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1

Separation Of Non-Periodic And Periodic 2D Profile Features Using B-Spline Functions

Janecki D., Cedro L., Zwierzchowski J.

Metrology and Measurement Systems

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2015

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Vol. 22, nr 2

289--302

EN

The form, waviness and roughness components of a measured profile are separated by means of digital filters. The aim of analysis was to develop an algorithm for one-dimensional filtering of profiles using approximation by means of B-splines. The theory of B-spline functions introduced by Schoenberg and extended by Unser et al. was used. Unlike the spline filter proposed by Krystek, which is described in ISO standards, the algorithm does not take into account the bending energy of a filtered profile in the functional whose minimization is the principle of the filter. Appropriate smoothness of a filtered profile is achieved by selecting an appropriate distance between nodes of the spline function. In this paper, we determine the Fourier transforms of the filter impulse response at different impulse positions, with respect to the nodes. We show that the filter cutoff length is equal to half of the node-to-node distance. The inclination of the filter frequency characteristic in the transition band can be adjusted by selecting an appropriate degree of the B-spline function. The paper includes examples of separation of 2D roughness, as well as separation of form and waviness of roundness profiles.

2

A problem of optimal cylindricity profile matching

Janecki D., Zwierzchowski J., Cedro L.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2015

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Vol. 63, nr 3

771--779

EN

The bird-cage strategy used for measuring cylindricity is reported to be the most effective, as it provides the most detailed information about an analyzed object. The average values of profiles measured with the cross-section and the generatrix strategies may differ slightly, yet this may result from some design imperfections of the measurement instruments used. In this study, the problem of optimal profile matching is formulated and solved. As a result, the differences between the values of the registered profiles at the points of intersection of the scanning trajectories can be minimized.

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Metoda filtracji profilu walcowości zaobserwowanego metodą linii śrubowej

Janecki D., Zwierzchowski J.

Pomiary Automatyka Kontrola

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2013

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R. 59, nr 1

59--62

PL

W prezentowanej pracy opracowano sposób filtracji profilu walcowości w celu odseparowania kształtu badanego przedmiotu od pozostałych składowych geometrii powierzchni. Do oceny profilu walcowości zastosowano metodę linii śrubowej, dzięki której zaobserwowano powierzchnię walca. Metoda linii śrubowej, chociaż nie jest ujęta w normach to jest cytowana w literaturze. W rozważanej w pracy metodzie profil zaobserwowany składa się z trzech fragmentów: dwóch profili w przekrojach poprzecznych na obu brzegach analizowanej powierzchni i jednego profilu o kształcie linii śrubowej. W pracy zastosowano podejście funkcjonałowe, a dokładnie jego rozwinięcie i uogólnienie przedstawione w pracy [5]. Dodatkowo, aby uzyskać ciągłość profilu przefiltrowanego w punktach styku dwóch trajektorii w przekrojach poprzecznych i trajektorii linii śrubowej dołożono dodatkowe warunki równościowe. Profil przefiltrowany aproksymowany jest za pomocą sześciennych funkcji sklejanych. W pracy przedstawiono opis matematyczny zastosowanej metody filtracji oraz wyniki działania algorytmu filtracji dla rzeczywistych zaobserwowanych profilach walcowości zmierzonych metodą linii śrubowej.

XX

The aim of the work was to develop an algorithm for filtering a cylindricity profile measured by means of the helical line method. Although the helical line method is not included in the relevant standards, it is quoted in the literature on the subject. The method analyzed here assumes that a measured profile consists of three fragments: two cross-sectional profiles on both edges of the analyzed surface and one helical profile. A functional approach was applied to filtering the profile. To ensure continuity of the filtered profile at the point of contact of the two cross-sectional trajectories with the helical trajectory, additional equality conditions were used. The filtered profile was approximated by means of cubic spline functions. The work includes the mathematical description of the filtration method and the results of the filtration algorithm used for real cylindricity profiles measured with the helical line method. In the study there was also considered an important element affecting the filter properties. It is good selection of the distance between the nodes of approximating function.

4

Podstawy teoretyczne oceny profili walcowości obrotowych części maszyn metodą klatki Część III: Wyznaczanie linii środkowej walca zaobserwowanego

Janecki D., Zwierzchowski J.

Pomiary Automatyka Kontrola

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2010

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R. 56, nr 2

205-208

PL

W pracy zaproponowano algorytm wyznaczania linii środkowej walca zaobserwowanego metodą przekrojów wzdłużnych i metodą klatki pomiaru walcowości. W celu wyeliminowania wpływu składowych falistości zastosowano filtrację ocen punktów linii środkowej wykorzystującą podejście funkcjonałowe.

EN

The median line of a cylinder measured with the cross-section method is determined on the basis of the centre points of the mean circles of several subsequent cross-sections roundness profiles. Basically, the same principles could be used in the generatrix method. There may be, however, a large scatter of results due to a small number of measurement points with predetermined height coordinates and the occurrence of a waviness component in the cylindricity profile. The method for determination of the median line discussed in this paper involves filtering the measured points by applying the functional approach, as suggested by Krystek in Refs. [2, 3]. It is assumed that the coordinates of the measured median line points minimise the functional (13). One component of the functional assures appropriate approximation of the centres of the mean circles by means of the median line, and the other, termed the cost of curve bending, assures suitable smoothness of the median line eliminating the effect of the waviness components. A similar concept was employed in the bird-cage method, which is actually the combination of the cross-section and generatrix method. In this method, the median line points defined with the cross-section method are taken into account with appropriately greater weight than the points measured with the generatrix method (17). In the examples provided, the median line of a polished roller with a strong waviness component was determined for different cylindricity measurement methods.

5

Podstawy teoretyczne oceny profili walcowości obrotowych części maszyn metodą klatki Część II: Metoda filtracji wykorzystująca podejście funkcjonałowe

Janecki D., Zwierzchowski J.

Pomiary Automatyka Kontrola

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2009

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R. 55, nr 12

1017-1022

PL

W celu rozdzielenia składowych kształtu od składowych falistości i chropowatości stosuje się filtry cyfrowe dolnoprzepustowe. W pracy opracowano metodę filtracji profili walcowości zaobserwowanych metodą "klatki". Wykorzystuje ona tzw. podejście funkcjonałowe. Pozwala to na eliminację efektu brzegowego oraz zapewnia, że w punktach przecięcia trajektorii skanowania wartości promienia przefiltrowanych profili okrągłości i prostoliniowości są równe.

EN

It is assumed that roughness, waviness and form constitute the geometrical surface structure. The components of form with the longest wavelengths can now be extracted using digital low-pass filters. In the case of cylindrical surfaces, the assessment of form requires that a profile be filtered with the aim of eliminating the waviness and roughness constituents. The filtration process selected for a profile registered with the bird-cage method should provide filtration both roundness and straightness profiles. Additionally, it is logical requiring the values of the radii of the filtered roundness and straightness profiles be equal at the points of intersection of the scanning trajectories. The profiles under study were filtered using the so called functional approach, which was introduced to surface metrology by Krystek [2, 3] and further developed by other re-searchers [7]. According to the approach, a filtered profile is modeled as a spline, which assumes a required smooth form due to local stresses proportional to the value of a deviation. The major advantage of the functional approach is the elimination of the boundary effect, which occurs during filtration of straightness profiles. Although it is relatively simple to directly generalize the functional approach to the filtration of a cylindricity profile registered with the bird-cage method (26), the problem is very difficult to solve because of its dimension. Therefore, simplified approach has been applied. First, we filtered each profile separately using the pre-determined methods for roundness and straightness profiles. Subsequently, we filtered the profiles by introducing additional equality conditions (33), which assured profile continuity at the points of intersection of the cylinder scanning trajectories. The proposed two-step filtration algorithm can be combined with the procedure of profile matching discussed in Ref. [1].

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Podstawy teoretyczne oceny profili walcowości obrotowych części maszyn metodą klatki Część I: Zagadnienie optymalnego dopasowania profili

Janecki D., Zwierzchowski J.

Pomiary Automatyka Kontrola

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2009

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R. 55, nr 9

715-722

PL

Metoda klatki pomiaru profili walcowości dostarcza najwięcej informacji o mierzonym przedmiocie. Jednak w wyniku pewnych niedokładności przyrządów pomiarowych wartości średnie profili zaobserwowanych metodą przekrojów poprzecznych i wzdłużnych mogą się nieznacznie różnić. W pracy sformułowano i rozwiązano zagadnienie optymalnego dopasowania profili, co pozwala zminimalizować różnice wartości promienia profilu w punktach przecięcia trajektorii skanowania przedmiotu.

EN

The bird-cage method applied to measurement the cylindricity of rotary objects combines the principles of the cross-section and the generatrix methods, see Fig 1. Cylindricity measurement results obtained by means of the bird-cage method, show that the values of the profile radius at the points of intersection of scanning trajectories at the cross and longitudinal sections are slightly different. The difference may be due to the occurrence of measurement noise and instrument vibrations or the design imperfections of the sensor system. Note that the measurement conditions for the cross-section method are different from those for the generatrix method. This causes different distribution of forces acting on the sensor tip. As a result, the profile achieved by the cross-section method can be slightly shifted in relation to the profile achieved by the generatrix method (see Fig. 2 and Table 1). In order to reduce these errors a problem of optimal profile matching is formulated and solved. Values of the profiles in each section are shifted in such a way that the difference in the radii at the points of intersection of the scanning trajectories is the smallest as possible. To do this, a square index of profile matching (20) is defined and the required values of profile shifts (16)-(19) are derived from necessary conditions of optimality. The observed profiles are modified in accordance with the formulae (25)-(26). Additionally, it is also shown that it is possible to correct the difference in the rotation axis position during roundness measurements with the spindle in rotary motion and during straightness measurements with the spindle at standstill. The effects of the application of the profile matching algorithms are analyzed basing on the measurements of two rollers with a diameter of 52 mm and a height of 100 mm, each. One specimen was polished and the other was grounded. Results of the experiments show that due to the optimal profile matching, the root-mean-square of the difference in the radii at the points of intersection of the scanning trajectories may be decreased from several to several dozen times depending on the level of the waviness component (see Figures in Section 3).