Stiffness Of Ball Bearing Research Articles

A nonlinear dynamic approach for analyzing dynamic stiffness of ball bearings at different raceway diameters and guiding clearances

In this work, the stiffness matrix model is integrated into the dynamic model of ball bearings, in which the dynamic forces real-timely affect the stiffness calculation to form the dynamic stiffness, which is a novel solution for the dynamic performance analysis of ball bearings under different loads, assemblies and bearing structures. Then, the main influencing factors for the magnitude of bearing stiffness are analyzed. Next, the fluctuation mechanism of bearing stiffness is revealed. Subsequently, the effects of raceway diameters and guiding clearances on the dynamic stiffness of ball bearings are investigated. The obtained results suggest the ball diameter and the number of balls at various raceway diameters alter contact loads, contact angles, contact deformations and offset distances to change the bearing stiffness, and the combinations of inner and outer raceway diameters should be compromised to attain the favorable magnitude and fluctuation of bearing stiffness. Moreover, guiding clearance should be adjusted to achieve the desired trajectory of cage to degrade the fluctuation of bearing stiffness.

Influence of various balls and groove curvature radii on dynamic stiffness of ball bearings based on nonlinear dynamic model

In the context of this study, a novel dynamic model is integrated with the stiffness matrix model so that the dynamic forces effect the stiffness calculation real-timely to form the dynamic stiffness, which is a novel solution for the dynamic performance analysis of rolling bearings under different loads, assemblies and bearing structures. Based on this model, the influence of ball quantity and groove curvature radii on the dynamic stiffness of ball bearings is explored. The optimal ball's quantity and groove curvature radii combination were determined to attain the desired magnitude and fluctuation of bearing stiffness. The results show that favorable stiffness can be achieved when the number of balls is decreased by one or two, the radius of outer groove curvature approximates the ball radius, and the radius of inner groove curvature is appropriately larger than the ball radius. This configuration offers theoretical backing for optimizing the structural dimensions of ball bearings.

Особенности применения теории силовых смещений к расчету контактной жесткости опоры шпиндельного радиально-упорного шарикоподшипника

Contact stiffness is an important parameter for describing the interaction of plenty precise engineering solutions, for example, for machine-tool manufacture, the most important element of the carrier system is the spindle assembly supports, largely determining the accuracy of machining parts on the machine. The article considers an example of a system approach to describing the variable stiffness of a high-precision spindle bearing, combining an analytical approach to calculating the variable quasi-static stiffness of an angular contact ball bearing and a methodological approach that takes into account the contact normal and tangential force displacements of the upper bearing ring while considering the friction force for tangential interactions. The proposed system approach to calculating the stiffness of a ball bearing support is based on fairly simple models, in particular, the apparatus proposed by Professor P.M. Chernyanskiy, describing the phenomena of changing the spindle supports stiffness. These changes were experimentally obtained and weren’t described in earlier works. It is shown that the experimental stiffness consists of elastic and contact components, depending on the friction forces, therefore, the concept of apparent stiffness is introduced for data that are not taken into account in the processes under study. The effect of the axial displacement of the spindle is explained on the basis of contact interactions in the support of the spindle unit. The resulting analytical system model was tested on well-known bearing supports and the obtained stiffness solutions were compared with validated experimental data from open sources

Analytical Determination and Influence Analysis of Stiffness Matrix of Ball Bearing under Different Load Conditions

Bearing stiffness, as one of the most important service characteristics for ball bearing, plays a crucial role in the bearing design and rotor dynamic analysis. To rapidly and accurately calculate the stiffness matrix of ball bearing under the arbitrary load conditions, a 5-DOF analytical model for bearing stiffness matrix analysis has been established by the ball–raceway contact analysis, implicit/explicit differential method, and matrix operations. The model has been validated comparing with the previous methods and experimental results. Based on this, the model has been used to investigate the influences of the load and operation conditions, the structural parameters variation on stiffness of ball bearing. The results show that property increasing axial preload can inhibit the attenuation of speed-varying stiffness, and the contact states between balls and raceways also have significant influence on the change in the stiffness of ball bearings. Besides, a larger curvature coefficient of inner raceway and a small curvature coefficient of outer raceway can effectively improve the stiffness of ball bearing at high speed. Therefore, the proposed method can be a useful tool in bearing optimize design and performance analysis of ball and rotor system under various load conditions.

Parametric studies on dynamic stiffness of ball bearings supporting a flexible rotor

Most of the researchers in the field of dynamics of the rolling element bearing have considered bearing stiffness as time invariant and/or not related to dynamics of the bearing. In the present paper, the bearing stiffness has been taken as function of dynamic response at every time step of numerical simulation and a detailed parametric study is performed to investigate the effect of flexibility of the rotor shaft, rotational speed, and internal radial clearance on the instantaneous and average value of dynamic stiffness of the ball bearing. The mathematical formulation is based on the Timoshenko beam finite element theory. Gravity and bearing forces are considered as external forces acting on a free-free flexible shaft. A stable Newmark-β numerical integration scheme coupled with Newton–Raphson method is used for numerical integration and for convergence to an accurate value of bearing stiffness. The results showing the variation of different components of bearing stiffness as a function of time-invariant parameters has improved the understanding of the dynamic behavior of the bearing during motion. The variation pattern of bearing stiffness coefficients is observed to be sensitive to direction of rotation. The amplitude of periodic change of these coefficients increases with the increase of the stiffness ratio of shaft and the decrease of radial clearance.

볼 베어링의 강성과 스퍼기어 계의 모드 특성

Abstract : This study focuses on a detailed ball-bearing model for spur gears. The nonlinear ball-bearing stiffness with radial clearance is derived and calculated. The bearing stiffness is used to the 3-degree-of- freedom (DOF) spur gear system model. The mode characteristics of the gear system model are analyzed and verified by performing finite-element analysis (FEA). From the results, the bearing stiffness on the radial clearance was more sensitive under low-load conditions compared to high-load conditions. The bearing stiffness significantly affected the gear dynamics in the low-frequency region, while the mesh stiffness affected the high natural frequency. †Corresponding Author, pci@gwnu.ac.krⒸ 2016 The Korean Society of Mechanical Engineers - 기호설명 -  : 베어링 반경방향 틈새  : 기어 감쇠  : 축과 베어링의 조합 감쇠  : 볼지름  : 내륜지름  : 외륜지름  : 베어링 피치 지름e:정적 전달오차E:탄성계수EI : 굽힘정수  : 베어링에 작용하는 외력H:하중을 받는 베어링 구름요소수   : 베어링강성   : 내륜 접촉강성   : 외륜 접촉강성  : 내륜-전동체-외륜 접촉 유효 강성  : 기어 치강성  : 축과 베어링의 조합 강성n: 베어링 하중변위관계 지수

Rapid-convergence iterative method for Hertzian point contact theory

Point contact theory between two bodies with constant curvature was built by H. Hertz. His theory is now widely known as "Hertzian contact theory" and it is widely applied to the calculation of the stiffness of ball bearings and CVJs. Calculation of the theory, however, has some difficulties: numerical calculation of the complete elliptic integrals and convergence calculation for ellipticity parameter are required. Some fitting equations have been already suggested to resolve the difficulties. They are convenient for hand calculation, but they are too rough to calculate with the latest high-spec computers. So more accurate algorithm for Hertzian contact theory is required. In this paper, rapid-convergence iterative method is suggested for computing the theoretical solution of Hertzian contact theory. The method can be programmed simply and converge rapidly because the method is based on "self-convergence algorithm" which is also suggested in this paper. As 10-digit accurate values can be get by 3 iterations, the method is fast enough to apply to dynamic simulations, in which point contact has to be evaluated very frequently.

Theoretical and Experimental Study of Spindle Ball Bearing Nonlinear Stiffness

ABSTRACTThis paper concentrates on theoretical and experimental nonlinear stiffness study of milling machine tool spindle angle contact ball bearing. The theoretical study allows us to build an analytical model to define nonlinear stiffness of angle contact ball bearings based on geometrical and physical parameters. Modifications were done on literature's models (e.g. Balls deformations) having positives impacts on conformity of models to experimental results.FEM model using ANSYS is constructed to analyze the different parameters affecting the nonlinear stiffness of ball bearing. Among those parameters are physical including the geometry, friction coefficient and the boundary conditions of the model and Numerical parameters such as mesh density and penetration.Experimental tests were done on the spindle ball bearing 7014, to measure the rigidity. Universal tensile testing machine is used to achieve load displacement curves. The developed theoretical model, constructed finite element model and experimental results showed good conformity.

Forced Vibration Analysis of an IPM Motor for Electrical Vehicles due to Magnetic Force

This paper discusses a method to analyze the forced vibration of an interior permanent magnet (IPM) motor due to magnetic force. A structural finite element (FE) model of the IPM motor that includes stiffness of ball bearings and laminated effect of core is developed and verified by comparing the simulated natural frequencies and mode shapes with experimental data. A magnetic FE model is also developed and both the radial and tangential magnetic forces are calculated in the air-gap. These forces are subsequently transformed into equivalent nodal forces and applied to the structural FE model to investigate the forced vibration characteristics of the IPM motor. It was found that the magnetically induced vibration of a stator mainly results from the contribution of the dominant harmonics of the magnetic force and structural resonance. In addition, the magnetically induced vibration of the rotor mainly appeared as rigid body modes due to the flexibility of ball bearings.

Study of milling stability with Hertz contact stiffness of ball bearings

This present work examines the stability and nonlinear responses of a spindle milling system supported by ball bearings. A shaft finite element based on Timoshenko beam theory is employed to model the spindle, and modal reduction method is therefore adopted for saving the numerical calculating time. The issues of evaluating the effects of the ball bearing Hertz contact stiffness are consequently addressed. It is found that suitable constant bearing stiffness can be adopted to replace the nonlinear nonsmooth Hertz stiffness in prediction of the critical cutting depth of the milling system in certain bearing configuration conditions. For the constant bearing stiffness can be obtained by experiment, this replacement will undoubtedly simplify the spindle-bearing milling system. But with the increase in the bearing clearance, the spindle milling system will present obvious nonlinear behaviors, and the nonlinear Hertz contact bearing stiffness will take over. Isolated islands of chatter vibration, which are induced by the nonlinear nonsmooth bearing Hertz stiffness, can be found exist in milling processes in large bearing clearance conditions.

A simple and compact model of defects and non-linear dynamic stiffness of a ball bearing

This paper presents a modelling study of a ball bearing dynamic behaviour with different defects types, which is part of an investigation related to the modelling of machinery rotating components. Our contribution in this work is a proposition of a five-degree of freedom model describing the nonlinear dynamic behaviour of a ball bearing. The aim is a parametric formulation of the ball bearing stiffness which allows the introduction of the defects characteristics. This approach is a result of an intrinsic structural behaviour making it different from methods which introduce external impulsion to simulate defects. Hence, a more realistic dynamic ball bearing defect simulation is obtained for better use in design and maintenance domain. This simulation can be formulated by two methods. The first, partial contact method, is based on elimination of some ball stiffness. The second one imposes displacement in the system response. Obtained results give response forms similar to standards and to different author’s results (theoretical and experimental) found in the literature. The developed model is simple, compact as compared to existing ones and we can express our satisfaction about this promising model.

Rotordynamic Analysis of a Turbopump with the Casing Structural Flexibility

A critical speed analysis is performed for a 30-ton thrust demonstrator turbopump considering the casing structural flexibility. A full three-dimensional finite-element method including rotor and casing is used to predict rotordynamic behavior. The rotor-alone model and the rotor-casing-coupled model with fixed-fixed and free-free boundary conditions are calculated to investigate the effects of the casing structural flexibility. The stiffness of ball bearings is applied as unloaded and loaded values to consider rotor operating conditions in a vacuum and real engine, respectively. From the results of the numerical analyses, it is found that the effect of the casing structural flexibility reduces the critical speeds of the turbopump. Especially, the loaded rotor condition with higher bearing stiffness is affected dramatically more than the unloaded rotor condition with lower bearing stiffness.

케이징 구조 유연성을 고려한 터보펌프 임계 속도 해석

케이징 구조 유연성을 고려한 30톤 추력급 터보펌프에 대한 임계 속도 해석이 수행되었다. 로터와 케이징을 포함한 전체 모델에 대한 3차원 유한요소법이 회전체 동역학 특성을 예측하기 위하여 적용되었다. 케이징 구조 유연성의 영향을 고찰하기 위하여 로터 단독 모델과 고정-고정 및 자유-자유 경계조건을 갖는 로터-케이징 연계 모델에 대한 수치 해석이 이루어 졌다. 무부하 하중 조건과 부하 하중 조건에서의 볼 베어링 강성이 각각 진공 조건과 실제 엔진 장착 조건에서의 작동을 모사하기 위하여 사용되었다. 수치해석 결과로부터 케이징 구조 유연성의 영향이 터보펌프의 임계 속도를 감소시킴을 확인하였다. 특히, 높은 베어링 강성을 갖는 부하 로터 조건에서 낮은 베어링 강성을 갖는 무부하 로터 조건에 비하여 더 많은 영향을 받음을 알 수 있었다. A critical speed analysis is performed for a 30 ton thrust turbopump considering the casing structural flexibility. A full three-dimensional finite element method including rotor and casing is used to predict rotordynamic behavior. Rotor alone model and rotor-casing coupled model with fixed-fixed and free-free boundary conditions are calculated to investigate the effects of the casing structural flexibility. The stiffness of ball bearings are applied as unloaded and loaded values to consider rotor operating conditions in vacuum and real engine respectively. From the results of the numerical analyses, it is found that the effect of the casing structural flexibility reduces the critical speeds of the turbopump. Especially, the loaded rotor condition with higher bearing stiffness is affected dramatically rather than the unloaded rotor condition with lower bearing stiffness.

Ball Bearing Stiffness Incorporating Bearing Ring Flexibility.

An investigation was carried out to explain the ball bearing stiffness which took into account the bearing ring flexibility, and the relationship between the deformation of the bearing ring and the stiffness of the ball bearing under an axial load. In this study, first, the axial stiffness of the ball bearing under the axial load and the deformation of the outer ring were measured. Second, the calculation method of the bearing stiffness incorporating the bearing ring flexibility was proposed. Third, the validity of the presented calculation method was discussed. Fourth, the deformation of the bearing ring was discussed. From the results of experiments and analysis, the conclusions were obtained as follows: (1) The calculation method of the bearing stiffness incorporating the bearing ring flexibility was proposed. The axial stiffness of the ball bearing under the axial load can be estimated using the presented calculation method. (2) The outside surface of the outer ring of the bearing under the axial load causes a deformation in the pyramid with the polygonal number n. The polygonal number n is equal to the number Z of balls. The bottom face of the pyramid is where the axial load is applied. (3) The out of roundness of the periphery of the outer ring of the ball bearing under the axial load changes with the axial position of the outer ring. The axial position where the out of roundness is maximized coincides with the axial position of the contact point between the ball and the outer ring.

Reduced Weight Design of a Flexible Rotor with Ball Bearing Stiffness Characteristics Varying with Rotational Speed and Load

This paper presents an effective design approach for reducing the weight of a flexible rotor in ball bearings with rotational speed and load dependent stiffness characteristics under constraints on the system eigenvalues and bearing fatigue life. Design variables are chosen to be the inner radii of shaft elements, the positions of ball bearings, and the preloads on the bearings. The stiffness characteristics of high speed ball bearings are completely described as functions of applied loads and spin speed, and applied to the dynamic behavior analysis of a rotor-bearing system. A transfer matrix method is used to obtain eigenvalues of the system and an augmented Lagrange multiplier (ALM) method is employed as an optimization technique. A multi-stepped rotor supported by two angular contact ball bearings is analyzed and designed to show the speed and load dependent stiffness effect on the system dynamic behavior and to demonstrate the effectiveness of the proposed optimum design approach. The results show that the effect of the stiffness on the system dynamic behavior is noticeable and that the suggested design approach is effective. [S0739-3717(00)00803-5]

STRUCTURAL INTENSITY CALCULATIONS FOR COMPLIANT PLATE–BEAM STRUCTURES CONNECTED BY BEARINGS

A computational strategy, based on component mobility and modal synthesis approaches, is described to calculate structural power flow through multi-dimensional connections such as rolling element bearings and joints. Research issues are discussed in the context of narrow band frequency analysis methods for vibration energy transmission and dissipation. Through the example case of a beam (shaft), ball bearings and an elastic machinery casing plate, the structural intensity calculation procedure is illustrated. A new pre-synthesis algorithm is outlined which is utilized to determine the effective stiffness of ball bearings while accounting for the compliance of the neighbouring structure. The finite element method is used to facilitate computations and to generate structural intensity results in the post-synthesis mode. Sample results are included along with a discussion of various research issues.

Stiffness of deep-groove ball bearings

Ball bearing stiffness is an important parameter in the design of machine tool spindles because of its effect on the performance of the spindle system. As bearing stiffness is given in manufacturers' catalogues and handbooks it would be useful to derive an equation to predict this parameter. Such an equation for predicting the stiffness of deep-groove ball bearings is derived and expressed in terms of the available bearing dimensions. Experimental verification showed that the predicted results are satisfactory.

Vibration of Rotor Based on Ball Bearing : 2nd Report, Static Stiffness of Ball Bearing Containing a Small Number of Balls

An analysis of static stiffness of a ball bearing which contains three and/or four balls is made as an extension of the former report. Though the bearing is far from being useful, it provides us with one of the most important models in examining the spring property inherent in this mechanism. All the effects due to the irregularity usually arising in the course of manufacturing and assembling are taken up for consideration, while the exciting forms of oscillation, as suggested in the former report, are actually represented by this ball bearing which is the most simple practical one. The excitation in the direction of rotor axis is motivated variously : (a) by simple force due to inner ring rotation, (b) by displacement due to rotation of inner ring and balls, (c) by parametric as a Mathieu system due to ball rotation and (d) by variable clearance due to inner ring rotation. Correspondence of exciting form to its cause was disclosed clearly by means of an approximated analytical expression. For instance, it is possible that a mounting error, e.g. an obliqueness of rings, constitutes an important origin of oscillation, especially in the direction of rotor axis, i.e. the so-called abnormal vibration.

Vibration of Rotor Based on Ball Bearing : 1st Report, Static Stiffness of Ball Bearing

That a ball bearing forming a link of an oscillatory system presents a strange and complicated character, when seen as one of the springs, is quite probable from its mechanism, and some of its features have been already pointed out. The first step in the investigation of the vibration of the rotor supported on ball bearings is to know its spring property in detail, but there are few papers that have pursued that question sufficiently. Here is reported an analysis which is simple but effective for inspecting the character of spring due to Hertz contact of elements in the bearing. It proposes a simple method of estimation with some considerations on the effects of production and assembling accuracy, which were given little consideration as spring theory and an actual calculation on the case containing only two balls which is a simple but important model with which to penetrate the essential feature of this mechanism. The authors believe that this theory gives a key to the causes of vibration especially in the direction of revolving axis, which is a curious trouble among the rotors constructed comparatively rigid e.g. an induction motor.

Vibration of Rotor Based on Ball Bearing : 1st Report, Static Stiffness of Ball Bearing

Vibration of Rotor Based on Ball Bearing : 1st Report, Static Stiffness of Ball Bearing