NASA/TMp2002-211829

Transient Vibration Prediction for Rotors on

Ball Bearings Using Load-Dependent

Non-Linear Bearing Stiffness

David P. Fleming

Glenn Research Center, Cleveland, Ohio

J.V. Poplawski

J.V. Poplawski and Associates, Bethlehem, Pennsylvania

August 2002
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NASA / TM--2002-211829

Transient Vibration Prediction for Rotors on

Ball Bearings Using Load-Dependent

Non-Linear Bearing Stiffness

David P. Fleming

Glenn Research Center, Cleveland, Ohio

J.V. Poplawski

J.V. Poplawski and Associates, Bethlehem, Pennsylvania

Prepared for the

IFToMM Sixth International Conference on Rotor Dynamics

sponsored by the National Committee of Applied Mechanics of The Institution of Engineers

Sydney, Australia, September 30--October 3, 2002

National Aeronautics and

Space Administration

Glenn Research Center

August 2002
NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

Available from

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov
Transient Vibration Prediction for Rotors on Ball Bearings

Using Load-Dependent Non-Linear Bearing Stiffness

David P. Fleming

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio

J.V. Poplawski

J.V. Poplawski and Associates

Bethlehem, Pennsylvania

Abstract

Rolling-element bearing forces vary nonlinearly with bearing deflection. Thus an accurate

rotordynamic transient analysis requires bearing forces to be determined at each step of the

transient solution. Analyses have been carried out to show the effect of accurate bearing

transient forces (accounting for non-linear speed and load dependent bearing stiffness) as

compared to conventional use of average rolling-element bearing stiffness. Bearing forces

were calculated by COBRA-AHS (Computer Optimized Ball and Roller Bearing Analysis -

Advanced High Speed) and supplied to the rotordynamics code ARDS (Analysis of

RotorDynamic Systems) for accurate simulation of rotor transient behavior. COBRA-AHS is

a fast-running 5 degree-of-freedom computer code able to calculate high speed rollingelement bearing load-displacement data for radial and angular contact ball bearings and also

for cylindrical and tapered roller bearings. Results show that use of nonlinear bearing

characteristics is essential for accurate prediction of rotordynamic behavior.

Introduction

Rotordynamics analyses are now fairly mature, yielding accurate predictions of rotor behavior as long

as the input data are accurate. Modem codes can almost universally be run on personal computers,

making them very portable; an example is ARDS [1, 2]. The challenge nowadays is to furnish the

rotordynamics code with accurate input data such as bearing stiffness.

Response of all but very flexible rotors depends strongly on bearing properties. When bearings are

nonlinear, accurate rotor response calculations demand use of bearing properties for the precise conditions

encountered, rather than average properties. Fluid film bearings have been studied for over 100 years and

their properties can be calculated fairly well; moreover, for many uses these bearings behave in a

reasonably linear manner. Rolling-element bearings, on the other hand, not only have a somewhat shorter

analytical history, but also have a much less linear force-displacement relationship.

Analyses of load-stress relationships for rolling element bearings were published by Lundberg and

Palmgren in the 1940s [3]. These relationships were not easily usable until the analytical additions of

Jones [4] led to the marketing of his computer code in the 1960s. Jones' code, which is still the most

widely used rolling-element bearing analysis tool, is primarily for calculation of bearing stress and fatigue

NASA/TM--2002-211829 1
life, withload-deflectiondataonlyabyproduct.ThecodeCOBRA-AHS(ComputerOptimizedBalland

RollerBearingAnalysis---AdvancedHighSpeed)hasbeendevelopedrecentlyundera NASA Small

BusinessInnovativeResearch(SBIR)contract[5].COBRA-AHS is intended to be a major improvement

over earlier rolling-element bearing analytical tools; use of up-to-date stress/life data, accurate stress

calculations considering bearing installation press fits, and an interactive front end are some of its

characteristics. It accounts for 5 degrees of freedom in the bearing, and can calculate load-displacement

data for high speed radial and angular contact ball beatings and also for cylindrical and tapered roller

bearings. It executes rapidly; thus it was natural to consider making it callable by a rotordynamics code

for accurate transient calculations. The rotordynamics code chosen is ARDS (Analysis of RotorDynamic

Systems) [1, 2], a versatile public-domain code.

Analytical System and Procedure

Figure 1 is an ARDS drawing of the shaft system. It depicts a fairly stiff shaft with concentrated

masses (which may represent compressor or turbine wheels) at stations 3 and 4. In total, 6 stations and 5

elements were used in the model, with 3 of the elements made up of 2 or more subelements.

Loo_-d_.peneent beorln_j stiffness fro_ COBRA

30,000 rp_. 8 oz-;n _ode Io=s

I I

0.0 2.0 4 0 6,0

CONFIGURATION SCALE

03/12/02

13:51

T I

* I t

80 10.0 _2.0

BEARINGS

LINEAR

STN1 STN2

2 o

5 0

NONLINF_AR

STN1 STN2

2 g

5 o

Figure 1. ARDS drawing of rotor

160000

14OOOO

120000 E

- lOO000

8COOO

0 6OOOO

0 4OOOO

2OO0O

-- First critical speed

.......... Second critical speed

--- Operating speed

...'

. ,, ...,-'"'"

/

/

A B C

i i i

10 100 10(]0

Bearing stiffness, MN/m

Figure 2. Critical speeds for varying bearing stiffness

Radial (deep groove) ball bearings of

25 mm bore diameter are at stations 2 and 5.

The shaft material is steel; total mass of the

shaft system is 4.6 kg.

Figure 2 shows the first two system

critical speeds as a function of bearing

stiffness. Letters A, B, and C denote specific

stiffness values to be discussed later. For

stiffness up to about 200 MN/m

(approximately at point C), critical speeds

rise rapidly with stiffness, indicating

significant bearing participation in the rotor

motion. Above 200 MN/m, critical speeds

do not increase as rapidly, indicating that

NASA/TM--2002-211829 2
increasingamountsof motionaredueto shaftbendingratherthanbearingdeflection.Mode shapes

calculatedat criticalspeeds,shownin figure3 for two valuesof bearingstiffness,confinnthis. Fora

bearingstiffnessof 18MN/m,mostof theelasticdeflectiontakesplacein thebearings.At thehigher

stiffnessof 180MN/m,ontheotherhand,nearlyall of theelasticdeflectionis in theshaftat thefirst

criticalspeed,whileatthesecondcriticalspeeddeflectionisaboutevenlydividedbetweenbearingsand

shaft.

n10

0.5

00

-0 5

-1 0

/2_ 200_n _'---.._

(first critical)

" centerline

44 100 rpm

• (second critical)

L h i i J i

5 10 15 20 25 30

Axial distance, cm

(a) Bearing stiffness = 18 MN/m

10

05

E

O0

-05

......... " _ 50 100rpm

.... 2--2272

10 15 2O 25

Axial distance, cm

(b) Bearing stiffness = 180 MN/m

-1 o

0 5 30

Figure 3. Critical speed mode shapes

Shaft transient response to suddenly applied

imbalance was studied at 30000 rpm. The

imbalance magnitudes were those that could have

resulted from loss of a blade in a turbomachine.

COBRA-AHS could have been called at each time

step of the analysis to supply bearing force at the

current displacement. However, for the present

studies with only one bearing variable (radial

displacement: speed is constant) it was more

efficient to run COBRA-AHS once to generate load

versus deflection data for the bearings. A fourthorder curve was then fitted to these data and the

coefficients supplied to ARDS. Figure 4 shows two

stiffness curves from the bearing data. The upper

curve is the incremental stiffness, that is, the slope

400

3O0

g

Z

200

cO

100

incremental

0

000 001 002 003 004

Radial deflection, mm

Figure 4. Ball bearing stiffness

005

of the load-deflection curve. This stiffness would be used if one were studying motion about an off-center

steady-state point. The lower curve is the load divided by the deflection, or average stiffness; it is this

value that was used by ARDS for calculation of the instantaneous bearing force. Note that, for both

curves, there is more than an order of magnitude increase over the range of the figure. This illustrates the

significant nonlinearity of the ball bearing used, and is representative of rolling-element bearings.

For the low deflection expected at normal (well-balanced) operating conditions, the bearing stiffness is

about 20 MN/m. Reference to figure 2 (stiffness point A) shows that this places the initial operating

condition between the first and second critical speeds. For higher stiffness resulting from higher bearing

loads due to rotor imbalance, the operating regime of the rotor could well change to subcritical. Thus, as

will be seen, having accurate bearing characteristics for the particular operating condition is vital for

prediction of rotor behavior.

NASA/TM--2002-211829 3
Results

Transient orbit plots for suddenly applied imbalances of 36, 144, and 576 g cm and a speed of

30 000 rpm are shown as figure 5 using the ball bearing properties calculated by COBRA-AHS. The

imbalance was applied at station 4 (the location of one of the concentrated masses), and the orbits are

those of station 5, the right bearing location. One hundred time steps per revolution were used by the

transient analysis, and five revolutions are plotted. The plotting code superimposes a small circle on the

orbit trace at the completion of each revolution; these are numbered 1 through 5 in the plots. The leftmost

plot, for 36 g cm imbalance, shows the amplitude increasing more or less smoothly to its steady-state

value of 9 lain, with only a small overshoot. In the middle plot, for 144 g cm imbalance, the orbit is not as

smooth, and the amplitude of 18 Hm at the end of five revolutions is noticeably less than the maximum of

23 _tm seen in the plot. The rightmost plot, for 576 g cm imbalance, is somewhat more ragged looking,

and again the maximum amplitude reached, 49 p,m, is greater than the 39 lam seen at the end of five

revolutions. The amount by which the amplitude overshoots the steady-state value depends on bearing

damping. While damping in rolling-element bearings is known to be low, precise values are not readily

available. A value of 2 kN/m was used in the analyses; this is close to the estimates reported in [6].

Z TRA_ TION -10 "_

(a) Imbalance 36 g cm (b) Imbalance 144 g cm (c) Imbalance 576 g cm

Figure 5. Transient orbits for load-dependent bearing stiffness

3  ̧

| z :

dc

(a) Stiffness 18 MNIm (b) Stiffness 53 MNIm (c) Stiffness 180 MNIm

Figure 6. Transient orbits for constant bearing stiffness. Imbalance 144 g cm

NASA:TM-----2002-211829 4
Parts a, b, and c of figure 5 each use different scales, namely 5, 10, and 20 microns per division

10.0002, 0.0004, and 0.0008 inches per division); thus the orbit sizes are roughly in the ratio of 1, 2, and

4. However, the imbalances are in the ratio 1,4, and 16; thus the orbit size for these three cases is roughly

proportional to the square root of the imbalance. This is not a surprising result, since the bearing stiffness

increases with the load applied.

For comparison, orbit plots for three values of linear bearing stiffness (18, 53, and 180 MN/m) and an

imbalance of 144 g cm are shown as figure 6. Again, numbers are placed on the plots denoting the

number of completed revolutions. For these cases it suffices to consider only one level of imbalance,

since, for linear bearings, amplitudes are directly proportional to imbalance. The left and center orbits are

fairly smooth, and all the orbits show a maximum amplitude that is significantly larger than the amplitude

at the end of 5 revolutions. Reference to figure 2 (stiffness point ,4) shows that the operating condition for

the left plot is between the first and second critical speeds, while for the center and right plots

(corresponding to points B and C in fig. 2) operation is entirely subcritical. Surprisingly, the left and

center orbits (plotted to the same scale of 50 _tm (0.002 in) per division) are nearly the same size, and

much larger than the orbit of figure 5(b) for the same imbalance. However, the right orbit (plotted at a

magnified scale of 10 lam (0.0004 in) per division) is much smaller. One may reasonably conclude from

figures 5 and 6 that estimation of average linear bearing stiffness, and consequently accurate

rotordynamic prediction with a linear analysis, is difficult at best.

ARDS was allowed to run for a longer time (40 revolutions) in order to discern the steady state that

would be attained following the imposition of imbalance. Figure 7 summarizes the results, obtained from

measurements on the orbit plots made using variable bearing stiffness from COBRA-AHS. As seen

previously from the orbits (fig. 5), shaft peak amplitude is higher than the eventual steady state value.

50

4O

£

._ 30

E

•_- 20

Q.

E

<

10

0

0

-- Steady state

i i J i i

1oo 200 300 400 500 600

Imbalance, gcm

Figure 7. Station 5 amplitude after sudden imbalance

160

140

120

0 100

__

E

_" 80

_.- 6o

E

4o

-- Steady state

0

0 100 200 300 400 500 600

Imbalance, gcm

Figure 8. Station 4 amplitude after sudden imbalance

Figure 8 displays the amplitude at station 4, the location of the imbalance. The trend as imbalance

changes is similar to figure 7, but the amplitudes are larger, and they increase at a greater rate than do

amplitudes at the bearings. Figure 9 shows the reason for this. This figure shows the mode shape assumed

by the rotor as the motion approaches steady state. Even though the rotor speed is well below the critical

speeds calculated for the effective bearing stiffness, there is significant rotor bending. The rotor bending

causes vibration amplitude to be considerably larger over the rotor midspan than at the bearings.

Moreover, the amount of bending increases with imbalance, producing a greater disparity between

bearing and midspan amplitudes.

NASA/q'M--2002-211829 5
Figure10plotstheeffectivestiffnessof thebearingat station5 fortherangeof imbalancestudied;it

is calculatedasthebearingloaddividedbythevibrationamplitude.Thestiffnesscurvelabeledpeak is

naturally higher than that for steady state because of the higher peak amplitude. Both peak and steadystate stiffness increase with the amount of imbalance, again because of the corresponding higher

amplitudes.

The effective stiffness of 170 MN/m for steady state conditions with 576 g cm imbalance is close to

the stiffness of 180 MN/m used for figure 3 (b). There is a similarity in mode shapes between figures 3(b)

and 9, with the shape of figure 9 being that of the first mode of fig. 3 (b) modified by the second mode.

This similarity of mode shapes occurs despite the operating speed of 30 000 rpm being well below the

first critical speed of 50 100 rpm shown in figure 3(b). This implies that the observed mode will

correspond to the critical speed mode shapes for the effective bearing stiffness, even if the operating

speed is well removed from the critical speed. A further implication is that the beating stiffness of

significance is that at the operating condition: this stiffness may be considerably different than at design

condition. Furthermore, identical beatings at different locations on the rotor may have different

stiffnesses, as determined by the beating deflection; note, from figure 9, that the amplitudes are not the

same at the two bearing locations. The imbalance was applied closer to the tight-hand bearing, leading to

greater amplitude on the right side of the rotor.

012 -

010-

008

E

E 0O6

_. OO4

o 02

000

-0 02

1// 576 g ¢m imbalance

/

/

/

/

/

/

!

//

; ! 44 ...........

\

\

\

\

i I

. \

/ 36 k

/ \

'-_- ......... Toff_nT_.Tne- ........ 2I---

bearing ring

5 10 15 20 25 30

Axial distance, cm

Figure 9. Rotor steady state mode shapes

220 ,

200 ,

180

Z

160

14o

.=_

>_ 120

100

:1=

LLJ

8O

60

/

40

-- Steady state

i Peak

100 200 3013 400 500

Imbalance, g cm

Figure 10. Station S effective bearing stiffness

600

Concluding Remarks

Transient rotordynamic calculations were made for suddenly applied imbalance on a ball bearing

supported rotor. Accurate beating forces were obtained at each time step from data generated by COBRAAHS. For the bearings employed, effective bearing stiffness increased substantially with bearing

deflection. As a result, beating vibration increased less than linearly with the amount of imbalance.

Vibration amplitude at rotor midspan was larger than at the bearings because of rotor bending, but again

the rate of vibration increase with imbalance was less than linear. The rotor assumed a mode shape

corresponding to the critical speed mode shapes, even though the operating speed was well removed from

any critical speed. Results show that it is essential to use amplitude-dependent beating properties in order

to obtain accurate rotordynamic predictions.

NASA/TM----2002-211829 6
References

. Nelson, H.D.; Meacham, W.L.: Fleming, D.P.: and Kascak, A.F. (1983): Nonlinear Analysis of

Rotor-Bearing Systems Using Component Mode Synthesis. Journal of Engineering ,)Cot"Power,

vol. 105, no. 3, pp. 606-614.

-) Fleming, David P. (1996): Rotordynamics on the PC: Transient Analysis with ARDS. Rotating

Machinetq'----1996 (Proceedings of the Sixth International Symposium on Transport Phenomena and

Dynamics of Rotating Machinery), Vol. 1, D.C. Ham S.T. Ro, and J.H. Kim, eds., Pacific Center of

Thermal-Fluids Engineering, Kihei, Maui, Hawaii, U.S.A., pp. 367-375.

3. Lundberg, G., and Pahngren, A. (1947): Dynamic Capaci_ of Rolling Bearings. Acta Polytechnica

(Mechanical Engineering Series), vol. 1, no. 3.

. Jones, A.B. (1960): A General Theory for Elastically Constrained Ball and Radial Roller Bearings

Under Arbitrary Load and Speed Conditions. Trans. ASME, J. Basic Eng., vol. 82, no. 2,

pp. 309-320.

. Poplawski, J.V.; Rumbarger, J.H.: Peters, S.M.; Flower, R.: and Galaitis, H. (2002): Advanced

Analysis Package for High Speed Multi-Bearing Shaft Systems: COBRA-AHS. Final Report, NASA

Contract NAS3-00018.

6. Lewis, P.; and Malanoski, S.B. (1965): Rotor-Bearing Dynamics Design Technology; Part IV: Ball

Bearing Design Data. AFAPL-TR-65-45, Part IV.

NASA/TM-----2002-211829 7
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August 2002 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Transient Vibration Prediction for Rotors on Ball Bearings Using

Load-Dependent Non-Linear Bearing Stiffness

6. AUTHOR(S)

David P. Fleming and J.V. Poplawski

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135 - 3191

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National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-728-30-10-00

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REPORT NUMBER

E-13523

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NASA TMI2002-211829

11. SUPPLEMENTARY NOTES

Prepared for the IFToMM Sixth International Conference on Rotor Dynamics sponsored by the National Committee of

Applied Mechanics of The Institution of Engineers, Sydney, Australia, September 30-October 3, 2002. David P. Fleming,

NASA Glenn Research Center, and J.V. Poplawski, J.V. Poplawski and Associates, Bethlehem, Pennsylvania. Responsible

person. David P. Fleming, organization code 5950, 216--433-6013.

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

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Available electronically at httr_://_ltrs._rc.nasa.oov

This publication is available from the NASA Center for AeroSpace Information. 301--621 _)390.

13. ABSTRACT (Maximum 200 words)

Rolling-element bearing forces vary nonlinearly with bearing deflection. Thus an accurate rotordynamic transient analysis

requires bearing forces to be determined at each step of the transient solution. Analyses have been carried out to show the

effect of accurate bearing transient forces (accounting for non-linear speed and load dependent beating stiffness) as

compared to conventional use of average rolling-element bearing stiffness. Bearing forces were calculated by

COBRA-AHS (Computer Optimized Ball and Roller Bearing Analysis - Advanced High Speed) and supplied to the

rotordynamics code ARDS (Analysis of Rotor Dynamic Systems) for accurate simulation of rotor transient behavior.

COBRA-AHS is a fast-running 5 degree-of-freedom computer code able to calculate high speed rolling-element bearing

load-displacement data for radial and angular contact ball bearings and also for cylindrical and tapered roller bearings.

Results show that use of nonlinear bearing characteristics is essential for accurate prediction of rotordynamic behavior.

14. SUBJECT TERMS

Rotordynamics; Rolling-element bearing stiffness; Transient dynamic analysis;

Rolling-element bearings: Bearing life

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