The effect of a footprint on perceived surface roughness

1986 ◽  
Vol 405 (1829) ◽  
pp. 303-327 ◽  
Keyword(s):  
Surface Roughness ◽  
Weighting Function ◽  
Sampling Interval ◽  
Asymptotic Convergence ◽  
Reference Surface ◽  
Two Dimensional ◽  
Random Surface ◽  
Spectral Density Functions ◽  
Continuous Theory ◽  
Discrete Theory

A two-dimensional homogeneous random surface { y ( X )} is generated from another such surface { z ( X )} by a process of smoothing represented by y ( X ) = ∫ ∞ d u w ( u – X ) z ( u ), where w ( X ) is a deterministic weighting function satisfying certain conditions. The two-dimensional autocorrelation and spectral density functions of the smoothed surface { y ( X )} are calculated in terms of the corresponding functions of the reference surface { z ( X )} and the properties of the ‘footprint’ of the contact w ( X ). When the surfaces are Gaussian, the statistical properties of their peaks and summits are given by the continuous theory of surface roughness. If only sampled values of the surface height are available, there is a corresponding discrete theory. Provided that the discrete sampling interval is small enough, profile statistics calculated by the discrete theory should approach asymptotically those calculated by the continuous theory, but it is known that such asymptotic convergence may not occur in practice. For a smoothed surface { y ( X )} which is generated from a reference surface { z ( X )} by a ‘good’ footprint of finite area, it is shown in this paper that the expected asymptotic convergence does occur always, even if the reference surface is ideally white. For a footprint to be a good footprint, w ( X ) must be continuous and smooth enough that it can be differentiated twice everywhere, including at its edges. Sample calculations for three footprints, two of which are good footprints, illustrate the theory.

A unified theory of surface roughness

1984 ◽  
Vol 393 (1804) ◽  
pp. 133-157 ◽  
Keyword(s):  
Surface Roughness ◽  
Rough Surface ◽  
Random Noise ◽  
Sampling Interval ◽  
The Other ◽  
Real Surface ◽  
Unified Theory ◽  
Two Dimensional ◽  
Minor Degree ◽  
A Minor

The properties of the peaks and summits of a rough surface are predicted on the assumption that the surface is two-dimensional random noise. The important result is that, in non-dimensional form, the answers depend only to a minor degree on the parameters describing the surface or on the sampling interval used: on the other hand the absolute values are strongly dependent on the sampling interval. Experimental results on a real surface agree remarkably well with the predictions.

scholarly journals Investigation of Two-Dimensional Ultrasonic Surface Burnishing Process on 7075-T6 Aluminum

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Zhenyu Zhou ◽  
Qiuyang Zheng ◽  
Cong Ding ◽  
Guanglei Yu ◽  
Guangjian Peng ◽  
...  
Keyword(s):  
Surface Roughness ◽  
Parameter Optimization ◽  
Wear Mechanism ◽  
Feed Rate ◽  
Sample Surface ◽  
Original Sample ◽  
Two Dimensional ◽  
Delamination Wear ◽  
Treated Sample ◽  
Burnishing Process

AbstractA novel two-dimensional ultrasonic surface burnishing process (2D-USBP) is proposed. 7075-T6 aluminum samples are processed by a custom-designed 2D-USBP setup. Parameter optimization of 2D-USBP is conducted to determine the best processing strategy of 7075-T6 aluminum. A uniform design method is utilized to optimize the 2D-USBP process. U13(133) and U7(72) tables are established to conduct parameter optimization. Burnishing depth, spindle speed, and feed rate are taken as the control parameters. The surface roughness and Vickers hardness are taken as the evaluation indicators. It establishes the active control models for surface quality. Dry wear tests are conducted to compare the wear-resistance of the 2D-USBP treated sample and the original sample. Results show that the machining quality of 2D-USBP is best under 0.24 mm burnishing depth, 5000 r/min spindle speed, and 25 mm/min feed rate. The surface roughness Sa of the sample is reduced from 2517.758 to 50.878 nm, and the hardness of the sample surface is improved from 167 to 252 HV. Under the lower load, the wear mechanism of the 2D-USBP treated sample is mainly abrasive wear accompanied by delamination wear, while the wear mechanism of the original sample is mainly delamination wear. Under the higher load, the accumulation of frictional heat on the sample surface transforms the wear mechanisms of the original and the 2D-USBP treated samples into thermal wear.

Acoustic receptivity and evolution of two-dimensional and oblique disturbances in a Blasius boundary layer

2001 ◽  
Vol 432 ◽  
pp. 69-90 ◽  
Author(s):  
RUDOLPH A. KING ◽  
KENNETH S. BREUER
Keyword(s):  
Boundary Layer ◽  
Surface Roughness ◽  
Acoustic Wave ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Surface Waviness ◽  
S Wave ◽  
Boundary Layer Instability ◽  
Blasius Boundary Layer ◽  
Tollmien Schlichting

An experimental investigation was conducted to examine acoustic receptivity and subsequent boundary-layer instability evolution for a Blasius boundary layer formed on a flat plate in the presence of two-dimensional and oblique (three-dimensional) surface waviness. The effect of the non-localized surface roughness geometry and acoustic wave amplitude on the receptivity process was explored. The surface roughness had a well-defined wavenumber spectrum with fundamental wavenumber kw. A planar downstream-travelling acoustic wave was created to temporally excite the flow near the resonance frequency of an unstable eigenmode corresponding to kts = kw. The range of acoustic forcing levels, ε, and roughness heights, Δh, examined resulted in a linear dependence of receptivity coefficients; however, the larger values of the forcing combination εΔh resulted in subsequent nonlinear development of the Tollmien–Schlichting (T–S) wave. This study provides the first experimental evidence of a marked increase in the receptivity coefficient with increasing obliqueness of the surface waviness in excellent agreement with theory. Detuning of the two-dimensional and oblique disturbances was investigated by varying the streamwise wall-roughness wavenumber αw and measuring the T–S response. For the configuration where laminar-to-turbulent breakdown occurred, the breakdown process was found to be dominated by energy at the fundamental and harmonic frequencies, indicative of K-type breakdown.

An Evaluation of the Average Flow Model [1] for Surface Roughness Effects in Lubrication

1980 ◽  
Vol 102 (3) ◽  
pp. 360-366 ◽  
Author(s):  
J. L. Teale ◽  
A. O. Lebeck
Keyword(s):  
Surface Roughness ◽  
Flow Model ◽  
Two Dimensional ◽  
Roughness Effects ◽  
Important Conclusion ◽  
Average Flow ◽  
One Dimensional ◽  
Dimensional Flow ◽  
Two Dimensional Flow ◽  
Average Flow Model

The average flow model presented by Patir and Cheng [1] is evaluated. First, it is shown that the choice of grid used in the average flow model influences the results. The results presented are different from those given by Patir and Cheng. Second, it is shown that the introduction of two-dimensional flow greatly reduces the effect of roughness on flow. Results based on one-dimensional flow cannot be relied upon for two-dimensional problems. Finally, some average flow factors are given for truncated rough surfaces. These can be applied to partially worn surfaces. The most important conclusion reached is that an even closer examination of the average flow concept is needed before the results can be applied with confidence to lubrication problems.

On the effects of two-dimensional Reynolds roughness in hydrodynamic lubrication

1978 ◽  
Vol 364 (1716) ◽  
pp. 89-106 ◽  
Keyword(s):  
Surface Roughness ◽  
Numerical Computation ◽  
Autocorrelation Function ◽  
Reynolds Equation ◽  
Hydrodynamic Lubrication ◽  
The Other ◽  
Roughness Height ◽  
Two Dimensional ◽  
Roughness Effects ◽  
Dimensional Surface

The hydrodynamic lubrication of rough surfaces is analysed with the Reynolds equation, whose application requires the roughness spacing to be large, and the roughness height to be small, compared with the thick­ness of the fluid film. The general two-dimensional surface roughness is considered, and results applicable to any roughness structure are obtained. It is revealed analytically that two types of term contribute to roughness effects: one depends on the shape of the autocorrelation function and the other does not. The former contribution was neglected by previous workers. The numerical computation of an example shows that these two contributions are comparable in magnitude.

Field Computation for Two-Dimensional Array Transducers with Limited Diffraction Array Beams

2005 ◽  
Vol 27 (4) ◽  
pp. 237-255 ◽  
Author(s):  
Jian-Yu Lu ◽  
Jiqi Cheng
Keyword(s):  
Continuous Wave ◽  
Weighting Function ◽  
Near Field ◽  
Bessel Beam ◽  
Two Dimensional ◽  
Axially Symmetric ◽  
Computationally Efficient ◽  
Annular Array ◽  
Array Transducer ◽  
Analytical Expressions

A method is developed for calculating fields produced with a two-dimensional (2D) array transducer. This method decomposes an arbitrary 2D aperture weighting function into a set of limited diffraction array beams. Using the analytical expressions of limited diffraction beams, arbitrary continuous wave (cw) or pulse wave (pw) fields of 2D arrays can be obtained with a simple superposition of these beams. In addition, this method can be simplified and applied to a 1D array transducer of a finite or infinite elevation height. For beams produced with axially symmetric aperture weighting functions, this method can be reduced to the Fourier-Bessel method studied previously where an annular array transducer can be used. The advantage of the method is that it is accurate and computationally efficient, especially in regions that are not far from the surface of the transducer (near field), where it is important for medical imaging. Both computer simulations and a synthetic array experiment are carried out to verify the method. Results (Bessel beam, focused Gaussian beam, X wave and asymmetric array beams) show that the method is accurate as compared to that using the Rayleigh-Sommerfeld diffraction formula and agrees well with the experiment.

scholarly journals Effects of Surface Roughness, Freestream Turbulence, and Incidence Angle on the Performance of a 2-D Compressor Cascade

1990 ◽  
Author(s):  
W. C. Elrod ◽  
P. I. King ◽  
E. M. Poniatowski
Keyword(s):  
Surface Roughness ◽  
Total Pressure ◽  
Incidence Angle ◽  
Two Dimensional ◽  
Freestream Turbulence ◽  
Compressor Cascade ◽  
Suction Surface ◽  
Linear Cascade ◽  
Pressure Loss Coefficient ◽  
Total Pressure Loss Coefficient

The effects of surface roughness, freestream turbulence, and incidence angle on the performance of a two-dimensional compressor cascade were investigated. The test section consisted of seven NACA 65-A506 airfoils arranged in a linear cascade. Four different surface roughness conditions were applied to the first 25 percent chord on the suction surface of each of the five middle blades in the cascade. Freestream turbulence levels of approximately one and seven percent were used. Incidence angles of −3, zero and +3 degrees were investigated. Of the three parameters tested, freestream turbulence exerted the largest influence on blade performance. The total pressure loss coefficient increased with increased roughness and was reduced for large turbulence. Changes in flow incidence had a lesser effect on the performance of the blade.

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