111 Key Analytical  Considerations  

Mechanisms that implement r olling- element bearings typically rely on  certain performance parameters from  the bearings . The most common parameters are load capability, stiffness, torque,  and operational life.  As  it turns out , all of these  parameters are dependent on the in ternal load distributions —meaning normal loads  and associated contact angles —within each bearing defined in the system.  Given accurate loaded contact  angles of each element within a system , Hertzian contact mechanics provide the elliptical contact areas  that are the basis for solving stresses, stiffness, primary torque components, and the fatigue life. Hence,  the importance of accurately predicting the internal load distributions of a system of bearings shoul d be  foremost . 

A point often overlooked in the analysis of bearings, which is particularly  true for aerospace mechanisms  due to their  extremely high launch load environment,  is that external  loads are  typically derived by analytical  means , such as finite element modeling, and thus the bearing element stiffness is required as an input to  the finite element model . This creates a circular process where the bearing stiffness is needed in order to  predict  the loads on the bearings. Additional complications aris e due to the fact that finite element models  require a linear stiffness yet  bearing stiffness is nonlinear with load . Typically, the  best that can be done  here is to determine a linear approximation of the bearing stiffness in the vicinity of the load reco very. This  may require more than one iteration to establish.    Abstract model s of real system s are based on assumptions of the phenomena modeled and it is therefore  prudent to review such assumptions . Let’s first discuss the  inherent assumptions of the  mathematical  theories  derived by A. B. Jones . The Jones model can be summarized as follows. E ach raceway ’s center  of curvature is represented with a fixed diameter circle.  The outer raceway circles  are fixed in space and a  local coordinate frame is attached at  the center of the inner raceway  circle s. Inner raceway displacements,  which are relative to the fixed outer rings, create differences to the normal approach between the inner and outer circles. The location of the ball center, relative to the fixed outer raceways, and solved at each ball  station, is determined by ensuring the ball is in quasi -static equilibrium. A Newton -Raphson search routine  then iterates on displacements to the m oveable coordinate inner rings . Given the normal approach of the  ball center relative to each raceway , classical Hertzian contact analysis is utilized to determine normal loads  and elliptical contact areas  at each raceway  contact point . The vector sum of all ball loads then provides  the resultant force reacting on the shaft for a given set of inner ring displacements. Complete bearing  equilibrium is achieved when the shaft reaction forces are equal , or sufficiently close,  and opposite to the  externally applied loads.    The Jones  method, which is a very elegant derivation, does rely on the assumption that raceway circles  maintain a fixed radius  (a.k.a. fixed ring assumption) . Should the raceways expand or contract it would  change the normal approach to the ball center and could have a profound effect to the internal load distribution. Most traditional bearing materials are intentionally selected to be very hard and stiff  and typical   normal approach dimensions of the contacting bodies are on the order of 5 micromet ers (0.0002 inch). This  implies very small changes to the normal approac h can have a profound effect on contact angles and  normal loads . Such changes should therefore be included in the predictive model to enhance accuracies.    Items such as interference fitting, ring clamping, preloading and thermal expansions will all effect the  bearing ring diameters. Most of which, if not all, are inherent in every mechanism containing rolling- element  bearings. Before continuing our discussion on key model assumptions lets illustrate some common  mounting influences with an example. Figure 4 illustrates a duplex pair of angular contact bearings that are  oriented back -to-back . Although not shown in the figure we shall assume the bearings are hard preloaded  by some means (perhaps a clamp plate or jam nut).  It is well known that such systems as this can be quite  sensitive to the mating fits of the bearing. Also,  since the bearings are mounted in structure with a higher  coefficient of thermal expansion than the bearing material (e.g. , aluminum shaft and housing versus 440C  bearings), we expect that t emperature excursions  will also affect the internal load distribution of the  mounted bearings.