A Dynamic Analysis of a Flexible Rotor in Ball Bearings with Nonlinear Stiffness Characteristics

This paper presents an effective analysis approach for a flexible rotor in ball bearings with 
nonlinear stiffness characteristics to obtain realistic dynamic behavior results. The ball bearing 
is modeled in five degrees of freedom and the nonlinear stiffness characteristics of the 
bearing are completely described as functions of combined loads and spin speed. For dynamic 
behavior analysis of the nonlinear rotor-bearing system, a transfer-matrix method is 
iteratively used until the bearing displacements and the shaft displacements at every bearing 
location converge to the same values. The results show that the nonlinear stiffness characteristics 
of ball bearings significantly influence system dynamic behaviors and the proposed 
analysis approach for the nonlinear rotor-bearing system is effective.


INTRODUCTION
High-Speed rotating machinery with ball bearings, such as aircraft engines and small capacity gas tur- bines, has been widely used because of easy mainte- nance, little rotordynamic instability, and low cost.
Many authors have shown that a large number of pa- rameters, which include system geometry, disk properties, stiffness distribution of a rotating shaft, and bearing stiffnesses can influence the dynamic behav- ior of a rotor-bearing system.Amongst them, the ball bearing stiffnesses in particular possess nonlinear characteristics which vary with applied loads and spin speed (Harris 1991 ]).Recently, many research- ers have studied on the analysis of a rotor-bearing system for design and diagnosis.However, their anal- ysis results are difficult to apply because they usually considered the bearing stiffness in radial direction only and disregarded bearing stiffness variation with spin speed and applied loads.
The objective of this study is to propose an effec- tive way of handling the nonlinear stiffness character- istics of ball bearings in the context of the system dynamic behavior analysis to obtain realistic results.*Corresponding author.Tel.: 82-2-290-0433.Fax." 82-2-296-1710.74   D.-S.LEE and D.-H.CHOI Two major components necessary to achieve the ob- jective are a ball bearing stiffnesses analyzer and a system dynamic behavior analyzer.
For the analysis of ball bearing stiffnesses, while [1979] calculated radial stiffness by using finite ap- proximation, Gargiulo [1980] presented bearing stiff- ness formula in radial and axial directions, and Lim and Singh [1990| developed a numerical scheme to compute complete bearing stiffnesses.However, all of them neglected high speed effects such as centrif- ugal forces and gyroscopic moments of ball elements.Jones [19601 and De Mulet al. [19891 presented gen- eral analysis theories for the ball bearing under arbi- trary load and speed conditions.In their analyses, however, the inner-and outer-raceway load-deflection factors, centrifugal force, and gyroscopic mo- ment of every ball were assumed to have constant values for all the ball locations.These values, how- ever, actually depend on contact angle values of each ball.This may often cause some errors in ball bearing analysis results under applied load and high speed conditions since contact angle values become quited- ifferen from ball location to ball location.To correct this defect, we devise an iterative ball bearing analysis scheme based on Jones' general theory, and then apply it in this study.
For the dynamic behavior analysis of a nonlinear rotor-bearing system, a transfer-matrix method (Murphy and Vance [19831;Rao [1983]; Vance [19881) and the proposed bearing analysis scheme are directly integrated and iteratively used until the bearing dis- placements and the shaft displacements converge to the same values within a specified tolerance at every bearing location.
The next section presents an analysis scheme for obtaining complete nonlinear ball bearing stiffnesses.The numerical procedure for the dynamic behavior analysis of a nonlinear rotor-bearing system is de- scribed in section 3.In section 4, numerical results are presented to illustrate the nonlinear stiffness characteristics of a high speed ball bearing, the ef- fect of the nonlinear characteristics on the dynamic behavior of a rotor-bearing system, and the useful- ness of the proposed analysis approach for a flexible rotor in ball bearings.Finally, concluding remarks are mentioned.

BALL BEARING ANALYSIS
The relationship between the bearing loads F {F Fy, Fz, My, Mz} T and the bearing displacements bT g) {g), g,),, g), 0,),, 0=}, as shown in Figure 1, has to be determined for ball bearing analysis.In this study, an iterative bearing analysis algorithm based on Jones' theory [1960] is devised and used to calcu- late a complete stiffness matrix, and a brief flow chart of the algorithm is shown in Figure 2.
As shown in Figure 2, the input data such as bear- ing geometry, material, applied loads, and spin speed are specified, and the values of bearing displacements and inner-and outer-raceway contact angles at every ball location (or,.( and ot(Z:.)are initially assumed at first.
In this study, we use the free contact angle value for the initial contact angles.Next, we calculate the in- ner-and outer-raceway load-deflection factors at each ball location (K(i and K(:i) as well as the centrifugal force and gyroscopic moment of each ball element (CF.i,GMi), and their values are fixed as constant during the process of the inner loop.In the inner loop, the contact angles and deformations at each ball location are computed by solving a system of nonlin- ear algebraic equations (which consists of two com- patibility equations and two ball equilibrium equations) with the /alues of bearing displacements as- sumed.Having obtained these values, the bearing displacements (_6) can be computed by solving a sys- tem of five bearing equilibrium equations.This pro- cess of the inner loop is repeated until the displace- ments are converged.Now, the differences between the current contact angle values obtained at the end of the inner loop (%1/ and o,:/) and the previous ones oi/and o,:/) are tested against a specified tolerance () for every ball location.If they are not within the tol- erance, the contact angle values are set to the current ones, and the outer loop process is repeated until con- ?!.....72 FIGURE 3 Schematic diagram of a rotor-bearing system.
vergence.With the converged results, a complete stiffness matrix is finally calculated by differentiating bearing equilibrium equations.

NONLINEAR ROTOR-BEARING SYSTEM ANALYSIS
A rotor system supported by ball bearings can be modeled as an assemblage of disks, flexible shaft el- ements, and bearings as shown in Figure 3.The dynamic behavior of the system is nonlinear since bear- ing stiffnesses vary with spin speed, applied preload, and transmitted loads due to unbalance.For the anal- ysis of the nonlinear rotor-bearing system, an overall procedure is shown in Figure 4.As shown in Figure 4, bearing and rotor data are specified and a spin speed is given at first.Bearing loads (_FI',), which include an applied preload and loads transmitted from the rotor, and bearing dis- placements (_8,/',) are also initially assumed at every bearing location.Having the assumed bearing loads  FIGURE 4 Overall procedure tbr the dynamic analysis of a nonlinear rotor-bearing system.
D.-S.LEE and D.-H.CHOI and displacements, bearing stiffnesses (k,,) and dis- ') placements (_,, of every bearing are calculated in the bearing analysis module.The calculated bearing stiff- nesses (k,,) are fed into the unbalance response anal- ysis module to compute the shaft displacements (__8)'I) and internal shaft forces (FI'I) at every bearing lo.cation.Then, the differences between the bearing dis- placements (8_,',) and shaft displacements (8__)'i) are com- pared against a given tolerance for every bearing.If they are not within the tolerance, the bearing loads and displacements (_F,',, 8_)are updated to have the values of the shaft internal forces and displacements (F,,, _,,), and the above process is repeated until con- vergence.Finally, using the bearing stiffnesses (k,,, n m) converged, the eigenvalues of the system at the given spin speed are computed in the eigen- value analysis module.In this study, the scheme de- scribed in the previous section is used for ball bearing analysis and a transfer matrix method (Murphy and  Vance [1983]; Rao [1983]; Vance [1988]) for the eigenvalue and unbalance response analyses.
For the dynamic behavior analysis of a nonlinear rotor-bearing system in a range of spin speeds, the above procedure can be repeatedly used to sweep the range.

NUMERICAL RESULTS
To evaluate the usefulness of the suggested dynamic analysis approach for a nonlinear rotor-bearing sys- tem, a computer program is developed to implement the numerical procedures described in sections 2 and 3.In the following, the numerical results are pre- sented to illustrate nonlinear stiffness characteristics of a ball bearing and the effect of the nonlinear char- acteristics on the dynamic behavior of a rotor-bearing system.

Ball Bearing Stiffness Analysis
The specifications of the angular contact ball bearing used in this study are listed in Table I.To investigate the effects of spin speed and loads on bearing stiff- nesses, we simulated the variation of bearing stiff- nesses within the speed range of 0-40000 rpm under the loads ofF 1500 N, F,. 1000 N, My 10 N.m.Simulation results are shown in Figure 5.The vari- ations of radial, coupling, and rotational stiffnesses are plotted as a function of spin speed.Further, the stiffness results analyzed by the proposed scheme are compared with those analyzed by Jones' scheme.As can be seen in Figure 5, the effect of spin speed on bearing stiffnesses is significant and the stiffness characteristic is anisotropic due to the force in the z direction and the moment in the y direction.The dis- crepancy between the results analyzed by the two schemes increases as spin speed increases, because the proposed iterative scheme well considers the ef- fects of centrifugal force and gyroscopic moment of each ball element on bearing stiffness characteristics.

Dynamic Behavior Analysis of a Nonlinear
Rotor-Bearing System In this study, the ball bearing analysis module, the system unbalance response analysis module, and the system eigenvalue analysis module are directly inte- grated and applied to the dynamic behavior analysis of a multi-stepped rotor which is supported by two angular contact ball bearings as shown in Figure 6.The shaft with varying cross-sectional area is divided into 18 elements and the bearings of Table under the preload of 1500 N in axial direction are located at stations 11 and 15.One disk with a fixed mass is located at station 5.The details of rotor configuration data are listed in Table II.
To investigate the effect of nonlinear bearing stiff- ness characteristics on system critical speeds, the un- balance response with constant bearing stiffnesses and that with nonlinear stiffnesses are obtained and the results are shown in Figure 7 and Figure 8, re- spectively.The constant bearing stiffness values used here are the ones calculated at the spin speed of 22000 rpm, and the unbalance at the disk is assumed as 1.0 10kg.m in both cases.Comparison of Figure 7 and Figure 8 shows that the discrepancies of the first and the second critical speeds of two results are 14% and 10%, respectively.Figure 9 shows that the whirl orbit results with nonlinear bearing stiff- nesses at 19000 rpm, 22000 rpm, and 24400 rpm are circular.This implies that the isotropic characteristics of the bearings are hardly affected since the forces and moments transmitted by the unbalance are not noticeable compared with the applied preload.
To examine the effect of unbalance magnitude on the dynamic behavior of the nonlinear rotor-bearing system, we analyze system eigenvalues, unbalance response, and whirl orbits with the unbalance of 1.0 10 .6 kg.m, and the simulation results are shown in Figure 10, Figure 11, and Figure 12, respectively.The Response at the right bearing).
comparison of Figure 8 and Figure reveals that the effect of unbalance magnitude on the first and the second forward critical speeds is not noticeable.With relatively large unbalance, however, backward criti- cal speeds appear as shown in Figure 10 and Figure 11.The comparison of Figure 9 and Figure 12 shows that the whirl orbits near the critical speeds become quite elliptical as unbalance magnitude increases.
Response at the left bearing

CONCLUDING REMARKS NOMENCLATURE
This paper proposes a new approach of directly inte- grating a ball bearing analyzer and a dynamic behav- ior analyzer for the dynamic analysis of a nonlinear rotor-bearing system.At first, a numerical scheme for calculating the speed-and load-dependent stiffnesses of a ball bearing is developed in which the depen- dency of load-deflection factors, centrifugal force, and gyroscopic moment on contract angle values is well regarded.Then, an iterative procedure for ana- lyzing the dynamic behavior of a nonlinear rotor- bearing system in a truly integrated fashion is devel- oped and applied to the analysis of a multi-stepped rotor supported by two angular contract ball bearings.
Numerical results demonstrate the importance of including the nonlinear bearing stiffness characteris- tics for dynamic behavior analysis by showing the discrepancy between the analysis results with nonlin- ear bearing stiffnesses and those with constant bear- ing stiffnesses.Comparison of the results also illus- trates that the effect of unbalance magnitude on system dynamic behavior can be remarkable.
Even though the suggested analysis approach is ap- plied only to the rotor system supported by ball bear- ings in this study, we believe that the same approach can be applied to the rotor system supported by slid- ing bearings.

FIGUREFIGURE 2
FIGURELoads and displacements of a ball bearing.

FIGURE 9
FIGURE 9 Whirl orbits of nonlinear rotor-bearing system (Unbalance 1.0 10kg.m: ER RG GY Y M MA AT TE ER RI IA AL LS S Materials Science & Engineering for Energy SystemsEconomic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.S SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y