Flexible Rotor Ball Bearings with Nonlinear Stiffness Characteristicsdownloads.hindawi.com/journals/ijrm/1997/302073.pdf · 2019. 8. 1. · modeled as an assemblage ofdisks, flexible

International Journal of Rotating Machinery

1997, Vol. 3, No. 2, pp. 73-80

Reprints available directly from the publisherPhotocopying permitted by license only

(C) 1997 OPA (Overseas Publishers Association)Amsterdam B.V. Published in The Netherlands

under license by Gordon and Breach Science Publishers

Prin,ed in Singapore

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

DONG-SOO LEE and DONG-HOON CHOI*

Department of Mechanical Design & Production Engineering Hanyang University, Seoul 133-791, Korea

(Received 21 March 1996)

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

Keywords: Rotor-bearing system, ball bearing, nonlinear bearing stiffness, critical speed, unbalanceresponse, whirl orbit

1 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 prop-erties, stiffness distribution of a rotating shaft, andbearing stiffnesses can influence the dynamic behav-ior of a rotor-bearing system. Amongst them, the ballbearing stiffnesses in particular possess nonlinear

characteristics which vary with applied loads andspin speed (Harris 1991 ]). Recently, many research-ers have studied on the analysis of a rotor-bearingsystem for design and diagnosis. However, their anal-ysis results are difficult to apply because they usuallyconsidered 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 systemdynamic behavior analysis to obtain realistic results.

*Corresponding author. Tel.: 82-2-290-0433. Fax." 82-2-296-1710.

73

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 Limand Singh [1990| developed a numerical scheme to

compute complete bearing stiffnesses. However, allof 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-deflec-tion 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 eachball. This may often cause some errors in ball bearinganalysis results under applied load and high speedconditions since contact angle values become quited-ifferen from ball location to ball location. To correct

this defect, we devise an iterative ball bearing analy-sis scheme based on Jones’ general theory, and thenapply it in this study.

For the dynamic behavior analysis of a nonlinear

rotor-bearing system, a transfer-matrix method (Mur-phy and Vance [19831; Rao [1983]; Vance [19881)and the proposed bearing analysis scheme are directlyintegrated and iteratively used until the bearing dis-

placements and the shaft displacements converge to

the same values within a specified tolerance at everybearing 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 resultsare presented to illustrate the nonlinear stiffness

characteristics of a high speed ball bearing, the ef-fect of the nonlinear characteristics on the dynamicbehavior of a rotor-bearing system, and the useful-ness of the proposed analysis approach for a flexible

rotor in ball bearings. Finally, concluding remarksare mentioned.

2 BALL BEARING ANALYSIS

The relationship between the bearing loads F {FFy, Fz, My, Mz}T and the bearing displacements

bTg) {g), g,),, g), 0,),, 0=}, as shown in Figure 1, hasto be determined for ball bearing analysis. In this

study, an iterative bearing analysis algorithm basedon Jones’ theory [1960] is devised and used to calcu-late a complete stiffness matrix, and a brief flow chartof 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 speedare specified, and the values of bearing displacementsand inner- and outer-raceway contact angles at everyball location (or,.( and ot(Z:.) are initially assumed at first.

In this study, we use the free contact angle value forthe initial contact angles. Next, we calculate the in-

ner- and outer-raceway load-deflection factors at eachball location (K(i and K(:i) as well as the centrifugalforce 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 balllocation are computed by solving a system of nonlin-

ear algebraic equations (which consists of two com-

patibility equations and two ball equilibrium equa-tions) with the /alues of bearing displacements as-

sumed. Having obtained these values, the bearingdisplacements (_6) can be computed by solving a sys-

FIGURE Loads and displacements of a ball bearing.

A FLEXIBLE ROTOR IN BALL BEARINGS 75

Calculate K,. o: ), Ko a ),

OUTER -I..OOP CalcuIate contact angles mdderogations at each ball Dcation

NRLOOP

FIGURE 2 Flow chart fl)r ball bearing analysis.

tem of five bearing equilibrium equations. This pro-cess of the inner loop is repeated until the displace-ments are converged. Now, the differences betweenthe current contact angle values obtained at the end ofthe 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 completestiffness matrix is finally calculated by differentiatingbearing equilibrium equations.

3 NONLINEAR ROTOR-BEARING SYSTEMANALYSIS

A rotor system supported by ball bearings can bemodeled as an assemblage of disks, flexible shaft el-ements, and bearings as shown in Figure 3. The dy-namic 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 overallprocedure is shown in Figure 4.As shown in Figure 4, bearing and rotor data are

specified and a spin speed is given at first. Bearingloads (_FI’,), which include an applied preload andloads transmitted from the rotor, and bearing dis-

placements (_8,/’,) are also initially assumed at everybearing location. Having the assumed bearing loads

System dataSpin speed :nInitial guesses for bearingloads and displacements:, a__.:,

,,,J

Bearing analysismodule

(kb.and _#., n 1,m)

Unbalance respo’sanalysis module(_ and, n 1,m)

Eig;rvaluemoduleanalysis[

FIGURE 4 Overall procedure tbr the dynamic analysis of a nonlinear rotor-bearing system.


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.ca-tion. Then, the differences between the bearing dis-

placements (8_,’,) and shaft displacements (8__)’i) are com-pared against a given tolerance for every bearing. Ifthey are not within the tolerance, the bearing loadsand displacements (_F,’,, 8_)are updated to have thevalues 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 systemat 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 bearinganalysis and a transfer matrix method (Murphy andVance [1983]; Rao [1983]; Vance [1988]) for theeigenvalue and unbalance response analyses.

For the dynamic behavior analysis of a nonlinear

rotor-bearing system in a range of spin speeds, theabove procedure can be repeatedly used to sweep the

range.

4 NUMERICAL RESULTS

To evaluate the usefulness of the suggested dynamicanalysis approach for a nonlinear rotor-bearing sys-

TABLE Ball bearing specifications

Number of ball elements 13

Pitch diameter (mm)Ball diameter (mm)Young’s modulus (MPa)Initial contact angle (deg.)Outer groove curvature radius (mm)Inner groove curvature radius (mm)

46.09.526

20600040.04.9504.950

tem, a computer program is developed to implementthe numerical procedures described in sections 2 and3. 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-bearingsystem.

4.1 Ball Bearing Stiffness Analysis

The specifications of the angular contact ball bearingused in this study are listed in Table I. To investigatethe 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 underthe loads ofF 1500 N, F,. 1000 N, My 10N.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, thestiffness results analyzed by the proposed scheme are

compared with those analyzed by Jones’ scheme. As

180,

0 10000 20000 30000 40000 0Spin speed (rpra)

(a) (b)

10000 20000 30000 40000

Spin speed (rpm) Spin speed (rpm)

FIGURE 5 Nonlinear stiffness characteristics of the angular contact ball bearing (--: proposed scheme Jones’ scheme).


iz

0.3581 m

FIGURE 6 A multi-stepped rotor-bearing system.

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 zdirection and the moment in the y direction. The dis-

crepancy between the results analyzed by the two

schemes increases as spin speed increases, becausethe proposed iterative scheme well considers the ef-fects of centrifugal force and gyroscopic moment ofeach ball element on bearing stiffness characteristics.

4.2 Dynamic Behavior Analysis of a Nonlinear

Rotor-Bearing System

In this study, the ball bearing analysis module, thesystem unbalance response analysis module, and thesystem eigenvalue analysis module are directly inte-

grated and applied to the dynamic behavior analysisof 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 thepreload 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 configurationdata 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 andthe results are shown in Figure 7 and Figure 8, re-

spectively. The constant bearing stiffness values usedhere are the ones calculated at the spin speed of22000 rpm, and the unbalance at the disk is assumedas 1.0 10- kg.m in both cases. Comparison ofFigure 7 and Figure 8 shows that the discrepancies ofthe first and the second critical speeds of two resultsare 14% and 10%, respectively. Figure 9 shows thatthe whirl orbit results with nonlinear bearing stiff-

nesses at 19000 rpm, 22000 rpm, and 24400 rpm are

circular. This implies that the isotropic characteristicsof the bearings are hardly affected since the forcesand 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-bearingsystem, we analyze system eigenvalues, unbalance

response, and whirl orbits with the unbalance of 1.010.6 kg.m, and the simulation results are shown in

Figure 10, Figure 11, and Figure 12, respectively. The

TABLE II Multi-stepped rotor configuration data

Element Node Bearing Outer Inner Element Node Bearing Outer Innernode location & radius radius node location & radius radiusno. (m) disk (m) (m) no. (m) disk (m) (m)

2345678910

-0.1790-0.1663-0.1282-0.1028-0.0901-0.0774-0.0723-0.0647-0.0520-0.0444

Disk

0.0051 0.01390.0 02 12 0.01150.0076 13 0.04960.0203 14 0.08770.0203 0.0152 15 O. 10800.0330 16 O. 12580.0330 0.0152 17 0.13600.0254 0.0178 18 O. 16640.0254 19 0.17910.0127

Bearing

Bearing

0.01270.0152 0.00520.0152 0.00520.01270.01270.03810.0203 0.01520.0203 0.0152

Disk: Location (m)0.0901

Mass (kg) Polar inertia (kg.m) Diametral inertia (kg.m)1.401 0.0020 0.00136


3xlOq

, 2xlO"1-

lxlO"10

01.SxlO 1.8xlO 2. lxlO 2.4x10 2.7x10 3.0x10

Spin speed (rpm)

FIGURE 7 Unbalance response results with constant bearingstiffnesses (Unbalance 1.0 10- kg.m: Response at theleft bearing Response at the right bearing).

3xlOqo

,, 2xlO"-

lxlO"10-

01.SxlO 1.8xlO 2. lxlO 2.4x104 2.7x104 3.0xlO

Spin speed (rpm)

FIGURE 8 Unbalance response results with nonlinear bearingstiffnesses (Unbalance 1.0 10- kg.m: Response at theleft bearing Response at the right bearing).

comparison of Figure 8 and Figure reveals that theeffect of unbalance magnitude on the first and thesecond forward critical speeds is not noticeable. With

relatively large unbalance, however, backward criti-cal speeds appear as shown in Figure 10 and Figure11. The comparison of Figure 9 and Figure 12 showsthat the whirl orbits near the critical speeds become

quite elliptical as unbalance magnitude increases.

4"0x104"

"’,,3.0x104- "’ 1F

1.OxlO4

0.0..0.0 1.0x104 2.0x104 3.0x104 4.0x104

Spin speed (rpm)

FIGURE 10 Whirl speed map of a multi-stepped rotor-bearingsystem.

3.0xlO"6

, 2.0xlO"s

{ l.OxlO

0.01.Sx104 1.8x104 2. lxlO 2.4x104 2.7x104 3.0xlO

Spin speed (rpm)

FIGURE 11 Unbalance response results with nonlinear bearingstiffnesses (Unbalance 1.0 10- kg.m" Response at theleft bearing Response at the right bearing).

This reflects that the bearing stiffness characteristics

become anisotropic due to the loads transmitted fromthe rotating shaft.

4xlOq

0 Y<j’ (m)

_4xlO-o_4xlO-1 0 4xlO1

4xlOdo

0 ’: :’’: ;’ (Ym)

.4xlO-O.4xlO- 4xlO

q

lxlO-9

0 Y(m)

_1x10-9olxlO-9 lxlO"9

(a) 19000 rpm (b) 22000 rpm (c) 24400 rpm

FIGURE 9 Whirl orbits of nonlinear rotor-bearing system (Unbalance 1.0 10- kg.m:Response at the right bearing).

Response at the left bearing


4x10"Z(m)

4x10"Z(m)

6x10.

Ym) 0 :’ Y"-" (m)

..4x10-6..4xlO-6 0 4x10"6

.4x10-6.4x10" 0 4x10"6

.6X10-6_6xlO"6 0 6xlO6

(a) 19000 rpm (b) 22000 rpm (c) 24400 rpm

FIGURE 12 Whirl orbits of the nonlinear rotor-bearing system (Unbalance 1.0 10-6 kg.m:Response at the right bearing).

Response at the left bearing

5 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 forcalculating 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-steppedrotor supported by two angular contract ball bearings.

Numerical results demonstrate the importance ofincluding the nonlinear bearing stiffness characteris-tics for dynamic behavior analysis by showing thediscrepancy 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 sys-tem 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 approachcan be applied to the rotor system supported by slid-

ing bearings.

CFFGMKk

kpq

centrifugal force IN]load vector

gyroscopic moment [N.m]load-deflection factor [N/m3/2]stiffness matrix

stiffness in p direction due to the load in qdirection

number of ball bearingsnumber of ball elementsspin speed [rpm]displacement vector

contact angle [degree]specified tolerance for convergence

Superscript

b bearingI initial guesss shaft

Subscript

"inner-racewayj ball azimuth locationn ball bearing location

o outer-raceway

Acknowledgements

This study is supported by Turbo and Power Machin-

ery Research Center in Korea Science and Engineer-ing Foundation.

ReferencesDe Mul, J. M., Vree, J. M., and Maas, D. A., 1989. Equilibrium

and Associated Load Distribution in Ball and Roller BearingsLoaded in Five Degrees of Freedom While Neglecting Fric-

8O D.-S. LEE and D.-H. CHOI

tion--Part I: General theory and Application to Ball Bearings,Journal of Tribology, Vol. 111, pp. 142-148.

Gargiulo, E. E, 1980. A Simple Way to Estimate Bearing Stiffness,Machine Design, Vol. 52, pp. 107-1 0.

Harris, T. E., 1991. Rolling Bearing Analysis: 3zl Edition, JohnWiley & Sons Publishing, New York, pp. 395-399.

Jones, A. B., 1960. A General Theory for Elastically ConstrainedBall and Roller Bearing Under Arbitrary Load and Speed Con-ditions, Journal of Basic" Engineering, Vol. 82, pp. 309-320.

Lira, T. C. and Singh, R., 1990. Vibration Transmission ThroughRolling Element Bearing, Part I: Bearing Stiffness Formulation,Journal of Sound and Vibration, Vol. 139, No. 2, pp. 179-199.

Murphy, B. T. and Vance, J. M., 1983. An Improved Method tk)r

Calculating Critical Speeds and Rotor dynamics Stability of Tur-bomachinery, Journal of Engineering for Power, Vol. 105, pp.591-595.

Rao, J. S., 1983. Rotor Dynamics, John Wiley & Sons Publishing,New York, pp. 51-85.

Vance, J. M., 1988. Rotordynamics of Turbomachine3’, John Wiley& Sons Publishing, New York.

While, M. E, 1979. Rolling Element Bearing Vibration TransferCharacteristics: Effect of stiffness, Journal of Applied Mecha-nisms, Vol. 46, pp. 677-684.

EENNEERRGGYY MMAATTEERRIIAALLSSMaterials Science & Engineering for Energy Systems

Economic and environmental factors are creating ever greater pressures for theefficient generation, transmission and use of energy. Materials developments arecrucial to progress in all these areas: to innovation in design; to extending lifetimeand maintenance intervals; and to successful operation in more demandingenvironments. Drawing together the broad community with interests in theseareas, Energy Materials addresses materials needs in future energy generation,transmission, utilisation, conservation and storage. The journal covers thermalgeneration and gas turbines; renewable power (wind, wave, tidal, hydro, solar andgeothermal); fuel cells (low and high temperature); materials issues relevant tobiomass and biotechnology; nuclear power generation (fission and fusion);hydrogen generation and storage in the context of the ‘hydrogen economy’; andthe transmission and storage of the energy produced.

As well as publishing high-quality peer-reviewed research, Energy Materialspromotes discussion of issues common to all sectors, through commissionedreviews and commentaries. The journal includes coverage of energy economicsand policy, and broader social issues, since the political and legislative contextinfluence research and investment decisions.

SSUUBBSSCCRRIIPPTTIIOONN IINNFFOORRMMAATTIIOONNVolume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM3 member rates please emailssuubbssccrriippttiioonnss@@mmaanneeyy..ccoo..uukk

EEDDIITTOORRSSDDrr FFuujjiioo AAbbeeNIMS, Japan

DDrr JJoohhnn HHaalldd, IPL-MPT,Technical University ofDenmark, Denmark

DDrr RR VViisswwaannaatthhaann, EPRI, USA

FFoorr ffuurrtthheerr iinnffoorrmmaattiioonn pplleeaassee ccoonnttaacctt::Maney Publishing UKTel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: [email protected] Publishing North AmericaTel (toll free): 866 297 5154 Fax: 617 354 6875 Email: [email protected]

For further information or to subscribe online please visitwwwwww..mmaanneeyy..ccoo..uukk

CCAALLLL FFOORR PPAAPPEERRSSContributions to the journal should be submitted online athttp://ema.edmgr.com

To view the Notes for Contributors please visit:www.maney.co.uk/journals/notes/ema

Upon publication in 2006, this journal will be available via theIngenta Connect journals service. To view free sample contentonline visit: wwwwww..iinnggeennttaaccoonnnneecctt..ccoomm//ccoonntteenntt//mmaanneeyy

NNEEWW

FFOORR 22000066

Maney Publishing on behalf of the Institute of Materials, Minerals and Mining

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2010

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Documents

Flexible Rotor Ball Bearings with Nonlinear Stiffness Characteristicsdownloads.hindawi.com/journals/ijrm/1997/302073.pdf · 2019. 8. 1. · modeled as an assemblage ofdisks, flexible