On the construction of the curve of a support surface
Abstract Generated abstract
The paper addresses the construction of support surface curves used to characterize the microgeometry of solid surface layers, with attention to differences between isotropic roughness and surfaces with directional machining marks. It argues that, for directionally rough surfaces, the material distribution over height can be modeled by moving a transverse support curve along a longitudinal profilogram, yielding a cylindrical surface representation. Using power-law forms for transverse and longitudinal support curves, the authors derive an equation for this surface and show how it can be used to compute material sectional area at a prescribed level, contrasting the results with a previously proposed multiplication-based method.
Full Text
ON THE CONSTRUCTION OF THE CURVE OF A SUPPORT SURFACE
B. P. MITROFANOV, G. A. RAKHVALOVA, K. N. TSUKUBILINA
(Presented by the scientific seminar of the Department of Strength of Materials)
One of the common methods for evaluating the microgeometry of the surface layer of solids is the construction of the curve of the support surface. This curve characterizes the degree to which the zones of a normal section of a solid body, lying between parallel lines drawn along the profile of the body, are filled with material [1].
The most objective characteristic is the curve of the dependence of the filling with material of the zones of the surface of a solid body formed by planes parallel to its surface. In [2] a method is set forth for experimentally obtaining such a dependence. This method provides for constructing transverse and longitudinal curves of support surfaces. The abscissas of the curves obtained that correspond to one and the same level are multiplied, thereby obtaining the values of the sectional areas. Let us note that the profilogram of the longitudinal surface is obtained by probing the surface with a blade having a width of 2 mm.
It is known that, depending on the method of machining, there is either a definite directionality in the arrangement of projections and in their shape, or an isotropic geometrical structure for all directions.
A clearly expressed directionality is observed in turning, planing, milling, grinding with an abrasive wheel, and certain other methods of mechanical machining. An isotropic distribution of projections is obtained in electropolishing, electro-spark machining, and shot blasting.
The curves of support surfaces with an isotropic distribution of projections are the same for all directions, and the sectional area of the material at a given level will be equal to the square of the abscissa of the curve of the support surface.
For a rough surface with a definite directionality, the dependence of the sectional area on height can be obtained on the basis of the following considerations.
Let us define the distribution of material in the transverse direction by the curve of the support surface. If in the longitudinal direction the rough surface did not change, then it could be represented as bounded by a cylindrical surface, for which the profilogram in the transverse direction would be the generatrix and a line segment the directrix. The actual directrix can be obtained by taking a longitudinal profilogram with the aid of a blade [2]. Then, in order to obtain the dependence of the distribution of material of the rough—
of this layer over height, a cylindrical surface may be used, obtained by moving the transverse curve of the support surface along the longitudinal one (or conversely). Such a surface is shown in Fig. 1. The equations of the curves of the support surfaces in their initial part may be represented in the form [3]
\[ \begin{aligned} x &= b_1 z^{\nu_1},\\ y &= b_2 z^{\nu_2}, \end{aligned} \tag{1} \]
where \(b_1,\ \nu_1,\ b_2,\ \nu_2\) are the parameters of the curves of the support surfaces for the transverse and longitudinal roughnesses. Taking (1) into account, the equation of the cylindrical surface is obtained in the form
\[ z=\frac{1}{b_1^{1/\nu_1}}x^{1/\nu_1} +\frac{1}{b_2^{1/\nu_2}}y^{1/\nu_2}. \tag{2} \]
Fig. 1. Scheme for constructing the support curve.
\(1\) — transverse support curve,
\(2\) — longitudinal support curve
Table
| \(\nu_1\) | \(\nu_2\) | \(\dfrac{A_{r2}}{A_{r1}}\) |
|---|---|---|
| 1 | 1 | 2 |
| 2 | 1 | 3 |
| 3 | 1 | 4 |
| 2 | 2 | 5.87 |
Dependence (2) makes it possible to compute the sectional area of the material at a prescribed level \(z\). In the table given below, the values of the sectional areas of the material computed by formula (2)—\(A_{r1}\)—and by the method proposed in [2]—\(A_{r2}\)—are compared.
REFERENCES
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P. E. Dyachenko, N. N. Tolkacheva, K. N. Goryunov. Determination of the area of actual contact of surfaces. In: Study of Wear of Machine Parts by Means of Radioactive Isotopes. Publishing House of the Academy of Sciences of the USSR, 1957.
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N. B. Demkin. Actual Contact Area of Solid Surfaces. Publishing House of the Academy of Sciences of the USSR, 1962.