Full report

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NASA Technical Memorandum 108532 Modeling of Rolling Element Bearing Mechanics_Computer Program Updates S.G. Ryan March 1997

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NASA Technical Memorandum 108532 Modeling of Rolling Element Bearing Mechanics_Computer Program Updates S.G. Ryan Marshall Space Flight Center • MSFC, Alabama National Aeronautics and Space Administration Marshall Space Flight Center • MSFC, Alabama 35812 March 1997

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TABLE OF CONTENTS Page INTRODUCTION ................................................................................................................................... 1 THEORETICAL MANUAL UPDATES ................................................................................................. 1 Solution Methodology (Section 2.1) .............................................................................................. 1 Constraints and Solution Procedure (Sections 7.1-7.3) ................................................................. 2 Programming Implementation (Section 8) ..................................................................................... 6 USER' S MANUAL UPDATES ......................................... .................................................................... 13 Effects of Functional Changes ...................................................................................................... 14 Effects of Theoretical Changes ... .................................................................................................. 18 Additional User's Manual Changes .............................................................................................. 20 GENERAL COMMENTS ON ASSUMPTIONS AND INHERENT ERRORS ................................... 21 APPENDIX ........................................................................................................................................... 23 ..° 111

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LIST OF ILLUSTRATIONS Figure Title Page la. FEREBA subroutine calling tree ....................................................................................... 8 lb. FEREBA subroutine calling tree (continued) .................................................................... 9 lc. FEREBA subroutine calling tree (continued) .................................................................. 10 2. FEREBA flow chart ......................................................................................................... 11 3. UPDATE flow chart ......................................................................................................... 12 4. Overall schematic of REBANS operation ....................................................................... 13 5. Example force/deflection curve ....................................................................................... 15 6. Gap 2 constraint forces for ball locations 1, 2, and 3 ...................................................... 16 7. Outer race curvature center deflections polar plot ........................................................... 17 8. Example of FEREBA status information output ............................................................. 19 iv

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TECHNICAL MEMORANDUM MODELING OF ROLLING ELEMENT BEARING MECHANICS-- COMPUTER PROGRAM UPDATES INTRODUCTION The Rolling Element Bearing Analysis System (REBANS) was developed for NASA's Marshall Space Flight Center (MSFC) by Aerojet. The initial version (version 1.10), delivered in December 1994, contained several errors and exhibited numerous convergence difficulties. The program has been modified in house at MSFC to correct the errors and greatly improve the convergence (version 2.0). The most significant changes are related to the improved convergence and will be described in the Solution Methodology section and the Constraints and Solution Procedure section. The other changes, corrections, and enhancements will be discussed in the Programming Implementation section. THEORETICAL MANUAL UPDATES The changes affecting convergence address areas discussed in sections 2.1, 7, and 8 of the original Theoretical Manual. In this discussion, equations and figures in the original manual will be referenced by the original numbers, preceded by an uppercase letter "A." The original nomenclature will be used, as well. Many aspects of the original theory remain applicable after the changes; therefore, the reader should be familiar with the initial theory before examining the updates. The introductory material of section 7 (pages 7-1 through the beginning of 7-3) remains valid. Likewise, sections 7.4 and 8.4 remain valid. The material which follows modifies the remainder of sections 2.1, 7, and 8. Solution Methodology (Section 2.1) The program can operate in two modes. The primary mode is to accept user-specified shaft/inner race displacements and calculate the resulting loads. The other mode is to accept forces and moments as input and determine the corresponding shaft/inner mode with an outer iteration loop around it. race displacements. This mode is essentially the first For either of the solution modes described above, the fundamental problem to be solved is this: given a position of the shaft/inner race, determine rolling element positions and loads and race elastic deflections. The resulting rolling element loads are then summed to yield the net forces and moments on the shaft. This fundamental problem is the area addressed by the changes.

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Theoriginal approachutilized sequentialconvergenceof nestedloopsto achievefinal convergence. This approachprovedto be seriouslydeficientin robustness.Convergencewasmoretheexceptionthan therule.The approachwaschangedto iterateall variablessimultaneously.This hastheadvantageof usingknowledgeof the effect of eachvariableoneachothervariable(via thesystemJacobian)when determiningthe incrementalchanges.The stepsareasfollows: . The outer ring is fixed at its current position (begins at initial guess). The inner race is at a prescribed position. Using classical quasi-static analysis procedures, the rolling element positions and loads are determined. . Calculated loads are applied to the outer and inner races and elastic deflections calculated. Nonlinear gap closures are identified and constraint forces determined. The constraint forces are now modeled using artificial, stiff springs at closed gaps. The partial derivatives of rolling element and constraint forces with respect to race elastic deflections are determined using finite perturbations. These data are used to calculate incremental changes in race elastic deflections. This process is repeated until convergence is attained. . If the program is running in the force input mode, a series of inner race displacement perturbations is made in order to calculate the partial derivatives of forces and moments with respect to inner race displacements. These data are used to calculate increments in the inner race displacements, and the process is repeated until convergence is achieved. Constraints and Solution Procedure (Sections 7.1-7.3) The original approach to defining constraint forces was to force gap deflections to identically equal clearances at locations where contact occurs. For a given set of assumed closure locations, the contact forces required to enforce the constraints can be calculated as described in section 7.3. The difficulty with this approach is that the calculated constraints then change the race deflections and generate a new set of closure locations. This requires an iterative solution for each stage of the larger iteration loop involving rolling element forces. The current approach treats the contact forces using artificial stiff springs. The formulation begins using equation (A7-2) for the inner race, yielding: Uli and augmenting it with the corresponding equation (1) o 0 C0_2 Fbi

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The subscripts and superscripts o and i have been added to refer to the outer and inner race, respectively. The outer race rolling element force vector is a function of both the inner and outer race curvature center deflections: Fbo = Fbo (Ulo , Uli ) (2) The inner race rolling element force vector is a function of both, as well: Fbi = Fbi (Ulo, Uli ) (3) The outer race constraint force vector is a function the outer race and carrier: only of the deflections of potential contact points on f-Kcontac t (U3rac e -U3carrier-clearance); for closed gaps Fc Fc(U3)= [o (4) ; for open gaps where Kcontact is set to a value much larger than any actual stiffness in the system. In terms of the gap variable defined in equation (A7-9) (equation (A7-14) for negative faces), equation (4) for closed gaps becomes: (5) Fck = Kcontac t (U3(Ck)+ Gk -U3(O k)) = Kcontac t Ag (k) Equation (1) can be written more concisely as u = CF (u) (6) This implicit equation can be iteratively solved using Newton's method. Applying a first order perturbation to equation (6) yields: U o+6U=CF(U0)+C which can be rearranged to: II C 0F ] (7) 5U _Uo (8) - _-Uu0J b'U=CF(U O)-U 0

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Equation (8) now forms the basis for the iterative solution. Substituting the nth iterate for the equilibrium value (subscript 0) results in: I - C cgF u n ] bUn = CF(U n)- U n = R n (9) where Rn is the residual vector which should ideally be zero when a solution is found. The Jacobian (--) and the residual must be evaluated at every iteration point. Once this is done, equation (9)can be solved for SUn using standard linear algebra techniques. Convergence is achieved when the error norm, defined by: IIRnll (lo) err n -]lUn[[ is less than a specified tolerance. Equation (9) can be expanded to the notation of equation (1) yielding: - O3Fbo OFbo - c_Ulo auli Ic% co o ] i-Lc % c%0 C0_2o 1 C)Fbi _Fbi c% c% c% c% 0 0 4 0 {5Ulo1 Uli J n 0 Un. o] (11) -[_I ILi n

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The solution estimates are updated as follows: [ '1ol : I<o/ (12) tuiU3iln+l tu1U3iln LC_UliJn The process is repeated until convergence is attained. Examination of equation (11) reveals that there are two types of partial derivative evaluations required. The first is the partial of inner and outer race rolling element forces with respect to inner and outer race curvature center elastic deflections. These partials are determined by numerically perturbing each curvature center variable independently and forming finite difference approximations to the partial derivatives. The second is the partial of the constraint force with respect to deflections of the outer race and carder gap variables. It can be seen from equation (5) that, for closed gaps, this partial is simply Kcontac t with the appropriate sign based on whether the face is positive or negative and whether the variable is on the race or the carrier. For open gaps, it must first be determined whether the gap corresponds to an unloaded preload spring. This is correctly addressed in section 7.4. If the gap does not correspond to an unloaded preload spring, the partial is zero. If it does, the partial is equal to Kprei (equation (A7-21)) with the appropriate sign based on whether the face is positive or negative and whether the variable is on the race or the carder.

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Programming Implementation (Section 8) This section will describe the program logic and subroutine hierarchy required to implement the previously defined theory. In addition to the theoretical changes discussed, some additional functional and minor theoretical changes have been made in the implementation: . FEREBA will now write the reduced flexibility matrices to a disk file. This enables the user to make subsequent analyses which use the same race model without having to recompute these matrices. . For the displacement input mode, multiple sets of inner ring displacements can be read from a file. These will be solved sequentially. The initial guess for a new solution set will be the solution values from the previous set. This facilitates the determination of load-deflection curves for a bearing. . The displacement input mode can be run with force iteration in the axial direction only. This allows the user to specify a fixed axial load while determining the reactions to the lateral displacements and rotations. This is also useful to determine the initial axial displacement needed to yield a certain initial preload for the more typical displacement input mode. This feature is primarily useful for simplex bearing arrangements. 4. FEREBA will calculate an average "rigid body" tilt for the outer ring based on weighted averages of the elastic deflections of the finite element model node locations. . Several additional output files are now generated for each run. For the multiple case displacement input, the displacements and loads are written in a tabular format to an ASCII file and to a binary file. These files facilitate interpretation and plotting of results. The outer ring constraint forces are also written to a binary file. In addition, the capability has been added to graphically display the outer race curvature center (ORCC) deflections as a distorted ring. The values defining these curves are written to another binary file for plotting. . For simplex bearing configurations, the user can specify a lateral stiffness between the race and carrier at the preload spring contact face. This is used to represent the lateral load path which can occur due to friction between the spring and the cartier and race axial faces. . A set of external forces can be applied to the outer race independently of the preload and internal reaction forces. One interesting use of this capability is to apply a moment to force the race to cock or tilt within the deadband resulting in "corner-to-corner" deadband contact or gap closure. This would be used to partially simulate a cocked and hung bearing. Since friction forces are not included, the race would still be free to slide axially. 8. The original code required two radial master degrees-of-freedom (DOF's) on the outer diameter of the outer race. This introduced a significant error due to the large spatial resolution in the axial 6

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direction.The codehasbeenchangedto requirethreeradial masterDOF's on this surface. This allows for "comer" nodes to properly carry tilt loads and a central node to account for "bridging" of the race. The bearing support configuration descriptions in section 3.1 of the Theoretical Manual have been applied updated for cases affected by this change (IBSCOR=2, 3, 5, 6, 8, 9). These updates are included in the appendix. 9. With the new solution approach, an initial guess (call to subroutine INDEL) must be made for the outer race deflections, even in the displacement input mode. 10. The constraint transformation matrices Ts defined in equation (A7-17) and Tfdefined in equation (A7-20) provide notational convenience but are computationally inefficient. A more efficient implementation has been applied that uses index arrays and indirect addressing of array components to perform the same functions. 11. Duplex bearing analysis previously required an ANSYS model of the inner race-shaft in order to determine initial offsets needed for a specified preload. A subroutine has been added (CIRIG) that generates an inner race compliance matrix corresponding to two rigid rings connected by "pseudorigid" (i.e., very stiff) springs. This allows duplex bearing analysis without requiring a full finite element model of the inner race. 12. For duplex bearing analysis, there was previously a convergence error in the iterative calculation of the preload induced contact angle (equation (A5-32)). This value is used in the calculation of the preload-induced axial elastic displacement of the inner rings (equation (A5-35)). The function, f((Xp), and its derivative,f ((Xp), contain a singularity at the value (Xp = 0_°. The original approach used 0¢° as the initial guess for the iteration and, consequently, it never converged. The initial guess was modified to 1.10_ ° to avoid this singularity. Starting at a value greater than a° is appropriate since axial preload will always increase the angle from its unloaded value. 13. The rolling element orbital speed used in the centrifugal force calculation was based on the initial estimate of rolling element positions from subroutine GUESS. The algorithm has been revised to update the speeds based on the positions obtained from an initial solution by DNEQNF (same as the original final solution). With the updated DNEQNF is called again to yield a revised orbital speeds (and resulting centrifugal forces), solution. This process could theoretically be iterated until convergence is achieved; however, the speed change was found to be insignificant after only one update so no looping logic was coded. The subroutine calling hierarchy is shown in figures la through lc. The subroutines listed in bold type are newly created to implement the program changes. The subroutines listed in italic type are not used by FEREBA but are included for anticipated future functionality. The logic flow is presented in the flow chart of figures 2 and 3. Figure 2 is the overall the UPDATE subroutine. The variable list of section program flow chart and figure 3 is the flow chart for 8.4 in the original manual remains valid. 7

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FEREBA --ERSET IMSL routineto control responseto error conditions (normally commented out) DBLSET sets double precision arraysto a specifiedconstant(used in many routines) mGETARG returns command line arguments(system specific routine) READAF readsanalysisvariables and bearing datafrom PREBANgeneratedinput file DATOUT prints analysis summary--geometry, loads,fits, materials,temperatures -- FDATE writes analysistime anddate to output file (system specific routine) ROLDAT prints dataon rolling elementsspecific to ball or roller bearing mCALCON calculatesgeneralconstants relevantto all bearingtypes --TMPFIT calculateschangein clearancesdueto differentialtemperature effects --BCON defines parametersspecific to ball bearings • I-. ABDEL calculatesdimensions and elasticconstantsfor point contact --CRCON defines parametersspecificto roller bearings ABDEL calculatesdimensions and elasticconstantsfor line contact SLICES calculatesslice constantsand crown drops for cylindrical roller bearings SETGAP sets gapbetweenroller flangeand roller enddependingon flangetype " --SPRING determinesapproximate spring compliancesfor axial, radial, andangular loads --INDEL makesinitial guess on outerring elastic deflectionsand shaft/inner ring global displacements -- SYSCON prints nonzeroarray elementsin the SYSTEMcommon block (used in many routines) --FEMFLX readsfinite elementsubstructures and convertsto reducedflexibility matrices --READMT reads previously written reducedflexibility matrices (avoids rerunning FEMFLX) _RD8ASC readsASCIIformat ANSYSsubstructure matrices, storing stiffness only --TRANAF reordersDOF'sin stiffness matricesto FEREBAsystem --MATPRT prints out matrices 12 columns per page(for diagnostics) --ADDSTF adds preloadsprings and 'dummy' springs for deadband --MATRED condensesDOFout of cylindrical stiffness matrices I.- SHIFTS shifts rows and columns of stiffness matricesover removed DOF --DLFTRG IMSL routine that computes LUfactorization of a matrix --DLFIRG IMSL routineto obtain solution to linear equationset --WRITMT writes reduced flexibility matricesfor usewith subsequentreruns --CODIM sets dimensionsof outer ring/carrierflexibility matrix --COPART stores requiredpartitions of outer ring/carrierflexibility matrices _CIDIM sets dimensionsof inner ring/shaft flexibility matrix --CIPART stores required partitions of inner ring/shaft flexibility matrices --CIRIG generatespseudorigid inner ring flexibility matrix for duplexcase without ANSYSmodel -- ETIME elapsedtime counter (system specific routine, usedin many routines) --QSTATE main driver of quasi-static equilibrium problem --GAPDEF setsvalues of deadbandgapsdependingon ring configurations --PRELOD determinesaxialdisplacement of inner ring dueto axialpreload--used onlyfor duplexcase I NRSOLV determinesaxial preloadcontact angle using a Newton-Raphsoniteration DEFLOP calculatesaverageelastic ORCCdeflectionsdueto axial preload CNSTIF computes constraint transformation matrix (for gap closures) _. MEMSET sets single precision or integer arraysto a constantvalue (used in manyroutines) DLINRG IMSL routine to invert constrainedflexibility matrix DEFLIP calculatesaverage linearelastic IRCCdeflectionsdueto axial preload continues FIGURE 1 a.--FEREBA subroutine calling tree.

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from QSTATE L.for all specifiedinner racedisplacements (multiple run feature) - DO ITRF= 1 NITRF (applied load iteration loop) - DO ITR = 1, NITRD (elastic deflectioniteration loop) - SLVBEQ directs solution of rolling elementload and position equilibrium equations --GUESS provides initial guessor updatesof rolling elementunknowns E GROLLB sets roller positions (optional), rotationaland orbital speedsGBALLB sets ball positions (optional), rotational and orbital speeds GUESCG estimatescage-ringrelative motion and assigns cageDOF's --BRGFOR determines brg. loadsfor current shaftdisplacementsand ring elastic deflections m DO IRE= 1, NRE(loop for each rolling element) - PREPAR initializerolling element,cagevariables,and arrays --BALLIN calculatescurvaturecenterdistances(ball bearing) -- UNLODB determinesif ball is out of contact with inner race -- ROLLIN transforms coordinatesfor roller/racewayinterferencecalculations LFLNDEF calculatesroller/flange interference -- UNLODR determinesif roller is out of contactwith inner race LTPNORM calculatesloads about roller CG -- MAXMIN sets max/minvalues for cageandrolling elementvariables - DNEQNF IMSL routineto solve nonlinear equationswith numerical partials I=. BEAF_EQsets up equilibrium equationsdependingon bearingtype --BALLEQ defines ball bearingequilibrium equations . MIX calculatesfrictional load components BDRAG calculatesdrag coefficient --ROLLIN ** follows samesubroutine hierarchyas previouscall ROLLEQ definesroller bearingequilibrium equations FLNORM determinesflange loads TPNORM calculatesloadsabout roller CGFMIXRcalculatesfrictional load components FLMIX calculatesflangefrictional loads -- BRGAX calculatesm/lingelementaccelerationterms -- CAGEEQdefines cage-elementinteraction -6UESSl provides initial guessor updatesof unknowns for single rolling element .- 6BALL1 sets positions (optional), rotational and orbital speedsfor single ball 6ROLL1 sets positions (optional), rotational and orbital speedsfor single roller - DNEQNF IMSL routine to solve nonlinearequationswith numerical partials ** follows samesubroutinehierarchy as previous call =end loop on each rolling element -- CALFOR forms outer raceload vector from rolling elementforces and angles/moments -- SUMFIR sums forces and moments acting on inner ring mGUESS ** follows same subroutine hierarchyas previouscall (called for update) A B C FIGURE lb.--FEREBA subroutine calling tree (continued). 9

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A--Continue--"for all specified inner race displacements" loop B--Continue--"DO ITRF= 1, NITRF" loop C--Continue--"DO ITR = 1, NITRD" loop | _CNSTIF computes constraint transformation matrix (for gap closures) | -LFTOFF modifies constraint forces to null tension in preload springs UPDATE increments race deflection variables for next iteration IF FUUSUB calculates partials of rolling element forces with respect to curvature centerdeflections Lfor all race DOF's I// ,-oo-,,::follows samesubroutine hierarchy as previous call (called for initial guess) I/ / It" PREPAR ** follows same subroutine hierarchy as previous call || | _ DNEQNF follows same subroutine hierarchy as previous call I/ / It" GUESS1 _ follows same subroutine hierarchy as previous call (called for update) || | It- DNEQNF follows same subroutine hierarchy as previous call I/ L "-'- PERFOR forms race load vectors from rolling element forces and angles/moments niterationtolerancecheck"elastiDLSARGio/MsLrOutinetosOIvefOrrace-c deflectdeflection increments (equation 10) -DO IDOF = 1, NDOF (perturbations to form Jacobian for applied load iteration loop) _,-SLVBEO. ** follows same subroutine hierarchy as previous call _,-UPDATE ** follows same subroutine hierarchy as previous call I DO ITR = 1, NITRD (elastic deflection iteration loop) "-elastic deflection iteration tolerance check Jacobian formed for applied load iteration loop -DLSARG IMSL routine to solve for inner ring displacement increments in applied load loop -applied load iteration tolerance check LIFE calculates bearing fatigue life I,-- FLMFAC lubrication film factor for life adjustment writes solution data to files m RELOUT writes rolling element output -,-,SETUP routine to setup piecewise cubic interpolation of ORCCdeflections for graphing TRPLAT piecewise cubic interpolation routine REFLOT writes output for roller bearing flanges elapsed time counter (system specific routine, used in many routines) finish reading user specified inner racedisplacements (multiple run feature) FIGURE lc.--FEREBA subroutine calling tree (continued). 10

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READAF Read/EchoData OATOUT Constants andCalculatentaDeflections ICALCON I Loads and Ring ReadRace Stiffness Deflecti°n S°luti°n l'i- - - -t i....{Do Matrices; Reduce FEMFLX and Invert; Partition ITR=I,NITR ) ,YLVBEO EstimateRolling Element Positions No , I IS°'veEquiIibr,Quas'-Stat'cumIosrArE I ! ! I ! ! I DefineGapgeadbandIsA"DE# and Speeds I CheckInitialize Variables,for Unloading IPREpA/t Element Positions I Solvefor Rolling 0NEONF and Forces I Deflection SolutionLoads and Ring I { I Calculate DuplexPreloadOffset PREL00 i I ElementSpeedsUpdate Rolling GUESSl I FormPartialDiscreteDerivs.I ID Element Positions Re-solvefor Rolling ONEONF and Forces --( n°,oo Looo) 1 Form RaceLoad Vector OALFOR I DisplacementsUpdate Shaft I OLSAne i No EndITRF Loop Sum Forceson I Inner Race ISUMFIR LI I ElementSpeedsUpdate Rolling IGUE88 I Update RaceDeflections IupoArE I OutputCalculateLife,Results OUTRSPLIFE End '_ FIGURE 2.1FEREBA flow chart. 11

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I Form Constraint ForceTransformation CNSTIF Matrix Calculate Relative Gap Deflections and Identify Closed Gaps CalculateConstraint Forcesand Transform to Gap DOFVector (Equation 5) Update Constraint Forcesfor Unloaded LFTOFF Preload Spring Segments Define Partials of Constraint Forces w.r.t. Gap Deflections Vector and I Calculate ResidualNormalized Error + CalculatePartials of Ball Forces w.r.t. Curvature Center Deflections Form Jacobian; Solve for Race OLSARG Deflection Increments (Equation 10) Deflections T -I_[o IRDOF=I,NRDO] Center DOF# IRDOF Element Positions GUESS1 and Speeds I Checkfor UnloadingInitialize Variables, PREPAR Solvefor Rolling I Element Positionsand Forces 10NEONF ElementSpeedsUpdate Rolling 10UESS1 Re-solvefor Rolling Element Positions ONEONF and Forces Form Race Load I Vectors IPERFOR CalculatePartials of all Ball Forces w.r.t. Curvature Center DOF# IRDOF I I I ! i... _v incre n ace(Equation 11)I Return FIGURE 3.--UPDATE flow chart. 12

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USER'S MANUAL UPDATES The theoretical and functional changes described in the previous section impact the user primarily through additional input and output files. This is illustrated in the revised schematic of figure 4 (replaces fig. A2-1). The additional files are the result of the functional changes rather than the theoretical changes. The theoretical changes alter some of the output in the existing output file. These effects will be discussed later. The modifications functional changes will be discussed first. These and additional files required/generated by the will be referenced to the numbering used in the Programming Implementation subsection of the Theoretical Manual Updates section. M_Structure I I ISubstructureFile I Analysis File I I SubstructureFileI Auxiliary Input Files ] I (multiple displacementrun, I outer race external force, I preload spring lateral stiffness) J User Edit [ |_ Inverted Flexibility I -]FE.,B,i_ .e°uce°an°t [ Analysis Results I Analysis Results I HEREBANS FIGURE 4.--Overall schematic of REBANS operation. 13

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Effects of Functional Changes 1. FEREBA will write the reduced flexibility matrices to a file whose name is based on the original substructure file name. The name is generated by prefixing the original substructure file name with "red." to signify that it contains the "reduced" flexibility matrices. If FEREBA finds a file by this name it will use it instead of reading the substructure file and reducing and inverting the stiffness matrix; therefore, if changes are made to the ANSYS model and the same file name is used, the "red." file must be moved or deleted. This file also contains the axial preload spring stiffness so that if it is changed in the analysis file the substructure file will be reread and reduced. If the user attempts to add or change a lateral preload spring stiffness (functional change 6), the change will not be recognized matrices rather than the substructure stiffness 2. The multiple sets of inner ring displacements since FEREBA will read the reduced flexibility matrix. are supplied by the user in an ASCII file read from FORTRAN unit number 51. Each record of this file should contain four comma-separated values. These values correspond to: Y-Axis Translation, Z-Axis Translation, XZ-Plane Rotation (About Y), XY-Plane Rotation (About Z). Multiple axial displacements are not currently provided for. Each case will be solved sequentially until an end-of-file is reached. The file will be read if appropriate linkage exists between it and FORTRAN unit number 51 (procedures for connecting a file to a FORTRAN unit number are operating system-dependent). If the program is running in force input mode, this file is ignored. . In order to run the displacement input mode with force iteration in the axial direction only, the user must edit the analysis file (.dat). Beginning with an analysis file that is set up for displacement input mode, insert a "LOADS" set immediately prior to the "DISPL" data. This set should contain the desired axial preload in the first record and zero for the remaining four records. The axial translation in the "DISPL" LOADS 1000.0000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 0.0000000E+00 DISPL 0.0000000E+00 2.2000000E-03 0.0000000E-03 0.0000000E-07 0.0000000E-07 set will be ignored. For example, the records preload force X-Axis (Axial) Translation Y-Axis Translation Z-Axis Translation XZ-Plane Rotation (About Y) XY-Plane Rotation (About Z) will iterate axial translation to achieve 1,000 pounds of axial load while the Y-axis translation is fixed at 0.0022 inch. 14

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1. Thecalculatedaverage"rigid body" tilt for the outerring is written to theASCII outputfile and to anadditionalbinary file for plotting (seeitem 5). 5. The displacementsandloadsfor the multipledisplacementcasesarewritten to FORTRANunit number62 with abinary write andto unit number61 with a freeformatwrite. Eachrecordis structuredasfollows: X, Fsumx , Yavg' Fsumy' Zavg' Fsumz' OaVgy, Msumy' ) avg z ' MSUmz' (Tilti, i= 1, nbrg), (Fx, Fy, Fz, My, Mz)i, i= 1, nbrg. The sum and avg subscripts refer to summing and averaging values for both bearings of a duplex set. The designation nbrg is 2 for a duplex bearing and 1 otherwise. Figure 5 shows an example plot from this file. The outer ring constraint forces are written to FORTRAN unit number 64 with a binary write. Each record is structured as follows: Yavg' Zavg' O avgy' O aVgz' Fci , i = 1, number of gaps. 3-1 _ 4,000 3,500 "¢3 3,000 f- 0 2,500 0 U. 0 2,000 .m *d m ee. 1,500 .tn 1,000 / 500 4-1 / / / / i I I i i i I 0.0000 0.0010 0.0020 0.0030 0.0040 0.0005 0.0015 0.0025 0.0035 ShaftY AxisDisplacement(inches) FIGURE 5.--Example force/deflection curve. 15

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Figure6 showsanexampleplot from this file. The polarplot for the ORCCdeflectionsis written to FORTRANunit number63 with a binarywrite. Eachrecordis structuredasfollows: Y*+Ayl,Z*+Azl,Y*,Z*,Y*+Ay2,Z*+ Az2 • The values of y* and z* correspond to a circle with a radius of 2.5 times the radial gap value. The subscripts 1 and 2 refer to bearings 1 and 2 in a duplex set (if it is duplex). The y axis deflection of the race at ball position 1 has been subtracted from the values of Ay 1 to remove the "rigid body" translation bias from the plot (for loading in the +y direction). This facilitates the comparison of the deflected plot with the undeflected plot. Figure 7 shows an example plot from this file. The dotted line represents the undeflected plot and the solid line the deflected plot. For multiple displacement cases this plot is of marginal utility. Each successive deflected ring is drawn over the previous ones so that the deflected shapes are usually not visible. 1-1 6-1 >4-)4 11-1 _ 16-1 ,00 _ 00 t-. -300 (#J ° -400 .= m .., -500 -600 -700 -800 ' ' ' 0.0000 0.0010 ", \ ,X, \ I ' ' ' ' 0.0020 0.0030 0.0040 0.0005 0.0015 0.0025 0.0035 ShaftY AxisDisplacement(inches) FIGURE 6.--Gap 2 constraint forces for ball locations 1, 2, and 3. 16

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1-I 2-1 0.004 _ 0.003 0.002 0.001 - A / 0.000 - t-- .N N -0.001 - -0.002 %%% -0.003 -0.004 i I _ I i I -0.004 -0.002 3-2 4-2 1 J S J r I i I i I i q p 0.000 0.002 0.004 -0.003 -0.001 0.001 0.003 Y (inches) FIGURE 7.--Outer race curvature center deflections polar plot. . The lateral stiffness between the race and carrier at the preload spring contact face is supplied by the user in an ASCII file read from FORTRAN unit number 53. This value should be on the first record beginning in the first column. . The set of external forces applied to the outer race are supplied by the user in an ASCII file read from FORTRAN unit number 52. The number of such forces must be the only value on the first record (beginning in the first column) of this file. Each remaining record should contain two comma-separated values. The first value is an integer number designating the gap DOF index number corresponding to the location of force application. The second value is the force magnitude at that location. As an example, 3 1,50,0. 26,-250. 31,-250. will apply 500 pounds in the axial direction for IBSCOR=2 with 11 balls, the records at ball location 1 and -250 pounds in the axial direction at each of ball locations 6 and 7. To determine the gap DOF numbering the user must refer to the appendix (revises section 3 of the original Theoretical Manual due to the additional radial master DOF). 17

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. The additional radial master DOF requires a change to the ANSYS model. The outer race configuration descriptions of section 3.2 of the User's Manual should be modified according to the Theoretical Manual updates provided in the appendix. The example ANSYS input file of section 3.4 should be modified where the master DOF's are defined. Lines 240 and 241 should be replaced with the following: M, 1023, UX, 1029, 6, UY, UZ MGEN, NRE, MINC, 1023, 1029, 6 M, 1024, UX, 1028, 2, UY, UZ MGEN, NRE, MINC, 1024, 1028, 2 * Master nodes 1023 and 1029 * Define similar nodes as masters * Master nodes 1024, 1026, and 1028 * Define similar nodes as masters. Similar changes should be made to models of other configurations. Effects of Theoretical Changes The changes made in the theoretical development only affect the program output. This occurs primarily in the definition of "error" and in the messages related to convergence failure. Due to the new iteration approach, the error norm (defined by equation (9)) is no longer an "average" of various displacement errors as before. There are also no longer separate errors for the inner and outer races. Deflection errors for both races are contained in the residual vector Rn. Referring to the User's Manual, this will change the sample printout of FEREBA status information (section 5.2, page A5-3) as shown in figure 8. Also, the iteration approach discussed in section 5.0 is modified in the same manner as was discussed for the Theoretical Manual--Solution Methodology (section 2.1). The discussion of convergence problems (section 5.3) is significantly modified by the new solution procedure. It remains true that for an iterative, nonlinear analysis such as that performed by FEREBA, absolute convergence cannot be guaranteed. The revised procedure, however, has greatly reduced the frequency of occurrence of nonconvergence with this program. The original approach required four iteration loops for a solution. The revised procedure eliminates the outer ring/carrier gap iteration loop and moves the force iteration outside the elastic deflection iteration. The possible errors associated with each of the three iteration loops are discussed in the following: ERROR FROM DNEQNF--The number of calls to FCN has exceeded ITMAX* (N+I). ERROR FROM DNEQNF--The iteration has not made good progress. These errors are from the rolling element equilibrium iteration (DNEQNF is called from subroutine BRGFOR). The accuracy and iteration limits are initially set internally. The user-supplied accuracy and iteration limits for the shaft/inner race force iteration loop are substituted here if they call for a more precise solution (smaller accuracy, larger iteration limit). These values can be altered if this error occurs. 18

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R E B A N S Rolling Element Bearing Analysis System - R E B A N S FFFFFFF EEEEEEE RRRRRR FF EE RR RR FFFFFF EEEEEE RRRRRR FF EE RR RR FF EEEEEE RR RR Flexibility Enhanced Rolling Final Release Version 2.00 Title Of Analysis Being Run: EEEEEEE BBBBBB AAA/kA EE BB BB AA AA EEEEEE BBBBBB AAAAAAA EE BB BB AA AA EEEEEE BBBBBB AA AA Element Bearing Analysis - Dated 01 August 1996 Test Case with flexible outer ring, IBSCOR=2, 1 mil radial deadband Name of Input Analysis File: Name of Output Results File: Forming Compliance Matrices for Outer Ring reading previously reduced and inverted red.bor2ms3.asc /usr/people/ryansg/rebans/testl.dat testls.out outer race file: Compliance Matrices Created for Outer Ring, Elapsed Time = 0.41 secs Now Iterating to Determine Quasi-Static Equilibrium Solution Elastic deformation iteration 1 complete, Error = 6.5675E+03% Elastic deformation iteration 2 complete, Error = 2.0154E+07% Elastic deformation iteration 3 complete, Error = 4.7745E+03% Elastic deformation iteration 4 complete, Error = 4.6094E+05% Elastic deformation iteration 5 complete, Error = 8.1675E+01% Elastic deformation iteration 6 complete, Error = 1.0520E+01% Elastic deformation iteration 7 complete, Error = 3.0943E-02% Elastic deformation iteration 8 complete, Error = 9.6991E-05% Elastic deformation iteration 9 complete, Error = 3.0534E-07% Quasi-Static Iteration Finished - Elapsed Execution Time = 8.48 secs FEREBA Execution Completed, Total CP Time = 8.96 secs Printed Output Written to File: testls.out FIGURE 8.--Example of FEREBA status information output. 19

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Activating print optionlevel 3 will displaythefinal iterationvalues.Option level4 will displayeach valueduringtheiterationprocess.Theusercanmodify the actiontakenwhentheseerrorsoccurby enabling(uncommenting)the call to theroutineERSET.This call is madein the mainprogram FEREBA. This will allow the userto testthevalueof theIMSL functionIERCD aftercalling DNEQNF andspecifyactionaccordingly.Fortunately,the occurrenceof theseerrorsis the exceptionratherthan the rule. ***** WARNING--Elastic deformationsunconvergedafter ??loops--WARNING ***** Error is = XXXXXXXX% Requestedmaximumiterationerroris = YYYYYYYY% ***** WARNING--Solution maybe acceptable--examinecarefully--WARNING ***** This warningoccursin the elasticdeflectioniterationloop.If the solutionis converging,thenumber of iterationsallowedmay be increased.If theerroris smallenough,therequiredaccuracymaybe increased.Activating print option level 2 will display information about gap constraint forces to aid in debugging, if necessary. ***** WARNING--Force iteration unconverged after ?? loops--WARNING ***** Requested maximum iteration Error is = XXXXXXXX% error is = YYYYYYYY% ***** WARNING--Solution may be acceptable--examine carefully--WARNING ***** This warning occurs in the force iteration loop. If the solution is converging, the number of iterations allowed may be increased. If the error is small enough, the required accuracy may be increased. Activating print option level 1 will display information about assumed inner race displacements and calculated inner race forces to aid in debugging, if necessary. Additional User's Manual Changes The boundary conditions for outer race configuration 4 (illustrated on page 3-5) may not be representative of a user's configuration. It is not necessary to constrain the duplex bearing in the axial direction as shown. It must be restrained radially similarly to the primary bearing. While it is not necessarily erroneous to constrain the duplex bearing in the manner shown, it is not required as the text implies. The user should determine which constraints analyzed. 20 are appropriate for the configuration being

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GENERAL COMMENTS ON ASSUMPTIONS AND INHERENT ERRORS The assumptions required to formulate a mathematical representation of a physical device inevitably introduce errors. There are three particular errors of this nature in the REBANS analysis. Two of these are caused by the addition of race/support flexibility and the way it is modeled. The other error is not unique to this enhancement and probably exists in other bearing analysis programs. The error that is not unique to REBANS is related to the summation of roiling element forces. When rolling element forces are summed on the inner race and the outer race, the two sums are not equal. This result is not physically possible. The difference is due to the neglect of rolling element azimuth angle deviations and tangential acceleration forces. In the classical quasi-static bearing analysis, the rolling element speeds are calculated based on the individual element kinematics (contact angles, etc.). In order for ball #1 to have a different orbital speed than ball #2, the ball must accelerate or decelerate in the tangential direction. The inertia force from this acceleration must be reacted on the raceways. In addition, the ball will make an excursion from the evenly spaced azimuth position usually assumed in the analysis. For a classical fixed outer race analysis, incorporating the acceleration forces and the azimuth angle deviations negates the difference in the sum of forces on the inner and outer races. This cannot be easily corrected in the REBANS analysis. The ball forces can only be applied at the fixed azimuth positions defined by the finite element model node locations. The present form of the program also reduces out tangential DOF's from the race model so that the acceleration forces cannot be applied. The magnitude of this error is on the order of 1-2 percent. The two REBANS unique errors are related to the finite element model grid. The first of these is related to ball bearing contact angles. The ball/race contact loads must be applied to a fixed node location on the race. This point is typically placed at either the tinloaded contact angle or the estimated loaded contact angle. The actual contact angles vary from ball to ball and will not, in general, equal the assumed value used to define the finite element model grid. While the loads are applied at the correct angle, their line of application is displaced slightly to match the node location. The magnitude of this error is difficult to quantify; however, an analysis of a rigid ring (including a tilt DOF) with the same error simulated resulted in 1-2 percent error. The second REBANS unique error is related to race/support contact. The race contacts the support with circumferentially and axially distributed contact loads. These distributed loads are represented by discrete forces applied to the structures. In the circumferential direction, there are as many locations as rolling elements. In the axial direction, there are three locations. The placement of the axial locations will have a direct effect on the tipping moment which can be reacted at the race corner. Chamfering and local flexibilities will alter where the "effective" reaction force should be placed to most accurately represent the actual mechanics of this interaction. The axial spacing error has been greatly reduced by incorporating the additional radial master DOF for the outer race. The error due to circumferential spacing requires further analysis to determine the degree of spatial resolution required to adequately represent the race/support contact mechanics. 21

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APPENDIX . Single ball bearing, flexible outer ring with deadband, carrier assumed rigid (IBSCOR=2). This model is nonlinear due to deadband contact, which is dependent on ball loads applied to the outer race. There are 5 gaps between the outer ring and rigid housing, and at each azimuthal position, 7 master nodes are required. After reduction in FEREBA, 9 DOF's remain at each ball location (4 axial, 5 radial). 2_ 3_ 4z_ F Preload Spring 1,, Active DOF's ANSYS 21 "n (X, Y, Z) FEREBA 9"n (X, R) Active Nodes (DOF's) X: 1,2,3,7(1,4,7,19) R: 1,2,4,5,6(2,5,11,14,17) Reduced DOF's 3*, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21 * DOF 3 is deleted . Single ball bearing with flexible outer ring and carrier with deadband (IBSCOR=3). This model is nonlinear due to deadband contact, which is dependent on ball loads applied to the outer race, and also due to the preload spring (nodes 7-12), which can bottom. There are 5 gaps between the outer ring and cartier, and at each azimuthal position, 12 master nodes are required (7 on the outer ring, 5 on the carrier). After reduction in FEREBA, 14 DOF's remain at each ball location (6 axial, 8 radial). 9 10 11 Active DOF's ANSYS 36"n (X, Y, Z) FEREBA 14"n (X, R) Active Nodes (DOF's) X: 1,2,3,7,8,12 (1, 4, 7, 19, 22, 34) R: 1,2,4,5,6,9,10,11 -- Preload Spring 2 3_ 4_ I" (2, 5, 11, 14, 17, 26, 29, 32) Reduced DOF's x 3*, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 25, 27, 28, 30, 31, 33, 35, 36 * DOF 3 is deleted 23

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. Duplex ball bearing set with flexible outer rings with deadband and with carrier assumed rigid (IBSCOR=5). The analysis is restricted to having n equal in both bearings. This model is nonlinear due to deadband contact and the possibility that the preload spring (7-10) could bottom. There are 9 gaps between the outer ring and rigid housing, and at each azimuthal plane, 14 master nodes are required (7 on each bearing). After reduction in FEREBA, 18 DOF's remain at each ball location (8 axial, 10 radial). F Preload Spring £4 ; Bq "11 1"2 135 x Active DOF's ANSYS 42"n (X, Y, Z) FEREBA 18"n (X, R) Active Nodes (DOF's) X: 1, 2, 3, 7, 8, 9,10,14 .. (1, 4, 7, 19, 22, 25, 28, 40) R: 1,2,4,5,6,8,9,11,12,13 (2, 5, 11, 14, 17, 23, 26, 32, 35, 38) Reduced DOF's 3*, 6, 8, 9, 10, 12, 13, 15, 16, 18, 20, 21, 24", 27, 29, 30, 31, 33, 34, 36, 37, 39, 41, 42 * DOF's 3 and 24 are deleted The relative positions of the "primary" and "duplex" bearings are the same as configuration 4. . Duplex ball bearing set with flexible outer rings and carrier with deadband (IBSCOR=6). The analysis is restricted to having n equal in both bearings. This model is nonlinear due to deadband contact and the possibility that the preload spring (7-10) could bottom. There are 9 gaps between the outer ring and carrier, and at each azimuthal position, 22 master nodes are required (7 on each bearing and 8 on the carrier). After reduction in FEREBA, 26 DOF's remain at each ball location (10 axial, 16 radial). z_ 16 17 18 z_/];Prel°adS21ring 21 I ; 4101 ;2 i --T x 24 Active DOF's ANSYS 66"n (X, Y, Z) z_ FEREBA 26"n (X, R) Active Nodes (DOF's) I X: 1,2,3,7,8,9,10,14,15,22(1,4,7,19, 22, 25, 28, 40, 43, 64) R: 1,2,4,5,6,8,9,11,12,13,16,17,18, 19, 20, 21 (2, 5, 11, 14, 17, 23, 26, 32, 35, 38, 47, 50, 53, 56, 59, 62)

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Reduced DOF's 3",6,8,9,10,12,13,15,16,18,20,21, 24",27,29,30,31,33,34,36,37,39,41, 42,44,45,46,48,49,51,52,54,55,57,58, 60,61,63,65,66 * DOF's 3 and 24 are deleted The relative positions of the "primary" and "duplex" bearings are the same as configuration 4. o Cylindrical roller bearing with flexible outer ring with deadband and with carrier assumed rigid (IBSCOR=8). This model is nonlinear due to deadband contact. There are 3 gaps between the outer ring and rigid housing, and at each azimuthal position, 5 master nodes are required. After reduction in FEREBA, 5 radial DOF's remain at each roller location. 1_ 2 __ 3_ R _ 1 L_. I x Roller i Active DOF's ANSYS 15"n (X, Y, Z) FEREBA 5"n (R) Active Nodes (DOF's) X: None R: 1, 2, 3, 4, 5 (2, 5, 8, 11, 14) •Reduced DOF's 1,3,4,6,7,9, 10, 12, 13, 15 . Cylindrical roller bearing with flexible outer ring and carrier with deadband (IBSCOR = 9). This model is nonlinear due to deadband contact. There are 3 gaps between the outer ring and carrier, and at each azimuthal position, 8 master nodes are required (5 on the bearing and 3 on the carrier). After reduction in FEREBA, 8 radial DOF's remain at each roller location. _ I I ' _ 3 2 4 _ _ LJ/ i - 1 2 Rq I ° ° i _ I R0110r I x Active DOF's RANSYS 24"n (X, Y, Z) X:FEREBANone 8"n(R) Active Nodes (DOF's) R: 1,2,3,4,5,6,7,8 (2, 5, 8, 11, 14, 17, 20, 23) Reduced DOF's 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24 25

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APPROVAL MODELING OF ROLLING ELEMENT BEARING MECHANICSm COMPUTER PROGRAM UPDATES Stephen Ryan The information in this report has been reviewed for technical content. Review of any information concerning Department of Defense or nuclear energy activities or programs has been made by the MSFC Security Classification Officer. This report, in its entirety, has been determined to be unclassified. Director, Structurest_/id D_a'a_r_s Laboratory

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REPORT DOCUMENTATION PAGE FormApproved OMB No. 0704-0188 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data soumes, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operation and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, aad to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503 1. AGENCY USE ONLY (Leave Blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED March 1997 Technical Memorandum 4. TITLE AND SUBTITLE Modeling of Rolling Element Bearing Mechanics-- Computer Program Updates 6. AUTHORS S.G. Ryan 7. PERFORMINGORGANIZATIONNAMES(S)ANDADDRESS(ES) George C. Marshall Space Flight Center Marshall Space Flight Center, Alabama 35812 5. FUNDING NUMBERS 8. PERFORMING ORGANIZATION REPORT NUMBER ADDRESS(ES) 10. SPONSORING/MONITORING 9. SPONSORING/MONITORINGAGENCYNAME(S)AND National Aeronautics and Space Administration Washington, DC 20546--0001 11, SUPPLEMENTARY NOTES AGENCY REPORT NUMBER NASA TM-108532 Prepared by Structures and Dynamics Laboratory, Science and Engineering Directorate 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclas sified-Unlimited 13. ABSTRACT (Maximum 200 words) 12b. DISTRIBUTION CODE The Rolling Element Bearing Analysis System (REBANS) extends the capability available with traditional quasi-static bearing analysis programs by including the effects of bearing race and support flexibility. This tool was developed under contract for NASA-MSFC. The initial version delivered at the close of the contract contained several errors and exhibited numerous convergence difficulties. The program has been modified in-house at MSFC to correct the errors and greatly improve the convergence. The modifications consist of significant changes in the problem formulation and nonlinear convergence procedures. The original approach utilized sequential convergence for nested loops to achieve final convergence. This approach proved to be seriously deficient in robustness. Convergence was more the exception than the rule. The approach was changed to iterate all variables simultaneously. This approach has the advantage of using knowledge of the effect of each variable on each other variable (via the system Jacobian) when determining the incremental changes. This method has proved to be quite robust in its convergence. This technical memorandum documents the changes required for the original Theoretical Manual and User's Manual due to the new approach. 14. SUBJECT TERMS 15. NUMBER OF PAGES 32 rolling element, bearing, stiffness, deadband, flexible race, quasi-static 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION OF REPORT OF THIS PAGE Unclassified Unclassified NSN 7540-01-280-5500 16. PRICE CODE NTIS 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF ABSTRACT Unclassified Unlimited Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. 239-18 298-102

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National Aeronautics and Space Administration Code JTT Washington, DC 20546-0001 Official Business Penalty for Private Use, $300 Postmaster: If Undeliverable (Section 158 Postal Manual), Do Not Return