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117
y^K 539.62
Contact force resulting from rolling on a self-affine fractal
rough surface
M. Popov
Freie Universitat Berlin, Berlin, 14195, Germany
The time dependence of the contact force resulting from the rolling of a sphere on self-affine fractal rough surfaces was investigated using the method of reduction of dimensionality. The main focus was on the part of the spectrum corresponding to the short wavelength corrugation (with a wavelength much smaller than the nominal contact radius). It has been shown that in this domain, the spectral density of the normal force is a power function of frequency. This result can be used for calculation of high frequency acoustic emission of rolling contacts in various technical systems as well as for development of diagnostic systems.
Keywords: method of reduction of dimensionality, diagnostic systems, contact problem, surface roughness
1. Introduction
The dynamics of rolling on rough surfaces is of great interest from the viewpoint of functionality, noise production, and for the diagnostics of many technical systems, such as gears, ball bearings, wheel/rail, tire/road systems, and many others. Vertical dynamics in rolling contacts have been investigated intensively since the 1970’s in the context of noise produced by wheel/rail contacts [1, 2]. For the problem of noise in a wheel/rail contact, it is widely agreed that the dominant excitation is due to surface roughness of wavelengths equal to or larger than the size of the contact. The dynamics caused by corrugations of shorter wavelength have not been studied in detail earlier. In the present paper, we treat this unsolved problem of the vertical dynamics of rolling bodies due to short wavelength roughness.
We consider a rough elastic sphere rolling on a nominally planar rough surface, while its center of mass is restricted to planar movement, and calculate the contact force acting on it. It is the purpose of this paper to characterize the spectrum of the contact force as a function of the spectrum of the surface roughness. This work may lead to a better understanding of vibration and acoustic emission in applications such as ball bearings or gears. To the knowledge of the author, the above problem has not previously been solved, mostly owing to the fact that the contact forces between rough surfaces have been very computationally expensive to evaluate. This has changed with the recent introduction of the method of reduction of dimensionality [3]
and its generalization to randomly self-affine surfaces [4]. The validity of the method has been proven exactly for arbitrary bodies of revolution [5] and has been verified recently for self-affine rough surfaces in a wide range of fractal dimensions [6]. The method allows us to evaluate the contact forces in a one-dimensional substitute model, vastly reducing computational requirements.
2. Computational model
We consider an elastic rough sphere of radius R, mass m, Young’s modulus Es and Poisson’s ratio vs, rolling on an elastic body with Young’s modulus Ep and Poisson’s ratio vp and having a nominally planar randomly rough surface. For both the normal contact problem and the rolling contact without propulsion, one can consider the equivalent problem of a contact between a rough elastic sphere and a plane rigid surface. The roughness of the sphere should be defined as the combined roughness of both bodies and the sphere characterized by the effective elastic modulus [7]
1
E*
1 -vp 1 -v2
Ep Es
(1)
We will assume that the combined surface roughness is a random self-affine fractal with the power spectrum [8]
C2d(<?) = C
qo
(2)
© Popov M., 2012
where q is the wave number, q0 is arbitrary reference wave number, C is a constant determining the amplitude of the roughness, H is the Hurst exponent, and qmin and qmax are the cut-off wave vectors. As our focus is on the investigation of corrugations with small wavelength, we will choose
_ 2n 2n
qmin , qmax >> ,
a a
a being the (macroscopic) radius of the contact area (without considering roughness). These conditions guarantee that the corrugations with wavelength in the interval 2n/qmax < <X« a are present in the power spectrum.
The rules of generating an equivalent 1D system according to the method of reduction of dimensionality are described in [7]. The three-dimensional profile of a sphere in the vicinity of the contact point z = r2/(2R) is replaced by a one-dimensional profile modified by the factor 2 and moving to the right with velocity v0
Ak = E Ax, (9)
where Ax is the spacing between adjacent springs [4].
At a given point in time, all of the points satisfying the condition ztot (xt) < 0 will be in contact with the base plane. Their vertical displacements from the reference state will (3) be equal to uz (xt) = -ztot( xt) and the corresponding elastic forces, fz (xt) = -E ztot (xt)Ax. The total contact force is given by the sum over all points in the contact Nc:
F0(Z) = - E * A x £ z tot(x,). (10)
(4)
while the two-dimensional roughness with power spectrum (2) is replaced by a one-dimensional rough line with the power spectrum
-2 H-1
CiD(q) = nqC2D(q) = nC-
q0
-2 H-2 '
(5)
We start with the generation of a randomly rough line with the power spectrum (5) according to the rule
z(x) = X Bid (q) exP(i( qx + q))), (6)
q
where ^(q) is a random phase, l2n
B1D(q) = ^ LC1D(q) = BlD(-q), (7)
and L = 2nR. This rough line is periodically extended giving an infinite rough line zrough( x). The geometry of the system under consideration is finally given by the superposition
ztot(x) Z + zrough (x) +
(8)
where Z is the coordinate of the center of mass of the sphere
(Fig. 1).
According to the method of reduction of dimensionality, this profile is considered to be a one-dimensional elastic foundation with independent springs, each having the stiffness
3. Simulation results and discussion
The numerical simulation of the horizontal motion of the sphere produces a time series for the contact force, as new asperites come into contact with the plane and others come free again. The Fourier transformation of the contact force (in time) divided by the mass of the wheel provides the power spectrum of the acceleration as a function of frequency. This power spectrum was smoothed with a Gaussian filter with a standard deviation of six wave numbers. The resulting power spectrum for a rough surface with the Hurst exponent H = 0.7 is represented in double logarithmic presentation in Fig. 2.
We see that in the medium region of frequencies, the dependence of the logarithm of spectral density on the logarithm of frequency can be approximated with a linear dependence, thus, suggesting a power dependence of the spectral density on frequency with the exponent a.
In Fig. 3, the dependence of this power on the Hurst exponent of the roughness is shown. A linear dependence can clearly be seen.
More detailed investigation of the power spectrum of the contact force on loading conditions, rolling velocity, and power spectrum of the surface roughness will be published in a separate paper.
4. Conclusion
In the present paper, the initial results on the high frequency changes in the contact force of a sphere rolling on a rough surface are presented. Using the method of reduc-
Fig. 1. Schematic presentation of the considered system
Fig. 2. Power spectrum of contact force for H = 0.7
Fig. 3. Power a as a function of the Hurst exponent H
tion of dimensionality, we could solve the macroscopic nonlinear and multiscale microscopic contact problem which would have been prohibitively computationally expensive until now. It was shown that the spectrum of the contact force due to self-affine fractal rough surface is a power function of the frequency. More detailed investigation of the parameter space as well as consideration of roughness on both microscopic and macroscopic scale is an important topic of future work.
The author is thankful to B. Grzemba for providing the program generating fractal lines and to A.E. Filippov, R. Pohrt, V.L. Popov and L. Voll for support in the initial stages of this work and valuable discussions.
References
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5. Hefi M. Uber die exakte Abbildung ausgewahlter dreidimensionaler Kontakte auf Systeme mit niedrigerer raumlicher Dimension. - Gottingen: Cuvillier-Verlag, 2011. - 172 p.
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7. Popov V.L. Contact Mechanics and Friction. Physical Principles and Applications. - Berlin: Springer-Verlag, 2010. - 362 p.
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Sci. Rep. - 2006. - V. 61. - P. 201-227.
Поступила в редакцию 10.06.2012 г.
Сведения об авторе
Popov Mikhail, BSc, Institut fur Informatik, Freie Universitat Berlin, Germany, [email protected]
