevitable influencing factor for the VC vibration. This paper focuses on the VC resonance characteristics in a ball bearing rotor system. An analytical model is established, and the vibration responses of the system are calculated in a large speed range with the consideration of different ball numbers and different rotor eccentricities. The theoretical results show that the VC vibration is clearer in low-speed range where the VC resonance exist, while it is suppressed in high-speed range. In general, the intensity of the VC resonance decreases with the increase of ball numbers and is not sensible to the rotor eccentricities in low-speed range. Finally, a ball bearing rotor experiment system is setup, the VC resonance is clearly detected, and the high-quality bearing vibration information is obtained. The experimental results qualitatively agree with the theoretical results.

Advantages of pulse force model over geometrical boundary model in a rigid rotor–ball bearing system

For a local defect modeling in a defective bearing–rotor system, there mainly exist two types of commonly-used models. One is a description of the geometrical boundary simulating a real local defect (GB model), while the other one is a depiction of the consequence —the pulse force produced by a rolling element striking the local defect (PF model). Dynamic equations of defective ball bearing–rotor system based on the two local defect models are established. The defective systems depending on the two defect models reflect the similar dynamic characteristics such as the defect frequencies, the primary resonance and the super-harmonic resonance. However, compared to the GB model, the PF model based governing equation is more suitable to be processed by the classical method for nonlinearities which exists in a defective bearing–rotor system. A frequency-domain method —harmonic balance (HB) method which can obtain the semi-analytical solution is utilized to solve the PF model based governing equation and prove the calculation efficiency. Besides, a rigid rotor–ball bearing experiment system is established, the phenomena such as the defect frequencies, resonance and super-harmonic resonances in theoretical analysis reflected together by two defect models are verified qualitatively. Therefore, it can be concluded that the PF model has an advantage over the GB model for the dynamic analysis. Moreover, the discoveries in this paper may supply some clues for diagnosis of a defective rolling bearing system.

Nonlinear Dynamic Modeling of the Cracked Rotor Ball Bearing System With Emphasis on Damage Detection Capabilities

It is confirmed experimentally that in case of a rotor with crack, multiple harmonics are generated when the rotor revolves at a particular frequency. Only few modeling techniques successfully predict this particular behavior of the cracked rotor. It is observed in this research that modeling cracked rotors using conventional finite element methods cannot predict this particular behavior successfully. A nonlinear dynamic model of the flexible rotor with ball bearings is developed using discrete mass spring damper elements combined with an existing model of the crack to truly predict this confirmed experimental behavior. Certain crack detection techniques based on the steady-state response work well on this basic concept of the multiharmonics generation due to nonlinearities caused by cracks in the rotor. The presence of ball bearings, rotor-coupling misalignment, rotor-stator rub, and rotor bow can also cause significant nonlinearities in the overall system. These additional nonlinearities render these crack detection techniques to lose their effectiveness. Our work justifies through simulations that the Jeffcott rotors are the over simplified version of real-life rotor-bearing systems. Hence, these crack detection techniques cannot be efficiently applied for condition monitoring of real-life rotor-bearing systems. The proposed model also helps to understand that the presence of flexible bearing supports affects the dynamics of the system considerably and negatively affects the effectiveness of these crack detection techniques.

Study for ball bearing outer race characteristic defect frequency based on nonlinear dynamics analysis

The characteristic defect frequencies are widely used for diagnosing the local defect of the ball bearing. The varying compliance (VC) frequency of a fault-free rotor–bearing system equals to the BPFO (ball bearing outer race defect frequency) due to the internal kinematic relationship of a bearing assembly. In order to indicate this issue, a semi-analytical method—the harmonic balance method with alternating frequency/time domain technique—is exploited to obtain the solutions of rotor–ball bearing systems with /without an outer race defect. The solutions and the features of a rotor–ball bearing system with essentially nonlinear parametric excitation are analyzed. We prove the VC frequency equals the BPFO and explain the reasons that the harmonics of the characteristic defect frequency generally appear in the frequency domain. The VC, BPFO as well as their harmonics affected by the primary and super-harmonic resonance of the system are found out. Finally, a test rig of a rigid rotor–bearing system is established to verify the theoretical analysis qualitatively by presenting the performance of VC, BPFO and their harmonics in the frequency domain. In addition, the tests are accomplished in a cycle of running up and down to reveal the primary and super-harmonic resonance characteristics. On the basis of the theoretical and experimental results, the basic BPFO is not enough to judge an outer race defect. The discussion on frequency spectrum, the primary and super-harmonic resonance provides a more reliable way to elucidate the characteristic defect frequencies.

Experiments and Numerical Results for Varying Compliance Contact Resonance in a Rigid Rotor–Ball Bearing System

This paper investigates the nonlinear characteristics of varying compliance contact resonance in a rotor–bearing system and takes into consideration the Hertzian contact deformation and internal radial clearance. We created an experimental rig of a rigid rotor supported by rolling element bearings. In the course of the rotational speed run up and down, the frequency–amplitude curves of the varying compliance vibrations were observed during experiments using different radial loads and compared with the results of our numerical simulations. The experimental and numerical results indicate that the varying compliance contact resonance in the vertical direction presents the soft spring characteristic, while the soft and hard spring characteristics coexist for the horizontal resonance.

Nonlinear vibration analysis of a cracked rotor-ball bearing system during flight maneuvers

This paper focuses on the nonlinear responses of a cracked rotor-ball bearing system caused by aircraft flight maneuvers. The equations of motion of the system are formulated with the consideration of the breathing mechanism of a transverse crack and the maneuver load of a climbing-diving flight. The fourth order Runge-Kutta method is employed to detect the nonlinear responses of the system, which are reflected by bifurcation diagrams, power spectrums, maximum Lyapunov exponent, phase portraits and Poincaré sections. It is shown that the super-harmonic responses of the system are affected significantly by the maneuver load under sub-critical speeds. Plenty of quasi-periodic motions are obtained, and a variety of complex nonlinear behaviors including bifurcations and jumping phenomenon are observed near 1/4, 1/3, 2/5 and 1/2 critical speeds when the maneuver load increases from 0 to 10g. The nonlinear responses of the system influenced by crack stiffness, bearing clearance and rotor eccentricity are also investigated. Chaotic motions are demonstrated when the crack stiffness or the bearing clearance increases across a critical value. However, the responses maintain quasi-periodic when the rotor eccentricity changes. The results will contribute to a better understanding of the nonlinear dynamic behaviors of cracked rotors in flight maneuvers.

A New Dynamic Model of Ball-Bearing Rotor Systems Based on Rigid Body Element

Ball-bearing rotor systems are key components of rotating machinery. In this work, a new dynamic modeling method for ball-bearing rotor systems is proposed based on rigid body element (RBE). First, the concept of RBE is given, and then the rotor is divided into several discrete RBEs. Every two adjacent RBEs are connected by imaginary springs, whose stiffness is calculated according to properties of the RBEs. Second, all the parts of rolling ball bearings (i.e., outer ring, inner ring, ball, and cage) are considered as RBEs, and Gupta's model is employed to model bearings which include radial clearance, waviness, pedestal effect, etc. Finally, the rotor and all the rolling ball bearings are coupled to develop a dynamic model of the ball-bearing rotor system. The vibration responses of the ball-bearing rotor system can be calculated by solving dynamic equations of each RBE. The proposed method is verified with both simulation and experiment. The RBE model of the rotor is compared with its finite element (FE) model first, and numerical simulation shows the validity of the RBE model. Then, experiments are conducted on a rotor test rig which is supported with two rolling ball bearings as well. Good agreements between measurement and simulation show the ability of the model to predict the dynamic behavior of ball-bearing rotor systems.

A General Method for the Dynamic Modeling of Ball Bearing–Rotor Systems

A general dynamic modeling method of ball bearing–rotor systems is proposed. Gupta's bearing model is applied to predict the rigid body motion of the system considering the three-dimensional motions of each part (i.e., outer ring, inner ring, ball, and rotor), lubrication tractions, and bearing clearances. The finite element method is used to model the elastic deformation of the rotor. The dynamic model of the whole ball bearing–rotor system is proposed by integrating the rigid body motion and the elastic vibration of the rotor. An experiment is conducted on a test rig of rotor supported by two angular contact ball bearings. The simulation results are compared with the measured vibration responses to validate the proposed model. Good agreements show the accuracy of the proposed model and its ability to predict the dynamic behavior of ball bearing–rotor systems. Based on the proposed model, vibration responses of a two bearing–rotor system under different bearing clearances were simulated and their characteristics were discussed. The proposed model may provide guidance for structural optimization, fault diagnosis, dynamic balancing, and other applications.

Dynamic Analysis of Rotor-Ball Bearing System of Air Conditioning Motor of Electric Vehicle

Nowadays, electric vehicles (EV) present a promising solution to reduce greenhouse gas emissions. They are considered zero emission vehicles. Rotor bearing system is important part of air conditioning motor of EV. The aim of this research is to develop a numerical model to investigate the structural dynamic response of the rigid rotor supported on deep groove ball bearings. The numerical model considers rotor imbalance that varies with speed, as well as sources of nonlinearity such as Hertzian contact force, ball clearance and varying compliance vibration. This is very important on the design point of view. The 4th order Runge-Kutta numerical integration technique has been applied. The results are presented in form of time displacement response, frequency spectra, and Poincare map. The analysis demonstrates that the number of balls is one of the key factors affecting on the dynamic characteristics of rotor bearing system. The model can also be used as a tool for predicting nonlinear dynamic behavior of rotor system of air conditioning motor of electric vehicle under different operating conditions. Moreover, the study may contribute to a further understanding of the nonlinear dynamics of rotor bearing system.

Parametric representation of stationary points in rotor-ball bearing system

The frequencies of the stationary points in rotor-ball bearing system are difficult to determine due to the nonlinear and non-smooth ball supporting effects. This paper presents a linearized discriminant to calculate the frequency of the stationary points of the system. The influences of system parameters on the stationary points with the Timoshenko beam and the Bernoulli–Euler beam models are also discussed. The comparison between the linear and nonlinear results indicates that the underlying assumptions are reasonable for the range of parameters considered. With the help of linearized discriminant, the frequency values of the stationary points can be found conveniently. Taking the frequency values of stationary points into the linearized discriminant, the corresponding system parameters can also be obtained by setting the linear determinant to be zero.

Global nonlinear dynamic characteristics of rod-fastening rotor supported by ball bearings

Flexible rod-fastening rotor-ball bearing system (FRRBS) is usually adopted in aircraft engines. In this paper, a simplified model for the FRRBS is proposed, and the global nonlinear dynamic characteristics of FRRBS are investigated. The FRRBS has two kinds of special structural features: rods and interfaces. The circumferentially distributed rods are modelled as a constant stiffness matrix with an additional moment (when unequal pre-tightening occurs) without mass effect. The interfaces between disks are modelled as stiffness matrices with normal and tangential contact stiffness, which are determined by pre-tightening force. Then, the shaft is discretized by Timoshenko elements and supported by ball bearings with radial internal clearance. Periodic motions and bifurcation sets are obtained by shooting method. The result shows that the insufficient pre-tightening brings about abnormal vibrations through quasi-periodic instability in high speed region, and the unequal pre-tightening leads to severe vibrations in stable periodic operation state. Moreover, the mass eccentricity would be the most nonlinear source for the nonlinear instabilities. Generally, this paper presents a method for modelling a FRRBS to study the nonlinear dynamic characteristics of the FRRBS.