Power Spectral Density of Rough Surfaces

Power Spectral Density of Rough Surfaces

<p style="text-align: justify;">Roughness is known to be an important factor in friction, wear and related performance of the machine parts. It was shown recently that Power Spectrum Density (PSD) can be used as a metric to describe the surface from tribological point of view. Besides the field of tribology, roughness determines surface properties, such as hydrophulicity, adhesion, fatigue resistance, etc. Some of the analytical models of contact mechanics predict contact area, stiffness and other important parameters based on the PSD. Therefore, knowledge of PSD of a surface can bring valuable information about its (tribological) properties.</p> <p style="text-align: justify;">Power spectral density reveals presence and contribution of different spatial frequencies (wavelengths, wavevectors) in the surface. For example, if the surface roughness is composed of two sine waves of different wavelengths, its PSD will have only two peaks (see the figure below). These peaks indicate presence of two wavelengths in the surface.</p> [caption id="attachment_21776" align="aligncenter" width="278"]<img class="wp-image-21776 size-full" src="http://www.tribonet.org/wp-content/uploads/2018/07/PSD.jpg" alt="Power Spectral Density" width="278" height="384" /> Power Spectral Density of a surface composed of two sine waves of different wavelength[/caption] <p style="text-align: justify;">Power spectral density of a randomly rough surfaces was shown to frequently look like in the figure below:</p> [caption id="attachment_21777" align="aligncenter" width="308"]<img class="size-full wp-image-21777" src="http://www.tribonet.org/wp-content/uploads/2018/07/PSD-Regular.jpg" alt="PSD of a rough surface" width="308" height="290" /> Power spectral density of randomly rough surface[/caption] <p style="text-align: justify;">In this case, the PSD reveals a self-affine surface, which is not always the case (if not self-affine, it may look curved and not straight line).</p> <p style="text-align: justify;">There are various definitions of the power spectral density, which differ typically with a constant. Here we follow [1] and define the PSD as follows:</p> <p style="text-align: justify;">[math] C^{2D}_{q_x,q_y} = \frac{1}{A}*\abs{\widetilde{h}_{q_x,q_y}}^2 [/math],</p> <p style="text-align: justify;">where [math] C^{2D}_{q_x,q_y} [/math] is the 2 dimensional PSD, A is the area of the surface, [math] q_x, q_y [/math] are wavevectors in x and y directions. Fourier transform of the surface is given as follows:</p> <p style="text-align: justify;">[math] \widetilde{h}_{q_x,q_y} = \int_{A} h_{x,y} e^{-i(q_x+q_y)} dxdy[/math].</p> <p style="text-align: justify;">Equivalent definition of the PSD is used by B. Persson [2]:</p> <p style="text-align: justify;">[math] C^{2D}_{q_x,q_y} = \int_{A}  &lt;h_{x,y} h(0,0)&gt; e^{-i(q_x+q_y)} dxdy [/math],</p> <p style="text-align: justify;">where the &lt;..&gt; denotes average of the ensemble. These definitions are equivalent due to Wiener-Khintchine theorem [2].</p> <p style="text-align: justify;">The definition of the power spectral density in terms of Fourier transform seems to be more convenient, since it also shows a direct way of calculating it for a given surface.</p> <p style="text-align: justify;">References</p> <p style="text-align: justify;">[1] <a href="http://iopscience.iop.org/article/10.1088/2051-672X/aa51f8/meta">Quantitative characterization of surface topography using spectral analysis</a>,<span data-authors=""> <span class="nowrap">Tevis Jacobs</span>, <span class="nowrap">Till Junge </span>and <span class="nowrap">Lars Pastewka.</span></span></p> <p style="text-align: justify;">[2] <a href="https://arxiv.org/pdf/cond-mat/0502419.pdf">On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion</a>, Bo Persson, O Albohr, U Tartaglino, A I Volokitin and E Tosatti</p>