Description of the rolling friction process by acoustic modelling

33

Wear, 167 (1993) 3340

Description D. G. Evseev,

of the rolling friction process by acoustic modelling B. M. Medvedev,

G. G. Grigoryan

and 0. A. Ermolin

Moscow Railroad Transport Institute, Obrastsova str. 15, Moscow (Russian Federation)

(Received

August 19, 1992; in revised form January 14, 1993)

Abstract A model is developed to describe the spectral density of the acoustic signal accompanying the rolling friction process. A simple case of elastic contact interaction between cylinder and plane is considered. The model is based on the Greenwood-Williamson elastic microcontact model, and on a description of the acoustic signal as Poisson’s flow of impulses. Theoretical relations between the spectral density and various factors such as rolling speed, load, surface roughness parameters and lubricant film thickness are obtained and analysed.

1. Introduction It is known that rolling bearings in operation generate broad-band (up to 1 MHz) acoustic signals [l]. These signals are generated basically in the contact zones between the rolling elements and raceways. The acoustic signals in the range up to 20 kHz are created by macro-inhomogeneities of contacting surfaces of the rolling elements (e.g. geometric imperfections, waviness, pitting and various surface defects) and load oscillations. The dominant sources of acoustic signals at high frequencies (above 20 kHz) are the impacts of the asperity summits of contacting surfaces. The acoustic impulses generated by these sources have short duration and steep fronts. Thus corresponding frequency spectra range up to 1 MHz. Although the energy contribution in the total acoustic signal from these sources is less than that from low-frequency vibrations, the information content is often more important. Owing to the high area1 density of asperities of the contacting surfaces, a high activity of the acoustic signals (i.e. a large number of impulses per unit time) takes place. Individual impulses are superimposed, and a continuous signal is formed. It has a stochastic nature and is subjected to low-frequency modulations, which are produced by, for example, macrodefects, the nonstationary nature of the load and periodic variations of the transfer function of the medium from a moving source to a stationary receiver. By recording this signal in real time during the operation of the bearing, it is possible to reveal dangerous surface defects of the rolling elements of the bearing and damage to its

0043-1648/93/$6.00

condition at an early stage of evolution, e.g. wear, lubricant film losses and other failures that adversely affect the lifetime of a bearing [2]. Existing methods of acoustic bearing diagnostics can be used for these problems [3-71, but the data are listed for relatively narrow ranges of frequencies. No results of theoretical and experimental studies on acoustic signal parameters are available in a wide frequency range. In the authors’ opinion, such studies are necessary to obtain a more detailed acoustic picture of bearing condition. In order to work out the efficiency criteria for the rolling bearings on the basis of acoustic information, it is necessary to have information about the laws by which acoustic signal parameters are varied. This is also important from a diagnostic point of view. The authors’ task is to describe the theoretical approach to the problem of recording and analysis of acoustic information. On the basis of this theory, methods for diagnostics of bearings with the help of multiband filtration of acoustic signals in a wide frequency range can be elucidated. The work reported here was aimed at the characterization of the rolling friction process using an acoustic signal model. A step towards the solution of the abovementioned problem is the concept of the power spectral density of an acoustic signal. The goal is to relate this acoustic signal parameter to basic mechanical characteristics of the rolling friction process. This work is a part of the authors’ research in the field of contact phenomena during interactions between rough surfaces of machine parts in the process of friction and wear

PI0 1993

- Elsevier

Sequoia. All rights reserved

34

D. G. Evseev ei al. I Acoustic

2. Theoretical background and formulation of the model Consider the following model. A local source at some point of an isotropic medium? generates acoustic signals in response to force x(r, t) at the instant t (where X(f, t) #O only when r=?,,,, t >O). A local receiver at some point of the medium i, registers the surface displacement y(rl, t) caused by an elastic wave propagating from the source. Propagation of the signal from source to receiver is described by the well-known equation [9]

_2(1+v)(l-2V)

grad div F(
where F((r, t) is the field of displacements, p is the material density of the medium and E is the elastic modulus of the medium. Putting J?(?, t) =x(t) and p(‘(r,t) = y(‘ct)gives the solution for y(t) in the form of Duhamel’s integral [lo] Y(t) =

j;r

(7)X(r - 7) d7

(2a)

0

or Y(t) =h(t)*X(t)

(2b)

where h(t) is the unit impulse response function of the medium in which the acoustic signal is propagated and * denotes a conjunction. In many practical cases it is necessary to carry out a spectral analysis of the process examined. So, performing the direct Fourier transformation of equation (2), we obtain S,(w) = H(w) *S(o)

(3)

where SJw), S,,(w) are the Fourier transformations of emitted and received signals respectively, H(w) is the transfer function and w is the cyclic frequency. In terms of the power spectral density used in analysing the random processes G,(w) = G,(w). PW12

(4)

where G,(o) = I&(#,

G,.(o) = t$(~)12

(5)

These are called the functions of source and receiver respectively in the following. Difficult conditions under which the acoustic signals from numerous moving contact zones are propagated, complex geometry of the rolling bearing and the presence of junctions will result in a significant distortion of

modelling

of rolling friction

initial acoustic information. However, if these factors are permanent, the natural behaviour of G,(o) will be retained. The calculation of the signal propagation is outside the scope of this paper: attention is focused on the description of power spectral density G(w) = G,(w). On the basis of an elastic model for the contact of two surfaces, the formula for G(w) can be derived as a function of the parameters that influence its behaviour in the contact processes. In order to simulate the process of contact interaction between the rolling element (e.g. roller) and the raceway in the bearing, the following model is used. Consider a cylinder of radius R and length b under a normal load F rolling without sliding with a speed V on a plane surface of a half-infinite space. Surfaces of the cylinder and plane are assumed to have isotropic roughness formed by abrasive machining; cylinder and plane surface are assumed to be made of the same elastic material. Thus a contact zone arises between the contacting surfaces. It is known that the upper limit of the frequency range of the acoustic signals from the rolling bearings is 500 kHz. Up to this frequency the acoustic wavelength is much greater than the size of the contact zone in the direction of rolling, therefore the contact zone is regarded as a point source of acoustic signals. Furthermore, since the speed of the rolling elements moving on the raceway is much lower than the sound speed, the influence of the speed of the source moving the stationary receiver relatively on the frequency spectrum (the Doppler effect) can be neglected. It is assumed that the asperity length in the direction of rolling is such that the boundary effect can be neglected, and for these contacting bodies the Hertz solution exists. Thus within the contact zone there is an elliptical load distribution. To carry out our calculation, this distribution is approximated by dividing it and the contact area into sectors. It is supposed that within each of them a contact of two plane surfaces under a constant load takes place. Furthermore, since the surface roughness of rolling elements in bearings is quite low and the load is not very large, the elastic interactions between the asperities can be said to predominate. In order to model the elastic contact of two surfaces, the Greenwood and Williamson microcontact model (the GW model) [ll] is used. The model is based on the assumptions that (1) the rough surfaces are isotropic, (2) asperities have spherically capped summits of constant radius R irrespective of their height, (3) asperities are far apart and there is no interaction between them - the load they support depends only on their height and not on the load supported by neighbouring asperities, (4) there is no bulk deformation - only the

D. G. Evseev et al. / Acoustic modelling of rohg

asperities are deformed during contact, (5) asperities are deformed elastically in accordance with the hertzian relations between deflection, load and contact area, and (6) summit height expressed as a deviation from the mean plane of the summits is a random variable, and follows a gaussian probability distribution with standard deviation u. The model requires three input parameters: 77, the surface density of summits; a, the standard deviation of the probability distribution of summit heights; and R, the deterministic (non-random) radius of the spherical summit caps. These values are determined on the basis of spectral moments for given surface traces according to McCool

WI* In the present model the case of the contact of two rough surfaces within any sector of the contact area is reduced to the case of the contact between rigid smooth flat and rough plane surfaces. The roughness parameters of the rough surface are combinations of roughness parameters of the original rough surfaces (see ref. 12). According to the GW model, the coordinate z of asperity height is measured from the mean plane of the asperity heights, and d is the distance between the flat and the mean plane. (z-d) is the interference of an asperity, and only asperities with positive interference have a contact (see Fig. 1). Elastic contact load of an asperity summit is P(z -d) = ; E ‘R”*(z - d)3”

(6)

where E’ is the effective elastic modulus of the joint

friction

35

vl, V, and E,, E, are Poisson’s ratios and Young’s moduli respectively of the two surfaces 1 and 2 forming the joint. Contact pressure corresponding to elastically deformed asperity summits of height z > d can be expressed according to the GW theory by the integral over all asperity summits in contact as m p(d) = ;

.qE ‘RID (z - d)3Rcp(z) dz s d

where

dz)=

-- 2 (2T;‘“uexP ( 1 2d

From assumption (3) of the GW model it follows that any event of the elastic interaction between the asperity summit and the rigid flat is the statistically independent event. Thus this event makes its own contribution to the total Poisson’s flow of impulses as the impulse of P(t). Power spectral density of this flow can be expressed according to ref. 13 as G(o) =&9-(P(t)))’

(IO)

where Z? is the count rate of the impulse flow, 98@(t)) is the Fourier transformation of the load of an asperity and the angled brackets denote the average of a set of observations. An individual time dependence of this function takes place for each asperity. It is convenient to use the dimensionless height of asperity in equation (6), u=z/d. Therefore equation (6) b ecomes m

(7)

P(d) = ; 7E ‘R”%-=

s

(u -d/a)3Rcp(u) du

(11)

dlo

where

(12)

SYOOTH PLANE

;“=d

m B

zi

-I-

YEAN OF ASPERITY HEIGHTS

-36 c/ (a) Fig. 1. asperity number contact

(b)

A PUlElX

Consider the range of UE [ -3, 31 and divide it into M equal intervals. Assume that within each of these intervals all asperities have a constant height Ui=-3+AU(i-l), i=l, .... M+l, Au=6/M. Therefore the number of asperity summits of height entering into contact with the flat surface per unit time is

P(Z)

rir,= @rp(u,)

I

’

(c)

(a) Model of contacting joint. (b) Distribution law of heights (see eqn. (9)): shaded area corresponds to the of asperities in contact. (c) Distribution law of elastic loads of asperity summits (see eqn. (6)).

Au

(13)

A cylinder of length b and radius R, rolling on a plane in the x direction has a rectangular contact zone of area 2ub. In such a case the size 2u of the contact zone depends on the load F normal to the plane. It is known from elasticity theory that

36

D. G. Evseev et al. I Acoustic modeliing of rollingfriction

(14) and the contact direction is

~(4 =pmax I1-

zone pressure

(hertzian)

in the x

pi, = i E ‘R1f2~3’2(~j - u~)~'~, j> i

or Pij= i E’Rlna3”

(15)

(x2/aZ)11R

where 2F

P max= -

(19)

&Q-i)‘n,

jai

(20)

An

example of the graphic illustration of these contact load impulses is shown in Fig. 3. Taking the function evenness into account, the Fourier transformation of the contact load is

nnb

and p is a constant at a given X. For this reason we consider the pressure distribution at some cross-section parallel to the rolling direction, i.e. the x-axis (see Fig. 2). Because there are M groups of asperities of height Ui,every point of thex-axis corresponds to some number of the asperities entering into the contact. The x-axis is divided in such a way that every new Xi corresponds to the ith group of the asperity summits leaving the contact. Since every new group of the asperity summits enters into the contact when pi is reached, where

57(P(t)) = 2jP(r)

cos wt dt

(21)

0

Greater computational accuracy with a smaller number of divisions of x can be obtained as follows. Consider the contact load for some jth group of the asperity heights as a piecewise linear function of X. The contribution from any segment [Xi-I,Xi]to the total spectrum is I‘ s li-I

(4 -Bt) cos wt dt

(22)

where m

pi =p(u,)

=

;

qE ‘R~‘=u~~ 0, - Ui)“dY) s ui

and y is the current coordinate height, the distribution of x is:

dY

(17)

of the dimensionless

xi = a ( 1 -p:/p,ax2)‘n

A= B=

Pi_~xi-Pixi-, xi-xi-1

(23)

pi-l-pi

(24)

xj-xi-1

(18)

In this case only K,,,,, groups of the asperity summits will enter into contact with a flat whenp,,, is reached. Thus, a double division exists for groups of the asperity summits ui and xi. Now each value xi corresponds to its own pressurep, and to the groups of asperity summits uj>ui. In accordance with the Hertz solution for the elastic contact of a ball of radius R with a plane, for each asperity summit we have a contact load

Fig. 2. Pressure rolling direction

distribution at some cross-section (see eqn. (15)).

parallel

to Fig. 3. Shapes of contact

load impulses (eqn. (19))

for K,,,,=6.

D. G. Evseev et al. / Acoustic modelhg

Then the Fourier transformation of the contact load for the jth group of the asperity heights takes the form

where Pi is the magnitude of the contact load in the ith group of asperities at point x. The following relation between x and t holds: t=x/K Thus, taking equations (10) and (21) into account, we have the following formula for the power spectral density (10)

where ‘=

64 ?‘b Au4dE”R 9(2m-)lR ‘I

3. Calculated

(27)

results and discussion

The formula for the source function G(o) obtained above gives an opportunity to estimate the effects of the various factors on it. This is of considerable interest from the friction process analysis point of view. It is assumed that cylinder and plane surface are made of the same bearing steel lOOCr6. This steel has Young’s modulus E = 2.1 x 101’ Pa and Poisson’s ratio Y= 0.3. It is also assumed that the surfaces have been subjected to the same method of machining and finishing, and thus their roughnesses are the same. This fact is used in the calculation of the roughness parameters for the equivalent plane rough surface. So for this contact pair the effective Young’s modulus is E’ = 1.5 x 1O’l Pa. The values used in calculations were R, = 10m2 m, b = 10v2 m. These values and values of the influence factors given below are quite arbitrary, but they are within the characteristic range for rolling bearings. Influence factors are surface roughness, rolling speed of cylinder, load and lubricant film thickness. The effect of each of these factors on power spectral density is estimated when the others remain invariant. The frequency range from 1 to 500 kHz was divided into 500 equal intervals, each of width 1 kHz. The range of ui variables was divided into 60 equal intervals. The optimal relation between the accuracy and the calculation rate is thus provided. Variables xi were defined as a result of the calculation of load P (eqn.

of rolling friction

37

(14)) according to eqn. (15). Therefore the interval x,-x,_, is decreased in the range of the loads as the load impulse front steepness is increased. In this case the counting accuracy is higher than for uniform division.

3.1. Roughness effects For the study of the effects of surface roughness on the source function, two kinds of abrasive machining of the cylinder and the plane - fine grinding and superfinishing - were considered. The roughness parameters for equivalent plane surface were calculated, and the following values were used. For fine grinding: a=0.74X low6 m, R= 10m5 m, n = 0.36 x 10” mm2. F or super-finishing: (T= 0.03 X 10e6 m, R = 5 x lo-’ m, 7)= 6 X lOlo me2. Cylinder speed and load are invariant: V= 10 m s-‘, F= 10 N, and it is assumed that the rolling friction process with no lubrication takes place. The calculated results are shown in Fig. 4. G(w) functions are shown in double logarithmic scale. The shape of the curve of the source function for a super-finished surface is the same as that for a fineground surface because‘the rolling speed and contact area are the same in both cases. The distinction between the total levels of these curves corresponds well to the experimental data with respect to the fact that increasing refinement of the surface finishing reduces the noise level [ 141. If the lubricant film were available the distinctions in level mentioned above would be more marked, because the presence of lubricant would reduce the effect of low-lying asperities on function G(o). The effect of various groups of asperity summits on G(w) is shown in Fig. 5. In this calculation 38 groups of asperity summits out of 60 groups were in contact. The most projected asperity summits have the most important effect on the power spectral density of the acoustic signal in spite of their small number relative to the total number of asperities. It is known that the form of an asperity and the space configuration of surface microtopography depend on the type of abrasive machining: grinding, superfinishing or honing. Thus these types of machining produce different roughness parameters, a, R and 17, and distribution functions of asperity heights, which can differ from the gaussian one. Data on a, R, 37 and cp(z) can be computed directly from a given profile trace. By inputting these values into our model, it would be possible to estimate the effect of the type and regime parameters of abrasive machining on G(o). This gives the possibility of getting qualitative and quantitative estimate of the efficiency of each kind of surface finishing examined. The following criterion can be used: the higher the surface finish that

D. G. Evseev et al. / Acoustic

Fine

modelling of roiling ftiction

grinding

Superfinishing

I

Frequency

,

Fig. 4. Effect of surface finish on power spectral

10

2 2

-’

10

-2

10

-‘-

j

=

36

j

=

2

I

I

10

10

2

10

’

kHz

density.

10-4. .

10

-‘-

4.

s

H 10 ,o

-‘-

z

lo*.

g

lo4

3,

o -to_

& B 10

-‘I-

1

,o

-‘I-

10

-“-

I 10 Frequency

,

1 10 t

I 10

’

kHz

Fig. 5. Effect of asperity height groups on power spectral

density.

is reached, the lower is the level of the power spectral density in the whole frequency range.

3.2 Effects of rolling speed and load The following variations of speed and load were used in calculations: (1) V= 1 m s-l; Y= 10 m SC’, F= 100 N; (2) F = 100 N, F= 1000 N, V= 10 m SC’. The following values of the surface roughness parameters were used: o=7X10-7m,R=5X10-5m,~=109m-2.Thesewere invariant in the calculations. Effects of lubricant were not taken into account. The results of calculations are shown in Figs. 6 and 7. From the analysis of these curves, the following conclusions are drawn. As the speed is increased by a certain factor, the period of the spectrum curve is increased by the same factor (see Fig. 6). Also, the spectrum level is increased within the high-frequency range. These phenomena take place due to a reduction in interaction time between

asperities and to an increase in the steepness of acoustic signal fronts respectively. As the load P is increased, the period of the spectrum curve is reduced proportionally to (P)ln, and the curve level within the high-frequency range is changed (see Fig. 7). This is related to the fact that as the load is increased, the asperity interaction time is increased due to the expansion of the contact area according to eqn. (14). Furthermore, because there is greater correspondence between contacting surfaces, more asperities are involved in contact.

3.3 Effect of lubricant film thickness The lubricant film between the contacting surfaces of cylinder and plane is treated as a layer of thickness L which is not changed as the contact conditions vary. It was convenient to make L proportional to the interval of division of the asperity heights Au.

D. G. Evseev et al. / Acoustic 10

modelling of rolling fiction

39

-l

I: lo-* > z

-’


.=

E 8 loz 2 g

lo-'

3 t lo: L 10 -’ 10

-aI

Fig. 6. Effect

of rolling speed

‘9

F

.

IV

.

.

= 1000

.

.

spectral

10'

kHz

f

lo=

density.

N

.

,

of load of a cylinder

.

Frequency

on power

10

1

Fig. 7. Effect

I

1

10

1

Frequency

on power

.

.

...I,o’

.;;,,

kHz

spectral

density.

1 N 4 z

10“ 10-a

_

lo-’

.2 $g lo-’ 2 73 2 :: cn

10 -’ 10-w lo-’

28 1 10‘* 2 10 -*

10 -‘I

10

1

Fig. 8. Effect

of lubricant

Frequency

film thickness

,

on power

kHz

spectral

10’

density.

The source functions for lubricant layers of various thickness L from OAu to 35Au are shown in Fig. 8, and correspond to a range of asperity heights from 0 to 2.45 X low6 m.

As L is increased, the curve envelope moves down in the high-frequency range more abruptly than in the low-frequency range. Such behaviour of the power spectral density corresponds to alteration of the fre-

40

D. G. Evseev

et al. / Acoustic

quency spectrum from pink noise to red noise. The envelope slopes of the curves are described by l/f”. Thus, for ranges of O-35 layers the it exponent of these envelopes changes from -2.5 to -4.0.

model&g

of rolling friction

13 G. A. Kom and T. M. Kom, Mathematical Handbook for Scientists and Engineers, Nauka, Moscow, 1974, p. 602 (in Russian). 14 K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985.

Appendix: Nomenclaturt 4. Conclusions Using the power spectral density as a parameter of acoustic signals generated by contact spots between asperity summits in elastic contact zones of contacting bodies, a method for quantitative study of the contact processes in rolling friction has been developed. It can potentially specify the efficiency of different kinds of abrasive machining from the optimal acoustic parameter point of view. With modification of the model, it would be possible to predict the processes of running-in and wear in rolling bearings. The results of this work are the theoretical basis for the analysis of experimental data gathered in studying the acoustic signals from bearing in operation. They also can be used to develop test methods for condition monitoring of rolling element bearings.

x P P max P Inax R RC r

References

ro, rl

T. Igarashi and J. Kato, Bull. JSME, 28 (1985) 492-499. T. E. Tallian, Trans. ASME I. Lubric. Technol., IO3 (1981) 509-516.

H. Schicker, Proc. 4 Konf: Instandhalt., Budapest, May 11-13, 1982, pp. 99-102. P. J. Brown, Proc. Confi Bearings: the Search for Longer Life, Cheltenham, 17 October 1984, Bury St. Edmunds, UK pp. 39-47. T. Yoshioka and T. Fujiwara, Wear, 113 (1986) 291-294. T. Konishi, Plant Eng., 19 (9) (1987) 8-13. Y. Furusawa, Jpn. .I. Paper Technol., 32 (6) (1989) 374. D. G. Evseev, B. M. Medvedev and G. G. Grigoryan, Wear, 150 (1991) 7!+-88. 9 L. D. Landau and E. M. Lifshitz,

Elasticiv

Theory,

Nauka,

Moscow, 1965, p. 203 (in Russian). 10 J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement Procedures, Wiley, New York, 1971. 11 W. R. Chang, I. Etsion and D. B. Bogy, J. Tribal. 109 (1987) 257-263. 12 J. I. McCool,

L

u4, w4 t U

V x x, i!

y

Greek letters

44 77 Y (T

P .J. Tribal.,

109 (1987)

2-275.

Size of the contact zone in the rolling direction Length of the cylinder Separation between two contacting surfaces Young’s modulus of bodies 1 and 2 respectively Effective elastic modulus of joint Normal force on the contact Fourier’s transformation Power spectral densities of source and receiver, respectively Unit impulse response function Fourier transformation of h, or transfer function Lubricant film thickness Exponent Count rate of acoustic impulse flow Elastic contact pressure Elastic contact load Maximum contact pressure Asperity radius of curvature Cylinder radius Three-dimensional vector in medium Three-dimensional vectors of source and receiver respectively Fourier transformations of X(t) and Y(f) respectively Time Dimensionless height of the asperity summit Rolling speed of cylinder Coordinate in the direction of rolling Force and response respectively Height of asperity measured from mean of asperity heights

0=29j

Distribution function of asperity heights Area1 density of asperity Poisson’s ratio Standard deviation of asperity heights Material density of medium Cyclic frequency