experimental study of a precision, hydrodynamic wheel...

Experimental study of aprecision, hydrodynamicwheel spindle for submicroncylindrical grinding

J. F. Tu,* M. Corless,† M. J. Gehrich,‡ and A. J. Shih‡*School of Industrial Engineering, and†School of Aeronautics and Astronautics Engineering, Purdue University,West Lafayette, IN, USA‡Cummins Engine Company, Columbus, IN, USA

Hydrodynamic journal bearings have been widely used in various types ofrotating machinery, ranging from heavy duty, high-impact applications,such as the crank shaft of an internal combustion engine and turbine rotor,to high-precision, light load applications, such as precision spindles incylindrical grinding machines. Although extensive theoretical and experi-mental results have been presented for hydrodynamic bearings, the avail-able literature seems to be limited for precision hydrodynamic bearingspindles. In this study, practical methods have been developed to quantifythe performance of a hydrodynamic wheel spindle operating in the hori-zontal mode to produce precision parts with submicron roundness toler-ance and very fine surface finish. These methods can easily and cost effec-tively be implemented on various machines in an actual productionenvironment for effective predictive maintenance. The main experimentalresults show that the long-term drift of the spindle at steady state is less than1 mm vertically and 0.2 mm horizontally, and the radial error motion of thespindle based on unfiltered data is less than 1.6 mm for all the speeds tested.It is also found that the shaft center position (vertical lift and horizontalshift) at the cold condition is substantially different from that in the steady-state warm condition. From the results, an optimal spindle speed is recom-mended. © 1998 Elsevier Science Inc.

Keywords: hydrodynamic bearing, spindle, radial error motion, cylindri-cal grinding roundness, radial error motion, whirl, spindle rotatingaccuracy

Introduction

Hydrodynamic journal bearings have been widelyused in various types of rotating machinery, rang-ing from heavy-duty, high-impact applications,such as the crank shaft of an internal combustionengine and turbine rotor, to high-precision, light

load applications, such as precision spindles incylindrical grinding machines. Raimondi andBoyd1 reported one of the first systematic analysesof hydrodynamic journal bearings. Their resultsare still widely used as powerful design and analy-sis tools. More complex bearing geometries andgeometry qualities related to alignment and sur-face roughness have been investigated.2–9 Re-cently, attention has focused on the thermal as-pects of hydrodynamic bearings. An extensivereview can be found in Pinkus.10 Attempts have

Address reprint requests to Dr. J. F. Tu, School of IndustrialEngineering, 1287 Grissom Hall, West Lafayette, IN, 47907-1287,USA. E:mail: [email protected]

Precision Engineering 22:43–57, 1998© 1998 Elsevier Science Inc. All rights reserved. 0141-6359/98/$19.00655 Avenue of the Americas, New York, NY 10010 PII S0141-6359(98)00003-8

been made to predict the steady-state temperaturedistribution and dynamic characteristics.11–19

Some interesting work demonstrated that a signif-icant temperature difference existed across a syn-chronously whirling journal.20,21 Transient ther-mal problems were studied by Paranjpe andHan.22

Considerable experimental work also hasbeen conducted to measure dynamic characteris-tics of hydrodynamic bearings.23–32 An extensivehydrodynamic journal bearing test rig was devel-oped by Flack et al.,33 which is capable of measur-ing operating eccentricity, temperature distribu-tion, continuous circumferential pressure, andfilm thickness profiles, as well as dynamic stiffnessand damping coefficients.

However, for precision hydrodynamic bear-ings used to produce precision parts with submi-cron roundness tolerance, the available literatureis limited. In particular, there is a need to developa practical and cost-effective method to evaluatethe performance of precision hydrodynamic spin-dles ready for production. A major constraint forstudying this type of spindle is that very few inva-sive instrumentations such as those developed byFlack et al.33 can be attached to the spindle. Inother words, none of the spindle components canbe altered permanently (e.g., drilling a hole orgrinding a groove). Although techniques exist toevaluate spindle running accuracy,34–39 most re-quire a precisely aligned precision arbor, multiplesensors, and invasive installations. In practice, it isusually unfeasible to install a precision arbor toevery production spindle, and the fixtures neededfor multiple sensors also present great difficul-ties.

In this paper, practical methods are devel-oped to study the operating performance of ahydrodynamic journal bearing, wheel spindle usedfor submicron precision cylindrical grinding. Inparticular, the spindle of interest is a wheel spin-dle for a large cylindrical grinder, operated in thehorizontal mode. The weight of the spindle shaft/wheel assembly functions as the primary load. Theeffects of lubricant, speed, and temperature onthe spindle running accuracy, thermal drift, dy-namic characteristics, and stability are examined.

This paper begins with a discussion of the prac-tical methods developed to assess spindle runningaccuracy followed by a description of the experi-mental apparatus and related signal processing. Ex-perimental results are then presented regardinghow lubricant, temperature growth, and speed af-fect the steady-state temperature, center position,and spindle error motion. The results are com-pared with classical theoretical results. A summary

of the results and concluding remarks complete thepaper.

Spindle metrology

The rotating accuracy of a machine tool spindledirectly affects the roundness of machinedparts.34–36,40–44 Commonly, a precision arbor(e.g., a precision sphere or cylinder) and a probeare used to inspect the spindle axis error motionfor the case of a fixed sensitive direction,35,36,45–52

or two probes are used for the case of a rotatingsensitive direction.53–56 Only one probe is neededfor the fixed sensitive direction case, because onlyone direction (along the sensitive direction) of thespindle motion is critical to the machining quality.On the other hand, when the sensitive direction isrotating, the axis motion in both directions (e.g., xand y directions) are important; therefore, twoprobes are needed. When a reference precisionarbor is unavailable or when the spindle errormotion is in the same order of magnitude as theaccuracy of the reference arbor, it is imperative toseparate the roundness error of the reference ar-bor from the spindle error. According to the sum-mary by Whitehouse57 and Gao et al.,37 there aretwo types of error separation methods. One isknown as the multiorientation method, which in-cludes the Donaldson method58 and the multistepmethod.59–62 The multiorientation method is sim-ple and effective if the spindle error is repeatable.The other method is the multiprobe method. Thismethod is more suitable for on-machine measure-ment. One of the most widely used multiprobemethods is the three-probe method,37,63,64 whichuses three probes mounted at three different an-gles to obtain the roundness shape of the refer-ence arbor and the two-dimensional (2-D) spindleerror motion simultaneously. Because of its capa-bility of obtaining 2-D spindle motion, it is suitablefor cases of both a fixed sensitive direction and arotation sensitive direction.

In this study, several practical techniques aredeveloped to study long-term thermal drifts andthe radial error motion of a hydrodynamic wheelspindle ready for production without using a pre-cision arbor or the three-probe method. Specifi-cally, these techniques are able to measure theaverage journal center position and the radial er-ror motion of the spindle.

Center position

In hydrodynamic lubrication, the shaft (or jour-nal) is floated and separated from the bearingsleeve by a layer of lubrication film. This film isproduced entirely by fluid pressure generated by

Tu et al.: Precision, hydrodynamic wheel spindle

44 JANUARY 1998 VOL 22 NO 1

the relative motion between the shaft and thebearing sleeve. The separation mechanism is dif-ferent from that of hydrostatic lubrication, inwhich a highly pressurized fluid is introduced intothe bearing. The film thickness affects the positionof the shaft and the pressure distribution of thelubricant, which in turn affect the running accu-racy and loading capacity of the shaft. Many fac-tors affect the film thickness; mainly, the lubricantviscosity m, the rotating speed n, and the bearingunit load P, defined as load W divided by bearingprojected area (shaft diameter D times bearinglength L). Generally speaking, the higher the vis-cosity, the lower the rotating speed needed to floatthe shaft at a given load; the higher the rotatingspeed, the lower the viscosity needed to float theshaft at a given load; and the lower the bearingunit load, the lower the rotating speed and viscos-ity needed to float the shaft. At very high speeds,an instability can occur in hydrodynamic bearingsknown as “whirl.” When whirl occurs, the shaft isorbiting around the sleeve center at a frequencyapproximately one-half the spindle frequency, andthe spindle stiffness is reduced to zero. It is impor-tant to detect the occurrence of the whirl and toavoid it at all times.

In this paper, the film thickness is measuredby determining the displacement of the shaftcenter from the sleeve center. The center posi-

tion defined here is not the instaneous driftingof the center position within one spindle revolu-tion but the center of a perfect circle with theaverage radius for each revolution. The insta-neous drifts of the spindle center are studied bycharacterizing the spindle radial error motion inthe next section. To determine the actual centerdisplacement, the form error and thermal expan-sion of the reference arbor must be compensatedfor. With the grinding wheel hub of the spindleused as an inexpensive reference arbor, its formerror is compensated by taking averages of theradial measurements to define the shaft center asthe center of a perfect circle with the averagedradius. As shown in Figure 1, the stationary posi-tion, represented by x0 and y0, is determined byrotating the spindle by hand to twenty-fourequal-spaced angular positions. After being ro-tated to an angular position, the shaft is allowedto “sink-in” to establish its stationary position atthat angle before measurements are taken, andthe spindle is rotated to the next angular posi-tion.

Compensation of the arbor’s thermal expan-sion is considered next. In Figure 1, as the grindingwheel center moves from O to O9 while the radiusof the wheel hub changes from R to R9, the mea-surements by capacitance probes #1 and #2 are

Figure 1 Capacitance probe lo-cations and radial motion mea-surements


45PRECISION ENGINEERING

x1 5 R 1 x0 1 x 2 R9 cosu

fx 5 x1 2 x0 1 (R9 cosu 2 R) (1)

y1 5 y0 1 R 2 R9 cosf 2 y

fy 5 y0 2 y1 2 (R9 cosf 2 R) (2)

Therefore, according to Equation (1), the appar-ent displacement of the center in the x-direction isdetermined as x1 2 x0. However, although theactual displacement x is zero, the increase of thegrinding wheel hub radius attributable to thermalexpansions will reduce the value of x1 2 x0; thus,the shaft appears to move to the left gradually asthe grinding wheel hub diameter increases. Simi-larly, according to Equation 2, the shaft will ap-pears to move upward as the grinding wheel hubheats up. This thermal effect must be consideredwhen one attempts to interpret data for shaft cen-ter displacement.

To completely isolate the effect of thermalexpansion, two more capacitance probes, #3 and#4, can be added as shown in Figure 1. The mea-surements of the additional two probes can beexpressed as

x4 5 R 1 x3 1 x 2 R9 cosu

fx 5 x4 2 x3 1 (R9 cosu 2 R) (3)

y4 5 y3 1 R 2 R9 cosf 2 y

fy 5 y3 2 y4 2 (R9 cosf 2 R) (4)

Therefore, using Equations (1)–(4), the four un-knowns x, y, , R9 cosu, and R9 cos f can be deter-mined uniquely. Note that the center positionthus obtained is only an average center positionfor revolution. It is different from instantaneouscenter positions versus different angular positionsdescribed in a polar chart obtained by standardshaft center measurement techniques with a pre-cision arbor.35 For the purpose of investigatinglong-term thermal drifts, film thickness, and tran-sient lifts, the knowledge of average center is suf-ficient. However, it is not cost effective to use fourprobes and, in practice, sufficient space may notexist for mounting four probes properly and rig-idly. In the experiments conducted in this study,only two probes are used. The thermal expansionof the wheel hub is compensated for by the fol-lowing procedures. First, the center displacementis only measured very shortly (within a few sec-onds) after the spindle is started from its coldcondition to a constant speed, based on the as-sumption that the thermal expansion is negligiblein the very beginning. After the spindle reaches itsthermal steady state at a specific speed, the center

positions at other speeds are then calibrated bymeasuring the relative center shifts immediatelyafter the speed change. Additional details are dis-cussed in the results section.

With the center displacement determined, theminimum film thickness d is then determined as

d 5 c 2 Î(c9 2 y)2 1 x2 (5)

where c is the radial clearance of the bearing at thecold condition, and c9 is new clearance at the cur-rent temperature condition. As indicated in Equa-tion (5), if the horizontal shift x is small comparedwith c9 2 y and c > c9, the minimum film thicknesscan be approximated as the vertical lift y.

Spindle radial error motion

A new algorithm is developed to determine thespindle radial error motion and to detect the oc-currence of whirl error motion at one-half thespindle frequency. Considering a data array ofradial motion measured by a capacitance probe,

m(i) 5 mw(i) 1 mf(i) 1 mr(i) 1 m0,

i 5 1, . . . , M z N (6)

where mw is the error motion caused by whirl, mf isthe form error of the reference arbor, mr is theerror motion caused by other effects mainly ran-dom in nature, m0 is a constant related to thenominal gap between the probe and the referencearbor, N is the total samples per revolution, and Mis the number of revolutions of the data array. Asdefined in ANSI/ASME B89.3.4,34 both the errormotion caused by whirl and the random motionare parts of asynchronous error motion. In thispaper, the error motion caused by whirl is singledout from the asynchronous error motion, and theremaining components of asynchronous error mo-tion including measurement noise are lumped to-gether and referred to as random error motion,although they may not be truely random in na-ture. Rearranging the data array into a matrixform with each column representing the samplesof one revolution, we have

m(i, j) 5 [m1(i), m2(i), . . . , mM(i)]

5 [mj(i)],

i 5 1, . . . , N, j 5 1, . . . , M (7)

where mj(i) represents a column data vector forthe jth revolution. We can further group the col-umns in Equation (7) into odd-numbered andeven-numbered columns and obtain their corre-sponding averages:



m# odd(i) 52M (

k51

M/2

m2k21(i), i 5 1, . . . , N

(8)

m# even(i) 52M (

k51

M/2

m2k(i), i 5 1, . . . , N

(9)

The error motion terms in Equation (6) become

m# odd(i) 5 mw(i)uodd 1 mf(i) 1 m# r(i)uodd 1 m0,

i 5 1, . . . , N (10)

m# even(i) 5 mw(i)ueven 1 mf(i) 1 m# r(i)ueven 1 m0,

i 5 1, . . . , N (11)

Notice that the error motions caused by the formerror are the same for Equations (10) and (11);whereas, the whirl and the random error motionterms are different, because they are not synchro-nized with the spindle frequency. Without loss ofgenerality, assuming that the shaft is orbitingaround the sleeve center with a radius of rw and atone-half spindle frequency, the whirl error motionduring the odd-numbered and even-numberedrevolutions can be expressed as

mw(i)uodd 5 rw cosS2pf2

i DtD5 rw cosSp

iND ,

i 5 1, . . . , N (12)

mw(i)ueven 5 2rw cosS2pf2

i DtD5 2rw cosSp

iND ,

i 5 1, . . . , N (13)

Introducing Equations (12) and (13) into Equa-tions (10) and (11), respectively, and using thereversal principle, we can obtain the spindle errormotion me(i) and the arbor form error mf(i) simul-taneously:

me(i) 512

[m# odd(i) 2 m# even(i)]

5 rw cos(pi/N) 112

(m# r(i)uodd

2 m# r(i)ueven),

i 5 1, . . . , N (14)

mf(i) 512

[m# odd(i) 1 m# even(i)] 2 m0

212Sm# r(i)uodd 1 m# r(i)uevenD ,

i 5 1, . . . , N (15)

Note that the form error has been eliminated inEquation (14). The spindle radial error motionobtained by Equation (14) can be plotted in polarcharts to indicate the running characteristics ofthe spindle. The radial error motion for each dataarray [m(i)] is calculated as

er 5 maxi

me(i) 2 mini

me(i),

i 5 1, . . . , N (16)

This simpler index is preferred here, because it isessentially impractical to record the data array ofeach spindle revolution during long-term moni-toring. The radial error motion er is then calcu-lated on-line to study the effects of spindle speedson the spindle radial error motion.

In Equation (14), if half-frequency whirl doesnot occur, the measured error motion me is mainlycaused by the random error motion. Because themotion is mainly random in nature, m# r(i)uodd andm# r(i)ueven will not cancel each other. If whirl oc-curs, rw cos(pi/N) will appear and is usually dom-inant in Equation (14). Using Equations (14) and(16), the spindle radial motion is investigated fortwo purposes: to detect whirl motion and, if nowhirl motion, to determine radial error motion ofthe spindle. The form error of the arbor can alsobe determined simultaneously by Equation (15)assuming the random error motion is compara-tively much smaller than the form error of thereference arbor. This is generally true in accessinghigh-precision spindles. On the other hand, if it isnot, form error is not a concern.

Experimental setup

Motor, spindle, and test table

The main experimental setup includes a rebuilthydrodynamic spindle used for a high-precisioncylindrical grinder, a DC brush motor and itsdrive, and a vibration-free table capable of damp-ing out frequency over 4 Hz. The complete setupis capable of running the spindle over 4,000 rpm.However, the maximum speed is limited to 2,600rpm because of the safety concern of the grindingwheel at high speeds. This speed limit, however, isapproximately 850 rpm higher than the recom-mended speed by the spindle manufacturer.



Thermocouples

One key objective of this study is to characterizethe thermal behavior of the spindle. For this pur-pose, up to 25 thermocouples are mounted insideand outside the spindle at many locations. Afterseveral initial tests, 13 representative thermocou-ples are chosen for the main experiments. Thelocations and the addresses of these thermocou-ples are shown in Figure 2. All the thermocouplesare of T type. Note that point #29 measures theinlet oil temperature to the front bearing and thetemperature at the back of the front bearingsleeve is not measured, because there is no passageto attach a thermocouple there as at the rear bear-ing (#18).

Capacitance probes

Capacitance sensors are used to determine theaverage center position and the radial error mo-tion of the spindle. The measurement range of thesensors is 0.75 mm. Using a 16-bit A/D converter,these sensors have a resolution of 1.0 nm, but therepeatability is limited to 0.25 mm because of noiseand nonlinearity. Two sensors are mounted tomeasure the radial position of the grinding wheel,and the measurements are then calibrated to shaftcenter positions and spindle error motion, as dis-cussed in the previous sections.

Data acquisition systems

The data acquisition system includes a multiplex-er/pre-amplifier box for accepting thermocouplesignals and regular voltage signals ranging from25 to 15 V. The A/D converter is of 16-bit reso-lution. A set of programs are developed to accessthe box and the A/D card within the platform ofMATLAB. A master data acquisition program isthen developed to automate the experimentalprocedures and to display and record data on-line.The program can easily be expanded and inte-grated for other purposes.

Form error separation

The reversal principle proposed by Donaldson58 isused to determine the form error of the referencedisk of the grinding wheel. This form error wascompared with the form error obtained by thealgorithm developed above. As shown in Figure 3,there are very good agreements between these twomethods.

Signal processing

The radial motion measurement has been sampledat a sampling rate of 100 K sample/second. A con-tinuous sampling record is shown in Figure 4. Notethe downward spikes, which reflect the index pointon the wheel hub. The long record of the measure-ment shown in Figure 4 is then separated into short

Figure 2 Thermocouple locations



records of data for each revolution with the indexspikes removed. These short records are then pro-cessed according to the algorithm developed aboveto determine the whirl error motion at one-halfspindle frequency. Currently, no filtering is applied.

The downward spikes shown in Figure 4 are alsouseful in indicating whether or not the radial mo-tion measurement system is of sufficient dynamiccapability. If these spikes cannot be detected, thenthe sampling rate may not be fast enough. If the sizeof the spikes are not the full size (down to 0.25

mm), then the dynamic bandwidth may not be suf-ficient, which, in turn, will result in underestimationof accessing the radial error motion.

Results and discussions

Lubricant effect

Two lubricants, Velocite 3 and Velocite 6, are com-pared in terms of their effects on the spindle tem-perature. The results show that, at 1,750 rpm, thesteady-state temperature of the spindle measuredon the housing is 67°C with Velocite 6, which is20°C in excess of the temperature for which Ve-locite 3 is used. Therefore, it is determined thatVelocite 6 generates too much heat because of itshigher viscosity and is unsuitable for the requiredhigh-speed application. In this study, only Velocite3 is used for further investigations.

Temperature growth

Figure 5 presents the temperature growth measuredat various locations of the spindle. The measure-ment locations are defined in Figure 2. In this test,the speed was maintained constant at 1,780 rpm for2,000 minutes (over 33 hours). During this period,the temperatures grew, reached their steady states,and remained constant. The maximum steady-statetemperatures at the back of the rear bearing sleeve(point #18 in Figure 2) and at the housing on frontbearing side (#17) are 46°C and 40°C respectively.The temperatures measured at both edges of thebearings (#20–27) are the same within 0.5°C and

Figure 3 Form error separation

Figure 4 Long, raw record of theradial motion measurement



equal to the oil temperature at the reservoir. There-fore, only temperature #20 is plotted in Figure 5. Alltemperature measurements demonstrate similargrowth patterns. The settling time of the spindle isfound to be 1,350 minutes when the temperaturereaches within 1% of the final steady-state temper-ature. The corresponding time constant is then cal-culated to be 270 minutes, which is equivalent to 4hours and 30 minutes. Because of this long settling

time, it is recommended not to shut the spindle offbetween shifts to maintain the steady-state thermalcondition.

Steady-state temperature

Several tests were conducted to determine thesteady-state temperatures at speeds ranging from1,400 rpm to 2,500 rpm. Figure 6 represents the

Figure 5 Temperature growthat a constant speed of 1,780rpm

Figure 6 Speed versus steady-state temperature



steady state-temperatures measured at the housing(#17) versus speed. This temperature location ischosen because of its easy access and is justified,because it still captures the transient behavior ofthe spindle credibly, as indicated by Figure 5. It isclear that the correlation between the steady-statetemperature, and the speed is quite linear. Thischart is useful in defining the normal steady-statetemperatures for speeds between 1,400 rpm and2,450 rpm. If the steady-state temperature be-comes several degrees higher than the one de-fined by this chart, it may indicate, for example,dirty lubricant or faulty bearings. This result hasthe potential to be developed into a predictivemaintenance tool by alerting the operator aboutthe potential abnormal conditions. Additionallong-term tests in the actual production line areneeded to characterize fully the signatures ofthese potential problems.

Spindle error motion

Several experiments were conducted to study therelationship between spindle error motion and

spindle speed. This study is important to evaluatethe merits of increasing the grinding speed. Theradial error motion of the spindle is obtainedbased on the algorithm developed above. No fil-tering is applied in calculating radial error mo-tion.

In Figure 7, the spindle had been runningovernight at 1,800 rpm to ensure that the spindlehad reached its steady-state condition. The speedwas then raised to 2,000 rpm, 2,200 rpm, 2,450rpm, and then down to 1,600 rpm, 1,400 rpm, andwas shut down at the end. Each speed was main-tained for at least 80 minutes before changes. It isnoticed that the radial error motion is lowest at2,450 rpm, and when the speed was reduced from2,450 rpm to 1,600 rpm, a distinct increase inradial error motion (both the mean and the fluc-tuations) was noticed. The above correlation be-tween the radial error motion and the speed isconfirmed by several other similar experiments.

The mean and standard deviations of the ra-dial error motion measurements at differentspeeds are listed in Table 1. According to Table 1,

Figure 7 Radial errormotion versus speed

Table 1 Mean and standard deviation of the radial error motion measurements

1,400 rpm 1,600 rpm 1,800 rpm 2,000 rpm 2,200 rpm 2,450 rpm

Mean, mm 0.0016 0.0016 0.0015 0.0015 0.0014 0.0012Standard deviation, mm 1.7 3 1024 3.5 3 1024 1.7 3 1024 3.5 3 1024 1.9 3 1024 1.6 3 1024



operating speeds above 2,200 rpm are recom-mended because of their higher spindle runningaccuracy.

Whirl error motion. For all tests conducted up2,600 rpm, whirl was not observed based on Equa-tion 14. Higher speeds were not tested because ofsafety concerns regarding the grinding wheel. Thestability margin and dynamic stiffness of the spin-dle at different speeds are determined in the nextsection based on the measured minimum filmthickness.

Center position drifts

In this section, the center position of the shaft ismeasured to examine its thermal drifts and theminimum film thickness. The center position isdetermined for both the initial cold condition andthe steady-state warm condition. The measuredfilm thickness is used to assess spindle character-istics and compared with the theoretical resultsfrom Raimondi and Boyd,1 Lund,65 and Ham-rock.66

Thermal drift. Several experiments were con-ducted to verify the long-term thermal drift of thespindle. One of the typical results is shown inFigure 8, which record the center drifts for over 33hours. The numbers labeled on the curve in thefigure correspond to the time in minutes since thespindle was started. The results show that the mea-sured “drift” is mainly attributable to the thermalexpansion of the grinding wheel hub. Note that

the drift appears to move upward and to the leftbecause of the thermal expansion of the wheelhub, as discussed in the Center position section.Note that the actual center position may be mov-ing to the opposite direction. After 180 minutes(i.e., approximately one time constant) the spin-dle center appears to drift only slightly within arectangular boundary measured approximately0.7 mm horizontally and 3 mm vertically (Figure 8).As the effect of arbor thermal expansion subdues,the shaft center appears to shift gradually down-ward and to the left. However, the actual centerdrift is unclear, because the difficulties in separat-ing the effects of arbor expansion and the centerdrift. After 1,400 minutes, the drift appears tostabilize in the lower, right corner of the rectanglein Figure 8; the measured horizontal drift is only0.2 mm, and the vertical drift is 1.0 mm. It is,therefore, highly beneficial to arrange the sensi-tive direction of the grinding to be in the horizon-tal direction of the spindle to take advantage of itsextremely low drift in that direction.

Transient lift. Several experiments were con-ducted to investigate the transient lift of the spin-dle shaft under cold conditions. The room tem-perature of the laboratory was maintained to bewithin 20 6 1°C. For the cold lift, the spindle wasstarted at a temperature close to the room tem-perature, and the speed was raised quickly to ap-proximately 1,800 rpm. The center position of theshaft was recorded along with the acceleratingspeed. The sampling period was set to 10 seconds

Figure 8 Long-term thermaldrifts



and up to five samples were recorded before thespindle was shut off and allowed to cool down formore than 10 minutes before the next test wasconducted.

The cold lift test results are shown in Figure 9.The first sampled lift position of each test is la-beled with two numbers; the first number repre-sents the speed in rpm and the second numberthe temperature in °C. For example, 1150, 21,stand for 1,150 rpm and 21°C. The last sampledlift position is labeled only with the speed. Thetemperature is essentially the same because of theshort duration between the first and last samples.In the cold lift test, the vertical lifts are consis-tently over 11 mm, with an average of 13 mm. Mostshifts in x-direction remained in the positive x-di-rection as expected from classical hydrodynamiclubrication theory for a counterclockwise rotation,except for one set of data. However, the x-direc-tion shifts appear to be rather small comparedwith the vertical lift. This may be attributable tothe higher viscosity of the lubricant in the coldcondition. The kinematic viscosity of Velocite 3 at20°C is about 3.8 centistoke and about 2.4 centist-oke at 50°C according to a datasheet provided bythe manufacturer. The center position is likely tobecome lower and to the right as the spindlewarms up, as indicated by the steady-state drift inFigure 8. However, it is unclear how to separate thethermal expansion of the wheel hub from thecenter position shift in a warm condition using

only two probes, as pointed out in the Centerposition section.

Attempts were also made to determine thetransient lift of a spindle already at its thermalsteady state. This was done by stopping the spindlepromptly, recalibrating the stationary center, re-starting the spindle, and recording the transientlift shortly thereafter. However, as shown in Figure9, the results were not repeatable in the x-direc-tion and showed unlikely low values of vertical liftsand negative x shift. This may be caused by signif-icant errors in calibrating the stationary position.The stationary position calibration during theshort shut-off period was prone to larger errors,because the calibration had to be rushed to savetime, and the temperature of the spindle was in arapid cooling down period. In particular, by rush-ing through calibration, the shaft was not allowedto “sink-in” as in the cold condition calibration.This may result in the less repeatable stationaryposition in the x-direction, because x-directiondoes not have the favor of the spindle weight tohold the position. In the next section, a differentmethod was used to investigate the center posi-tions at different speeds to overcome the difficultyin separating the thermal expansion of the grind-ing wheel hub in the warm condition.

Center positions versus speeds. Instead ofmeasuring the transient lift in the cold conditionas done in the previous section, the spindle wasfirst run overnight to achieve steady states. Two

Figure 9 Cold and warmlifts



speeds were used, 1,800 rpm and 2,600 rpm. Atthe steady state of 1,800 rpm, the speed was thensuddenly changed to a different speed and back to1,800 rpm. The relative center shifts were imme-diately measured after the speed change and be-fore the thermal condition changed. After thespeed was brought back to 1,800 rpm, the spindlewas allowed to run for several minutes to recoverto the steady state before the next speed change.Six different speeds were tested ranging from1,050 to 2,200 rpm. At the steady state of 2,600rpm, the speed was quickly brought down in ap-proximately 200 rpm a step to 750 rpm and backto 2,600 rpm before significant temperaturechanges occurred. The relative shift at each stepwas measured. The relative shifts were then reca-librated with respect to that of 1,800 rpm for bothtests. To determine the absolute positions, an ab-solute center position for a specific speed isneeded.

An absolute center position at 2,600 rpm isassumed based on the theoretical result from Rai-mondi and Boyd.1 This results in an absolute cen-ter at 2,600 rpm very close to the sleeve center.With this assumption, the center position at eachspeed is then calculated according to its relativeshift measurement. As shown in Figure 10, the cen-ter positions independently determined in bothtests for 2,000 and 2,200 rpm are in good agree-ment, which provides a high confidence level ofthis method. The steady-state center position at

1,800 rpm is then determined to be 9.0 mm invertical lift and 4.5 mm in horizontal shift. In otherwords, there is a downward shift about 4 mm and apositive x-shift about 4 mm from its cold positionafter the spindle has reached its steady state. Thisresult seems to be reasonable because of the lowerviscosity of the lubricant at the warm steady state.

The center positions appear to be shiftingalong a 45° line as the speed changes. This isconfirmed by several other tests. Another observa-tion of Figure 10 is that the center position atspeeds lower than 1,700 rpm do not demonstrateclear center shifting patterns. This behavior maybe related to the larger radial error motions mea-sured at these speeds (Figure 7).

Minimum film thickness. The center positionsshown in Figure 10 are used to determine the min-imum film thickness and the corresponding ec-centricity ratio using Equation (5) by assumingthat c 5 c9. The results are compared with theclassical numerical results from Raimondi andBoyd1 using corresponding Sommerfeld numbers,S 5 (r/c)2 mn/P, where r 5 45 mm is the bearingradius, c 5 0.014 mm is the radial clearance, n isspindle speed in revolution per second, m is theabsolute viscosity in MPa, and P is determined bythe dead weight of the shaft plus the wheel (36Kg) divided by the projected bearing area, whichis equal to bearing length (L 5 105 mm) timesbearing diameter (D 5 90 mm).

Figure 10 Center positions at differentspeeds



In Figure 11, the numerical results from Rai-mondi and Boyd1 are reproduced for L/D 5 1/4,1/2, 1, and `. The investigated spindle has anL/D ratio of 2.33, which is considered as a longbearing with characteristics close to that ofL/D 5 `. However, as shown in Figure 11, thedatapoints spread broadly and do not appear tofollow any characteristic line with respect to anyconstant L/D value.

This may be attributable to many assumptionsmade in the theoretical results. First, the end leak-age is obviously not zero as assumed in the longbearing theory, although the L/D ratio is 2.33 inthis case. Substantial end flow must exist, becausethere is a built-in gear pump used to circulate oilthrough the bearing. Second, because of the longaxial bearing length, an axial temperature gradientmay exist. Both long and short bearing theoriesassume constant temperature distributions. Unfor-tunately, because of the constraint of noninvasiveinstrumentation allowed, thermocouples cannot be

mounted along the bearing sleeve to measure pos-sible temperature distributions. Finally, the posi-tions of the datapoints are not determined for theircorresponding steady-state conditions. As indicatedby the drift and cold lift tests, the actual lifts atdifferent speeds can be higher than those deter-mined in Figure 10 because of their lower steady-state temperatures and the resulting higher lubri-cant viscosity. This will move most of the datapointsslightly upward. To verify this, the four-probemethod discussed in the Center Position sectioncan be applied to determine the center positionwithout the adverse effect of the reference hub.However, this is not feasible in this study because ofthe lack of facility and difficulties in fixturing of thefourth capacitance probe. Nevertheless, even withthe above effect considered, the bearing still be-haves more like a short bearing than a long bearing.Therefore, the spindle stability margin and dynamiccharacteristics are accessed using the short bearingtheory from Lund65 and Hamrock.66

Figure 11 Comparison be-tween theoretical and exper-imental results

Table 2 Steady-state and dynamic parameters at different speeds

Speed, rpm e0 (Ma)cr (V# r)cr Kxx Kxy Kyx Kyy Bxx Bxy Byx Byy

1,800 0.5 6.46 0.515 2.92 3.98 20.857 2.21 6.62 2.25 2.25 3.052,000 0.38 6.52 0.524 2.14 4.04 21.69 2.34 7.06 2.36 2.36 4.402,200 0.26 6.95 0.516 1.66 5.29 23.67 2.45 9.86 2.46 2.46 8.052,400 0.15 7.36 0.506 1.40 5.26 24.32 2.51 10.1 2.51 2.51 9.062,600 0.04 7.62 0.501 1.28 32.5 232.2 2.54 64.8 2.54 2.54 64.5



Stability and dynamic characteristics. Table 2lists the several dimensionless parameters relatedto the spindle stability and dynamic characteris-tics, which include eccentricity e0 5 (c 2 d)/c,dimensionless critical mass parameter (Ma)cr, di-mensionless speed parameter, (V# r)cr, dimension-less stiffness coefficients (Kxx, Kxy, Kyx, Kyy), anddimensionless damping coefficients (Bxx, Bxy, Byx,Byy). Please refer to Hamrock66 for detailed defi-nitions of these dimensionless variables.

The dimensionless mass of the system must belarger than the critical mass to render the systemstable (without whirl). Therefore, the increasedvalues of the dimensionless critical mass indicate aslimmer stability margin as the spindle speed in-creases. On the other hand, the whirl frequencybecomes closer to one-half the spindle frequencyas the speed becomes higher.

As pointed out earlier, the spindle investi-gated in this paper operates in the horizontalmode, and its eccentricity is mainly produced bythe weight of the spindle shaft/wheel assembly.The working load is usually applied transverse tothe primary load. As defined in this paper, theprimary load is, therefore, in the y-direction, andthe main grinding load and the sensitive directionare in the x-direction. Therefore, it is useful tochoose a spindle speed that is of higher values inKxx, Kyx, Bxx, and Byx of spindle dynamic stiffness.According to Table 2, both 2,000 and 2,200 rpmare preferred. If the radial error motion measure-ments of Table 1 are also considered, 2,200 rpmseems to be a best choice.

Conclusions

Practical methods have been developed and im-plemented to study the performance of a high-precision hydrodynamic spindle. The results areused to select an optimal operating condition fora precision cylindrical grinding process. Thesemethods can be easily and cost effectively imple-mented on various machines in an actual produc-tion environment for effective predictive mainte-nance.

One advantage of the proposed error motionalgorithm is its ability to remove the form error ofa reference arbor, obtain the error motion, anddetect the whirl simultaneously by using only oneprobe. It does not require installing a ultrapreci-sion reference arbor and multiple sensors. Usuallythe spindle nose, such as the grinding wheel hub,can be used as the reference arbor. This algorithmcan easily be extended to isolate error motionsoccurring at different fractions of the spindle fre-quency.

Experimental results show that the long-termdrift of the spindle at steady state is less than 1 mmvertically and 0.2 mm horizontally and the radialerror motion is less than 1.6 mm for all the speeds.The shaft center position (vertical lift and horizon-tal shift) at the cold condition is substantially dif-ferent from the one in the steady-state warm con-dition. The center position appears to shift alonga 45° line as the speed is promptly changed. Al-though the L/D ratio is 2.33, the bearing behavesmore as a short bearing than as a long bearingaccording to Figure 11.

Future work will focus on the spindle perfor-mance evaluation and product quality monitoringin a practical production environment subject toadverse factors such as lubricant contaminationand coolant effects.

References1 Raimondi, A. A. and Boyd, J. “A solution for the finite

journal bearing and its application to analysis and design,”Parts I, II, and III, Trans ASLE, 1958, 1, 159–209

2 Choy, F. K., Braun, M. J. and Hu, Y. “Nonlinear study of amisaligned hydrodynamic journal bearing,” Tribol Trans,1993, 36, 421–431

3 El-Gamal, H. A. “Analysis of the steady-state performanceof a wedge-shaped hydrodynamic journal bearing,” Wear,1995, 184, 111–117

4 Guha, S. K. “Analysis of dynamic characteristics of hydro-dynamic journal bearings with isotropic roughness effects,”Wear, 1993, 167, 173–179

5 Leung, P. S., Craighed, I. A. and Wilkinson, T. S. “An anal-ysis of the steady-state and dynamic characteristics of aspherical hydrodynamic journal bearing,” J Tribol, 1989,111, 459–467

6 Malik, M. “A comparative study of some two-lobed journalbearing configurations,” ASLE Trans, 1983, 26, 118–124

7 Mehta, N. P. and Rattan, S. S. “Stability analysis of three-lobe bearings with pressure dams,” Wear, 1993, 167, 181–185

8 Shaw, M. C. and Macks, E. F. Analysis and Lubrication ofBearings. New York: McGraw-Hill, 1949

9 Szeri, A. Z. Tribology, Friction, Lubrication and Wear. NewYork: McGraw-Hill, 1980, pp. 55–101

10 Pinkus, O. Thermal Aspects of Fluid Film Tribology, NewYork: ASME Press, 1990

11 Booser, E. R., Ryan, F. D. and Linkinhoker, C. L. “Maximumtemperature for hydrodynamic bearings under steadyload,” Lubrication Eng, 1995, 51, 61–67

12 Dowson, D., Hudson, J. D., Hunter, B. and March, C. N. “Anexperimental investigation of the thermal equilibrium ofsteadily loaded journal bearings,” Proc Inst Mech Eng,1966–67a, 181, 70–80

13 Dowson, D. and March, C. N. “A thermohydrodynamic anal-ysis of journal bearings,” Proc Inst Mech Eng, 1966–67b,181, 117–126

14 Freund, N. O. and Tieu, A. K. “Thermo-elasto-dydrody-namic study of journal bearing with controlled deflection,”J Tribol, 1993, 115, 550–556

15 Knight, J. D. and Barrett, L. E. “An approximate solutiontechnique for multilobe journal bearings including thermaleffects with comparison to experiment,” ASLE Trans, 1983,26, 501–508

16 Lund, J. W. and Hansen, P. K. “An approximate analysis ofthe temperature conditions in a journal bearing. Part I: The-ory,” J Tribol, 1984, 106, 228–236

17 Lund, J. W. and Tonnesen, J. “An approximate analysis of



the temperature conditions in a journal bearing, Part II:Application,” J Tribol, 1984, 106, 237–245

18 Mitsui, J. “A study of the lubricant film characteristics ofjournal bearings, Part 2: Effects of various design parame-ters on thermal characteristics of journal bearing,” BullJSME, 1982, 25, 2010–2017

19 Tonnesen, J. and Hansen, P. K. “Some experiments on thesteady state characteristics of a cylindrical fluid–film bear-ing considering thermal effects,” J Lubrication Technol,1981, 103, 107–114

20 Keogh, P. S. and Morton, P. G. “Journal bearing differentialheating evaluation with influence on rotor dynamic behav-ior,” Proc Roy Soc Lond, A, 1993, 441, 527–548

21 Tucker, P. G. and Keogh, P. G. “On the dynamic thermalstate in a hydrodynamic bearing with whirling journal usingCFD techniques,” J Tribol, 1995, 1–8 (ASME TP 95-TRIB-34)

22 Paranjpe, R. S. and Han, T. “A transient thermohydrody-namic analysis including mass conserving cavitation fordynamically loaded journal bearings,” Proceedings of theSTLE/ASME Tribology Conf., Hawaii, Paper 94-Trib-36, 1994

23 Burrows, C. R. and Stanway, R. “Identification of journalbearing characteristics,” J Dyn Syst, Measurement Control,1977, 99, 167–173

24 Glieniche, J. “Experimental investigation of the stiffnessand damping coefficients of turbine bearings and their ap-plication to instability prediction,” Proc Inst Mech Eng,1966–1967, 181, 116–129

25 Glieniche, J., Han, D. C. and Leonhard, M. “Practical deter-mination and use of bearing dynamic coefficients,” Trib Int,1980, 13, 297–309

26 Hagg, A. C. and Sankey, G. O. “Dome dynamic properties ofoil film journal bearings with reference to the unbalancevibration of rotors,” J Appl Mech, 1956, 78, 302–306

27 Kostrzewsky, G. J. and Falck, R. D. “Accuracy evaluation ofexperimentally derived dynamic coefficients of fluid filmbearings, Part II: Case studies,” Tribol Trans, 1990, 33, 1,115–121

28 Morton, P. G. “The derivation of bearing characteristics bymeans of transient excitation applied directly to a rotatingshaft,” GECJ Sci Tech, 1975, 42, 37–47

29 Parkins, D. W. “Measured characteristics of a journal bear-ing oil film,” J Lubrication Technol, 1981, 103, 120–125

30 Parkins, D. W. “Theoretical and experimental determinationof the dynamic characteristics of a hydrodynamic journalbearing,” J Lubrication Technol, 1979, 101, 129–139

31 Read, L. J. and Flack, R. D. “Temperature, pressure, and filmthickness measurements for an offset half bearing,” Wear,1987, 117, 197–210

32 Stanway, R. “Journal bearing identification under operat-ing conditions,” J Dyn Syst, Measurement Control, 1984,106, 178–182

33 Flack, R. D., Kostrzewsky, G. J. and Taylor, D. V. “A hydro-dynamic journal bearing test rig with dynamic measure-ment capabilities,” Tribol Trans, 1993, 36, 497–512

34 ANSI/ASME B5.54, Methods for Performance Evaluation ofCNC Machining Centers. New York: American Society ofMechanical Engineers, 1992

35 ANSI/ASME B89.3.4, Axes of Rotation, Methods for Speci-fying and Testing. New York: American Society of Mechan-ical Engineers, 1985

36 Bryan, J. B. and Varherck, P. “Unification of terminologyconcerning the error motion of axes of rotation,” Ann CIRP,1975, 24, 555–562

37 Gao, W., Kiyono, S. and Nomura, T. “A new multiprobemethod of roundness measurements,” Prec Eng, 1996, 19,37–45

38 Ozawa, N., Takeuchi, K. and Sugano, T. “Development of anew measurement method of central position of spindlerotation (1st Report: Steel ball method),” Intl J JSME, 1993,36, 89–92

39 Zhang, G. X. “Four-point method of roundness and spindleerror measurements,” Ann CIRP, 1993, 42, 593–596

40 Bryan, J. B., Clouser, R. W. and Holland, E. “Spindle Accu-racy,” Amer Machinist, 1967, Dec., 149–164

41 BS 4656: Part 30, Accuracy of Machine Tools and Methodsof Test, British Standards Institute, 1992

42 Mitsui, K. and Sato, H. “A study on the effect of the struc-tural vibration of a machine tool to the circumferential sur-face roughness,” Proceedings of the 19th MTDR Confer-ence, 1978, 391

43 Noguchi, S., Tsukada, T. and Sakamoto, A. “Evaluationmethod of radial accuracy for ultrahigh precision air spin-dle,” J Japan Soc Prec Eng, 1992, 58, 1053–1058

44 Sawabe, M. and Fujinuma, N. “Influence of radial motionon form error of workpiece in turning,” Ann CIRP, 1978, 27,

50545 Deleted in proof46 Figatner, A. M., Levin, A. and Fiskin, E. A. “Assessing spin-

dle rotation errors of precision machine tools,” Stanki iInstrum, 1990, 61, 19–21

47 Goddard, E. J., Cowley, A. and Burdekin, M. “A measuringsystem for the evaluation of spindle rotation accuracy,”Proceedings of the 13th MTDR Conference, 1972, 125–131

48 Holster, P. L., Jacobs, J. A. H. M. and Sastra, I. B. “Measur-ing the radial error of precision air bearing,” Philips TechRev, 1984, 11, 334–337

49 Murthy, T. S. R., Mallanna, C. and Visveswaran, M. E. “Newmethods of evaluating axis of rotation error,” Ann CIRP,1978, 27, 365–369

50 Shinno, H., Mitui, K., Tatsue, Y., Tanaka, N. and Tabata, T.“A new method for evaluating error motion of ultra preci-sion spindle,” Ann CIRP, 1987, 36, 381–384

51 Su, Y.-S. “A new instrument for axis rotation metrology,”Ann CIRP, 1985, 34, 439–444

52 Yasui, T., Shirase, K. and Hirao, M. “Simple testing methodfor spindle rotating accuracy of machine tool,” Proceedingsof the xth MTDR Conference, 1995, 276–282

53 Kakino, Y., Yamamoto, Y. and Ishii, N. “New measuringmethod of rotating accuracy of spindle,” Ann CIRP, 1977,26, 241–244

54 Martin, D. L., Tabenkin, A. N. and Parsons, F. G. “Precisionspindle and bearing error analysis,” Intl J Machine ToolsManufact, 1992, 35, 187–193

55 Tlusty, J. “System and methods of testing machine tools,”Microtechnic, 1959, 13, 162–178

56 Vanherck, P. and Peters, J. “An axis of rotation analyzer,”Proceedings of the 14th MTDR Conference, 1973, 299–305

57 Whitehouse, D. J. “Some theoretical aspects of error sepa-ration technique in surface metrology,” J Phys, E: Sci In-strum, 1976, 9, 531–536

58 Donaldson, R. R. “A simple method for separating spindleerror from test ball roundness error,” Ann CIRP, 1972, 21,

125–12659 Cao, L. “The measuring accuracy of the multistep method

in the error separation technique,” J Physics, E: Sci Instrum,1988, 22, 903–906

60 Chetwynd, D. G. and Siddall, G. J. “Improving the accuracyof roundness measurement,” J Phys, E: Sci Instrum, 1976,9, 537–544

61 Danil’chenko, Yu. M., Figatner, A. M., Bal’mont, V. B. andBondar, S. E. “Reducing rotation errors of high-speed spin-dle units in antifriction bearings,” Stanki i Instrum, 1987, 7,

16–1862 Nakamura, T., Funabashi, K. and Kosaka, K. “Measurement

of form errors by phase difference method,” J JSME, 1987,53, 716–720

63 Mitsui, K. “Development of a new measuring method forspindle rotation accuracy by three points method,” Pro-ceedings of the 23rd International MTDR, 1982, 115–121

64 Ozono, S. “The roundness in process measurement bythree-point method,” J Japan Soc Prec Eng, 1976, 503–504

65 Lund, J. W. “Review of the concept of dynamic coefficientsfor fluid film journal bearings,” J Tribol, 1987, 109, 37–41

66 Hamrock, B. J. Fundamentals of Fluid Film Lubrication. NewYork: McGraw-Hill, 1994



experimental study of a precision, hydrodynamic wheel...

Documents