An experimental study on oil-film dynamic coefficients

Tribology International 37 (2004) 245–253www.elsevier.com/locate/triboint

An experimental study on oil-film dynamic coefficients

Hua Zhou∗, Sanxing Zhao, Hua Xu, Jun ZhuTheory of Lubrication and Bearing Institute, Xi’an Jiaotong University, Xi’an Shannxi, 710049, China

Received 3 April 2003; received in revised form 20 June 2003; accepted 19 August 2003

Abstract

The oil-film force of hydrodynamic bearing is often characterized by a set of linear stiffness and damping coefficients. Thispaper presents an experimental method to recognize these coefficients and establishes their characteristics under varieties of operatingconditions. The fundamental test model is obtained from a Taylor series expansion of bearing reaction force. A delicate test rig isconstructed and experimental data are acquired under various testing conditions. The coefficients are evaluated by means of leastmean square in time domain. The experiments indicate that the linear oil-film dynamic coefficients are sensitive to the excitationforce. The sensitivity is varied for the different coefficients. From the investigation, it can be seen that the linear model is invalidunder condition of high excitation force. That is to say, the nonlinear components, which are ignored in traditional linear model,should be considered in those conditions. 2003 Elsevier Ltd. All rights reserved.

Keywords: Oil-film force; Linear stiffness coefficients; Linear damping coefficients; Linear model

1. Introduction

Journal bearings have been widely used in high-speedrotating machinery. The dynamic coefficients of oil-filmforce affect the machine unbalance response andmachine stability. With some ambiguous understandingson the oil-film bearing theory, such as the boundary con-ditions, cavitation and whirl phenomena, it is difficult tocalculate the dynamic coefficients accurately. Thereforeboth of the experimental and theoretical investigationson the dynamic coefficients of journal bearing are indis-pensable.

Some theoretical studies have been reported. Qiu andTieu [1] studied the effect of perturbation amplitudes oneight dynamic coefficients of the journal bearing. In theirstudy, the linear coefficients were calculated by the finiteperturbation method under different perturbation ampli-tudes. The nonlinear properties of the bearing dynamiccoefficients were studied by Choy et al.[2,3]. Theypresented the various orders (3rd, 5th and 7th power) of�X and�Y to approximate the nonlinear oil-film forcesand calculated the nonlinear stiffness through a set of

∗ Corresponding author. Tel.:+86-29-266-9083.E-mail address: [email protected] (H. Zhou).

0301-679X/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.triboint.2003.08.002

algebraic equations and the equivalent stiffness coef-ficients were achieved.

Some experimental methods have been proposed toidentify the linear dynamic coefficients of bearing. Tieuand Qiu[4] proposed a method to determine the dynamiccoefficients from two or more sets of unbalanceresponses. They utilized the synchronous unbalanceresponses to simplify the calculation of coefficients. Thismethod was convenient to estimate the coefficients oflarge journal bearing on-line. Another experimentalmethod to estimate dynamic coefficients is based onimpulse responses[5–8]. An impulse excitation coversa wide range of frequency characteristics, whichincreases the reliability of estimated coefficients.Kostrzewsky and Flack[9,10] used sinusoidal excitationto recognize dynamic coefficients. The main advantagesof this method are high energy in the specified frequencyand high signal noise ratio. This method is widely usedin experimental studies, including this paper.

This paper attempts to identify the linear coefficientsunder varieties of operating conditions and study theinfluence of perturbation amplitude on the linear coef-ficients. In this investigation, a method based on leastmean squares in time domain is presented. All experi-ments were executed on a special designed test rig withbearing of 152 mm diameter. The eight linear oil-film

246 H. Zhou et al. / Tribology International 37 (2004) 245–253

Nomenclature

c radial clearance, mmd1 linear damping coefficient matricesdXX, dXY, dYX, dYY linear damping coefficients, N/(m·sec�1)DXX, DXY, DYX, DYY

nondimensional linear damping coefficients, Di,j =di,j

mwL�cR�3

, (i,j = X,Y)

f1 f2, f3 load, N

F =fy2

mUL nondimensional forcek1 linear stiffness coefficient matriceskXX, kXY, kYX, kYY linear stiffness coefficients, N/mKXX, KXY, KYX, KYY

nondimensional linear stiffness coefficients, Ki,j =ki,j

mwL�cR�3

, (i,j = X,Y)

L bearing length, mmm mass of bearing housing, KgR journal radius, mmU tangential surface velocity of the journal, m/sx1, y1 journal’s relative movement, mmX, Y system coordinates, bearing housing’s absolute movementm kinetic viscosity, Pa·sw angular velocity of rotation shaft, rad/sy 1.5‰ = relative clearance

Subscripts

X direction of XY direction of Y

dynamic coefficients are identified and studied from theexperimental data. The influence of excitation amplitudeon these dynamic coefficients are observed and dis-cussed in this paper. The experimental procedures anddata processing techniques are presented.

The linear model used in the paper is derived from afirst-order Taylor series expansion of oil-film force for-mula. The oil-film force increment is a function of dis-placements (x,y) and velocities (x,y) to the static equilib-rium position (x0,y0), which can be represented asfollows:

�fX

fY� � k1�x

y� � d1�x

y� � �KXX KXY

KYX KYY��x

y� (1)

� �DXX DXY

DYX DYY��x

y�

where fX and fY are the oil-film force increments in thehorizontal and vertical directions. k1 is a matrix withfour linear stiffness coefficients. d1 is a matrix with fourlinear damping coefficients.

2. Experimental techniques

2.1. Test rig introduction

Fig. 1 illustrates a schematic diagram of the test rig.The test shaft (5) is supported by a pair of five pad tilt-ing-pad journal bearings (6) on both ends. Betweenthem, a test bearing (13), diameter 152 mm, length width88.3 mm, radial clearance 0.114 mm, is mounted. Thetest bearing together with the bearing housing (11) andthe metal gasket (16) sits on the bottom splint (17). Thebottom splint is dragged by four steel rods (9) which arefastened to the top splint (12). The top splint is hungover the spring-opposed bellows (14) which sits on astiff supporting framework (8). Using this suspensionstructure, which like a nacelle, the test bearing floatsover the test shaft. In order to measure the relative move-ment between the shaft and the bearing, displacementsensors are installed on the test bearing housing.

When high pressure air makes the spring-opposed bel-lows expand, the top splint is pushed up. Consequently,

247H. Zhou et al. / Tribology International 37 (2004) 245–253

Fig. 1. Outline of journal bearing test rig.

the test bearing is dragged up. By this means, the equiv-alent static load, which can be controlled by the airpressure and measured by a pressure sensor (10), can beapplied to the bearing. The static load can be up to 40kN. In order to ensure the test bearing’s parallel move-ment in the plane which is perpendicular to the axes ofthe shaft, eight smooth ground rollers (15) are employed.They are located at both ends of the test bearing. Theserollers prevent the test bearing from rotating, titling, andmoving in the axial direction. Four smooth ground steelballs are put between bottom splint (17) and metal gasket(16) to minimize friction.

The shaft is driven by an 80 kW direct current motor(1) through an infinitely variable transmission controllerand a gear box (3). The rotational frequency could bealtered by adjusting the controller and could reach 60Hz (3600 rpm). The transmission ratio of the gear boxis 2.8. Two separate oil supply systems were adopted tofeed the lubricant into the gear box (3) and test bearing(13). The lubricant is turbine oil 22#, kinetic viscosity28.25 × 10�3 Pa·s at 20 °C and 48.65 × 10�3 Pa·s at30 °C.

It can be seen from Figs. 2 and 3, the dynamic loadsystem comprises two electric exciters (Fig. 2-1) andconnecting components. The two exciters are positionedperpendicular to each other at 45° to the horizontal,pointing to the geometric center of the test bearing. Eachof the exciters is manipulated by an individual controllerand driver. Some sinusoidal excitation tests can be com-pleted by any of them. When the two controllers arecabled and preset correctly, they can generate sinusoidalforces in two directions simultaneously. The sinusoidalforce, provided by the exciters, can be up to 1.5 kN. Theexciters are connected to the bearing housing by thin-walled tubular connecters. The pressure sensors (Fig. 2-2), mounted in the middle of connecters, are used tomeasure the exciting force.

Fig. 2. Outline of loading system and sensor’s locations.

2.2. Oil-film force analysis

The analysis to oil-film force in the experimental rigis figured out in Fig. 4. f3 acts as a static load on thebearing, which is produced by pneumatic loading sys-tem. f1 and f2 are dynamic loads, which are produced bytwo exciters. In Fig. 4a, the X and Y are absolute systemcoordinates. In Fig. 4b the x1 and y1 are relative coordi-nates of the rotor. The oil-film forces (fX,fY) in X and Ydirections can be calculated using eq. (2).

�fX � f1cos45°�f2cos45° � mX

fY � f1sin45° � f2sin45° � f3�mg � mY(2)


Fig. 3. Overview of the test rig.

Fig. 4. Analysis of oil-film force of bearing. (a) forces acted on thebearing; (b) forces acted on the oil-film of journal.

where f1, f2 and f3 can be measured from pressure sensorsinstalled in the test rig. Each of f1, f2, f3, fX and fY canbe divided into two components: the static fi0, and thedynamic fi (i = 1, 2, 3, X, or Y). For f3, the static loadf30 is dominant, and the dynamic load f3, which resultsfrom the pneumatic loading system, is trivial. In the pro-cess of computing oil-film force increment (the dynamicoil-film force fY), the tiny dynamic component f3 is takeninto account. In the next section, these sensors and dataacquisition techniques are discussed in detail.

2.3. Data acquisition

Displacement and force sensors are employed toacquire testing data. The positions of these sensors areindicated in Fig. 2. Six electric eddy current sensors(Tsinghua 8500) are used to measure displacements.Four of them (12, 15) which measure the relative dis-placement between the bearing and the journal in thevertical and horizontal directions, are installed on bothends of the test bearing housing. The other two (3, 13)are installed on the framework to measure its absolutedisplacement in the horizontal and vertical directions. A

pressure sensor (Fig. 2-17, GKCT15-1A) is positionedunder the top splint to measure the pneumatic load. Theother two pressure sensors (Fig. 2-2, GKCT15-2C) areconnected to the exciters for dynamic load measurement.Additionally, a temperature sensor (10) based on ther-mistor Pt is placed 0.5 mm under the surface of the bear-ing to determine the temperature data.

In order to reach high accuracy in experiments, eachsensor is equipped with a separate amplifier and a highaccurate power supply. All of the signal cables are wellshielded and well grounded to avoid the electromagneticdisturbance from ambient apparatuses. All of the outputsfrom the amplifiers are regulated in the range of �10and 10 V. These 10 amplified signals are connected toa data acquisition card (ADlink NuDAQ PCI-bus Card9114). These types of data acquisition card are based onthe 32-bit PCI Bus architecture with 16-bit precision andsampling rate up to 100 KHz. The 10 tested signals aresampled and converted into digital data in turns, becausethe card can only handle one input channel at a time. Inall the experiments, 50,000 data points per channel werecaptured at a rate of 5000 samples per second.

2.4. Experimental steps

The experiments are executed as follows.

Run the DC motor and make the test shaft rotate at agiven speed. (300–1500 rps).Start the air pressure system and apply the static load(f3) to the test bearing at a given value (0–20 kN).Turn on the electric exciter and apply dynamic load (f1and f2) to the test bearing (0–1.5 kN, sinusoidal wave0–80 Hz).Run the data sampling software and acquire the datafrom the test rig. Then record the data in a computer as afile. The PC-based control software developed in VisualBasic 6.0 completes this task.

From the above steps, a set of data can be achievedunder a given operating condition. The data are savedin a file for further analysis. Repeating the above stepswith different rotating speeds, static and dynamic loads,the data under various conditions can be determined.

3. Results and discussion

Many sets of data are achieved through the experi-ments. They are converted into standard ASCII files bythe converting software developed in Visual C++ 6.0.The ASCCII file can be processed by Matlab, Fortranand other popular software.


3.1. Coefficients identification

In the time domain, the velocity of the bearing can bederived from the displacement by a differentiator (FIRdifferential filter). It is valid to reuse the differentiatorto obtain acceleration from velocity. With the measuredmass (m) of the bearing housing, the oil-film force canbe worked out by eq. (2). Hence, the oil-film force, dis-placement and velocity in eq. (1) become known para-meters. What we should do next is to identify theunknown linear oil-film dynamic coefficients by meansof the least mean square. Each set of data is processedas follows:

1. Feed the raw data to a low pass filter to eliminatehigh frequency noise.

2. Check the data of the eddy current sensor to find outwhether they are in the linear measurement range ornot. Although the sensors are calibrated and installedcarefully, their outputs may go out of the linearmeasurement range, in some cases. All of the data inthis file are considered as invalid if an eddy currentsensor’s output is out of its own linear range. Becausethe displacement data are fundamental to the next cal-culations, the linear measurement range of each eddycurrent sensor is defined by accurate calibration.

3. Convert every channel’s data (voltage) into a corre-sponding physical value (displacement, force andtemperature).

4. Owing to the character of the data acquisition card,input channels are not sampled simultaneously, butacquired one by one in turn. Consequently, phase dif-ferences among channels exist. An inner-insert filteris designed to eliminate the phase differences. Thisfilter completes the insertion of values, digital smoo-thing and data re-sampling. Using this inner-insert fil-ter, new groups of data are reconstructed.

5. Calculate the relative displacements and absolute dis-placements.

6. Calculate the velocity from displacement using a dif-ferential FIR filter. After that, derive the accelerationfrom velocity reusing the differential FIR filter.

7. Compute the oil-film forces of the bearing using eq.(2). Divide them into static parts (fX0,fY0) and dynamicparts (fX,fY).

8. Recognize the parameters in eq. (1) by means of leastmean square.

9. Non-dimensionalize the coefficients.

The oil-film force increment can be written as eq. (3)

�fi(k) � ki,1x1(k) � ki,2y1(k) � ki,3x1(k) (3)

� ki,4y1(k) (i � X or Y; k � 1�n)

where �fi(k) is the recognized oil-film force increment,n is the number of data points per channel, ki,1 and ki,2

are linear stiffness coefficients, ki,3 and ki,4 are lineardamping coefficients. It is assumed that fi(k) is the oil-film force increment (dynamic oil-film force) derived bymeasurement data using eq. (2). The least mean squarebetween �fi(k) and fi(k) can be described as eq. (4):

min(k,d)e2i � �m

k � 1

[�fi(k)�fi(k)]2 (4)

Then, the following equation is valid.

∂e2i∂ki,p

� 0 (p � 1 � 4 i � X,Y) (5)

Combining eq. (3) and eq. (5) yields:

��n

k � 1

x1(k)2 �n

k � 1

x1(k)y(k) �n

k � 1

x1(k)x1(k) �n

k � 1

x1(k)y1(k)

�n

k � 1

y1(k)x1(k) �n

k � 1

y1(k)2 �n

k � 1

y1(k)x1(k) �n

k � 1

y1(k)y1(k)

�n

k � 1

x1(k)x1(k) �n

k � 1

x1(k)y1(k) �n

k � 1

x1(k)2 �n

k � 1

x1(k)y1(k)

�n

k � 1

y1(k)x1(k) �n

k � 1

y1(k)y1(k) �n

k � 1

y1(k)x1(k) �n

k � 1

y1(k)2

��ki,1

ki,2

ki,3

ki,4

� (6)

� ��n

k � 1

x1(k)fi(k)

�n

k � 1

y1(k)fi(k)

�n

k � 1

x1(k)fi(k)

�n

k � 1

y1(k)fi(k)

�kX,1, kX,2, kY,1 and kY,2 correspond to linear stiffness coef-ficients KXX, KXY, KYX and KYY in eq. (1). kX,3, kX,4, kY,3

and kY,4 correspond to linear damping coefficients DXX,DXY, DYX and DYY in eq. (1). Therefore, the eight linearoil-film dynamic coefficients can be solved from eq. (6).

3.2. The effect of the load parameter on linearcoefficients

Under a given working condition, the load of bearingcan be calculated. When the load is nondimensionalizedby formula (7), a load parameter FY can be obtained.

fyFY

�mULy2 (7)

The load parameter is a comprehensive parameter inwhich bearing load, working temperature, viscosity oflubricant and the bearing parameters are involved.Therefore, the load parameter is suitable to characterizethe operating condition of the bearing. Figs. 5 and 6present the characteristics of linear stiffness and damp-ing coefficients under various load parameters.


Fig. 5. The linear stiffness coefficients vs. the load parameter.

Fig. 6. The linear damping coefficients vs. the load parameter.

There are 43 sets of coefficients in the figures, whichare identified by use of least mean square. The dashedfitting curve in each figure is computed by a quadraticpolynomial.

It has been proven that the nondimensional stiffnessand damping coefficients of the journal bearing changewith the load parameter. It can be observed from Figs.5 and 6 that the linear stiffness and damping coefficientsincrease with the growth of load parameter. Thisphenomenon coincides with the classic linear theory.

When FY is less than 0.5, most of the results are close

to the fitting curve. When FY is greater than 1, the resultsare scattered away from the fitting curve. This trendshows that with the growth of FY, the coefficientsbecome less stable. It indicates that nonlinear compo-nents of the oil-film force become larger, i.e. the effectof the nonlinear force is stronger in large FY than insmall FY. It is proven that the linear coefficients are notenough to characterize the oil film force in case of highload parameter.

Comparing the figures, it can be found that less scatteris presented in the stiffness coefficients than in the


Fig. 7. The linear stiffness coefficients vs. the excitation amplitude.

damping coefficients. The main reason for this phenom-enon is that the velocity is achieved by a FIR differentialfilter. The filter introduces some noise in the calculation.Accordingly, the velocity error becomes larger than thatof displacement. Additionally, the components of velo-city have more effect on damping coefficients than onstiffness coefficients.

3.3. The effect of excitation force on linearcoefficients

Figs. 7 and 8 present the stiffness and damping coef-ficients at different excitation amplitude, under the con-

Fig. 8. The linear damping coefficients vs. the excitation amplitude.

dition of FY = 0.35. The x-axis indicates the maximumnon-dimensional movement amplitude, which is definedas d = LK /c. LK = max(√x2

1 + y21), x1 and y1 are relative

displacements of the bearing in two coordinates. c is theradial clearance of bearing. By this means, the value ofexcitation amplitude can be represented by d.

In order to find out the relationship between eachcoefficient and d, a cubic polynomial is constructed tofit the coefficients, written as:

KD(d) � a1d3 � b1d2 � c1d � d1 (8)

where a1, b1, c1 and d1 are parameters of the fitting curve.When the perturbation acting on the rotor is sufficiently


small, the linear coefficients remain unchanged. There-fore, when d→0, each coefficient should be a constant.Accordingly, the following equation is valid.

∂KD∂d |d=0 � 0 (9)

Combining eqs. (8) and (9) yields c1 = 0, then eq. (8)can be rewritten as:

KD(d) � a1d3 � b1d2 � d1 (10)

While solving the parameters a1, b1 and d1, the leastmean square is also used. The dashed fitting curve ineach figure is established using this method. With thehelp of the curve, the influence of excitation amplitudeon these coefficients is easier to observe. But due to thedifferent scale range in every figure, it is still not easy todiscriminate the influence intensity of the d on differentcoefficients. A relative variation of the coefficient isintroduced to solve this problem. It is defined as:

RV �KD(d)�KD(0)

KD(0)(11)

Eight RVs correspond to eight linear coefficients ofbearing. The results are presented in Figs. 9 and 10.

It can be seen in Fig. 10a (FY = 1.3), at d = 0.05, therelative variation of stiffness coefficients exceed 20%(except for KXY); while in Fig. 9a (FY = 0.35), at d =0.1, the relative variation are not over 20%. A similartrend occurs on damping coefficients. When FY = 1.3(Fig. 10b), at d = 0.02 the variation of damping coef-ficients do not exceed 30%, while at FY = 0.35 (Fig.10b), d = 0.05, the variation are not over 25%. Both ofthe observations indicate that coefficients are more sensi-

Fig. 9. The relative variation of linear coefficient of bearing vs. theexcitation amplitude (FY = 0.35).

Fig. 10. The relative variation of linear coefficient of bearing vs. theexcitation amplitude (FY = 1.3).

tive to the excitation amplitude in the case of large loadparameters than in the case of small load parameters.

Analyzing these figures, more observations can befound. On the same condition, each coefficient’s RV hasdifferent sensitivity to excitation amplitude. KXY and DYX

are most sensitive among the eight coefficients. In brief,the linear oil-film coefficients are sensitive to the exci-tation amplitude. The sensitivity is varied for differentcoefficients. In case of large load parameters, the influ-ence of the excitation amplitude becomes larger.

The above observations indicate the nonlinear charac-ters of oil-film force. In the case of small excitationamplitude, the oil-film force can be simplified as a linearmodel. The experimental results prove that the linearmodel is valid under this condition. However, when theexcitation amplitude becomes large, the nonlinearcharacters of the oil-film force should not be ignored. Ifthe linear model is still used to describe the oil-filmforce, the recognized coefficients would change greatly.Therefore, the linear model is invalid and a new nonlin-ear model should be reconstructed in this case.

4. Conclusions

The following conclusions can be drawn from theabove experimental study.

1. The identified coefficients are repeatable and stableunder conditions of small load parameter.

2. Least mean square is a fast and effective algorithmon the parameter identification.

3. Because the velocity is acquired by a FIR differentialfilter, there is some noise involved.

4. The value of linear coefficients increase with the


growth of the load parameter. This is in accord withthe theoretical calculation.

5. In the case of small load parameter, the values of thecoefficients are stable. Under conditions of large loadparameter, the coefficients scatter to some extent.

6. Under the same working condition, the influence ofexcitation amplitude is varied for different coef-ficients.

7. Under conditions of the large excitation amplitude,the conventional linear model is invalid and shouldbe modified.

Acknowledgements

This project is supported by the National NaturalScience Foundation of China (Grant No. 90210007 andGrant No. 19990510).

References

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[3] Choy FK, Braun MJ, Hu Y. Nonlinear effects in a plain journalbearing: Part 2. Results. Journal of Tribology 1991;113:563–70.

[4] Tieu AK, Qiu ZL. Identification of sixteen dynamic coefficientsof two journal bearings from experimental unbalance responses.Wear 1994;177:63–9.

[5] Qiu ZL, Tieu AK. Identification of sixteen dynamic coefficientsof two journal bearings from impulse responses. Wear1997;212:206–12.

[6] Zhang YY, Xie YB, Qiu DM. Identification of linearized oil-filmcoefficients in a flexible rotor-bearing system, Part I: Model andsimulation. Journal of Sound and Vibration 1992;152:531–47.

[7] Zhang YY, Xie YB, Qiu DM. Identification of linearized oil-filmcoefficients in a flexible rotor-bearing system, Part II: Experi-ment. Journal of Sound and Vibration 1992;152:549–59.

[8] Jiang GD, Hu H, Xu W, Jin ZW, Xie YB. Identification of oilfilm coefficients of large journal bearings on a full scale journalbearing test rig. Tribology International 1997;30:789–93.

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An experimental study on oil-film dynamic coefficients