Abstract
The objectives of this paper are to demonstrate the viability, and to validate, in part, a multiscale method for calculating curvature tensors on measured surface topographies with two different methods of specifying the scale. The curvature tensors are calculated as functions of scale, i.e., size, and position from a regular, orthogonal array of measured heights. Multiscale characterization of curvature is important because, like slope and area, it changes with the scale of observation, or calculation, on irregular surfaces. Curvatures can be indicative of the topographically dependent behavior of a surface and, in turn, curvatures are influenced by the processing and use of the surface. Curvatures of surface topographies have not been well- characterized yet. Curvature has been used for calculations in contact mechanics and for the evaluation of cutting edges. Manufactured surfaces are studied for further validation of the calculation method because they provide certain expectations for curvatures, which depend on scale and the degree of curvature. To study a range of curvatures on manufactured surfaces, square edges are machined and honed, then rounded progressively by mass finishing; additionally, a set of surfaces was made by turning with different feeds. Topographic measurements are made with a scanning laser confocal microscope. The calculations use vectors, normal to the measured surface, which are calculated first, then the eigenvalue problem is solved for the curvature tensor. Plots of principal curvatures as a function of position and scale are presented. Statistical analyses show expected interactions between curvature and these manufacturing processes.
Highlights
The objectives of this paper are to demonstrate the viability, and to validate, in part, how surface topographies can be characterized by curvature tensors calculated from areal topographic measurements of manufactured surfaces
The richness of the multiscale tensor curvature characterizations suggest that they have a strong potential for many kinds of applications in engineering, forensics, paleontology, physical anthropology, and archaeology. The viability of these methods has been demonstrated for multiscale characterization of curvature tensors on measured topographies (z = z(x,y))
The analyses are based on calculating unit-normal vectors to the surface, at three proximal locations, and using an eigenvalue approach to the problem of calculating the curvature tensors
Summary
The objectives of this paper are to demonstrate the viability, and to validate, in part, how surface topographies can be characterized by curvature tensors calculated from areal topographic measurements of manufactured surfaces. Two methods for specifying the scale are studied. The machined, honed, and mass finished surfaces have regular and irregular topographic components. The second-order curvature tensors vary with scale, position, and orientation, i.e., direction. They are calculated from regular arrays of measured surface heights, producing multiscale characterizations that are both position- and orientation-specific. The validation is tested by comparing the results with expectations, based on the machining, honing, and finishing processes
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