RADA PROGRAMOWA / PROGRAM COUNCIL


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2 RADA PROGRAMOWA / PROGRAM COUNCIL PRZEWODNICZĄCY / CHAIRMAN: prof. dr hab. dr h.c. ult. Czesław CEMPEL Politechnika Poznańska REDAKTOR NACZELNY / CHIEF EDITOR: prof. dr hab. inż. Ryszard MICHALSKI UWM w Olsztynie CZŁONKOWIE / MEMBERS: prof. dr hab. inż. Jan ADAMCZYK AGH w Krakowie prof. Francesco AYMERICH University of Cagliari Italy prof. Jérôe ANTONI University of Technology of Copiegne France prof. dr. Ioannis ANTONIADIS National Technical University Of Athens Greece dr inż. Roan BARCZEWSKI Politechnika Poznańska prof. dr hab. inż. Walter BARTELMUS Politechnika Wrocławska prof. dr hab. inż. Wojciech BATKO AGH w Krakowie prof. dr hab. inż. Ada CHARCHALIS Akadeia Morska w Gdyni prof. Li CHENG The Hong Kong Polytechnic University China prof. dr hab. inż. Wojciech CHOLEWA Politechnika Śląska prof. dr hab. inż. Zbigniew DĄBROWSKI Politechnika Warszawska Prof. Charles FARRAR Los Alaos National Laboratory USA prof. Wiktor FRID Royal Institute of Technology in Stockhol Sweden dr inż. Toasz GAŁKA Instytut Energetyki w Warszawie prof. Len GELMAN Cranfield University England prof. Mohaed HADDAR National School of Engineers of Sfax Tunisia prof. dr hab. inż. Jan KICIŃSKI IMP w Gdańsku prof. dr hab. inż. Daniel KUJAWSKI Western Michigan University USA prof. Graee MANSON University of Sheffield UK prof. dr hab. Wojciech MOCZULSKI Politechnika Śląska prof. dr hab. inż. Stanisław RADKOWSKI Politechnika Warszawska prof. Bob RANDALL University of South Wales Australia prof. dr Raj B. K. N. RAO President COMADEM International England prof. Massio RUZZENE Georgia Institute of Technology USA prof. Vasily S. SHEVCHENKO BSSR Acadey of Sciences Mińsk Belarus prof. Menad SIDAHMED University of Technology Copiegne France Prof. Tadeusz STEPINSKI Uppsala University  Sweden prof. Wiesław TRĄMPCZYŃSKI Politechnika Świętokrzyska prof. dr hab. inż. Tadeusz UHL AGH w Krakowie prof. Vitalijus VOLKOVAS Kaunas University of Technology Lithuania prof. Keith WORDEN University of Sheffield UK prof. dr hab. inż. Andrzej WILK Politechnika Śląska dr Gajraj Singh YADAVA Indian Institute of Technology India prof. dr hab. inż. Bogdan ŻÓŁTOWSKI UTP w Bydgoszczy WYDAWCA: Polskie Towarzystwo Diagnostyki Technicznej ul. Narbutta Warszawa REDAKTOR NACZELNY: prof. dr hab. inż. Ryszard MICHALSKI SEKRETARZ REDAKCJI: dr inż. Sławoir WIERZBICKI CZŁONKOWIE KOMITETU REDAKCYJNEGO: dr inż. Krzysztof LIGIER dr inż. Paweł MIKOŁAJCZAK ADRES REDAKCJI: Redakcja Diagnostyki Katedra Budowy, Eksploatacji Pojazdów i Maszyn UWM w Olsztynie ul. Oczapowskiego, 736 Olsztyn, Poland tel.: , fax: eail: KONTO PTDT: Bank PEKAO SA O/Warszawa nr konta: NAKŁAD: 5 egzeplarzy
3 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ Spis treści / Contents Piotr KOHUT, Krzysztof HOLAK, Tadeusz UHL AGH University of Science and Technology in Kraków...3 Prototype of the vision syste for deflection easureents Prototyp systeu wizyjnego do poiarów ugięć Paweł MIKOŁAJCZAK UWM Olsztyn...3 Prediction of changes in the technical condition using discriinant analysis Predykcja paraetrów diagnostycznych z wykorzystanie analizy dyskryinacyjnej Mateusz MARZEC, Tadeusz UHL, Toasz BARSZCZ AGH University of Science and Technology in Kraków... Application of UML odelling for analysis of safety integrity level in railway traffic control systes Zastosowanie języka UML do badania poziou nienaruszalności bezpieczeństwa systeów sterowania ruche kolejowy Czesław CEMPEL Poznan University of Technology...7 Singular values of sypto observation atrix of a syste in operation as indicators of syste daage Wartości szczególne syptoowej acierzy obserwacji eksploatowanego systeu echanicznego jako wskaźniki zużycia Dariusz BRODA, Toasz BARSZCZ AGH University of Science and Technology in Kraków...39 Description of varying operational conditions in wind turbines Opis zienności warunków operacyjnych turbin wiatrowych Jędrzej MĄCZAK Warsaw University of Technology...47 Application of the local eshing plane in detecting assebly and anufacturing errors of gears Wykorzystanie lokalnej płaszczyzny przyporu w wykrywaniu błędów wykonania i ontażu przekładni zębatych Ada CHARCHALIS, Mirosław DERESZEWSKI Gdynia Maritie University...53 Monitoring of instantaneous angular speed of the crankshaft for control of the ship engine perforance changes Monitorowanie chwilowej prędkości kątowej wału korbowego w celu kontroli zian jakości pracy silnika okrętowego Maciej KLEMM, Maciej TABASZEWSKI Poznan University of Technology...59 Agent approach in achine diagnosis Podejście agentowe w diagnostyce aszyn Bogdan POJAWA Polish Naval Acadey...65 Tests of LM5 naval gas turbine with the objective of deterining its operating characteristics Badania okrętowego turbinowego silnika spalinowego LM5 w aspekcie wyznaczenia charakterystyk obrotowych Stanisław RADKOWSKI, Robert GUMIŃSKI Warsaw University of Technology...7 Use of Hough s transfor for designing the decisionaking odule of a proactive operations syste Wykorzystanie transforacji Hough a w projektowaniu odułu decyzyjnego systeu proaktywnej eksploatacji Warto przeczytać / Worth to read...77 Lista recenzentów artykułów opublikowanych w czasopiśie Diagnostyka w roku...78
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5 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents 3 PROTOTYPE OF THE VISION SYSTEM FOR DEFLECTION MEASUREMENTS Piotr KOHUT, Krzysztof HOLAK, Tadeusz UHL AGH University of Science and Technology, Departent of Robotics and Mechatronics Al. A. Mickiewicza 3, 359 Krakow, tel Suary The visionbased ethod of civil engineering construction s inplane deflection easureents was developed. Displaceent field of the analyzed structure resulting fro load was coputed by eans of digital iage correlation coefficient. The application of hoography apping enabled the deflection curve to be coputed fro two iages of the construction acquired fro two distinct points in space. The shape filter and rectangular arker detector were ipleented to provide higher level of autoation of the ethod. There are discussed developed ethodology, created architecture of software tool as well as experiental results obtained fro tests ade on lab setups. Keywords: digital iage correlation, iage registration, vision systes, deflection easureent. PROTOTYP SYSTEMU WIZYJNEGO DO POMIARÓW UGIĘĆ Streszczenie W pracy przedstawiono opracowany prototyp systeu wizyjnego do poiarów dwuwyiarowych deforacji konstrukcji. Pole przeieszczeń analizowanego obiektu, powstałe pod wpływe działających obciążeń, wyznaczono przy poocy znoralizowanego współczynnika korelacji. Zastosowanie przekształcenia hoograficznego uożliwiło wykonanie poiarów ugięcia konstrukcji na podstawie jej dwóch obrazów zarejestrowanych z dwóch różnych punktów przestrzeni. Zaipleentowany filtr kształtów oraz detektor znaczników referencyjnych uożliwił zwiększenie autoatyzacji procesu poiarowego. W artykule przedstawiono opracowaną etodykę, architekturę stworzonego oprograowania oraz wyniki testów eksperyentalnych systeu na stanowisku laboratoryjny. Słowa kluczowe: korelacja obrazów, nakładanie obrazów, systey wizyjne, poiary ugięcia.. INTRODUCTION Structural Health Monitoring ethods can be divided into two ain categories: local ethods and global ethods. The latter are applied if a global change in the geoetry or otion of a structure under the loads can be observed. On the other hand, local ethods ake use of the physical phenoena acting locally within a sall area of the construction. Visionbased techniques belong to the group of contactless global SHM ethods which enable global easureents of static deforations as well as dynaic processes to be carried out. They allow daage detection to be perfored by eans of a change in the geoetry of a structure analysis, such as deflection curve or ode shapes of vibrations. In diagnostics of civil engineering structures, displaceents easureents are the ajor aspect of constructions static states and dynaic characteristics evaluation. In this area, the analysis of deflection shapes of structures has becoe ore significant and accurate than other ethods of the analysis [6]. Nowadays, the increase of availability of vision systes for the easureent of otion and threediensional geoetry of objects is noticed on the world arkets. However their nuber is still sall in the field of easureent of deforations and lowfrequency vibration of structures. In this paper, the developed prototype of vision based syste, for inplane easureent of civil engineering structures displaceent fields is presented. The syste provides onitoring of static states of civil engineering constructions such as displaceents, deflections and deforations. The syste consists of one or ore high resolution digital caeras ounted on a head or on portable tripods, the software ebedded in MSWindows operating syste, lighting syste and the set of special arkers placed on the construction. Calibration patterns which enable coputation of the scale coefficient and lens distortions are also parts of the syste. Deflection curve is obtained fro two iages of the construction: reference one and the one acquired after application of a load. The principle of the
6 4 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents ethod is calculation of object s points displaceent by eans of noralized cross correlation coefficient. Iage registration techniques were introduced in order to increase flexibility and accuracy of the ethod. Perspective distortions of the construction s iage are reoved by eans of hoography apping, which allows two photographs of the object to be taken fro two distinct points in space. In order to calculate correspondences between atching features on both iages, new technique of arkers detection and shape filtering, as well as subpixel corner detection are introduced.. VISION MEASUREMENT SYSTEMS CURRENTLY AVAILABLE ON THE MARKET Nowadays, there are a lot of optical systes for threediensional structure and otion easureents. These systes allow for considerable shortening of a tie needed for carrying out phenoena analysis, as well as getting threediensional structures of inspected objects. In this paper, optical easuring systes available on the arket were listed, their ost iportant features were described and they were divided into groups on account of the principle based on which they operate. The ost iportant copanies offering vision easureent systes existing at present on the arket are: Correlated Solution, GOM, LIMESS Messtechnik & Software, Dantec Dynaics, Metris (Krypton 9) [74]. There are following systes for the easureent of three diensional otion available: PONTOS/TRITOP (GOM), Vic D/Vic 3D (Correlated Solution/LIMESS). Accuracies of these systes vary in the range fro /5 to / pixels, whereas sizes of easured object depend on the configuration of the syste: fro a few up to. Available frequencies of iages acquisition are in the range fro a few to a few hundred thousand fraes per second, depending on the used resolution and the field of view. For exaple in the PONTOS syste, carrying out easureents requires placing special arkers on exained structures. An acquisition of vibration of the structures with the help of the pair of highfrequency caeras is the next step of the easureent. A course of the displaceent of particular points of the object is the analysis result. The systes PONTOS ARAMIS/ (GOM), Vic D/Vic 3 D (Correlated Solution/LIMESS) or Q 4  II (Dantec Dynaics) can be applied to easureents of displaceents, deforations and stresses of the structure. ARAMIS (GOM) is an exaple of systes of this type, which enable analysis of the deforation and stresses of objects with coplicated geoetry, defored under the load. The syste uses the ethod of threediensional correlation and high resolution digital caeras. A stochastic or regular pattern is placed on the inspected structure. The object under the load is observed by one or ore digital caeras. The visualization of threediensional deforations of the structure is obtained as a result of the analysis. Systes ATOS, TRITOP (GOM), and 3 DCa (Correlated Solution/LIMESS) are used in the reverse engineering for the easureent of the threediensional shape of objects. ATOS is the active vision syste, which eans that a shape of the light pattern projected onto the exained surface is analyzed. The scanner works on the principle of a triangulation, two caeras are observing courses of stripes on the easured detail which enables threediensional coordinates of all points of interest to be deterined. Measureents requires placing round arkers with a known diaeter on the analyzed structure. In the next step, the prepared object is registered by a photograetric caera fro a few different points of view. The software allows for finding coordinates of all arkers. Threediensional easureents of geoetry using three caeras are carried out by the following devices: Krypton of K series (Metris) and OPTIGO (Cognitens). Krypton syste uses 3 linear caeras, additionally it has a possibility of tracking the arkers put on the structure, whereas OPTIGO uses 3 atrix caeras for the easureent of a geoetry. Active interferential techniques are offered by systes: ESPI SD3 / SDS systes (GOM); Shearwin NT (Correlated Solution /LIMESS), or Q8 (Dantec Dynaics) can be applied for the easureent of the deforation. Measureent ethods are based on phenoena of interference and they concern little objects [74]. 3. MEASUREMENT METHOD The ethod of noncontact easureent of civil engineering constructions inplane deflection consists of three ajor steps [4]. In the first step, a rectification [5],[6] of iages acquired fro distinct points of view, not coincident with the reference one which can be chosen by an user, is perfored by eans of hoography atrix H. In the following step, the deflection of a construction is calculated using the noralized cross correlation coefficient (NCC). Subpixel feature detection techniques were introduced in order to increase the accuracy of the easureent. In the final step, the scale coefficient is coputed with the help of a circular intensity pattern with a known diaeter. The developed algorith is presented on figure. Iage registration [3] is a ethod of stitching two or ore iages taken at different ties, fro distinct points of view or by using different iaging devices. In this work, hoography apping was introduced to align two iages acquired fro distinct points of view. Iage rectification [4],[5],[6] is a process of projective distortions reduction by eans of hoography transforation. Four pairs of coplanar corresponding points are sufficient for the
7 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents 5 coputation of atrix H if none three of the are collinear. Fig.. Developed algorith of the inplane deflection easureent The set of corresponding points used for the hoography coputation consist of vertices of rectangular arkers, which are placed on the structure. Markers ust be coplanar with the plane of the construction and can t change their position as it defors [4]. Coordinates of the corresponding points on both iages are calculated by autoatic corner detector. In the first step, rectangles are detected on iages by eans of contour processing and shape filtering ethods. Exact positions of each of arkers vertices are deterined by the subpixel iproveent of the detector. Vertices positions are expressed in polar coordinates in the coordinate frae with the origin at the center of ass of arkers set. Markers are sorted by coparison of their corners polar angles and distances fro the origin. As the alternative for the aforeentioned ethod of feature atching, iage patch correspondence atching based on binary codes recognition has been developed. When the hoography apping between two iages of a construction is calculated, projective distortions of the particular plane of the object are reoved fro the iage. The noralized cross correlation coefficient [5,6] (NCC) is applied for the coputation of the inplane displaceent field. In the developed ethod, the reference iage of unloaded construction is divided into intensity patterns whose position are coputed by eans of the NCC coefficient. The displaceent vector for each of the easureent points is coputed as a difference between positions of the pattern on two iages of the construction: taken before and after application of a load. [4]. In order to express a deflection curve in etric units, calibration of the syste is necessary. It is perfored by a circular intensity pattern with a known diaeter. Optionally, full caera calibration is perfored in order to obtain intrinsic paraeters, which are necessary for reduction of radial and tangential lens distortions [4]. 4. FEATURE POINT DETECTION AND MATCHING The higher level of autoation was provided by developent of novel arkers detection and atching algorith. Two sets of rectangles are detected on two iages by eans of binary iage processing, contour detection and shape filtering. The set of corresponding points positions necessary for hoography coputation, consisting of vertices of rectangles, are calculated by Harris corner detector with a subpixel iproveent of the accuracy. Markers are expressed in polar coordinates and sorted with respect to arker set s center of the ass. The binary iage I of resolution M by N is the iage which consist of two kind of pixel areas: A the foreground and B the background where A and B are two sets of pixels defined as [79]: A = {( x, y) : x M, y N and I( x, y) = } () B = {( x, y) : x M, y N and I( x, y) = } Let D 8 be the 8neighbourhood [79] of a pixel p i = (x,y). A closed contour (or a boundary) of a foreground region A on a binary iage is a set of pixels defined as follows: cc p A : ( p p ( B D ( p )) and p = p = { i=,..., n i g i i N )}() The first step of the algorith is binarization of a grayscale iage. The threshold value is obtained by analysis of an intensity histogra of an iage. In the next step, contours enclosing all foreground object are detected [79] on both of iages. Contours are transfored to the chain polygon representation in which only endpoints of the line eleents approxiating the contour are stored. In the case of the ipleentation of the ethod in described software, the set of points consist of vertices of rectangular arkers. Contours are filtered by the shape filter whose response is the strongest for convex, rectangular contours with user defined ranges of: area enclosed by the boundary, width to height ratio and angle between sides of the quadrilateral. Obtained vertices positions are refined by Harris corner detection algorith with the subpixel accuracy iproveent. The exaple of application of the ethod is illustrated on figure. It is assued that there is no rotation about the optical axis of the caera coordinate frae fro which the reference iage was obtained. In the first step, the center of ass of set of arkers vertices is coputed. The calculated point becoes the origin of the new coordinate frae. All of the points have to be expressed in this new coordinate frae in the polar representation. Next, the sorting of points is perfored. Points polar angles and radial distances fro the origin are input to the coparison function passed to the sorting algorith. The sorting is
8 6 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents carried out on sets of arkers on both iages of the construction. Fig.. Exaple of the application of rectangle detection algorith In the second ethod, the position and orientation of the caera is not constrained by the requireent of no rotation described above. The iage patches which are atched on two iages are coded arkers. The arker consists of N rows and N coluns of sall squares arranged in chessboardlike pattern. Each of the squares can be black which represents logic or white which represent logic. The innerost x pattern of the arker is the sae for all arkers and reseble letter L (see figure 3). The outer part of the chessboard pattern is different for each arker and encodes the nuber. The position of each square arker on the iage is detected by eans of algorith described previously. Next, the hoography apping is applied to reove the perspective distortions fro the iage of the arker. Iage patch is rectified using data fro arker odel which is specified by the user. The arker orientation is decoded fro the innerost L shaped pattern of the arker. In the last step, the iage patch is encoded as N x N array of logical values. In the pattern atching step, actual iages of arkers are not copared with each other, but instead their code representations are. The process is uch faster than iage pattern atching ethods. Fig. 3. Exaple of arker used for pattern atching and set of arkers on real photograph The atching of arkers by eans of their code representations coparison can be applied in probles like iage rectification, iage registration ethod (iage stitching, osaicing) as well as in 3D structure and otion reconstruction techniques based on epipolar geoetry [5,6] and fundaental atrix (corresponding set of points needed for F atrix coputation can be encoded as the chessboard pattern arkers). 5. DEVELOPED SOFTWARE The ain purpose of the software is construction s continuous onitoring and diagnostics. As a standalone syste (operating in online ode) can iediately evaluate changes of static states of structure and send the to a diagnostic center. Application provides advanced operations on caera devices like live preview ode or reote odification of caera paraeters. Syste's IO Handler odule allows ultiple picture acquisition devices to operate in the realtie. Application supports popular SLR caeras with available driver libraries used for device anageent fro syste. Calibration odule (figure 4) allows calculating calibration data based on iages of a planar chessboard pattern with a known geoetry (odd row and even colun count). Calibration process results contains intrinsic and extrinsic caera paraeters. Calibration data is necessary for identification of caera's position and orientation with respect to the exained object. The intrinsic caera paraeters can be used for reduction of radial and tangential distortions fro acquired iages. Fig. 4. Software odules: ain window, live preview, caera paraeter setting odule and caera calibration odule Caera configuration is accessed and odified fro configuration window. Coon odifiable caera paraeters consists of shutter speed, aperture value and ISO speed. Configuration window provides also specific caera paraeters: live view zoo and inforation about battery level. Enabling live view ode allows quick check on how odified paraeters affect caera s work. It s possible to store caera configuration. Such stored configuration can be loaded into caera anually or autoatically after connection (depending on settings). The developed software tool provides a high level of easureent process autoation, accoplished by autoated operations like iage acquisition, iage preprocessing and algorith calculation (Figures 4,5). Although the syste is suitable for online work using data acquisition
9 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents 7 devices, it s also possible to perfor offline easureents using existing iages. Correlation algorith provided with software tool is fully custoizable: user can odify start/end point for easured line on calculated iages, search window size and window count. During configuration it s possible to test algorith on easureent iages in order to finetune algorith paraeters. User can ake algorith work autoatically whenever new iage has been added to easureent. This feature cobined with possibility of creating scheduled sequences of picture taking akes it possible for software to autoatically execute data acquisition and calculation. Rectification algorith provides functionality on generating perspective transforation for iages based on iage arkers location on iage. In rectification window user can odify rectification paraeters and see preview iages: original iage, original iage with drawn iage arkers locations, transfored iage and reference iage. Rectification paraeters are stored in the easureent configuration. Scale algorith is used for converting pixel values into real length unit. In order to calculate scale coefficient user has to load iage containing circular arker and provide inforation about its size. After scale coefficient has been calculated, analyzed iages results will be shown in illietres instead of pixels. The syste warns the user when a critical level of easureent estiates has been exceeded. A essage is sent by eail or text essage. It's possible to send data to external onitoring and diagnostics systes using TCP/IP. Data generated by the application can be exported to external data sheets in popular forats e.g. Excel spreadsheet, PDF and HTML The software architecture is shown in figure 6. The drivers (DRV) are the odules that provide data acquisition fro caeras: configuration, reote picture taking and downloading. Drivers are coordinated by an IO Handler (IO HDLR) responsible for device data transfer synchronization to ensure no conflicts in the driver's work. The data processor (DATA PROC) perfors all iage calculations such as vision algoriths and preprocessing ethods. The database server (DBS) is used for storing various project data downloaded iages, algorith results and configuration. The liit checked odule (ALM) checks if easureent estiates are within allowed liits. The essenger (MSGR) is responsible for sending essages when the estiate crosses liit values. The configuration server (CFGS) is a separate storage odule used for storing syste configuration. The watchdog (WDG) controls a syste's perforance. Fig. 5. Software odules: Measureent odule, rectification odule, iage correlation odule and result browsing odule Fig.6. Software Tool Architecture 6. LAB TESTS OF THE DEVELOPED SOFTWARE In the first exaination, the syste perforance was tested on the labsetup consisting of steel frae fixed at one end, loaded by the single weight (Figure 7,8). The point of application of the force could be oved along the length of the vertical part of the frae which provided variable loading conditions. Two digital SLR caeras Canon EOS 5D Mark II with a lens with Canon 47 f/,8l zoo lens with 5 focal length adjustent were placed in two points on the scene. The first one was positioned in such a way that its optical axis was perpendicular to the construction s plane. The second one was placed at the sae distance fro the construction as the first one, but its orientation was changed the angle between its optical axis and the direction perpendicular to the frae s plane was 5 degrees. During the investigation, the load was oved along the length of the vertical part of the frae. The set of its positions (d []) has been presented in table. For each of the load positions, 3 iages were captured by both caeras. The ean value of the easured axiu deflection (in free endpoint of the bea) and its standard deviation were coputed for deflection curves obtained by both caeras, for each of the positions of the load. The difference between the corresponding values of statistical paraeters (ean value of axiu deflection) calculated fro the data captured by the first caera and the second one was a easure of the error
10 8 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents introduced to the syste by the rectification algorith. Additionally, the noise of the ethod resulting fro the lighting conditions as well as inaccuracy of correlation algorith were investigated. The scale coefficient value in the exaination was.74 /pixel. The position of the caera with respect to the construction as well as its orientation were changed during the investigation. The easureent of the noise of the ethod was carried out to exaine the influence of the lighting conditions on the ethod. The digital SLR caera Canon EOS 5D Mark II with Canon 47 f/,8l zoo lens with 45 focal length adjustent was used. There were no artificial lights present on the scene. The easureent points fro which iages were acquired are shown on figure. Fig. 7. The lab setup d d P P P Fig.. The points of space fro which iages were acquired Fig. 8. The photograph of the first lab setup In next experient, the developed software was tested by perforing the continuous vision onitoring of the structure under the load. The wooden bea of the length of 8 c supported at two ends was inspected construction (figure 9). The bea was loaded by a point force acting in the iddle of its length. There was no artificial speckle pattern on the plane of the structure, natural texture of the aterial was used as an intensity pattern in correlation ethod. The easureent was verified with the help of the laser distance sensor Disto D5. The conducted experient was divided into continuous easureent sessions of 3 photographs, separated by inutes breaks. The continuous change of the deflection in the point of axiu deflection during investigation was observed (figure 3). It was confired on the basis of the easureent with the laser distance sensor. Fig. 9. The photograph of the second lab setup 7. RESULTS OF THE EXAMINATION In the first exaination, the ean value of the noise of the ethod for the natural lighting conditions was. (.5%). For the iages captured fro one point in space, the standard deviation of calculated displaceent value in the point of axiu deflection (repeatability of the ethod) was not affected by the change of position of the load and had value about.4. Introduction of the rectification algorith increased the error of the ethod. The axiu value of the difference between displaceent calculated fro the reference point and fro the second position was about.5. The relative error induced by the rectification calculated as the ratio between difference Δx (table ) and the displaceent coputed fro the reference iage was in a range between.% to.%. The results of the exaination are shown in the table. The exaples of curves of deflection for different positions of load are presented on figure. In the second experient, when there was no artificial speckle pattern on the object because of that value of the easureent noise (its standard deviation obtained fro a series of 3 iages) of the ethod induced by variable illuination condition turned out to be.4 (Fig. ). The standard deviation of the easured value of the deflection in the point of axiu deflection was.3 for the case of the iages acquired fro the sae point in space, however the standard deviation of the easureent after application of rectification reached value of.9.
11 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents 9 Table. The results of exaination. The first colun position of a load with respect to the fixed end of the frae (Figure ). The coluns  5 results (ean value fro 3 iages and standard deviation) of deflection coputation in point of axiu deflection for caera and. The colun 6 difference between results obtained fro two caeras, in, colun 7 the relative difference with respect to the first caera, in percents. Caera Caera d[] ean std ean std Δx Δx% Results of easureent are shown on figure. In figure 3, an increase of the average value of bea s displaceent in the axiu point of deflection resulting fro the increasing deforation can be noticed. In figure 4, the change of the standard deviation resulting fro application of the rectification algorith was shown. Fig.. Deflection curve faily (fro 3 iages) calculated fro iages acquired fro the Point P Fig. 3. Increase of the ean value of the displaceent in the point of axiu deflection. Tie interval between each point is inutes. Fig.. Deflection courses for different cases of the loading (table ). Results obtained fro the reference caera Fig. 4. Increase of the standard deviation of the displaceent values calculated fro the series of 3 iages after application of rectification. Tie interval between each point is inutes 8. CONCLUSIONS Fig..The noise of the ethod in the case of object without artificial speckle intensity pattern SHM visionbased easureent syste enables structure static and quasistatic states assessents during inspections and online continuous state onitoring to be perfored by eans of analysis of changes in the geoetric properties of the structure, such as a shape of the deflection curve. The introduction of iage registration techniques has iproved the flexibility, universality and accuracy of this ethod. The technique of inplane deflection easureent, with application of iage registration ethods, presented in this article enables iages of the construction to be taken fro different points of view during exaination. The lab tests revealed that the influence of the lighting on the perforance of
12 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents the ethod was negligible (easureent noise value.3%). The standard deviation of the deflection coputation, in the case of iages obtained fro the sae point in space aounted to.4. When the rectification algorith was applied, the standard deviation of axiu deflection was located in the range between.3 to.5 for speciens with artificial speckle pattern and up to. for the objects without special texture. The developed techniques of arker detection and atching ake it possible to create an application with fully autoatic vision structure's online onitoring systes in which the construction can be exained during its everyday use. On the other hand, the developed ethods can be applied in an userfriendly software which can help one to quickly assess the state of the construction during inspection. The syste has eployed easily available digital SLR caera as the easuring sensor for the easureent of static or quasistatic states of structures. It can be applied to the structures with an artificial texture in the for of the optical noise, natural texture of aterials and when special geoetric arkers are available. The syste has provided high easureent density without use of active optical ethods. It can be eployed to various civil engineering structures like bridges, footbridges, chineys, viaducts, girders, ceilings, halls, asts, wind turbines, buildings, achines and devices. ACKNOWLEDGEMENTS Reported research realized within confines of OPIE project No:..3/8 cofinancing by ERDF within a frae of Operational Progra Innovative Econoy. BIBLIOGRAPHY [] Kohut P., Holak K., Uhl T.: Application of iage correlation for SHM of steel structures, Structural Health Monitoring 8: proceedings of the fourth European workshop, Cracow, Poland,, July 4 8, pp [] Uhl T.: SHM of civil structures ethods, tools and applications, Proceedings of the international conference on Experiental Vibration Analysis for Civil Engineering Structures EVACES 9, Dolnośląskie Wydawnictwo Edukacyjne, Wrocław 9 [3] Uhl T., Kohut P., Holak K.: Construction deflection exaination by eans of correlation coefficient, Proceedings of the 3th School of Modal Analysis,Cracow, Poland 9, [4] Uhl T., Kohut.P, Szwedo M., Holak K.: Static and dynaic optical easureent in SHM of civil structures, Proceeding of the 7th international workshop on Structural Health Monitoring, DESTech publications, Lancaster 9 [5] Kohut P., Kurowski P.: The integration of vision based easureent syste and odal analysis for detection and localization of daage, Engineering achieveents across the global village, The International Journal of INGENIUM, Cracow, 5, pp [6] Kohut P., Kurowski P.: The integration of vision syste and odal analysis for SHM applications, Proc. Of the IMACXXIV a conference and exposition on structural dynaics, January 3February St. Louis, 6, pp. 8 [7] Lee, M., Precht, M., Tyson, J., Schidt, T.: Application of 3D Measureent Syste with CCD Caera in Microelectronics, Advanced Packaging, Noveber 3. [8] Schidt, T., Tyson, J., Galanulis, K.: Full Field Dynaic Displaceent and Strain Measureent Using Advanced 3D Iage Correlation Photograetry, Experiental Techniques, Vol.7, No.3, pp.444, May/June 3. [9] Johnson, E., Woodard, N.: LowCost Electrical Resistance Spot Welding Technique for Micro Optical Coponent Mounting with SubMicron Tolerance, Proceedings of Photonics Packaging Syposiu, San Francisco, CA, Aug. 3. [] Schidt, T., Tyson, J., Galanulis, K.: Dynaic Strain Measureent Using Advanced 3D Photograetry, Proceedings of IMAC XXI, Orlando, FL, Feb. 3 [] Tyson, J., Schidt, T., Shahinpoor, M., Galanulis, K.: Bioechanics Deforation and Strain Measureents with 3D Iage Correlation, Experiental Techniques, Vol.6, No.5, pp.394, Sept./Oct. [] Tyson, J., Schidt, T., Galanulis, K.: Advanced Photograetry for Robust Deforation and Strain Measureent, Proceedings of SEM. Annual Conference, Milwaukee, WI, June [3] Hotz, W., Hänggi, P., Deterination of Foring Liit Curves,. Bergann, D., Ritter, R., 3D Deforation Measureent in sall Areas based on Grating Method and Photograetry, 998 [4] Alpers, B., Bergann, D., Galanulis, K., Winter, D.: Advanced Deforation Measureent in Sheet Metal Foring, 999 [5] Steinchen, W. & Yang, L.: Digital Shearography, Theory and Application of Digital Speckle Pattern Shearing Interferoetry, SPIE Press, 3 [6] Schidt, T., Tyson, J.: Pulsed Laser Systes for Wide Field Vibration Analysis, ASNT Fall Conference, Oct. [7] Tyson, J.: NonContact Fullfield Strain Measureent with 3D ESPI, Sensors, Vol. 7 No. 5, pp. 67, May
13 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ KOHUT, HOLAK, UHL, Prototype of the vision syste for deflection easureents [8] Tyson, J.: Dynaic Deforation Analysis with Pulsed ESPI, Society of Experiental Mechanics Conference, June 999 [9] Tyson, J.: Fullfield Vibration and Strain Measureent with 3D ESPI, Sensors, Vol 6 No. 6, pp. 6 , June 999 [] Tyson, J.: Advanced Brake Squeal Analysis and Vibration Measureent, 4th International Brake Colloquiu, Graado, Brazil, April 5, 999 []Tyson, John: Portable Interferoetry Systes for Rapid NDI, USAF Aircraft Structural Integrity Conference, Dec. 998 [] Tyson, John: Practical Coposite Repair Evaluation, Proceedings of 4st Int l SAMPE Syposiu, Vol. 4, Book, pp , March 996. [3] Tyson, John: Shearography NDE of Rocket Motor Coponents, Joint AryNavyNASA Air Force Nondestructive Evaluation Subcoittee Meeting, July 46, 99 [4] Chu T. C., Ranson W. F., Sutton M. A., Peters W. H.: Application of digitaliage correlation techniques to experiental echanics. Exp Mech. 5, Septeber 985, pp.3 4 [5] Hartley R., Zisseran A.: Multiple View Geoetry in Coputer Vision, Cabridge University Press, 4 [6] Ma Y., Soatto S., Kosetska J., Sastry S.: An Invitation to 3D Coputer Vision, Springer Verlag, New York 4 [7] Gonzales R., Woods R.: Digital Iage Processing, Prentice Hall [8] Russ J.: Iage Processing Handbook, CRC Press 998 [9] Shapiro L., Coputer Vision, Prentice Hall, [3] Zitova B., Flusser J.: Iage registration ethods: A survey, Iage and Vision Coputing, Elsevier, 3 [3] Holak K, Kohut P., Cieślik J.: Application of threediensional vision techniques in echanical vibrations easureent (in polish), Proceedings of the 8th Workshop of Mechatronics Design, 8, pp [3] Uhl T., Kohut P., Holak K.: Iage correlation techniques in structures deflection easureent, Proceedings of the international conference on Experiental Vibration Analysis for Civil Engineering Structures EVACES 9, Dolnośląskie Wydawnictwo Edukacyjne, Wrocław 9 [33] Bradsky G., Kaebler A.: Learning OpenCV, O Reilly 8 Tadeusz UHL, Prof. is the head of the Departent of Robotics and Mechatronics of AGH University of Science and Technology in Cracow. His scientific interests are diagnostics and Structural Health Monitoring of constructions, dynaics of constructions, odal analysis, control systes and echatronics. He is the author of 6 onographs and over 5 scientific papers. Piotr KOHUT, Ph.D. is an adjunct professor at the Departent of Robotics and Mechatronics of AGH University of Science and Technology in Cracow. His scientific interests focus on echatronics, vision systes, ethods of iage processing and analysis and 3D easureent techniques. Krzysztof HOLAK, M.Sc. is Ph.D. student at the Departent of Robotics and Mechatronics of AGH University of Science and Technology in Cracow. His works are connected with iage processing and analysis and vision
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15 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ MIKOŁAJCZAK, prediction of changes in the technical condition using discriinant analysis 3 PREDICTION OF CHANGES IN THE TECHNICAL CONDITION USING DISCRIMINANT ANALYSIS Paweł MIKOŁAJCZAK Katedra Budowy, Eksploatacji Pojazdów i Maszyn. Wydział Nauk Technicznych. UWM w Olsztynie Suary Discriinant analysis can be used for identification of variables which identify (discriinate) two or ore naturally eerging groups of f.ex. diagnostic syptos or factors influencing developent of particular type of achine part use. The goal in this case is searching for rules for assignent of ultidiensional objects to one of any populations of known paraeters, at the lowest classification istake level possible. The idea of discriinant analysis is definition if the groups differ on account of a ean of a variable, and using this variable for appurtenance to a group predicting (f.ex. new cases of diagnostic syptos, factors influencing the use level). In the following paper, an exaple of discriinant analysis application to diagnostic paraeters choosing and identification of external factors influencing intensity of rolling bearings use in rotary achines is introduced. Keywords: utual easures, syptos, sensitivity analysis, discriinant analysis, technical diagnostics. PREDYKCJA ZMIAN STANU TECHNICZNEGO Z WYKORZYSTANIEM ANALIZY DYSKRYMINACYJNEJ Streszczenie Analizę dyskryinacyjną stosuje się do rozstrzygania, które zienne wyróżniają (dyskryinują) dwie lub więcej naturalnie wyłaniających się grup np. syptoów diagnostycznych lub czynników wpływających na rozwój danego rodzaju zużycia części aszyn. Stawiany cele w ty przypadku jest poszukiwanie reguł postępowania ającego na celu przyporządkowanie wielowyiarowych obiektów do jednej z wielu populacji o znanych paraetrach przy ożliwie inialnych błędach klasyfikacji. Główna idea leżąca u podstaw analizy dyskryinacyjnej to rozstrzyganie, czy grupy różnią się ze względu na średnią pewnej ziennej, a następnie wykorzystanie tej ziennej do przewidywania przynależności do grupy (np. nowych przypadków syptoów diagnostycznych, czynników wpływających na wartość zużycia). W pracy przedstawiono przykład zastosowania analizy dyskryinacyjnej do wyboru paraetrów diagnostycznych i predykcji stanu łożysk tocznych wentylatorów w zależności od czynników wyuszających zianę tego stanu. Słowa kluczowe: iary wzajene, ocena wrażliwości syptoów, analiza dyskryinacyjna, diagnostyka techniczna.. MUTUAL MEASURES Technical diagnostics of objects is, aong others, coparing two states current and standard. Hence, in quality or quantity assessent it is necessary to identify how uch the signal analyzed is siilar to standard signal. The proble is then creating signals estiators and checking for their utual siilarity. The proble can be solved by creating separate easures for the signal analyzed and the signal standard, and coparing the easures values in diagnostic process [9]. Another solution is creating shared easures, definition of technical condition with shared easure is assessent of siilarity of standard signal to the signal analyzed or coparing separate easures of both signals. Shared easure describing relation between signal value x(t)in the oent t and the value of the second signal y(t) in the oent t+τ is the function of utual correlation (intercorrelation): T R = li x( t) y( t + τ ) dt Rˆ ( τ ) = xy xy T T () T x( t) y( t + τ ) dt = x( t) y( t + τ ) T where: Rˆ xy ( τ )  estiator of utual correlation function Noralized utual correlation function is described with the forula: Rxy ( τ ) x( t) y( t + τ ) ρ xy ( τ ) = = () R () R () x ( t) y ( t) x y
16 4 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ MIKOŁAJCZAK, prediction of changes in the technical condition using discriinant analysis Value of the function is in the range described with the forula ρ xy ( τ ) and it is equal to / when the signals are identical. Noralized correlation function can be used for localization of vibroacoustic signal sources and for definition of signal track. Noralized utual correlation function is a local easure of siilarity of signal in lag tie, which eans it shows siilarity level between signals separately for each τ. Disadvantage noralized utual correlation function is that it needs to be identified for signals generated concurrently, i.e. for the sae value of object exploitation tie Θ. Because of its local character diagnostic state assessent requires not a nuber but function or a set of nubers defining utual correlation function [7]. Hence, there is also necessity to identify easures of signals siilarity for the signals generated in various tie of objects exploitation and given with a one nuber. The easure discussed is a noralized utual correlation function described with the forula []: R( τ ) R ( τ + β ) dτ K ( β ) = R( τ ) dτ R ( τ ) dτ (3) Square of the forula presented above, for (β=), gives the following result: K R ( τ ) R ( τ ) dτ = R( τ ) dτ R( τ ) dτ (4) where: R (τ), R (τ) functions of vibroacoustic signals correlation in the tie Θ and Θ of technical object exploitation. The last forula presents global easure of signals siilarity for various tie of exploitation, thus it is a global easure of object exploitation state siilarity. Using plexus characteristics and Wiener Chinczyn relation and applying a local kind of noralization in frequency range, noralized utual correlation function can be described with the following forula []: Value of coherence function ay vary in the range given with the forula γ xy ( f ). For linear configurations with constant paraeters γ xy ( f ) =, which eans signals x(t) and y(t) are copletely coherent. In the case when for a given frequency γ xy ( f ) =, signal x(t) and y(t) are incoherent. If signals x(t) and y(t) are stochastically independent, γ xy ( f ) = for all frequencies. When coherence function value is in the range given γ xy ( f ) easures results include disturbances, which eans that the output signal is influenced not only by the input signal, but also other signals or a configuration cobining signals x(t) and y(t) is not linear. Daage in kineatic couple of technical object set causes changes in vibroacoustic signal and consequently in coherence function. Thus coherence function easured for various tie of object s life detects daage, changes in transittance and relative changes of signal caused by daage. coherence function, like utual correlation function, is a local easure of siilarity of signals of vibration sources separately for each frequency. It identifies relative quantity of inforation concerning input (priary) signal in output signal. Global easure of vibroacoustic signals generated in the sae oent of object lifetie, in all frequency ranges siilarity is so called sources siilarity level S, because at any transittance of a configuration S xy = []: (3) Rxy ( τ ) dτ Gxy ( f ) df (6) S xy = = R ( τ ) R ( τ ) dτ G ( f ) G ( f ) df G ( ) xy f γ ( f ) = (5) (4) xy Gxx( f ) Gyy ( f ) Gxy ( f ) cosϑdf + Gxy ( f ) sinϑdf where: (7) ρ xy = G xx (f), G yy (f) functions of density of signals Gxx f df Gyy f df x(t) and y(t) power; ( ) ( ) G xy (f) function of utual density of signals The forula presented above is a su of x(t) and y(t) power; square utual power, active and passive, γ xy ( f )  squared coherence function. noralized to a product of x(t) and y(t) signals power. xy yy Whereas when there are no input or output disturbances S xy <, these characteristics can be used in diagnostics for identification of vibration and noise sources. Using the utual correlation function defined as opposite Fourier s transforation of utual spectra power of signals density and the definition of correlation function shifted by 9 for one of the signals, the following result can be obtained: xy xx yy
17 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ MIKOŁAJCZAK, prediction of changes in the technical condition using discriinant analysis 5 ρ Analysis of the characteristics of xy needs to the conclusion that this is the global easure of signals siilarity and it can be used in diagnostics for easuring siilarity of processes perfored in identical configuration. Taking noralized function of utual correlation into consideration, the global easure of siilarity of vibroacoustic signals in given frequency generated in various tie of object exploitation K can be defined as []: K = G ( f ) G G ( f ) df G ( f ) df ( f ) df (8) where: G (f), G (f) spectral density of power of vibroacoustic signals defining technical states of an object in exploitation tie Θ and Θ. The easure enables assessent of changes in condition of the object in its exploitation tie. Mutual easures enable coparing two states or syptos, analyzed and standard, however they do not facilitate choosing one of the diagnostic syptos fro a large group of syptos. One can also apply the physical probabilistic odels in task of decresing of uncertainty of reise and evolution of failure [] or ethods of autoatic data classification [],[6].. METHODS OF DIAGNOSTIC SYMPTOMS SENSITIVITY TO CHANGES OF TECHNICAL STATE ASSESSMENT There are the following ethods of diagnostics paraeters choice presented in the literature [], [9]:  Maxiu relative changes in diagnostic paraeters ethod,  Maxiu variation of diagnostic paraeter ethod,  Maxiu sensitivity of diagnostic paraeter to changes in technical state ethod,  Maxiu inforation capacity of diagnostic paraeter ethod. Maxiu relative changes in diagnostic paraeters ethod in this ethod, a diagnostic paraeter which has the greatest value of k j indicator is chosen. It takes average speed of paraeters changes in tie. It is calculated with the following forula: bj k j = b j j = b j = K K ( v y ( v j i+ ) y ( v ) v ) y ( v ) y i= i+ i i i j* g i i (9) where: K the nuber of eleents of tie series in (v, v b ) interval. Diagnostic paraeter y * is chosen with the following forula: y * * = y *, j,..., Λk * = ax( k j ), j =,..., j j () Maxiu variation of diagnostic paraeter ethod diagnostic paraeters analyzed should presented a sufficient level of variability in achine exploitation tie. The paraeters with the greatest value of variability indicator g j are chosen fro the set of final results: S j g j = () S S j = K K j y ( v j = i= i+ j i+ ) y j ( vi ) v v i () Where: K the nuber of eleents of tie series in (v, v b ) interval. The paraeter y * is chosen with the following forula: y * * = y *, j,..., Λg * = ax( g j ), j =,..., j j (3) Maxiu sensitivity of diagnostic paraeter to changes in technical state ethod the idea of the ethod is to choose, fro the set of output paraeters of a configuration or an object set, a paraeter that has the greatest value of a j indicator, which includes paraeters dependence on achine state: k a = M ( i, j); i =,,,k; j =,, j i= (4) where: M(i, j) [M(i, j)] kx an eleent of binary diagnostic atrix of a technical object. The paraeter y * then is chosen for the set of diagnostic paraeters by choosing y j with axiu value of a j indicator: * * y = y *, j,..., Λa * = ax( a j ), j =,..., j j (5) Maxiu inforation capacity of diagnostic paraeter ethod based on the choice of the paraeter that provides the ost inforation on achine s technical state. The
18 6 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ MIKOŁAJCZAK, prediction of changes in the technical condition using discriinant analysis diagnostic paraeter is the ore iportant for definition of object s technical state, the ore it is connected to the object and the less it is connected to the other diagnostics paraeters. The relation discussed above is represented by the indicator of integral capacity of diagnostics paraeter h j introduced with the forula: h j j = + r i, j= r i, j (6) where: r j = r(w,y j ); j =,, linear correlation between variables coefficient, W state of an object set, r i,j  linear correlation between y i and y j variables coefficient. The paraeter y * is chosen to the set of diagnostics paraeters by axiization of h j indication with the following forula: y * * = y *, j,..., Λh * = ax( h j ), j =,..., j j (7) Each of the ethods diagnostic paraeters choice presented above considers changes in paraeters values in technical state of an object analyzed function on a different level, however, they do not enable siultaneous assessent of nuerous diagnostic syptos and exploitation conditions influence on track of technical object use value. In the next chapter the exaple of discriinant analysis use for forecasting changes in rolling bearings state with respect to the set of diagnostic syptos and external enforceents given is presented. 3. DISCRIMINANT ANALYSIS Discriinant analysis is a statistical analysis which allows to identify differences between two or ore groups, analyzing several variables at the sae tie. Variables used for groups distinguishing are called discriinant variables. Practically, discriinant analysis is a general ter referring to several related statistical procedures. Siplifying, the procedures can be distinguished into [5], [7], []:  Descriptive procedures and intergroup differences identifying procedures. The procedures explain: if it is possible, with a set of several variables, distinguish one group fro another? How well discriinant variables distinguish groups given? Which of the variables are the best in discriinating?  Procedures of cases classification, which is definition of characteristics values, based on observation or experience, that enable decision to which of the groups a new case should be included. It is connected with definition of one or ore functions classifying the cases analyzed to the right groups. The exaple of discriinant analysis to be presented is based on exploitation data concerning the group of fans introduced in the table. The purpose of the analysis is showing, how the use progress of rolling bearings can be predicted with the diagnostics paraeters analyzed. Table.. Coparison of diagnostics and exploiation paraeters for the analyzed group of fans Fan v(fo) v(fo) T [ Choking C] nuber [/s] [/s] [%] State of the rolling bearings w 3,6 5, 65 5 inadissible w 7,5 3 7 inadissible w inadissible w4 8,5,6 6 inadissible w5 4, 3, 84 3 inadissible w6 3,5 4, 65 4 sufficient w7 5,, 63 sufficient w8 3,9 3,5 6 3 sufficient w9 5,5, sufficient w 4 3,7 7 3 sufficient w 4,3, 6 3 good w,5,9 74 good w3,8, good w4,5,6 6 good w5, good Sybols used in the table: v(fo) the first haronica of vibrations, v(fo) the second haronica of vibrations, T teperature of rolling bearing cover, Choking percentage of flow choking.
19 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ MIKOŁAJCZAK, prediction of changes in the technical condition using discriinant analysis 7 Analysis starts with definition of canonical discriinant function distinguishing the groups analyzed. Any function can be used for the definition, however, the ost coonly used are linear functions. The functions to be used are described with the following forula [5], [7], []: D kj = β + β x β x kj p pkj (8) where: p nuber of dicriinant variables (in the following exaple p = 4), n size of the saple (in the following exaple n = 5), g nuber of groups (in the following exaple g = 3), D kj  canonical discriinant function for kth case in jth group value, k=l,...,n and j=l,...,g, x ikj value of ith discriinant variable for kth case in jth group, i=,...,p, β i coefficients of canonical discriinant functions defined with characteristics of the function. The proble of β i coefficients definition is definition is solving atrix equation given with the forula [3], []: (MλW)β= (9) where: W intragroup atrix of squares and ixed products, M intergroup atrix of squares and ixed products, β  vector of canonical discriinant functions coefficients, λ  so called atrix value. In the table there are calculated, raw and standardized values of discriinant functions coefficients. The calculations were perfored with STATISTICA PL software, version 9.. Function (raw coefficients) Table. Coefficients of discriinant functions Function Function Function (raw (standardized (standardized coefficients) coefficients) coefficients) v(fo) [/s] , ,76 ,373 ,74 v(fo) [/s] ,39 ,3739 ,5 ,39777 T [ C] ,955,583 ,7398, Choking [%],588 ,934,57 , constant, ,974 8,373,6546 value 8,373,6546,936, cuulated ratio of cleared variation,936, ,373 ,74 Canonical discriinant functions are described with the following forulas: D =, ,84973v(fo) ,39v(fo) ,955T +,588Choking D =  5,974 ,76(fo) ,3739V(fo) +,583 ,93Choking The first function is responsible for 93% of cleared variance (while the second for 7% only), which eans the first function is the ost iportant. Raw coefficients of discriinant function can be used with the data fro observation ainly for calculating function value, but they cannot be used for interpretation. One of the reasons is the fact that coordinates beginning is not equal to the central point represented by average values of discriinant variables for all the points. To eliinate this proble standardization of coefficients is used [3], [4], [5]: where: β = β n g i β = β x () i i p i= xi  average value of i discriinant variable, β i  standardized coefficients of canonical discriinant function, the rest as described above. To identify how each discriinant function distinguishes the groups, the diagra presented in the figure can be used. i i
20 8 DIAGNOSTYKA  DIAGNOSTICS AND STRUCTURAL HEALTH MONITORING 4(6)/ MIKOŁAJCZAK, prediction of changes in the technical condition using discriinant analysis 5 Pierw wz. pierw 4 3 W Pierw Pierw Fig.. The diagra of canonical values dispersion inadissible sufficient good Fig. verifies the results of variation cleared. In the diagra, the fans with bearings in good, sufficient and inadissible are distinguished with the first discriinant function (the vertical lines can be drawn to identify these groups 93% of cleared variation). The one exception is the point identified for the fan nuber. The reason for that is the fact that despite good dynaic condition (low vales of the first and the second haronica) there is a great interaction and a loadof the fan. Thus, there is a risk of changing the condition, fro the good level to the sufficient level (which eans increasing the distance fro good group centroid, arked with the black point). Another easure in ultidiensional variables space can be used Mahalanobis distance [4], [7], []. It is a distance between each case and a group centroid. It is worth entioning, that the saller Mahalanobis distance, the ore certain appurtenance to the group. In the figure, square Mahalanobis distance for the case analyzed is presented. Square of Mahalanobis distance inadissible sufficient good Fun nuber Fig.. Mahalanobis distance colun graph Analysis of the figure leads to the conclusion, that predefined functions discriinate three groups of rolling bearings states analyzed very well. In the first five fans, condition of rolling bearings was inappropriate, and in these cases coluns labeled with inadissible are the sallest. The group of five fans with rolling bearings of sufficient condition is the one in the iddle, in this group the sallest values are represented by the coluns labeled sufficient. The last group is also easy to be distinguished, coluns labeled as good are the sallest.
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