POINT DISTRIBUTED STATIC LOAD OF A ROUGH ELASTIC CONTACT CAN CAUSE BROADBAND VIBRATIONS DURING ROLLING

POINT DISTRIBUTED STATIC LOAD OF A ROUGH ELASTIC CONTACT CAN CAUSE BROADBAND VIBRATIONS DURING ROLLING

Mechanical Systems and Signal Processing (2002) 16(2–3), 285–302 doi:10.1006/mssp.2001.1461, available online at http://www.idealibrary.com on POINT ...

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Mechanical Systems and Signal Processing (2002) 16(2–3), 285–302 doi:10.1006/mssp.2001.1461, available online at http://www.idealibrary.com on

POINT DISTRIBUTED STATIC LOAD OF A ROUGH ELASTIC CONTACT CAN CAUSE BROADBAND VIBRATIONS DURING ROLLING J. Feldmann Technische Universita.t Berlin, Institut fu.r Technische Akustik, Einsteinufer 25, D-10587 Berlin, Germany. E-mail: [email protected] (Received 16 August 2000, accepted 26 September 2001) The paper is concerned with questions about high-frequency vibration excitation caused by time-variant elastic load reaction forces of rough treads in rolling contact. The described theoretical approach is based on well-known elastic rough contact models in connection with some aspects of signal processing. Data which have to be calculated for instance are the number of mean single contacts or the mean contact time. Because in a statistical sense all the mean contacts are identical, the signal processing is that of a pseudo-random binary process, whereby the term pseudo has its origin in the period which will be created by the global contact dimension divided by the rolling speed. The results give a confirmation that such a contact yields broadband excitation up to the ultra-sonic range. But beside some plausible results a parameter study shows that the approach defies experimental results in two points: (i) the resulting force amplitudes alone are not large enough to explain measured vibrations in general and (ii) the speed exponent of 3–4, that is an increase in the excitation level of 10–12 dB per doubling the rolling speed could not be verified. Possible causes of such disadvantages will be discussed. # 2002 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

The subject can be incorporated into questions of rolling noise generation mechanisms. It connects well-known statistical approaches of elastic rough contact mechanics with some aspects of signal processing. Earlier investigations at the wheel/rail system [1, 2] have shown, that the rolling process can create high vibration amplitudes up to some hundred kilo-Hertz, similar effects are known from rolling bearings [3–5]. The known and commonly used theory of rolling noise generation in the audible frequency range is based on a direct displacement excitation caused by the resulting tread roughness [6]. Because of effects of low-pass wavelength filtering due to the size of the contact patch, this approach is restricted more or less to the waviness of the treads and, therefore, it is not suited to explain high-frequency excitation. Other mechanisms which were investigated in the past like global load reactions [7], parameter excitation [8] or wheel squeal [9] also can be excluded. The model which will be described in the present paper starts from the assumption that the source of broadband excitation has its origin in the multiple distributed static contact load due to discrete elastic-deformed asperities within the contact patch. This effect leads to short and steep time-variant force impulses which move through the global contact patch during rolling. The existance of such pulses was discussed first in [10] in a more qualitative sense. A serious calculation model was developed in [5], here the elastic contact region of a cylinder on a smooth plane with its typical Hertzian global pressure distribution is divided into small strips with a corresponding constant partial 0888–3270/02/+$35.00/0

# 2002 Elsevier Science Ltd. All rights reserved.

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J. FELDMANN

pressure. These strips are matched stepwise in the rolling direction with groups of actual contacting asperities of the roughness of the plane. For every step a static force equilibrium is created, based on the known elastic contact theory of flat rough surfaces [11]. The resulting force signals are interpreted as a Poisson flow of impulses created by the rolling body on the rough surface. The parameter study leads to some plausible results, although the calculated power spectral density of the force depends on the size of the choosen groups of asperity heights, and some influences of parameters on the force spectrum are not convincing, for example, the invariance of the surface roughness on the frequency content. The model serves the purpose to explain the effects of rolling friction, it is of some interest that [12] also refers to the effect of surmounting the irregularities of a rough surface by a rolling body as a source of energy loss concerning the rolling resistance. The aim of the present paper was to modify and extend the ideas expressed in [5] with regard to the application as a vibration excitation mechanism, looking for the possibility to include the Hertzian elastic impact theory into the approach, and additionally, to relate the results to an experimentally investigated set-up which consist of a rolling cylinder on a flat beam. Because of the broadband character of the expected effects they are also relevant in view of their importance in the audio-frequency range.

2. THEORETICAL BACKGROUND

2.1. SPECIFICATIONS OF INPUT ROUGHNESS PROFILE With respect to the input of the model one needs parameter of the normally measured surface roughness profile. In a three-dimensional case it is convenient to assume an isotropic, that means a directional independent distribution of asperities, so it is sufficient to take only one representative profile from the rough surface. The remaining difficulty lies in the fact that the deducible measures depend on the length of the trace, which means they depend on the picked-up wavelength content. Therefore, in the past some authors [13, 14] have introduced the so-called ‘functional filtering’, a tool which makes possible the consideration of only those roughness wavelength which are relevant to the specific application. With regard to the case in question the lower limit should be set by the nominal size of the contact patch, which cannot ‘see’ wavelengths much longer than its longest dimension, so the cut-off of a high-pass filtering is khp ¼

2p 2p ¼ lhp 2ceff

ð1Þ

where 2ceff is the width of the global contact patch, lhp is the wavelength limit, and khp is the corresponding wavenumber. This kind of high-pass filter stands in contrast to the socalled contact patch low-pass filter which is used in rolling noise models, mentioned in Section 1 in context with direct displacement excitation, where wavelengths which are shorter than the contact size are reduced. In the present case the upper limit in the form of a low-pass filtering with a cut-off, klp, should be set by the onset of plasticity. Generally, the shorter the wavelengths the sharper the asperities and the larger the probability to deform plastically even under light loads. The transition range between elastic and plastic deformations can be calculated with the aid of material hardness [14]. In practice, there are two reasons why such an additional low-pass filtering is not necessary: first, normally one applies surfaces under running-in conditions, where elastic deformations dominate, second, a kind of low-pass filtering occurs by the finite size of the roughness measurement stylus. Therefore, in the present investigation the input roughness profile results from a

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287

normally worn steel surface, picked-up with a stylus radius of 5 mm, only filtered with a four-pole-high-pass filter with a cut-off, khp, defined above. The used input profile compared with the original one, can be seen in Fig. 1, represented in the wavenumber range. 2.2. MODEL OF ELASTIC ROUGH CONTACT The present analysis is based on the well-known basic work of [11, 15–18, see also 12, 14]. In these references the Hertzian static contact theory of smooth elastic surfaces [19], Fig. 2, is extended to the statistically distributed rough elastic contact under certain conditions, see Fig. 3. It is not the purpose of the paper either to repeat the basic theories of these authors or to prove basic mathematical relationships, this is well documented in the mentioned references. Only those basic definitions should be repeated which are important in understanding the author. The preconditions in all the approaches are, see again Fig. 3: (i) (ii) (iii)

On an average contacting asperities are identical, they have spherical capped tips with a mean radius, ri, independent of their heights. Asperities are mechanically independent, that means no interactions between them. Single mean asperities deform elastically corresponding to the Hertzian contact theory in the following form: single contact area: Ai ¼ pri di

ðm2 Þ

ð2Þ

and single elastic force 4 1=2 3=2 ðNÞ Fi ¼ E 0 ri di 3 where E 0 is the resulting elasticity (Young’s) modulus 1 1  m21 1  m22 ¼ þ 0 E E1 E2

ðm2 =NÞ

ð3Þ

ð4Þ

consist of the single moduli E1,2 and the single Poisson ratios m1;2 , respectively, the

Power spectral density (dB re 1 m3)

_110 _130 _150 _170 _190 _210 _230 _250 150

1000

10 000 Wavenumber k (1/m)

100 000 350 000

Figure 1. Power spectral density vs wavenumber (k ¼ 2p=l, l is the wavelength) of the used roughness profile , roughness origin; , roughness input (high-pass filtered). referred to one tread.

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J. FELDMANN

R Load pmax

E2, 2 p (x)

z x

H 2 . aH

E1, 1

Figure 2. Classical Hertzian elastic contact between a sphere and a plane. Half contact width a2H ¼ RdH , 0 3=2 pressure distribution pðxÞ ¼ pmax ð1  x2 =a2H Þ1=2 with pmax ¼ 4=3E R1=2 dH , contact area AH ¼ pa2H , dH is the deformation, E1,2, m1,2 are the single elasticity moduli, the Poisson ratios, respectively, R is the global radius.

(iv)

(v)

subscripts 1,2 refer to the two corresponding contacting bodies, di is the elastic compression of one single mean contact. Summit heights expressed as a deviation from the mean plane of the summits is a random variable, and follows a Gaussian probability distribution with a standard deviation ss . In the case of a flat contact it is not necessary to consider bulk deformation, that means only the asperities are deformed elastically during contact. The situation is more complicated in the case of contacting spheres or cylinders, where the separation of the nominal surfaces is a function of the distance from the origin of contact caused by both the geometric shape and the elastic global bulk deformation, in other words there exist an interaction between macro- and micro-contact geometries.

In the past, some effort was made to improve the simplificative assumptions. Some authors introduced different (random) radii of asperities, other developed an elastic contact model that treated asperities as elliptical paraboloids with a random axis orientation and aspect ratio [14, 20–22]. In [23] different models are compared with the result, that the simple approach leads to good order-of-magnitude results in relation to both the numerical effort and the simplicity to use. One needs three parameters for the characterisation of the input roughness profile: (i) (ii) (iii)

Z the surface density of summits, ss the standard deviation of the probability distribution of summit heights, ri the deterministic (non-random) radius of the mean spherical summit caps.

In [23, 24] it is shown that these three quantities can be derived with the aid of the socalled (spectral) moments of a roughness profile z(x):   (i) m 0 ¼ E z2 ð5Þ ðm2 Þ (

(ii) m2 ¼ E

2 )

dz dx

ðdimensionlessÞ

ð6Þ

POINT DISTRIBUTED STATIC LOAD

289

Figure 3. Elastic micro-contact model. The global contact patch consists of a certain number of statistically distributed on average similar contacts. The classical Hertzian theory is assumed to hold at each single microcontact (see details). ceff is now an effective half-contact width, which, in general, is larger than the corresponding measure of a total smooth contact according to the Hertzian theory.

(

(iii) m4 ¼ E

d2 z dx2

2 )

ð1=m2 Þ

ð7Þ

where m0 is the mean square surface height (squared rms roughness amplitude), m2 is the mean square slope, m4 is called kurtosis or fourth moment, and E{ } denotes the statistical expectation. In [23] it is pointed out how to treat two rough surfaces with different moments. With regard to anisotropic rough surfaces in [23] it is also specified how to handle with the moments, and especially, how the present model approach also can be used in such a case. Now, under the assumption that z(x) is a Gaussian random variable, the three necessary input parameter can be calculated in the following way [24]: m4 Z¼ ð8Þ m2  32:65 1=2

ss ¼ ð1  0:8968=aÞ1=2 m0 

p ri ¼ 0:375 m4

ð9Þ

1=2 :

ð10Þ

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J. FELDMANN

The expressions include the fact that the model originally is based only on the summits, whereby available roughness data and, therefore, the moments normally are generated from a complete profile. The parameter, a, sometimes called ‘bandwidth parameter’, is defined as a ¼ m0 m4 =m22 :

ð11Þ

This parameter indicates that, in general, the summits have a lower variance than the surface profile as a whole. According to Fig. 3, if the separation of the nominal surface at the position of a particular asperity is u, there will be a contact at that asperity if the height, z, is larger than u. The probability of this event is determined by the probability density function, FðzÞ, of the distribution of asperity heights. The expected value of a summit contact area, A% i , is thus Z 1 A% i ¼ pri ðz  uÞFðzÞ dz ð12Þ u

the expected force, F% i , is F% i ¼

Z

1 u

4 0 1=2 E ri ðz  uÞ3=2 FðzÞ dz 3

ð13Þ

with z  u ¼ d (compare equations (2) and (3)). Most of the mentioned authors work with standardised variables referring to ss . The integralterms then are expressed by Z 1 0 In ðu Þ ¼ ðz0  u0 Þn F * ðz0 Þ dz0 ð14Þ u0

with u0 ¼ u=ss and z0 ¼ z=ss . For a standardised Gaussian distribution one can write 1 02 F * ðz0 Þ ¼ pffiffiffiffiffiffie1=2z 2p

ð15Þ

Values of In can be obtained by numerical integration, for instance [23]. If now the global radius, R, of the contacting bodies is large compared with the dimensions of both the asperities and the contact patch, one can assume a partial plane contact within an element, dA0, of a nominal (global) contact area, A0, so the expected number of contacts, dn, the expected real area of contacts, dAc, and the expected elastic force, dF, can be calculated as dn ¼ ZI0 ðu0 Þ dA0

ð16Þ

dAc ¼ Zri pss I1 ðu0 Þ dA0

ð17Þ

4 1=2 0 dF ¼ Z E 0 r1 s3=2 s I3=2 ðu Þ dA0 3

ð18Þ

and for the corresponding partial pressure can be written as p¼

dF : dA0

ð19Þ

Another useful measure sometimes is the normal incremental contact stiffness, Kn, of a rough contact [25], in general defined as kn ¼

dF dd

ðN=mÞ:

ð20Þ

POINT DISTRIBUTED STATIC LOAD

291

u0 ¼ z0  d=ss

ð21Þ

dF du0 : du0 dd

ð22Þ

Since

one can write kn ¼

Differentiation of equation (18) with respect to u0 , and differentiation of equation (21) with respect to d lead to the wanted expression: Z kn ¼ 2ZE 0 ðri ss Þ1=2 I1=2 ðu0 Þ dA0 : ð23Þ A0

In the case of a nominal flat contact the integration leads to the area A0 itself, one gets simply the results for the expected number of contacts per unit area, n/A0, the expected real contact area as a fraction of the nominal area, Ac/A0, or the total load per unit area supported by asperities, F/A0. In the cases under consideration, the integration is much more complicated, because the separation, u, now is a function of the shape of the nominal surfaces, either this can be a contact of rough spheres [17] or the contact of rough cylinders [18]. Determined by an existing experimental set-up data the following representation is focused on the latter mentioned case. Then the calculations can be carried out in a twodimensional way in terms of cylinder length, ly, and the surface element dA0 can be treated as 1 dr that means a reduction of the surface integration to the dimension, r, of the contact area width, compared in Fig. 4. In a first step one has to define the relations between the separation, u, and the global geometry of the bodies in elastic contact. The dependency of u from the coordinate, r, generally consist of two terms: (i)

the pure geometrical shape term of the nominal surfaces: uð0Þ þ r2 =2R

ð24Þ

R1 Load

E1, 1 Nominal surfaces of contacting bodies

Real surface (2)

u(0)

w(r) u(r) x r

R2 ⇒ ∞

ceff 0 E2, 2

Figure 4. Illustration of the used nomenclature for the calculation of the separation, u(r), regarding the elastic contact between a cylinder and a rough plane.

292

J. FELDMANN

with R¼

(ii)

R1 R2 R1 þ R2

ð25Þ

where u(0) is the minimum separation in the origin, R denotes the resulting radius, and R1,2 are the corresponding single radii of the two bodies, and the term which considers the elastic bulk deformation, w: wð0Þ  wðrÞ:

ð26Þ

The index (0) also refers to the origin of contact, see Fig. 4, thus, the total separation can be described as follows: uðrÞ ¼ uð0Þ þ r2 =2R  ½wð0Þ  wðrÞ :

ð27Þ

For the solution of the problem the following three expressions with respect to the contact width coordinate, r, are necessary: (i) the pressure distribution according to equations (18), (19) and (27)

dF 4 0 1=2 3=2 uð0Þ þ r2 =2R  ½wð0Þ  wðrÞ pðrÞ ¼ ¼ Z E r1 ss I3=2 ð28Þ dr 3 ss (ii)

(iii)

the well-known elastic deformation solution of a semi-infinite plane, for instance [12] Z 1 x  r 2 ð29Þ pðxÞ ln wð0Þ  wðrÞ ¼ dx pE 0 1 x which has to be subjected to a pressure distribution corresponding to equation (28), where, x, is the current coordinate, and according to equation (27), finally one gets Z 1 x  r 4 uðrÞ ¼ uð0Þ þ r2 =2R  0 pðxÞ ln ð30Þ dx: pE 0 x

After standardisation of the single variables, in [18] an approximate way of solving equation (30) is shown. In contrast to the well-defined smooth cylinder contact in the rough case the expected pressure distribution at a point far from the origin is only small, but not necessarily zero, therefore, a global pressure distribution can be assumed, which has more a bell shape with zero at infinity. Since in practice the expected nominal rough contact width cannot be infinite, a kind of effective width, ceff, is defined. In the case of very rough surfaces in connection with light loads this effective width can be much larger than the width of a corresponding smooth contact, only in the case of less roughness and higher loads the effective width equals the Hertzian one [17]. The measure ceff gives an orientation for the choice of the practical integration range with respect to the local distance from the centre of the contact, additionally, 2ceff should be used for calculating the functional filter limits (compare Section 2.1). But because ceff is a priori unknown, in a strict sense also the functional filter process has to be performed on an iterative way [26]. In a further step one needs information about the elastic separation, u(0), at the centre of contact. Without a large error one can apply a nomogram created in [17, 26], which represents the dependency of a dimensionless surface number, n, against a dimensionless loadnumber, X, with both u0 (0) and w0 (0) as parameters: pffiffiffiffiffiffiffiffiffi 8 u ¼ Zss 2ri R 3

ðdimensionlessÞ

ð31Þ

293

POINT DISTRIBUTED STATIC LOAD

and X¼

2F0 pffiffiffiffiffiffiffiffiffiffiffi ðdimensionlessÞ 2Rss

ð32Þ

ss E 0

with F0= the total static load of the global contact. Besides, X ¼ 200 represents approximately the limit value for the actually discussed rough contact theory in context p with affi smooth contact. ffiffiffiffiffiffiffiffiffiffi Knowing u0 (0) and c0 ¼ ceff = 2Rss from an approximation one can calculate the pffiffiffiffiffiffiffiffiffiffiffi standardised separation u0 ðr0 Þ for every distance, r0 ¼ r= 2Rss , from the centre of contact according to a series expansion of equation (30) given in [18]. Thus, it is now possible to perform a numerical integration with the effective integration limit c0 regarding the following still unknown measures: (i)

the real contact area, Ac/ly, as a fraction of the nominal area, A0/ly, pffiffiffiffiffiffiffiffiffiffiffi Z c0 I1 ðu0 Þ dr0 ðm2 =mÞ Ac =ly ¼ 2pZri ss 2Rss

ð33Þ

0

(ii)

the total number of mean asperities in contact, n/ly, pffiffiffiffiffiffiffiffiffiffiffi Z c0 n=ly ¼ 2Z 2Rss I0 ðu0 Þ dr0

ð1=mÞ

ð34Þ

0

and (iii), if desired, the contact stiffness, kn/ly, Z 0 pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffi c kn =ly ¼ 2ZE 0 ri ss 2Rss I1=2 ðu0 Þ dr0

ðN=m mÞ

ð35Þ

0

(iv)

whereas the results, as can be seen, referred to the length, ly, of the cylinder (remember that In is also an integral, and u0 is a function of r0 ). Other needful measures are: the nominal contact area, A0, A0 =ly ¼ 2ceff

(v)

ð36Þ

ðm2 Þ

ð37Þ

the area of one single mean contact spot, Ai, Ai ¼ Ac =n

(vi)

ðm2 =mÞ

with its diameter, 2 ai, 2ai ¼ 2

pffiffiffiffiffiffiffiffiffiffi Ai =p ðmÞ

ð38Þ

and (vii) the single static contact force per spot, Fn, Fn ¼ F0 =n ðNÞ:

ð39Þ

The results show a complicated interaction with respect to the several parameter as discussed in the mentioned references. 2.3. TREATMENT OF THE ROLLING PROCESS When a (rough) elastic cylinder or sphere rolls over a rough elastic surface, the global static load has to be supported by a certain number of asperities within the nominal contact patch during any time increment. In other words, the force equilibrium between the contacting rough solids has to be maintained by a set of individual, per definition, non-

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J. FELDMANN

interacting elastic reaction forces transmitted by an array of discrete contact spots. Because of the continuous process of formation and decomposition of such single contacts during the rolling process, in principal, this array is non-stationary. But it seems to be not very efficient to try to calculate every detail in a moving contact patch due to an actual real random-like roughness input, therefore, in the authors opinion it is more sensible to assume a periodically repeated quasistationary process based on the statistical properties of such a contact, as presented in the foregoing section. This approach is supported by the fact that normally the contact dimensions are small compared with both the global dimensions of the contacting bodies, and the corresponding wavelengths of structural vibrations, so the force array can be interpreted as an exciting point source. The period is defined by the nominal effective contact width divided by the rolling speed, U, T ¼ 2ceff =U

ðsÞ

ð40Þ

because per definition all the mean single contact spots are identical, see Fig. 5: (i)

the single contact time, ti, controlled by the deformed mean single contact area is [see equation (38)] ti ¼ 2ai =U

ðsÞ

ð41Þ

and (ii) all the single mean force amplitudes, Fn, [compare equation (39)] are also identical. n Identical contact spots U

ly

x

lx Identical force pulses Fn, single duration time ti

Rolling direction n = ny nx T = lx / U ti = clock

ny

ti nx ti T

Figure 5. Treatment of the rolling process as a pseudo-random binary process. ‘Pseudo’ means the repeating with the period, T, defined by the global contact patch width divided by the rolling speed, ‘random’ is related to the certain number, n, of statistically distributed mean contact spots within the global contact patch, ‘binary’ decribes the fact that both the amplitude, Fn, and the contact clock, ti, are similar for all splitted single forces, so within the raster of the clock there are only two states possible: contact or no contact.

POINT DISTRIBUTED STATIC LOAD

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The shape of the single force pulses are assumed to be rectangular, a non-imperative approximation which results from the generally very small contact time. Thus, the system is determined by only one time increment, so the single contact time can be seen as a kind of clock within the whole contact. Before the resulting force spectrum can be calculated two questions still have to be discussed. Firstly, in the common literature, for instance [27], force pulses as a randomly occurring series of similar events are interpreted as a Poisson process generated by periodic sampling on the basis of a clock. If the pulse time is very short this process can be handled simply as a counting process of state changes within a given interval, defined by the so-called count-rate, g, that is the expected number of events, n, per second nno g¼E ð1=sÞ: ð42Þ T The larger the count-rate the larger the generated amplitudes and the higher the frequency content. Such a simple treatment with regard to the present model defies practical experience, because a relatively smooth contact can show a large number of single contacts } i.e. a large count-rate } but generates low excitation power with low frequency content. So one has to modify the description of the present random process. This was done on the basis of a pseudo-random binary process as described in [28], a process which also considers statistically distributed identical events with respect to their amplitudes and their duration within a certain periodical interval. Secondly, one gets down to the question, in what way the rolling dynamics can influence the force generation, because as mentioned the described rough contact model originally is based on a static contact. Some estimations were carried out based on the assumption that at every identical contact underlies the principles of an incomplete elastic impact according to the well-known Hertzian elastic impact theory [19, 29]. The impact is called incomplete because in contrast with a free impact the elastic compression of a single mean contact is limited by the relation between force and contact stiffness [compare equation (20)]. This limitation leads to relatively short impact times connected to inacceptable and unrealistic large force amplitudes due to the inverse proportionality between the squared impact time and the resulting impact force. For the present this approach was dropped. On the other hand some authors [30, 31] found out that the Hertzian static elastic contact is not so sensitive regarding movements or vibrations as long as only the normal direction of contact is relevant and not the tangential one. A third aspect could not be cleared sufficiently, that is the question about the influence of the structural vibrations on the described form of contact. According to [32] it can be assumed that those interactions are negligible as long as the wavelengths are large compared with the real contact size. The next step is connected to the question how can the two-dimensional statistically distributed number of contact spots be handled with respect to the one-dimensional rolling direction. It will be assumed that a number of individual non-interacting contact spots which are distributed bivariately within the generally rectangular contact patch of a cylindrical contact. Let, x, be the coordinate of the width, lx, and, y, the coordinate of the length, ly, then the partial number of contacts, nx resp. ny, in the corresponding direction can be obtained simply by weighting the total contact number with the length to width relation in the following way: qffiffiffiffiffiffiffiffiffiffiffiffi nx ¼ nlx =ly ð43Þ and ny ¼

qffiffiffiffiffiffiffiffiffiffiffiffi nly =lx

ð44Þ

296

J. FELDMANN

and therefore n ¼ nx ny :

ð45Þ

Besides, in the case pffiffiof ffi the rotational symmetry of a spherical contact, nx and ny would be equal to a value n. Regarding the existence of a mean contact time a consideration of limit cases is helpful. If the nominal contact patch has a width of one single spot}i.e. nx=1 and ny=n}all the contacts are lying in a row in the y-direction perpendicular to the defined rolling direction, x, the mean contact time is equal to the single time, ti, and the total power amplitude of the force must be nFn2 . In the other case, if the length, ly, is similar to one spot}i.e. nx=n and ny=1}all single contacts are lying in a row in the rolling direction, so the mean contact time in a quasistationary approach would be nti. Thus, one comes to the conclusion that it is sensible to introduce a mean contact time, tc, which has to be tc ¼ nx ti :

ð46Þ

Finally, according to Fig. 6, the expression of the resulting power spectrum of the force can be derived from the well-known Fourier-transformed autocorrelation function of a pseudo-random binary process [28]:   tc sinðftc =2Þ 2 PWRðf Þ ¼ ny Fn2 ðN2 Þ ð47Þ ftc =2 T with f= frequency in Hz. 3. APPLICATION}PARAMETER STUDY}DISCUSSION

The described micro-contact model of rough elastic bodies was verified on the basis of a well-investigated experimental set-up, consisting of a steel cylinder rolling on a steel beam [33], with the following input data: speed U=5 m/s, load F0=58.86 N, global radius R=0.123 m, cylinder length ly=0.01 m, elasticity-modulus E=2.1  1011 N/m2, the Poisson ratio m ¼ 0:33, rms- roughness of the beam tread (measured and functional filtered, compare with Section 2.1) s=0.617  106 m. The roughness of the cylinder tread was assumed to be in the same order. The model yields all the necessary data such as, the total number of mean contacts n= 548, respectively, nx=8, ny=68.5, the total real contact area Ac=1.21  108 m2, the effective nominal contact area A0=1.154  105 m2 or the width of the contact ceff=5.77  104 m. For comparison the Hertzian contact width

Figure 6. Autocorrelation function (ACF) and corresponding power spectrum (PWR) of the pseudo-random binary process described in Fig. 5, tc is a mean contact time caused by the quasistationarity due to the period T (see also [28]).

297

POINT DISTRIBUTED STATIC LOAD

0 _10

2

Power spectrum (dB re 1 N )

_ 20 _30 _ 40 _50 _60 _70 _80

2000 000

1 000 000

100 000

10000

_100

3000

_90

Frequency (Hz)

Figure 7. Calculated force excitation power spectra. Parameter is the rolling speed, U. Other data are: , U=2.5 m/s; , m=6 kg, R=0.123 m, ly=0.01 m, E=2.1  1011 N/m2, m=0.33, srms=0.6169  106 m. , U=10 m/s; , U=20 m/s; , U=40 m/s. U=5 m/s;

would be aH= 8.844  105 m. For a single force one gets Fn=0.1073 N. The signalprocessing approach according to equation (47) leads to spectra of the resulting force excitation which can be seen in Fig. 7. Regarding the speed 5 m/s one has a maximum of 18.39 dB re 1 Nrms with a first minimum at 118.615 kHz, the corresponding relation tc/T is 3.65  102. Comparing with demands on a comprehensive excitation model, first of all one has to state that in the case under consideration the amplitudes are not large enough, they show approximately 30 dB lower values than one needs to explain usually measured vibrations in the audible frequency range. This does not change essentially if one introduce a kind of dynamic amplification factor in the order of 1.2–2 [29]. To get more information about the general significance it is helpful to submit the model approach to a parameter study. The results can be seen in Figs. 7–12, they are summarised in Figs. 13 and 14 with regard to both the maximum amplitude and the first minimum of the corresponding spectra, at which the results from the mentioned input data were used as a reference (default case). Most of the calculated effects seem to coincide with practical requirements. For instance: (i) an increase of the radius, R (global geometry) leads to a decrease of both the amplitudes, and the frequency content. (ii) An increase of the length, ly, also shows lower amplitudes, but an unchanged frequency range. At this point the model is not suited to explain the rolling resistance which tends to larger values for an increased dimension perpendicular to the rolling direction. (iii) Increasing the mass (load) of the rolling body, m, creates higher amplitudes, but a lower-frequency content. (iv) A softer contacting material decreases both the amplitudes and the first minimum of the spectrum. (v) If one separately increases the roughness amplitude, s, both the force amplitudes and the frequency content also increase. This effect is more significant regarding the amplitudes if one simultaneously increases the three roughness moments in an appropriate way. The calculated effects of increasing rolling speed, U, not with respect to the frequency range, but with respect to the power spectra amplitudes, do not lead to the expected

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Figure 8. Calculated force excitation power spectra. Parameter is the mass (load) of the rolling body, m. Other , m=6 kg; data are: U=5 m/s, R=0.123 m, ly=0.01 m, E=2.1  1011 N/m2, m=0.33, srms=0.6169  106 m. , m=12 kg; , m=24 kg; , m=48 kg; , m=96 kg.

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Figure 9. Calculated force excitation power spectra. Parameter is the global radius of the rolling body, R. , Other data are: U=5 m/s, m=6 kg, ly=0.01 m, E=2.1  1011 N/m2, m=0.33, srms=0.6169  106 m. , R=0.123 m; , R=0.246 m; , R=0.492 m; , R=0.984 m. R=0.0615 m;

behaviour (Fig. 7). Here the results coincide with those in [5]. The investigations on the wheel/rail system in the ultrasonic range [1] show a speed exponent of 3–4, that is a level increase of 10–12 dB per doubling the speed. Such dependencies can only be obtained with an elastic impact approach (see the remark in Section 2.3) or can be caused by other

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Figure 10. Calculated force excitation power spectra. Parameter is the rolling body length, ly. Other data are: , ly=0.005 m; , U=5 m/s, m=6 kg, R=0.123 m, E=2.1  1011 N/m2, m=0.33, srms=0.6169  106 m. , ly=0.02 m; , ly=0.04 m; , ly=0.08 m. ly=0.01 m;

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Figure 11. Calculated force excitation power spectra. Parameter is the elasticity modulus, E. Other data are: , Emod=4.2  1011; , U=5 m/s, m=6 kg, R=0.123 m, ly=0.01, m=0.33, srms=0.6169  106 m. , Emod=1.05  1011; , Emod=0.525  1011; , Emod=0.2625  1011. Emod=2.1  1011;

additional but unknown mechanisms. So, as a consequence of the results the decribed mechanism can explain most of the important parameters of broadband rolling vibration excitation in coincidence with practical experience, excepting the behaviour of the amplitudes of the force respect to the speed dependency. This is true not only in the audio

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Figure 12. Calculated force excitation power spectra. Parameter is the root mean square of the roughness amplitude, srms (or sqrt(m0)). Other data are: U=5 m/s, m=6 kg, R=0.123 m, ly=0.01 m, E=2.1  1011 N/m2, , sqrt(m0)=0.30845  106; , sqrt(m0)=0.6169  106; , sqrt(m0)=1.2338  106; , m=0.33. , sqrt(m0)=4.9352  106. sqrt(m0)=2.4676  106;

Figure 13. Summarised characteristics of single power spectra of the force. Parameter-dependent presentation of the maximal value of the power level as a function of a doubling of the corresponding parameter. Reference , U/5 m/s; , U/6 kg; , R/0.123 m; belongs to the abscissa value one, and the ordinate value zero dB. , sigma/0.6169e-6 m; , ly/0.01 m; , 2.1e11 N/m2/E-Mod.

range where other mechanisms seem to be more dominant [6, 33, 34], but also in the range of most interest in the paper, the ultrasonic range. If the investigated mechanism is not the only existing one it can gain a certain significance only if other mechanisms are suppressed by technical means. This is a problem of an experimental proof. Nevertheless, the results call for further investigations.

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Figure 14. Summarised characteristics of single power spectra of the force. Parameter-dependent presentation of the first frequency minimum of the power spectrum as a function of the doubling of the corresponding , U/5 m/s; , U/6 kg; , R/0.123 m; , parameter. Reference belongs to the coordinate values one. , ly/0.01 m; , 2.1e11 N/m2/E-Mod. sigma/0.6169e-6 m;

4. CONCLUSIONS

This paper presents an approach which combines well-known theories of static rough elastic contact with some quasistationary aspects of analytical random signal processing. The multiple distributed static load of contacting elastic rough bodies caused by a certain number of single elastic contacts within the contact patch is seen as a source of vibration excitation during a rolling process mainly in the ultrasonic frequency range. The significance of such a mechanism is discussed. In a first step a short repetition of the important theoretical aspects is carried out, followed by a description of the necessary input data. In a further step the known theories are upgraded, and are expanded to a treatment as a pseudo-random binary process. After assumptions regarding both the twodimensional distribution of the whole contact number within the contact area and considerations about a mean contact time are carried out, an expression of the mean exciting force power spectrum is defined. In the last step a parameter study relates the model approach to real data from an experimental set-up. It can be shown that most of the results are plausible with respect to experience in rolling vibration generation. Only the calculated force amplitudes with their rolling speed dependency are not in full agreement with the expected results. A treatment of the single contacts such as elastic Hertzian impacts certainly can lead to the correct speed dependency, but lead to unrealistic heights of amplitudes. A possible interaction between the contact and the structural dynamics of the contacting bodies remains an unknown aspect. Based on the obtained results further investigations are necessary. REFERENCES 1. Ch. Michalke and J. Feldmann 1989 DAGA’89 Fortschritte der Akustik, 615–618. Bad Honnef: DPG-Verlag, Ultraschallverhalten einer Eisenbahnschiene. 2. K. Werner 1986 ZEV-Glasers Annalen 110, 353–360. Diskrete Riffelabst.ande und die Suche nach Ursachen der Schienenriffeln. 3. R. C. Drutowski 1959 Transactions of the ASME, Series D 81, 233–238. Energy losses of balls rolling on plates. 4. S. Y. Poon and F. P. Wardle 1983 Chartered Mechanical Engineer July/August 1983, 36–40. Rolling bearing noise}cause and cure.

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5. D. G. Evseev, B. M. Medvedev, G. G. Grigoryan and O. A. Ermolin 1993 Wear 167, 33–40. Description of the rolling friction process by acoustic modelling. 6. D. J. Thompson 1993 Journal of Sound and Vibration 161, 387–482. Wheel–rail noise generation, part I–V. 7. M. L. Munjal and M. Heckl 1982 Journal of Sound and Vibration 81, 477–489. Some mechanisms of excitation of a railway wheel. 8. A. Nordborg 1998 ACUSTICA 84, 854–859. Parametrically excited rail/wheel vibrations due to track–support irregularities. 9. U. Fingberg 1990 Journal of Sound and Vibration 143, 365–377. A model of wheel–rail squealing noise. 10. R. S. Sayles and S. Y. Poon 1981. Journal of Precision Engineering 3, 137–144. Surface topography and rolling element vibration. 11. J. A. Greenwood and J. B. P. Williamson 1966 Proceedings of the Royal Society London Series A 295, 300–319. Contact of nominally flat surfaces. 12. K. L. Johnson 1985 Contact Mechanics, pp. 311. Cambridge: Cambridge University Press. 13. T. R. Thomas and R. S. Sayles 1978 Tribology International 11, 8–9. Some problems in the tribology of rough surfaces. 14. T. R. Thomas (ed.) 1982 Rough Surfaces. NY: Longman Group Ltd. 15. J. A. Greenwood 1967 Transactions of the ASME, Journal of Lubrication Technology 89F, 81– 91. The area of contact between rough surfaces and flats. 16. J. A. Greenwood and J. H. Tripp 1970 Proceedings of the Institution Mechanical Engineers 185, 625–633. The contact of two nominally flat rough surfaces. 17. J. A. Greenwood and J. H. Tripp 1967 Transactions of the ASME, Journal of Applied Mechanics 34E, 153–159. The elastic contact of rough spheres. 18. C. C. Lo 1969 International Journal Mechanical Science 11, 105–115. Elastic contact of rough cylinders. 19. K. L. Johnson 1982 Proceedings of the Institution of Mechanical Engineers 196, 363–378. One hundred years of Hertz contact. 20. R. A. Onions and J. F. Archard 1973 Journal of Physics D: Applied Physics 6, 289–304. The contact of surfaces having a random structure. 21. D. J. Whitehouse and J. F. Archard 1970 Proceedings of the Royal Society of London Series A 316, 97–121. The properties of random surfaces of significance in their contacts. 22. A. W. Bush, R. D. Gibson and T. R. Thomas 1975 Wear 35, 87–111. The elastic contact of a rough surface. 23. J. I. Mccool 1986 Wear 107, 37–60. Comparison of models for the contact of rough surfaces. 24. J. I. Mccool 1987 Transactions of the ASME, Journal of Tribology 109, 264–270. Relating profile instrument measurements to the functional performance of rough surfaces. 25. H. A. Sherif 1991 Wear 145, 113–121. Parameters affecting contact stiffness of nominally flat surfaces. 26. T. R. Thomas 1979 American Society of Lubrication Engineers Transactions 22, 184–189. Calculation of elastic contact stresses for rough-curved surfaces. 27. G. A. Korn and T. M. Korn 1968 Mathematical Handbook for Scientists and Engineers, Chapters 18–11. New York: McGraw-Hill. 28. D. E. Newland 1987 An Introduction to Random Vibrations and Spectral Analysis, 2nd edn. UK: Longman Group Ltd. 29. W. Goldsmith 1960 Impact. London: Edward Arnold Ltd. 30. J. Feldmann and F. Zimmermann 1989 Proceedings of the Workshop on Rolling Noise Generation, Techn. Universita.t Berlin, Inst. f. Techn. Akustik, 89–101. The incremental contact stiffness}a comparison between theory and experiment. 31. M. Wang 1995 Doctoral Thesis in Fortschr.-Ber. VDI Reihe 11 (217). Untersuchungen u. ber hochfrequente Kontaktschwingungen zwischen rauhen Oberfl.achen. 32. S. C. Hunter 1957 Journal of Mechanics and Physics of Solids 5, 162–171. Energy absorbed by elastic waves during impact. 33. J. Feldmann 1987 Journal of Sound and Vibration 116, 527–543. A theoretical model for structure-borne excitation of a beam and a ring in rolling contact. 34. M. Heckl, et al. 1998. Hochfrequenter Rollkontakt der Fahrzeugra.der Results from SFB: . bersicht zur Technische Universit.at Berlin/ DFG, K. Knothe and F. Bo. hm, (eds). Wiley-VCH. U Ger.auschentstehung beim Rollen.