Analyses of contact forces and vibration response for a defective rolling element bearing using an explicit dynamics finite element model

This paper provides insights into the physical mechanism by which defect-related impulsive forces, and consequently, vibrations are generated in defective rolling element bearings. A dynamic nonlinear finite element model of a rolling element bearing with an outer raceway defect was numerically solved using the explicit dynamics finite element software package, LS-DYNA. A hypothesis was developed to explain the numerical noise observed in the predicted vibrations and contact forces, and the noise frequencies were analytically estimated. In-depth analyses of the numerically estimated dynamic contact forces between the rolling elements and the raceways of a bearing, which are not measured in practice, and have not been reported previously, are presented in this paper. Several events associated with the traverse of the rolling elements through the outer raceway defect are elaborated, and the impulsive force generating mechanism is explained. It was found that the re-stressing of the rolling elements that occurs near the end of a raceway defect generates a burst of multiple short-duration force impulses. The modelling results also highlight that much higher contact forces and accelerations are generated on the exit of the rolling elements out of defect compared to when they strike the defective surface. A bearing with a machined outer raceway defect was tested in a controlled experiment; the measured acceleration response compared favourably with the numerically modelled acceleration results, thereby, validating the low- and high-frequency characteristics of the de-stressing and re-stressing of the rolling elements, respectively.     

滚动轴承声信号特征提取和诊断试验研究

为解决振动检测方法不能有效识别低速旋转机械滚动轴承故障问题,利用声发射检测方法,建立了滚动轴承低速声发射信号采集试验装置,对模拟人工缺陷滚动轴承声发射信号进行了采集,进而对滚动轴承声发射信号进行总体平均经验模式分解,结合能量矩及相关系数法综合判断分解后各模态分量的真伪,据此提取出特征信号并做出其局部Hilbert边际谱,最后对滚动轴承各种故障模式进行诊断。试验结果表明该诊断方法能准确识别滚动轴承声发射信号故障频率,依据特征频率及幅值大小可对低速滚动轴承故障进行有效诊断。     

Bearing condition diagnosis and prognosis using applied nonlinear dynamical analysis of machine vibration signal

In this paper, we introduce a modified form of the correlation integral developed by Grassberger and Procaccia referred to as the partial correlation integral, which can be computed in real time. The partial correlation integral algorithm is then used to analyze machine vibration data obtained throughout a life test of a rolling element bearing. From the experimental results, the dimensional exponent (an approximation of the correlation dimension) as computed from the partial correlation integral algorithm tends to increase as time progresses and the useful remaining life of the bearing is decreasing. The dimensional exponents of a healthy bearing and a bearing close to failure are statistically different. We also propose a computational scheme for bearing condition monitoring (diagnosis and prognosis) using the dimensional exponent integrated with a surrogate data testing technique. As a result, we can characterize the condition of the bearing from the results of the surrogate data test and furthermore, we provide some preliminary evidence that the dimensional exponent can be used to predict the failure of rolling element bearings in rotating machinery from real-time vibration data.     

Vibrations of balanced fault-free ball bearings

This paper investigates the vibrations of balanced fault-free ball bearings. A lumped mass-damper-spring model is adopted including the use of the Hertzian contact theory to represent the stiffness of the bearing rolling elements. We found that the equilibrium point of the bearing undergoes a supercritical pitchfork bifurcation as the bearing internal clearance increases. We developed closed-form expressions for the frequency-response functions of the horizontal and vertical motions of bearings with small internal clearance (below the bifurcation point). We also developed a chaos map to describe the locations and intensity of chaos in the internal clearance-shaft speed parameter space for bearings with larger internal clearance (beyond the bifurcation point).     

Roller bearing acoustic signature extraction by wavelet packet transform,applications in fault detection and size estimation

Continuous online monitoring of rotating machines is necessary to assess real-time health conditions so as to enable early detection of operation problems and thus reduce the possibility of downtime. Rolling element bearings are crucial parts of many machines and there has been an increasing demand to find effective and reliable health monitoring technique and advanced signal processing to detect and diagnose the size and location of incipient defects. Condition monitoring of rolling element bearings, comprises four main stages which are, statistical analysis, fault diagnostics, defect size calculation, and prognostics. In this paper the effect of defect size, operating speed, and loading conditions on statistical parameters of acoustic emission (AE) signals, using design of experiment method (DOE), have been investigated to select the most sensitive parameters for diagnosing incipient faults and defect growth on rolling element bearings. A modified and effective signal processing algorithm is designed to diagnose localized defects on rolling element bearings components under different operating speeds, loadings, and defect sizes. The algorithm is based on optimizing the ratio of Kurtosis and Shannon entropy to obtain the optimal band pass filter utilizing wavelet packet transform (WPT) and envelope detection. Results show the superiority of the developed algorithm and its effectiveness in extracting bearing characteristic frequencies from the raw acoustic emission signals masked by background noise under different operating conditions. To experimentally measure the defect size on rolling element bearings using acoustic emission technique, the proposed method along with spectrum of squared Hilbert transform are performed under different rotating speeds, loading conditions, and defect sizes to measure the time difference between the double AE impulses. Measurement results show the power of the proposed method for experimentally measuring size of different fault shapes using acoustic emission signals.     

Dynamics and stability of rigid rotors levitated by passive cylinder-magnet bearings and driven/supported axially by pointwise contact clutch

A stable rotor—supported laterally by passive magnetic bearings and longitudinally by magnetic forces and a clutch—loses suddenly its contact to the clutch and executes abruptly longitudinal movements away from its original equilibrium position as a result of small increases in angular velocity. Such an abrupt unstable behaviour and its reasons are thoroughly theoretically as well as experimentally investigated in this work. In this context, this paper gives theoretical as well as experimental contributions to the problem of two dimensional passive magnetic levitation and one dimensional pointwise contact stability dictated by mechanical–magnetic interaction. Load capacity and stiffness of passive multicylinder magnetic bearings (MCMB) are thoroughly investigated using two theoretical approaches followed by experimental validation. The contact dynamics between the clutch and the rotor supported by MCMB using several configurations of magnet distribution are described based on an accurate nonlinear model able to reliably reproduce the rotor-bearing dynamic behaviour. Such investigations lead to: (a) clear physical explanation about the reasons for the rotor's unstable behaviour, losing its contact to the clutch and (b) an accurate prediction of the threshold of stability based on the nonlinear rotor-bearing model, i.e. maximum angular velocity before the rotor misses its contact to the clutch as a function of rotor, bearing and clutch design parameters.     

Rolling bearing vibrations—The effects of geometrical imperfections and wear

The geometrical shape and surface properties of the components of rolling bearings will always deviate to some extent from their theoretical design. For bearings of standard tolerances these deviations are large enough to cause measurable levels of vibrations when the bearing is in operation. The purpose of this paper is to show in some detail how these surface irregularities are related to the vibration characteristics of the bearing. The study is restricted to radial bearings, having a radial load and a positive clearance. The approximate methods used render the results useful mainly for lightly loaded bearings operating at low and moderate speeds. Attention has been focused on the effects of inner ring waviness and non-uniform diameters of the rolling elements. A mixed theoretical and experimental impedance approach has been used to treat the bearing when fitted in a simple machine structure, thereby showing how resulting vibrations of the bearing pedestal can be calculated, with account taken of the effects of bearing, rotor and foundation properties. During operation bearings undergo progressive surface and subsurface deterioration. These alterations of geometrical and surface properties of bearing components will always be accompanied by some degree of change of the vibrations characteristics of the bearing. Two common modes of surface deterioration—spalling fatigue and abrasive wear—have been studied, the practical objective being to highlight some possible methods of condition monitoring and prediction of impending bearing failure.     

Vibration analysis of a rotating system due to the effect of ball bearing waviness

This research presents an analytical model to investigate vibration due to ball bearing waviness in a rotating system supported by two or more ball bearings, taking account of the centrifugal force and gyroscopic moment of the ball. The waviness of rolling elements is modelled by the sinusoidal function, and it is incorporated into the position vectors of the race curvature center. The Hertzian contact theory is applied to calculate the elastic deflection and non-linear contact force, while the rotor has translational and angular motions. Both the centrifugal force and gyroscopic moment of the ball and the waviness of the rolling elements are included in the kinematic constraints and force equilibrium equations of a ball to derive the non-linear governing equations of the rotor, which are solved by using the Runge–Kutta–Fehlberg algorithm to determine the new position of the rotor. The proposed model is validated by the comparison of the results of the prior researchers. This research shows that the centrifugal force and gyroscopic moment of the ball plays the important role in determining the bearing frequencies, i.e., the principal frequencies, their harmonics and the sideband frequencies resulting from the waviness of the rolling elements of ball bearing. It also shows that the bearing vibration frequencies are generated by the waviness interaction not only between the rolling elements of one ball bearing, but also between those of two or more ball bearings constrained by the rotor.     

Angular dependence of the differential cross section of hadronic exclusive processes at asymptotic energies

The angular dependence of the differential cross section is derived for meson-baryon and baryon-baryon elastic scattering and for some related processes, in the framework of the constituent interchange model, assuming the validity of dimensional scaling laws and using the Mandelstam representation of the amplitude.     

Influence of the alignment of load and oscillation on the frictional shakedown of an elastic rolling contact with Coulomb friction

We examine frictional shakedown of a three dimensional elastic rolling contact. Slight oscillatory rolling of one contacting body varies the normal pressure distribution. In turn this causes incremental sliding processes and a macroscopic rigid body motion. We consider two settings: tangential force and rolling direction aligned parallel and perpendicular to each other. In both cases, the slip ceases after the first few periods and a safe shakedown occurs if the oscillation is sufficiently small. Otherwise ratcheting occurs and the accumulated slip leads to a continuing rigid body motion.Numerical simulations with Kalker’s and Vollebregt’s software CONTACT show that the rolling direction leads to differences in the contact region and the traction distribution. Using the method of dimensionality reduction we derive the analytical shakedown limits for the tangential load and the oscillation amplitude. The results show strong agreement with experimental data and allow the accurate prediction of the shakedown displacement and the maximum tangential load capacity in the shakedown state. It shows that a perpendicular alignment of force and rolling direction increases the final displacement in case of shakedown as well as the incremental shift in case of ratcheting.     

Generalized formulation of the interactions between soft spheres

The goal of this paper is to identify the most general formulation that consistently links the different degrees of freedom in a contact between spherical soft particles. These contact laws have two parts: a set of “generalized contact velocities” that characterize the relative motion of the two particles, and a set of “generalized contact forces” that characterize the interparticle forces. One well known constraint on contact models is that the contact velocities must be objective. This requirement fixes the number of linearly independent contact velocities. We also present a previously unnoticed (in this context) constraint, namely, that the velocities and forces must be related in such a way that the stiffness matrix is symmetric. This constraint also places restrictions on the coupling between the contact forces. Within our generalized contact model, we discuss the expression for rolling velocity that need to be used in the calculation of rolling resistance, and the risk or producing perpetual mobile when other expressions of rolling velocity are using instead.     

Vibration quenching in a large scale rotor-bearing system using journal bearings with variable geometry

In large scale rotating machinery the resonance amplitude during the passage through resonance is a matter of consideration because of its influence in the surrounding environment of the rotational system and foundation. In this paper, a variable geometry journal bearing (VGJB), recently patented, is applied for the mounting of a large scale rotor bearing system operating at the range of medium speed. The simulation of the rotor-bearing system incorporates a recent method for simulation of a multi-segment continuous rotor in combination with nonlinear bearing forces. The use of the current bearing gives results that encourage the use of such a bearing in rotating machinery since the vibration amplitude during the passage through the critical speed can be reduced up to 60–70%. In the presented study, the developed amplitude and the rotor stresses are severely reduced compared to those of the system with normal cylindrical journal bearings during a virtual start up of the system.     

A Bearing Fault Diagnosis Method Based on PAVME and MEDE

When rolling bearings have a local fault, the real bearing vibration signal related to the local fault is characterized by the properties of nonlinear and nonstationary. To extract the useful fault features from the collected nonlinear and nonstationary bearing vibration signals and improve diagnostic accuracy, this paper proposes a new bearing fault diagnosis method based on parameter adaptive variational mode extraction (PAVME) and multiscale envelope dispersion entropy (MEDE). Firstly, a new method hailed as parameter adaptive variational mode extraction (PAVME) is presented to process the collected original bearing vibration signal and obtain the frequency components related to bearing faults, where its two important parameters (i.e., the penalty factor and mode center-frequency) are automatically determined by whale optimization algorithm. Subsequently, based on the processed bearing vibration signal, an effective complexity evaluation approach named multiscale envelope dispersion entropy (MEDE) is calculated for conducting bearing fault feature extraction. Finally, the extracted fault features are fed into the k-nearest neighbor (KNN) to automatically identify different health conditions of rolling bearing. Case studies and contrastive analysis are performed to validate the effectiveness and superiority of the proposed method. Experimental results show that the proposed method can not only effectively extract bearing fault features, but also obtain a high identification accuracy for bearing fault patterns under single or variable speed.     

A novel methodology for online detection of bearing health status for naturally progressing defect

A methodology is proposed for the online detection of health status of rolling element bearing into various damage stages for naturally progressing defect. Various damage identification parameters are derived from processing vibration data in time domain, frequency domain and time–frequency domain. The parameters are fused into a single parameter, Mahalanobis distance, by application of Gram–Schmidt Orthogonalization process. Chebyshev?s inequality is applied to the Mahalanobis distance for online monitoring and damage stage detection. A simulation study is first carried out to show working of the proposed methodology in presence of varying trends of damage identification parameters. The proposed methodology is then validated on experimental data. The first validation is on the vibration data acquired from a bearing having seeded defect. Later, two accelerated life tests are conducted on a specially designed test rig at different load and speed combinations on the bearings for ensuring naturally induced and progressed defects. The methodology is successfully verified on the vibration data acquired from the naturally induced and progressed defect experiments.     

Continuous-scale mathematical morphology-based optimal scale band demodulation of impulsive feature for bearing defect diagnosis

Integer-scale structuring element is usually used in the traditional mathematical morphology (MM) for signal processing. When applied for impulsive feature demodulation of vibration signal of rolling element bearings, the integer-scale MM (ISMM) may lead to low resolution result and thus undermines its defect diagnosis capability. For this reason, this paper proposes a continuous-scale MM (CSMM) scheme by interpolation and re-sampling to improve scale resolution for more reliable fault signature extraction. Based on the frequency domain kurtosis criterion, a narrowband merging operation is employed to locate the optimal scale band that best reflects the impulsive feature from the CSMM analysis results. The demodulated components in the optimal scale band are employed to detect the existence of the bearing fault. The proposed optimal CSMM demodulation technique is evaluated using both simulated and experimental bearing vibration signals. The results show that, the CSMM is capable of generating demodulation signals with higher resolution, and the optimal scale band demodulation based on the CSMM can reliably extract impulsive features for bearing defect diagnosis.     

Hierarchical Amplitude-Aware Permutation Entropy-Based Fault Feature Extraction Method for Rolling Bearings

In order to detect the incipient fault of rolling bearings and to effectively identify fault characteristics, based on amplitude-aware permutation entropy (AAPE), an enhanced method named hierarchical amplitude-aware permutation entropy (HAAPE) is proposed in this paper to solve complex time series in a new dynamic change analysis. Firstly, hierarchical analysis and AAPE are combined to excavate multilevel fault information, both low-frequency and high-frequency components of the abnormal bearing vibration signal. Secondly, from the experimental analysis, it is found that HAAPE is sensitive to the early failure of rolling bearings, which makes it suitable to evaluate the performance degradation of a bearing in its run-to-failure life cycle. Finally, a fault feature selection strategy based on HAAPE is put forward to select the bearing fault characteristics after the application of the least common multiple in singular value decomposition (LCM-SVD) method to the fault vibration signal. Moreover, several other entropy-based methods are also introduced for a comparative analysis of the experimental data, and the results demonstrate that HAAPE can extract fault features more effectively and with a higher accuracy.     

A flexible multi-body approach for frictional contact in spur gears

In the present paper, a large rotational approach for dynamic contact problems with friction is proposed. The approach is used for modelling a spur gear pair with shafts and bearings. The model is obtained by superposing small displacement elasticity on rigid-body motions, and postulating tribological laws on the gear flanks. The finite element method is used to model the elastic properties of the gear pair. Shafts and bearings are represented by linear springs. The tribological laws of the contact interface are Signorini's contact law and Coulomb's law of friction. An important feature of the approach is that the difficulties of impacting mass nodes are avoided. The governing equations of the model are numerically treated by use of the augmented Lagrangian approach. In such manner the geometry of the gear flanks are well represented in the numerical simulations. It is possible to study accurately the consequences of different types of profile modifications as well as flank errors. In this work, the dynamic transmission error is studied. For instance, it turns out that the effect from profile modification is less significant for the transmission error when frictional effects are included.     

Rotating machinery dynamics simulation. I. Rigid systems with ball bearing nonlinearities and outer ring ovality under rotating unbalance excitation

The radial clearance in rolling bearing systems, required to compensate for dimensional changes associated with thermal expansion of the various parts during operation, may cause dimensional attrition and comprise bearing life, if unloaded operation occurs and balls skid [D. Childs and D. Moyer, ASME J. Eng. Gas Turb. Power 107, 152-159 (1985)]. Also, it can cause jumps in the response to unbalance excitation. These undesirable effects may be eliminated by introducing two or more loops into one of the bearing races so that at least two points of the ring circumference provide a positive zero clearance [D. Childs, Handbook of Rotordynamics, edited by F. Ehrich (McGraw-Hill, NY, 1992)]. The deviation of the outer ring with two loops, known as ovality, is one of the bearing distributed defects. Although this class of imperfections has received much work, none of the available studies has simulated the effect of the outer ring ovality on the dynamic behavior of rotating machinery under rotating unbalance with consideration of ball bearing nonlinearities, shaft elasticity, and speed of rotation. To fill this gap, the equations of motion of a rotor-ball bearing system are formulated using finite-elements (FE) discretization and Lagrange's equations. The analyses are specialized to a rigid-rotor system, by retaining the rigid body modes only in the FE solution. Samples of the results are presented in both time domain and frequency domain for a system with and without outer ring ovality. It is found that with ideal bearings (no ovality), the vibration spectrum is qualitatively and quantitatively the same in both the horizontal and vertical directions. When the ring ovality is introduced, however, the spectrum in both orthogonal planes is no longer similar. And magnitude of the bearing load has increased in the form of repeated random impacts, between balls and rings, in the horizontal direction (direction of maximum clearance) compared to a continuous contact along the vertical direction (direction of positive zero clearance). This underlines the importance of the vibration measuring probe's direction, with respect to the outer ring axes, to capture impact-induced vibrations. Moreover, when the harmonic excitation is increased for a system with ideal bearings, the spectral peaks above forcing frequency have shifted to a higher-frequency region, indicating some sort of a hard spring mechanism inherent in the system. Another observation, is that for the same external excitation, vibration amplitude at forcing frequency in the bearing force spectrum is the same for systems with or without outer ring ovality.     

Intelligent Online Monitoring of Rolling Bearing: Diagnosis and Prognosis

This paper suggests a new method to predict the Remaining Useful Life (RUL) of rolling bearings based on Long Short Term Memory (LSTM), in order to obtain the degradation condition of the rolling bearings and realize the predictive maintenance. The approach is divided into three parts: the first part is the clustering to detect the damage state by the density-based spatial clustering of applications with noise. The second one is the health indicator construction which could give a better reflection of the bearing degradation tendency and is selected as the input for the prediction model. In the third part of the RUL prediction, the LSTM approach is employed to improve the accuracy of the prediction. The rationale of this work is to combine the two methods—the density-based spatial clustering of applications with noise and LSTM—to identify the abnormal state in rolling bearings, then estimate the RUL. The suggested method is confirmed by experimental data of bearing life cycle, and the RUL prediction results of the model LSTM are compared with the nonlinear au-regressive model with exogenous input model. In addition, the constructed health indicator is compared with the spectral kurtosis feature. The results demonstrated that the suggested method is more appropriate than the nonlinear au-regressive model with exogenous input model for the prediction of bearing RUL.     

The Wearing of Bearings Studied by Means of High Resolution Microautoradiography)

The high resolution microautoradiography has been originally developed as a sensitive method for the examination of cell structures and cell metabolism. The present state of autoradiography in biology can be denoted as an autoradiography of the molecules within the cells. The employment of microautoradiography is restricted not only to the field of the cell or molecular biolgy, but after a modification of the procedure it can be applied in quite different branches.

In our case microautoradiography has been used as a complementary method in running studies of the wearing out of the rolling bearings (Klime? 1975) for a localization of adhesive wear²) on the different parts of the bearings. If high resolution microautoradiography is used there is a presumption of exact identification of the contact path in the contact surface with qualitative differentiation of rolling and/or sliding contact. These facts give rise to the qualitatively new pieces of knowledge.