Dariusz Janecki

Sphericity measurements by the radial method: I. Mathematical fundamentals

Traditionally, form errors of spherical components have been assessed on the basis of roundness profiles measured in several randomly selected cross-sections. However, such evaluation is superficial, especially if there are significant local irregularities. A new concept was thus developed at the Kielce University of Technology to enable measurement of spherical specimens along some predefined paths so that the surface is densely covered with a grid of points. This approach assumes that measurements can be performed using a typical radial roundness measuring instrument equipped with a special mechanism for controlled positioning of a measured element. This work discusses the assumptions of the new concept and describes a mathematical model of sphericity measurement by the radial method.

A method for determining the median line of measured cylindrical and conical surfaces

The paper presents a novel method for the determination of the median line of the cylindrical and conical surfaces. This method can be incorporated into virtually any cylindricity measurement strategy, including the bird-cage strategy and the helical line strategy. In the study, a median line was determined by minimizing the functional made up of two components. The form of the first component results from the classic definition of the median line provided in the corresponding standard. The other, termed the bending energy, is responsible for ensuring appropriate smoothness of the median line. In order to solve this variational problem, the median line was approximated by means of linear combination of cubic B-spline functions. A simulation and experiments were conducted to establish the suitability of the algorithm developed for the determination of the median line using the helical-line and the cross-section measurement strategy.

Sphericity measurements by the radial method: II. Experimental verification

The new concept of sphericity measurements enables accurate measurement of spherical elements. This concept assumes that measurements can be performed using a typical radial roundness measuring instrument equipped with a special mechanism for controlled positioning of a measured element. The concept requires solving numerous theoretical problems, and this was described in the previous companion paper. This second paper discusses the measuring equipment and the results of the experimental verification of the concept.

Problems of Mathematical Modelling of Rotary Elements

Rotary elements constitute a large and significant group of machine parts. Such parts are used in various branches of industry, for example in bearing, automotive, paper, shipping or power industry. Most of rotary machine parts are cylindrical ones but sometimes other types of elements are used, as spheres, cones or barrel-shaped and saddle-shaped elements. Measurements of form deviations of such elements require performing a number of mathematical operations on measurement data, including calculation of reference features. The basis for such operations is to apply an appropriate mathematical model of the surface. Scientific literature and standards refer mainly to measurements of form errors of cylindrical shapes. Apart from the cylindrical surfaces, this paper discusses also the problem of mathematical modelling of cones, spheres and barrel-shaped elements.