108 Laboratories. The study compares the resolved mounted preload state on a hard preloaded duplex pair of  bearings for variations to the bearing fits, contact angles, curvature, temperature and free preload.  

The final section presents case studies , based on real bearing analyses found in a literature search, where  published results are compared with ORBIS predictions . The first case study is from the 22nd Aerospace  Mechani sms Sy mposia proceedings  titled  “Two Gimbal Bearing Case Studies: Some Lessons Learned” by  S. H. Loewenthal . The second case study is from NASA/TP- 2014- 217906, titled “Analysis of Space Station  Centrifuge Rotor Bearing Systems: A Case Study” by J. V. Poplawski, S. H. Loewenthal, F. B. Oswald, E. V. Zaretski, W. Morales, and K. W. Street.  

Technical Model  Overview    ORBIS uses numerical techniques to solve the nonline ar elastic behavior of the user -defined system of one  or more bearing rows. The model considers eac h ball -to-race contact for all bearing rows defined in the  system,  resulting in complete knowledge of the element load distributions and their raceway attitudes. A  solution to the system is achieved when the sum of all bearing row reaction forces is suffic iently close to  the external appli ed forces (system equilibrium).    The system model follows the same process necessary to assemble a rotational system:   initial conditions  are defined, the bearings are fit into the assembly, preload is applied to the beari ngs, and external loading  is finally applied to the mounted and preloaded bearing system. The parameters describing relative ring displacements and internal clearance changes are tracked at each step of the process; ultimately leading to the final state of  the bearing system.    Since bearings typically operate with very small internal clearances,  it is important to consider the effects  of boundary conditions on the races when establishing the mounted state of the bearing system. ORBIS implements a compliance model based on classical thick ring theories to determine the final mounted and  preloaded state of the bearing system. The compliance model is also used for assessing thermal strains.  Once all mounting, preloading and thermal strains have been considered,  the traditional fixed ring theories,  as developed by Jones, are applied to assess effects due to external loading.  

The compliance model makes a key assumption that  in the local vicinity of the bearing, the housing, shaft  and bearing rings can be expressed with a series of nested rings. Thick ring theory assumes the representative rings have uniform constant wall thickness and all deflections remain within the linear elastic region of the material. Since the groove side of a bearing race clearly has a non- constant cross section,  an  equivalent diameter has been developed to represent bearing rings. The equivalence model is shown in ( 1)  and (2), and has been correlated to detailed finite element models . A key benefit to the equivalence model  is that it is based on the geometry defining the racewa y depth and therefore can account for the general  class of raceways . The compliance algorithms  also track  various sudden changes in the boundary   conditions of each nested ring;  such as when the fit between a bearing outer diameter ( O.D.)  and housing  inner diameter ( I.D.) transitions from clearance to interference due to preload expansion or thermal  expansion.  

 20.68( ) ( ) race racel IR IR l d l d IRhD d hh hhd= + +− −   (1) 

 20.68( ) ( ) race racel OR OR l d l d ORhD d hh hhd= − ++ −   (2)