Optimal Design of Tapered Roller Bearings Based on Multi⁃Physics Objectives Using Evolutionary Algorithms

Rolling element bearing is the most common machine element in rotating machinery.An extended life is among the foremost imperative standards in the optimal design of rolling element bearings,which confide on the fatigue failure,wear,and thermal conditions of bearings.To fill the gap,in the current work,all three objectives of a tapered roller bearing have been innovatively considered respectively,which are the dynamic capacity,elasto-hydrodynamic lubrication(EHL)minimum film⁃thickness,and maximum bearing temperature.These objective function formulations are presented,associated design variables are identified,and constraints are discussed.To solve complex non⁃linear constrained optimization formulations,a best⁃practice design procedure was investigated using the Artificial Bee Colony(ABC)algorithms.A sensitivity analysis of several geometric design variables was conducted to observe the difference in all three objectives.An excellent enhancement was found in the bearing designs that have been optimized as compared with bearing standards and previously published works.The present study will definitely add to the present experience based design followed in bearing industries to save time and obtain assessment of bearing performance before manufacturing.To verify the improvement,an experimental investigation is worthwhile conducting.

Nonlinear Dynamics Behaviors of a Rotor Roller Bearing System with Radial Clearances and Waviness Considered

A rotor system supported by roller beatings displays very complicated nonlinear behaviors due to nonlinear Hertzian contact forces, radial clearances and bearing waviness. This paper presents nonlinear bearing forces of a roller bearing under four-dimensional loads and establishes 4-DOF dynamics equations of a rotor roller bearing system. The methods of Newmark-β and of Newton-Laphson are used to solve the nonlinear equations. The dynamics behaviors of a rigid rotor system are studied through the bifurcation, the Poincar è maps, the spectrum diagrams and the axis orbit of responses of the system. The results show that the system is liable to undergo instability caused by the quasi-periodic bifurcation, the periodic-doubling bifurcation and chaos routes as the rotational speed increases. Clearances, outer race waviness, inner race waviness, roller waviness, damping, radial forces and unbalanced forces-all these bring a significant influence to bear on the system stability. As the clearance increases, the dynamics behaviors become complicated with the number and the scale of instable regions becoming larger. The vibration frequencies induced by the roller bearing waviness and the orders of the waviness might cause severe vibrations. The system is able to eliminate non-periodic vibration by reasonable choice and optimization of the parameters.

THE 3-D BOUNDARY ELEMENT METHOD OF ROLLER BEARING BY PLATE ELEMENT ANALOGUE

The mill roller bearing is made up of an internal ring, middlerolls and an external ring, the analysis of which is a multi-bodiescontact problem. In this paper, based on the three-dimensionalelastic contact BEM without friction, and using the structuralcharacteristics of roller bearings, middle rolls are de- scribed byelastic plate units of different shapes, which is placed on theinternal ring.

A method combining refined composite multiscale fuzzy entropy with PSO-SVM for roller bearing fault diagnosis

Combining refined composite multiscale fuzzy entropy(RCMFE)and support vector machine(SVM)with particle swarm optimization(PSO)for diagnosing roller bearing faults is proposed in this paper.Compared with refined composite multiscale sample entropy(RCMSE)and multiscale fuzzy entropy(MFE),the smoothness of RCMFE is superior to that of those models.The corresponding comparison of smoothness and analysis of validity through decomposition accuracy are considered in the numerical experiments by considering the white and 1/f noise signals.Then RCMFE,RCMSE and MFE are developed to affect extraction by using different roller bearing vibration signals.Then the extracted RCMFE,RCMSE and MFE eigenvectors are regarded as the input of the PSO-SVM to diagnose the roller bearing fault.Finally,the results show that the smoothness of RCMFE is superior to that of RCMSE and MFE.Meanwhile,the fault classification accuracy is higher than that of RCMSE and MFE.

Bending stress of rolling element in elastic composite cylindrical roller bearing

A new structure design method of elastic composite cylindrical roller bearing is proposed, in which PTFE is embedded into a hollow cylindrical rolling element, according to the principle of creative combinations and through innovation research on cylindrical roller bearing structure. In order to systematically investigate the inner wall bending stress of the rolling element in elastic composite cylindrical roller bearing, finite element analysis on different elastic composite cylindrical rolling elements was conducted. The results show that, the bending stress of the elastic composite cylindrical rolling increases along with the increase of hollowness with the same filling material. The bending stress of the elastic composite cylindrical rolling element decreases along with the increase of the elasticity modulus of the material under the same physical dimension. Under the same load, on hollow cylindrical rolling element, the maximum bending tensile stress values of the elastic composite cylindrical rolling element after material filling at 0° and 180° are 8.2% and 9.5%, respectively, lower than those of the deep cavity hollow cylindrical rolling element. In addition, the maximum bending-compressive stress value at 90° is decreased by 6.1%.

Temperature Distribution and Scuffing of Tapered Roller Bearing

In the field of aerospace, high-speed trains and automobile, etc, analysis of temperature filed and scuffing failure of tapered roller bearings are more important than ever, and the scuffing failure of elements of such rolling bearings under heavy load and high speed still cannot be effectively predicted yet. A simplified model of tapered roller bearings consisted of one inner raceway, one outer raceway and a tapered roller was established, in which the interaction of several heat sources is ignored. The contact mechanics model, temperature model and model of scuffing failure are synthesized, and the corresponding computer programs are developed to analyze the effects of bearings parameters, different material and operational conditions on thermal performance of bearings, and temperature distribution and the possibility of surface scuffing are obtained. The results show that load, speed, thermal conductivity and tapered roller materials influence temperature rise and scuffing failure of bearings. Ceramic material of tapered roller results in the decrease of scuffing possibility of bearings to a high extent than the conventional rolling bearing steel. Compared with bulk temperature, flash temperature on the surfaces of bearing elements has a little influence on maximum temperature rise of bearing elements. For the rolling bearings operated under high speed and heavy load, this paper proposes a method which can accurately calculate the possibility of scuffing failure of rolling bearings.

FAULT DIAGNOSIS APPROACH FOR ROLLER BEARINGS BASED ON EMPIRICAL MODE DECOMPOSITION METHOD AND HILBERT TRANSFORM

Based upon empirical mode decomposition (EMD) method and Hilbert spectrum, a method for fault diagnosis of roller bearing is proposed. The orthogonal wavelet bases are used to translate vibration signals of a roller bearing into time-scale representation, then, an envelope signal can be obtained by envelope spectrum analysis of wavelet coefficients of high scales. By applying EMD method and Hilbert transform to the envelope signal, we can get the local Hilbert marginal spectrum from which the faults in a roller bearing can be diagnosed and fault patterns can be identified. Practical vibration signals measured from roller bearings with out-race faults or inner-race faults are analyzed by the proposed method. The results show that the proposed method is superior to the traditional envelope spectrum method in extracting the fault characteristics of roller bearings.

Mechanical behaviors of cross roller bearings with raceway roundness error

Taking the raceway roundness error into account,mechanical characteristics of cross roller bearings(CRBs)were investigated.A static analysis model of CRBs considering the raceway roundness error was established.Based on this model,the rotational accuracy and load distribution of CRBs under constraints of geometry and external loads were derived.The fatigue life of CRBs with roundness error was calculated by applying Palmgren-Miner linear cumulative damage theory.The influence of inner and outer raceway roundness error on the performance of the CRBs,such as rotational accuracy,load distribution,and fatigue life,was studied through the analysis of examples.The results indicate that the influence of roundness error on the rotating inner raceway is more significant than that of roundness error on the nonrotating outer raceway.The roundness error on the rotating inner raceway always degrades the performance of CRBs.However,a proper roundness error on the nonrotating outer raceway can reduce the loads acting on the rollers and thus improve the fatigue life of CRBs.The effect of the roundness error amplitude on the bearing performance is ordinal,whereas the effect of the roundness order on the bearing performance is not in order.

A Roller Bearing Fault Diagnosis Method Based on Improved LMD and SVM

Aiming at the non-stationary feattwes of the roller bearing fault vibration signal, a roller bearing fault diagnosis methtxt based on improved Local Mean Decomposition （LMD） and Support Vector Machine （SVM） is proposed. In this paper, firstly, the wavelet analysis is introduced to the signal decomposition and reconstruction; secondly, the LMD method is used to decompose the recomtnion signal obtained by the wavelet analysis into a ntmaber of Product Ftmctions （PFs） that include main fault characteristics, thus, the initial feattwe vector matrixes could be formed automatically; Thirdly, by applying the Singular Valueition （SVD） techniques to the initial feature vector matrixes, the singular values of the matrixes can be obtained, which can be used as the fault feature vectors of the roller bearing and serve as the input vectors of the SVM classifier; Finally, the recognition results can be obtained from the SVM output. The results of analysis show that the propsed method can be applied to roller beating fault diagnosis effectively.

Elastohydrodynamic Lubrication Studies on Effects of Crowning Value in Roller Bearings

The purpose of this paper is to present a comparison of numerical calculations and experiment results of optical interferometry in finite line contact for the elastohydrodynamic lubrication(EHL) problem of Lundberg's profiled cylindrical roller under the conditions of flooded state, moderate load and material parameter. It shows clearly the effects of crowning value on the variations of oil film shape and thickness. The agreement between numerical analysis and experiment results is very good. The results indicate there must be an optimum crowning value that will induce the thickest and most even oil film in EHL state for a given working condition, and this value is larger than the design value in dry contact state for the same working conditions.

An Improved Kernel K-Mean Cluster Method and Its Application in Fault Diagnosis of Roller Bearing

For the kernel K-mean cluster method is run in an implicit feature space, the initial and iterative cluster centers cannot be defined explicitly. Against the deficiency of the initial cluster centers selected in the original space discretionarily in the existing methods, this paper proposes a new method for ensuring the clustering center that virtual clustering centers are defined in the feature space by the original classification as the initial cluster centers and the iteration clustering centers are ensured by the further virtual classification. The improved method is used for fault diagnosis of roller bearing that achieves a good cluster and diagnosis result, which demonstrates the effectiveness of the proposed method.

Influence of the Foundation on the Threshold of Stability for Rotating Machines with Roller Bearings—A Theoretical Analysis

The paper presents a mathematical model for analyzing the threshold of stability for rotating machines, where the rotor is linked to the stator by roller bearings, bearing housings and end-shields and where the stator feet are mounted on a soft foundation. The internal (rotating) damping of the rotor is the only source of instability, which is considered in the paper. After the mathematical coherences of the multibody model are described, a procedure is presented for deriving the threshold of stability. Additionally, a numerical example is shown, where the threshold of stability is calculated for different boundary conditions. It could be demonstrated, that the stiffness of the foundation—even if the foundation stiffness is isotropic—can help stabilizing this kind of vibration system in the same way as orthotropic bearing stiffness or orthotropic bearing housing and end-shield stiffness for a rigid foundation.

A research on the contact stress of roller bearing based on crowning analysis

According to elastic contact theory, contact model between roller and race is established. Compared with the Hertz results, the results are proved, based on which contact stress distribution of different crowning and initial contact length is given, then the appropriate value is derived. On the basis, inertia force and different radial force is given into consideration. Via analysis, it con- cludes that under balanced pure radial load condition the largest contact stress between roller and race increases along with crowning value increasing. With the same crowning value, the largest contact stress between roller and race decreases in the first and increases at the end along with initial contact length increasing. Contact stress between roller and outer race increases along with revolution speed increasing.

A Method for Incipient Fault Diagnosis of Roller Bearings Based on the Wavelet Transform Correlation Filter and Hilbert Transform

Noise is the biggest obstacle that makes the incipient fault diagnosis results of roller bearings uncorrected; a new method for diagnosing incipient fault of roller bearings based on the Wavelet Transform Correlation Filter and Hilbert Transform was proposed. First, the weak fault information features are picked up from the roller bearings fault vibration signals by use of a de-noising characteristic of the Wavelet Transform Correlation Filter as the preprocessing of the Hilbert Envelope Analysis. Then, in order to get fault features frequency, de-noised wavelet coefficients of high scales which represent high frequency signal were analyzed by Hilbert Envelope Spectrum Analysis. The simulation signals and diagnosing examples analysis results reveal that the proposed method is more effective than the method of direct wavelet coefficients-Hilbert Transform in de-noising and clarifying roller bearing incipient fault.

Stiffness of high-speed cylindrical roller bearing subjected to radial force and bending moment

Presents a practical way of calculation for stiffness of cylindrical roller bearing based on traditional quasistatic analytical methods subject to radial force and bending moment,which take into consideration the effect of such factors as thickness of oil elastohydrodynamic film,pressure,shear force,friction force of cage,centrifugal force and slides,taken into consideration and presents the results of stiffness calculation with a typical example with an aeroengine cylindrical roller bearing were given in the paper.And,discusses the effect of major parameters’influence on stiffness.

Relationship of rolling bearing stiffness with diameter of roller

The theoretical formula of roller bearing stiffness is induced and compared with its empirical formula. In the experience formula the stiffness of roller bearing has nothing to do with the roller diameter. The relationship of roller bearing stiffness with roller diameter was studied using Hz contacting theory. It is concluded that conclusion in experience formula is only approximate result of data processing under special conditions, and the relation between stiffness of roller bearing and roller diameter must be taken into consideration while designing or selecting roller bearings.

Robust Optimization Method of Cylindrical Roller Bearing by Maximizing Dynamic Capacity Using Evolutionary Algorithms

Optimization of cylindrical roller bearings(CRBs)has been performed using a robust design.It ensures that the changes in the objective function,even in the case of variations in design variables during manufacturing,have a minimum possible value and do not exceed the upper limit of a desired range of percentage variation.Also,it checks the feasibility of design outcome in presence of manufacturing tolerances in design variables.For any rolling element bearing,a long life indicates a satisfactory performance.In the present study,the dynamic load carrying capacity C,which relates to fatigue life,has been optimized using the robust design.In roller bearings,boundary dimensions(i.e.,bearing outer diameter,bore diameter and width)are standard.Hence,the performance is mainly affected by the internal dimensions and not the bearing boundary dimensions mentioned formerly.In spite of this,besides internal dimensions and their tolerances,the tolerances in boundary dimensions have also been taken into consideration for the robust optimization.The problem has been solved with the elitist non-dominating sorting genetic algorithm(NSGA-II).Finally,for the visualization and to ensure manufacturability of CRB using obtained values,radial dimensions drawing of one of the optimized CRB has been made.To check the robustness of obtained design after optimization,a sensitivity analysis has also been carried out to find out how much the variation in the objective function will be in case of variation in optimized value of design variables.Optimized bearings have been found to have improved life as compared with standard ones.

Condition Monitoring of Roller Bearing by K-star Classifier andK-nearest Neighborhood Classifier Using Sound Signal

Most of the machineries in small or large-scale industry have rotating elementsupported by bearings for rigid support and accurate movement. For proper functioning ofmachinery, condition monitoring of the bearing is very important. In present study soundsignal is used to continuously monitor bearing health as sound signals of rotatingmachineries carry dynamic information of components. There are numerous studies inliterature that are reporting superiority of vibration signal of bearing fault diagnosis.However, there are very few studies done using sound signal. The cost associated withcondition monitoring using sound signal (Microphone) is less than the cost of transducerused to acquire vibration signal (Accelerometer). This paper employs sound signal forcondition monitoring of roller bearing by K-star classifier and k-nearest neighborhoodclassifier. The statistical feature extraction is performed from acquired sound signals. Thentwo-layer feature selection is done using J48 decision tree algorithm and random treealgorithm. These selected features were classified using K-star classifier and k-nearestneighborhood classifier and parametric optimization is performed to achieve the maximumclassification accuracy. The classification results for both K-star classifier and k-nearestneighborhood classifier for condition monitoring of roller bearing using sound signals werecompared.

Influence of the Number of Rollers on a Tapered Roller Bearing

In roller bearings, the outer ring is usually fixed and the inner ring has the rolling motion. Concerning TRB （tapered roller bearings）, this motion generates forces that are transmitted to the outer ring by the tapered rollers. Thus, contact stresses occur and the number of rollers plays a major role with respect to the load distribution. This influence is analyzed in this study by the FEM （finite element method） with commercial code ABAQUS, where two models were evaluated： a common TRB and the same one but with fewer rollers. As an application, a manual automotive transmission was considered for the input loads.

Friction moment calculation method for tapered roller bearings under combined loads

Tapered roller bearings(TRBs) can withstand axial loads, radial loads, and overturning moments. The performance, safety, and efficiency of rotating machinery are directly influenced by the friction moments within the TRBs. However, most current research has relied on empirical formulas that focus on axial loads. Additionally, the friction coefficient between the rollers and the inner ring rib has been defined using simple empirical methods. In actual applications, the loads on TRBs are not purely axial or radial, and simple empirical friction coefficients do not adequately account for the varying lubrication conditions. To address this challenge, this study proposes an improved method for calculating the friction moments of TRB under combined axial and radial loads. This study employs a calculation method for sliding friction coefficients that can model dry, boundary, elastohydrodynamic, and mixed lubrication conditions. To demonstrate the advantages of the proposed method, the friction moments obtained using the existing and proposed methods are compared. Additionally, the influence of TRB structural parameters on the friction moment is discussed. An experimental study is conducted to validate the effectiveness of the proposed method. The findings provide valuable insights for designing TRB structural parameters to minimize friction moments.