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In engineering practice, anuloid surfaces are produced that are subject to strict requirements in terms of accuracy (e.g., in the bearing industry). Roundness is also often measured. The accuracy of the roundness measurement does not only depend on the accuracy of a particular metrological system. The setting of the component on the measuring device can also have a significant influence on the accuracy of the measurement. The axis of the measured surface must be perpendicular to the section in which the roundness is measured. Tilting the measured surface causes a systematic measurement error, which depends on the size of the tilt angle and the parameters of the measured surface. This study presents mathematical relations for the calculation of this error when measuring roundness on an anuloid surface. Mathematical relations for the internal and external surfaces formed by the inner and outer parts of the anuloid are derived herein. There is also a theoretical analysis of the influence of inclination and the parameters of the measured surface on the size of this error. Theoretically calculated values were compared with practically measured values of roundness. The error had a negative effect on the accuracy of the roundness measurement. It affected the roundness value, the second harmonic component, and also the roundness profile itself. [ABSTRACT FROM AUTHOR]

In engineering practice, anuloid surfaces are produced that are subject to strict requirements in terms of accuracy (e.g., in the bearing industry). Roundness is also often measured. The accuracy of the roundness measurement does not only depend on the accuracy of a particular metrological system. The setting of the component on the measuring device can also have a significant influence on the accuracy of the measurement. The axis of the measured surface must be perpendicular to the section in which the roundness is measured. Tilting the measured surface causes a systematic measurement error, which depends on the size of the tilt angle and the parameters of the measured surface. This study presents mathematical relations for the calculation of this error when measuring roundness on an anuloid surface. Mathematical relations for the internal and external surfaces formed by the inner and outer parts of the anuloid are derived herein. There is also a theoretical analysis of the influence of inclination and the parameters of the measured surface on the size of this error. Theoretically calculated values were compared with practically measured values of roundness. The error had a negative effect on the accuracy of the roundness measurement. It affected the roundness value, the second harmonic component, and also the roundness profile itself.

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Copyright of Technisches Messen is the property of De Gruyter and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The modern industry has a constantly increasing demand toward vibration-free rotating machines. Large rotor systems are commonly operated in the subcritical speed range. In the subcritical speed range, the bearings can be a significant source of vibration excitation. The bearing inner ring excites the rotor system at a frequency, which is the rotating frequency multiplied by the waviness component number in the roundness profile of the bearing inner ring. Consequently, subcritical resonance peaks can be observed when the bearing excitation frequency coincides with the rotor natural frequency. The present study utilized a novel compensative 3D grinding to manufacture the bearing installation shaft into a geometry, which minimized the roundness error of the installed bearing inner ring. The decreased roundness error reduced the bearing based excitations to the rotor system. The successful grinding operation was confirmed with roundness measurements. The relevance of the method and the study was proven with rotor dynamic measurements. The results clearly suggest that the compensative 3D grinding reduced the roundness error in both bearing inner rings of the rotor system. The decreased roundness error led to a significantly improved rotor dynamic response, which was observed as reduced amplitudes of the subcritical resonance peaks in the typical operating rotating frequency range. The present study includes also a comparison between the proposed method and a previous steel strip method. The comparison shows that the proposed compensative 3D grinding method produced increasingly better roundness profiles and thus also better rotor dynamic responses. [ABSTRACT FROM AUTHOR]

A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series are known as harmonic analysis. It is a useful way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. This paper deals with the mathematical basics of Fourier series using trigonometric functions. This is the basic for a discrete Fourier transform. It allows transforming the discrete data to the frequency data or vice versa, i.e. transforming the frequency data to the discrete data. The most important part of the article is the application of the Fourier series and the Fourier transform to metrology, specifically on the roundness profile. The mathematical relationships for the practical use of harmonic analysis and the detailed method of determining the actual phase were described. General relationships do not give accurate results, due to the phase shift quadrant. The results of the harmonic analysis were applied graphically by the authors on a concrete example of a roundness profile. The individual harmonic components are shown in the linear and polar graphs as well as the resulting roundness profile. The Fourier analysis knowledge will contribute to a better analysis of the roundness profiles measured on the drawn tubes that will be investigated in the research project. [ABSTRACT FROM AUTHOR]

Fourier model–based multi-probe-error-separation is an error separation method that is useful for precise measurement of the rotation error motion of spindles or roundness profile of rotating machine parts. However, it is not yet widely used, because it suffers from the critical problem of harmonic distortion. This study explores the phenomenon of harmonic distortion in detail and analytically clarifies the principles behind its occurrence. The position at which harmonic distortion occurs can be calculated exactly and easily using principles related not only to probe arrangement angles, but also to measurement conditions such as data length and undulation range of interest. Based on these principles, we propose multi-probe system design guidelines for the effective selection of probe arrangement angles and avoidance of harmonic distortion. The reliability and usability of the proposed design method are verified by simulation tests; stable multi-probe-error-separation can easily be achieved using the proposed design technique. We conduct experimental tests using a special measuring system based on four probes composed of two different probe arrangement sets and verify that this multi-probe-error-separation method can acquire rotation error motion or roundness profile precisely without harmonic distortion. [ABSTRACT FROM AUTHOR]

Recently, a fractal analysis has been used as a useful tool in the characterization of the geometry. Especially in fractal analysis, a fractal interpolation method has been widely used to estimate and analyse experimental geometry. Due to the chaotic nature of the dynamic roundness profile in round shape machining, a more desirable method must be used for the analysis of experimental data which is natural to sequential data. A fractal method which is analysed in this paper focused its objects on the scope of the fractal interpolation and fractal dimension. Also for the calculation of fractal dimension, two methods for computing the fractal dimension have been introduced and discussed. These two methods can calculate the fractal dimensions of dynamic roundness profile according to the number of data points in which the fixed data are generally lower than 120 data points. This fractal analysis shows that it is possible to analysis fractal characteristics of roundness profile that have some different geometric properties. The fractal parameters were calculated and analysed using the measured profile of workpiece after turning. [ABSTRACT FROM AUTHOR]