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Mathematical Modelling of Load and Stress Distribution in Ball Bearing
External load of ball bearing is transferred from one ring to the other one through the balls. Load distribution between balls is unequal. Degree of load distribution inequality depends on internal geometry of bearing (number of balls, internal radial clearance) and external load. The new mathematical model of load distribution is developed introducing new value – the load distribution factor into the classic theory of ball bearings. Developed mathematical model includes all relevant influences on ball bearing load distribution (bearing internal geometry and load). The new load distribution model is used for description of stress distribution in the bearing.
Mathematical Model of Load Distribution in Rolling Bearing
FME Transactions
External load of rolling bearing is transferred from one ring to the other one through the rolling elements. Load distribution between rolling elements is unequal. Degree of load distribution unequality depends on internal geometry of bearing and magnitude of external load. Two boundary load distributions in radially loaded ball bearing were defined and discussed in this paper. These are ideally equal and extremely unequal load distribution. Real load distribution is between these boundary cases. The new mathematical model of load distribution is developed respecting classic rolling bearing theory and by introduction of new, original value defined as load distribution factor. Developed mathematical model includes all main influences on load distribution in rolling bearing (number of rolling elements, internal radial clearance and external load).
CONTACT STRESSES AND DEFORMATIONS IN THRUST BALL BEARING
machine design, 2018
The aim of this paper is to determine deformations and stresses in the statically loaded thrust ball bearing subjected to a centric external axial load. In this case, all balls in the bearing are equally engaged in the transmission of the operational load. The influence of the load on the stresses and deformations in the bearings of different series has been analysed. The obtained results are the basis for further research of the load distribution in an eccentrically loaded axial ball bearing in order to determine the reduction of the static load bearing capacity in relation to the values determined in appropriate standard and also to the values prescribed by manufacturers’ catalogues.
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BALL BEARING STIFFNESS. A NEW APPROACH OFFERING ANALYTICAL EXPRESSIONS
Space mechanisms use preloaded ball bearings in order to withstand the severe vibrations during launch. The launch strength requires the calculation of the bearing stiffness, but this calculation is complex. Nowadays, there is no analytical expression that gives the stiffness of a bearing. Stiffness is computed using an iterative algorithm such as Newton-Raphson, to solve the nonlinear system of equations. This paper aims at offering a simplified analytical approach, based on the assumption that the contact angle is constant. This approach gives analytical formulas of the stiffness of preloaded ball bearing. Notations a Semimajor axis of contact ellipse b Semiminor axis of contact ellipse B = f e + f i – 1Total curvature of the bearing D Ball diameter d m Bearing pitch diameter E Modulus of elasticity e Axial deflection due to preload f f e = r e /D Dimensionless parameter F(κ) Elliptic integral of the first kind K n Ball stiffness k a Axial stiffness of paired bearing k r Radial stiffness of paired bearing p H Hertzian pressure P Bearing preload Q Ball normal load r i r e Raceway groove curvature radius R x R y Equivalent curvature radius S(κ) Elliptic integral of the second kind Z Ball complement α Contact angle δa Axial deflection δn Normal approach along the line of contact δr Radial deflection ε = 0,5 [1+ (δa/δr) tanα] γ = D cosα /d m Dimensionless parameter Γ Curvature difference κ = a /b Elongation of elliptic contact area ν Poisson's ratio ρ = R x / R y
Numerical Results of a Quasi-Static Analysis on Hybrid Ball Bearings
A comparative numerical quasi-static analysis has been made between an angular steel ball bearing and an angular hybrid ball bearing, having the same external and internal geometry (7007C series). Though a quasi-static program offers only a solution independent of time regarding the contact forces, it is useful to appreciate the load distribution and the fatigue life for a rolling bearing. To reduce the maximum load values and to increase the bearing life the results indicate the need of geometry optimisation for the inner race/ ball contacts in the hybrid rolling bearing. KEYWORDS: hybrid rolling bearings, quasi-static analysis 1. INTRODUCTION The rolling bearings' application at ultra-high speeds is limited, because in such conditions the outer ring race is extra-charged due to the centrifugal forces acting upon the rolling elements. Lightweight balls must be used for decreasing the centrifugal forces effects. There are two solutions to obtain lighter balls: (i) the decreasing...
Impact affects the dynamic characteristics of mechanical multi-body systems and damages those rotating parts, such as the joint rolling element bearings, which are high-precision, defect intolerant components. Based on multi-body dynamic theory, Hertzian contact theory, and a continuous contact model, this study proposed a modelling method that can describe the dynamic behaviour of planar mechanical multi-body systems containing a rolling ball bearing joint under impact. In this method, the rigid bodies and bearing joint were connected according to their joint force constraints; the impact constraint between the multi-body system and the target rigid body was constructed using a continuous contact force model. Based on this method, the reflection relationship between the external impacts of the mechanical multi-body system and the variation law governing the dynamic load on the rolling bearing joint were revealed. Subsequently, an impact multi-body system, which was composed of a sliding-crank mechanism containing a rolling ball bearing joint and the target rigid body with an elastic support, was analysed to explore the dynamic response of such a complex discontinuous dynamic system andthe relevant relationship governing the dynamic load on the rolling bearing joint. In addition, a multi-body dynamic simulation software was used to build a virtual prototype of the impact slider-crank system. Compared with the theoretical model, the prototype had an additional deep groove ball bearing. That is to say, the prototype model took account of the specific geometric structural characteristics and the complex contact relationship of the inner and outer races, rolling balls, and bearing cage. Finally, the effectiveness of the theoretical method proposed in this study was verified by comparative analysis of the results. The results suggested that the external impact of a mechanical multi-body system was prone to induce sudden changes in the equivalent reaction force on its bearing joint and the dynamic load carried on its rolling balls. This study provided an effective method for exploring the distribution characteristics of dynamic loads on rolling ball bearing joints under working impact load conditions. Moreover, it offered support for the parameter optimi-sation of geometric structure, performance evaluation, and dynamic design of the rolling ball bearings.
Influence of Abrasive Wear on Ball Bearing Internal Geometry
When the machine environment is contaminated with dust, sand or minerals, small solid abrasive particles could be involved into rolling bearing lubricant. Sharp hard particles cause intensive abrasive wear of bearing parts. This process leads to change of rolling bearing internal macro geometry and contact surfaces micro geometry. Consequently, the bearing looses its operational ability and do not achive predicted service life. An influence of abrasive particles size and concentration on the rolling bearing internal micro-and macro geometry is analyzed in this paper. The presented results are emphasizing the importance of both sealing and lubrication system development keeping rolling bearing operational ability during its predicted service life.